Subgroups — #123×C₇₅₇ #37564271

Order 24224.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₇₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×C₇₅₇	C₂	1	2	yes	maximal, complemented
4	C₂²⋊C₄×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×C₇₅₇	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇₅₇	C₂²	1	1	yes	characteristic
7	C₂³×C₇₅₇	C₂²	1	1	yes	characteristic
8	D₄×C₇₅₇	—	2	2	no	—
9	C₂×C₄×C₇₅₇	—	2	2	no	—
10	C₂×C₄×C₇₅₇	C₂²	1	2	yes	—
11	C₂×C₄×C₇₅₇	C₂²	1	2	yes	—
12	C₂×C₄×C₇₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇₅₇	—	2	2	no	—
14	C₂²×C₇₅₇	—	2	1	no	—
15	C₂²×C₇₅₇	—	2	2	no	—
16	C₂²×C₇₅₇	D₄	1	2	yes	—
17	C₂²×C₇₅₇	C₂³	1	1	yes	characteristic
18	C₄×C₇₅₇	—	2	2	no	—
19	C₄×C₇₅₇	—	2	1	no	—
20	C₄×C₇₅₇	—	2	2	no	—
21	C₂×C₇₅₇	—	2	2	no	—
22	C₂×C₇₅₇	—	4	1	no	—
23	C₂×C₇₅₇	C₂×D₄	1	1	yes	characteristic
24	C₂×C₇₅₇	#41	1	2	yes	—
25	C₇₅₇	#123	1	1	yes	characteristic, direct product factor, Sylow
26	#123	C₇₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂×D₄	C₂×C₇₅₇	1	1	yes	characteristic
28	C₂²×C₄	C₂×C₇₅₇	1	1	yes	characteristic, complemented
29	C₄⋊C₄	C₂×C₇₅₇	1	2	yes	complemented
30	C₂²⋊C₄	C₂×C₇₅₇	1	1	yes	characteristic, complemented
31	C₂²⋊C₄	C₂×C₇₅₇	1	2	yes	complemented
32	C₂³	C₂²×C₇₅₇	1	1	yes	characteristic
33	C₂³	C₂²×C₇₅₇	1	1	yes	characteristic
34	D₄	—	2	2	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	C₂²×C₇₅₇	1	2	yes	—
37	C₂×C₄	C₂²×C₇₅₇	1	2	yes	—
38	C₂×C₄	C₂²×C₇₅₇	1	1	yes	characteristic
39	C₂²	—	2	2	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	2	2	no	—
42	C₂²	D₄×C₇₅₇	1	2	yes	—
43	C₂²	C₂³×C₇₅₇	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	—	2	1	no	—
46	C₄	—	2	2	no	—
47	C₂	—	2	2	no	—
48	C₂	—	4	1	no	—
49	C₂	C₂×D₄×C₇₅₇	1	1	yes	characteristic
50	C₂	#41×C₇₅₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₅₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇₅₇
prime radical	C₂×D₄×C₇₅₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₇₅₇	C₂	semidirect
3. C₄⋊C₄×C₇₅₇	C₂	semidirect
4. C₂²⋊C₄×C₇₅₇	C₂	semidirect
5. C₂²⋊C₄×C₇₅₇	C₂	semidirect
25. C₇₅₇	#123	direct (N × H)
26. #123	C₇₅₇	direct (N × H)
28. C₂²×C₄	C₂×C₇₅₇	semidirect
29. C₄⋊C₄	C₂×C₇₅₇	semidirect
30. C₂²⋊C₄	C₂×C₇₅₇	semidirect
31. C₂²⋊C₄	C₂×C₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₇₅₇	2
C₂²×C₄×C₇₅₇	2
C₄⋊C₄×C₇₅₇	2
C₂²⋊C₄×C₇₅₇	2
#123	757

Subgroups — #123×C757 #37564271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #123×C₇₅₇ #37564271