Subgroups — #11716498

Order 800.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₂₅	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂₅	C₂²	1	2	yes	—
6	D₄×C₂₅	—	2	2	no	—
7	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
8	C₂×Dic₂₅	C₂²	1	2	yes	—
9	C₂×Dic₂₅	C₂²	1	2	yes	—
10	Dic₅₀	—	2	2	no	—
11	#4082	—	5	1	no	maximal
12	C₂²×C₂₅	—	2	2	no	—
13	C₂²×C₂₅	—	2	4	no	—
14	C₂²×C₂₅	C₂³	1	1	yes	characteristic
15	C₄×C₂₅	D₄	1	2	yes	—
16	Dic₂₅	—	2	2	no	—
17	Dic₂₅	—	2	2	no	—
18	C₂×C₅×D₄	D₅	1	1	yes	characteristic
19	C₂×Dic₁₀	—	5	1	no	—
20	C₂²⋊Dic₅	—	5	4	no	—
21	C₄×Dic₅	—	5	1	no	—
22	C₂×C₂₅	—	4	2	no	—
23	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
24	C₂×C₂₅	#41	1	2	yes	—
25	C₂³×C₅	C₂×D₅	1	2	yes	—
26	C₅×D₄	—	2	2	no	—
27	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
28	C₂×Dic₅	—	5	2	no	—
29	C₂×Dic₅	—	5	2	no	—
30	Dic₁₀	—	10	2	no	—
31	#124	—	25	1	no	Sylow
32	C₂₅	#124	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₅	—	2	4	no	—
34	C₂²×C₅	—	2	2	no	—
35	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
36	C₄×C₅	C₂²⋊D₅	1	2	yes	—
37	Dic₅	—	10	2	no	—
38	Dic₅	—	10	2	no	—
39	C₂×Q₈	—	25	1	no	—
40	C₂×D₄	D₂₅	1	1	yes	characteristic
41	C₂²⋊C₄	—	25	4	no	—
42	C₄²	—	25	1	no	—
43	C₂×C₅	—	4	2	no	—
44	C₂×C₅	#711	1	2	yes	—
45	C₂×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
46	C₂³	C₂×D₂₅	1	2	yes	—
47	Q₈	—	50	2	no	—
48	D₄	—	2	2	no	—
49	C₂×C₄	—	25	2	no	—
50	C₂×C₄	—	25	2	no	—
51	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
52	C₅	#4082	1	1	yes	characteristic
53	C₂²	—	2	4	no	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂²×D₂₅	1	1	yes	characteristic
56	C₄	—	50	2	no	—
57	C₄	—	50	2	no	—
58	C₄	C₂²⋊D₂₅	1	2	yes	—
59	C₂	—	4	2	no	—
60	C₂	#87746	1	2	yes	—
61	C₂	C₂×C₂²⋊D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂³
Frattini, socle	C₂²×C₅	C₂²×D₅
Fitting	C₂×D₄×C₂₅	C₂
prime radical	C₂×C₅×D₄	D₅
hypercenter	C₂×D₄	D₂₅
hypocenter	C₂₅	#124
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅₀	C₂	semidirect
3. C₂²⋊Dic₂₅	C₂	semidirect
4. C₄×Dic₂₅	C₂	semidirect
32. C₂₅	#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₂₅	2
C₂×Dic₅₀	2
C₂²⋊Dic₂₅	2
C₄×Dic₂₅	2
#4082	5