Subgroups — C₆₇×#21236 #30700487

Order 17152.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 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×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1972	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
15	C₂×AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
16	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	8	yes	complemented
17	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#428×C₆₇	—	2	4	no	—
26	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
28	#418×C₆₇	—	2	4	no	—
29	#417×C₆₇	—	2	4	no	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	C₄×M₁₆×C₆₇	—	2	4	no	—
35	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
36	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
37	#391×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#391×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#390×C₆₇	—	2	2	no	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#390×C₆₇	C₂²	1	2	yes	—
42	#390×C₆₇	C₂²	1	1	yes	characteristic, complemented
43	#390×C₆₇	C₂²	1	1	yes	characteristic, complemented
44	#386×C₆₇	—	2	2	no	—
45	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
46	C₂⁵×C₆₇	—	2	1	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
53	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	—	2	2	no	—
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	2	no	—
58	C₂²×D₄×C₆₇	—	2	1	no	—
59	C₂²×D₄×C₆₇	—	2	2	no	—
60	C₂²×D₄×C₆₇	—	2	2	no	—
61	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	—	2	1	no	—
65	C₂³×C₄×C₆₇	—	2	1	no	—
66	C₂³×C₄×C₆₇	D₄	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
69	C₂×SD₁₆×C₆₇	—	2	4	no	—
70	C₂×SD₁₆×C₆₇	—	2	4	no	—
71	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
72	C₂×D₈×C₆₇	—	2	4	no	—
73	C₂×D₈×C₆₇	—	2	4	no	—
74	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
75	C₂×M₁₆×C₆₇	—	2	2	no	—
76	C₂×M₁₆×C₆₇	—	2	2	no	—
77	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
79	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
80	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
83	#117×C₆₇	—	2	2	no	—
84	#117×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
97	C₂×C₄²×C₆₇	—	2	2	no	—
98	C₂×C₄²×C₆₇	—	2	2	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	D₄	1	2	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	#107×C₆₇	—	2	4	no	—
104	#106×C₆₇	—	2	4	no	—
105	#103×C₆₇	—	2	4	no	—
106	#102×C₆₇	—	2	4	no	—
107	#102×C₆₇	—	2	4	no	—
108	#102×C₆₇	—	2	4	no	—
109	#97×C₆₇	—	2	4	no	—
110	C₄×C₈×C₆₇	—	2	4	no	—
111	#95×C₆₇	—	2	2	no	—
112	#95×C₆₇	—	2	2	no	—
113	C₂⁴×C₆₇	—	2	4	no	—
114	C₂⁴×C₆₇	—	2	1	no	—
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	—	4	4	no	—
117	C₂⁴×C₆₇	—	4	4	no	—
118	C₂⁴×C₆₇	—	4	2	no	—
119	C₂⁴×C₆₇	—	4	2	no	—
120	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
121	#41×C₆₇	—	2	8	no	—
122	#41×C₆₇	—	2	8	no	—
123	#41×C₆₇	—	4	4	no	—
124	#41×C₆₇	—	4	4	no	—
125	C₂×Q₈×C₆₇	—	2	1	no	—
126	C₂×Q₈×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	1	no	—
130	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂×D₄×C₆₇	—	2	4	no	—
132	C₂×D₄×C₆₇	—	2	8	no	—
133	C₂×D₄×C₆₇	—	2	8	no	—
134	C₂×D₄×C₆₇	—	2	4	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	1	no	—
137	C₂×D₄×C₆₇	—	2	1	no	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	2	1	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	1	no	—
145	C₂×D₄×C₆₇	—	4	2	no	—
146	C₂×D₄×C₆₇	—	4	2	no	—
147	C₂×D₄×C₆₇	—	4	4	no	—
148	C₂×D₄×C₆₇	—	4	2	no	—
149	C₂×D₄×C₆₇	—	4	4	no	—
150	C₂×D₄×C₆₇	—	4	2	no	—
151	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	4	2	no	—
161	C₂²×C₄×C₆₇	—	4	2	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	—	4	4	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	#41	1	2	yes	—
173	C₂²×C₄×C₆₇	#41	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	SD₁₆×C₆₇	—	2	16	no	—
176	D₈×C₆₇	—	2	16	no	—
177	M₁₆×C₆₇	—	2	8	no	—
178	M₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	—	2	2	no	—
181	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
182	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
183	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
184	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	1	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	1	no	—
191	C₄⋊C₄×C₆₇	—	4	2	no	—
192	C₂²⋊C₄×C₆₇	—	4	4	no	—
193	C₂²⋊C₄×C₆₇	—	4	8	no	—
194	C₂²⋊C₄×C₆₇	—	4	2	no	—
195	C₂²⋊C₄×C₆₇	—	4	4	no	—
196	C₂²⋊C₄×C₆₇	—	4	8	no	—
197	C₄²×C₆₇	—	2	4	no	—
