Subgroups — #37543584

Order 23168.

376 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	#26041909	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041758	C₂	1	1	yes	maximal, characteristic
3	#26041585	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041583	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041581	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041569	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041567	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041565	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041563	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041320	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041304	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041277	C₂	1	1	yes	maximal, characteristic
14	#26041272	C₂	1	1	yes	maximal, characteristic
15	#478×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668448	C₂²	1	1	yes	characteristic
17	C₂×#15668432	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
19	#19153816	C₂²	1	1	yes	characteristic
20	#19153815	—	2	1	no	—
21	#19153815	C₂²	1	1	yes	characteristic
22	#19153807	C₂²	1	1	yes	characteristic
23	#19153806	—	2	1	no	—
24	#19153796	—	2	1	no	—
25	#19153794	—	2	1	no	—
26	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
27	#19153781	C₂²	1	1	yes	characteristic
28	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₁₈₁	—	2	1	no	—
30	#19153776	—	2	1	no	—
31	#19153774	C₂²	1	1	yes	characteristic
32	#19153773	C₂²	1	1	yes	characteristic
33	#19153772	—	2	1	no	—
34	C₂×#15668428	—	2	1	no	—
35	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
37	#19153749	C₂²	1	1	yes	characteristic
38	#19153748	C₂²	1	1	yes	characteristic
39	#19153748	C₂²	1	1	yes	characteristic
40	#19153747	C₂²	1	1	yes	characteristic
41	#19153747	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	C₄⋊#13089678	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153745	C₂²	1	1	yes	characteristic
46	#19153741	C₂²	1	1	yes	characteristic
47	#19153741	C₂²	1	1	yes	characteristic
48	#19153739	C₂²	1	1	yes	characteristic
49	#19153739	C₂²	1	1	yes	characteristic
50	#19153737	C₂²	1	1	yes	characteristic
51	#19153737	C₂²	1	1	yes	characteristic
52	#125×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×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	#102×C₁₈₁	C₂²	1	1	yes	characteristic
58	#102×C₁₈₁	C₂²	1	1	yes	characteristic
59	#15668448	—	2	2	no	—
60	#15668448	Q₈	1	4	yes	complemented
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
65	C₂²×Dic₁₈₁	—	2	1	no	—
66	C₂²×Dic₁₈₁	—	2	1	no	—
67	C₂×C₄×D₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	—	2	1	no	—
69	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
70	#15668432	—	2	2	no	—
71	C₂×#13089678	C₂³	1	1	yes	characteristic
72	C₂×#13089678	C₂³	1	1	yes	characteristic
73	C₈×D₁₈₁	—	2	2	no	—
74	C₄⋊Dic₁₈₁	—	2	1	no	—
75	C₄⋊Dic₁₈₁	—	2	1	no	—
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	—	2	1	no	—
84	#15668428	—	2	1	no	—
85	#15668428	D₄	1	1	yes	characteristic
86	#15668428	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₈₁	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	D₄	1	1	yes	characteristic
90	#15668426	D₄	1	1	yes	characteristic
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₄×Dic₁₈₁	—	2	1	no	—
