Subgroups — C₆₇×#22443 #30701694

Order 17152.

1622 conjugacy classes of subgroups.

The subgroup lattice has 884 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#440×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2341	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1972	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1880	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1880	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1879	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1879	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1809	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1753	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
19	C₂×AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
20	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	8	yes	complemented
21	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
31	#440×C₆₇	—	2	4	no	—
32	C₄×SD₁₆×C₆₇	—	2	4	no	—
33	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	#390×C₆₇	—	2	1	no	—
47	#390×C₆₇	—	2	1	no	—
48	#390×C₆₇	C₂²	1	1	yes	characteristic
49	#390×C₆₇	C₂²	1	1	yes	characteristic
50	#390×C₆₇	C₂²	1	1	yes	characteristic, complemented
51	#390×C₆₇	C₂²	1	1	yes	characteristic, complemented
52	#386×C₆₇	—	2	1	no	—
53	#386×C₆₇	—	2	1	no	—
54	#380×C₆₇	—	2	1	no	—
55	#380×C₆₇	C₂²	1	1	yes	characteristic
56	#380×C₆₇	C₂²	1	1	yes	characteristic
57	#379×C₆₇	—	2	1	no	—
58	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
59	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
60	#340×C₆₇	C₂²	1	1	yes	characteristic, complemented
61	#340×C₆₇	C₂²	1	1	yes	characteristic, complemented
62	#336×C₆₇	C₂²	1	1	yes	characteristic, complemented
63	#336×C₆₇	C₂²	1	1	yes	characteristic, complemented
64	C₂⁵×C₆₇	—	2	1	no	—
65	C₂×#41×C₆₇	—	2	2	no	—
66	C₂×#41×C₆₇	—	2	2	no	—
67	C₂×#41×C₆₇	—	2	2	no	—
68	C₂×#41×C₆₇	—	4	1	no	—
69	C₂×#41×C₆₇	—	4	1	no	—
70	C₂×#41×C₆₇	C₂×C₄	1	4	yes	complemented
71	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₆₇	—	2	2	no	—
73	C₂²×D₄×C₆₇	—	2	2	no	—
74	C₂²×D₄×C₆₇	—	2	1	no	—
75	C₂²×D₄×C₆₇	—	2	1	no	—
76	C₂²×D₄×C₆₇	—	2	2	no	—
77	C₂²×D₄×C₆₇	—	4	1	no	—
78	C₂²×D₄×C₆₇	—	4	1	no	—
79	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
80	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	—	2	1	no	—
84	C₂³×C₄×C₆₇	—	2	1	no	—
85	C₂³×C₄×C₆₇	—	2	1	no	—
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
88	C₂×SD₁₆×C₆₇	—	2	2	no	—
89	C₂×SD₁₆×C₆₇	—	2	2	no	—
90	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
91	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
92	C₂×SD₁₆×C₆₇	D₄	1	4	yes	—
93	C₂×SD₁₆×C₆₇	D₄	1	4	yes	—
94	C₂×D₈×C₆₇	—	2	2	no	—
95	C₂×D₈×C₆₇	—	2	2	no	—
96	C₂×D₈×C₆₇	—	2	2	no	—
97	C₂×D₈×C₆₇	—	2	2	no	—
98	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
99	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
100	C₂×M₁₆×C₆₇	—	2	2	no	—
101	C₂×M₁₆×C₆₇	—	2	2	no	—
102	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
103	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄×Q₈×C₆₇	—	2	2	no	—
108	C₄×Q₈×C₆₇	—	2	2	no	—
109	C₄×Q₈×C₆₇	—	4	1	no	—
110	C₄×Q₈×C₆₇	—	4	1	no	—
111	C₄×D₄×C₆₇	—	2	2	no	—
112	C₄×D₄×C₆₇	—	2	2	no	—
113	C₄×D₄×C₆₇	—	4	1	no	—
114	C₄×D₄×C₆₇	—	4	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
127	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
128	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
129	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
130	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
131	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
132	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
133	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
134	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