198	C₄²×C₆₇	—	2	1	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₄²×C₆₇	—	2	1	no	—
202	C₄²×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	2	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	2	4	no	—
207	C₂³×C₆₇	—	4	16	no	—
208	C₂³×C₆₇	—	4	1	no	—
209	C₂³×C₆₇	—	4	2	no	—
210	C₂³×C₆₇	—	4	1	no	—
211	C₂³×C₆₇	—	4	2	no	—
212	C₂³×C₆₇	—	4	4	no	—
213	C₂³×C₆₇	—	4	2	no	—
214	C₂³×C₆₇	—	4	2	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	8	no	—
217	C₂³×C₆₇	—	8	4	no	—
218	C₂³×C₆₇	—	8	4	no	—
219	C₂³×C₆₇	—	8	8	no	—
220	C₂³×C₆₇	C₄×D₄	1	2	yes	—
221	C₂³×C₆₇	#121	1	2	yes	—
222	C₂³×C₆₇	#127	1	2	yes	—
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	2	no	—
225	Q₈×C₆₇	—	2	2	no	—
226	Q₈×C₆₇	—	4	1	no	—
227	Q₈×C₆₇	—	4	1	no	—
228	D₄×C₆₇	—	2	2	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	2	2	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	2	no	—
234	D₄×C₆₇	—	4	1	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	4	1	no	—
237	D₄×C₆₇	—	4	8	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	4	8	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	8	no	—
248	C₂×C₄×C₆₇	—	4	1	no	—
249	C₂×C₄×C₆₇	—	4	1	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	—	8	4	no	—
255	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
256	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
257	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
258	C₂×C₄×C₆₇	#121	1	4	yes	—
259	C₂×C₄×C₆₇	#121	1	2	yes	—
260	C₂×C₄×C₆₇	#124	1	4	yes	—
261	C₂×C₄×C₆₇	#127	1	2	yes	—
262	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	complemented
263	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
269	C₈×C₆₇	—	2	4	no	—
270	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	complemented
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	4	no	—
274	C₂²×C₆₇	—	2	4	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	8	no	—
277	C₂²×C₆₇	—	4	2	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	8	16	no	—
280	C₂²×C₆₇	—	8	1	no	—
281	C₂²×C₆₇	—	8	2	no	—
282	C₂²×C₆₇	—	8	1	no	—
283	C₂²×C₆₇	—	8	2	no	—
284	C₂²×C₆₇	—	8	2	no	—
285	C₂²×C₆₇	—	8	4	no	—
286	C₂²×C₆₇	#390	1	4	yes	—
287	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
288	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
289	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
290	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
291	C₂²×C₆₇	#575	1	2	yes	—
292	C₄×C₆₇	—	2	4	no	—
293	C₄×C₆₇	—	4	2	no	—
294	C₄×C₆₇	—	4	2	no	—
295	C₄×C₆₇	—	8	1	no	—
296	C₄×C₆₇	—	8	2	no	—
297	C₄×C₆₇	—	8	1	no	—
298	C₄×C₆₇	#390	1	4	yes	—
299	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
300	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
301	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
302	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
303	#21236	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂×C₆₇	—	2	4	no	—
305	C₂×C₆₇	—	8	2	no	—
306	C₂×C₆₇	—	8	4	no	—
307	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
308	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
309	C₂×C₆₇	#3055	1	2	yes	—
310	C₂×C₆₇	#3151	1	1	yes	characteristic
311	C₂×C₆₇	#3153	1	1	yes	characteristic
312	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
313	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
314	C₂×#417	C₂×C₆₇	1	1	yes	characteristic, complemented
315	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#2297	C₂×C₆₇	1	1	yes	characteristic, complemented
318	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#1973	C₂×C₆₇	1	2	yes	complemented
320	#1972	C₂×C₆₇	1	2	yes	complemented
321	#1970	C₂×C₆₇	1	2	yes	complemented
322	#1968	C₂×C₆₇	1	2	yes	complemented
323	C₆₇	#21236	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
325	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×AGL(1,ℤ/8)	—	2	4	no	—
327	C₂×AGL(1,ℤ/8)	—	2	4	no	—
328	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	8	yes	complemented
329	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
330	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
331	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
332	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
334	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
335	C₂²×C₂²⋊C₄	—	2	2	no	—
336	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
337	#428	—	2	4	no	—
338	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
339	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
340	#418	—	2	4	no	—
341	#417	—	2	4	no	—
342	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
