Subgroups — C₆₇×#61406 #30740657

Order 17152.

1986 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3506	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3403	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3397	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3396	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3393	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3391	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3155	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3045	C₂	1	4	yes	maximal, complemented
10	C₆₇×#3036	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2951	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
15	C₂×#135×C₆₇	C₂²	1	2	yes	—
16	C₂×#135×C₆₇	C₂²	1	2	yes	—
17	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	#559×C₆₇	C₂²	1	8	yes	complemented
19	#556×C₆₇	C₂²	1	4	yes	complemented
20	#555×C₆₇	—	2	2	no	—
21	#553×C₆₇	—	2	4	no	—
22	#550×C₆₇	C₂²	1	4	yes	complemented
23	#547×C₆₇	—	2	2	no	—
24	#538×C₆₇	—	2	4	no	—
25	#535×C₆₇	C₂²	1	4	yes	—
26	#533×C₆₇	C₂²	1	1	yes	characteristic
27	#532×C₆₇	C₂²	1	2	yes	—
28	#525×C₆₇	—	2	4	no	—
29	#520×C₆₇	C₂²	1	4	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	#496×C₆₇	C₂²	1	8	yes	—
32	#495×C₆₇	C₂²	1	4	yes	—
33	#495×C₆₇	C₂²	1	2	yes	complemented
34	#495×C₆₇	C₂²	1	2	yes	complemented
35	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#493×C₆₇	C₂²	1	4	yes	—
37	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#492×C₆₇	C₂²	1	2	yes	complemented
39	#464×C₆₇	C₂²	1	8	yes	complemented
40	#463×C₆₇	C₂²	1	8	yes	complemented
41	#463×C₆₇	C₂²	1	8	yes	complemented
42	#461×C₆₇	C₂²	1	4	yes	complemented
43	#460×C₆₇	C₂²	1	8	yes	complemented
44	#459×C₆₇	C₂²	1	4	yes	complemented
45	#459×C₆₇	C₂²	1	8	yes	complemented
46	#446×C₆₇	C₂²	1	2	yes	complemented
47	#446×C₆₇	C₂²	1	2	yes	complemented
48	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
49	C₈×Q₈×C₆₇	C₂²	1	2	yes	complemented
50	#440×C₆₇	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
52	C₄×D₈×C₆₇	C₂²	1	2	yes	—
53	C₄×D₈×C₆₇	C₂²	1	2	yes	—
54	C₄×D₈×C₆₇	C₂²	1	2	yes	—
55	C₄×D₈×C₆₇	C₂²	1	2	yes	—
56	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
57	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
58	#424×C₆₇	C₂²	1	2	yes	—
59	#423×C₆₇	C₂²	1	1	yes	characteristic
60	#417×C₆₇	C₂²	1	4	yes	—
61	C₂×#102×C₆₇	C₂²	1	2	yes	—
62	#405×C₆₇	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
64	C₂×#41×C₆₇	—	2	4	no	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	—	2	2	no	—
68	AGL(1,ℤ/8)×C₆₇	—	4	8	no	—
69	#135×C₆₇	—	2	8	no	—
70	#135×C₆₇	—	4	4	no	—
71	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
76	C₂×D₈×C₆₇	—	2	4	no	—
77	C₂×D₈×C₆₇	C₂³	1	2	yes	—
78	C₂×D₈×C₆₇	C₂³	1	2	yes	—
79	C₂×D₈×C₆₇	C₂³	1	2	yes	—
80	C₂×D₈×C₆₇	C₂³	1	2	yes	—
81	C₂×D₈×C₆₇	C₂³	1	8	yes	—
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
86	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
88	#127×C₆₇	C₂³	1	2	yes	—
89	#127×C₆₇	C₂³	1	8	yes	—
90	#126×C₆₇	—	2	8	no	—
91	#125×C₆₇	D₄	1	2	yes	—
92	#124×C₆₇	—	2	8	no	—
93	#124×C₆₇	C₂³	1	4	yes	—
94	#124×C₆₇	C₂³	1	8	yes	—
95	#123×C₆₇	—	2	8	no	—
96	#123×C₆₇	—	2	8	no	—
97	C₂²⋊Q₈×C₆₇	—	2	4	no	—
98	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
99	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
100	#121×C₆₇	—	2	2	no	—
101	#121×C₆₇	—	2	2	no	—
102	#121×C₆₇	—	2	4	no	—
103	#121×C₆₇	—	2	4	no	—
104	#121×C₆₇	—	2	4	no	—
105	#121×C₆₇	—	2	8	no	—
106	#120×C₆₇	—	2	8	no	—
107	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
108	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