135	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	—	2	1	no	—
137	C₂×C₄²×C₆₇	—	2	1	no	—
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	#107×C₆₇	—	2	2	no	—
141	#106×C₆₇	—	2	2	no	—
142	#103×C₆₇	—	2	2	no	—
143	#103×C₆₇	—	2	2	no	—
144	#102×C₆₇	—	2	2	no	—
145	#102×C₆₇	—	2	2	no	—
146	#102×C₆₇	—	4	1	no	—
147	#102×C₆₇	—	4	1	no	—
148	#102×C₆₇	—	4	1	no	—
149	#102×C₆₇	—	4	1	no	—
150	C₂²⋊C₈×C₆₇	—	2	2	no	—
151	#97×C₆₇	—	2	2	no	—
152	C₄×C₈×C₆₇	—	2	2	no	—
153	#95×C₆₇	—	2	1	no	—
154	#95×C₆₇	—	2	1	no	—
155	#95×C₆₇	D₄	1	1	yes	characteristic
156	#95×C₆₇	D₄	1	1	yes	characteristic
157	C₂⁴×C₆₇	—	2	1	no	—
158	C₂⁴×C₆₇	—	2	1	no	—
159	C₂⁴×C₆₇	—	2	1	no	—
160	C₂⁴×C₆₇	—	2	4	no	—
161	C₂⁴×C₆₇	—	4	2	no	—
162	C₂⁴×C₆₇	—	4	2	no	—
163	C₂⁴×C₆₇	—	4	2	no	—
164	C₂⁴×C₆₇	—	4	2	no	—
165	C₂⁴×C₆₇	—	8	1	no	—
166	C₂⁴×C₆₇	—	8	1	no	—
167	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
168	#41×C₆₇	—	2	8	no	—
169	#41×C₆₇	—	4	4	no	—
170	#41×C₆₇	—	4	4	no	—
171	#41×C₆₇	—	4	4	no	—
172	C₂×Q₈×C₆₇	—	2	1	no	—
173	C₂×Q₈×C₆₇	—	2	1	no	—
174	C₂×Q₈×C₆₇	—	2	1	no	—
175	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
176	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
177	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
178	C₂×D₄×C₆₇	—	2	8	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	2	1	no	—
182	C₂×D₄×C₆₇	—	2	1	no	—
183	C₂×D₄×C₆₇	—	2	1	no	—
184	C₂×D₄×C₆₇	—	2	1	no	—
185	C₂×D₄×C₆₇	—	2	1	no	—
186	C₂×D₄×C₆₇	—	2	1	no	—
187	C₂×D₄×C₆₇	—	2	1	no	—
188	C₂×D₄×C₆₇	—	2	1	no	—
189	C₂×D₄×C₆₇	—	2	1	no	—
190	C₂×D₄×C₆₇	—	2	1	no	—
191	C₂×D₄×C₆₇	—	2	1	no	—
192	C₂×D₄×C₆₇	—	2	1	no	—
193	C₂×D₄×C₆₇	—	2	1	no	—
194	C₂×D₄×C₆₇	—	4	4	no	—
195	C₂×D₄×C₆₇	—	4	4	no	—
196	C₂×D₄×C₆₇	—	4	4	no	—
197	C₂×D₄×C₆₇	—	4	2	no	—
198	C₂×D₄×C₆₇	—	4	2	no	—
199	C₂×D₄×C₆₇	—	4	2	no	—
200	C₂×D₄×C₆₇	—	4	2	no	—
201	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
202	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
203	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
204	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
205	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
206	C₂²×C₄×C₆₇	—	2	1	no	—
207	C₂²×C₄×C₆₇	—	2	1	no	—
208	C₂²×C₄×C₆₇	—	2	1	no	—
209	C₂²×C₄×C₆₇	—	2	1	no	—
210	C₂²×C₄×C₆₇	—	2	1	no	—
211	C₂²×C₄×C₆₇	—	2	4	no	—
212	C₂²×C₄×C₆₇	—	2	1	no	—
213	C₂²×C₄×C₆₇	—	2	1	no	—
214	C₂²×C₄×C₆₇	—	2	1	no	—
215	C₂²×C₄×C₆₇	—	2	1	no	—
216	C₂²×C₄×C₆₇	—	4	2	no	—
217	C₂²×C₄×C₆₇	—	4	2	no	—
218	C₂²×C₄×C₆₇	—	4	2	no	—
219	C₂²×C₄×C₆₇	—	4	2	no	—
220	C₂²×C₄×C₆₇	—	4	2	no	—
221	C₂²×C₄×C₆₇	—	4	2	no	—
222	C₂²×C₄×C₆₇	—	8	1	no	—
223	C₂²×C₄×C₆₇	—	8	1	no	—
224	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
225	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
227	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
228	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
229	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
230	C₂²×C₄×C₆₇	#41	1	2	yes	—
231	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
232	SD₁₆×C₆₇	—	2	8	no	—
233	SD₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
234	D₈×C₆₇	—	2	8	no	—
235	D₈×C₆₇	—	2	8	no	—
236	M₁₆×C₆₇	—	2	8	no	—
237	C₂×C₈×C₆₇	—	2	1	no	—
238	C₂×C₈×C₆₇	—	2	1	no	—
239	C₂×C₈×C₆₇	—	2	1	no	—
240	C₂×C₈×C₆₇	—	2	1	no	—
241	C₂×C₈×C₆₇	—	2	1	no	—
242	C₂×C₈×C₆₇	—	2	1	no	—
243	C₂×C₈×C₆₇	—	2	1	no	—
244	C₂×C₈×C₆₇	—	2	1	no	—
245	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
246	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
247	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
248	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
249	C₄⋊C₄×C₆₇	—	2	1	no	—
250	C₄⋊C₄×C₆₇	—	2	1	no	—
251	C₄⋊C₄×C₆₇	—	2	1	no	—
252	C₄⋊C₄×C₆₇	—	2	1	no	—
253	C₄⋊C₄×C₆₇	—	2	1	no	—
254	C₄⋊C₄×C₆₇	—	2	1	no	—
255	C₄⋊C₄×C₆₇	—	2	1	no	—
256	C₄⋊C₄×C₆₇	—	2	1	no	—
257	C₄⋊C₄×C₆₇	—	4	2	no	—
