Subgroups — #37564395

Order 24224.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131088	C₂	1	1	yes	maximal, characteristic
2	#27131071	C₂	1	2	yes	maximal, complemented
3	C₂×#20258493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×D₇₅₇	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₅₁₄	C₄	1	1	yes	characteristic
7	D₃₀₂₈	C₄	1	1	yes	characteristic
8	C₂²⋊D₇₅₇	C₄	1	2	yes	—
9	C₄×D₇₅₇	C₂²	1	2	yes	—
10	#20258493	C₂²	1	1	yes	characteristic
11	#20258493	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₅₇	C₂²	1	1	yes	characteristic
13	C₈×C₇₅₇	C₂²	1	2	yes	—
14	C₂²×C₇₅₇	C₂×C₄	1	1	yes	characteristic
15	C₂×D₇₅₇	C₂×C₄	1	2	yes	—
16	C₄×C₇₅₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₇₅₇	C₂³	1	1	yes	characteristic
18	Dic₇₅₇	C₂×C₄	1	2	yes	—
19	C₂×C₇₅₇	—	2	1	no	—
20	C₂×C₇₅₇	C₂²×C₄	1	1	yes	characteristic
21	D₇₅₇	—	2	2	no	—
22	C₇₅₇	#131	1	1	yes	characteristic, complemented, Sylow
23	#131	—	757	1	no	maximal, Sylow
24	#41	—	757	1	no	—
25	M₁₆	—	757	2	no	—
26	M₁₆	D₇₅₇	1	1	yes	characteristic, complemented
27	C₂×C₈	—	757	1	no	—
28	C₂×C₈	—	757	2	no	—
29	Q₈	—	757	1	no	—
30	D₄	—	757	2	no	—
31	D₄	—	757	1	no	—
32	C₂×C₄	—	757	2	no	—
33	C₂×C₄	C₂×D₇₅₇	1	1	yes	characteristic
34	C₈	—	757	1	no	—
35	C₈	—	757	1	no	—
36	C₈	C₂×D₇₅₇	1	2	yes	—
37	C₂²	—	757	2	no	—
38	C₂²	C₄×D₇₅₇	1	1	yes	characteristic
39	C₄	—	757	2	no	—
40	C₄	C₄×D₇₅₇	1	1	yes	characteristic
41	C₄	C₂²×D₇₅₇	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	—	1514	2	no	—
44	C₂	C₂×C₄×D₇₅₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₇₅₇
commutator, socle	C₂×C₇₅₇	C₂²×C₄
Fitting	M₁₆×C₇₅₇	C₂
prime radical	#27131088	C₂
cosocle	C₄×C₇₅₇	C₂³
hypercenter	M₁₆	D₇₅₇
hypocenter	C₇₅₇	#131
LCS	G ▷ C₂×C₇₅₇ ▷ C₇₅₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #27131071	C₂	semidirect
3. C₂×#20258493	C₂	semidirect
4. C₈×D₇₅₇	C₂	semidirect
5. M₁₆×C₇₅₇	C₂	semidirect
22. C₇₅₇	#131	semidirect
26. M₁₆	D₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
#27131088	2
#27131071	2
C₂×#20258493	2
C₈×D₇₅₇	2
M₁₆×C₇₅₇	2
#131	757