Subgroups — #37550495

Order 23168.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042358	C₂	1	1	yes	maximal, characteristic, complemented
2	#26042357	C₂	1	1	yes	maximal, characteristic, complemented
3	#26042326	C₂	1	2	yes	maximal, complemented
4	#26042324	C₂	1	2	yes	maximal, complemented
5	#26042322	C₂	1	2	yes	maximal, complemented
6	#26042321	C₂	1	2	yes	maximal, complemented
7	#26042294	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₈₁⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19153880	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#10624382	C₂	1	1	yes	maximal, characteristic, complemented
11	#26042075	C₂	1	1	yes	maximal, characteristic
12	#19153904	C₂²	1	2	yes	complemented
13	#19153903	—	2	1	no	—
14	#19153903	—	2	1	no	—
15	#19153903	—	2	2	no	—
16	#19153903	C₂²	1	2	yes	complemented
17	#19153903	C₂²	1	2	yes	complemented
18	#19153903	C₂²	1	2	yes	complemented
19	C₂×#15668459	—	2	1	no	—
20	C₂×#15668459	C₂²	1	1	yes	characteristic
21	D₁₈₁⋊M₁₆	—	2	1	no	—
22	D₁₈₁⋊M₁₆	C₂²	1	1	yes	characteristic
23	D₁₈₁⋊M₁₆	C₂²	1	1	yes	characteristic
24	C₂×#15668458	—	2	2	no	—
25	C₂×C₄×#10624382	C₂²	1	1	yes	characteristic
26	#19153887	C₂²	1	2	yes	complemented
27	#19153886	C₂²	1	2	yes	complemented
28	#19153881	C₂²	1	1	yes	characteristic, complemented
29	#19153881	C₂²	1	1	yes	characteristic, complemented
30	#19153880	C₂²	1	2	yes	complemented
31	#19153880	C₂²	1	2	yes	complemented
32	#19153880	C₂²	1	1	yes	characteristic, complemented
33	#19153880	C₂²	1	1	yes	characteristic, complemented
34	#19153870	C₂²	1	2	yes	complemented
35	C₈×#10624382	C₂²	1	2	yes	complemented
36	C₂×#15668448	C₂²	1	1	yes	characteristic
37	#41×D₁₈₁	C₂²	1	1	yes	characteristic
38	#19153849	C₄	1	1	yes	characteristic, complemented
39	#19153846	C₄	1	1	yes	characteristic, complemented
40	#19153845	C₄	1	2	yes	complemented
41	#19153842	C₄	1	1	yes	characteristic, complemented
42	#19153840	C₄	1	2	yes	complemented
43	#19153837	C₂²	1	2	yes	—
44	#19153835	C₂²	1	2	yes	—
45	M₁₆×D₁₈₁	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₁₈₁	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10624382	—	2	1	no	—
48	#15668459	—	2	2	no	—
49	#15668459	—	2	1	no	—
50	#15668459	D₄	1	1	yes	characteristic
51	#15668459	D₄	1	1	yes	characteristic
52	#15668459	C₂³	1	2	yes	—
53	#15668459	C₂³	1	1	yes	characteristic
54	#15668459	C₂³	1	1	yes	characteristic
55	#15668458	—	2	2	no	—