258	C₄⋊C₄×C₆₇	—	4	2	no	—
259	C₂²⋊C₄×C₆₇	—	2	2	no	—
260	C₂²⋊C₄×C₆₇	—	4	1	no	—
261	C₂²⋊C₄×C₆₇	—	4	2	no	—
262	C₂²⋊C₄×C₆₇	—	4	2	no	—
263	C₂²⋊C₄×C₆₇	—	4	2	no	—
264	C₂²⋊C₄×C₆₇	—	4	2	no	—
265	C₂²⋊C₄×C₆₇	—	4	2	no	—
266	C₂²⋊C₄×C₆₇	—	4	2	no	—
267	C₂²⋊C₄×C₆₇	—	4	2	no	—
268	C₂²⋊C₄×C₆₇	—	8	1	no	—
269	C₂²⋊C₄×C₆₇	—	8	1	no	—
270	C₂²⋊C₄×C₆₇	—	8	1	no	—
271	C₂²⋊C₄×C₆₇	—	8	1	no	—
272	C₂²⋊C₄×C₆₇	—	8	1	no	—
273	C₄²×C₆₇	—	2	1	no	—
274	C₄²×C₆₇	—	2	1	no	—
275	C₄²×C₆₇	—	2	1	no	—
276	C₄²×C₆₇	—	2	1	no	—
277	C₄²×C₆₇	—	4	2	no	—
278	C₄²×C₆₇	—	4	2	no	—
279	C₂³×C₆₇	—	2	2	no	—
280	C₂³×C₆₇	—	2	2	no	—
281	C₂³×C₆₇	—	2	2	no	—
282	C₂³×C₆₇	—	2	1	no	—
283	C₂³×C₆₇	—	2	1	no	—
284	C₂³×C₆₇	—	2	1	no	—
285	C₂³×C₆₇	—	2	1	no	—
286	C₂³×C₆₇	—	2	1	no	—
287	C₂³×C₆₇	—	2	1	no	—
288	C₂³×C₆₇	—	2	2	no	—
289	C₂³×C₆₇	—	2	2	no	—
290	C₂³×C₆₇	—	4	8	no	—
291	C₂³×C₆₇	—	4	8	no	—
292	C₂³×C₆₇	—	4	1	no	—
293	C₂³×C₆₇	—	4	1	no	—
294	C₂³×C₆₇	—	4	1	no	—
295	C₂³×C₆₇	—	4	1	no	—
296	C₂³×C₆₇	—	4	1	no	—
297	C₂³×C₆₇	—	4	1	no	—
298	C₂³×C₆₇	—	4	1	no	—
299	C₂³×C₆₇	—	4	1	no	—
300	C₂³×C₆₇	—	4	1	no	—
301	C₂³×C₆₇	—	4	1	no	—
302	C₂³×C₆₇	—	4	1	no	—
303	C₂³×C₆₇	—	4	1	no	—
304	C₂³×C₆₇	—	4	1	no	—
305	C₂³×C₆₇	—	4	1	no	—
306	C₂³×C₆₇	—	4	1	no	—
307	C₂³×C₆₇	—	4	1	no	—
308	C₂³×C₆₇	—	8	4	no	—
309	C₂³×C₆₇	—	8	4	no	—
310	C₂³×C₆₇	—	8	4	no	—
311	C₂³×C₆₇	—	8	4	no	—
312	C₂³×C₆₇	—	8	4	no	—
313	C₂³×C₆₇	C₄×D₄	1	2	yes	—
314	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
315	Q₈×C₆₇	—	2	2	no	—
316	Q₈×C₆₇	—	2	2	no	—
317	Q₈×C₆₇	—	2	2	no	—
318	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	complemented
319	D₄×C₆₇	—	2	2	no	—
320	D₄×C₆₇	—	2	2	no	—
321	D₄×C₆₇	—	2	2	no	—
322	D₄×C₆₇	—	2	2	no	—
323	D₄×C₆₇	—	2	2	no	—
324	D₄×C₆₇	—	2	2	no	—
325	D₄×C₆₇	—	2	2	no	—
326	D₄×C₆₇	—	4	8	no	—
327	D₄×C₆₇	—	4	8	no	—
328	D₄×C₆₇	—	4	1	no	—
329	D₄×C₆₇	—	4	1	no	—
330	D₄×C₆₇	—	4	1	no	—
331	D₄×C₆₇	—	4	1	no	—
332	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
333	C₂×C₄×C₆₇	—	2	2	no	—
334	C₂×C₄×C₆₇	—	2	1	no	—
335	C₂×C₄×C₆₇	—	2	2	no	—
336	C₂×C₄×C₆₇	—	2	2	no	—
337	C₂×C₄×C₆₇	—	2	1	no	—
338	C₂×C₄×C₆₇	—	2	1	no	—
339	C₂×C₄×C₆₇	—	2	1	no	—
340	C₂×C₄×C₆₇	—	2	2	no	—
341	C₂×C₄×C₆₇	—	2	2	no	—
342	C₂×C₄×C₆₇	—	2	2	no	—
343	C₂×C₄×C₆₇	—	2	2	no	—
344	C₂×C₄×C₆₇	—	2	1	no	—
345	C₂×C₄×C₆₇	—	2	1	no	—
346	C₂×C₄×C₆₇	—	2	1	no	—
347	C₂×C₄×C₆₇	—	2	1	no	—
348	C₂×C₄×C₆₇	—	2	1	no	—
349	C₂×C₄×C₆₇	—	2	1	no	—
350	C₂×C₄×C₆₇	—	2	1	no	—
351	C₂×C₄×C₆₇	—	2	1	no	—
352	C₂×C₄×C₆₇	—	2	1	no	—
353	C₂×C₄×C₆₇	—	4	8	no	—
354	C₂×C₄×C₆₇	—	4	1	no	—
355	C₂×C₄×C₆₇	—	4	1	no	—
356	C₂×C₄×C₆₇	—	4	1	no	—
357	C₂×C₄×C₆₇	—	4	1	no	—
358	C₂×C₄×C₆₇	—	4	1	no	—
359	C₂×C₄×C₆₇	—	4	1	no	—
360	C₂×C₄×C₆₇	—	4	1	no	—
361	C₂×C₄×C₆₇	—	4	1	no	—
362	C₂×C₄×C₆₇	—	4	1	no	—
363	C₂×C₄×C₆₇	—	4	1	no	—
364	C₂×C₄×C₆₇	—	4	1	no	—
365	C₂×C₄×C₆₇	—	4	1	no	—
366	C₂×C₄×C₆₇	—	8	4	no	—
367	C₂×C₄×C₆₇	—	8	4	no	—
368	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
369	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	complemented
370	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
371	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
372	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
373	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
374	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
375	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
376	C₈×C₆₇	—	2	2	no	—
377	C₈×C₆₇	—	4	1	no	—
378	C₈×C₆₇	—	4	1	no	—
379	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	complemented
380	C₂²×C₆₇	—	2	2	no	—
381	C₂²×C₆₇	—	2	4	no	—
382	C₂²×C₆₇	—	2	2	no	—
383	C₂²×C₆₇	—	2	2	no	—
384	C₂²×C₆₇	—	4	2	no	—
385	C₂²×C₆₇	—	4	2	no	—
386	C₂²×C₆₇	—	4	2	no	—
387	C₂²×C₆₇	—	4	1	no	—
388	C₂²×C₆₇	—	4	1	no	—
389	C₂²×C₆₇	—	4	2	no	—
390	C₂²×C₆₇	—	4	2	no	—
391	C₂²×C₆₇	—	4	1	no	—
392	C₂²×C₆₇	—	4	1	no	—
393	C₂²×C₆₇	—	4	2	no	—