109	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
110	C₄×D₄×C₆₇	—	2	8	no	—
111	C₄×D₄×C₆₇	—	2	8	no	—
112	C₄×D₄×C₆₇	—	2	2	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	C₂³	1	4	yes	—
117	C₄×D₄×C₆₇	C₂³	1	8	yes	—
118	C₄×D₄×C₆₇	C₂³	1	4	yes	—
119	#117×C₆₇	—	2	4	no	—
120	#117×C₆₇	—	2	4	no	—
121	#117×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	#107×C₆₇	C₂³	1	2	yes	—
127	#107×C₆₇	C₂³	1	2	yes	—
128	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
129	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
130	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
131	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
132	#103×C₆₇	C₂³	1	8	yes	—
133	#102×C₆₇	C₂³	1	8	yes	—
134	#102×C₆₇	C₂³	1	4	yes	—
135	#102×C₆₇	C₂³	1	4	yes	—
136	#97×C₆₇	C₂³	1	4	yes	—
137	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₆₇	C₂³	1	4	yes	—
139	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
142	C₂⁴×C₆₇	—	2	2	no	—
143	#41×C₆₇	—	2	8	no	—
144	#41×C₆₇	—	2	8	no	—
145	#41×C₆₇	—	8	8	no	—
146	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
147	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	8	no	—
153	C₂×D₄×C₆₇	—	2	2	no	—
154	C₂×D₄×C₆₇	—	2	8	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
158	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	8	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	8	4	no	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₆₇	—	2	4	no	—
169	Q₁₆×C₆₇	—	4	2	no	—
170	SD₁₆×C₆₇	—	2	8	no	—
171	SD₁₆×C₆₇	—	4	4	no	—
172	SD₁₆×C₆₇	—	4	8	no	—
173	D₈×C₆₇	—	2	4	no	—
174	D₈×C₆₇	—	2	4	no	—
175	D₈×C₆₇	—	2	4	no	—
176	D₈×C₆₇	—	4	2	no	—
177	D₈×C₆₇	—	4	8	no	—
178	M₁₆×C₆₇	—	2	4	no	—
179	C₂×C₈×C₆₇	—	2	2	no	—
180	C₂×C₈×C₆₇	#41	1	4	yes	complemented
181	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
184	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	8	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
193	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₄²×C₆₇	—	2	8	no	—
206	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
209	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₄²×C₆₇	C₂⁴	1	4	yes	—
211	C₂³×C₆₇	—	2	4	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	4	4	no	—
214	C₂³×C₆₇	—	8	2	no	—
215	C₂³×C₆₇	—	8	2	no	—
216	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	Q₈×C₆₇	—	2	4	no	—
218	Q₈×C₆₇	—	2	4	no	—
219	Q₈×C₆₇	—	2	4	no	—
220	Q₈×C₆₇	—	4	2	no	—
221	D₄×C₆₇	—	2	4	no	—
222	D₄×C₆₇	—	2	4	no	—
223	D₄×C₆₇	—	2	4	no	—
224	D₄×C₆₇	—	2	4	no	—
225	D₄×C₆₇	—	4	8	no	—
226	D₄×C₆₇	—	4	4	no	—
227	D₄×C₆₇	—	4	4	no	—
228	D₄×C₆₇	—	8	2	no	—
229	D₄×C₆₇	—	8	8	no	—
230	D₄×C₆₇	—	8	8	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	8	no	—
236	C₂×C₄×C₆₇	—	4	8	no	—
237	C₂×C₄×C₆₇	—	8	4	no	—
238	C₂×C₄×C₆₇	—	8	8	no	—
239	C₂×C₄×C₆₇	—	8	2	no	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
248	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
249	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
250	C₈×C₆₇	—	2	1	no	—
251	C₈×C₆₇	—	2	1	no	—
252	C₈×C₆₇	—	2	4	no	—
253	C₈×C₆₇	C₂×#41	1	2	yes	—
254	C₈×C₆₇	C₂×#41	1	2	yes	—
255	C₂²×C₆₇	—	2	1	no	—
256	C₂²×C₆₇	—	2	1	no	—
257	C₂²×C₆₇	—	4	4	no	—
258	C₂²×C₆₇	—	4	4	no	—
259	C₂²×C₆₇	—	4	4	no	—
260	C₂²×C₆₇	—	4	4	no	—
261	C₂²×C₆₇	—	4	4	no	—
262	C₂²×C₆₇	—	4	4	no	—
263	C₂²×C₆₇	—	8	8	no	—
264	C₂²×C₆₇	#550	1	2	yes	—
265	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₆₇	—	2	1	no	—
267	C₄×C₆₇	—	2	1	no	—
268	C₄×C₆₇	—	2	4	no	—
269	C₄×C₆₇	—	2	1	no	—
270	C₄×C₆₇	—	2	4	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	—	4	4	no	—
273	C₄×C₆₇	#550	1	2	yes	—