56	#15668458	—	2	2	no	—
57	#15668458	—	2	2	no	—
58	#15668458	—	2	1	no	—
59	#15668458	D₄	1	2	yes	—
60	D₁₈₁⋊C₈	—	2	2	no	—
61	D₁₈₁⋊C₈	—	2	1	no	—
62	D₁₈₁⋊C₈	C₂³	1	2	yes	—
63	C₂×#13089685	—	2	2	no	—
64	C₂×#13089685	—	2	2	no	—
65	C₂×#13089685	—	2	2	no	—
66	C₂×#13089685	—	2	1	no	—
67	C₄×#10624382	—	2	1	no	—
68	C₄×#10624382	C₂³	1	1	yes	characteristic
69	C₄×#10624382	C₂³	1	1	yes	characteristic
70	#15668450	C₂³	1	1	yes	characteristic
71	#15668449	—	2	1	no	—
72	#15668449	C₂×C₄	1	1	yes	characteristic
73	#15668448	—	2	1	no	—
74	#15668448	—	2	1	no	—
75	#15668448	—	2	1	no	—
76	#15668448	C₂×C₄	1	2	yes	—
77	#15668448	C₂³	1	1	yes	characteristic
78	#15668448	C₂³	1	2	yes	—
79	C₂×Dic₃₆₂	C₂×C₄	1	1	yes	characteristic
80	Q₈×D₁₈₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
82	D₄×D₁₈₁	—	2	1	no	—
83	D₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₈₁	—	2	1	no	—
85	C₂×C₄×D₁₈₁	—	2	1	no	—
86	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
87	Dic₇₂₄	C₂×C₄	1	2	yes	complemented
88	#15668439	C₂×C₄	1	2	yes	complemented
89	#15668438	C₂×C₄	1	2	yes	complemented
90	#15668437	C₂×C₄	1	2	yes	complemented
91	#15668437	C₂×C₄	1	2	yes	complemented
92	#15668434	C₂×C₄	1	2	yes	complemented
93	#15668433	C₂×C₄	1	1	yes	characteristic
94	#15668432	C₂×C₄	1	2	yes	—
95	C₈×D₁₈₁	C₂³	1	2	yes	—
96	#41×C₁₈₁	C₂×C₄	1	1	yes	characteristic
97	C₂×D₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
98	SD₁₆×C₁₈₁	C₂×C₄	1	2	yes	complemented
99	D₈×C₁₈₁	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
101	C₂×#10624382	—	2	1	no	—
102	C₂×#10624382	—	2	1	no	—
103	C₂×#10624382	—	4	2	no	—
104	#13089685	—	2	1	no	—
105	#13089685	—	2	1	no	—
106	#13089685	—	2	2	no	—
107	#13089685	C₂×D₄	1	2	yes	—
108	#13089685	C₂×D₄	1	2	yes	—
109	C₂²×D₁₈₁	—	2	1	no	—
110	C₂²×D₁₈₁	#41	1	1	yes	characteristic
111	Dic₃₆₂	—	2	1	no	—
112	Dic₃₆₂	—	2	1	no	—
113	Dic₃₆₂	C₂²×C₄	1	1	yes	characteristic
114	Dic₃₆₂	C₂²×C₄	1	2	yes	—
115	D₇₂₄	—	2	1	no	—
116	D₇₂₄	C₂²×C₄	1	1	yes	characteristic
117	C₂²⋊D₁₈₁	—	2	1	no	—
118	C₂²⋊D₁₈₁	—	2	2	no	—
119	C₂²⋊D₁₈₁	—	2	1	no	—
120	C₂²⋊D₁₈₁	—	2	2	no	—
121	C₂²⋊D₁₈₁	—	2	1	no	—
122	C₂×Dic₁₈₁	—	2	1	no	—
123	C₂×Dic₁₈₁	—	2	2	no	—
124	C₂×Dic₁₈₁	—	2	1	no	—
125	C₂×Dic₁₈₁	—	2	2	no	—
126	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₈₁	—	2	1	no	—
128	C₄×D₁₈₁	—	2	1	no	—
129	C₄×D₁₈₁	—	2	1	no	—
130	C₄×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₈₁	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₈₁	#41	1	1	yes	characteristic
