Subgroups — C₂²⋊Q₈×C₇₅₇ #37564270

Order 24224.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₇₅₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇₅₇
prime radical	C₂³×C₇₅₇	C₂²
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
1. C₂×Q₈×C₇₅₇	C₂	semidirect
3. C₄⋊C₄×C₇₅₇	C₂	semidirect
4. C₄⋊C₄×C₇₅₇	C₂	semidirect
15. C₂²×C₇₅₇	Q₈	semidirect
25. C₇₅₇	C₂²⋊Q₈	direct (N × H)
26. C₂²⋊Q₈	C₇₅₇	direct (N × H)
27. C₂×Q₈	C₂×C₇₅₇	semidirect
29. C₄⋊C₄	C₂×C₇₅₇	semidirect
30. C₄⋊C₄	C₂×C₇₅₇	semidirect
41. C₂²	Q₈×C₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₇₅₇	2
C₂²×C₄×C₇₅₇	2
C₄⋊C₄×C₇₅₇	2
C₂²⋊C₄×C₇₅₇	2
C₂²⋊Q₈	757

Subgroups — C22⋊Q8×C757 #37564270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊Q₈×C₇₅₇ #37564270