Subgroups — #30839095

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22571551	C₂	1	2	yes	maximal
2	#22569143	C₂	1	1	yes	maximal, characteristic
3	#22559712	C₂	1	4	yes	maximal, complemented
4	#22559610	C₂	1	2	yes	maximal, complemented
5	#22559600	C₂	1	2	yes	maximal, complemented
6	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
7	#22558899	C₂	1	2	yes	maximal
8	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
9	#16863953	C₂²	1	1	yes	characteristic
10	#16863691	—	2	2	no	—
11	#16863688	C₂²	1	4	yes	—
12	#16863635	—	2	4	no	—
13	#16863606	—	2	2	no	—
14	#16863603	C₂²	1	4	yes	—
15	#16863526	—	2	1	no	—
16	#16863526	C₂²	1	2	yes	—
17	C₈×Dic₁₃₄	—	2	1	no	—
18	C₈×Dic₁₃₄	C₂²	1	2	yes	—
19	#16863422	—	2	1	no	—
20	#16863420	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
22	#16863351	—	2	1	no	—
23	#16863149	C₂²	1	2	yes	—
24	#16863149	C₂²	1	2	yes	—
25	#16863145	C₂²	1	4	yes	—
26	#16863142	C₂²	1	4	yes	—
27	#533×C₆₇	C₂²	1	2	yes	—
28	#500×C₆₇	C₂²	1	1	yes	characteristic
29	#500×C₆₇	C₂²	1	1	yes	characteristic
30	#499×C₆₇	—	2	1	no	—
31	#499×C₆₇	—	2	1	no	—
32	#498×C₆₇	C₂²	1	2	yes	—
33	#427×C₆₇	—	2	1	no	—
34	#427×C₆₇	—	2	1	no	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#11761618	—	2	4	no	—
37	C₂×#11761612	—	2	1	no	—
38	C₂²×C₆₇⋊C₈	—	2	2	no	—
39	#14524043	D₄	1	1	yes	characteristic
40	#14524043	D₄	1	1	yes	characteristic
41	#14524035	—	2	1	no	—
42	#14524022	—	2	1	no	—
43	#14524022	D₄	1	2	yes	—
44	C₄×Dic₁₃₄	—	2	1	no	—
45	C₄×Dic₁₃₄	C₂³	1	2	yes	—
46	#14524002	—	2	1	no	—
47	#14523976	—	2	4	no	—
48	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
49	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
51	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
52	Dic₆₇⋊C₈	—	2	2	no	—
53	Dic₆₇⋊C₈	—	2	2	no	—
54	#14523967	—	2	4	no	—
55	#14523966	—	2	4	no	—
56	#14523953	—	2	2	no	—
57	C₈×Dic₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
62	#125×C₆₇	—	2	2	no	—
63	C₂²⋊Q₈×C₆₇	—	2	2	no	—
64	C₂²⋊Q₈×C₆₇	—	2	2	no	—
65	C₄×Q₈×C₆₇	—	2	2	no	—
66	C₄×Q₈×C₆₇	—	2	2	no	—
67	#117×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	—	2	1	no	—
71	#107×C₆₇	—	2	1	no	—
72	#107×C₆₇	—	2	1	no	—
73	#106×C₆₇	—	2	2	no	—
74	#106×C₆₇	—	2	2	no	—
75	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₃₄	—	2	1	no	—
80	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
81	#11761618	—	8	4	no	—
82	#11761612	—	4	2	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₂×C₆₇⋊C₈	—	2	4	no	—
85	C₂×C₆₇⋊C₈	—	8	2	no	—
86	C₄⋊Dic₆₇	—	2	1	no	—
87	C₄⋊Dic₆₇	—	2	1	no	—
88	C₄⋊Dic₆₇	—	2	1	no	—
89	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
90	Dic₆₇⋊C₄	—	2	2	no	—
91	Dic₆₇⋊C₄	—	2	2	no	—
92	C₂²⋊Dic₆₇	—	2	2	no	—
93	C₄×Dic₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	Q₁₆	1	4	yes	complemented
101	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	#41	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	#121	1	1	yes	characteristic
118	Q₈×C₆₇	—	4	2	no	—
119	Q₈×C₆₇	—	4	2	no	—
120	Q₈×C₆₇	—	8	2	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₆₇	#137	1	2	yes	—
131	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₆₇	—	2	2	no	—
135	C₂×Dic₆₇	—	2	2	no	—
136	Dic₁₃₄	—	2	2	no	—
137	Dic₁₃₄	—	2	2	no	—
138	Dic₁₃₄	—	4	2	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
142	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
143	C₆₇⋊C₈	—	4	2	no	—
144	C₆₇⋊C₈	—	4	4	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	#462	1	2	yes	—
148	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#462	1	2	yes	—
155	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
156	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
157	C₄×C₆₇	#576	1	1	yes	characteristic
158	C₄×C₆₇	#578	1	1	yes	characteristic
159	Dic₆₇	—	4	2	no	—
160	Dic₆₇	—	4	2	no	—
161	#19211	—	67	1	no	maximal, Sylow
162	C₂×C₆₇	—	2	2	no	—
163	C₂×C₆₇	C₂×#462	1	1	yes	characteristic
164	C₂×C₆₇	#3061	1	1	yes	characteristic
165	C₂×C₆₇	#3067	1	1	yes	characteristic
166	#3249	—	67	2	no	—
167	#3160	D₆₇	1	1	yes	characteristic
168	#2960	—	67	1	no	—
169	#1692	—	67	4	no	—
170	#1676	—	67	2	no	—
171	#1674	—	67	2	no	—
172	C₂×#335	—	67	1	no	—
173	#1550	—	67	2	no	—
