Subgroups — #30822379

Order 17152.

988 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
3	#22558630	C₂	1	4	yes	maximal, complemented
4	#22558612	C₂	1	4	yes	maximal, complemented
5	#22558604	C₂	1	4	yes	maximal, complemented
6	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863453	—	2	1	no	—
8	#16863453	—	2	1	no	—
9	#16863449	—	2	1	no	—
10	#16863449	—	2	1	no	—
11	#16863449	C₂²	1	2	yes	—
12	#16863449	C₂²	1	2	yes	—
13	#16863449	C₂²	1	2	yes	—
14	#16863449	C₂²	1	2	yes	—
15	#16863420	—	2	1	no	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	#16863360	—	2	1	no	—
18	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
19	C₄×#11761612	C₂²	1	2	yes	—
20	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
21	Q₈⋊(C₆₇⋊C₈)	C₂²	1	16	yes	complemented
22	#533×C₆₇	—	2	1	no	—
23	#533×C₆₇	—	2	1	no	—
24	#533×C₆₇	—	2	1	no	—
25	#533×C₆₇	—	2	1	no	—
26	#533×C₆₇	C₄	1	2	yes	—
27	#533×C₆₇	C₄	1	2	yes	—
28	#533×C₆₇	C₂²	1	2	yes	—
29	#533×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#125×C₆₇	—	2	1	no	—
33	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
34	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
35	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
36	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
37	C₂×#117×C₆₇	—	2	1	no	—
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#11761612	—	2	2	no	—
40	C₂²×C₆₇⋊C₈	—	2	2	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
47	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
48	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
49	#14523954	—	2	2	no	—
50	C₄×C₆₇⋊C₈	—	2	2	no	—
51	C₄×C₆₇⋊C₈	—	2	2	no	—
52	C₄×C₆₇⋊C₈	C₂³	1	4	yes	—
53	C₂²×Q₈×C₆₇	—	2	1	no	—
54	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
58	C₄⋊Q₈×C₆₇	—	2	1	no	—
59	C₄⋊Q₈×C₆₇	—	2	1	no	—
60	C₄⋊Q₈×C₆₇	—	2	1	no	—
61	C₄⋊Q₈×C₆₇	—	2	1	no	—
62	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
64	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
66	#125×C₆₇	—	2	2	no	—
67	#125×C₆₇	—	2	2	no	—
68	#125×C₆₇	—	2	2	no	—
69	#125×C₆₇	—	2	2	no	—
70	C₂²⋊Q₈×C₆₇	—	2	2	no	—
71	C₂²⋊Q₈×C₆₇	—	2	2	no	—
72	C₂²⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	2	no	—
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	—	2	2	no	—
76	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
77	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₆₇	—	2	2	no	—
79	C₄×Q₈×C₆₇	—	2	2	no	—
80	C₄×Q₈×C₆₇	—	2	2	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	2	no	—
83	C₄×Q₈×C₆₇	—	2	2	no	—
84	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
85	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
86	#117×C₆₇	—	2	2	no	—
87	#117×C₆₇	—	2	2	no	—
88	#117×C₆₇	—	2	2	no	—
89	#117×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₄²×C₆₇	—	2	2	no	—
98	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	#11761612	—	4	4	no	—
111	C₂×C₆₇⋊C₈	—	2	4	no	—
112	C₂×C₆₇⋊C₈	—	2	4	no	—
113	C₂×C₆₇⋊C₈	—	4	4	no	—
114	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	—	2	8	no	—
119	C₂×Q₈×C₆₇	—	4	4	no	—
120	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	4	2	no	—
126	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	4	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
156	C₂²⋊C₄×C₆₇	—	2	4	no	—
157	C₂²⋊C₄×C₆₇	—	2	4	no	—
158	C₄²×C₆₇	—	2	4	no	—
159	C₄²×C₆₇	—	2	4	no	—
160	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
161	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
162	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
163	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
164	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
165	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
166	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	4	1	no	—
179	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	8	no	—
183	Q₈×C₆₇	—	4	4	no	—
184	Q₈×C₆₇	—	4	4	no	—
185	Q₈×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	4	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	#131	1	4	yes	—
216	C₂×C₄×C₆₇	#137	1	4	yes	—
217	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₆₇⋊C₈	—	4	4	no	—
221	C₆₇⋊C₈	—	4	4	no	—
222	C₂²×C₆₇	—	2	1	no	—
223	C₂²×C₆₇	—	2	1	no	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	1	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	2	1	no	—
229	C₂²×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	#408	1	2	yes	—
231	C₂²×C₆₇	#419	1	2	yes	—
232	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	1	no	—
236	C₄×C₆₇	—	2	1	no	—
237	C₄×C₆₇	—	2	1	no	—
238	C₄×C₆₇	—	2	1	no	—
239	C₄×C₆₇	—	2	1	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	—	4	4	no	—
242	C₄×C₆₇	#408	1	2	yes	—
243	C₄×C₆₇	#419	1	2	yes	—
244	C₄×C₆₇	#421	1	2	yes	—
245	C₄×C₆₇	C₂×#131	1	2	yes	—
246	C₄×C₆₇	C₂×#137	1	2	yes	—
247	#17464	—	67	1	no	maximal, Sylow
248	C₂×C₆₇	—	2	2	no	—
249	C₂×C₆₇	—	2	2	no	—
250	C₂×C₆₇	—	4	1	no	—
251	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
252	C₂×C₆₇	C₂×#419	1	1	yes	characteristic
253	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
254	C₂×#533	D₆₇	1	1	yes	characteristic
255	#2920	—	67	1	no	—
256	C₂×#431	—	67	1	no	—
257	#1485	—	67	4	no	—
258	#1481	—	67	4	no	—
259	#1479	—	67	4	no	—
260	C₆₇	#17464	1	1	yes	characteristic, complemented, Sylow
261	#533	—	2	1	no	—
262	#533	—	2	1	no	—
263	#533	—	2	1	no	—
264	#533	—	2	1	no	—
265	#533	Dic₆₇	1	2	yes	—
266	#533	Dic₆₇	1	2	yes	—
267	#533	C₂×D₆₇	1	2	yes	—
268	#533	C₂×D₆₇	1	2	yes	—
269	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
271	C₂×#125	—	2	1	no	—
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₂²⋊Q₈	Dic₆₇	1	2	yes	—
274	C₂×C₄×Q₈	—	2	1	no	—
275	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
276	C₂×#117	—	2	1	no	—
277	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
278	#432	—	134	1	no	—
279	#432	—	134	1	no	—
280	#431	—	67	2	no	—
281	#431	—	67	2	no	—
282	#431	—	67	2	no	—
283	#431	—	67	2	no	—
284	#431	—	134	1	no	—
285	#431	—	134	1	no	—
286	#423	—	134	1	no	—
287	C₂×C₄⋊C₈	—	67	2	no	—
288	#407	—	134	1	no	—
289	C₂×C₂²⋊C₈	—	134	1	no	—
290	C₄×M₁₆	—	67	2	no	—