133	C₄×D₁₈₁	C₂⁴	1	1	yes	characteristic
134	#13089678	C₂²×C₄	1	2	yes	—
135	C₂³×C₁₈₁	—	2	1	no	—
136	Q₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
137	D₄×C₁₈₁	—	2	1	no	—
138	D₄×C₁₈₁	—	2	1	no	—
139	D₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
140	D₄×C₁₈₁	C₂²×C₄	1	2	yes	—
141	C₂×C₄×C₁₈₁	—	2	1	no	—
142	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
143	C₈×C₁₈₁	C₂²×C₄	1	2	yes	—
144	C₂²×C₁₈₁	—	2	1	no	—
145	C₂²×C₁₈₁	—	2	2	no	—
146	C₂²×C₁₈₁	—	8	1	no	—
147	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
148	C₂×D₁₈₁	—	2	1	no	—
149	C₂×D₁₈₁	—	4	1	no	—
150	C₂×D₁₈₁	—	8	1	no	—
151	C₂×D₁₈₁	C₄×D₄	1	1	yes	characteristic
152	C₂×D₁₈₁	C₂×#41	1	1	yes	characteristic
153	#10624382	—	2	2	no	—
154	#10624382	—	4	1	no	—
155	C₄×C₁₈₁	—	2	1	no	—
156	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
157	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
158	Dic₁₈₁	—	2	1	no	—
159	Dic₁₈₁	—	2	2	no	—
160	Dic₁₈₁	C₄×D₄	1	1	yes	characteristic
161	Dic₁₈₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₁₈₁	—	2	1	no	—
163	C₂×C₁₈₁	—	4	1	no	—
164	C₂×C₁₈₁	—	4	2	no	—
165	C₂×C₁₈₁	C₂×C₄×D₄	1	1	yes	characteristic
166	D₁₈₁	—	2	1	no	—
167	D₁₈₁	—	2	1	no	—
168	D₁₈₁	—	4	1	no	—
169	C₁₈₁	#2956	1	1	yes	characteristic, complemented, Sylow
170	#2956	—	181	1	no	maximal, Sylow
171	#575	—	181	1	no	—
172	#568	—	181	1	no	—
173	#568	—	181	1	no	—
174	#444	—	181	2	no	—
175	#444	—	181	2	no	—
176	#443	—	181	2	no	—
177	#443	—	181	2	no	—
178	#430	—	181	1	no	—
179	C₂×#104	—	181	1	no	—
180	C₂×#104	—	181	1	no	—
181	C₄×M₁₆	—	181	1	no	—
182	C₂×#41	—	181	1	no	—
183	C₂×#41	—	181	1	no	—
184	#137	—	181	1	no	—
185	#137	—	181	2	no	—
186	#137	—	181	1	no	—
187	AGL(1,ℤ/8)	—	181	1	no	—
188	AGL(1,ℤ/8)	—	181	2	no	—
189	AGL(1,ℤ/8)	#10624382	1	1	yes	characteristic, complemented
190	#135	—	181	2	no	—
191	#135	—	181	2	no	—
192	#131	—	362	2	no	—
193	#131	—	362	1	no	—
194	#131	—	362	1	no	—
195	#131	—	181	2	no	—
196	#131	—	181	2	no	—
197	#131	—	181	2	no	—
198	#131	—	181	2	no	—
199	C₂×M₁₆	—	362	2	no	—
200	C₂×M₁₆	—	362	1	no	—
201	C₂×M₁₆	—	362	1	no	—
202	C₂×M₁₆	—	181	1	no	—
203	C₂×M₁₆	—	181	1	no	—
204	C₂×M₁₆	—	181	1	no	—
205	C₂×M₁₆	—	181	1	no	—
206	C₂×C₄²	—	181	1	no	—
207	#108	—	181	2	no	—
208	#108	—	181	2	no	—
209	#104	—	181	2	no	—
210	#104	—	181	1	no	—
211	#104	—	181	1	no	—
212	#104	—	181	2	no	—
213	#104	—	181	1	no	—
