Subgroups — #37546634

Order 23168.

340 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₆₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041985	C₂	1	1	yes	maximal, characteristic
3	#26041689	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041686	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041683	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041588	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041572	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041568	C₂	1	1	yes	maximal, characteristic
9	#26041560	C₂	1	1	yes	maximal, characteristic
10	#26041558	C₂	1	1	yes	maximal, characteristic
11	#26041408	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041382	C₂	1	1	yes	maximal, characteristic
13	#26041377	C₂	1	1	yes	maximal, characteristic
14	#26041364	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668436	C₂²	1	1	yes	characteristic
17	#19153816	—	2	1	no	—
18	#19153816	C₂²	1	1	yes	characteristic
19	#19153815	—	2	1	no	—
20	#19153815	—	2	1	no	—
21	#19153815	C₂²	1	1	yes	characteristic
22	#19153815	C₂²	1	1	yes	characteristic
23	#19153815	C₂²	1	1	yes	characteristic
24	#19153807	C₂²	1	1	yes	characteristic
25	#19153796	—	2	1	no	—
26	#19153794	—	2	1	no	—
27	#19153794	—	2	1	no	—
28	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
29	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₆₂	—	2	1	no	—
31	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
32	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
33	#19153778	C₂²	1	1	yes	characteristic
34	#19153778	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₈₁	—	2	1	no	—
36	#19153750	C₂²	1	1	yes	characteristic
37	#19153750	C₂²	1	1	yes	characteristic
38	#19153749	C₂²	1	1	yes	characteristic
39	#19153749	C₂²	1	1	yes	characteristic
40	#19153749	C₂²	1	1	yes	characteristic
41	#19153747	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	#19153745	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153740	C₂²	1	1	yes	characteristic
46	#19153738	C₂²	1	1	yes	characteristic
47	#19153738	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153727	C₂²	1	1	yes	characteristic
50	#19153726	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	#125×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×C₁₈₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
57	#103×C₁₈₁	C₂²	1	1	yes	characteristic
58	#103×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	—	2	1	no	—
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
64	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	C₂×C₄×D₁₈₁	—	2	1	no	—
68	#15668436	Q₈	1	4	yes	complemented
69	C₂×#13089678	C₂³	1	1	yes	characteristic
70	C₂×#13089678	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₈₁	—	2	1	no	—
72	C₄⋊Dic₁₈₁	—	2	1	no	—
73	C₄⋊Dic₁₈₁	—	2	1	no	—
74	C₄⋊Dic₁₈₁	—	2	1	no	—
75	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	D₄	1	1	yes	characteristic
84	#15668428	D₄	1	1	yes	characteristic
85	#15668426	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	C₄×Dic₁₈₁	—	2	1	no	—
89	C₄×Dic₁₈₁	—	2	1	no	—
90	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
