Subgroups — C₆₇×#18876 #30698127

Order 17152.

976 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3245	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3206	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1646	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1644	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1624	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1622	C₂	1	2	yes	maximal, complemented
9	C₂×#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1480	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1477	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₆₇	—	2	1	no	—
13	#556×C₆₇	—	2	1	no	—
14	#555×C₆₇	C₂²	1	2	yes	—
15	#554×C₆₇	C₂²	1	2	yes	—
16	#553×C₆₇	—	2	2	no	—
17	#548×C₆₇	—	2	2	no	—
18	#547×C₆₇	—	2	1	no	—
19	#542×C₆₇	—	2	2	no	—
20	#537×C₆₇	—	2	1	no	—
21	#533×C₆₇	—	2	1	no	—
22	#532×C₆₇	—	2	1	no	—
23	#532×C₆₇	C₂²	1	1	yes	characteristic
24	#532×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	#520×C₆₇	—	2	1	no	—
27	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
28	#492×C₆₇	C₂²	1	2	yes	—
29	#487×C₆₇	C₂²	1	2	yes	—
30	#474×C₆₇	—	2	1	no	—
31	#474×C₆₇	C₂²	1	2	yes	—
32	#466×C₆₇	—	2	2	no	—
33	#464×C₆₇	—	2	1	no	—
34	#463×C₆₇	—	2	1	no	—
35	#462×C₆₇	C₂²	1	1	yes	characteristic
36	#461×C₆₇	—	2	2	no	—
37	#461×C₆₇	C₂²	1	1	yes	characteristic
38	#460×C₆₇	C₂²	1	1	yes	characteristic
39	#459×C₆₇	—	2	1	no	—
40	#459×C₆₇	C₂²	1	1	yes	characteristic
41	#459×C₆₇	C₂²	1	2	yes	—
42	#448×C₆₇	—	2	2	no	—
43	#431×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	#416×C₆₇	—	2	1	no	—
47	#333×C₆₇	C₂²	1	2	yes	—
48	#333×C₆₇	C₂²	1	2	yes	—
49	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
50	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
51	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
52	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	—	2	1	no	—
55	C₂×#41×C₆₇	—	2	1	no	—
56	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
58	#135×C₆₇	—	4	4	no	—
59	C₂×Q₁₆×C₆₇	—	2	1	no	—
60	C₂×SD₁₆×C₆₇	—	2	1	no	—
61	C₂×SD₁₆×C₆₇	—	2	1	no	—
62	C₂×SD₁₆×C₆₇	—	2	4	no	—
63	C₂×D₈×C₆₇	—	2	1	no	—
64	C₂×D₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	—	2	1	no	—
66	C₂²×C₈×C₆₇	—	2	1	no	—
67	C₄⋊Q₈×C₆₇	—	2	1	no	—
68	C₄⋊Q₈×C₆₇	—	2	1	no	—
69	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
70	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
73	#127×C₆₇	D₄	1	1	yes	characteristic
74	#127×C₆₇	C₂³	1	1	yes	characteristic
75	#126×C₆₇	—	2	4	no	—
76	#125×C₆₇	—	2	2	no	—
77	#124×C₆₇	—	2	2	no	—
78	#124×C₆₇	—	2	4	no	—
79	#124×C₆₇	—	2	1	no	—
80	#124×C₆₇	—	2	1	no	—
81	#123×C₆₇	—	2	4	no	—
82	C₂²⋊Q₈×C₆₇	—	2	2	no	—
83	C₂²⋊Q₈×C₆₇	—	2	2	no	—
84	C₂²⋊Q₈×C₆₇	—	2	2	no	—
85	C₂²⋊Q₈×C₆₇	—	2	4	no	—
86	#121×C₆₇	—	2	2	no	—
87	#121×C₆₇	—	2	4	no	—
88	#121×C₆₇	—	2	2	no	—
89	#121×C₆₇	—	2	2	no	—
90	#121×C₆₇	—	2	1	no	—
91	#121×C₆₇	—	2	1	no	—
92	#121×C₆₇	—	2	1	no	—
93	#121×C₆₇	D₄	1	2	yes	—
94	C₄×Q₈×C₆₇	—	2	2	no	—
95	C₄×Q₈×C₆₇	—	2	2	no	—
96	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₄×D₄×C₆₇	—	2	2	no	—
99	C₄×D₄×C₆₇	—	2	4	no	—
100	C₄×D₄×C₆₇	—	2	2	no	—
101	C₄×D₄×C₆₇	—	2	1	no	—
102	C₄×D₄×C₆₇	—	2	1	no	—
103	C₄×D₄×C₆₇	—	2	1	no	—
104	C₄×D₄×C₆₇	—	2	1	no	—
105	C₄×D₄×C₆₇	—	2	1	no	—
106	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
107	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₄×D₄×C₆₇	C₂³	1	2	yes	—
109	#117×C₆₇	—	2	2	no	—
110	#117×C₆₇	—	2	2	no	—
111	#117×C₆₇	—	2	1	no	—
112	#117×C₆₇	D₄	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₄²×C₆₇	—	2	1	no	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	#107×C₆₇	—	2	2	no	—
119	C₄⋊C₈×C₆₇	—	2	1	no	—
120	C₄⋊C₈×C₆₇	—	2	1	no	—
121	C₄⋊C₈×C₆₇	—	2	1	no	—
122	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
123	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
124	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
125	#103×C₆₇	—	2	4	no	—
