Subgroups — #13098341

Order 1600.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#125	C₂	1	1	yes	maximal, characteristic
2	#11716431	C₂	1	1	yes	maximal, characteristic
3	#11716357	C₂	1	4	yes	maximal
4	#11716351	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₀	C₂²	1	2	yes	—
6	C₄⋊C₄×C₂₅	—	2	2	no	—
7	C₄⋊C₄×C₂₅	C₂²	1	2	yes	—
8	C₄²×C₂₅	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₅	—	2	2	no	—
10	C₂×#6014	C₂²	1	2	yes	—
11	#65358	—	5	1	no	maximal
12	C₂×C₄×C₂₅	—	2	2	no	—
13	C₂×C₄×C₂₅	D₄	1	2	yes	—
14	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
15	C₂×Dic₂₅	—	2	2	no	—
16	Dic₅₀	—	2	4	no	—
17	#6014	—	2	2	no	—
18	C₅×#125	D₅	1	1	yes	characteristic
19	#4015	—	5	1	no	—
20	#3942	—	5	4	no	—
21	#3935	—	5	1	no	—
22	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
23	C₄×C₂₅	—	2	2	no	—
24	C₄×C₂₅	—	4	2	no	—
25	C₄×C₂₅	#41	1	2	yes	—
26	Dic₂₅	—	4	2	no	—
27	C₂×Dic₁₀	—	5	2	no	—
28	C₅×C₄⋊C₄	—	2	2	no	—
29	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
30	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
31	C₄⋊Dic₅	—	10	2	no	—
32	C₂×#168	—	5	2	no	—
33	#491	—	25	1	no	Sylow
34	C₂×C₂₅	#124	1	1	yes	characteristic
35	C₂×C₂₅	#137	1	2	yes	—
36	C₂×C₄×C₅	—	2	2	no	—
37	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
38	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
39	C₂×Dic₅	—	10	2	no	—
40	Dic₁₀	—	10	4	no	—
41	#168	—	10	2	no	—
42	C₄⋊Q₈	—	25	1	no	—
43	#125	D₂₅	1	1	yes	characteristic
44	#103	—	25	4	no	—
45	#97	—	25	1	no	—
46	C₂₅	#491	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
48	C₄×C₅	—	2	2	no	—
49	C₄×C₅	—	4	2	no	—
50	C₄×C₅	#713	1	2	yes	—
51	Dic₅	—	20	2	no	—
52	C₂×Q₈	—	25	2	no	—
53	C₂×C₈	—	25	2	no	—
54	C₄⋊C₄	—	2	2	no	—
55	C₄⋊C₄	—	50	2	no	—
56	C₄⋊C₄	C₂×D₂₅	1	2	yes	—
57	C₄²	C₂×D₂₅	1	1	yes	characteristic
58	C₂×C₅	#4093	1	1	yes	characteristic
59	C₂×C₅	#4097	1	2	yes	—
60	Q₈	—	50	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	50	2	no	—
63	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
64	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
65	C₈	—	50	2	no	—
66	C₅	#65358	1	1	yes	characteristic
67	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	4	2	no	—
70	C₄	—	100	2	no	—
71	C₄	#87748	1	2	yes	—
72	C₂	#11716509	1	1	yes	characteristic
73	C₂	#11716511	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#125	C₂
socle, prime radical	C₂²×C₅	C₂×C₂²⋊D₅
hypercenter	#125	D₂₅
hypocenter	C₂₅	#491
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
46. C₂₅	#491	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#125	2
#11716431	2
#11716357	2
#11716351	2
#65358	5