Subgroups — #49145804

Order 36832.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₁₁₅₁	C₂	1	2	yes	maximal, complemented
2	C₂²⋊Dic₁₁₅₁	C₂	1	2	yes	maximal
3	C₄×Dic₁₁₅₁	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₁₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₁₅₁	C₂²	1	4	yes	—
6	C₂³×C₁₁₅₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₁₅₁	C₄	1	4	yes	complemented
8	C₂×C₄×C₁₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₁₅₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₁₅₁	—	2	2	no	—
11	C₂²×C₁₁₅₁	C₂×C₄	1	2	yes	—
12	C₂²×C₁₁₅₁	C₂³	1	1	yes	characteristic
13	C₄×C₁₁₅₁	C₂×C₄	1	2	yes	—
14	C₄×C₁₁₅₁	C₂×C₄	1	2	yes	—
15	Dic₁₁₅₁	—	2	4	no	—
16	C₂×C₁₁₅₁	—	2	2	no	—
17	C₂×C₁₁₅₁	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₁₁₅₁	#41	1	2	yes	—
19	C₁₁₅₁	#117	1	1	yes	characteristic, complemented, Sylow
20	#117	—	1151	1	no	maximal, Sylow
21	C₂²×C₄	D₁₁₅₁	1	1	yes	characteristic
22	C₄⋊C₄	—	1151	2	no	—
23	C₂²⋊C₄	—	1151	2	no	—
24	C₄²	—	1151	2	no	—
25	C₂³	C₂×D₁₁₅₁	1	1	yes	characteristic
26	C₂×C₄	—	1151	4	no	—
27	C₂×C₄	Dic₁₁₅₁	1	4	yes	complemented
28	C₂×C₄	C₂×D₁₁₅₁	1	1	yes	characteristic
29	C₂×C₄	C₂×D₁₁₅₁	1	1	yes	characteristic
30	C₂²	—	2	2	no	—
31	C₂²	C₂×Dic₁₁₅₁	1	2	yes	—
32	C₂²	C₂²×D₁₁₅₁	1	1	yes	characteristic
33	C₄	—	2302	4	no	—
34	C₄	C₂×Dic₁₁₅₁	1	2	yes	—
35	C₄	C₂×Dic₁₁₅₁	1	2	yes	—
36	C₂	—	2	2	no	—
37	C₂	C₂²×Dic₁₁₅₁	1	1	yes	characteristic
38	C₂	#32914076	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₅₁
commutator	C₂×C₁₁₅₁	C₂²×C₄
Frattini	C₂²	C₂²×D₁₁₅₁
Fitting	C₂²×C₄×C₁₁₅₁	C₂
socle, cosocle	C₂²×C₁₁₅₁	C₂³
prime radical	C₂³×C₁₁₅₁	C₂²
hypercenter	C₂²×C₄	D₁₁₅₁
hypocenter	C₁₁₅₁	#117
LCS	G ▷ C₂×C₁₁₅₁ ▷ C₁₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊Dic₁₁₅₁	C₂	semidirect
3. C₄×Dic₁₁₅₁	C₂	semidirect
7. C₂×C₄×C₁₁₅₁	C₄	semidirect
19. C₁₁₅₁	#117	semidirect
27. C₂×C₄	Dic₁₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Dic₁₁₅₁	2
C₂²⋊Dic₁₁₅₁	2
C₄×Dic₁₁₅₁	2
C₂²×C₄×C₁₁₅₁	2
#117	1151