126	#103×C₆₇	—	2	1	no	—
127	#103×C₆₇	—	2	1	no	—
128	#102×C₆₇	—	2	2	no	—
129	#102×C₆₇	—	2	1	no	—
130	#102×C₆₇	—	2	2	no	—
131	#102×C₆₇	—	2	1	no	—
132	C₂²⋊C₈×C₆₇	—	2	1	no	—
133	C₄×C₈×C₆₇	C₂³	1	2	yes	—
134	#41×C₆₇	—	2	2	no	—
135	#41×C₆₇	—	2	2	no	—
136	#41×C₆₇	—	4	2	no	—
137	#41×C₆₇	—	4	2	no	—
138	#41×C₆₇	—	4	2	no	—
139	#41×C₆₇	—	4	2	no	—
140	#41×C₆₇	—	8	4	no	—
141	C₂×Q₈×C₆₇	—	2	1	no	—
142	C₂×Q₈×C₆₇	—	2	2	no	—
143	C₂×Q₈×C₆₇	—	2	1	no	—
144	C₂×Q₈×C₆₇	—	8	2	no	—
145	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	4	1	no	—
156	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	1	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	4	1	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	C₂×D₄	1	1	yes	characteristic
168	Q₁₆×C₆₇	—	4	2	no	—
169	SD₁₆×C₆₇	—	4	2	no	—
170	SD₁₆×C₆₇	—	4	2	no	—
171	SD₁₆×C₆₇	—	8	4	no	—
172	D₈×C₆₇	—	4	2	no	—
173	D₈×C₆₇	—	8	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	4	2	no	—
179	C₂×C₈×C₆₇	—	8	1	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	1	no	—
189	C₄⋊C₄×C₆₇	—	2	1	no	—
190	C₄⋊C₄×C₆₇	—	2	1	no	—
191	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
192	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
193	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
194	C₂²⋊C₄×C₆₇	—	2	4	no	—
195	C₂²⋊C₄×C₆₇	—	2	4	no	—
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	1	no	—
198	C₂²⋊C₄×C₆₇	—	2	1	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	1	no	—
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₂²⋊C₄×C₆₇	—	4	1	no	—
203	C₂²⋊C₄×C₆₇	—	4	1	no	—
204	C₄²×C₆₇	—	2	4	no	—
205	C₄²×C₆₇	—	2	1	no	—
206	C₄²×C₆₇	—	2	1	no	—
207	C₄²×C₆₇	—	2	1	no	—
208	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₂³×C₆₇	—	2	1	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	4	1	no	—
215	C₂³×C₆₇	#120	1	1	yes	characteristic
216	Q₈×C₆₇	—	2	1	no	—
217	Q₈×C₆₇	—	2	1	no	—
218	Q₈×C₆₇	—	4	4	no	—
219	Q₈×C₆₇	—	4	4	no	—
220	Q₈×C₆₇	—	4	2	no	—
221	Q₈×C₆₇	—	4	1	no	—
222	Q₈×C₆₇	—	4	1	no	—
223	D₄×C₆₇	—	2	1	no	—
224	D₄×C₆₇	—	2	1	no	—
225	D₄×C₆₇	—	2	2	no	—
226	D₄×C₆₇	—	2	2	no	—
227	D₄×C₆₇	—	4	2	no	—
228	D₄×C₆₇	—	4	2	no	—
229	D₄×C₆₇	—	4	2	no	—
230	D₄×C₆₇	—	4	4	no	—
231	D₄×C₆₇	—	4	1	no	—
232	D₄×C₆₇	—	4	1	no	—
233	D₄×C₆₇	—	4	1	no	—
234	D₄×C₆₇	—	4	1	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	8	4	no	—
237	D₄×C₆₇	—	8	2	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	2	no	—
247	C₂×C₄×C₆₇	—	4	2	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	1	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	—	4	2	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	8	2	no	—
254	C₂×C₄×C₆₇	—	8	4	no	—
255	C₂×C₄×C₆₇	—	8	2	no	—
256	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
257	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
258	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
259	C₂×C₄×C₆₇	#135	1	2	yes	—
260	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
264	C₈×C₆₇	—	4	2	no	—
265	C₈×C₆₇	—	4	1	no	—
266	C₈×C₆₇	—	4	1	no	—
267	C₈×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	4	1	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	4	1	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	1	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	8	1	no	—
277	C₂²×C₆₇	—	8	1	no	—
278	C₂²×C₆₇	—	8	1	no	—
279	C₂²×C₆₇	#449	1	2	yes	—
280	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
281	C₄×C₆₇	—	2	1	no	—
282	C₄×C₆₇	—	2	1	no	—
283	C₄×C₆₇	—	2	1	no	—
284	C₄×C₆₇	—	4	1	no	—
285	C₄×C₆₇	—	4	2	no	—
286	C₄×C₆₇	—	4	1	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	4	no	—
289	C₄×C₆₇	—	8	1	no	—
290	C₄×C₆₇	#449	1	2	yes	—
291	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
292	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₄×C₆₇	#577	1	1	yes	characteristic
294	C₄×C₆₇	#578	1	1	yes	characteristic
295	#18876	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	2	no	—