394	C₂²×C₆₇	—	4	2	no	—
395	C₂²×C₆₇	—	4	2	no	—
396	C₂²×C₆₇	—	4	2	no	—
397	C₂²×C₆₇	—	4	2	no	—
398	C₂²×C₆₇	—	4	2	no	—
399	C₂²×C₆₇	—	4	2	no	—
400	C₂²×C₆₇	—	8	16	no	—
401	C₂²×C₆₇	—	8	1	no	—
402	C₂²×C₆₇	—	8	1	no	—
403	C₂²×C₆₇	—	8	1	no	—
404	C₂²×C₆₇	—	8	1	no	—
405	C₂²×C₆₇	—	8	1	no	—
406	C₂²×C₆₇	—	8	1	no	—
407	C₂²×C₆₇	—	8	1	no	—
408	C₂²×C₆₇	—	8	1	no	—
409	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
410	C₂²×C₆₇	D₄²	1	1	yes	characteristic
411	C₂²×C₆₇	D₄²	1	1	yes	characteristic
412	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
413	C₂²×C₆₇	#575	1	1	yes	characteristic
414	C₂²×C₆₇	#576	1	1	yes	characteristic
415	C₂²×C₆₇	#577	1	1	yes	characteristic
416	C₄×C₆₇	—	2	2	no	—
417	C₄×C₆₇	—	2	4	no	—
418	C₄×C₆₇	—	2	2	no	—
419	C₄×C₆₇	—	4	1	no	—
420	C₄×C₆₇	—	4	1	no	—
421	C₄×C₆₇	—	4	1	no	—
422	C₄×C₆₇	—	8	1	no	—
423	C₄×C₆₇	—	8	1	no	—
424	C₄×C₆₇	—	8	1	no	—
425	C₄×C₆₇	—	8	1	no	—
426	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
427	C₄×C₆₇	#518	1	1	yes	characteristic
428	C₄×C₆₇	#546	1	1	yes	characteristic
429	C₄×C₆₇	#546	1	1	yes	characteristic
430	#22443	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
431	C₂×C₆₇	—	4	2	no	—
432	C₂×C₆₇	—	4	4	no	—
433	C₂×C₆₇	—	8	2	no	—
434	C₂×C₆₇	—	8	2	no	—
435	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
436	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
437	C₂×C₆₇	#2946	1	1	yes	characteristic
438	C₂×C₆₇	#3265	1	1	yes	characteristic
439	C₂×C₆₇	#3268	1	1	yes	characteristic
440	C₂×C₆₇	#3274	1	1	yes	characteristic
441	C₂×C₆₇	#3275	1	1	yes	characteristic
442	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
443	C₂×#440	C₂×C₆₇	1	1	yes	characteristic, complemented
444	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
445	#2341	C₂×C₆₇	1	1	yes	characteristic, complemented
446	D₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
447	#1972	C₂×C₆₇	1	1	yes	characteristic, complemented
448	#1970	C₂×C₆₇	1	1	yes	characteristic, complemented
449	#1880	C₂×C₆₇	1	1	yes	characteristic, complemented
450	#1880	C₂×C₆₇	1	1	yes	characteristic, complemented
451	#1879	C₂×C₆₇	1	1	yes	characteristic, complemented
452	#1879	C₂×C₆₇	1	1	yes	characteristic, complemented
453	#1809	C₂×C₆₇	1	1	yes	characteristic, complemented
454	#1788	C₂×C₆₇	1	1	yes	characteristic, complemented
455	#1786	C₂×C₆₇	1	1	yes	characteristic, complemented
456	#1753	C₂×C₆₇	1	1	yes	characteristic, complemented
457	C₆₇	#22443	1	1	yes	characteristic, direct product factor, Sylow
458	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
459	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
460	C₂×AGL(1,ℤ/8)	—	2	4	no	—
461	C₂×AGL(1,ℤ/8)	—	2	4	no	—
462	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	8	yes	complemented
463	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
464	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
465	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
466	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
467	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
468	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
469	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
470	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
471	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
472	C₂²×C₂²⋊C₄	—	2	1	no	—
473	#440	—	2	4	no	—
474	C₄×SD₁₆	—	2	4	no	—
475	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
476	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic, complemented
477	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
478	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic, complemented
479	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
480	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
481	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