214	#104	—	181	1	no	—
215	#97	—	181	2	no	—
216	C₄×C₈	—	181	2	no	—
217	#41	—	362	1	no	—
218	#41	—	362	1	no	—
219	#41	—	362	1	no	—
220	#41	—	362	1	no	—
221	#41	—	181	1	no	—
222	#41	—	181	2	no	—
223	#41	—	181	1	no	—
224	#41	—	181	2	no	—
225	#41	—	181	1	no	—
226	#41	C₂×#10624382	1	1	yes	characteristic
227	C₂×Q₈	—	181	1	no	—
228	C₂×Q₈	—	181	1	no	—
229	C₂×D₄	—	362	1	no	—
230	C₂×D₄	—	362	1	no	—
231	C₂×D₄	—	181	1	no	—
232	C₂×D₄	C₂×#10624382	1	1	yes	characteristic
233	C₂²×C₄	—	362	1	no	—
234	C₂²×C₄	—	362	1	no	—
235	C₂²×C₄	—	362	1	no	—
236	C₂²×C₄	—	181	1	no	—
237	Q₁₆	—	181	2	no	—
238	Q₁₆	—	181	2	no	—
239	SD₁₆	—	181	2	no	—
240	SD₁₆	—	181	2	no	—
241	SD₁₆	—	181	2	no	—
242	SD₁₆	C₂×#10624382	1	2	yes	complemented
243	D₈	—	181	2	no	—
244	D₈	C₂×#10624382	1	2	yes	complemented
245	M₁₆	—	362	1	no	—
246	M₁₆	—	362	1	no	—
247	M₁₆	—	362	2	no	—
248	M₁₆	—	362	2	no	—
249	M₁₆	—	362	2	no	—
250	M₁₆	—	362	2	no	—
251	M₁₆	—	181	2	no	—
252	M₁₆	—	181	1	no	—
253	M₁₆	—	181	1	no	—
254	M₁₆	—	181	2	no	—
255	M₁₆	—	181	1	no	—
256	M₁₆	—	181	1	no	—
257	M₁₆	—	181	1	no	—
258	M₁₆	—	181	2	no	—
259	M₁₆	C₂×#10624382	1	1	yes	characteristic
260	C₂×C₈	—	362	1	no	—
261	C₂×C₈	—	362	1	no	—
262	C₂×C₈	—	362	2	no	—
263	C₂×C₈	—	362	2	no	—
264	C₂×C₈	—	362	2	no	—
265	C₂×C₈	—	362	2	no	—
266	C₂×C₈	—	181	2	no	—
267	C₂×C₈	—	181	2	no	—
268	C₄²	—	362	1	no	—
269	C₄²	—	181	1	no	—
270	C₄²	—	181	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	362	1	no	—
273	C₂³	—	181	1	no	—
274	Q₈	—	362	1	no	—
275	Q₈	—	362	1	no	—
276	Q₈	—	181	1	no	—
277	Q₈	—	181	2	no	—
278	Q₈	C₂²×#10624382	1	1	yes	characteristic
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	362	1	no	—
282	D₄	—	362	1	no	—
283	D₄	—	362	2	no	—
284	D₄	—	362	2	no	—
285	D₄	—	362	1	no	—
286	D₄	—	362	1	no	—
287	D₄	—	181	1	no	—
288	D₄	C₂²×#10624382	1	2	yes	—
289	D₄	C₂²×#10624382	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	362	1	no	—
292	C₂×C₄	—	362	1	no	—
293	C₂×C₄	—	362	1	no	—
294	C₂×C₄	—	362	1	no	—
295	C₂×C₄	—	362	1	no	—
296	C₂×C₄	—	362	2	no	—
297	C₂×C₄	—	362	2	no	—
298	C₂×C₄	—	362	1	no	—
299	C₂×C₄	—	362	1	no	—
300	C₂×C₄	—	181	1	no	—
301	C₂×C₄	—	181	1	no	—
302	C₂×C₄	—	181	1	no	—
303	C₂×C₄	—	181	1	no	—
304	C₂×C₄	—	181	1	no	—
305	C₂×C₄	—	724	2	no	—
306	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
307	C₈	—	362	2	no	—
308	C₈	—	362	1	no	—
309	C₈	—	362	1	no	—
310	C₈	—	181	2	no	—
311	C₈	—	181	2	no	—
312	C₈	—	181	2	no	—
