Subgroups — #13097908

Order 1600.

105 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11716498	C₂	1	1	yes	maximal, characteristic
2	#11716417	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×C₂₅	C₄	1	1	yes	characteristic
4	C₂×Dic₅₀	C₄	1	1	yes	characteristic
5	C₂×#6016	—	2	2	no	—
6	C₂²⋊Dic₂₅	—	2	2	no	—
7	C₄×Dic₂₅	C₂²	1	1	yes	characteristic
8	#64921	—	5	1	no	maximal
9	C₂³×C₂₅	—	2	1	no	—
10	D₄×C₂₅	—	4	1	no	—
11	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
12	C₂×Dic₂₅	—	2	1	no	—
13	C₂×Dic₂₅	D₄	1	2	yes	—
14	Dic₅₀	—	4	1	no	—
15	#6016	—	2	4	no	—
16	#4082	—	5	1	no	—
17	#4002	—	5	2	no	—
18	C₂²×C₂₅	—	4	1	no	—
19	C₂²×C₂₅	—	4	2	no	—
20	C₂²×C₂₅	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₅	—	2	1	no	—
22	Dic₂₅	—	2	2	no	—
23	Dic₂₅	—	4	1	no	—
24	C₂×C₅×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₂×Dic₁₀	—	5	1	no	—
26	C₂×#170	—	10	2	no	—
27	C₂²⋊Dic₅	—	10	2	no	—
28	C₄×Dic₅	—	5	1	no	—
29	#329	—	25	1	no	Sylow
30	C₂×C₂₅	—	8	1	no	—
31	C₂×C₂₅	#100	1	1	yes	characteristic
32	C₂×C₂₅	#104	1	2	yes	—
33	C₂³×C₅	—	2	1	no	—
34	C₅×D₄	—	4	1	no	—
35	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₂×Dic₅	—	5	2	no	—
37	C₂×Dic₅	—	10	1	no	—
38	Dic₁₀	—	20	1	no	—
39	#170	—	10	4	no	—
40	#124	—	25	1	no	—
41	#97	—	25	2	no	—
42	C₂₅	#329	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₅	—	4	2	no	—
44	C₂²×C₅	—	4	1	no	—
45	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₄×C₅	—	2	1	no	—
47	Dic₅	—	10	2	no	—
48	Dic₅	—	20	1	no	—
49	C₂×Q₈	—	25	1	no	—
50	C₂×D₄	#1035	1	1	yes	characteristic
51	C₂×C₈	—	50	2	no	—
52	C₂²⋊C₄	—	50	2	no	—
53	C₄²	—	25	1	no	—
54	C₂×C₅	—	8	1	no	—
55	C₂×C₅	#4008	1	2	yes	—
56	C₂×C₅	#4013	1	1	yes	characteristic
57	C₂³	—	2	1	no	—
58	Q₈	—	100	1	no	—
59	D₄	—	4	1	no	—
60	C₂×C₄	—	25	2	no	—
61	C₂×C₄	—	50	1	no	—
62	C₂×C₄	C₂×#1035	1	1	yes	characteristic
63	C₈	—	50	4	no	—
64	C₅	#64921	1	1	yes	characteristic
65	C₂²	—	4	2	no	—
66	C₂²	—	4	1	no	—
67	C₂²	C₂²⋊#1035	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	50	2	no	—
70	C₄	—	100	1	no	—
71	C₂	—	8	1	no	—
72	C₂	#11716423	1	2	yes	—
73	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	C₂×C₅×D₄	AGL(1,𝔽₅)
cosocle	C₄×Dic₂₅	C₂²
hypercenter	C₂×D₄	#1035
hypocenter	C₂₅	#329
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. #11716417	C₂	semidirect
42. C₂₅	#329	semidirect

Maximal subgroups

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

subgroup	index
#11716498	2
#11716417	2
#64921	5