93	C₄×Dic₁₈₁	—	2	1	no	—
94	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₈₁	—	2	1	no	—
97	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₈₁	—	2	1	no	—
100	C₄⋊C₄×C₁₈₁	—	2	1	no	—
101	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
105	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
107	Dic₃₆₂	—	2	1	no	—
108	Dic₃₆₂	—	8	1	no	—
109	Dic₃₆₂	—	8	1	no	—
110	Dic₃₆₂	C₂×Q₈	1	2	yes	—
111	C₂²⋊D₁₈₁	—	2	4	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	2	4	no	—
114	C₂×Dic₁₈₁	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	—	8	1	no	—
118	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
121	C₄×D₁₈₁	—	8	1	no	—
122	C₄×D₁₈₁	—	8	1	no	—
123	C₄×D₁₈₁	C₂×Q₈	1	2	yes	—
124	C₄×D₁₈₁	#41	1	2	yes	—
125	#13089678	—	2	1	no	—
126	#13089678	—	2	1	no	—
127	C₂³×C₁₈₁	—	2	1	no	—
128	D₄×C₁₈₁	—	2	1	no	—
129	D₄×C₁₈₁	C₂×Q₈	1	2	yes	—
130	C₂×C₄×C₁₈₁	—	2	1	no	—
131	C₂×C₄×C₁₈₁	—	2	1	no	—
132	C₂×C₄×C₁₈₁	—	2	1	no	—
133	C₂×C₄×C₁₈₁	—	8	1	no	—
134	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₈₁	—	2	1	no	—
138	C₈×C₁₈₁	—	2	1	no	—
139	C₂²×C₁₈₁	—	2	2	no	—
140	C₂²×C₁₈₁	—	4	1	no	—
141	C₂²×C₁₈₁	—	4	1	no	—
142	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₈₁	—	2	1	no	—
144	C₂×D₁₈₁	—	2	1	no	—
145	C₂×D₁₈₁	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₁₈₁	—	2	1	no	—
147	C₄×C₁₈₁	—	2	1	no	—
148	C₄×C₁₈₁	—	4	1	no	—
149	C₄×C₁₈₁	—	4	1	no	—
150	C₄×C₁₈₁	—	4	1	no	—
151	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
153	Dic₁₈₁	—	2	2	no	—
154	Dic₁₈₁	—	2	1	no	—
155	Dic₁₈₁	—	2	1	no	—
156	Dic₁₈₁	—	4	1	no	—
157	Dic₁₈₁	—	4	1	no	—
158	Dic₁₈₁	—	4	1	no	—
159	Dic₁₈₁	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₁₈₁	—	4	2	no	—
161	C₂×C₁₈₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₈₁	#577	1	1	yes	characteristic
163	C₂×C₁₈₁	#577	1	1	yes	characteristic
164	D₁₈₁	—	2	2	no	—
165	C₁₈₁	#3084	1	1	yes	characteristic, complemented, Sylow
166	#3084	—	181	1	no	maximal, Sylow
167	#544	—	181	1	no	—
168	#520	—	181	1	no	—
169	#479	—	181	1	no	—
170	#479	—	181	1	no	—
171	#478	—	181	1	no	—
172	#478	D₁₈₁	1	1	yes	characteristic, complemented
173	#476	—	181	1	no	—
174	#476	—	181	1	no	—
175	#475	—	181	1	no	—
176	#475	—	181	1	no	—
177	#428	—	181	1	no	—
178	C₂×#106	—	181	1	no	—
179	#424	—	181	1	no	—
180	#417	—	181	1	no	—
181	#416	—	181	1	no	—
182	C₂×#41	—	181	1	no	—
183	C₂×M₁₆	—	181	1	no	—
184	C₂²×C₈	—	181	1	no	—
185	C₄⋊Q₈	—	362	1	no	—
186	C₄⋊Q₈	—	181	1	no	—
187	C₄⋊Q₈	—	181	1	no	—
188	#125	—	362	1	no	—
189	#125	—	181	1	no	—
190	#125	C₂×D₁₈₁	1	1	yes	characteristic
191	C₂²⋊Q₈	—	362	1	no	—
192	C₂²⋊Q₈	—	362	1	no	—
193	C₄×Q₈	—	181	1	no	—
194	C₄×Q₈	—	181	1	no	—
195	C₄×D₄	—	362	1	no	—
196	C₄×D₄	—	362	1	no	—
197	C₄×D₄	—	181	1	no	—
198	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
199	#117	—	362	1	no	—
200	#117	—	181	1	no	—
201	#117	—	181	1	no	—
202	C₂×C₄⋊C₄	—	362	1	no	—
203	C₂×C₄⋊C₄	—	181	1	no	—
204	C₂×C₄⋊C₄	—	181	1	no	—
205	#107	—	181	1	no	—
206	#107	C₂×D₁₈₁	1	1	yes	characteristic
207	#106	—	181	1	no	—
208	#106	—	181	1	no	—
209	#106	—	181	1	no	—
210	#106	—	181	1	no	—
211	#106	—	181	1	no	—
212	#106	C₂×D₁₈₁	1	1	yes	characteristic
213	C₄⋊C₈	—	181	1	no	—
214	C₄⋊C₈	—	181	1	no	—
215	C₄⋊C₈	—	181	1	no	—
216	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