174	C₆₇	#19211	1	1	yes	characteristic, complemented, Sylow
175	#533	—	67	1	no	—
176	#533	C₂×D₆₇	1	2	yes	—
177	#500	C₂×D₆₇	1	1	yes	characteristic
178	#500	C₂×D₆₇	1	1	yes	characteristic
179	#499	—	2	1	no	—
180	#499	—	2	1	no	—
181	#498	C₂×D₆₇	1	2	yes	—
182	#477	—	67	4	no	—
183	#477	—	134	2	no	—
184	#467	—	134	4	no	—
185	#462	—	67	4	no	—
186	#462	—	134	2	no	—
187	#446	—	67	2	no	—
188	#446	—	134	1	no	—
189	C₈×Q₈	—	67	2	no	—
190	C₈×Q₈	—	134	1	no	—
191	#427	—	2	1	no	—
192	#427	—	2	1	no	—
193	#424	—	134	1	no	—
194	#423	—	67	1	no	—
195	C₂×C₄⋊C₈	—	67	2	no	—
196	#405	—	134	1	no	—
197	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
198	#335	—	67	2	no	—
199	#335	—	67	2	no	—
200	#333	—	67	4	no	—
201	#332	—	67	4	no	—
202	C₂×Q₁₆	—	134	4	no	—
203	C₂×M₁₆	—	134	1	no	—
204	C₂²×C₈	—	134	2	no	—
205	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
207	C₄⋊Q₈	—	67	1	no	—
208	C₄⋊Q₈	—	67	1	no	—
209	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
210	C₄⋊Q₈	C₂²×D₆₇	1	2	yes	—
211	#125	—	2	2	no	—
212	#125	—	134	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	67	2	no	—
216	C₂²⋊Q₈	—	134	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	67	2	no	—
220	C₄×Q₈	—	134	1	no	—
221	#117	—	2	2	no	—
222	#117	—	134	1	no	—
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	134	4	no	—
229	#106	—	2	2	no	—
230	#106	—	2	2	no	—
231	C₄⋊C₈	—	67	2	no	—
232	C₄⋊C₈	—	67	4	no	—
233	C₄⋊C₈	—	134	2	no	—
234	C₄⋊C₈	—	134	2	no	—
235	C₄⋊C₈	—	134	2	no	—
236	C₄⋊C₈	—	134	1	no	—
237	#103	—	134	4	no	—
238	#103	—	134	4	no	—
239	#97	—	134	2	no	—
240	C₄×C₈	—	134	2	no	—
241	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
242	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	67	2	no	—
248	C₂×Q₈	—	134	1	no	—
249	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
251	C₂²×C₄	#11761627	1	1	yes	characteristic
252	Q₁₆	—	536	4	no	—
253	M₁₆	—	268	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	536	2	no	—
256	C₂×C₈	—	134	4	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	#11761618	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	67	2	no	—
270	C₄⋊C₄	—	134	1	no	—
271	C₄⋊C₄	—	134	1	no	—
272	C₄⋊C₄	—	134	2	no	—
273	C₄⋊C₄	—	134	1	no	—
274	C₄⋊C₄	—	134	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	134	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	134	2	no	—
279	C₄²	D₄×D₆₇	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
281	C₄²	#11761627	1	1	yes	characteristic
282	C₄²	C₂³×D₆₇	1	1	yes	characteristic
283	C₂³	#14524014	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	268	2	no	—
288	Q₈	—	134	2	no	—
289	Q₈	—	134	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	134	2	no	—
296	C₂×C₄	—	134	2	no	—
297	C₂×C₄	#14524014	1	1	yes	characteristic
298	C₂×C₄	#14524014	1	1	yes	characteristic
299	C₂×C₄	#14524014	1	1	yes	characteristic
300	C₂×C₄	C₂×#11761618	1	2	yes	—
301	C₂×C₄	#14524074	1	2	yes	—
302	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	268	2	no	—
308	C₈	—	268	4	no	—
309	C₈	C₂×#11761618	1	2	yes	—
310	C₈	C₂×#11761618	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	#16863603	1	2	yes	—
314	C₂²	C₂×#14524014	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	268	2	no	—
321	C₄	—	268	2	no	—
322	C₄	#16863603	1	2	yes	—
323	C₄	C₂²×#11761618	1	1	yes	characteristic
324	C₄	C₂×#14524074	1	1	yes	characteristic
325	C₄	#16864198	1	1	yes	characteristic
326	C₄	#16864210	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	C₂×#16863603	1	1	yes	characteristic
329	C₂	#22569875	1	1	yes	characteristic
330	C₂	#22569925	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14524014
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂³×C₆₇	#121
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#19211
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #22559712	C₂	semidirect
4. #22559610	C₂	semidirect
5. #22559600	C₂	semidirect
100. C₂×C₈×C₆₇	Q₁₆	semidirect
174. C₆₇	#19211	semidirect
258. C₂×C₈	#11761618	semidirect

Maximal subgroups

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

subgroup	index
#22571551	2
#22569143	2
#22559712	2
#22559610	2
#22559600	2
C₂×#16863149	2
#22558899	2
C₆₇×#3160	2
#19211	67