Subgroups — #11716459

Order 800.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#87720	C₂	1	2	yes	maximal
5	C₄⋊Dic₂₅	C₂	1	2	yes	maximal, complemented
6	C₂²×D₂₅	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₂₅	C₂²	1	2	yes	—
8	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₅	C₂²	1	2	yes	—
11	C₄×D₂₅	—	2	2	no	—
12	Dic₅₀	—	2	2	no	—
13	#4043	—	5	1	no	maximal
14	C₂²×C₂₅	C₂³	1	1	yes	characteristic
15	C₂×D₂₅	—	2	1	no	—
16	C₂×D₂₅	—	2	1	no	—
17	C₂×D₂₅	Q₈	1	2	yes	complemented
18	C₄×C₂₅	—	2	2	no	—
19	C₄×C₂₅	D₄	1	2	yes	—
20	Dic₂₅	—	2	2	no	—
21	Dic₂₅	—	2	1	no	—
22	C₂×C₄×D₅	—	5	1	no	—
23	C₂×Dic₁₀	—	5	1	no	—
24	C₅×C₄⋊C₄	D₅	1	1	yes	characteristic
25	#685	—	5	2	no	—
26	C₄⋊Dic₅	—	5	2	no	—
27	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
28	C₂×C₂₅	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₂₅	#41	1	1	yes	characteristic
30	D₂₅	—	2	2	no	—
31	C₂²×D₅	—	5	1	no	—
32	C₂×C₄×C₅	C₂×D₅	1	2	yes	—
33	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
34	C₂×Dic₅	—	5	1	no	—
35	C₂×Dic₅	—	5	2	no	—
36	C₄×D₅	—	10	2	no	—
37	Dic₁₀	—	10	2	no	—
38	C₂²⋊Q₈	—	25	1	no	Sylow
39	C₂₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
41	C₂×D₅	—	5	2	no	—
42	C₂×D₅	—	10	1	no	—
43	C₂×D₅	—	10	1	no	—
44	C₄×C₅	—	2	2	no	—
45	C₄×C₅	D₂₀	1	2	yes	—
46	Dic₅	—	10	2	no	—
47	Dic₅	—	10	1	no	—
48	C₂×Q₈	—	25	1	no	—
49	C₂²×C₄	—	25	1	no	—
50	C₄⋊C₄	—	25	2	no	—
51	C₄⋊C₄	D₂₅	1	1	yes	characteristic, complemented
52	C₂²⋊C₄	—	25	2	no	—
53	C₂×C₅	C₂×D₂₀	1	1	yes	characteristic
54	C₂×C₅	#711	1	1	yes	characteristic
55	C₂×C₅	Q₈×D₅	1	1	yes	characteristic
56	D₅	—	10	2	no	—
57	C₂³	—	25	1	no	—
58	Q₈	—	50	2	no	—
59	C₂×C₄	—	25	2	no	—
60	C₂×C₄	—	25	1	no	—
61	C₂×C₄	—	50	2	no	—
62	C₂×C₄	C₂×D₂₅	1	2	yes	—
63	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
64	C₅	#4043	1	1	yes	characteristic
65	C₂²	—	25	2	no	—
66	C₂²	—	50	1	no	—
67	C₂²	—	50	1	no	—
68	C₂²	C₂²×D₂₅	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	50	1	no	—
71	C₄	—	50	2	no	—
72	C₄	D₁₀₀	1	2	yes	—
73	C₂	—	50	2	no	—
74	C₂	C₂×D₁₀₀	1	1	yes	characteristic
75	C₂	#87746	1	1	yes	characteristic
76	C₂	Q₈×D₂₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₅
commutator, cosocle	C₂²×C₂₅	C₂³
Frattini, socle	C₂²×C₅	C₂²×D₅
Fitting	C₄⋊C₄×C₂₅	C₂
prime radical	C₂²×D₂₅	C₂²
hypercenter	C₄⋊C₄	D₂₅
hypocenter	C₂₅	C₂²⋊Q₈
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅₀	C₂	semidirect
3. C₄⋊C₄×C₂₅	C₂	semidirect
5. C₄⋊Dic₂₅	C₂	semidirect
17. C₂×D₂₅	Q₈	semidirect
39. C₂₅	C₂²⋊Q₈	semidirect
51. C₄⋊C₄	D₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₂₅	2
C₂×Dic₅₀	2
C₄⋊C₄×C₂₅	2
#87720	2
C₄⋊Dic₂₅	2
#4043	5