Subgroups — #11716497

Order 800.

117 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	C₂²×Dic₂₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₂₅	C₂	1	2	yes	maximal, complemented
5	#87718	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂₅	C₂²	1	1	yes	characteristic
7	C₂³×C₂₅	C₂²	1	1	yes	characteristic
8	D₄×C₂₅	—	2	2	no	—
9	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₅	—	2	2	no	—
11	C₂×Dic₂₅	C₂²	1	2	yes	—
12	C₂×Dic₂₅	C₂²	1	2	yes	—
13	#4081	—	5	1	no	maximal
14	C₂²×C₂₅	—	2	2	no	—
15	C₂²×C₂₅	—	2	1	no	—
16	C₂²×C₂₅	—	2	2	no	—
17	C₂²×C₂₅	D₄	1	2	yes	—
18	C₂²×C₂₅	C₂³	1	1	yes	characteristic
19	C₄×C₂₅	—	2	1	no	—
20	Dic₂₅	—	2	2	no	—
21	Dic₂₅	—	2	2	no	—
22	C₂×C₅×D₄	D₅	1	1	yes	characteristic
23	C₂²×Dic₅	—	5	1	no	—
24	C₂²⋊Dic₅	—	5	1	no	—
25	C₂²⋊Dic₅	—	5	2	no	—
26	#683	—	5	2	no	—
27	C₂×C₂₅	—	2	2	no	—
28	C₂×C₂₅	—	4	1	no	—
29	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
30	C₂×C₂₅	#41	1	2	yes	—
31	C₂³×C₅	C₂×D₅	1	1	yes	characteristic
32	C₂³×C₅	C₂×D₅	1	1	yes	characteristic
33	C₅×D₄	—	2	2	no	—
34	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
35	C₂×Dic₅	—	5	2	no	—
36	C₂×Dic₅	—	5	2	no	—
37	C₂×Dic₅	—	10	2	no	—
38	#123	—	25	1	no	Sylow
39	C₂₅	#123	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₅	—	2	2	no	—
41	C₂²×C₅	—	2	2	no	—
42	C₂²×C₅	—	2	1	no	—
43	C₂²×C₅	C₂²⋊D₅	1	2	yes	—
44	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
45	C₄×C₅	—	2	1	no	—
46	Dic₅	—	10	2	no	—
47	Dic₅	—	10	2	no	—
48	C₂×D₄	D₂₅	1	1	yes	characteristic
49	C₂²×C₄	—	25	1	no	—
50	C₄⋊C₄	—	25	2	no	—
51	C₂²⋊C₄	—	25	1	no	—
52	C₂²⋊C₄	—	25	2	no	—
53	C₂×C₅	—	2	2	no	—
54	C₂×C₅	—	4	1	no	—
55	C₂×C₅	#711	1	2	yes	—
56	C₂×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
57	C₂³	C₂×D₂₅	1	1	yes	characteristic
58	C₂³	C₂×D₂₅	1	1	yes	characteristic
59	D₄	—	2	2	no	—
60	C₂×C₄	—	25	2	no	—
61	C₂×C₄	—	25	2	no	—
62	C₂×C₄	—	50	2	no	—
63	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
64	C₅	#4081	1	1	yes	characteristic
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	1	no	—
68	C₂²	C₂²⋊D₂₅	1	2	yes	—
69	C₂²	C₂²×D₂₅	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	50	2	no	—
72	C₄	—	50	2	no	—
73	C₂	—	2	2	no	—
74	C₂	—	4	1	no	—
75	C₂	#87746	1	2	yes	—
76	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	#123
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. C₂²×Dic₂₅	C₂	semidirect
3. C₂²⋊Dic₂₅	C₂	semidirect
4. C₂²⋊Dic₂₅	C₂	semidirect
5. #87718	C₂	semidirect
39. C₂₅	#123	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
C₂²×Dic₂₅	2
C₂²⋊Dic₂₅	2
#87718	2
#4081	5