313	C₈	C₂²×#10624382	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	362	1	no	—
318	C₂²	—	1448	1	no	—
319	C₂²	—	181	1	no	—
320	C₂²	—	181	1	no	—
321	C₂²	—	724	1	no	—
322	C₂²	D₄×#10624382	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	362	2	no	—
325	C₄	—	362	1	no	—
326	C₄	—	362	2	no	—
327	C₄	—	181	1	no	—
328	C₄	—	181	1	no	—
329	C₄	—	724	1	no	—
330	C₄	D₄×#10624382	1	1	yes	characteristic
331	C₄	C₂³×#10624382	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	1	no	—
335	C₂	—	362	1	no	—
336	C₂	—	362	1	no	—
337	C₂	—	724	1	no	—
338	C₂	C₂×D₄×#10624382	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10624382
commutator	C₄×C₁₈₁	C₂³×C₄
Frattini	C₄	C₂³×#10624382
Fitting	AGL(1,ℤ/8)×C₁₈₁	C₄
socle	C₂×C₁₈₁	C₂×C₄×D₄
prime radical	#26042075	C₂
cosocle	C₄×D₁₈₁	C₂⁴
hypercenter	AGL(1,ℤ/8)	#10624382
hypocenter	C₁₈₁	#2956
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #26042358	C₂	semidirect
2. #26042357	C₂	semidirect
3. #26042326	C₂	semidirect
4. #26042324	C₂	semidirect
5. #26042322	C₂	semidirect
6. #26042321	C₂	semidirect
7. #26042294	C₂	semidirect
8. D₁₈₁⋊#104	C₂	semidirect
9. C₂×#19153880	C₂	semidirect
10. M₁₆×#10624382	C₂	semidirect
12. #19153904	C₂²	semidirect
16. #19153903	C₂²	semidirect
17. #19153903	C₂²	semidirect
18. #19153903	C₂²	semidirect
26. #19153887	C₂²	semidirect
27. #19153886	C₂²	semidirect
28. #19153881	C₂²	semidirect
29. #19153881	C₂²	semidirect
30. #19153880	C₂²	semidirect
31. #19153880	C₂²	semidirect
32. #19153880	C₂²	semidirect
33. #19153880	C₂²	semidirect
34. #19153870	C₂²	semidirect
35. C₈×#10624382	C₂²	semidirect
38. #19153849	C₄	semidirect
39. #19153846	C₄	semidirect
40. #19153845	C₄	semidirect
41. #19153842	C₄	semidirect
42. #19153840	C₄	semidirect
46. AGL(1,ℤ/8)×C₁₈₁	C₄	semidirect
87. Dic₇₂₄	C₂×C₄	semidirect
88. #15668439	C₂×C₄	semidirect
89. #15668438	C₂×C₄	semidirect
90. #15668437	C₂×C₄	semidirect
91. #15668437	C₂×C₄	semidirect
92. #15668434	C₂×C₄	semidirect
98. SD₁₆×C₁₈₁	C₂×C₄	semidirect
99. D₈×C₁₈₁	C₂×C₄	semidirect
169. C₁₈₁	#2956	semidirect
189. AGL(1,ℤ/8)	#10624382	semidirect
242. SD₁₆	C₂×#10624382	semidirect
244. D₈	C₂×#10624382	semidirect

Maximal subgroups

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

subgroup	index
#26042358	2
#26042357	2
#26042326	2
#26042324	2
#26042322	2
#26042321	2
#26042294	2
D₁₈₁⋊#104	2
C₂×#19153880	2
M₁₆×#10624382	2
#26042075	2
#2956	181