Subgroups — #11716500

Order 800.

141 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, complemented
2	C₂×C₂²⋊D₂₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₂₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₂₅	C₂	1	2	yes	maximal, complemented
5	C₄⋊Dic₂₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂₅	C₂²	1	2	yes	—
7	C₂²×D₂₅	C₂²	1	1	yes	characteristic
8	D₄×C₂₅	—	2	2	no	—
9	C₂×C₄×C₂₅	C₂²	1	1	yes	characteristic
10	C₂²⋊D₂₅	—	2	4	no	—
11	C₂×Dic₂₅	C₂²	1	2	yes	—
12	C₂×Dic₂₅	C₂²	1	1	yes	characteristic
13	C₄×D₂₅	—	2	2	no	—
14	#4084	—	5	1	no	maximal
15	C₂²×C₂₅	—	2	2	no	—
16	C₂²×C₂₅	—	2	2	no	—
17	C₂²×C₂₅	—	2	2	no	—
18	C₂²×C₂₅	C₂³	1	1	yes	characteristic
19	C₂×D₂₅	—	2	1	no	—
20	C₂×D₂₅	—	2	1	no	—
21	C₂×D₂₅	D₄	1	2	yes	complemented
22	C₄×C₂₅	D₄	1	2	yes	complemented
23	Dic₂₅	—	2	1	no	—
24	Dic₂₅	—	2	2	no	—
25	C₂×C₅×D₄	D₅	1	1	yes	characteristic
26	C₂×C₂²⋊D₅	—	5	2	no	—
27	C₂×C₄×D₅	—	5	1	no	—
28	C₂²⋊Dic₅	—	5	2	no	—
29	C₄⋊Dic₅	—	5	1	no	—
30	C₂×C₂₅	—	4	2	no	—
31	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
32	C₂×C₂₅	C₂×D₄	1	1	yes	characteristic
33	C₂×C₂₅	#41	1	1	yes	characteristic
34	D₂₅	—	2	2	no	—
35	C₂³×C₅	C₂×D₅	1	2	yes	—
36	C₂²×D₅	—	5	1	no	—
37	C₅×D₄	—	2	2	no	—
38	C₂×C₄×C₅	C₂×D₅	1	1	yes	characteristic
39	C₂²⋊D₅	—	10	4	no	—
40	C₂×Dic₅	—	5	2	no	—
41	C₂×Dic₅	—	5	1	no	—
42	C₄×D₅	—	10	2	no	—
43	#121	—	25	1	no	Sylow
44	C₂₅	#121	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₅	—	2	2	no	—
46	C₂²×C₅	—	2	2	no	—
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
49	C₂×D₅	—	5	2	no	—
50	C₂×D₅	—	10	1	no	—
51	C₂×D₅	—	10	1	no	—
52	C₄×C₅	C₂²⋊D₅	1	2	yes	—
53	Dic₅	—	10	2	no	—
54	Dic₅	—	10	1	no	—
55	C₂×D₄	—	25	2	no	—
56	C₂×D₄	D₂₅	1	1	yes	characteristic, complemented
57	C₂²×C₄	—	25	1	no	—
58	C₄⋊C₄	—	25	1	no	—
59	C₂²⋊C₄	—	25	2	no	—
60	C₂×C₅	—	4	2	no	—
61	C₂×C₅	D₄×D₅	1	1	yes	characteristic
62	C₂×C₅	#711	1	1	yes	characteristic
63	C₂×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
64	D₅	—	10	2	no	—
65	C₂³	—	25	1	no	—
66	C₂³	C₂×D₂₅	1	2	yes	—
67	D₄	—	2	2	no	—
68	D₄	—	50	4	no	—
69	C₂×C₄	—	25	1	no	—
70	C₂×C₄	—	25	2	no	—
71	C₂×C₄	—	50	2	no	—
72	C₂×C₄	C₂×D₂₅	1	1	yes	characteristic
73	C₅	#4084	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	25	2	no	—
78	C₂²	—	50	1	no	—
79	C₂²	—	50	1	no	—
80	C₂²	C₂²×D₂₅	1	1	yes	characteristic
81	C₄	—	50	2	no	—
82	C₄	—	50	1	no	—
83	C₄	C₂²⋊D₂₅	1	2	yes	complemented
84	C₂	—	4	2	no	—
85	C₂	—	50	2	no	—
86	C₂	D₄×D₂₅	1	1	yes	characteristic
87	C₂	#87746	1	1	yes	characteristic
88	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
hypercenter	C₂×D₄	D₂₅
hypocenter	C₂₅	#121
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
1. C₂×D₄×C₂₅	C₂	semidirect
2. C₂×C₂²⋊D₂₅	C₂	semidirect
3. C₂×C₄×D₂₅	C₂	semidirect
4. C₂²⋊Dic₂₅	C₂	semidirect
5. C₄⋊Dic₂₅	C₂	semidirect
21. C₂×D₂₅	D₄	semidirect
22. C₄×C₂₅	D₄	semidirect
44. C₂₅	#121	semidirect
56. C₂×D₄	D₂₅	semidirect
83. C₄	C₂²⋊D₂₅	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×C₂²⋊D₂₅	2
C₂×C₄×D₂₅	2
C₂²⋊Dic₂₅	2
C₄⋊Dic₂₅	2
#4084	5