Subgroups — C₂²⋊C₄×D₇₅₇ #37564337

Order 24224.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₇₅₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#27131065	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₄×C₇₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₅₇	—	2	4	no	—
7	C₂²×D₇₅₇	C₄	1	4	yes	complemented
8	C₂²×D₇₅₇	C₂²	1	1	yes	characteristic
9	C₂²×D₇₅₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₇₅₇	C₂²	1	2	yes	complemented
11	C₄×D₇₅₇	—	2	4	no	—
12	C₂³×C₇₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇₅₇	C₂²	1	2	yes	complemented
14	C₂²×C₇₅₇	—	2	2	no	—
15	C₂²×C₇₅₇	C₂×C₄	1	2	yes	complemented
16	C₂²×C₇₅₇	C₂³	1	1	yes	characteristic
17	C₂×D₇₅₇	—	2	2	no	—
18	C₂×D₇₅₇	—	2	8	no	—
19	C₂×D₇₅₇	C₂×C₄	1	2	yes	—
20	C₂×D₇₅₇	C₂×C₄	1	2	yes	complemented
21	C₂×D₇₅₇	D₄	1	4	yes	—
22	C₄×C₇₅₇	—	2	2	no	—
23	Dic₇₅₇	—	2	2	no	—
24	C₂×C₇₅₇	—	2	2	no	—
25	C₂×C₇₅₇	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₇₅₇	C₂×D₄	1	2	yes	—
27	D₇₅₇	—	2	2	no	—
28	D₇₅₇	C₂²⋊C₄	1	4	yes	direct product factor
29	C₇₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₂²⋊C₄	—	757	1	no	maximal, Sylow
31	C₂⁴	—	757	1	no	—
32	C₂²×C₄	—	757	2	no	—
33	C₂²⋊C₄	—	757	1	no	—
34	C₂²⋊C₄	—	757	2	no	—
35	C₂²⋊C₄	D₇₅₇	1	1	yes	characteristic, direct product factor
36	C₂³	—	1514	4	no	—
37	C₂³	—	757	4	no	—
38	C₂³	—	757	1	no	—
39	C₂³	—	757	1	no	—
40	C₂³	C₂×D₇₅₇	1	1	yes	characteristic
41	C₂×C₄	—	1514	4	no	—
42	C₂×C₄	—	757	2	no	—
43	C₂×C₄	C₂×D₇₅₇	1	2	yes	complemented
44	C₂²	—	2	2	no	—
45	C₂²	—	1514	8	no	—
46	C₂²	—	1514	2	no	—
47	C₂²	—	757	2	no	—
48	C₂²	—	757	2	no	—
49	C₂²	—	757	4	no	—
50	C₂²	C₄×D₇₅₇	1	2	yes	complemented
51	C₂²	C₂²×D₇₅₇	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	—	1514	2	no	—
54	C₂	—	2	2	no	—
55	C₂	—	1514	2	no	—
56	C₂	—	757	4	no	—
57	C₂	C₂×C₄×D₇₅₇	1	1	yes	characteristic
58	C₂	D₄×D₇₅₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₇₅₇	C₂	semidirect
3. C₂²⋊Dic₇₅₇	C₂	semidirect
4. #27131065	C₂	semidirect
5. C₂²⋊C₄×C₇₅₇	C₂	semidirect
7. C₂²×D₇₅₇	C₄	semidirect
10. C₂×Dic₇₅₇	C₂²	semidirect
13. C₂×C₄×C₇₅₇	C₂²	semidirect
15. C₂²×C₇₅₇	C₂×C₄	semidirect
20. C₂×D₇₅₇	C₂×C₄	semidirect
28. D₇₅₇	C₂²⋊C₄	direct (N × H)
29. C₇₅₇	C₂×C₂²⋊C₄	semidirect
35. C₂²⋊C₄	D₇₅₇	direct (N × H)
43. C₂×C₄	C₂×D₇₅₇	semidirect
50. C₂²	C₄×D₇₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₇₅₇	2
C₂×C₄×D₇₅₇	2
C₂²⋊Dic₇₅₇	2
#27131065	2
C₂²⋊C₄×C₇₅₇	2
C₂×C₂²⋊C₄	757

Subgroups — C22⋊C4×D757 #37564337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×D₇₅₇ #37564337