297	C₂×C₆₇	—	8	1	no	—
298	C₂×C₆₇	—	8	2	no	—
299	C₂×C₆₇	—	16	1	no	—
300	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
301	C₂×C₆₇	#3013	1	1	yes	characteristic
302	C₂×C₆₇	#3019	1	1	yes	characteristic
303	#3511	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#3245	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#3206	C₂×C₆₇	1	1	yes	characteristic, complemented
306	#3035	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1646	C₂×C₆₇	1	2	yes	complemented
308	#1644	C₂×C₆₇	1	2	yes	complemented
309	#1624	C₂×C₆₇	1	2	yes	complemented
310	#1622	C₂×C₆₇	1	2	yes	complemented
311	C₂×#333	C₂×C₆₇	1	1	yes	characteristic, complemented
312	#1480	C₂×C₆₇	1	1	yes	characteristic, complemented
313	#1477	C₂×C₆₇	1	1	yes	characteristic, complemented
314	C₆₇	#18876	1	1	yes	characteristic, direct product factor, Sylow
315	C₂×#135	—	2	1	no	—
316	#556	—	2	1	no	—
317	#555	C₂²×C₆₇	1	2	yes	—
318	#554	C₂²×C₆₇	1	2	yes	—
319	#553	—	2	2	no	—
320	#548	—	2	2	no	—
321	#547	—	2	1	no	—
322	#542	—	2	2	no	—
323	#537	—	2	1	no	—
324	#533	—	2	1	no	—
325	#532	—	2	1	no	—
326	#532	C₂²×C₆₇	1	1	yes	characteristic
327	#532	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
329	#520	—	2	1	no	—
330	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
331	#492	C₂²×C₆₇	1	2	yes	—
332	#487	C₂²×C₆₇	1	2	yes	—
333	#474	—	2	1	no	—
334	#474	C₂²×C₆₇	1	2	yes	—
335	#466	—	2	2	no	—
336	#464	—	2	1	no	—
337	#463	—	2	1	no	—
338	#462	C₂²×C₆₇	1	1	yes	characteristic
339	#461	—	2	2	no	—
340	#461	C₂²×C₆₇	1	1	yes	characteristic
341	#460	C₂²×C₆₇	1	1	yes	characteristic
342	#459	—	2	1	no	—
343	#459	C₂²×C₆₇	1	2	yes	—
344	#459	C₂²×C₆₇	1	1	yes	characteristic
345	#448	—	2	2	no	—
346	#431	C₂²×C₆₇	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
349	#416	—	2	1	no	—
350	#333	C₂²×C₆₇	1	2	yes	—
351	#333	C₂²×C₆₇	1	2	yes	—
352	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
353	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
354	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
355	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
360	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
361	#135	—	4	4	no	—
362	C₂×Q₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	4	no	—
366	C₂×D₈	—	2	1	no	—
367	C₂×D₈	—	2	2	no	—
368	C₂²×C₈	—	2	1	no	—
369	C₂²×C₈	—	2	1	no	—
370	C₄⋊Q₈	—	2	1	no	—
371	C₄⋊Q₈	—	2	1	no	—
372	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
373	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
374	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
376	#127	D₄×C₆₇	1	1	yes	characteristic
377	#127	C₂³×C₆₇	1	1	yes	characteristic
378	#126	—	2	4	no	—
379	#125	—	2	2	no	—
380	#124	—	2	2	no	—
381	#124	—	2	4	no	—
382	#124	—	2	1	no	—
383	#124	—	2	1	no	—
384	#123	—	2	4	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	—	2	4	no	—
389	#121	—	2	2	no	—
390	#121	—	2	4	no	—
391	#121	—	2	2	no	—
392	#121	—	2	2	no	—
393	#121	—	2	1	no	—
394	#121	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	D₄×C₆₇	1	2	yes	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
400	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	4	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
410	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₄×D₄	C₂³×C₆₇	1	2	yes	—
412	#117	—	2	2	no	—
413	#117	—	2	2	no	—
414	#117	—	2	1	no	—
415	#117	D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
421	#107	—	2	2	no	—
422	C₄⋊C₈	—	2	1	no	—
423	C₄⋊C₈	—	2	1	no	—
424	C₄⋊C₈	—	2	1	no	—
425	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
426	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
427	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
428	#103	—	2	4	no	—
429	#103	—	2	1	no	—
430	#103	—	2	1	no	—
431	#102	—	2	2	no	—
432	#102	—	2	1	no	—
433	#102	—	2	1	no	—
434	#102	—	2	2	no	—
435	C₂²⋊C₈	—	2	1	no	—
436	C₄×C₈	C₂³×C₆₇	1	2	yes	—
437	#41	—	2	2	no	—
438	#41	—	2	2	no	—