482	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
483	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
484	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic, complemented
485	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
486	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic, complemented
487	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
488	#390	—	2	1	no	—
489	#390	—	2	1	no	—
490	#390	C₂²×C₆₇	1	1	yes	characteristic
491	#390	C₂²×C₆₇	1	1	yes	characteristic
492	#390	C₂²×C₆₇	1	1	yes	characteristic, complemented
493	#390	C₂²×C₆₇	1	1	yes	characteristic, complemented
494	#386	—	2	1	no	—
495	#386	—	2	1	no	—
496	#380	—	2	1	no	—
497	#380	C₂²×C₆₇	1	1	yes	characteristic
498	#380	C₂²×C₆₇	1	1	yes	characteristic
499	#379	—	2	1	no	—
500	C₄×C₂²⋊C₄	—	2	1	no	—
501	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
502	#340	C₂²×C₆₇	1	1	yes	characteristic, complemented
503	#340	C₂²×C₆₇	1	1	yes	characteristic, complemented
504	#336	C₂²×C₆₇	1	1	yes	characteristic, complemented
505	#336	C₂²×C₆₇	1	1	yes	characteristic, complemented
506	C₂⁵	—	2	1	no	—
507	C₂×#41	—	2	2	no	—
508	C₂×#41	—	2	2	no	—
509	C₂×#41	—	2	2	no	—
510	C₂×#41	—	4	1	no	—
511	C₂×#41	—	4	1	no	—
512	C₂×#41	C₂×C₄×C₆₇	1	4	yes	complemented
513	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
514	C₂²×D₄	—	2	2	no	—
515	C₂²×D₄	—	2	2	no	—
516	C₂²×D₄	—	2	1	no	—
517	C₂²×D₄	—	2	1	no	—
518	C₂²×D₄	—	2	2	no	—
519	C₂²×D₄	—	4	1	no	—
520	C₂²×D₄	—	4	1	no	—
521	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
522	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
523	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
524	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
525	C₂³×C₄	—	2	1	no	—
526	C₂³×C₄	—	2	1	no	—
527	C₂³×C₄	—	2	1	no	—
528	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
529	AGL(1,ℤ/8)	—	2	32	no	—
530	C₂×SD₁₆	—	2	2	no	—
531	C₂×SD₁₆	—	2	2	no	—
532	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
533	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
534	C₂×SD₁₆	D₄×C₆₇	1	4	yes	—
535	C₂×SD₁₆	D₄×C₆₇	1	4	yes	—
536	C₂×D₈	—	2	2	no	—
537	C₂×D₈	—	2	2	no	—
538	C₂×D₈	—	2	2	no	—
539	C₂×D₈	—	2	2	no	—
540	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
541	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
542	C₂×M₁₆	—	2	2	no	—
543	C₂×M₁₆	—	2	2	no	—
544	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
545	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
546	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
547	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
548	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
549	C₄×Q₈	—	2	2	no	—
550	C₄×Q₈	—	2	2	no	—
551	C₄×Q₈	—	4	1	no	—
552	C₄×Q₈	—	4	1	no	—
553	C₄×D₄	—	2	2	no	—
554	C₄×D₄	—	2	2	no	—
555	C₄×D₄	—	4	1	no	—
556	C₄×D₄	—	4	1	no	—
557	C₂×C₄⋊C₄	—	2	1	no	—
558	C₂×C₄⋊C₄	—	2	1	no	—
559	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
560	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
561	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
562	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
563	C₂×C₂²⋊C₄	—	2	1	no	—
564	C₂×C₂²⋊C₄	—	2	1	no	—
565	C₂×C₂²⋊C₄	—	2	1	no	—
566	C₂×C₂²⋊C₄	—	2	1	no	—
567	C₂×C₂²⋊C₄	—	2	1	no	—
568	C₂×C₂²⋊C₄	—	2	1	no	—
569	C₂×C₂²⋊C₄	—	2	1	no	—
570	C₂×C₂²⋊C₄	—	2	1	no	—
571	C₂×C₂²⋊C₄	—	2	1	no	—
572	C₂×C₂²⋊C₄	—	2	1	no	—
573	C₂×C₂²⋊C₄	—	2	1	no	—
574	C₂×C₂²⋊C₄	—	2	1	no	—
575	C₂×C₂²⋊C₄	—	4	4	no	—
576	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
577	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
578	C₂×C₄²	—	2	1	no	—
579	C₂×C₄²	—	2	1	no	—
580	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
581	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
582	#107	—	2	2	no	—
