Subgroups — #11717235

Order 800.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	#87902	C₂	1	1	yes	maximal, characteristic
3	C₅²×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#87812	C₂	1	1	yes	maximal, characteristic, complemented
5	#87810	C₂	1	1	yes	maximal, characteristic, complemented
6	#87802	C₂	1	1	yes	maximal, characteristic, complemented
7	#87799	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅²×Q₈	C₂²	1	1	yes	characteristic
9	C₅²×D₄	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
11	C₂×#1039	—	2	1	no	—
12	C₄×C₅⋊D₅	—	2	1	no	—
13	C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
14	#6045	—	2	1	no	—
15	#6045	C₂²	1	1	yes	characteristic
16	#6045	C₂²	1	1	yes	characteristic
17	C₅²×C₈	C₂²	1	1	yes	characteristic
18	#6029	C₂²	1	1	yes	characteristic
19	#4061	—	5	6	no	maximal
20	C₂²×C₅²	—	2	1	no	—
21	C₂×C₅⋊D₅	D₄	1	1	yes	characteristic
22	C₄×C₅²	—	2	1	no	—
23	C₄×C₅²	C₂³	1	1	yes	characteristic
24	#1039	—	2	1	no	—
25	#1039	—	2	1	no	—
26	#1039	D₄	1	1	yes	characteristic
27	Q₈×D₅	—	5	6	no	—
28	#711	—	5	6	no	—
29	C₅×SD₁₆	D₅	1	6	yes	complemented
30	#689	—	5	6	no	—
31	#687	—	5	6	no	—
32	Dic₂₀	—	5	6	no	—
33	#676	—	5	6	no	—
34	C₂×C₅²	—	4	1	no	—
35	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
36	C₅⋊D₅	—	2	1	no	—
37	C₅×Q₈	C₂×D₅	1	6	yes	—
38	C₅×D₄	C₂×D₅	1	6	yes	—
39	C₂²⋊D₅	—	10	6	no	—
40	C₂×Dic₅	—	10	6	no	—
41	C₄×D₅	—	5	6	no	—
42	C₄×D₅	—	10	6	no	—
43	Dic₁₀	—	5	6	no	—
44	Dic₁₀	—	5	6	no	—
45	Dic₁₀	—	10	6	no	—
46	C₅×C₈	C₂×D₅	1	6	yes	—
47	#168	—	5	6	no	—
48	#137	—	25	1	no	Sylow
49	C₅²	#137	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₅	—	2	6	no	—
51	C₂×D₅	—	5	6	no	—
52	C₄×C₅	—	2	6	no	—
53	C₄×C₅	C₂²×D₅	1	6	yes	—
54	Dic₅	—	5	6	no	—
55	Dic₅	—	10	6	no	—
56	Dic₅	—	10	6	no	—
57	#41	—	25	1	no	—
58	C₂×Q₈	—	25	1	no	—
59	Q₁₆	—	25	1	no	—
60	Q₁₆	—	25	1	no	—
61	SD₁₆	—	25	1	no	—
62	SD₁₆	C₅⋊D₅	1	1	yes	characteristic, complemented
63	M₁₆	—	25	1	no	—
64	C₂×C₅	—	4	6	no	—
65	C₂×C₅	D₄×D₅	1	6	yes	—
66	D₅	—	10	6	no	—
67	Q₈	—	25	1	no	—
68	Q₈	—	25	1	no	—
69	Q₈	—	50	1	no	—
70	Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
71	D₄	—	50	1	no	—
72	D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
73	C₂×C₄	—	25	1	no	—
74	C₂×C₄	—	50	1	no	—
75	C₂×C₄	—	50	1	no	—
76	C₈	—	25	1	no	—
77	C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
78	C₅	#4061	1	6	yes	complemented
79	C₂²	—	2	1	no	—
80	C₂²	—	25	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	25	1	no	—
83	C₄	—	50	1	no	—
84	C₄	—	50	1	no	—
85	C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
86	C₂	—	4	1	no	—
87	C₂	—	50	1	no	—
88	C₂	D₄×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₅⋊D₅
commutator, cosocle	C₄×C₅²	C₂³
Frattini	C₄	C₂²×C₅⋊D₅
Fitting	C₅²×SD₁₆	C₂
socle	C₂×C₅²	C₂×D₄
prime radical	#87902	C₂
hypercenter	SD₁₆	C₅⋊D₅
hypocenter	C₅²	#137
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×C₅⋊D₅	C₂	semidirect
3. C₅²×SD₁₆	C₂	semidirect
4. #87812	C₂	semidirect
5. #87810	C₂	semidirect
6. #87802	C₂	semidirect
7. #87799	C₂	semidirect
29. C₅×SD₁₆	D₅	semidirect
49. C₅²	#137	semidirect
62. SD₁₆	C₅⋊D₅	semidirect
78. C₅	#4061	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₅⋊D₅	2
#87902	2
C₅²×SD₁₆	2
#87812	2
#87810	2
#87802	2
#87799	2
#4061	5