Subgroups — C₂×C₄⋊C₄×C₇₅₇ #37564264

Order 24224.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₅₇	C₂²
commutator	C₂	C₂²×C₄×C₇₅₇
Frattini, cosocle	C₂²	C₂³×C₇₅₇
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
3. C₄⋊C₄×C₇₅₇	C₂	direct (N × H)
6. C₂×C₄×C₇₅₇	C₄	semidirect
14. C₄×C₇₅₇	C₂×C₄	semidirect
15. C₂×C₇₅₇	C₄⋊C₄	direct (N × H)
19. C₇₅₇	C₂×C₄⋊C₄	direct (N × H)
20. C₂×C₄⋊C₄	C₇₅₇	direct (N × H)
23. C₄⋊C₄	C₂×C₇₅₇	direct (N × H)
26. C₂×C₄	C₄×C₇₅₇	semidirect
34. C₄	C₂×C₄×C₇₅₇	semidirect
35. C₂	C₄⋊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
C₂²×C₄×C₇₅₇	2
C₄⋊C₄×C₇₅₇	2
C₂×C₄⋊C₄	757

Subgroups — C2×C4⋊C4×C757 #37564264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊C₄×C₇₅₇ #37564264