Subgroups — C₆₇×#61367 #30740618

Order 17152.

1826 conjugacy classes of subgroups.

The subgroup lattice has 864 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3389	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3388	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3385	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3384	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3381	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3380	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3166	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3165	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3080	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3079	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#3077	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#3077	C₂	1	2	yes	maximal, complemented
15	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
17	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	2	yes	maximal
18	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
19	Q₈²×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	#557×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	#557×C₆₇	C₂²	1	2	yes	complemented
22	#554×C₆₇	—	2	1	no	—
23	#554×C₆₇	—	2	2	no	—
24	#554×C₆₇	—	2	1	no	—
25	#554×C₆₇	—	2	1	no	—
26	#554×C₆₇	—	2	2	no	—
27	#554×C₆₇	C₂²	1	2	yes	—
28	#554×C₆₇	C₂²	1	2	yes	—
29	#554×C₆₇	C₂²	1	4	yes	—
30	#554×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#554×C₆₇	C₂²	1	2	yes	complemented
32	D₄×Q₈×C₆₇	—	2	1	no	—
33	D₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#533×C₆₇	—	2	2	no	—
35	#533×C₆₇	—	2	1	no	—
36	#533×C₆₇	C₂²	1	1	yes	characteristic
37	#533×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#125×C₆₇	C₂²	1	2	yes	—
40	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
41	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
42	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×#41×C₆₇	C₂²	1	2	yes	—
44	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
45	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
49	#501×C₆₇	C₂²	1	4	yes	complemented
50	#501×C₆₇	C₂²	1	4	yes	complemented
51	#501×C₆₇	C₂²	1	4	yes	complemented
52	#500×C₆₇	C₂²	1	2	yes	complemented
53	#499×C₆₇	C₂²	1	4	yes	complemented
54	#499×C₆₇	C₂²	1	4	yes	complemented
55	#498×C₆₇	C₂²	1	2	yes	complemented
56	#479×C₆₇	C₂²	1	4	yes	complemented
57	#479×C₆₇	C₂²	1	4	yes	complemented
58	#478×C₆₇	C₂²	1	4	yes	—
59	#478×C₆₇	C₂²	1	4	yes	—
60	#478×C₆₇	C₂²	1	4	yes	—
61	#478×C₆₇	C₂²	1	4	yes	complemented
62	#478×C₆₇	C₂²	1	4	yes	complemented
63	#477×C₆₇	C₂²	1	2	yes	complemented
64	#476×C₆₇	C₂²	1	2	yes	—
65	#476×C₆₇	C₂²	1	4	yes	—
66	#476×C₆₇	C₂²	1	2	yes	complemented
67	#475×C₆₇	C₂²	1	2	yes	complemented
68	#474×C₆₇	C₂²	1	2	yes	—
69	#474×C₆₇	C₂²	1	4	yes	—
70	#474×C₆₇	C₂²	1	2	yes	complemented
71	#446×C₆₇	C₂²	1	2	yes	complemented
72	#446×C₆₇	C₂²	1	4	yes	complemented
73	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
74	#442×C₆₇	C₂²	1	4	yes	—
75	#442×C₆₇	C₂²	1	2	yes	—
76	#440×C₆₇	C₂²	1	4	yes	—
77	#440×C₆₇	C₂²	1	2	yes	—
78	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
79	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
80	C₄×D₈×C₆₇	C₂²	1	2	yes	—
81	C₄×D₈×C₆₇	C₂²	1	4	yes	—
82	#428×C₆₇	C₂²	1	2	yes	—
83	#428×C₆₇	C₂²	1	2	yes	—
84	#428×C₆₇	C₂²	1	2	yes	—
85	#428×C₆₇	C₂²	1	2	yes	—
86	#428×C₆₇	C₂²	1	1	yes	characteristic
87	#423×C₆₇	C₂²	1	1	yes	characteristic
88	#423×C₆₇	C₂²	1	2	yes	—
89	#418×C₆₇	C₂²	1	1	yes	characteristic
90	#418×C₆₇	C₂²	1	2	yes	—
91	#417×C₆₇	C₂²	1	2	yes	—
92	#417×C₆₇	C₂²	1	1	yes	characteristic
93	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
94	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
95	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×Q₈×C₆₇	—	2	1	no	—
97	C₂²×Q₈×C₆₇	—	2	1	no	—
98	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
99	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂³×C₄×C₆₇	—	2	1	no	—
101	C₂³×C₄×C₆₇	—	2	2	no	—
102	AGL(1,ℤ/8)×C₆₇	Q₈	1	8	yes	complemented
103	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
104	C₂×D₈×C₆₇	C₂³	1	2	yes	—
