Subgroups — #11716488

Order 800.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Dic₂₅	C₂	1	2	yes	maximal
3	C₄⋊Dic₂₅	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₂₅	C₂	1	2	yes	maximal, complemented
5	C₂³×C₂₅	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₂₅	C₄	1	4	yes	complemented
7	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₅	C₂²	1	4	yes	—
10	#4072	—	5	1	no	maximal
11	C₂²×C₂₅	—	2	2	no	—
12	C₂²×C₂₅	C₂×C₄	1	2	yes	—
13	C₂²×C₂₅	C₂³	1	1	yes	characteristic
14	C₄×C₂₅	C₂×C₄	1	2	yes	—
15	C₄×C₂₅	C₂×C₄	1	2	yes	—
16	Dic₂₅	—	2	4	no	—
17	C₂²×C₄×C₅	D₅	1	1	yes	characteristic
18	C₂²⋊Dic₅	—	5	2	no	—
19	C₄⋊Dic₅	—	5	2	no	—
20	C₄×Dic₅	—	5	2	no	—
21	C₂×C₂₅	—	2	2	no	—
22	C₂×C₂₅	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₂₅	#41	1	2	yes	—
24	C₂³×C₅	C₂×D₅	1	1	yes	characteristic
25	C₂×C₄×C₅	Dic₅	1	4	yes	—
26	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
27	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
28	C₂×Dic₅	—	5	4	no	—
29	#117	—	25	1	no	Sylow
30	C₂₅	#117	1	1	yes	characteristic, complemented, Sylow
31	C₂²×C₅	—	2	2	no	—
32	C₂²×C₅	C₂×Dic₅	1	2	yes	—
33	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
34	C₄×C₅	C₂×Dic₅	1	2	yes	—
35	C₄×C₅	C₂×Dic₅	1	2	yes	—
36	Dic₅	—	10	4	no	—
37	C₂²×C₄	D₂₅	1	1	yes	characteristic
38	C₄⋊C₄	—	25	2	no	—
39	C₂²⋊C₄	—	25	2	no	—
40	C₄²	—	25	2	no	—
41	C₂×C₅	—	2	2	no	—
42	C₂×C₅	#709	1	2	yes	—
43	C₂×C₅	C₂²×Dic₅	1	1	yes	characteristic
44	C₂³	C₂×D₂₅	1	1	yes	characteristic
45	C₂×C₄	—	25	4	no	—
46	C₂×C₄	Dic₂₅	1	4	yes	complemented
47	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
48	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
49	C₅	#4072	1	1	yes	characteristic
50	C₂²	—	2	2	no	—
51	C₂²	C₂×Dic₂₅	1	2	yes	—
52	C₂²	C₂²×D₂₅	1	1	yes	characteristic
53	C₄	—	50	4	no	—
54	C₄	C₂×Dic₂₅	1	2	yes	—
55	C₄	C₂×Dic₂₅	1	2	yes	—
56	C₂	—	2	2	no	—
57	C₂	#87744	1	2	yes	—
58	C₂	C₂²×Dic₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₅
commutator	C₂×C₂₅	C₂²×C₄
Frattini, socle	C₂²×C₅	C₂²×D₅
Fitting	C₂²×C₄×C₂₅	C₂
prime radical	C₂³×C₅	C₂×D₅
cosocle	C₂²×C₂₅	C₂³
hypercenter	C₂²×C₄	D₂₅
hypocenter	C₂₅	#117
LCS	G ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊Dic₂₅	C₂	semidirect
4. C₄×Dic₂₅	C₂	semidirect
6. C₂×C₄×C₂₅	C₄	semidirect
30. C₂₅	#117	semidirect
46. C₂×C₄	Dic₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₂₅	2
C₂²⋊Dic₂₅	2
C₄⋊Dic₂₅	2
C₄×Dic₂₅	2
#4072	5