Subgroups — #13099551

Order 1600.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87802	C₂	1	1	yes	maximal, characteristic
2	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#11716686	C₂	1	1	yes	maximal, characteristic
4	C₂×#6045	—	2	1	no	—
5	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
6	#87804	—	2	1	no	—
7	#87802	—	2	1	no	—
8	#87802	C₄	1	2	yes	—
9	C₅²×C₁₆	—	2	1	no	—
10	#64737	—	5	6	no	maximal
11	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
12	C₂×#1039	—	4	1	no	—
13	#6045	—	2	2	no	—
14	#6045	—	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×Dic₂₀	—	5	6	no	—
19	C₅×M₃₂	D₅	1	6	yes	—
20	#3951	—	5	6	no	—
21	C₂²×C₅²	SD₁₆	1	1	yes	characteristic
22	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅²	D₈	1	1	yes	characteristic
24	#1039	—	4	2	no	—
25	C₂×Dic₁₀	—	10	6	no	—
26	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
27	#681	—	10	6	no	—
28	Dic₂₀	—	5	12	no	—
29	Dic₂₀	—	10	6	no	—
30	C₅×C₁₆	—	2	6	no	—
31	#362	—	25	1	no	Sylow
32	C₂×C₅²	—	2	1	no	—
33	C₂×C₅²	#102	1	1	yes	characteristic
34	C₂×C₄×C₅	D₂₀	1	6	yes	—
35	C₂×Dic₅	—	20	6	no	—
36	Dic₁₀	—	10	12	no	—
37	Dic₁₀	—	20	6	no	—
38	C₅×C₈	C₄×D₅	1	6	yes	—
39	C₅×C₈	C₂²⋊D₅	1	6	yes	—
40	#168	—	20	6	no	—
41	C₂×Q₁₆	—	25	1	no	—
42	M₃₂	C₅⋊D₅	1	1	yes	characteristic
43	#108	—	25	1	no	—
44	C₅²	#362	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₅	#677	1	6	yes	—
46	C₄×C₅	D₄₀	1	6	yes	—
47	C₄×C₅	#685	1	6	yes	—
48	Dic₅	—	20	12	no	—
49	C₂×Q₈	—	50	1	no	—
50	Q₁₆	—	25	2	no	—
51	Q₁₆	—	50	1	no	—
52	M₁₆	—	50	1	no	—
53	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₂×C₅	—	2	6	no	—
56	C₂×C₅	#3953	1	6	yes	—
57	Q₈	—	50	2	no	—
58	Q₈	—	100	1	no	—
59	C₂×C₄	—	100	1	no	—
60	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
61	C₈	—	100	1	no	—
62	C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
63	C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
64	C₅	#64737	1	6	yes	complemented
65	C₂²	#87800	1	1	yes	characteristic
66	C₄	—	100	2	no	—
67	C₄	#87801	1	1	yes	characteristic
68	C₄	#87808	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	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	C₂²×C₅²	SD₁₆
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	#362
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
44. C₅²	#362	semidirect
64. C₅	#64737	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87802	2
C₅²×M₃₂	2
#11716686	2
#64737	5