105	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
107	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
108	C₄⋊Q₈×C₆₇	—	2	1	no	—
109	C₄⋊Q₈×C₆₇	—	2	2	no	—
110	C₄⋊Q₈×C₆₇	—	2	1	no	—
111	C₄⋊Q₈×C₆₇	—	2	2	no	—
112	C₄⋊Q₈×C₆₇	—	2	1	no	—
113	C₄⋊Q₈×C₆₇	—	2	1	no	—
114	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
115	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
116	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
117	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
118	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
119	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
120	#125×C₆₇	—	2	2	no	—
121	#125×C₆₇	—	2	2	no	—
122	#125×C₆₇	—	2	2	no	—
123	#125×C₆₇	—	2	2	no	—
124	#125×C₆₇	—	2	2	no	—
125	#125×C₆₇	C₂³	1	2	yes	—
126	#125×C₆₇	C₂³	1	2	yes	—
127	#125×C₆₇	C₂³	1	2	yes	—
128	#125×C₆₇	C₂³	1	4	yes	—
129	#125×C₆₇	C₂³	1	4	yes	—
130	C₂²⋊Q₈×C₆₇	—	2	2	no	—
131	C₂²⋊Q₈×C₆₇	—	2	2	no	—
132	C₂²⋊Q₈×C₆₇	—	2	4	no	—
133	C₂²⋊Q₈×C₆₇	—	2	1	no	—
134	C₂²⋊Q₈×C₆₇	—	2	2	no	—
135	C₂²⋊Q₈×C₆₇	—	2	1	no	—
136	C₂²⋊Q₈×C₆₇	—	2	2	no	—
137	C₂²⋊Q₈×C₆₇	—	2	1	no	—
138	C₂²⋊Q₈×C₆₇	—	2	2	no	—
139	C₂²⋊Q₈×C₆₇	—	2	4	no	—
140	C₂²⋊Q₈×C₆₇	—	2	4	no	—
141	C₂²⋊Q₈×C₆₇	—	2	4	no	—
142	C₂²⋊Q₈×C₆₇	—	2	4	no	—
143	C₂²⋊Q₈×C₆₇	—	2	1	no	—
144	C₂²⋊Q₈×C₆₇	—	2	2	no	—
145	C₂²⋊Q₈×C₆₇	—	2	2	no	—
146	C₂²⋊Q₈×C₆₇	—	2	2	no	—
147	C₂²⋊Q₈×C₆₇	—	2	2	no	—
148	C₄×Q₈×C₆₇	—	2	1	no	—
149	C₄×Q₈×C₆₇	—	2	1	no	—
150	C₄×Q₈×C₆₇	—	2	2	no	—
151	C₄×Q₈×C₆₇	—	2	2	no	—
152	C₄×Q₈×C₆₇	—	2	1	no	—
153	C₄×Q₈×C₆₇	—	2	2	no	—
154	C₄×Q₈×C₆₇	—	2	1	no	—
155	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
156	C₄×Q₈×C₆₇	D₄	1	2	yes	—
157	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
158	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
159	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
160	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
161	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
162	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
163	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
164	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
165	C₄×D₄×C₆₇	—	2	1	no	—
166	C₄×D₄×C₆₇	—	2	2	no	—
167	C₄×D₄×C₆₇	—	2	1	no	—
168	C₄×D₄×C₆₇	—	2	2	no	—
169	C₄×D₄×C₆₇	—	2	2	no	—
170	C₄×D₄×C₆₇	—	2	1	no	—
171	C₄×D₄×C₆₇	—	2	2	no	—
172	C₄×D₄×C₆₇	—	2	1	no	—
173	C₄×D₄×C₆₇	C₂³	1	2	yes	—
174	C₄×D₄×C₆₇	C₂³	1	4	yes	—
175	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
176	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
177	C₄×D₄×C₆₇	C₂³	1	4	yes	—
178	C₄×D₄×C₆₇	C₂³	1	2	yes	—
179	C₄×D₄×C₆₇	C₂³	1	2	yes	—
180	C₄×D₄×C₆₇	C₂³	1	2	yes	—
181	#117×C₆₇	—	2	2	no	—
182	#117×C₆₇	—	2	1	no	—
183	#117×C₆₇	C₂³	1	1	yes	characteristic
184	#117×C₆₇	C₂³	1	2	yes	—
185	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
192	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
194	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
195	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
196	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
197	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
198	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₂×C₄²×C₆₇	—	2	2	no	—
200	C₂×C₄²×C₆₇	—	2	1	no	—
201	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
202	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
203	#107×C₆₇	C₂³	1	4	yes	—
204	#107×C₆₇	C₂³	1	4	yes	—
205	#107×C₆₇	C₂³	1	4	yes	—
206	#107×C₆₇	C₂³	1	2	yes	—
207	#107×C₆₇	C₂³	1	4	yes	—
208	#106×C₆₇	C₂³	1	4	yes	—
209	#106×C₆₇	C₂³	1	4	yes	—
210	#106×C₆₇	C₂³	1	4	yes	—
211	#106×C₆₇	C₂³	1	4	yes	—
212	#106×C₆₇	C₂³	1	2	yes	—
213	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
214	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
215	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
216	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
217	#103×C₆₇	C₂³	1	4	yes	—
218	#103×C₆₇	C₂³	1	2	yes	—
219	#102×C₆₇	C₂³	1	2	yes	—
220	#102×C₆₇	C₂³	1	4	yes	—
221	#102×C₆₇	C₂³	1	4	yes	—
222	#102×C₆₇	C₂³	1	4	yes	—
223	#102×C₆₇	C₂³	1	2	yes	—
224	#102×C₆₇	C₂³	1	2	yes	—
225	#97×C₆₇	C₂³	1	4	yes	—
226	#97×C₆₇	C₂³	1	2	yes	—
