Subgroups — #135×C₇₅₇ #37564283

Order 24224.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₅₇	C₂	1	2	yes	maximal, complemented
2	Q₁₆×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₁₆×C₇₅₇	C₂	1	2	yes	maximal, complemented
4	D₈×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₇₅₇	C₂²	1	2	yes	—
7	D₄×C₇₅₇	—	2	2	no	—
8	D₄×C₇₅₇	C₂²	1	2	yes	—
9	C₂×C₄×C₇₅₇	—	2	2	no	—
10	C₂×C₄×C₇₅₇	C₂²	1	1	yes	characteristic
11	C₈×C₇₅₇	C₂²	1	1	yes	characteristic
12	C₈×C₇₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇₅₇	—	2	2	no	—
14	C₂²×C₇₅₇	D₄	1	1	yes	characteristic
15	C₄×C₇₅₇	—	2	2	no	—
16	C₄×C₇₅₇	D₄	1	1	yes	characteristic
17	C₄×C₇₅₇	C₂³	1	1	yes	characteristic
18	C₂×C₇₅₇	—	2	1	no	—
19	C₂×C₇₅₇	—	4	2	no	—
20	C₂×C₇₅₇	C₂×D₄	1	1	yes	characteristic
21	C₇₅₇	#135	1	1	yes	characteristic, direct product factor, Sylow
22	#135	C₇₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	#41	C₂×C₇₅₇	1	2	yes	complemented
24	Q₁₆	C₂×C₇₅₇	1	1	yes	characteristic, complemented
25	SD₁₆	C₂×C₇₅₇	1	2	yes	complemented
26	D₈	C₂×C₇₅₇	1	1	yes	characteristic, complemented
27	C₂×C₈	C₂×C₇₅₇	1	1	yes	characteristic, complemented
28	Q₈	C₂²×C₇₅₇	1	2	yes	—
29	D₄	—	2	2	no	—
30	D₄	C₂²×C₇₅₇	1	2	yes	—
31	C₂×C₄	—	2	2	no	—
32	C₂×C₄	C₂²×C₇₅₇	1	1	yes	characteristic
33	C₈	C₂²×C₇₅₇	1	1	yes	characteristic
34	C₈	C₂²×C₇₅₇	1	1	yes	characteristic
35	C₂²	—	2	2	no	—
36	C₂²	D₄×C₇₅₇	1	1	yes	characteristic
37	C₄	—	2	2	no	—
38	C₄	D₄×C₇₅₇	1	1	yes	characteristic
39	C₄	C₂³×C₇₅₇	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	4	2	no	—
42	C₂	C₂×D₄×C₇₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₅₇	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇₅₇
socle	C₂×C₇₅₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₅₇ ◁ C₂×C₄×C₇₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #41×C₇₅₇	C₂	semidirect
2. Q₁₆×C₇₅₇	C₂	semidirect
3. SD₁₆×C₇₅₇	C₂	semidirect
4. D₈×C₇₅₇	C₂	semidirect
5. C₂×C₈×C₇₅₇	C₂	semidirect
21. C₇₅₇	#135	direct (N × H)
22. #135	C₇₅₇	direct (N × H)
23. #41	C₂×C₇₅₇	semidirect
24. Q₁₆	C₂×C₇₅₇	semidirect
25. SD₁₆	C₂×C₇₅₇	semidirect
26. D₈	C₂×C₇₅₇	semidirect
27. C₂×C₈	C₂×C₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₅₇	2
Q₁₆×C₇₅₇	2
SD₁₆×C₇₅₇	2
D₈×C₇₅₇	2
C₂×C₈×C₇₅₇	2
#135	757

Subgroups — #135×C757 #37564283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #135×C₇₅₇ #37564283