274	C₄×C₆₇	#576	1	2	yes	—
275	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
276	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
277	#61406	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂×C₆₇	—	2	2	no	—
279	C₂×C₆₇	—	8	4	no	—
280	C₂×C₆₇	—	8	4	no	—
281	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
282	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
283	C₂×C₆₇	#3584	1	1	yes	characteristic
284	#3506	C₂×C₆₇	1	2	yes	complemented
285	#3403	C₂×C₆₇	1	4	yes	complemented
286	#3397	C₂×C₆₇	1	4	yes	complemented
287	#3396	C₂×C₆₇	1	4	yes	complemented
288	#3393	C₂×C₆₇	1	2	yes	complemented
289	#3391	C₂×C₆₇	1	2	yes	complemented
290	#3155	C₂×C₆₇	1	2	yes	complemented
291	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
292	#3045	C₂×C₆₇	1	4	yes	complemented
293	#3036	C₂×C₆₇	1	2	yes	complemented
294	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
295	#2951	C₂×C₆₇	1	2	yes	complemented
296	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#61406	1	1	yes	characteristic, direct product factor, Sylow
298	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
299	C₂×#135	C₂²×C₆₇	1	2	yes	—
300	C₂×#135	C₂²×C₆₇	1	2	yes	—
301	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
302	#559	C₂²×C₆₇	1	8	yes	complemented
303	#556	C₂²×C₆₇	1	4	yes	complemented
304	#555	—	2	2	no	—
305	#553	—	2	4	no	—
306	#550	C₂²×C₆₇	1	4	yes	complemented
307	#547	—	2	2	no	—
308	#538	—	2	4	no	—
309	#535	C₂²×C₆₇	1	4	yes	—
310	#533	C₂²×C₆₇	1	1	yes	characteristic
311	#532	C₂²×C₆₇	1	2	yes	—
312	#525	—	2	4	no	—
313	#520	C₂²×C₆₇	1	4	yes	—
314	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
315	#496	C₂²×C₆₇	1	8	yes	—
316	#495	C₂²×C₆₇	1	4	yes	—
317	#495	C₂²×C₆₇	1	2	yes	complemented
318	#495	C₂²×C₆₇	1	2	yes	complemented
319	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
320	#493	C₂²×C₆₇	1	4	yes	—
321	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
322	#492	C₂²×C₆₇	1	2	yes	complemented
323	#464	C₂²×C₆₇	1	8	yes	complemented
324	#463	C₂²×C₆₇	1	8	yes	complemented
325	#463	C₂²×C₆₇	1	8	yes	complemented
326	#461	C₂²×C₆₇	1	4	yes	complemented
327	#460	C₂²×C₆₇	1	8	yes	complemented
328	#459	C₂²×C₆₇	1	4	yes	complemented
329	#459	C₂²×C₆₇	1	8	yes	complemented
330	#446	C₂²×C₆₇	1	2	yes	complemented
331	#446	C₂²×C₆₇	1	2	yes	complemented
332	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
333	C₈×Q₈	C₂²×C₆₇	1	2	yes	complemented
334	#440	C₂²×C₆₇	1	8	yes	complemented
335	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
336	C₄×D₈	C₂²×C₆₇	1	2	yes	—
337	C₄×D₈	C₂²×C₆₇	1	2	yes	—
338	C₄×D₈	C₂²×C₆₇	1	2	yes	—
339	C₄×D₈	C₂²×C₆₇	1	2	yes	—
340	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
341	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
342	#424	C₂²×C₆₇	1	2	yes	—
343	#423	C₂²×C₆₇	1	1	yes	characteristic
344	#417	C₂²×C₆₇	1	4	yes	—
345	C₂×#102	C₂²×C₆₇	1	2	yes	—
346	#405	C₂²×C₆₇	1	2	yes	—
347	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×#41	—	2	4	no	—
349	C₂×#41	C₂³×C₆₇	1	4	yes	—
350	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
351	C₂³×C₄	—	2	2	no	—
352	AGL(1,ℤ/8)	—	4	8	no	—
353	#135	—	2	8	no	—
354	#135	—	4	4	no	—
355	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
356	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
357	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
358	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
359	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
360	C₂×D₈	—	2	4	no	—
361	C₂×D₈	C₂³×C₆₇	1	2	yes	—
362	C₂×D₈	C₂³×C₆₇	1	2	yes	—
363	C₂×D₈	C₂³×C₆₇	1	2	yes	—
364	C₂×D₈	C₂³×C₆₇	1	2	yes	—
