Subgroups — #37550500

Order 23168.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042358	C₂	1	2	yes	maximal, complemented
2	#26042326	C₂	1	2	yes	maximal, complemented
3	#26042325	C₂	1	2	yes	maximal, complemented
4	#26042322	C₂	1	4	yes	maximal, complemented
5	C₂×#19153886	C₂	1	1	yes	maximal, characteristic, complemented
6	#26042276	C₂	1	2	yes	maximal, complemented
7	#26042242	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₁₈₁	C₂	1	1	yes	maximal, characteristic
9	#19153904	C₂²	1	4	yes	complemented
10	#19153903	—	2	4	no	—
11	#19153903	C₂²	1	4	yes	complemented
12	C₂×#15668459	—	2	2	no	—
13	C₂×#15668459	C₂²	1	2	yes	—
14	D₁₈₁⋊M₁₆	—	2	2	no	—
15	D₁₈₁⋊M₁₆	C₂²	1	1	yes	characteristic
16	#19153895	C₂²	1	1	yes	characteristic
17	#19153886	C₂²	1	2	yes	complemented
18	#19153886	C₂²	1	2	yes	complemented
19	#19153881	C₂²	1	4	yes	complemented
20	#19153880	C₂²	1	4	yes	complemented
21	#19153870	C₂²	1	2	yes	complemented
22	C₈×#10624382	C₂²	1	2	yes	complemented
23	#41×D₁₈₁	C₂²	1	2	yes	—
24	#19153839	C₂²	1	1	yes	characteristic
25	#19153838	C₄	1	1	yes	characteristic, complemented
26	#19153837	C₂²	1	2	yes	—
27	#19153836	C₄	1	2	yes	complemented
28	#19153835	C₂²	1	1	yes	characteristic
29	Q₁₆×D₁₈₁	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₁₈₁	C₄	1	2	yes	complemented
31	D₈×D₁₈₁	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
33	#135×C₁₈₁	C₄	1	1	yes	characteristic, complemented
34	#15668459	—	2	4	no	—
35	#15668459	—	2	2	no	—
36	#15668459	D₄	1	2	yes	—
37	#15668459	C₂³	1	4	yes	—
38	#15668458	—	2	4	no	—
39	#15668458	D₄	1	2	yes	—
40	D₁₈₁⋊C₈	—	2	4	no	—
41	D₁₈₁⋊C₈	C₂³	1	2	yes	—
42	C₂×#13089685	—	2	4	no	—
43	C₂×#13089685	—	2	2	no	—
44	C₄⋊#10624382	—	2	1	no	—
45	C₂²⋊#10624382	—	2	1	no	—
46	C₄×#10624382	C₂³	1	2	yes	—
47	#15668450	C₂³	1	2	yes	—
48	#15668449	—	2	2	no	—
49	#15668449	C₂×C₄	1	2	yes	—
50	#15668448	—	2	2	no	—
51	#15668448	C₂³	1	2	yes	—
52	Q₈×D₁₈₁	C₂×C₄	1	2	yes	—
53	D₄×D₁₈₁	—	2	2	no	—
54	D₄×D₁₈₁	C₂×C₄	1	2	yes	—
55	C₂×C₄×D₁₈₁	—	2	2	no	—
56	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
57	Dic₇₂₄	C₂×C₄	1	1	yes	characteristic
58	#15668439	C₂×C₄	1	2	yes	—
59	#15668438	C₂×C₄	1	2	yes	—
60	#15668437	C₂×C₄	1	2	yes	—
61	#15668436	C₂×C₄	1	2	yes	—
62	D₁₄₄₈	C₂×C₄	1	1	yes	characteristic
63	#15668434	C₂×C₄	1	2	yes	—
64	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
65	C₈×D₁₈₁	C₂×C₄	1	1	yes	characteristic
66	C₈×D₁₈₁	C₂×C₄	1	1	yes	characteristic
67	C₈×D₁₈₁	C₂³	1	1	yes	characteristic
68	C₈×D₁₈₁	C₂³	1	1	yes	characteristic
69	#41×C₁₈₁	C₂×C₄	1	2	yes	—
70	Q₁₆×C₁₈₁	C₂×C₄	1	1	yes	characteristic
71	SD₁₆×C₁₈₁	C₂×C₄	1	2	yes	—
72	D₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
74	C₂×#10624382	—	2	2	no	—
75	#13089685	—	2	4	no	—
76	#13089685	C₂×D₄	1	4	yes	—
77	C₂²×D₁₈₁	—	2	2	no	—
78	C₂²×D₁₈₁	#41	1	1	yes	characteristic
