Subgroups — #13098343

Order 1600.

165 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, complemented
2	#11716436	C₂	1	1	yes	maximal, characteristic
3	#11716358	C₂	1	4	yes	maximal, complemented
4	#11716351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₀₀	—	2	2	no	—
6	C₂×D₁₀₀	C₂²	1	2	yes	—
7	C₄⋊C₄×C₂₅	—	2	2	no	—
8	C₄⋊C₄×C₂₅	C₂²	1	2	yes	—
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	C₂×#6014	C₂²	1	2	yes	—
11	#65356	—	5	1	no	maximal
12	C₂²×D₂₅	—	2	2	no	—
13	C₂×C₄×C₂₅	—	2	2	no	—
14	C₂×C₄×C₂₅	D₄	1	2	yes	—
15	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
16	D₁₀₀	—	2	4	no	—
17	D₁₀₀	—	4	4	no	—
18	#6014	—	2	2	no	—
19	C₅×#125	D₅	1	1	yes	characteristic
20	#4020	—	5	1	no	—
21	#3941	—	5	4	no	—
22	#3935	—	5	1	no	—
23	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₅	—	4	2	no	—
25	C₂×D₂₅	—	4	4	no	—
26	C₄×C₂₅	—	2	2	no	—
27	C₄×C₂₅	—	4	2	no	—
28	C₄×C₂₅	#41	1	2	yes	—
29	C₂×D₂₀	—	5	2	no	—
30	C₂×D₂₀	—	10	2	no	—
31	C₅×C₄⋊C₄	—	2	2	no	—
32	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
33	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
34	C₂×#168	—	5	2	no	—
35	#490	—	25	1	no	Sylow
36	C₂×C₂₅	#124	1	1	yes	characteristic
37	C₂×C₂₅	AGL(1,ℤ/8)	1	2	yes	—
38	D₂₅	—	8	2	no	—
39	C₂²×D₅	—	10	2	no	—
40	C₂×C₄×C₅	—	2	2	no	—
41	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
42	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
43	D₂₀	—	10	4	no	—
44	D₂₀	—	20	4	no	—
45	#168	—	10	2	no	—
46	#127	—	25	1	no	—
47	#125	D₂₅	1	1	yes	characteristic, complemented
48	#102	—	25	4	no	—
49	#97	—	25	1	no	—
50	C₂₅	#490	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
52	C₂×D₅	—	20	2	no	—
53	C₂×D₅	—	20	4	no	—
54	C₄×C₅	—	2	2	no	—
55	C₄×C₅	—	4	2	no	—
56	C₄×C₅	#713	1	2	yes	—
57	C₂×D₄	—	25	2	no	—
58	C₂×D₄	—	50	2	no	—
59	C₂×C₈	—	25	2	no	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	C₂×D₂₅	1	2	yes	—
62	C₄²	C₂×D₂₅	1	1	yes	characteristic
63	C₂×C₅	#4093	1	1	yes	characteristic
64	C₂×C₅	#4095	1	2	yes	—
65	D₅	—	40	2	no	—
66	C₂³	—	50	2	no	—
67	D₄	—	50	4	no	—
68	D₄	—	100	4	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
71	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
72	C₈	—	50	2	no	—
73	C₅	#65356	1	1	yes	characteristic
74	C₂²	—	100	2	no	—
75	C₂²	—	100	4	no	—
76	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	4	2	no	—
79	C₄	#87748	1	2	yes	—
80	C₂	—	200	2	no	—
81	C₂	#11716509	1	1	yes	characteristic
82	C₂	#11716512	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	C₂²×C₅	C₂×C₂²⋊D₅
prime radical	#11716436	C₂
hypercenter	#125	D₂₅
hypocenter	C₂₅	#490
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
1. C₂₅×#125	C₂	semidirect
3. #11716358	C₂	semidirect
4. #11716351	C₂	semidirect
47. #125	D₂₅	semidirect
50. C₂₅	#490	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
#11716436	2
#11716358	2
#11716351	2
#65356	5