439	#41	—	4	2	no	—
440	#41	—	4	2	no	—
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	#41	—	8	4	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	8	2	no	—
448	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
449	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	4	1	no	—
459	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	4	1	no	—
468	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
471	Q₁₆	—	4	2	no	—
472	SD₁₆	—	4	2	no	—
473	SD₁₆	—	4	2	no	—
474	SD₁₆	—	8	4	no	—
475	D₈	—	4	2	no	—
476	D₈	—	8	2	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	4	2	no	—
482	C₂×C₈	—	8	1	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
495	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
496	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	2	1	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	4	1	no	—
506	C₂²⋊C₄	—	4	1	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
512	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
513	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
514	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
515	C₂³	—	2	1	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	4	1	no	—
518	C₂³	#120×C₆₇	1	1	yes	characteristic
519	Q₈	—	2	1	no	—
520	Q₈	—	2	1	no	—
521	Q₈	—	4	4	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	4	2	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	4	1	no	—
526	D₄	—	2	1	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	2	no	—
529	D₄	—	2	2	no	—
530	D₄	—	4	2	no	—
531	D₄	—	4	2	no	—
532	D₄	—	4	2	no	—
533	D₄	—	4	4	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	1	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	1	no	—
539	D₄	—	8	4	no	—
540	D₄	—	8	2	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	8	2	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
560	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
561	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
562	C₂×C₄	#135×C₆₇	1	2	yes	—
563	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
567	C₈	—	4	2	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	2	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	8	1	no	—
581	C₂²	—	8	1	no	—
582	C₂²	#449×C₆₇	1	2	yes	—
583	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
584	C₄	—	2	1	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	4	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	2	no	—
592	C₄	—	8	1	no	—
593	C₄	#449×C₆₇	1	2	yes	—
594	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
595	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
596	C₄	#577×C₆₇	1	1	yes	characteristic
597	C₄	#578×C₆₇	1	1	yes	characteristic
598	C₂	—	2	2	no	—
599	C₂	—	8	1	no	—
600	C₂	—	8	2	no	—
601	C₂	—	16	1	no	—
602	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
603	C₂	C₆₇×#3013	1	1	yes	characteristic
604	C₂	C₆₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3511	C₂	semidirect
2. C₆₇×#3245	C₂	semidirect
3. C₆₇×#3206	C₂	semidirect
4. C₆₇×#3035	C₂	semidirect
5. C₆₇×#1646	C₂	semidirect
6. C₆₇×#1644	C₂	semidirect
7. C₆₇×#1624	C₂	semidirect
8. C₆₇×#1622	C₂	semidirect
9. C₂×#333×C₆₇	C₂	semidirect
10. C₆₇×#1480	C₂	semidirect
11. C₆₇×#1477	C₂	semidirect
295. #18876	C₆₇	direct (N × H)
303. #3511	C₂×C₆₇	semidirect
304. #3245	C₂×C₆₇	semidirect
305. #3206	C₂×C₆₇	semidirect
306. #3035	C₂×C₆₇	semidirect
307. #1646	C₂×C₆₇	semidirect
308. #1644	C₂×C₆₇	semidirect
309. #1624	C₂×C₆₇	semidirect
310. #1622	C₂×C₆₇	semidirect
311. C₂×#333	C₂×C₆₇	semidirect
312. #1480	C₂×C₆₇	semidirect
313. #1477	C₂×C₆₇	semidirect
314. C₆₇	#18876	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3511	2
C₆₇×#3245	2
C₆₇×#3206	2
C₆₇×#3035	2
C₆₇×#1646	2
C₆₇×#1644	2
C₆₇×#1624	2
C₆₇×#1622	2
C₂×#333×C₆₇	2
C₆₇×#1480	2
C₆₇×#1477	2
#18876	67

Subgroups — C67×#18876 #30698127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18876 #30698127