79	Dic₃₆₂	—	2	2	no	—
80	Dic₃₆₂	C₂²×C₄	1	2	yes	—
81	D₇₂₄	—	2	2	no	—
82	D₇₂₄	C₂²×C₄	1	2	yes	—
83	C₂²⋊D₁₈₁	—	2	2	no	—
84	C₂²⋊D₁₈₁	—	2	2	no	—
85	C₂²⋊D₁₈₁	—	2	2	no	—
86	C₂×Dic₁₈₁	—	2	2	no	—
87	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
88	C₄×D₁₈₁	—	2	2	no	—
89	C₄×D₁₈₁	—	2	2	no	—
90	C₄×D₁₈₁	—	2	2	no	—
91	C₄×D₁₈₁	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₈₁	C₂×D₄	1	1	yes	characteristic
93	C₄×D₁₈₁	#41	1	1	yes	characteristic
94	C₄×D₁₈₁	C₂⁴	1	1	yes	characteristic
95	#13089678	C₂²×C₄	1	1	yes	characteristic
96	#13089678	C₂²×C₄	1	1	yes	characteristic
97	Q₈×C₁₈₁	C₂²×C₄	1	2	yes	—
98	D₄×C₁₈₁	—	2	2	no	—
99	D₄×C₁₈₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₁₈₁	—	2	2	no	—
101	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
102	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₁₈₁	—	2	2	no	—
105	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
106	C₂×D₁₈₁	—	2	2	no	—
107	C₂×D₁₈₁	—	4	1	no	—
108	C₂×D₁₈₁	—	8	2	no	—
109	C₂×D₁₈₁	C₄×D₄	1	1	yes	characteristic
110	C₂×D₁₈₁	C₂×#41	1	1	yes	characteristic
111	#10624382	—	4	2	no	—
112	C₄×C₁₈₁	—	2	2	no	—
113	C₄×C₁₈₁	C₄×D₄	1	1	yes	characteristic
114	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
115	Dic₁₈₁	—	2	2	no	—
116	Dic₁₈₁	C₄×D₄	1	1	yes	characteristic
117	Dic₁₈₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₁₈₁	—	2	1	no	—
119	C₂×C₁₈₁	—	4	2	no	—
120	C₂×C₁₈₁	C₂×C₄×D₄	1	1	yes	characteristic
121	D₁₈₁	—	2	1	no	—
122	D₁₈₁	—	2	1	no	—
123	D₁₈₁	—	4	2	no	—
124	C₁₈₁	#2957	1	1	yes	characteristic, complemented, Sylow
125	#2957	—	181	1	no	maximal, Sylow
126	C₂×#135	—	181	1	no	—
127	#568	—	181	2	no	—
128	#444	—	181	2	no	—
129	#444	—	181	2	no	—
130	#443	—	181	4	no	—
131	C₂×#108	—	181	1	no	—
132	#421	—	181	2	no	—
133	#405	—	181	1	no	—
134	C₂×#41	—	181	2	no	—
135	#135	—	181	2	no	—
136	#135	—	181	1	no	—
137	#135	—	181	2	no	—
138	#135	—	181	1	no	—
139	#135	—	181	1	no	—
140	#135	#10624382	1	1	yes	characteristic, complemented
141	C₂×Q₁₆	—	181	1	no	—
142	C₂×SD₁₆	—	181	2	no	—
143	C₂×D₈	—	181	1	no	—
144	#131	—	362	4	no	—
145	#131	—	181	4	no	—
146	#131	—	181	4	no	—
147	C₂×M₁₆	—	362	2	no	—
148	C₂×M₁₆	—	362	2	no	—
149	C₂×M₁₆	—	181	2	no	—
150	C₂×M₁₆	—	181	1	no	—
151	C₂²×C₈	—	181	1	no	—
152	#117	—	181	1	no	—
153	#108	—	181	2	no	—
154	#108	—	181	2	no	—
155	#104	—	181	4	no	—
156	#104	—	181	4	no	—
157	#97	—	181	2	no	—
158	C₄×C₈	—	181	2	no	—
159	#41	—	362	2	no	—
160	#41	—	362	2	no	—
161	#41	—	181	2	no	—
162	#41	—	181	2	no	—
163	#41	—	181	2	no	—
164	#41	C₂×#10624382	1	2	yes	—
165	C₂×Q₈	—	181	2	no	—
166	C₂×D₄	—	362	2	no	—
167	C₂×D₄	—	181	2	no	—
168	C₂²×C₄	—	362	2	no	—
169	C₂²×C₄	—	181	1	no	—
170	Q₁₆	—	181	2	no	—
171	Q₁₆	—	181	1	no	—
172	Q₁₆	C₂×#10624382	1	1	yes	characteristic
173	SD₁₆	—	181	2	no	—
174	SD₁₆	—	181	2	no	—
175	SD₁₆	—	181	2	no	—
176	SD₁₆	C₂×#10624382	1	2	yes	—