227	C₄×C₈×C₆₇	C₂³	1	2	yes	—
228	C₄×C₈×C₆₇	C₂³	1	4	yes	—
229	C₂⁴×C₆₇	—	2	1	no	—
230	#41×C₆₇	—	2	2	no	—
231	#41×C₆₇	C₂×Q₈	1	4	yes	—
232	C₂×Q₈×C₆₇	—	2	2	no	—
233	C₂×Q₈×C₆₇	—	2	1	no	—
234	C₂×Q₈×C₆₇	—	2	1	no	—
235	C₂×Q₈×C₆₇	—	2	2	no	—
236	C₂×Q₈×C₆₇	—	2	2	no	—
237	C₂×Q₈×C₆₇	—	2	2	no	—
238	C₂×Q₈×C₆₇	—	4	2	no	—
239	C₂×Q₈×C₆₇	—	4	4	no	—
240	C₂×Q₈×C₆₇	—	8	2	no	—
241	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
242	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
243	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
244	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
245	C₂×D₄×C₆₇	—	2	1	no	—
246	C₂×D₄×C₆₇	—	2	1	no	—
247	C₂×D₄×C₆₇	—	2	2	no	—
248	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
249	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
250	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
251	C₂²×C₄×C₆₇	—	2	1	no	—
252	C₂²×C₄×C₆₇	—	2	1	no	—
253	C₂²×C₄×C₆₇	—	2	2	no	—
254	C₂²×C₄×C₆₇	—	2	2	no	—
255	C₂²×C₄×C₆₇	—	2	1	no	—
256	C₂²×C₄×C₆₇	—	2	2	no	—
257	C₂²×C₄×C₆₇	—	2	2	no	—
258	C₂²×C₄×C₆₇	—	2	1	no	—
259	C₂²×C₄×C₆₇	—	2	2	no	—
260	C₂²×C₄×C₆₇	—	2	4	no	—
261	C₂²×C₄×C₆₇	—	2	4	no	—
262	C₂²×C₄×C₆₇	—	4	4	no	—
263	C₂²×C₄×C₆₇	—	8	4	no	—
264	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
265	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
266	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
267	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
268	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
269	SD₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
270	D₈×C₆₇	C₂×Q₈	1	8	yes	complemented
271	M₁₆×C₆₇	—	2	2	no	—
272	M₁₆×C₆₇	—	2	4	no	—
273	M₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
274	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
275	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
276	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
277	C₄⋊C₄×C₆₇	—	2	4	no	—
278	C₄⋊C₄×C₆₇	—	2	4	no	—
279	C₄⋊C₄×C₆₇	—	2	2	no	—
280	C₄⋊C₄×C₆₇	—	2	2	no	—
281	C₄⋊C₄×C₆₇	—	2	2	no	—
282	C₄⋊C₄×C₆₇	—	2	2	no	—
283	C₄⋊C₄×C₆₇	—	2	2	no	—
284	C₄⋊C₄×C₆₇	—	2	2	no	—
285	C₄⋊C₄×C₆₇	—	2	1	no	—
286	C₄⋊C₄×C₆₇	—	2	4	no	—
287	C₄⋊C₄×C₆₇	—	2	4	no	—
288	C₄⋊C₄×C₆₇	—	2	1	no	—
289	C₄⋊C₄×C₆₇	—	2	2	no	—
290	C₄⋊C₄×C₆₇	—	2	2	no	—
291	C₄⋊C₄×C₆₇	—	2	1	no	—
292	C₄⋊C₄×C₆₇	—	2	1	no	—
293	C₄⋊C₄×C₆₇	—	2	2	no	—
294	C₄⋊C₄×C₆₇	—	2	1	no	—
295	C₄⋊C₄×C₆₇	—	2	2	no	—
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	2	no	—
298	C₄⋊C₄×C₆₇	—	2	2	no	—
299	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
300	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
301	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
302	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
303	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
304	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
305	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
306	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
307	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
308	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
309	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
310	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
311	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
312	C₂²⋊C₄×C₆₇	—	2	1	no	—
313	C₂²⋊C₄×C₆₇	—	2	2	no	—
314	C₂²⋊C₄×C₆₇	—	2	2	no	—
315	C₂²⋊C₄×C₆₇	—	2	2	no	—
316	C₂²⋊C₄×C₆₇	—	2	4	no	—
317	C₂²⋊C₄×C₆₇	—	2	4	no	—
318	C₂²⋊C₄×C₆₇	—	2	2	no	—
319	C₂²⋊C₄×C₆₇	—	2	1	no	—
320	C₂²⋊C₄×C₆₇	—	2	1	no	—
321	C₂²⋊C₄×C₆₇	—	2	2	no	—
322	C₄²×C₆₇	—	2	2	no	—
323	C₄²×C₆₇	—	2	1	no	—
324	C₄²×C₆₇	—	2	1	no	—
325	C₄²×C₆₇	—	2	2	no	—
326	C₄²×C₆₇	—	2	2	no	—
327	C₄²×C₆₇	—	2	1	no	—
328	C₄²×C₆₇	C₂×D₄	1	2	yes	—
329	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
330	C₄²×C₆₇	C₂×D₄	1	2	yes	—
331	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
332	C₄²×C₆₇	C₂⁴	1	2	yes	—
333	C₄²×C₆₇	C₂⁴	1	2	yes	—
334	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
335	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
336	C₂³×C₆₇	—	2	4	no	—
337	C₂³×C₆₇	—	2	1	no	—
338	C₂³×C₆₇	—	2	2	no	—
339	C₂³×C₆₇	—	4	2	no	—
340	C₂³×C₆₇	—	4	2	no	—
