Finite Group Databasebeta

Subgroups — #37549293

Order 23168.

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042301	C₂	1	1	yes	maximal, characteristic
2	#26042298	C₂	1	1	yes	maximal, characteristic
3	#26042143	C₂	1	1	yes	maximal, characteristic, complemented
4	#26042142	C₂	1	1	yes	maximal, characteristic, complemented
5	#26042141	C₂	1	1	yes	maximal, characteristic, complemented
6	#26042140	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041331	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#13089685	C₂²	1	1	yes	characteristic
9	C₄⋊#13089685	C₂²	1	1	yes	characteristic
10	C₄⋊#13089685	C₂²	1	1	yes	characteristic
11	C₄⋊#13089685	C₂²	1	1	yes	characteristic
12	#19153873	—	2	1	no	—
13	#19153873	—	2	1	no	—
14	#19153871	—	2	1	no	—
15	C₄×#13089685	—	2	1	no	—
16	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
17	C₄⋊#13089678	C₄	1	1	yes	characteristic
18	#19153728	C₄	1	1	yes	characteristic
19	#19153728	C₄	1	1	yes	characteristic
20	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
21	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₈₁	C₄	1	1	yes	characteristic
23	C₂×#13089685	—	2	1	no	—
24	C₂×#13089685	—	2	1	no	—
25	C₂×#13089685	—	2	1	no	—
26	C₂×#13089685	—	2	1	no	—
27	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₈₁	Q₈	1	1	yes	characteristic
30	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
31	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
33	C₄×Dic₁₈₁	Q₈	1	1	yes	characteristic
34	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
38	#13089685	—	4	1	no	—
39	#13089685	—	4	1	no	—
40	#13089685	—	4	1	no	—
41	#13089685	—	4	1	no	—
42	C₂²×D₁₈₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₁₈₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₈₁	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₈₁	—	2	1	no	—
47	C₄×D₁₈₁	—	2	1	no	—
48	C₄×D₁₈₁	—	2	1	no	—
49	C₄×D₁₈₁	—	2	1	no	—
50	C₄×D₁₈₁	—	4	1	no	—
51	#13089678	—	2	1	no	—
52	#13089678	—	2	1	no	—
53	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
56	C₈×C₁₈₁	—	2	1	no	—
57	C₈×C₁₈₁	—	2	1	no	—
58	C₂²×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₁₈₁	—	2	1	no	—
60	C₂×D₁₈₁	—	2	1	no	—
61	C₂×D₁₈₁	—	2	1	no	—
62	C₄×C₁₈₁	—	2	1	no	—
63	C₄×C₁₈₁	—	2	1	no	—
64	C₄×C₁₈₁	#131	1	1	yes	characteristic
65	C₄×C₁₈₁	#131	1	1	yes	characteristic
66	Dic₁₈₁	—	2	1	no	—
67	Dic₁₈₁	—	2	1	no	—
68	Dic₁₈₁	#135	1	1	yes	characteristic
69	Dic₁₈₁	#135	1	1	yes	characteristic
70	C₂×C₁₈₁	#424	1	1	yes	characteristic
71	C₂×C₁₈₁	#427	1	1	yes	characteristic
72	C₂×C₁₈₁	#430	1	1	yes	characteristic
73	D₁₈₁	—	4	1	no	—
74	C₁₈₁	#1572	1	1	yes	characteristic, complemented, Sylow
75	#1572	—	181	1	no	maximal, Sylow
76	#432	—	181	1	no	—
77	#431	—	181	1	no	—
78	#431	—	181	1	no	—
79	#335	—	181	1	no	—
80	#335	—	181	1	no	—
81	#334	—	181	1	no	—
82	#334	—	181	1	no	—
83	C₂×C₄²	—	181	1	no	—
84	C₄⋊C₈	—	181	1	no	—
85	C₄⋊C₈	—	181	1	no	—
86	C₄⋊C₈	—	181	1	no	—
87	C₄⋊C₈	—	181	1	no	—
88	C₄⋊C₈	—	181	1	no	—
89	C₄⋊C₈	#10624382	1	1	yes	characteristic
90	C₂²⋊C₈	—	362	1	no	—
91	C₂²⋊C₈	—	362	1	no	—
92	C₂²⋊C₈	—	181	1	no	—
93	C₂²⋊C₈	—	181	1	no	—
94	#97	—	362	1	no	—
95	C₄×C₈	—	362	1	no	—
96	C₄×C₈	—	181	1	no	—
97	C₄×C₈	—	181	1	no	—
98	C₂²×C₄	—	181	1	no	—
99	C₂²×C₄	—	181	1	no	—
100	C₂²×C₄	—	181	1	no	—
101	C₂×C₈	—	362	1	no	—
102	C₂×C₈	—	362	1	no	—
103	C₂×C₈	—	362	1	no	—
104	C₂×C₈	—	362	1	no	—
105	C₂×C₈	—	181	1	no	—
106	C₂×C₈	—	181	1	no	—
107	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
108	C₂×C₈	C₂×#10624382	1	1	yes	characteristic
109	C₄²	—	181	1	no	—
110	C₄²	—	181	1	no	—
111	C₄²	—	181	1	no	—
112	C₄²	C₂×#10624382	1	1	yes	characteristic
113	C₂³	—	181	1	no	—
114	C₂×C₄	—	362	1	no	—
115	C₂×C₄	—	362	1	no	—
116	C₂×C₄	—	362	1	no	—
117	C₂×C₄	—	362	1	no	—
118	C₂×C₄	—	181	1	no	—
119	C₂×C₄	—	181	1	no	—
120	C₂×C₄	—	181	1	no	—
121	C₂×C₄	—	724	1	no	—
122	C₂×C₄	C₄⋊#10624382	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#10624382	1	1	yes	characteristic
124	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	362	1	no	—
128	C₈	—	362	1	no	—
129	C₈	—	724	1	no	—
130	C₈	—	724	1	no	—
131	C₈	—	724	1	no	—
132	C₈	—	724	1	no	—
133	C₂²	—	362	1	no	—
134	C₂²	—	362	1	no	—
135	C₂²	—	362	1	no	—
136	C₂²	C₂×C₄⋊#10624382	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	362	1	no	—
140	C₄	—	362	1	no	—
141	C₄	—	181	1	no	—
142	C₄	—	181	1	no	—
143	C₄	#19153903	1	1	yes	characteristic
144	C₄	#19153904	1	1	yes	characteristic
145	C₂	—	724	1	no	—
146	C₂	#26042286	1	1	yes	characteristic
147	C₂	#26042289	1	1	yes	characteristic
148	C₂	#26042294	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10624382
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#10624382
Fitting	C₄⋊C₈×C₁₈₁	C₄
socle	C₂²×C₁₈₁	C₂×C₄⋊C₄
prime radical	C₂²×D₁₈₁	C₄⋊C₄
cosocle	C₄×Dic₁₈₁	C₂³
hypercenter	C₄⋊C₈	#10624382
hypocenter	C₁₈₁	#1572
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #26042143	C₂	semidirect
4. #26042142	C₂	semidirect
5. #26042141	C₂	semidirect
6. #26042140	C₂	semidirect
74. C₁₈₁	#1572	semidirect

Maximal subgroups

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

subgroup	index
#26042301	2
#26042298	2
#26042143	2
#26042142	2
#26042141	2
#26042140	2
#26041331	2
#1572	181