Subgroups — #13099552

Order 1600.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717224	C₂	1	1	yes	maximal, characteristic
2	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716684	C₂	1	1	yes	maximal, characteristic, complemented
4	#87900	—	2	1	no	—
5	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
6	C₄⋊#1039	—	2	1	no	—
7	#87802	C₄	1	1	yes	characteristic, complemented
8	#87801	C₄	1	1	yes	characteristic, complemented
9	#87800	—	2	1	no	—
10	C₅²×C₁₆	—	2	1	no	—
11	#64738	—	5	6	no	maximal
12	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
13	C₂²⋊(C₅⋊D₅)	—	4	1	no	—
14	C₂×#1039	—	4	1	no	—
15	C₄⋊(C₅⋊D₅)	—	2	1	no	—
16	C₄×C₅⋊D₅	—	4	1	no	—
17	#6045	—	2	1	no	—
18	C₅²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₅²×C₈	D₄	1	1	yes	characteristic
20	#4050	—	5	6	no	—
21	C₅×M₃₂	D₅	1	6	yes	complemented
22	#3949	—	5	6	no	—
23	C₂²×C₅²	D₈	1	1	yes	characteristic
24	C₂×C₅⋊D₅	—	4	1	no	—
25	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₅²	SD₁₆	1	1	yes	characteristic
27	#1039	—	4	1	no	—
28	#1039	—	8	1	no	—
29	#709	—	10	6	no	—
30	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
31	C₄⋊Dic₅	—	10	6	no	—
32	Dic₂₀	—	5	6	no	—
33	D₄₀	—	5	6	no	—
34	#677	—	10	6	no	—
35	C₅×C₁₆	—	2	6	no	—
36	#360	—	25	1	no	Sylow
37	C₂×C₅²	—	2	1	no	—
38	C₂×C₅²	#102	1	1	yes	characteristic
39	C₅⋊D₅	—	8	1	no	—
40	C₂×C₄×C₅	D₂₀	1	6	yes	—
41	C₂²⋊D₅	—	20	6	no	—
42	C₂×Dic₅	—	20	6	no	—
43	D₂₀	—	10	6	no	—
44	C₄×D₅	—	20	6	no	—
45	Dic₁₀	—	10	6	no	—
46	C₅×C₈	C₄×D₅	1	6	yes	—
47	C₅×C₈	C₂²⋊D₅	1	6	yes	—
48	#135	—	25	1	no	—
49	M₃₂	C₅⋊D₅	1	1	yes	characteristic, complemented
50	#106	—	25	1	no	—
51	C₅²	#360	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₅	D₄₀	1	6	yes	—
53	C₂×D₅	—	20	6	no	—
54	C₄×C₅	#677	1	6	yes	—
55	C₄×C₅	#685	1	6	yes	—
56	Dic₅	—	20	6	no	—
57	Dic₅	—	40	6	no	—
58	#41	—	50	1	no	—
59	Q₁₆	—	25	1	no	—
60	SD₁₆	—	50	1	no	—
61	D₈	—	25	1	no	—
62	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
63	C₄⋊C₄	—	50	1	no	—
64	C₁₆	—	2	1	no	—
65	C₂×C₅	—	2	6	no	—
66	C₂×C₅	#3953	1	6	yes	—
67	D₅	—	40	6	no	—
68	Q₈	—	50	1	no	—
69	D₄	—	50	1	no	—
70	D₄	—	100	1	no	—
71	C₂×C₄	—	100	1	no	—
72	C₂×C₄	—	100	1	no	—
73	C₂×C₄	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
74	C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
75	C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
76	C₅	#64738	1	6	yes	complemented
77	C₂²	—	100	1	no	—
78	C₂²	#87801	1	1	yes	characteristic
79	C₄	—	100	1	no	—
80	C₄	—	200	1	no	—
81	C₄	#87800	1	1	yes	characteristic
82	C₄	#87808	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	200	1	no	—
85	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	#11717224	C₂
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	#360
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. #11716684	C₂	semidirect
7. #87802	C₄	semidirect
8. #87801	C₄	semidirect
21. C₅×M₃₂	D₅	semidirect
49. M₃₂	C₅⋊D₅	semidirect
51. C₅²	#360	semidirect
76. C₅	#64738	semidirect

Maximal subgroups

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

subgroup	index
#11717224	2
C₅²×M₃₂	2
#11716684	2
#64738	5