Subgroups — C₅²×#1712 #72221101

Order 3200.

1120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#493	C₂	1	6	yes	maximal, complemented
2	C₅²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅²×D₈	—	2	12	no	—
4	C₅²×#127	C₂²	1	4	yes	—
5	C₄×C₅²×C₈	C₂²	1	3	yes	—
6	C₅×#1712	C₅	1	6	yes	maximal, direct product factor
7	C₂×C₅²×D₄	—	2	12	no	—
8	C₅²×D₈	—	4	24	no	—
9	C₂×C₅²×C₈	D₄	1	6	yes	—
10	C₄²×C₅²	C₂³	1	1	yes	characteristic
11	C₅×#493	C₂×C₅	1	36	yes	complemented
12	C₅×C₈²	C₂×C₅	1	6	yes	complemented
13	C₂³×C₅²	—	4	4	no	—
14	C₅²×D₄	—	4	24	no	—
15	C₂×C₄×C₅²	C₂×D₄	1	3	yes	—
16	C₅²×C₈	D₈	1	12	yes	complemented
17	C₂×C₅×D₈	—	2	72	no	—
18	C₅×#127	C₂²×C₅	1	24	yes	—
19	C₄×C₅×C₈	C₂²×C₅	1	18	yes	—
20	#1712	C₅²	1	1	yes	characteristic, direct product factor, Sylow
21	C₂²×C₅²	—	8	12	no	—
22	C₂²×C₅²	#127	1	1	yes	characteristic
23	C₄×C₅²	C₂×D₈	1	6	yes	—
24	C₂×C₅×D₄	—	2	72	no	—
25	C₅×D₈	—	4	144	no	—
26	C₂×C₅×C₈	C₅×D₄	1	36	yes	—
27	C₄²×C₅	C₂³×C₅	1	6	yes	—
28	#493	C₂×C₅²	1	6	yes	complemented
29	C₈²	C₂×C₅²	1	1	yes	characteristic, complemented
30	C₂×C₅²	—	16	4	no	—
31	C₂×C₅²	#493	1	3	yes	—
32	C₂³×C₅	—	4	24	no	—
33	C₅×D₄	—	4	144	no	—
34	C₂×C₄×C₅	C₂×C₅×D₄	1	18	yes	—
35	C₅×C₈	C₅×D₈	1	72	yes	complemented
36	C₂×D₈	—	2	12	no	—
37	#127	C₂²×C₅²	1	4	yes	—
38	C₄×C₈	C₂²×C₅²	1	3	yes	—
39	C₅²	#1712	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₅	—	8	72	no	—
41	C₂²×C₅	C₅×#127	1	6	yes	—
42	C₄×C₅	C₂×C₅×D₈	1	36	yes	—
43	C₂×D₄	—	2	12	no	—
44	D₈	—	4	24	no	—
45	C₂×C₈	C₅²×D₄	1	6	yes	—
46	C₄²	C₂³×C₅²	1	1	yes	characteristic
47	C₂×C₅	—	16	24	no	—
48	C₂×C₅	C₅×#493	1	18	yes	—
49	C₂³	—	4	4	no	—
50	D₄	—	4	24	no	—
51	C₂×C₄	C₂×C₅²×D₄	1	3	yes	—
52	C₈	C₅²×D₈	1	12	yes	complemented
53	C₅	C₅×#1712	1	6	yes	direct product factor
54	C₂²	—	8	12	no	—
55	C₂²	C₅²×#127	1	1	yes	characteristic
56	C₄	C₂×C₅²×D₈	1	6	yes	—
57	C₂	—	16	4	no	—
58	C₂	C₅²×#493	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₅²	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₅²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₄²×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅²×#493	C₂	semidirect
2. C₅²×C₈²	C₂	semidirect
6. C₅×#1712	C₅	direct (N × H)
11. C₅×#493	C₂×C₅	semidirect
12. C₅×C₈²	C₂×C₅	semidirect
16. C₅²×C₈	D₈	semidirect
20. #1712	C₅²	direct (N × H)
28. #493	C₂×C₅²	semidirect
29. C₈²	C₂×C₅²	semidirect
35. C₅×C₈	C₅×D₈	semidirect
39. C₅²	#1712	direct (N × H)
52. C₈	C₅²×D₈	semidirect
53. C₅	C₅×#1712	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×#493	2
C₅²×C₈²	2
C₅×#1712	5

Subgroups — C52×#1712 #72221101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#1712 #72221101