Subgroups — #13099550

Order 1600.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87801	C₂	1	1	yes	maximal, characteristic
2	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716686	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
5	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
6	#87804	—	2	1	no	—
7	#87801	—	2	1	no	—
8	#87801	C₄	1	2	yes	—
9	C₅²×C₁₆	—	2	1	no	—
10	#64736	—	5	6	no	maximal
11	C₂²×C₅⋊D₅	—	4	1	no	—
12	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
13	C₄⋊(C₅⋊D₅)	—	2	2	no	—
14	C₄⋊(C₅⋊D₅)	—	4	1	no	—
15	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₅²×C₈	D₄	1	1	yes	characteristic
17	#6029	—	4	1	no	—
18	C₂×D₄₀	—	5	6	no	—
19	C₅×M₃₂	D₅	1	6	yes	complemented
20	#3951	—	5	6	no	—
21	C₂²×C₅²	SD₁₆	1	1	yes	characteristic
22	C₂×C₅⋊D₅	—	4	2	no	—
23	C₂×C₅⋊D₅	—	8	2	no	—
24	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₅²	D₈	1	1	yes	characteristic
26	C₂×D₂₀	—	10	6	no	—
27	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
28	#681	—	10	6	no	—
29	D₄₀	—	5	12	no	—
30	D₄₀	—	10	6	no	—
31	C₅×C₁₆	—	2	6	no	—
32	#361	—	25	1	no	Sylow
33	C₂×C₅²	—	2	1	no	—
34	C₂×C₅²	#102	1	1	yes	characteristic
35	C₅⋊D₅	—	8	2	no	—
36	C₂²×D₅	—	20	6	no	—
37	C₂×C₄×C₅	D₂₀	1	6	yes	—
38	D₂₀	—	10	12	no	—
39	D₂₀	—	20	6	no	—
40	C₅×C₈	C₄×D₅	1	6	yes	—
41	C₅×C₈	C₂²⋊D₅	1	6	yes	—
42	#168	—	20	6	no	—
43	C₂×D₈	—	25	1	no	—
44	M₃₂	C₅⋊D₅	1	1	yes	characteristic, complemented
45	#108	—	25	1	no	—
46	C₅²	#361	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	#677	1	6	yes	—
48	C₂×D₅	—	20	12	no	—
49	C₂×D₅	—	40	12	no	—
50	C₄×C₅	D₄₀	1	6	yes	—
51	C₄×C₅	#685	1	6	yes	—
52	C₂×D₄	—	50	1	no	—
53	D₈	—	25	2	no	—
54	D₈	—	50	1	no	—
55	M₁₆	—	50	1	no	—
56	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₂×C₅	—	2	6	no	—
59	C₂×C₅	#3953	1	6	yes	—
60	D₅	—	40	12	no	—
61	C₂³	—	100	1	no	—
62	D₄	—	50	2	no	—
63	D₄	—	100	1	no	—
64	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
65	C₈	—	100	1	no	—
66	C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
67	C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
68	C₅	#64736	1	6	yes	complemented
69	C₂²	—	100	2	no	—
70	C₂²	—	200	2	no	—
71	C₂²	#87800	1	1	yes	characteristic
72	C₄	#87801	1	1	yes	characteristic
73	C₄	#87808	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	200	2	no	—
76	C₂	#11716688	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716688
commutator	C₅²×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×C₅⋊D₅
Fitting	C₅²×M₃₂	C₂
socle	C₂×C₅²	#102
prime radical	C₂×#87801	C₂
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	#361
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₅²×M₃₂	C₂	semidirect
3. #11716686	C₂	semidirect
19. C₅×M₃₂	D₅	semidirect
44. M₃₂	C₅⋊D₅	semidirect
46. C₅²	#361	semidirect
68. C₅	#64736	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87801	2
C₅²×M₃₂	2
#11716686	2
#64736	5