Subgroups — #11717313

Order 800.

97 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#87840	C₂	1	1	yes	maximal, characteristic
2	#87836	C₂	1	2	yes	maximal
3	#87836	C₂	1	1	yes	maximal, characteristic
4	#87832	C₂	1	2	yes	maximal
5	#87830	C₂	1	1	yes	maximal, characteristic
6	C₂×#1044	C₂²	1	2	yes	—
7	C₂×#1044	C₂²	1	1	yes	characteristic
8	C₂×#1043	C₂²	1	1	yes	characteristic
9	#6036	C₂²	1	1	yes	characteristic
10	#6036	C₂²	1	2	yes	—
11	C₂×C₅⋊D₅	C₂³	1	1	yes	characteristic
12	#1044	—	2	2	no	—
13	#1044	Q₈	1	2	yes	—
14	#1043	—	2	1	no	—
15	C₅×Dic₅	—	2	1	no	—
16	C₅×Dic₅	—	2	2	no	—
17	C₄⋊AGL(1,𝔽₅)	—	10	2	no	—
18	C₄×AGL(1,𝔽₅)	—	10	1	no	—
19	C₂×C₅²	#41	1	1	yes	characteristic
20	C₅⋊D₅	C₂×Q₈	1	1	yes	characteristic
21	C₅⋊D₅	#41	1	1	yes	characteristic
22	C₂×AGL(1,𝔽₅)	—	10	1	no	—
23	C₂×AGL(1,𝔽₅)	—	10	1	no	—
24	C₂×AGL(1,𝔽₅)	—	10	2	no	—
25	C₂×AGL(1,𝔽₅)	—	10	2	no	—
26	C₄×D₅	—	10	1	no	—
27	C₄×D₅	—	10	2	no	—
28	#125	—	25	1	no	maximal, Sylow
29	C₅²	#125	1	1	yes	characteristic, complemented, Sylow
30	C₂×D₅	—	10	1	no	—
31	C₂×D₅	—	10	2	no	—
32	AGL(1,𝔽₅)	—	10	2	no	—
33	AGL(1,𝔽₅)	—	10	2	no	—
34	AGL(1,𝔽₅)	—	20	2	no	—
35	AGL(1,𝔽₅)	—	20	2	no	—
36	C₄×C₅	—	10	1	no	—
37	C₄×C₅	—	10	2	no	—
38	Dic₅	—	2	1	no	—
39	Dic₅	—	2	2	no	—
40	C₄⋊C₄	—	25	2	no	—
41	C₄⋊C₄	—	25	1	no	—
42	C₄⋊C₄	—	25	1	no	—
43	C₄⋊C₄	—	25	2	no	—
44	C₄²	—	25	1	no	—
45	C₂×C₅	—	2	1	no	—
46	C₂×C₅	—	2	2	no	—
47	D₅	—	10	1	no	—
48	D₅	—	10	2	no	—
49	D₅	—	10	1	no	—
50	D₅	—	10	2	no	—
51	C₂×C₄	—	25	1	no	—
52	C₂×C₄	—	25	2	no	—
53	C₂×C₄	—	25	1	no	—
54	C₂×C₄	—	25	1	no	—
55	C₂×C₄	—	25	2	no	—
56	C₅	—	2	1	no	—
57	C₅	—	2	2	no	—
58	C₂²	—	25	1	no	—
59	C₄	—	10	1	no	—
60	C₄	—	10	2	no	—
61	C₄	—	25	2	no	—
62	C₄	—	50	1	no	—
63	C₄	—	50	2	no	—
64	C₂	—	25	1	no	—
65	C₂	—	25	1	no	—
66	C₂	(D₅⋊AGL(1,𝔽₅))⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	(D₅⋊AGL(1,𝔽₅))⋊C₂
commutator, prime radical, cosocle	C₂×C₅⋊D₅	C₂³
Fitting, socle	C₂×C₅²	#41
hypocenter	C₅²	#125
LCS	G ▷ C₂×C₅⋊D₅ ▷ C₅²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
29. C₅²	#125	semidirect

Maximal subgroups

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

subgroup	index
#87840	2
#87836	2
#87832	2
#87830	2
#125	25