Subgroups — #13099549

Order 1600.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717221	C₂	1	1	yes	maximal, characteristic
2	C₅²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87756	C₂	1	1	yes	maximal, characteristic, complemented
4	#87900	C₄	1	1	yes	characteristic
5	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
6	#87804	C₄	1	1	yes	characteristic
7	#87799	—	2	1	no	—
8	C₈×C₅⋊D₅	—	2	1	no	—
9	C₅²×C₁₆	—	2	1	no	—
10	#87756	—	2	1	no	—
11	#64735	—	5	6	no	maximal
12	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
13	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
14	C₄⋊(C₅⋊D₅)	C₈	1	1	yes	characteristic
15	C₄×C₅⋊D₅	—	2	1	no	—
16	#6045	C₈	1	1	yes	characteristic
17	C₅²×C₈	D₄	1	1	yes	characteristic
18	C₅²×C₈	D₄	1	1	yes	characteristic
19	#6029	—	2	1	no	—
20	#4047	—	5	6	no	—
21	C₅×M₃₂	D₅	1	6	yes	complemented
22	C₂×#672	—	5	6	no	—
23	C₂²×C₅²	M₁₆	1	1	yes	characteristic
24	C₂×C₅⋊D₅	—	2	1	no	—
25	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₅²	C₂×C₈	1	1	yes	characteristic
27	#1039	—	2	1	no	—
28	#709	—	5	6	no	—
29	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
30	#681	—	5	6	no	—
31	#676	—	10	6	no	—
32	C₈×D₅	—	10	6	no	—
33	C₅×C₁₆	—	2	6	no	—
34	#672	—	10	6	no	—
35	#350	—	25	1	no	Sylow
36	C₂×C₅²	—	2	1	no	—
37	C₂×C₅²	C₂²⋊C₈	1	1	yes	characteristic
38	C₅⋊D₅	—	4	1	no	—
39	C₂×C₄×C₅	C₄×D₅	1	6	yes	—
40	C₂²⋊D₅	—	10	6	no	—
41	D₂₀	—	5	6	no	—
42	C₄×D₅	—	10	6	no	—
43	Dic₁₀	—	5	6	no	—
44	C₅×C₈	D₂₀	1	6	yes	—
45	C₅×C₈	C₂²⋊D₅	1	6	yes	—
46	#168	—	10	6	no	—
47	#131	—	25	1	no	—
48	M₃₂	C₅⋊D₅	1	1	yes	characteristic, complemented
49	C₂×C₁₆	—	25	1	no	—
50	C₅²	#350	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	#676	1	6	yes	—
52	C₂×D₅	—	10	6	no	—
53	C₄×C₅	C₈×D₅	1	6	yes	—
54	C₄×C₅	#685	1	6	yes	—
55	Dic₅	—	10	6	no	—
56	#41	—	25	1	no	—
57	M₁₆	—	25	1	no	—
58	M₁₆	—	50	1	no	—
59	C₂×C₈	—	50	1	no	—
60	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	50	1	no	—
63	C₂×C₅	—	2	6	no	—
64	C₂×C₅	#3952	1	6	yes	—
65	D₅	—	20	6	no	—
66	Q₈	—	25	1	no	—
67	D₄	—	25	1	no	—
68	D₄	—	50	1	no	—
69	C₂×C₄	—	50	1	no	—
70	C₂×C₄	C₄×C₅⋊D₅	1	1	yes	characteristic
71	C₈	—	50	1	no	—
72	C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
73	C₈	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
74	C₅	#64735	1	6	yes	complemented
75	C₂²	—	50	1	no	—
76	C₂²	#87799	1	1	yes	characteristic
77	C₄	—	50	1	no	—
78	C₄	C₈×C₅⋊D₅	1	1	yes	characteristic
79	C₄	#87808	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	100	1	no	—
82	C₂	#11716687	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊(C₅⋊D₅)
commutator	C₄×C₅²	C₂×C₈
Frattini	C₂×C₈	C₂×C₅⋊D₅
Fitting	C₅²×M₃₂	C₂
socle	C₂×C₅²	C₂²⋊C₈
prime radical	#87900	C₄
cosocle	C₂×C₅²×C₈	C₂²
hypercenter	M₃₂	C₅⋊D₅
hypocenter	C₅²	#350
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₈ ◁ 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. C₂×#87756	C₂	semidirect
21. C₅×M₃₂	D₅	semidirect
48. M₃₂	C₅⋊D₅	semidirect
50. C₅²	#350	semidirect
74. C₅	#64735	semidirect

Maximal subgroups

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

subgroup	index
#11717221	2
C₅²×M₃₂	2
C₂×#87756	2
#64735	5