365	C₂×D₈	C₂³×C₆₇	1	8	yes	—
366	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
367	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
369	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
370	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
372	#127	C₂³×C₆₇	1	2	yes	—
373	#127	C₂³×C₆₇	1	8	yes	—
374	#126	—	2	8	no	—
375	#125	D₄×C₆₇	1	2	yes	—
376	#124	—	2	8	no	—
377	#124	C₂³×C₆₇	1	4	yes	—
378	#124	C₂³×C₆₇	1	8	yes	—
379	#123	—	2	8	no	—
380	#123	—	2	8	no	—
381	C₂²⋊Q₈	—	2	4	no	—
382	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
383	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
384	#121	—	2	2	no	—
385	#121	—	2	2	no	—
386	#121	—	2	4	no	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	#121	—	2	8	no	—
390	#120	—	2	8	no	—
391	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
392	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
393	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	C₂³×C₆₇	1	4	yes	—
401	C₄×D₄	C₂³×C₆₇	1	4	yes	—
402	C₄×D₄	C₂³×C₆₇	1	8	yes	—
403	#117	—	2	4	no	—
404	#117	—	2	4	no	—
405	#117	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	—	2	4	no	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
410	#107	C₂³×C₆₇	1	2	yes	—
411	#107	C₂³×C₆₇	1	2	yes	—
412	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
413	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
414	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
415	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
416	#103	C₂³×C₆₇	1	8	yes	—
417	#102	C₂³×C₆₇	1	4	yes	—
418	#102	C₂³×C₆₇	1	4	yes	—
419	#102	C₂³×C₆₇	1	8	yes	—
420	#97	C₂³×C₆₇	1	4	yes	—
421	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₆₇	1	4	yes	—
426	C₂⁴	—	2	2	no	—
427	#41	—	2	8	no	—
428	#41	—	2	8	no	—
429	#41	—	8	8	no	—
430	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
432	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
442	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	8	4	no	—
449	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
452	Q₁₆	—	2	4	no	—
453	Q₁₆	—	4	2	no	—
454	SD₁₆	—	2	8	no	—
455	SD₁₆	—	4	4	no	—
456	SD₁₆	—	4	8	no	—
457	D₈	—	2	4	no	—
458	D₈	—	2	4	no	—
459	D₈	—	2	4	no	—
460	D₈	—	4	2	no	—
461	D₈	—	4	8	no	—
462	M₁₆	—	2	4	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	#41×C₆₇	1	4	yes	complemented
465	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
466	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
467	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
468	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
469	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
475	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
476	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
478	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
479	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
480	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
489	C₄²	—	2	8	no	—
490	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
491	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
492	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
493	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
494	C₄²	C₂⁴×C₆₇	1	4	yes	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	4	4	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
501	Q₈	—	2	4	no	—
502	Q₈	—	2	4	no	—
503	Q₈	—	2	4	no	—
504	Q₈	—	4	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	4	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	4	no	—
509	D₄	—	4	8	no	—
510	D₄	—	4	4	no	—
511	D₄	—	4	4	no	—
512	D₄	—	8	8	no	—
