Subgroups — C₂×C₂²⋊Dic₇₅₇ #37564339

Order 24224.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₇₅₇	C₂	1	2	yes	maximal, complemented
2	C₂²⋊Dic₇₅₇	C₂	1	4	yes	maximal, direct product factor
3	C₂⁴×C₇₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₅₇	—	2	4	no	—
5	C₂×Dic₇₅₇	C₂²	1	4	yes	complemented
6	C₂³×C₇₅₇	—	2	4	no	—
7	C₂³×C₇₅₇	C₄	1	4	yes	complemented
8	C₂³×C₇₅₇	C₂²	1	2	yes	—
9	C₂³×C₇₅₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇₅₇	—	2	8	no	—
11	C₂²×C₇₅₇	—	2	4	no	—
12	C₂²×C₇₅₇	C₂×C₄	1	2	yes	—
13	C₂²×C₇₅₇	C₂×C₄	1	4	yes	complemented
14	C₂²×C₇₅₇	D₄	1	4	yes	—
15	C₂²×C₇₅₇	C₂³	1	1	yes	characteristic
16	Dic₇₅₇	—	2	4	no	—
17	C₂×C₇₅₇	—	2	4	no	—
18	C₂×C₇₅₇	C₂²⋊C₄	1	4	yes	complemented
19	C₂×C₇₅₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₇₅₇	C₂×D₄	1	2	yes	—
21	C₇₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₂²⋊C₄	—	757	1	no	maximal, Sylow
23	C₂⁴	D₇₅₇	1	1	yes	characteristic
24	C₂²×C₄	—	757	2	no	—
25	C₂²⋊C₄	—	757	4	no	—
26	C₂³	—	2	4	no	—
27	C₂³	Dic₇₅₇	1	4	yes	complemented
28	C₂³	C₂×D₇₅₇	1	2	yes	—
29	C₂³	C₂×D₇₅₇	1	1	yes	characteristic
30	C₂×C₄	—	1514	4	no	—
31	C₂×C₄	—	757	4	no	—
32	C₂²	—	2	8	no	—
33	C₂²	—	2	4	no	—
34	C₂²	C₂×Dic₇₅₇	1	2	yes	—
35	C₂²	C₂×Dic₇₅₇	1	4	yes	complemented
36	C₂²	C₂²⋊D₇₅₇	1	4	yes	—
37	C₂²	C₂²×D₇₅₇	1	1	yes	characteristic
38	C₄	—	1514	4	no	—
39	C₂	—	2	4	no	—
40	C₂	C₂²⋊Dic₇₅₇	1	4	yes	direct product factor
41	C₂	C₂²×Dic₇₅₇	1	1	yes	characteristic
42	C₂	C₂×C₂²⋊D₇₅₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₇₅₇
commutator	C₂×C₇₅₇	C₂²×C₄
Frattini	C₂²	C₂²×D₇₅₇
Fitting, prime radical	C₂⁴×C₇₅₇	C₂
socle	C₂³×C₇₅₇	C₂²
cosocle	C₂²×C₇₅₇	C₂³
hypercenter	C₂⁴	D₇₅₇
hypocenter	C₇₅₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₇₅₇ ▷ C₇₅₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₇₅₇	C₂	semidirect
2. C₂²⋊Dic₇₅₇	C₂	direct (N × H)
5. C₂×Dic₇₅₇	C₂²	semidirect
7. C₂³×C₇₅₇	C₄	semidirect
13. C₂²×C₇₅₇	C₂×C₄	semidirect
18. C₂×C₇₅₇	C₂²⋊C₄	semidirect
21. C₇₅₇	C₂×C₂²⋊C₄	semidirect
27. C₂³	Dic₇₅₇	semidirect
35. C₂²	C₂×Dic₇₅₇	semidirect
40. C₂	C₂²⋊Dic₇₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₇₅₇	2
C₂²⋊Dic₇₅₇	2
C₂⁴×C₇₅₇	2
C₂×C₂²⋊C₄	757

Subgroups — C2×C22⋊Dic757 #37564339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊Dic₇₅₇ #37564339