341	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
342	Q₈×C₆₇	—	2	2	no	—
343	Q₈×C₆₇	—	2	2	no	—
344	Q₈×C₆₇	—	2	1	no	—
345	Q₈×C₆₇	—	4	2	no	—
346	Q₈×C₆₇	—	4	4	no	—
347	Q₈×C₆₇	—	4	2	no	—
348	Q₈×C₆₇	—	8	2	no	—
349	Q₈×C₆₇	—	8	1	no	—
350	Q₈×C₆₇	—	8	2	no	—
351	Q₈×C₆₇	—	8	2	no	—
352	Q₈×C₆₇	C₂²×Q₈	1	2	yes	—
353	D₄×C₆₇	—	2	4	no	—
354	D₄×C₆₇	—	2	4	no	—
355	D₄×C₆₇	—	2	2	no	—
356	D₄×C₆₇	—	2	1	no	—
357	D₄×C₆₇	C₂²×Q₈	1	2	yes	—
358	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
359	C₂×C₄×C₆₇	—	2	1	no	—
360	C₂×C₄×C₆₇	—	2	4	no	—
361	C₂×C₄×C₆₇	—	2	2	no	—
362	C₂×C₄×C₆₇	—	2	2	no	—
363	C₂×C₄×C₆₇	—	2	1	no	—
364	C₂×C₄×C₆₇	—	2	2	no	—
365	C₂×C₄×C₆₇	—	2	1	no	—
366	C₂×C₄×C₆₇	—	2	2	no	—
367	C₂×C₄×C₆₇	—	2	4	no	—
368	C₂×C₄×C₆₇	—	4	2	no	—
369	C₂×C₄×C₆₇	—	4	2	no	—
370	C₂×C₄×C₆₇	—	4	1	no	—
371	C₂×C₄×C₆₇	—	4	4	no	—
372	C₂×C₄×C₆₇	—	4	4	no	—
373	C₂×C₄×C₆₇	—	4	2	no	—
374	C₂×C₄×C₆₇	—	4	4	no	—
375	C₂×C₄×C₆₇	—	4	4	no	—
376	C₂×C₄×C₆₇	—	4	2	no	—
377	C₂×C₄×C₆₇	—	8	2	no	—
378	C₂×C₄×C₆₇	—	8	1	no	—
379	C₂×C₄×C₆₇	—	8	4	no	—
380	C₂×C₄×C₆₇	—	8	4	no	—
381	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
382	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
383	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
384	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
385	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
386	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
387	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
388	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
389	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
390	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
391	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
392	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
393	C₈×C₆₇	—	2	2	no	—
394	C₈×C₆₇	—	2	4	no	—
395	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
396	C₂²×C₆₇	—	2	1	no	—
397	C₂²×C₆₇	—	2	1	no	—
398	C₂²×C₆₇	—	2	2	no	—
399	C₂²×C₆₇	—	2	4	no	—
400	C₂²×C₆₇	—	4	8	no	—
401	C₂²×C₆₇	—	4	2	no	—
402	C₂²×C₆₇	—	4	1	no	—
403	C₂²×C₆₇	—	4	1	no	—
404	C₂²×C₆₇	—	4	2	no	—
405	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
406	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
407	C₄×C₆₇	—	2	2	no	—
408	C₄×C₆₇	—	2	2	no	—
409	C₄×C₆₇	—	2	1	no	—
410	C₄×C₆₇	—	2	1	no	—
411	C₄×C₆₇	—	2	2	no	—
412	C₄×C₆₇	—	2	1	no	—
413	C₄×C₆₇	—	2	2	no	—
414	C₄×C₆₇	—	2	2	no	—
415	C₄×C₆₇	—	4	4	no	—
416	C₄×C₆₇	—	4	2	no	—
417	C₄×C₆₇	—	4	1	no	—
418	C₄×C₆₇	—	4	2	no	—
419	C₄×C₆₇	—	4	2	no	—
420	C₄×C₆₇	—	4	1	no	—
421	C₄×C₆₇	D₄×Q₈	1	2	yes	—
422	C₄×C₆₇	#575	1	2	yes	—
423	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
424	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
425	#61367	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
426	C₂×C₆₇	—	2	2	no	—
427	C₂×C₆₇	—	4	2	no	—
428	C₂×C₆₇	—	4	4	no	—
429	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
430	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
431	C₂×C₆₇	#3585	1	1	yes	characteristic
432	Q₈×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
433	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
434	#3389	C₂×C₆₇	1	4	yes	complemented
435	#3388	C₂×C₆₇	1	4	yes	complemented
436	#3385	C₂×C₆₇	1	2	yes	complemented
437	#3384	C₂×C₆₇	1	2	yes	complemented
438	#3381	C₂×C₆₇	1	2	yes	complemented
439	#3380	C₂×C₆₇	1	2	yes	complemented
440	#3166	C₂×C₆₇	1	2	yes	complemented
441	#3165	C₂×C₆₇	1	1	yes	characteristic, complemented
442	#3080	C₂×C₆₇	1	2	yes	complemented
443	#3079	C₂×C₆₇	1	1	yes	characteristic, complemented
444	#3077	C₂×C₆₇	1	1	yes	characteristic, complemented
445	#3077	C₂×C₆₇	1	2	yes	complemented
446	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
447	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
448	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	—
449	C₆₇	#61367	1	1	yes	characteristic, direct product factor, Sylow
450	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
451	Q₈²	C₂²×C₆₇	1	1	yes	characteristic, complemented
452	#557	C₂²×C₆₇	1	1	yes	characteristic, complemented
453	#557	C₂²×C₆₇	1	2	yes	complemented
454	#554	—	2	1	no	—
455	#554	—	2	2	no	—