91	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₈₁	—	2	1	no	—
96	C₄⋊C₄×C₁₈₁	—	2	1	no	—
97	C₄⋊C₄×C₁₈₁	—	2	1	no	—
98	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₄²×C₁₈₁	—	2	1	no	—
102	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₂²×D₁₈₁	—	2	1	no	—
104	Dic₃₆₂	—	2	2	no	—
105	Dic₃₆₂	—	8	1	no	—
106	Dic₃₆₂	—	8	1	no	—
107	Dic₃₆₂	—	8	1	no	—
108	Dic₃₆₂	—	8	1	no	—
109	D₇₂₄	—	2	1	no	—
110	D₇₂₄	C₂×Q₈	1	2	yes	—
111	C₂×Dic₁₈₁	—	2	1	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	2	1	no	—
114	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₈₁	—	4	2	no	—
119	C₄×D₁₈₁	—	8	1	no	—
120	C₄×D₁₈₁	—	8	1	no	—
121	#13089678	—	2	1	no	—
122	#13089678	C₂×Q₈	1	2	yes	—
123	Q₈×C₁₈₁	—	2	1	no	—
124	Q₈×C₁₈₁	C₂×Q₈	1	2	yes	—
125	C₂×C₄×C₁₈₁	—	2	1	no	—
126	C₂×C₄×C₁₈₁	—	2	1	no	—
127	C₂×C₄×C₁₈₁	—	2	1	no	—
128	C₂×C₄×C₁₈₁	—	2	1	no	—
129	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
132	C₈×C₁₈₁	—	2	1	no	—
133	C₈×C₁₈₁	—	2	1	no	—
134	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₁₈₁	—	2	2	no	—
136	C₂×D₁₈₁	—	4	1	no	—
137	C₂×D₁₈₁	—	4	1	no	—
138	C₄×C₁₈₁	—	2	2	no	—
139	C₄×C₁₈₁	—	2	1	no	—
140	C₄×C₁₈₁	—	2	1	no	—
141	C₄×C₁₈₁	—	4	1	no	—
142	C₄×C₁₈₁	—	4	1	no	—
143	C₄×C₁₈₁	—	4	1	no	—
144	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
145	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₁₈₁	—	2	1	no	—
147	Dic₁₈₁	—	2	1	no	—
148	Dic₁₈₁	—	2	1	no	—
149	Dic₁₈₁	—	4	1	no	—
150	Dic₁₈₁	—	4	1	no	—
151	Dic₁₈₁	—	4	1	no	—
152	Dic₁₈₁	#137	1	2	yes	—
153	C₂×C₁₈₁	D₄×Q₈	1	1	yes	characteristic
154	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
155	C₂×C₁₈₁	#576	1	1	yes	characteristic
156	D₁₈₁	—	4	2	no	—
157	C₁₈₁	#3385	1	1	yes	characteristic, complemented, Sylow
158	#3385	—	181	1	no	maximal, Sylow
159	Q₈²	—	181	1	no	—
160	#554	—	181	1	no	—
161	#501	—	181	1	no	—
162	#501	—	181	1	no	—
163	#500	—	181	1	no	—
164	#479	—	181	1	no	—
165	#479	D₁₈₁	1	1	yes	characteristic, complemented
166	#477	—	181	1	no	—
167	#476	—	181	1	no	—
168	#474	—	181	1	no	—
169	#474	—	181	1	no	—
170	#446	—	181	1	no	—
171	#440	—	181	1	no	—
172	#440	—	181	1	no	—
173	C₄×SD₁₆	—	181	1	no	—
174	C₂×SD₁₆	—	181	1	no	—
175	C₄⋊Q₈	—	362	1	no	—
176	C₄⋊Q₈	—	362	1	no	—
177	C₄⋊Q₈	—	362	1	no	—
178	C₄⋊Q₈	—	181	1	no	—
179	C₄⋊Q₈	—	181	1	no	—
180	C₄⋊Q₈	—	181	1	no	—
181	C₄⋊Q₈	—	181	1	no	—
182	#125	—	181	1	no	—
183	#125	C₂×D₁₈₁	1	1	yes	characteristic
184	C₂²⋊Q₈	—	362	1	no	—
185	C₂²⋊Q₈	—	362	1	no	—
186	C₂²⋊Q₈	—	362	1	no	—
187	C₄×Q₈	—	362	1	no	—
188	C₄×Q₈	—	181	1	no	—
189	C₄×Q₈	—	181	1	no	—
190	C₄×Q₈	—	181	1	no	—
191	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
192	C₄×D₄	—	181	1	no	—
193	C₄×D₄	—	181	1	no	—
194	C₄×D₄	—	181	1	no	—
195	C₂×C₄⋊C₄	—	362	1	no	—
196	#107	—	181	1	no	—
197	#107	—	181	1	no	—
198	#107	—	181	1	no	—
199	#107	—	181	1	no	—
200	#107	—	181	1	no	—
201	#107	C₂×D₁₈₁	1	1	yes	characteristic
202	#106	—	181	1	no	—
203	#106	—	181	1	no	—
204	#106	C₂×D₁₈₁	1	1	yes	characteristic
205	C₄⋊C₈	—	181	1	no	—
206	C₄⋊C₈	—	181	1	no	—
207	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
208	#103	—	181	1	no	—