343	C₂×#102	C₂²×C₆₇	1	2	yes	—
344	C₂×#102	C₂²×C₆₇	1	2	yes	—
345	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
346	C₄×M₁₆	—	2	4	no	—
347	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
348	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
349	#391	C₂²×C₆₇	1	1	yes	characteristic, complemented
350	#391	C₂²×C₆₇	1	1	yes	characteristic, complemented
351	#390	—	2	2	no	—
352	#390	C₂²×C₆₇	1	2	yes	—
353	#390	C₂²×C₆₇	1	2	yes	—
354	#390	C₂²×C₆₇	1	1	yes	characteristic, complemented
355	#390	C₂²×C₆₇	1	1	yes	characteristic, complemented
356	#386	—	2	2	no	—
357	C₄×C₂²⋊C₄	—	2	2	no	—
358	C₂⁵	—	2	1	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	—	2	2	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	—	2	2	no	—
363	C₂×#41	—	2	2	no	—
364	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
365	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	—	2	2	no	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	—	2	2	no	—
370	C₂²×D₄	—	2	1	no	—
371	C₂²×D₄	—	2	2	no	—
372	C₂²×D₄	—	2	2	no	—
373	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
374	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
376	C₂³×C₄	—	2	1	no	—
377	C₂³×C₄	—	2	1	no	—
378	C₂³×C₄	D₄×C₆₇	1	2	yes	—
379	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
380	AGL(1,ℤ/8)	—	2	32	no	—
381	C₂×SD₁₆	—	2	4	no	—
382	C₂×SD₁₆	—	2	4	no	—
383	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
384	C₂×D₈	—	2	4	no	—
385	C₂×D₈	—	2	4	no	—
386	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
387	C₂×M₁₆	—	2	2	no	—
388	C₂×M₁₆	—	2	2	no	—
389	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
390	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
391	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
392	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
393	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
394	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
395	#117	—	2	2	no	—
396	#117	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	2	no	—
398	C₂×C₄⋊C₄	—	2	2	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	4	no	—
405	C₂×C₂²⋊C₄	—	2	4	no	—
406	C₂×C₂²⋊C₄	—	2	2	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	4	8	no	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	—	2	2	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	D₄×C₆₇	1	2	yes	—
413	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
415	#107	—	2	4	no	—
416	#106	—	2	4	no	—
417	#103	—	2	4	no	—
418	#102	—	2	4	no	—
419	#102	—	2	4	no	—
420	#102	—	2	4	no	—
421	#97	—	2	4	no	—
422	C₄×C₈	—	2	4	no	—
423	#95	—	2	2	no	—
424	#95	—	2	2	no	—
425	C₂⁴	—	2	4	no	—
426	C₂⁴	—	2	1	no	—
427	C₂⁴	—	2	2	no	—
428	C₂⁴	—	4	4	no	—
429	C₂⁴	—	4	4	no	—
430	C₂⁴	—	4	2	no	—
431	C₂⁴	—	4	2	no	—
432	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
433	#41	—	2	8	no	—
434	#41	—	2	8	no	—
435	#41	—	4	4	no	—
436	#41	—	4	4	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	8	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	—	4	2	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	4	no	—
462	C₂×D₄	—	4	2	no	—
463	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
464	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
481	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
484	C₂²×C₄	#41×C₆₇	1	2	yes	—
485	C₂²×C₄	#41×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	SD₁₆	—	2	16	no	—
488	D₈	—	2	16	no	—
489	M₁₆	—	2	8	no	—
490	M₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
491	C₂×C₈	—	2	2	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
494	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
495	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
496	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₂²⋊C₄	—	4	8	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₂²⋊C₄	—	4	2	no	—
507	C₂²⋊C₄	—	4	8	no	—
508	C₂²⋊C₄	—	4	4	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	4	2	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	4	16	no	—
520	C₂³	—	4	1	no	—
521	C₂³	—	4	2	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	4	2	no	—
524	C₂³	—	4	4	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	4	2	no	—
527	C₂³	—	4	4	no	—
528	C₂³	—	4	8	no	—
529	C₂³	—	8	8	no	—
530	C₂³	—	8	4	no	—
531	C₂³	—	8	4	no	—
532	C₂³	C₄×D₄×C₆₇	1	2	yes	—
533	C₂³	#121×C₆₇	1	2	yes	—
534	C₂³	#127×C₆₇	1	2	yes	—
535	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
536	Q₈	—	2	2	no	—
537	Q₈	—	2	2	no	—
538	Q₈	—	4	1	no	—
539	Q₈	—	4	1	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	2	no	—
