Subgroups — #11717395

Order 800.

113 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#87880	C₂	1	2	yes	maximal, complemented
2	C₂×#6034	C₂	1	1	yes	maximal, characteristic, complemented
3	#87839	C₂	1	1	yes	maximal, characteristic, complemented
4	#87838	C₂	1	2	yes	maximal, complemented
5	#87837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#1039	C₂²	1	1	yes	characteristic
7	C₅×C₂²⋊D₅	—	2	2	no	—
8	#6039	C₂²	1	2	yes	—
9	#6037	C₂²	1	2	yes	—
10	D₅×Dic₅	—	2	2	no	—
11	#6034	C₂²	1	1	yes	characteristic
12	#6034	C₂²	1	1	yes	characteristic
13	C₂²×C₅²	D₄	1	1	yes	characteristic
14	C₂×C₅×D₅	—	2	2	no	—
15	#1039	D₄	1	1	yes	characteristic
16	#1039	C₂³	1	1	yes	characteristic
17	C₅×Dic₅	—	2	2	no	—
18	#711	—	10	2	no	—
19	C₂×#170	—	10	1	no	—
20	C₂×C₅²	—	2	1	no	—
21	C₂×C₅²	C₂×D₄	1	1	yes	characteristic
22	C₅×D₅	—	4	2	no	—
23	C₅×D₄	—	10	2	no	—
24	C₂²⋊D₅	—	2	2	no	—
25	C₂²⋊D₅	—	10	2	no	—
26	C₂×Dic₅	—	10	1	no	—
27	C₂×Dic₅	—	10	2	no	—
28	C₂×Dic₅	—	10	2	no	—
29	C₄×D₅	—	10	2	no	—
30	Dic₁₀	—	10	2	no	—
31	#170	—	10	1	no	—
32	#170	—	10	1	no	—
33	#135	—	25	1	no	maximal, Sylow
34	C₅²	#135	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₅	—	2	1	no	—
36	C₂²×C₅	—	2	2	no	—
37	C₂²×C₅	—	10	2	no	—
38	C₂×D₅	—	2	2	no	—
39	C₄×C₅	—	10	2	no	—
40	Dic₅	—	2	2	no	—
41	Dic₅	—	10	2	no	—
42	Dic₅	—	10	1	no	—
43	Dic₅	—	10	1	no	—
44	Dic₅	—	10	2	no	—
45	#41	—	25	2	no	—
46	Q₁₆	—	25	1	no	—
47	SD₁₆	—	25	2	no	—
48	D₈	—	25	1	no	—
49	C₂×C₈	—	25	1	no	—
50	C₂×C₅	—	2	2	no	—
51	C₂×C₅	—	2	2	no	—
52	C₂×C₅	—	2	1	no	—
53	C₂×C₅	—	4	2	no	—
54	C₂×C₅	—	20	2	no	—
55	D₅	—	4	2	no	—
56	Q₈	—	25	2	no	—
57	D₄	—	10	2	no	—
58	D₄	—	25	2	no	—
59	C₂×C₄	—	25	1	no	—
60	C₂×C₄	—	50	2	no	—
61	C₈	—	25	1	no	—
62	C₈	—	25	1	no	—
63	C₅	—	2	2	no	—
64	C₅	—	2	1	no	—
65	C₂²	—	10	2	no	—
66	C₂²	D₅²⋊C₂	1	1	yes	characteristic
67	C₄	—	10	2	no	—
68	C₄	—	25	1	no	—
69	C₄	—	25	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	20	2	no	—
72	C₂	C₂×D₅²⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅²⋊C₂
commutator, cosocle	#1039	C₂³
Fitting	C₂²×C₅²	D₄
socle	C₂×C₅²	C₂×D₄
hypercenter	C₂²	D₅²⋊C₂
hypocenter	C₅²	#135
LCS	G ▷ #1039 ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. #87880	C₂	semidirect
2. C₂×#6034	C₂	semidirect
3. #87839	C₂	semidirect
4. #87838	C₂	semidirect
5. #87837	C₂	semidirect
34. C₅²	#135	semidirect

Maximal subgroups

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

subgroup	index
#87880	2
C₂×#6034	2
#87839	2
#87838	2
#87837	2
#135	25