291	C₂×C₄×C₈	—	134	1	no	—
292	Q₈⋊C₈	—	67	16	no	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
295	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
296	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
297	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
298	C₂×M₁₆	—	134	2	no	—
299	C₂²×C₈	—	134	2	no	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	—	2	1	no	—
302	C₄⋊Q₈	—	2	1	no	—
303	C₄⋊Q₈	—	2	1	no	—
304	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
305	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
306	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
307	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
308	#125	—	2	2	no	—
309	#125	—	2	2	no	—
310	#125	—	2	2	no	—
311	#125	—	2	2	no	—
312	C₂²⋊Q₈	—	2	2	no	—
313	C₂²⋊Q₈	—	2	2	no	—
314	C₂²⋊Q₈	—	2	2	no	—
315	C₂²⋊Q₈	—	2	2	no	—
316	C₂²⋊Q₈	—	2	2	no	—
317	C₂²⋊Q₈	—	2	2	no	—
318	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
319	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
320	C₄×Q₈	—	2	2	no	—
321	C₄×Q₈	—	2	2	no	—
322	C₄×Q₈	—	2	2	no	—
323	C₄×Q₈	—	2	2	no	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×Q₈	—	2	2	no	—
326	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
327	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
328	#117	—	2	2	no	—
329	#117	—	2	2	no	—
330	#117	—	2	2	no	—
331	#117	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
341	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
342	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
343	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
344	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
345	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
346	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
347	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
348	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
349	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
350	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
351	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
352	C₄⋊C₈	—	67	4	no	—
353	C₄⋊C₈	—	134	2	no	—
354	C₄⋊C₈	—	134	2	no	—
355	C₄⋊C₈	—	134	2	no	—
356	C₂²⋊C₈	—	134	2	no	—
357	C₂²⋊C₈	—	134	2	no	—
358	C₂²⋊C₈	—	134	2	no	—
359	C₂²⋊C₈	—	134	2	no	—
360	#97	—	134	2	no	—
361	C₄×C₈	—	67	4	no	—
362	C₄×C₈	—	134	2	no	—
363	C₄×C₈	—	134	2	no	—
364	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	2	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	8	no	—
369	C₂×Q₈	—	4	4	no	—
370	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	4	2	no	—
376	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
377	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
379	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
380	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
387	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
389	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
390	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
393	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
394	M₁₆	—	268	4	no	—
395	C₂×C₈	—	268	4	no	—
396	C₂×C₈	—	134	4	no	—
397	C₂×C₈	—	134	4	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₄²	—	2	4	no	—
413	C₄²	—	2	4	no	—
414	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
415	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
416	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
417	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
418	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
419	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
420	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
421	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
422	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
423	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
424	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
425	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
426	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
427	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
428	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
429	C₄²	C₂³×D₆₇	1	1	yes	characteristic
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	4	1	no	—
433	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
434	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
435	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
436	Q₈	—	2	8	no	—
437	Q₈	—	4	4	no	—
438	Q₈	—	4	4	no	—
439	Q₈	—	4	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	4	no	—
459	C₂×C₄	—	4	4	no	—
460	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
465	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
466	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
467	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
468	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
469	C₂×C₄	#14524050	1	4	yes	—
470	C₂×C₄	#14524074	1	4	yes	—
471	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
472	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
474	C₈	—	268	4	no	—
475	C₈	—	268	4	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	4	4	no	—
484	C₂²	#16863363	1	2	yes	—
485	C₂²	#16863405	1	2	yes	—
486	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
487	C₄	—	2	1	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	4	no	—
496	C₄	#16863363	1	2	yes	—
497	C₄	#16863405	1	2	yes	—
498	C₄	#16863413	1	2	yes	—
499	C₄	C₂×#14524050	1	2	yes	—
500	C₄	C₂×#14524074	1	2	yes	—
501	C₂	—	2	2	no	—
502	C₂	—	2	2	no	—
503	C₂	—	4	1	no	—
504	C₂	C₂×#16863363	1	1	yes	characteristic
505	C₂	C₂×#16863405	1	1	yes	characteristic
506	C₂	C₂×#16863413	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	#17464
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. #22558630	C₂	semidirect
4. #22558612	C₂	semidirect
5. #22558604	C₂	semidirect
21. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
260. C₆₇	#17464	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂×#16863449	2
#22558630	2
#22558612	2
#22558604	2
C₂×#533×C₆₇	2
#17464	67