Subgroups — #13097909

Order 1600.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716502	C₂	1	1	yes	maximal, characteristic
2	#11716419	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×C₂₅	C₄	1	1	yes	characteristic
4	C₂×C₂²⋊D₂₅	—	2	2	no	—
5	C₂×D₁₀₀	C₄	1	1	yes	characteristic
6	C₂×#6016	—	2	2	no	—
7	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
8	#64925	—	5	1	no	maximal
9	C₂³×C₂₅	—	2	1	no	—
10	C₂²×D₂₅	—	2	1	no	—
11	D₄×C₂₅	—	4	1	no	—
12	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
13	C₂²⋊D₂₅	—	4	2	no	—
14	C₂²⋊D₂₅	—	4	2	no	—
15	C₂×Dic₂₅	D₄	1	2	yes	—
16	D₁₀₀	—	4	1	no	—
17	#6016	—	4	2	no	—
18	#4086	—	5	1	no	—
19	#4004	—	5	2	no	—
20	C₂²×C₂₅	—	4	1	no	—
21	C₂²×C₂₅	—	4	2	no	—
22	C₂²×C₂₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₂₅	—	4	1	no	—
24	C₂×D₂₅	—	4	2	no	—
25	C₄×C₂₅	—	2	1	no	—
26	Dic₂₅	D₈	1	2	yes	—
27	Dic₂₅	SD₁₆	1	2	yes	—
28	C₂×C₅×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
29	C₂×C₂²⋊D₅	—	10	2	no	—
30	C₂×D₂₀	—	5	1	no	—
31	C₂×#170	—	10	2	no	—
32	C₄×Dic₅	—	5	1	no	—
33	#331	—	25	1	no	Sylow
34	C₂×C₂₅	—	8	1	no	—
35	C₂×C₂₅	#100	1	1	yes	characteristic
36	C₂×C₂₅	#102	1	2	yes	—
37	D₂₅	—	8	1	no	—
38	C₂³×C₅	—	2	1	no	—
39	C₂²×D₅	—	10	1	no	—
40	C₅×D₄	—	4	1	no	—
41	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂²⋊D₅	—	20	2	no	—
43	C₂²⋊D₅	—	20	2	no	—
44	C₂×Dic₅	—	5	2	no	—
45	D₂₀	—	20	1	no	—
46	#170	—	20	2	no	—
47	#127	—	25	1	no	—
48	C₄⋊C₈	—	25	2	no	—
49	C₂₅	#331	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₅	—	4	2	no	—
51	C₂²×C₅	—	4	1	no	—
52	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂×D₅	—	20	2	no	—
54	C₂×D₅	—	20	1	no	—
55	C₄×C₅	—	2	1	no	—
56	Dic₅	—	5	2	no	—
57	Dic₅	—	5	2	no	—
58	C₂×D₄	—	25	1	no	—
59	C₂×D₄	—	50	2	no	—
60	C₂×D₄	#1035	1	1	yes	characteristic
61	C₂×C₈	—	50	2	no	—
62	C₄²	—	25	1	no	—
63	C₂×C₅	—	8	1	no	—
64	C₂×C₅	#4007	1	2	yes	—
65	C₂×C₅	#4013	1	1	yes	characteristic
66	D₅	—	40	1	no	—
67	C₂³	—	2	1	no	—
68	C₂³	—	50	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	100	1	no	—
71	D₄	—	100	2	no	—
72	D₄	—	100	2	no	—
73	C₂×C₄	—	25	2	no	—
74	C₂×C₄	C₂×#1035	1	1	yes	characteristic
75	C₈	—	100	2	no	—
76	C₅	#64925	1	1	yes	characteristic
77	C₂²	—	4	2	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	100	2	no	—
80	C₂²	—	100	1	no	—
81	C₂²	C₂²⋊#1035	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	25	2	no	—
84	C₄	—	25	2	no	—
85	C₂	—	8	1	no	—
86	C₂	—	200	1	no	—
87	C₂	#11716424	1	2	yes	—
88	C₂	#11716428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#1035
commutator	C₂×C₄×C₂₅	C₂×C₄
Frattini	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)
Fitting	C₂×D₄×C₂₅	C₄
socle	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)
prime radical	#11716502	C₂
cosocle	C₄×Dic₂₅	C₂²
hypercenter	C₂×D₄	#1035
hypocenter	C₂₅	#331
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. #11716419	C₂	semidirect
49. C₂₅	#331	semidirect

Maximal subgroups

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

subgroup	index
#11716502	2
#11716419	2
#64925	5