456	#554	—	2	1	no	—
457	#554	—	2	1	no	—
458	#554	—	2	2	no	—
459	#554	C₂²×C₆₇	1	2	yes	—
460	#554	C₂²×C₆₇	1	2	yes	—
461	#554	C₂²×C₆₇	1	4	yes	—
462	#554	C₂²×C₆₇	1	1	yes	characteristic, complemented
463	#554	C₂²×C₆₇	1	2	yes	complemented
464	D₄×Q₈	—	2	1	no	—
465	D₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
466	#533	—	2	2	no	—
467	#533	—	2	1	no	—
468	#533	C₂²×C₆₇	1	1	yes	characteristic
469	#533	C₂²×C₆₇	1	2	yes	—
470	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
471	C₂×#125	C₂²×C₆₇	1	2	yes	—
472	C₂×C₂²⋊Q₈	—	2	1	no	—
473	C₂×C₂²⋊Q₈	—	2	2	no	—
474	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
475	C₄×#41	C₂²×C₆₇	1	2	yes	—
476	C₂×C₄×Q₈	—	2	1	no	—
477	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
478	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
479	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
480	C₂²×C₄⋊C₄	—	2	1	no	—
481	#501	C₂²×C₆₇	1	4	yes	complemented
482	#501	C₂²×C₆₇	1	4	yes	complemented
483	#501	C₂²×C₆₇	1	4	yes	complemented
484	#500	C₂²×C₆₇	1	2	yes	complemented
485	#499	C₂²×C₆₇	1	4	yes	complemented
486	#499	C₂²×C₆₇	1	4	yes	complemented
487	#498	C₂²×C₆₇	1	2	yes	complemented
488	#479	C₂²×C₆₇	1	4	yes	complemented
489	#479	C₂²×C₆₇	1	4	yes	complemented
490	#478	C₂²×C₆₇	1	4	yes	—
491	#478	C₂²×C₆₇	1	4	yes	—
492	#478	C₂²×C₆₇	1	4	yes	—
493	#478	C₂²×C₆₇	1	4	yes	complemented
494	#478	C₂²×C₆₇	1	4	yes	complemented
495	#477	C₂²×C₆₇	1	2	yes	complemented
496	#476	C₂²×C₆₇	1	2	yes	—
497	#476	C₂²×C₆₇	1	4	yes	—
498	#476	C₂²×C₆₇	1	2	yes	complemented
499	#475	C₂²×C₆₇	1	2	yes	complemented
500	#474	C₂²×C₆₇	1	2	yes	—
501	#474	C₂²×C₆₇	1	4	yes	—
502	#474	C₂²×C₆₇	1	2	yes	complemented
503	#446	C₂²×C₆₇	1	2	yes	complemented
504	#446	C₂²×C₆₇	1	4	yes	complemented
505	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
506	#442	C₂²×C₆₇	1	4	yes	—
507	#442	C₂²×C₆₇	1	2	yes	—
508	#440	C₂²×C₆₇	1	4	yes	—
509	#440	C₂²×C₆₇	1	2	yes	—
510	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
511	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	—
512	C₄×D₈	C₂²×C₆₇	1	2	yes	—
513	C₄×D₈	C₂²×C₆₇	1	4	yes	—
514	#428	C₂²×C₆₇	1	2	yes	—
515	#428	C₂²×C₆₇	1	2	yes	—
516	#428	C₂²×C₆₇	1	2	yes	—
517	#428	C₂²×C₆₇	1	2	yes	—
518	#428	C₂²×C₆₇	1	1	yes	characteristic
519	#423	C₂²×C₆₇	1	1	yes	characteristic
520	#423	C₂²×C₆₇	1	2	yes	—
521	#418	C₂²×C₆₇	1	2	yes	—
522	#418	C₂²×C₆₇	1	1	yes	characteristic
523	#417	C₂²×C₆₇	1	2	yes	—
524	#417	C₂²×C₆₇	1	1	yes	characteristic
525	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
526	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
527	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
528	C₂²×Q₈	—	2	1	no	—
529	C₂²×Q₈	—	2	1	no	—
530	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
531	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
532	C₂³×C₄	—	2	1	no	—
533	C₂³×C₄	—	2	2	no	—
534	AGL(1,ℤ/8)	Q₈×C₆₇	1	8	yes	complemented
535	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
536	C₂×D₈	C₂³×C₆₇	1	2	yes	—
537	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
538	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
539	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
540	C₄⋊Q₈	—	2	1	no	—
541	C₄⋊Q₈	—	2	2	no	—
542	C₄⋊Q₈	—	2	1	no	—
543	C₄⋊Q₈	—	2	2	no	—
544	C₄⋊Q₈	—	2	1	no	—
545	C₄⋊Q₈	—	2	1	no	—
546	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
547	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
548	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
549	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
550	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
551	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
552	#125	—	2	2	no	—
553	#125	—	2	2	no	—
554	#125	—	2	2	no	—
555	#125	—	2	2	no	—
556	#125	—	2	2	no	—
557	#125	C₂³×C₆₇	1	4	yes	—
558	#125	C₂³×C₆₇	1	4	yes	—
559	#125	C₂³×C₆₇	1	2	yes	—
560	#125	C₂³×C₆₇	1	2	yes	—
561	#125	C₂³×C₆₇	1	2	yes	—
562	C₂²⋊Q₈	—	2	2	no	—
563	C₂²⋊Q₈	—	2	2	no	—
564	C₂²⋊Q₈	—	2	4	no	—
565	C₂²⋊Q₈	—	2	1	no	—
566	C₂²⋊Q₈	—	2	2	no	—
567	C₂²⋊Q₈	—	2	1	no	—
568	C₂²⋊Q₈	—	2	2	no	—
569	C₂²⋊Q₈	—	2	1	no	—
570	C₂²⋊Q₈	—	2	2	no	—
571	C₂²⋊Q₈	—	2	4	no	—