583	#106	—	2	2	no	—
584	#103	—	2	2	no	—
585	#103	—	2	2	no	—
586	#102	—	2	2	no	—
587	#102	—	2	2	no	—
588	#102	—	4	1	no	—
589	#102	—	4	1	no	—
590	#102	—	4	1	no	—
591	#102	—	4	1	no	—
592	C₂²⋊C₈	—	2	2	no	—
593	#97	—	2	2	no	—
594	C₄×C₈	—	2	2	no	—
595	#95	—	2	1	no	—
596	#95	—	2	1	no	—
597	#95	D₄×C₆₇	1	1	yes	characteristic
598	#95	D₄×C₆₇	1	1	yes	characteristic
599	C₂⁴	—	2	1	no	—
600	C₂⁴	—	2	1	no	—
601	C₂⁴	—	2	1	no	—
602	C₂⁴	—	2	4	no	—
603	C₂⁴	—	4	2	no	—
604	C₂⁴	—	4	2	no	—
605	C₂⁴	—	4	2	no	—
606	C₂⁴	—	4	2	no	—
607	C₂⁴	—	8	1	no	—
608	C₂⁴	—	8	1	no	—
609	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
610	#41	—	2	8	no	—
611	#41	—	4	4	no	—
612	#41	—	4	4	no	—
613	#41	—	4	4	no	—
614	C₂×Q₈	—	2	1	no	—
615	C₂×Q₈	—	2	1	no	—
616	C₂×Q₈	—	2	1	no	—
617	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
618	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
619	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
620	C₂×D₄	—	2	8	no	—
621	C₂×D₄	—	2	4	no	—
622	C₂×D₄	—	2	4	no	—
623	C₂×D₄	—	2	1	no	—
624	C₂×D₄	—	2	1	no	—
625	C₂×D₄	—	2	1	no	—
626	C₂×D₄	—	2	1	no	—
627	C₂×D₄	—	2	1	no	—
628	C₂×D₄	—	2	1	no	—
629	C₂×D₄	—	2	1	no	—
630	C₂×D₄	—	2	1	no	—
631	C₂×D₄	—	2	1	no	—
632	C₂×D₄	—	2	1	no	—
633	C₂×D₄	—	2	1	no	—
634	C₂×D₄	—	2	1	no	—
635	C₂×D₄	—	2	1	no	—
636	C₂×D₄	—	4	4	no	—
637	C₂×D₄	—	4	4	no	—
638	C₂×D₄	—	4	4	no	—
639	C₂×D₄	—	4	2	no	—
640	C₂×D₄	—	4	2	no	—
641	C₂×D₄	—	4	2	no	—
642	C₂×D₄	—	4	2	no	—
643	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
644	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
645	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
646	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
647	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
648	C₂²×C₄	—	2	1	no	—
649	C₂²×C₄	—	2	1	no	—
650	C₂²×C₄	—	2	1	no	—
651	C₂²×C₄	—	2	1	no	—
652	C₂²×C₄	—	2	1	no	—
653	C₂²×C₄	—	2	4	no	—
654	C₂²×C₄	—	2	1	no	—
655	C₂²×C₄	—	2	1	no	—
656	C₂²×C₄	—	2	1	no	—
657	C₂²×C₄	—	2	1	no	—
658	C₂²×C₄	—	4	2	no	—
659	C₂²×C₄	—	4	2	no	—
660	C₂²×C₄	—	4	2	no	—
661	C₂²×C₄	—	4	2	no	—
662	C₂²×C₄	—	4	2	no	—
663	C₂²×C₄	—	4	2	no	—
664	C₂²×C₄	—	8	1	no	—
665	C₂²×C₄	—	8	1	no	—
666	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
667	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
668	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
669	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
670	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
671	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
672	C₂²×C₄	#41×C₆₇	1	2	yes	—
673	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
674	SD₁₆	—	2	8	no	—
675	SD₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
676	D₈	—	2	8	no	—
677	D₈	—	2	8	no	—
678	M₁₆	—	2	8	no	—
679	C₂×C₈	—	2	1	no	—
680	C₂×C₈	—	2	1	no	—
681	C₂×C₈	—	2	1	no	—
682	C₂×C₈	—	2	1	no	—
683	C₂×C₈	—	2	1	no	—
684	C₂×C₈	—	2	1	no	—
685	C₂×C₈	—	2	1	no	—
686	C₂×C₈	—	2	1	no	—
687	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
688	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
689	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
690	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
691	C₄⋊C₄	—	2	1	no	—
692	C₄⋊C₄	—	2	1	no	—
693	C₄⋊C₄	—	2	1	no	—
694	C₄⋊C₄	—	2	1	no	—
695	C₄⋊C₄	—	2	1	no	—
696	C₄⋊C₄	—	2	1	no	—
697	C₄⋊C₄	—	2	1	no	—
698	C₄⋊C₄	—	2	1	no	—
699	C₄⋊C₄	—	4	2	no	—
700	C₄⋊C₄	—	4	2	no	—
701	C₂²⋊C₄	—	2	2	no	—
702	C₂²⋊C₄	—	4	1	no	—
703	C₂²⋊C₄	—	4	2	no	—
704	C₂²⋊C₄	—	4	2	no	—
705	C₂²⋊C₄	—	4	2	no	—
706	C₂²⋊C₄	—	4	2	no	—
707	C₂²⋊C₄	—	4	2	no	—
708	C₂²⋊C₄	—	4	2	no	—
709	C₂²⋊C₄	—	4	2	no	—
710	C₂²⋊C₄	—	8	1	no	—
711	C₂²⋊C₄	—	8	1	no	—
712	C₂²⋊C₄	—	8	1	no	—
713	C₂²⋊C₄	—	8	1	no	—
714	C₂²⋊C₄	—	8	1	no	—
715	C₄²	—	2	1	no	—
716	C₄²	—	2	1	no	—
717	C₄²	—	2	1	no	—
718	C₄²	—	2	1	no	—
719	C₄²	—	4	2	no	—
720	C₄²	—	4	2	no	—