513	D₄	—	8	2	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	4	no	—
522	C₂×C₄	—	8	8	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
531	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
532	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
533	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	2	4	no	—
537	C₈	C₂×#41×C₆₇	1	2	yes	—
538	C₈	C₂×#41×C₆₇	1	2	yes	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	—	8	8	no	—
548	C₂²	#550×C₆₇	1	2	yes	—
549	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
550	C₄	—	2	1	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	4	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	4	no	—
557	C₄	#550×C₆₇	1	2	yes	—
558	C₄	#576×C₆₇	1	2	yes	—
559	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
560	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
561	C₂	—	2	2	no	—
562	C₂	—	8	4	no	—
563	C₂	—	8	4	no	—
564	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
565	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
566	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #533×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3506	C₂	semidirect
2. C₆₇×#3403	C₂	semidirect
3. C₆₇×#3397	C₂	semidirect
4. C₆₇×#3396	C₂	semidirect
5. C₆₇×#3393	C₂	semidirect
6. C₆₇×#3391	C₂	semidirect
7. C₆₇×#3155	C₂	semidirect
8. C₆₇×#3147	C₂	semidirect
9. C₆₇×#3045	C₂	semidirect
10. C₆₇×#3036	C₂	semidirect
11. C₆₇×#2960	C₂	semidirect
12. C₆₇×#2951	C₂	semidirect
13. C₂×C₄×D₈×C₆₇	C₂	semidirect
18. #559×C₆₇	C₂²	semidirect
19. #556×C₆₇	C₂²	semidirect
22. #550×C₆₇	C₂²	semidirect
33. #495×C₆₇	C₂²	semidirect
34. #495×C₆₇	C₂²	semidirect
35. #494×C₆₇	C₂²	semidirect
37. #493×C₆₇	C₂²	semidirect
38. #492×C₆₇	C₂²	semidirect
39. #464×C₆₇	C₂²	semidirect
40. #463×C₆₇	C₂²	semidirect
41. #463×C₆₇	C₂²	semidirect
42. #461×C₆₇	C₂²	semidirect
43. #460×C₆₇	C₂²	semidirect
44. #459×C₆₇	C₂²	semidirect
45. #459×C₆₇	C₂²	semidirect
46. #446×C₆₇	C₂²	semidirect
47. #446×C₆₇	C₂²	semidirect
48. C₈×Q₈×C₆₇	C₂²	semidirect
49. C₈×Q₈×C₆₇	C₂²	semidirect
50. #440×C₆₇	C₂²	semidirect
51. C₄×Q₁₆×C₆₇	C₂²	semidirect
56. C₄×D₈×C₆₇	C₂²	semidirect
180. C₂×C₈×C₆₇	#41	semidirect
277. #61406	C₆₇	direct (N × H)
284. #3506	C₂×C₆₇	semidirect
285. #3403	C₂×C₆₇	semidirect
286. #3397	C₂×C₆₇	semidirect
287. #3396	C₂×C₆₇	semidirect
288. #3393	C₂×C₆₇	semidirect
289. #3391	C₂×C₆₇	semidirect
290. #3155	C₂×C₆₇	semidirect
291. #3147	C₂×C₆₇	semidirect
292. #3045	C₂×C₆₇	semidirect
293. #3036	C₂×C₆₇	semidirect
294. #2960	C₂×C₆₇	semidirect
295. #2951	C₂×C₆₇	semidirect
296. C₂×C₄×D₈	C₂×C₆₇	semidirect
297. C₆₇	#61406	direct (N × H)
302. #559	C₂²×C₆₇	semidirect
303. #556	C₂²×C₆₇	semidirect
306. #550	C₂²×C₆₇	semidirect
317. #495	C₂²×C₆₇	semidirect
318. #495	C₂²×C₆₇	semidirect
319. #494	C₂²×C₆₇	semidirect
321. #493	C₂²×C₆₇	semidirect
322. #492	C₂²×C₆₇	semidirect
323. #464	C₂²×C₆₇	semidirect
324. #463	C₂²×C₆₇	semidirect
325. #463	C₂²×C₆₇	semidirect
326. #461	C₂²×C₆₇	semidirect
327. #460	C₂²×C₆₇	semidirect
328. #459	C₂²×C₆₇	semidirect
329. #459	C₂²×C₆₇	semidirect
330. #446	C₂²×C₆₇	semidirect
331. #446	C₂²×C₆₇	semidirect
332. C₈×Q₈	C₂²×C₆₇	semidirect
333. C₈×Q₈	C₂²×C₆₇	semidirect
334. #440	C₂²×C₆₇	semidirect
335. C₄×Q₁₆	C₂²×C₆₇	semidirect
340. C₄×D₈	C₂²×C₆₇	semidirect
464. C₂×C₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3506	2
C₆₇×#3403	2
C₆₇×#3397	2
C₆₇×#3396	2
C₆₇×#3393	2
C₆₇×#3391	2
C₆₇×#3155	2
C₆₇×#3147	2
C₆₇×#3045	2
C₆₇×#3036	2
C₆₇×#2960	2
C₆₇×#2951	2
C₂×C₄×D₈×C₆₇	2
#61406	67

Subgroups — C67×#61406 #30740657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61406 #30740657