572	C₂²⋊Q₈	—	2	4	no	—
573	C₂²⋊Q₈	—	2	4	no	—
574	C₂²⋊Q₈	—	2	4	no	—
575	C₂²⋊Q₈	—	2	1	no	—
576	C₂²⋊Q₈	—	2	2	no	—
577	C₂²⋊Q₈	—	2	2	no	—
578	C₂²⋊Q₈	—	2	2	no	—
579	C₂²⋊Q₈	—	2	2	no	—
580	C₄×Q₈	—	2	1	no	—
581	C₄×Q₈	—	2	1	no	—
582	C₄×Q₈	—	2	2	no	—
583	C₄×Q₈	—	2	2	no	—
584	C₄×Q₈	—	2	1	no	—
585	C₄×Q₈	—	2	2	no	—
586	C₄×Q₈	—	2	1	no	—
587	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
588	C₄×Q₈	D₄×C₆₇	1	2	yes	—
589	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
590	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
591	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
592	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
593	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
594	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
595	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
596	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
597	C₄×D₄	—	2	1	no	—
598	C₄×D₄	—	2	2	no	—
599	C₄×D₄	—	2	1	no	—
600	C₄×D₄	—	2	2	no	—
601	C₄×D₄	—	2	2	no	—
602	C₄×D₄	—	2	1	no	—
603	C₄×D₄	—	2	2	no	—
604	C₄×D₄	—	2	1	no	—
605	C₄×D₄	C₂³×C₆₇	1	2	yes	—
606	C₄×D₄	C₂³×C₆₇	1	4	yes	—
607	C₄×D₄	C₂³×C₆₇	1	4	yes	—
608	C₄×D₄	C₂³×C₆₇	1	2	yes	—
609	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
610	C₄×D₄	C₂³×C₆₇	1	2	yes	—
611	C₄×D₄	C₂³×C₆₇	1	2	yes	—
612	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
613	#117	—	2	2	no	—
614	#117	—	2	1	no	—
615	#117	C₂³×C₆₇	1	1	yes	characteristic
616	#117	C₂³×C₆₇	1	2	yes	—
617	C₂×C₄⋊C₄	—	2	2	no	—
618	C₂×C₄⋊C₄	—	2	1	no	—
619	C₂×C₄⋊C₄	—	2	2	no	—
620	C₂×C₄⋊C₄	—	2	2	no	—
621	C₂×C₄⋊C₄	—	2	4	no	—
622	C₂×C₄⋊C₄	—	2	2	no	—
623	C₂×C₄⋊C₄	—	2	1	no	—
624	C₂×C₄⋊C₄	—	2	2	no	—
625	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
626	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
627	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
628	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
629	C₂×C₂²⋊C₄	—	2	1	no	—
630	C₂×C₂²⋊C₄	—	2	2	no	—
631	C₂×C₄²	—	2	2	no	—
632	C₂×C₄²	—	2	1	no	—
633	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
634	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
635	#107	C₂³×C₆₇	1	2	yes	—
636	#107	C₂³×C₆₇	1	4	yes	—
637	#107	C₂³×C₆₇	1	4	yes	—
638	#107	C₂³×C₆₇	1	4	yes	—
639	#107	C₂³×C₆₇	1	4	yes	—
640	#106	C₂³×C₆₇	1	4	yes	—
641	#106	C₂³×C₆₇	1	2	yes	—
642	#106	C₂³×C₆₇	1	4	yes	—
643	#106	C₂³×C₆₇	1	4	yes	—
644	#106	C₂³×C₆₇	1	4	yes	—
645	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
646	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
647	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
648	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
649	#103	C₂³×C₆₇	1	4	yes	—
650	#103	C₂³×C₆₇	1	2	yes	—
651	#102	C₂³×C₆₇	1	4	yes	—
652	#102	C₂³×C₆₇	1	2	yes	—
653	#102	C₂³×C₆₇	1	4	yes	—
654	#102	C₂³×C₆₇	1	2	yes	—
655	#102	C₂³×C₆₇	1	2	yes	—
656	#102	C₂³×C₆₇	1	4	yes	—
657	#97	C₂³×C₆₇	1	2	yes	—
658	#97	C₂³×C₆₇	1	4	yes	—
659	C₄×C₈	C₂³×C₆₇	1	2	yes	—
660	C₄×C₈	C₂³×C₆₇	1	4	yes	—
661	C₂⁴	—	2	1	no	—
662	#41	—	2	2	no	—
663	#41	C₂×Q₈×C₆₇	1	4	yes	—
664	C₂×Q₈	—	2	2	no	—
665	C₂×Q₈	—	2	1	no	—
666	C₂×Q₈	—	2	1	no	—
667	C₂×Q₈	—	2	2	no	—
668	C₂×Q₈	—	2	2	no	—
669	C₂×Q₈	—	2	2	no	—
670	C₂×Q₈	—	4	2	no	—
671	C₂×Q₈	—	4	4	no	—
672	C₂×Q₈	—	8	2	no	—
673	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
674	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
675	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
676	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
677	C₂×D₄	—	2	1	no	—
678	C₂×D₄	—	2	1	no	—
679	C₂×D₄	—	2	2	no	—
680	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	complemented
681	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
682	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
683	C₂²×C₄	—	2	1	no	—
684	C₂²×C₄	—	2	1	no	—
685	C₂²×C₄	—	2	2	no	—
686	C₂²×C₄	—	2	2	no	—
687	C₂²×C₄	—	2	1	no	—
688	C₂²×C₄	—	2	2	no	—
689	C₂²×C₄	—	2	2	no	—
690	C₂²×C₄	—	2	1	no	—
691	C₂²×C₄	—	2	2	no	—
692	C₂²×C₄	—	2	4	no	—
693	C₂²×C₄	—	2	4	no	—
694	C₂²×C₄	—	4	4	no	—
695	C₂²×C₄	—	8	4	no	—
696	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
697	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