721	C₂³	—	2	2	no	—
722	C₂³	—	2	2	no	—
723	C₂³	—	2	2	no	—
724	C₂³	—	2	1	no	—
725	C₂³	—	2	1	no	—
726	C₂³	—	2	1	no	—
727	C₂³	—	2	1	no	—
728	C₂³	—	2	1	no	—
729	C₂³	—	2	1	no	—
730	C₂³	—	2	2	no	—
731	C₂³	—	2	2	no	—
732	C₂³	—	4	8	no	—
733	C₂³	—	4	8	no	—
734	C₂³	—	4	1	no	—
735	C₂³	—	4	1	no	—
736	C₂³	—	4	1	no	—
737	C₂³	—	4	1	no	—
738	C₂³	—	4	1	no	—
739	C₂³	—	4	1	no	—
740	C₂³	—	4	1	no	—
741	C₂³	—	4	1	no	—
742	C₂³	—	4	1	no	—
743	C₂³	—	4	1	no	—
744	C₂³	—	4	1	no	—
745	C₂³	—	4	1	no	—
746	C₂³	—	4	1	no	—
747	C₂³	—	4	1	no	—
748	C₂³	—	4	1	no	—
749	C₂³	—	4	1	no	—
750	C₂³	—	8	4	no	—
751	C₂³	—	8	4	no	—
752	C₂³	—	8	4	no	—
753	C₂³	—	8	4	no	—
754	C₂³	—	8	4	no	—
755	C₂³	C₄×D₄×C₆₇	1	2	yes	—
756	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
757	Q₈	—	2	2	no	—
758	Q₈	—	2	2	no	—
759	Q₈	—	2	2	no	—
760	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	complemented
761	D₄	—	2	2	no	—
762	D₄	—	2	2	no	—
763	D₄	—	2	2	no	—
764	D₄	—	2	2	no	—
765	D₄	—	2	2	no	—
766	D₄	—	2	2	no	—
767	D₄	—	2	2	no	—
768	D₄	—	4	8	no	—
769	D₄	—	4	8	no	—
770	D₄	—	4	1	no	—
771	D₄	—	4	1	no	—
772	D₄	—	4	1	no	—
773	D₄	—	4	1	no	—
774	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
775	C₂×C₄	—	2	2	no	—
776	C₂×C₄	—	2	1	no	—
777	C₂×C₄	—	2	2	no	—
778	C₂×C₄	—	2	2	no	—
779	C₂×C₄	—	2	1	no	—
780	C₂×C₄	—	2	1	no	—
781	C₂×C₄	—	2	1	no	—
782	C₂×C₄	—	2	2	no	—
783	C₂×C₄	—	2	2	no	—
784	C₂×C₄	—	2	2	no	—
785	C₂×C₄	—	2	2	no	—
786	C₂×C₄	—	2	1	no	—
787	C₂×C₄	—	2	1	no	—
788	C₂×C₄	—	2	1	no	—
789	C₂×C₄	—	2	1	no	—
790	C₂×C₄	—	2	1	no	—
791	C₂×C₄	—	2	1	no	—
792	C₂×C₄	—	2	1	no	—
793	C₂×C₄	—	2	1	no	—
794	C₂×C₄	—	2	1	no	—
795	C₂×C₄	—	4	8	no	—
796	C₂×C₄	—	4	1	no	—
797	C₂×C₄	—	4	1	no	—
798	C₂×C₄	—	4	1	no	—
799	C₂×C₄	—	4	1	no	—
800	C₂×C₄	—	4	1	no	—
801	C₂×C₄	—	4	1	no	—
802	C₂×C₄	—	4	1	no	—
803	C₂×C₄	—	4	1	no	—
804	C₂×C₄	—	4	1	no	—
805	C₂×C₄	—	4	1	no	—
806	C₂×C₄	—	4	1	no	—
807	C₂×C₄	—	4	1	no	—
808	C₂×C₄	—	8	4	no	—
809	C₂×C₄	—	8	4	no	—
810	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
811	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	complemented
812	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
813	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
814	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
815	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
816	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
817	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
818	C₈	—	2	2	no	—
819	C₈	—	4	1	no	—
820	C₈	—	4	1	no	—
821	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	complemented
822	C₂²	—	2	2	no	—
823	C₂²	—	2	4	no	—
824	C₂²	—	2	2	no	—
825	C₂²	—	2	2	no	—
826	C₂²	—	4	2	no	—
827	C₂²	—	4	1	no	—
828	C₂²	—	4	1	no	—
829	C₂²	—	4	2	no	—
830	C₂²	—	4	2	no	—
831	C₂²	—	4	1	no	—
832	C₂²	—	4	1	no	—
833	C₂²	—	4	2	no	—
834	C₂²	—	4	2	no	—
835	C₂²	—	4	2	no	—
836	C₂²	—	4	2	no	—
837	C₂²	—	4	2	no	—
838	C₂²	—	4	2	no	—
839	C₂²	—	4	2	no	—
840	C₂²	—	4	2	no	—
841	C₂²	—	4	2	no	—
842	C₂²	—	8	16	no	—
843	C₂²	—	8	1	no	—
844	C₂²	—	8	1	no	—
845	C₂²	—	8	1	no	—
846	C₂²	—	8	1	no	—
847	C₂²	—	8	1	no	—
848	C₂²	—	8	1	no	—
849	C₂²	—	8	1	no	—
850	C₂²	—	8	1	no	—
851	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
852	C₂²	D₄²×C₆₇	1	1	yes	characteristic
853	C₂²	D₄²×C₆₇	1	1	yes	characteristic
854	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
855	C₂²	#575×C₆₇	1	1	yes	characteristic
856	C₂²	#576×C₆₇	1	1	yes	characteristic
857	C₂²	#577×C₆₇	1	1	yes	characteristic
858	C₄	—	2	2	no	—
859	C₄	—	2	2	no	—
860	C₄	—	2	4	no	—
861	C₄	—	4	1	no	—
862	C₄	—	4	1	no	—
863	C₄	—	4	1	no	—
864	C₄	—	8	1	no	—
865	C₄	—	8	1	no	—
866	C₄	—	8	1	no	—
867	C₄	—	8	1	no	—
868	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
869	C₄	#518×C₆₇	1	1	yes	characteristic
870	C₄	#546×C₆₇	1	1	yes	characteristic