209	#103	C₂×D₁₈₁	1	1	yes	characteristic
210	#103	C₂×D₁₈₁	1	1	yes	characteristic
211	#102	—	181	1	no	—
212	#102	—	181	1	no	—
213	#102	—	181	1	no	—
214	#97	—	181	1	no	—
215	#97	—	181	1	no	—
216	C₄×C₈	—	181	1	no	—
217	C₂×Q₈	—	362	1	no	—
218	C₂×Q₈	—	362	1	no	—
219	C₂×Q₈	—	362	1	no	—
220	C₂×Q₈	—	362	1	no	—
221	C₂×Q₈	—	181	1	no	—
222	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
223	C₂×D₄	—	181	1	no	—
224	C₂²×C₄	—	362	1	no	—
225	C₂²×C₄	—	362	1	no	—
226	C₂²×C₄	—	362	1	no	—
227	SD₁₆	—	181	4	no	—
228	C₂×C₈	—	181	1	no	—
229	C₂×C₈	—	181	1	no	—
230	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	362	1	no	—
236	C₄⋊C₄	—	362	1	no	—
237	C₄⋊C₄	—	362	1	no	—
238	C₄⋊C₄	—	362	1	no	—
239	C₄⋊C₄	—	362	1	no	—
240	C₄⋊C₄	—	362	1	no	—
241	C₄⋊C₄	—	362	1	no	—
242	C₄⋊C₄	—	362	1	no	—
243	C₄⋊C₄	—	181	1	no	—
244	C₄⋊C₄	—	181	1	no	—
245	C₄⋊C₄	—	181	1	no	—
246	C₄⋊C₄	—	181	1	no	—
247	C₄⋊C₄	—	181	1	no	—
248	C₄⋊C₄	—	181	1	no	—
249	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
252	C₂²⋊C₄	—	362	1	no	—
253	C₂²⋊C₄	—	362	1	no	—
254	C₂²⋊C₄	—	362	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	362	1	no	—
257	C₄²	—	362	1	no	—
258	C₄²	—	181	1	no	—
259	C₄²	—	181	1	no	—
260	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
261	C₂³	—	362	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	—	362	2	no	—
264	Q₈	—	1448	1	no	—
265	Q₈	—	1448	1	no	—
266	Q₈	—	1448	1	no	—
267	Q₈	—	1448	1	no	—
268	Q₈	Q₈×D₁₈₁	1	2	yes	—
269	D₄	—	362	1	no	—
270	D₄	—	181	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	362	1	no	—
276	C₂×C₄	—	362	1	no	—
277	C₂×C₄	—	362	1	no	—
278	C₂×C₄	—	1448	1	no	—
279	C₂×C₄	—	1448	1	no	—
280	C₂×C₄	—	181	1	no	—
281	C₂×C₄	—	181	1	no	—
282	C₂×C₄	—	181	1	no	—
283	C₂×C₄	—	181	1	no	—
284	C₂×C₄	—	724	2	no	—
285	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
286	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	362	1	no	—
291	C₈	—	181	2	no	—
292	C₂²	—	362	2	no	—
293	C₂²	—	724	1	no	—
294	C₂²	—	724	1	no	—
295	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	362	1	no	—
303	C₄	—	362	1	no	—
304	C₄	—	362	1	no	—
305	C₄	—	181	2	no	—
306	C₄	—	724	1	no	—
307	C₄	—	724	1	no	—
308	C₄	—	724	1	no	—
309	C₄	C₂×Q₈×D₁₈₁	1	1	yes	characteristic
310	C₄	#19153863	1	1	yes	characteristic
311	C₂	—	724	2	no	—
312	C₂	#26041942	1	1	yes	characteristic
313	C₂	#137×D₁₈₁	1	1	yes	characteristic
314	C₂	#26042081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#479×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₇₂₄	C₂³
hypercenter	#479	D₁₈₁
hypocenter	C₁₈₁	#3385
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₆₂	C₂	semidirect
3. #26041689	C₂	semidirect
4. #26041686	C₂	semidirect
5. #26041683	C₂	semidirect
6. #26041588	C₂	semidirect
7. #26041572	C₂	semidirect
11. #26041408	C₂	semidirect
15. #479×C₁₈₁	C₂	semidirect
68. #15668436	Q₈	semidirect
157. C₁₈₁	#3385	semidirect
165. #479	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₆₂	2
#26041985	2
#26041689	2
#26041686	2
#26041683	2
#26041588	2
#26041572	2
#26041568	2
#26041560	2
#26041558	2
#26041408	2
#26041382	2
#26041377	2
#26041364	2
#479×C₁₈₁	2
#3385	181