698	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
699	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
700	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
701	SD₁₆	C₂×Q₈×C₆₇	1	8	yes	complemented
702	D₈	C₂×Q₈×C₆₇	1	8	yes	complemented
703	M₁₆	—	2	2	no	—
704	M₁₆	—	2	4	no	—
705	M₁₆	C₂×Q₈×C₆₇	1	4	yes	complemented
706	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
707	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
708	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
709	C₄⋊C₄	—	2	4	no	—
710	C₄⋊C₄	—	2	4	no	—
711	C₄⋊C₄	—	2	2	no	—
712	C₄⋊C₄	—	2	2	no	—
713	C₄⋊C₄	—	2	2	no	—
714	C₄⋊C₄	—	2	2	no	—
715	C₄⋊C₄	—	2	2	no	—
716	C₄⋊C₄	—	2	2	no	—
717	C₄⋊C₄	—	2	1	no	—
718	C₄⋊C₄	—	2	4	no	—
719	C₄⋊C₄	—	2	4	no	—
720	C₄⋊C₄	—	2	1	no	—
721	C₄⋊C₄	—	2	2	no	—
722	C₄⋊C₄	—	2	2	no	—
723	C₄⋊C₄	—	2	1	no	—
724	C₄⋊C₄	—	2	1	no	—
725	C₄⋊C₄	—	2	2	no	—
726	C₄⋊C₄	—	2	1	no	—
727	C₄⋊C₄	—	2	2	no	—
728	C₄⋊C₄	—	2	2	no	—
729	C₄⋊C₄	—	2	2	no	—
730	C₄⋊C₄	—	2	2	no	—
731	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
732	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
733	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
734	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
735	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
736	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
737	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
738	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
739	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
740	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
741	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
742	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
743	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
744	C₂²⋊C₄	—	2	1	no	—
745	C₂²⋊C₄	—	2	2	no	—
746	C₂²⋊C₄	—	2	2	no	—
747	C₂²⋊C₄	—	2	2	no	—
748	C₂²⋊C₄	—	2	4	no	—
749	C₂²⋊C₄	—	2	4	no	—
750	C₂²⋊C₄	—	2	2	no	—
751	C₂²⋊C₄	—	2	1	no	—
752	C₂²⋊C₄	—	2	1	no	—
753	C₂²⋊C₄	—	2	2	no	—
754	C₄²	—	2	2	no	—
755	C₄²	—	2	1	no	—
756	C₄²	—	2	1	no	—
757	C₄²	—	2	2	no	—
758	C₄²	—	2	2	no	—
759	C₄²	—	2	1	no	—
760	C₄²	C₂×D₄×C₆₇	1	2	yes	—
761	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
762	C₄²	C₂×D₄×C₆₇	1	2	yes	—
763	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
764	C₄²	C₂⁴×C₆₇	1	2	yes	—
765	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
766	C₄²	C₂⁴×C₆₇	1	2	yes	—
767	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
768	C₂³	—	2	4	no	—
769	C₂³	—	2	1	no	—
770	C₂³	—	2	2	no	—
771	C₂³	—	4	2	no	—
772	C₂³	—	4	2	no	—
773	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
774	Q₈	—	2	2	no	—
775	Q₈	—	2	2	no	—
776	Q₈	—	2	1	no	—
777	Q₈	—	4	2	no	—
778	Q₈	—	4	4	no	—
779	Q₈	—	4	2	no	—
780	Q₈	—	8	2	no	—
781	Q₈	—	8	2	no	—
782	Q₈	—	8	2	no	—
783	Q₈	—	8	1	no	—
784	Q₈	C₂²×Q₈×C₆₇	1	2	yes	—
785	D₄	—	2	4	no	—
786	D₄	—	2	4	no	—
787	D₄	—	2	2	no	—
788	D₄	—	2	1	no	—
789	D₄	C₂²×Q₈×C₆₇	1	2	yes	—
790	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
791	C₂×C₄	—	2	1	no	—
792	C₂×C₄	—	2	4	no	—
793	C₂×C₄	—	2	2	no	—
794	C₂×C₄	—	2	2	no	—
795	C₂×C₄	—	2	1	no	—
796	C₂×C₄	—	2	2	no	—
797	C₂×C₄	—	2	1	no	—
798	C₂×C₄	—	2	2	no	—
799	C₂×C₄	—	2	4	no	—
800	C₂×C₄	—	4	2	no	—
801	C₂×C₄	—	4	2	no	—
802	C₂×C₄	—	4	1	no	—
803	C₂×C₄	—	4	4	no	—
804	C₂×C₄	—	4	4	no	—
805	C₂×C₄	—	4	2	no	—
806	C₂×C₄	—	4	4	no	—
807	C₂×C₄	—	4	4	no	—
808	C₂×C₄	—	4	2	no	—
809	C₂×C₄	—	8	4	no	—
810	C₂×C₄	—	8	4	no	—
811	C₂×C₄	—	8	2	no	—
812	C₂×C₄	—	8	1	no	—
813	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
814	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
815	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
816	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
817	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
818	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
819	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
820	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
821	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
822	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
823	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