871	C₄	#546×C₆₇	1	1	yes	characteristic
872	C₂	—	4	2	no	—
873	C₂	—	4	4	no	—
874	C₂	—	8	2	no	—
875	C₂	—	8	2	no	—
876	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
877	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
878	C₂	C₆₇×#2946	1	1	yes	characteristic
879	C₂	C₆₇×#3265	1	1	yes	characteristic
880	C₂	C₆₇×#3268	1	1	yes	characteristic
881	C₂	C₆₇×#3274	1	1	yes	characteristic
882	C₂	C₆₇×#3275	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×AGL(1,ℤ/8)×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₂²⋊C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#440×C₆₇	C₂	semidirect
3. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
4. C₆₇×#2341	C₂	semidirect
5. D₄×C₂²⋊C₄×C₆₇	C₂	semidirect
6. C₆₇×#1972	C₂	semidirect
7. C₆₇×#1970	C₂	semidirect
8. C₆₇×#1880	C₂	semidirect
9. C₆₇×#1880	C₂	semidirect
10. C₆₇×#1879	C₂	semidirect
11. C₆₇×#1879	C₂	semidirect
12. C₆₇×#1809	C₂	semidirect
13. C₆₇×#1788	C₂	semidirect
14. C₆₇×#1786	C₂	semidirect
15. C₆₇×#1753	C₂	semidirect
20. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
26. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
27. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
33. C₂×#107×C₆₇	C₂²	semidirect
34. C₂×#106×C₆₇	C₂²	semidirect
35. C₂×#103×C₆₇	C₂²	semidirect
36. C₂×#103×C₆₇	C₂²	semidirect
41. C₂×#102×C₆₇	C₂²	semidirect
42. C₂×#102×C₆₇	C₂²	semidirect
44. C₂×#97×C₆₇	C₂²	semidirect
45. C₂×C₄×C₈×C₆₇	C₂²	semidirect
50. #390×C₆₇	C₂²	semidirect
51. #390×C₆₇	C₂²	semidirect
60. #340×C₆₇	C₂²	semidirect
61. #340×C₆₇	C₂²	semidirect
62. #336×C₆₇	C₂²	semidirect
63. #336×C₆₇	C₂²	semidirect
70. C₂×#41×C₆₇	C₂×C₄	semidirect
91. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
99. C₂×D₈×C₆₇	C₂×C₄	semidirect
102. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
233. SD₁₆×C₆₇	C₂²⋊C₄	semidirect
318. Q₈×C₆₇	C₂×C₂²⋊C₄	semidirect
369. C₂×C₄×C₆₇	AGL(1,ℤ/8)	semidirect
379. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
430. #22443	C₆₇	direct (N × H)
443. C₂×#440	C₂×C₆₇	semidirect
444. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
445. #2341	C₂×C₆₇	semidirect
446. D₄×C₂²⋊C₄	C₂×C₆₇	semidirect
447. #1972	C₂×C₆₇	semidirect
448. #1970	C₂×C₆₇	semidirect
449. #1880	C₂×C₆₇	semidirect
450. #1880	C₂×C₆₇	semidirect
451. #1879	C₂×C₆₇	semidirect
452. #1879	C₂×C₆₇	semidirect
453. #1809	C₂×C₆₇	semidirect
454. #1788	C₂×C₆₇	semidirect
455. #1786	C₂×C₆₇	semidirect
456. #1753	C₂×C₆₇	semidirect
457. C₆₇	#22443	direct (N × H)
462. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
468. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
469. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
475. C₂×#107	C₂²×C₆₇	semidirect
476. C₂×#106	C₂²×C₆₇	semidirect
477. C₂×#103	C₂²×C₆₇	semidirect
478. C₂×#103	C₂²×C₆₇	semidirect
483. C₂×#102	C₂²×C₆₇	semidirect
484. C₂×#102	C₂²×C₆₇	semidirect
486. C₂×#97	C₂²×C₆₇	semidirect
487. C₂×C₄×C₈	C₂²×C₆₇	semidirect
492. #390	C₂²×C₆₇	semidirect
493. #390	C₂²×C₆₇	semidirect
502. #340	C₂²×C₆₇	semidirect
503. #340	C₂²×C₆₇	semidirect
504. #336	C₂²×C₆₇	semidirect
505. #336	C₂²×C₆₇	semidirect
512. C₂×#41	C₂×C₄×C₆₇	semidirect
533. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
541. C₂×D₈	C₂×C₄×C₆₇	semidirect
544. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
675. SD₁₆	C₂²⋊C₄×C₆₇	semidirect
760. Q₈	C₂×C₂²⋊C₄×C₆₇	semidirect
811. C₂×C₄	AGL(1,ℤ/8)×C₆₇	semidirect
821. C₈	C₂×C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#440×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×#2341	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1972	2
C₆₇×#1970	2
C₆₇×#1880	2
C₆₇×#1879	2
C₆₇×#1809	2
C₆₇×#1788	2
C₆₇×#1786	2
C₆₇×#1753	2
#22443	67

Subgroups — C67×#22443 #30701694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22443 #30701694