544	D₄	—	4	8	no	—
545	D₄	—	4	2	no	—
546	D₄	—	4	1	no	—
547	D₄	—	4	2	no	—
548	D₄	—	4	1	no	—
549	D₄	—	4	8	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	8	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	8	4	no	—
567	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
568	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
569	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
570	C₂×C₄	#121×C₆₇	1	4	yes	—
571	C₂×C₄	#121×C₆₇	1	2	yes	—
572	C₂×C₄	#124×C₆₇	1	4	yes	—
573	C₂×C₄	#127×C₆₇	1	2	yes	—
574	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	complemented
575	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
576	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
579	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
580	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
581	C₈	—	2	4	no	—
582	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	complemented
583	C₂²	—	2	4	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	4	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	4	no	—
589	C₂²	—	4	8	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	8	16	no	—
592	C₂²	—	8	4	no	—
593	C₂²	—	8	2	no	—
594	C₂²	—	8	2	no	—
595	C₂²	—	8	1	no	—
596	C₂²	—	8	2	no	—
597	C₂²	—	8	1	no	—
598	C₂²	#390×C₆₇	1	4	yes	—
599	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
600	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
601	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
602	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
603	C₂²	#575×C₆₇	1	2	yes	—
604	C₄	—	2	4	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	8	2	no	—
608	C₄	—	8	1	no	—
609	C₄	—	8	1	no	—
610	C₄	#390×C₆₇	1	4	yes	—
611	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
612	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
613	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
614	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
615	C₂	—	2	4	no	—
616	C₂	—	8	4	no	—
617	C₂	—	8	2	no	—
618	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
619	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
620	C₂	C₆₇×#3055	1	2	yes	—
621	C₂	C₆₇×#3151	1	1	yes	characteristic
622	C₂	C₆₇×#3153	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 ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₆₇	C₂	semidirect
3. C₂×#417×C₆₇	C₂	semidirect
4. C₂×#418×C₆₇	C₂	semidirect
5. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
6. C₆₇×#2297	C₂	semidirect
7. C₂×#390×C₆₇	C₂	semidirect
8. C₆₇×#1973	C₂	semidirect
9. C₆₇×#1972	C₂	semidirect
10. C₆₇×#1970	C₂	semidirect
11. C₆₇×#1968	C₂	semidirect
16. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
26. C₂×#107×C₆₇	C₂²	semidirect
27. C₂×#106×C₆₇	C₂²	semidirect
30. C₂×#103×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
35. C₂×#97×C₆₇	C₂²	semidirect
36. C₂×C₄×C₈×C₆₇	C₂²	semidirect
37. #391×C₆₇	C₂²	semidirect
38. #391×C₆₇	C₂²	semidirect
42. #390×C₆₇	C₂²	semidirect
43. #390×C₆₇	C₂²	semidirect
71. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
74. C₂×D₈×C₆₇	C₂×C₄	semidirect
178. M₁₆×C₆₇	C₂²⋊C₄	semidirect
262. C₂×C₄×C₆₇	AGL(1,ℤ/8)	semidirect
270. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
303. #21236	C₆₇	direct (N × H)
313. C₂×#428	C₂×C₆₇	semidirect
314. C₂×#417	C₂×C₆₇	semidirect
315. C₂×#418	C₂×C₆₇	semidirect
316. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
317. #2297	C₂×C₆₇	semidirect
318. C₂×#390	C₂×C₆₇	semidirect
319. #1973	C₂×C₆₇	semidirect
320. #1972	C₂×C₆₇	semidirect
321. #1970	C₂×C₆₇	semidirect
322. #1968	C₂×C₆₇	semidirect
323. C₆₇	#21236	direct (N × H)
328. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
338. C₂×#107	C₂²×C₆₇	semidirect
339. C₂×#106	C₂²×C₆₇	semidirect
342. C₂×#103	C₂²×C₆₇	semidirect
345. C₂×#102	C₂²×C₆₇	semidirect
347. C₂×#97	C₂²×C₆₇	semidirect
348. C₂×C₄×C₈	C₂²×C₆₇	semidirect
349. #391	C₂²×C₆₇	semidirect
350. #391	C₂²×C₆₇	semidirect
354. #390	C₂²×C₆₇	semidirect
355. #390	C₂²×C₆₇	semidirect
383. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
386. C₂×D₈	C₂×C₄×C₆₇	semidirect
490. M₁₆	C₂²⋊C₄×C₆₇	semidirect
574. C₂×C₄	AGL(1,ℤ/8)×C₆₇	semidirect
582. 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×#428×C₆₇	2
C₂×#417×C₆₇	2
C₂×#418×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2297	2
C₂×#390×C₆₇	2
C₆₇×#1973	2
C₆₇×#1972	2
C₆₇×#1970	2
C₆₇×#1968	2
#21236	67

Subgroups — C67×#21236 #30700487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21236 #30700487