Subgroups — #37532020

Order 23168.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041330	C₂	1	3	yes	maximal
2	#26041001	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
5	#19153745	C₄	1	6	yes	—
6	#19153728	C₄	1	6	yes	—
7	C₄×#13089678	C₂²	1	3	yes	—
8	C₄×C₈×C₁₈₁	C₂²	1	3	yes	—
9	C₂×C₄×D₁₈₁	C₂×C₄	1	3	yes	—
10	C₂×#13089678	C₂×C₄	1	6	yes	—
11	C₄×Dic₁₈₁	C₂×C₄	1	3	yes	—
12	C₂×C₈×C₁₈₁	C₂×C₄	1	6	yes	—
13	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
14	C₂²×D₁₈₁	C₄²	1	1	yes	characteristic
15	C₂×Dic₁₈₁	C₄²	1	3	yes	—
16	C₄×D₁₈₁	—	2	6	no	—
17	#13089678	—	2	6	no	—
18	C₂×C₄×C₁₈₁	C₂²×C₄	1	3	yes	—
19	C₈×C₁₈₁	—	2	6	no	—
20	C₂²×C₁₈₁	C₂×C₄²	1	1	yes	characteristic
21	C₂×D₁₈₁	—	2	3	no	—
22	C₄×C₁₈₁	#131	1	6	yes	—
23	Dic₁₈₁	—	2	3	no	—
24	C₂×C₁₈₁	#405	1	3	yes	—
25	D₁₈₁	—	4	1	no	—
26	C₁₈₁	#1454	1	1	yes	characteristic, complemented, Sylow
27	#1454	—	181	1	no	maximal, Sylow
28	#431	—	181	3	no	—
29	#322	—	181	3	no	—
30	C₈²	D₁₈₁	1	1	yes	characteristic, complemented
31	C₂×C₄²	—	181	1	no	—
32	C₄⋊C₈	—	181	6	no	—
33	C₂²⋊C₈	—	181	6	no	—
34	C₄×C₈	—	181	3	no	—
35	C₄×C₈	C₂×D₁₈₁	1	3	yes	—
36	C₂²×C₄	—	181	3	no	—
37	C₂×C₈	—	181	6	no	—
38	C₂×C₈	C₄×D₁₈₁	1	6	yes	—
39	C₄²	—	181	3	no	—
40	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
41	C₂³	—	181	1	no	—
42	C₂×C₄	—	362	6	no	—
43	C₂×C₄	—	181	3	no	—
44	C₂×C₄	C₂×C₄×D₁₈₁	1	3	yes	—
45	C₈	—	2	6	no	—
46	C₈	—	362	6	no	—
47	C₂²	—	362	3	no	—
48	C₂²	C₄²×D₁₈₁	1	1	yes	characteristic
49	C₄	—	362	3	no	—
50	C₄	#19153822	1	6	yes	—
51	C₂	—	724	1	no	—
52	C₂	#26041233	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₈₁
commutator, socle	C₂²×C₁₈₁	C₂×C₄²
Fitting	C₈²×C₁₈₁	C₂
prime radical	C₂²×D₁₈₁	C₄²
cosocle	C₄²×C₁₈₁	C₂³
hypercenter	C₈²	D₁₈₁
hypocenter	C₁₈₁	#1454
LCS	G ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #26041001	C₂	semidirect
3. C₈²×C₁₈₁	C₂	semidirect
26. C₁₈₁	#1454	semidirect
30. C₈²	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041330	2
#26041001	2
C₈²×C₁₈₁	2
#1454	181