Subgroups — C₂×C₂²⋊C₄×C₇₅₇ #37564263

Order 24224.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₇₅₇	C₂²
commutator	C₂	C₂²×C₄×C₇₅₇
Frattini, cosocle	C₂²	C₂³×C₇₅₇
prime radical	C₂⁴×C₇₅₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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. C₂²⋊C₄×C₇₅₇	C₂	direct (N × H)
5. C₂³×C₇₅₇	C₄	semidirect
9. C₂×C₄×C₇₅₇	C₂²	semidirect
13. C₂²×C₇₅₇	C₂×C₄	semidirect
18. C₂×C₇₅₇	C₂²⋊C₄	direct (N × H)
21. C₇₅₇	C₂×C₂²⋊C₄	direct (N × H)
22. C₂×C₂²⋊C₄	C₇₅₇	direct (N × H)
24. C₂²×C₄	C₂×C₇₅₇	semidirect
25. C₂²⋊C₄	C₂×C₇₅₇	direct (N × H)
27. C₂³	C₄×C₇₅₇	semidirect
31. C₂×C₄	C₂²×C₇₅₇	semidirect
35. C₂²	C₂×C₄×C₇₅₇	semidirect
40. C₂	C₂²⋊C₄×C₇₅₇	direct (N × H)

Maximal subgroups

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

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

Subgroups — C2×C22⋊C4×C757 #37564263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊C₄×C₇₅₇ #37564263