824	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
825	C₈	—	2	2	no	—
826	C₈	—	2	4	no	—
827	C₈	C₂²×Q₈×C₆₇	1	4	yes	—
828	C₂²	—	2	1	no	—
829	C₂²	—	2	1	no	—
830	C₂²	—	2	2	no	—
831	C₂²	—	2	4	no	—
832	C₂²	—	4	8	no	—
833	C₂²	—	4	1	no	—
834	C₂²	—	4	2	no	—
835	C₂²	—	4	2	no	—
836	C₂²	—	4	1	no	—
837	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
838	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
839	C₄	—	2	1	no	—
840	C₄	—	2	2	no	—
841	C₄	—	2	2	no	—
842	C₄	—	2	2	no	—
843	C₄	—	2	2	no	—
844	C₄	—	2	1	no	—
845	C₄	—	2	1	no	—
846	C₄	—	2	2	no	—
847	C₄	—	4	2	no	—
848	C₄	—	4	2	no	—
849	C₄	—	4	1	no	—
850	C₄	—	4	4	no	—
851	C₄	—	4	2	no	—
852	C₄	—	4	1	no	—
853	C₄	D₄×Q₈×C₆₇	1	2	yes	—
854	C₄	#575×C₆₇	1	2	yes	—
855	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
856	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
857	C₂	—	2	2	no	—
858	C₂	—	4	4	no	—
859	C₂	—	4	2	no	—
860	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
861	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
862	C₂	C₆₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₆₇	C₂	semidirect
2. C₂×#554×C₆₇	C₂	semidirect
3. C₆₇×#3389	C₂	semidirect
4. C₆₇×#3388	C₂	semidirect
5. C₆₇×#3385	C₂	semidirect
6. C₆₇×#3384	C₂	semidirect
7. C₆₇×#3381	C₂	semidirect
8. C₆₇×#3380	C₂	semidirect
9. C₆₇×#3166	C₂	semidirect
10. C₆₇×#3165	C₂	semidirect
11. C₆₇×#3080	C₂	semidirect
12. C₆₇×#3079	C₂	semidirect
13. C₆₇×#3077	C₂	semidirect
14. C₆₇×#3077	C₂	semidirect
15. Q₈×M₁₆×C₆₇	C₂	semidirect
19. Q₈²×C₆₇	C₂²	semidirect
20. #557×C₆₇	C₂²	semidirect
21. #557×C₆₇	C₂²	semidirect
30. #554×C₆₇	C₂²	semidirect
31. #554×C₆₇	C₂²	semidirect
33. D₄×Q₈×C₆₇	C₂²	semidirect
49. #501×C₆₇	C₂²	semidirect
50. #501×C₆₇	C₂²	semidirect
51. #501×C₆₇	C₂²	semidirect
52. #500×C₆₇	C₂²	semidirect
53. #499×C₆₇	C₂²	semidirect
54. #499×C₆₇	C₂²	semidirect
55. #498×C₆₇	C₂²	semidirect
56. #479×C₆₇	C₂²	semidirect
57. #479×C₆₇	C₂²	semidirect
61. #478×C₆₇	C₂²	semidirect
62. #478×C₆₇	C₂²	semidirect
63. #477×C₆₇	C₂²	semidirect
66. #476×C₆₇	C₂²	semidirect
67. #475×C₆₇	C₂²	semidirect
70. #474×C₆₇	C₂²	semidirect
71. #446×C₆₇	C₂²	semidirect
72. #446×C₆₇	C₂²	semidirect
73. C₈×Q₈×C₆₇	C₂²	semidirect
102. AGL(1,ℤ/8)×C₆₇	Q₈	semidirect
248. C₂×D₄×C₆₇	C₂×Q₈	semidirect
269. SD₁₆×C₆₇	C₂×Q₈	semidirect
270. D₈×C₆₇	C₂×Q₈	semidirect
273. M₁₆×C₆₇	C₂×Q₈	semidirect
425. #61367	C₆₇	direct (N × H)
432. Q₈×#41	C₂×C₆₇	semidirect
433. C₂×#554	C₂×C₆₇	semidirect
434. #3389	C₂×C₆₇	semidirect
435. #3388	C₂×C₆₇	semidirect
436. #3385	C₂×C₆₇	semidirect
437. #3384	C₂×C₆₇	semidirect
438. #3381	C₂×C₆₇	semidirect
439. #3380	C₂×C₆₇	semidirect
440. #3166	C₂×C₆₇	semidirect
441. #3165	C₂×C₆₇	semidirect
442. #3080	C₂×C₆₇	semidirect
443. #3079	C₂×C₆₇	semidirect
444. #3077	C₂×C₆₇	semidirect
445. #3077	C₂×C₆₇	semidirect
446. Q₈×M₁₆	C₂×C₆₇	semidirect
449. C₆₇	#61367	direct (N × H)
451. Q₈²	C₂²×C₆₇	semidirect
452. #557	C₂²×C₆₇	semidirect
453. #557	C₂²×C₆₇	semidirect
462. #554	C₂²×C₆₇	semidirect
463. #554	C₂²×C₆₇	semidirect
465. D₄×Q₈	C₂²×C₆₇	semidirect
481. #501	C₂²×C₆₇	semidirect
482. #501	C₂²×C₆₇	semidirect
483. #501	C₂²×C₆₇	semidirect
484. #500	C₂²×C₆₇	semidirect
485. #499	C₂²×C₆₇	semidirect
486. #499	C₂²×C₆₇	semidirect
487. #498	C₂²×C₆₇	semidirect
488. #479	C₂²×C₆₇	semidirect
489. #479	C₂²×C₆₇	semidirect
493. #478	C₂²×C₆₇	semidirect
494. #478	C₂²×C₆₇	semidirect
495. #477	C₂²×C₆₇	semidirect
498. #476	C₂²×C₆₇	semidirect
499. #475	C₂²×C₆₇	semidirect
502. #474	C₂²×C₆₇	semidirect
503. #446	C₂²×C₆₇	semidirect
504. #446	C₂²×C₆₇	semidirect
505. C₈×Q₈	C₂²×C₆₇	semidirect
534. AGL(1,ℤ/8)	Q₈×C₆₇	semidirect
680. C₂×D₄	C₂×Q₈×C₆₇	semidirect
701. SD₁₆	C₂×Q₈×C₆₇	semidirect
702. D₈	C₂×Q₈×C₆₇	semidirect
705. M₁₆	C₂×Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₆₇	2
C₂×#554×C₆₇	2
C₆₇×#3389	2
C₆₇×#3388	2
C₆₇×#3385	2
C₆₇×#3384	2
C₆₇×#3381	2
C₆₇×#3380	2
C₆₇×#3166	2
C₆₇×#3165	2
C₆₇×#3080	2
C₆₇×#3079	2
C₆₇×#3077	2
Q₈×M₁₆×C₆₇	2
C₄×AGL(1,ℤ/8)×C₆₇	2
#61367	67

Subgroups — C67×#61367 #30740618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61367 #30740618