217	#103	—	181	1	no	—
218	#103	—	181	1	no	—
219	#103	—	181	1	no	—
220	#103	—	181	1	no	—
221	#102	—	181	1	no	—
222	#102	—	181	1	no	—
223	#102	C₂×D₁₈₁	1	1	yes	characteristic
224	#102	C₂×D₁₈₁	1	1	yes	characteristic
225	#41	—	362	2	no	—
226	#41	—	181	4	no	—
227	C₂×Q₈	—	362	1	no	—
228	C₂×Q₈	—	362	1	no	—
229	C₂×Q₈	—	181	1	no	—
230	C₂×D₄	—	362	1	no	—
231	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	362	1	no	—
234	C₂²×C₄	—	362	1	no	—
235	C₂²×C₄	—	362	1	no	—
236	C₂²×C₄	—	362	1	no	—
237	C₂²×C₄	—	181	1	no	—
238	M₁₆	—	362	2	no	—
239	C₂×C₈	—	362	2	no	—
240	C₂×C₈	—	181	1	no	—
241	C₂×C₈	—	181	1	no	—
242	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	362	1	no	—
247	C₄⋊C₄	—	362	1	no	—
248	C₄⋊C₄	—	362	1	no	—
249	C₄⋊C₄	—	362	1	no	—
250	C₄⋊C₄	—	362	1	no	—
251	C₄⋊C₄	—	362	1	no	—
252	C₄⋊C₄	—	362	1	no	—
253	C₄⋊C₄	—	362	1	no	—
254	C₄⋊C₄	—	181	1	no	—
255	C₄⋊C₄	—	181	1	no	—
256	C₄⋊C₄	—	181	1	no	—
257	C₄⋊C₄	—	181	1	no	—
258	C₄⋊C₄	—	181	1	no	—
259	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	362	1	no	—
264	C₂²⋊C₄	—	362	1	no	—
265	C₂²⋊C₄	—	181	1	no	—
266	C₂²⋊C₄	—	181	1	no	—
267	C₄²	—	362	1	no	—
268	C₄²	—	362	1	no	—
269	C₄²	—	362	1	no	—
270	C₄²	—	181	1	no	—
271	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	181	1	no	—
274	Q₈	—	362	1	no	—
275	Q₈	—	1448	1	no	—
276	Q₈	—	1448	1	no	—
277	Q₈	—	181	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	362	4	no	—
280	D₄	Q₈×D₁₈₁	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	362	1	no	—
286	C₂×C₄	—	362	4	no	—
287	C₂×C₄	—	362	1	no	—
288	C₂×C₄	—	362	1	no	—
289	C₂×C₄	—	362	1	no	—
290	C₂×C₄	—	1448	1	no	—
291	C₂×C₄	—	1448	1	no	—
292	C₂×C₄	—	1448	1	no	—
293	C₂×C₄	—	181	1	no	—
294	C₂×C₄	—	181	1	no	—
295	C₂×C₄	—	181	1	no	—
296	C₂×C₄	—	181	2	no	—
297	C₂×C₄	—	181	2	no	—
298	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	362	1	no	—
304	C₈	—	362	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	362	1	no	—
309	C₂²	—	362	1	no	—
310	C₂²	—	181	2	no	—
311	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	362	2	no	—
318	C₄	—	362	1	no	—
319	C₄	—	362	1	no	—
320	C₄	—	181	2	no	—
321	C₄	—	724	1	no	—
322	C₄	—	724	1	no	—
323	C₄	—	724	1	no	—
324	C₄	C₂×Q₈×D₁₈₁	1	1	yes	characteristic
325	C₄	#41×D₁₈₁	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	362	2	no	—
328	C₂	C₂²⋊Q₈×D₁₈₁	1	1	yes	characteristic
329	C₂	#26042086	1	1	yes	characteristic
330	C₂	#26042089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#478×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×#15668448	C₂²
hypercenter	#478	D₁₈₁
hypocenter	C₁₈₁	#3084
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #26041909	C₂	semidirect
3. #26041585	C₂	semidirect
4. #26041583	C₂	semidirect
5. #26041581	C₂	semidirect
6. #26041569	C₂	semidirect
7. #26041567	C₂	semidirect
8. #26041565	C₂	semidirect
9. #26041563	C₂	semidirect
10. #26041320	C₂	semidirect
11. #106×D₁₈₁	C₂	semidirect
12. #26041304	C₂	semidirect
15. #478×C₁₈₁	C₂	semidirect
60. #15668448	Q₈	semidirect
165. C₁₈₁	#3084	semidirect
172. #478	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041909	2
#26041758	2
#26041585	2
#26041583	2
#26041581	2
#26041569	2
#26041567	2
#26041565	2
#26041563	2
#26041320	2
#106×D₁₈₁	2
#26041304	2
#26041277	2
#26041272	2
#478×C₁₈₁	2
#3084	181