Subgroups — C₄⋊C₄×D₇₅₇ #37564340

Order 24224.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₇₅₇	C₂	1	2	yes	maximal
2	C₂×C₄×D₇₅₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#27131067	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₅₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₇₅₇	C₂²	1	2	yes	—
8	C₂×Dic₇₅₇	C₂²	1	1	yes	characteristic
9	C₄×D₇₅₇	—	2	4	no	—
10	C₄×D₇₅₇	C₄	1	4	yes	complemented
11	C₂×C₄×C₇₅₇	C₂²	1	2	yes	—
12	C₂×C₄×C₇₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇₅₇	C₂³	1	1	yes	characteristic
14	C₂×D₇₅₇	C₂×C₄	1	2	yes	—
15	C₂×D₇₅₇	D₄	1	2	yes	—
16	C₂×D₇₅₇	Q₈	1	2	yes	—
17	C₄×C₇₅₇	—	2	2	no	—
18	C₄×C₇₅₇	C₂×C₄	1	2	yes	complemented
19	Dic₇₅₇	—	2	2	no	—
20	Dic₇₅₇	C₂×C₄	1	2	yes	complemented
21	C₂×C₇₅₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₇₅₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₇₅₇	C₂×Q₈	1	1	yes	characteristic
24	D₇₅₇	C₄⋊C₄	1	4	yes	direct product factor
25	C₇₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₄⋊C₄	—	757	1	no	maximal, Sylow
27	C₂²×C₄	—	757	2	no	—
28	C₂²×C₄	—	757	1	no	—
29	C₄⋊C₄	—	757	2	no	—
30	C₄⋊C₄	—	757	1	no	—
31	C₄⋊C₄	D₇₅₇	1	1	yes	characteristic, direct product factor
32	C₂³	—	757	1	no	—
33	C₂×C₄	—	1514	4	no	—
34	C₂×C₄	—	757	2	no	—
35	C₂×C₄	—	757	1	no	—
36	C₂×C₄	—	757	4	no	—
37	C₂×C₄	C₂×D₇₅₇	1	2	yes	—
38	C₂×C₄	C₂×D₇₅₇	1	1	yes	characteristic
39	C₂²	—	757	2	no	—
40	C₂²	—	757	2	no	—
41	C₂²	—	757	2	no	—
42	C₂²	C₂²×D₇₅₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	1514	2	no	—
45	C₄	—	757	2	no	—
46	C₄	C₄×D₇₅₇	1	2	yes	complemented
47	C₂	—	757	4	no	—
48	C₂	C₂×C₄×D₇₅₇	1	1	yes	characteristic
49	C₂	D₄×D₇₅₇	1	1	yes	characteristic
50	C₂	Q₈×D₇₅₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₄⋊Dic₇₅₇	C₂	semidirect
4. #27131067	C₂	semidirect
5. C₄⋊C₄×C₇₅₇	C₂	semidirect
10. C₄×D₇₅₇	C₄	semidirect
18. C₄×C₇₅₇	C₂×C₄	semidirect
20. Dic₇₅₇	C₂×C₄	semidirect
24. D₇₅₇	C₄⋊C₄	direct (N × H)
25. C₇₅₇	C₂×C₄⋊C₄	semidirect
31. C₄⋊C₄	D₇₅₇	direct (N × H)
46. C₄	C₄×D₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₅₇	2
C₄⋊Dic₇₅₇	2
#27131067	2
C₄⋊C₄×C₇₅₇	2
C₂×C₄⋊C₄	757

Subgroups — C4⋊C4×D757 #37564340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×D₇₅₇ #37564340