177	D₈	—	181	2	no	—
178	D₈	—	181	1	no	—
179	D₈	C₂×#10624382	1	1	yes	characteristic
180	M₁₆	—	362	2	no	—
181	M₁₆	—	362	4	no	—
182	M₁₆	—	362	4	no	—
183	M₁₆	—	181	4	no	—
184	M₁₆	—	181	2	no	—
185	M₁₆	—	181	2	no	—
186	C₂×C₈	—	362	2	no	—
187	C₂×C₈	—	362	4	no	—
188	C₂×C₈	—	362	4	no	—
189	C₂×C₈	—	181	2	no	—
190	C₂×C₈	—	181	1	no	—
191	C₂×C₈	—	181	1	no	—
192	C₂×C₈	—	181	1	no	—
193	C₂×C₈	—	181	1	no	—
194	C₂×C₈	—	181	1	no	—
195	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
196	C₄⋊C₄	—	362	1	no	—
197	C₂²⋊C₄	—	362	1	no	—
198	C₄²	—	181	2	no	—
199	C₂³	—	362	2	no	—
200	C₂³	—	181	1	no	—
201	Q₈	—	362	2	no	—
202	Q₈	—	181	2	no	—
203	Q₈	C₂²×#10624382	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	362	2	no	—
206	D₄	—	362	2	no	—
207	D₄	—	362	2	no	—
208	D₄	—	362	2	no	—
209	D₄	—	181	2	no	—
210	D₄	C₂²×#10624382	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	362	2	no	—
213	C₂×C₄	—	362	2	no	—
214	C₂×C₄	—	362	2	no	—
215	C₂×C₄	—	362	2	no	—
216	C₂×C₄	—	362	2	no	—
217	C₂×C₄	—	181	1	no	—
218	C₂×C₄	—	181	1	no	—
219	C₂×C₄	—	181	1	no	—
220	C₂×C₄	—	181	1	no	—
221	C₂×C₄	—	181	1	no	—
222	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
223	C₈	—	362	4	no	—
224	C₈	—	181	4	no	—
225	C₈	—	181	1	no	—
226	C₈	—	181	1	no	—
227	C₈	C₂²×#10624382	1	1	yes	characteristic
228	C₈	C₂²×#10624382	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	362	2	no	—
231	C₂²	—	1448	2	no	—
232	C₂²	—	181	1	no	—
233	C₂²	—	181	1	no	—
234	C₂²	—	724	1	no	—
235	C₂²	D₄×#10624382	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	362	2	no	—
238	C₄	—	181	1	no	—
239	C₄	—	181	1	no	—
240	C₄	—	724	2	no	—
241	C₄	D₄×#10624382	1	1	yes	characteristic
242	C₄	C₂³×#10624382	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	362	1	no	—
246	C₂	—	362	1	no	—
247	C₂	—	724	2	no	—
248	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	#135×C₁₈₁	C₄
socle	C₂×C₁₈₁	C₂×C₄×D₄
prime radical	#135×D₁₈₁	C₂
cosocle	C₄×D₁₈₁	C₂⁴
hypercenter	#135	#10624382
hypocenter	C₁₈₁	#2957
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #26042358	C₂	semidirect
2. #26042326	C₂	semidirect
3. #26042325	C₂	semidirect
4. #26042322	C₂	semidirect
5. C₂×#19153886	C₂	semidirect
6. #26042276	C₂	semidirect
7. #26042242	C₂	semidirect
9. #19153904	C₂²	semidirect
11. #19153903	C₂²	semidirect
17. #19153886	C₂²	semidirect
18. #19153886	C₂²	semidirect
19. #19153881	C₂²	semidirect
20. #19153880	C₂²	semidirect
21. #19153870	C₂²	semidirect
22. C₈×#10624382	C₂²	semidirect
25. #19153838	C₄	semidirect
27. #19153836	C₄	semidirect
29. Q₁₆×D₁₈₁	C₄	semidirect
30. SD₁₆×D₁₈₁	C₄	semidirect
31. D₈×D₁₈₁	C₄	semidirect
33. #135×C₁₈₁	C₄	semidirect
124. C₁₈₁	#2957	semidirect
140. #135	#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
#26042326	2
#26042325	2
#26042322	2
C₂×#19153886	2
#26042276	2
#26042242	2
#135×D₁₈₁	2
#2957	181