Subgroups — #13097751

Order 1600.

129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#127	C₂	1	1	yes	maximal, characteristic
2	#11716382	C₂	1	2	yes	maximal, complemented
3	C₂×D₄×C₂₅	—	2	2	no	—
4	C₂×D₄×C₂₅	C₄	1	1	yes	characteristic, complemented
5	C₂×D₄×C₂₅	C₂²	1	1	yes	characteristic
6	C₄²×C₂₅	C₄	1	1	yes	characteristic, complemented
7	C₂²⋊Dic₂₅	—	2	2	no	—
8	#64765	—	5	1	no	maximal
9	C₂³×C₂₅	—	2	1	no	—
10	C₂³×C₂₅	D₄	1	2	yes	—
11	D₄×C₂₅	—	4	1	no	—
12	D₄×C₂₅	—	4	4	no	—
13	D₄×C₂₅	—	4	1	no	—
14	C₂×C₄×C₂₅	—	2	1	no	—
15	C₂×C₄×C₂₅	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₅	—	4	2	no	—
17	C₅×#127	D₅	1	1	yes	characteristic
18	#3966	—	5	2	no	—
19	C₂²×C₂₅	—	2	2	no	—
20	C₂²×C₂₅	—	4	2	no	—
21	C₂²×C₂₅	—	4	2	no	—
22	C₂²×C₂₅	—	4	1	no	—
23	C₂²×C₂₅	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₂₅	—	2	2	no	—
25	C₄×C₂₅	—	2	1	no	—
26	Dic₂₅	—	8	2	no	—
27	C₂×C₅×D₄	—	2	2	no	—
28	C₂×C₅×D₄	Dic₅	1	1	yes	characteristic
29	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
30	C₄²×C₅	Dic₅	1	1	yes	characteristic
31	C₂²⋊Dic₅	—	10	2	no	—
32	#353	—	25	1	no	Sylow
33	C₂×C₂₅	—	2	1	no	—
34	C₂×C₂₅	—	4	2	no	—
35	C₂×C₂₅	—	8	1	no	—
36	C₂×C₂₅	#99	1	1	yes	characteristic
37	C₂³×C₅	—	2	1	no	—
38	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
39	C₅×D₄	—	4	1	no	—
40	C₅×D₄	—	4	1	no	—
41	C₅×D₄	—	4	4	no	—
42	C₂×C₄×C₅	—	2	1	no	—
43	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
44	C₂×Dic₅	—	20	2	no	—
45	#127	D₂₅	1	1	yes	characteristic
46	#99	—	25	2	no	—
47	C₂₅	#353	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₅	—	2	2	no	—
49	C₂²×C₅	—	4	2	no	—
50	C₂²×C₅	—	4	2	no	—
51	C₂²×C₅	—	4	1	no	—
52	C₂²×C₅	C₂²⋊Dic₅	1	1	yes	characteristic
53	C₄×C₅	—	2	2	no	—
54	C₄×C₅	—	2	1	no	—
55	Dic₅	—	40	2	no	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	Dic₂₅	1	1	yes	characteristic, complemented
58	C₂×D₄	C₂×D₂₅	1	1	yes	characteristic
59	C₂²⋊C₄	—	50	2	no	—
60	C₄²	Dic₂₅	1	1	yes	characteristic, complemented
61	C₂×C₅	—	2	1	no	—
62	C₂×C₅	—	4	2	no	—
63	C₂×C₅	—	8	1	no	—
64	C₂×C₅	#3966	1	1	yes	characteristic
65	C₂³	—	2	1	no	—
66	C₂³	C₂²⋊D₂₅	1	2	yes	—
67	D₄	—	4	1	no	—
68	D₄	—	4	1	no	—
69	D₄	—	4	4	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	100	2	no	—
72	C₂×C₄	C₂×Dic₂₅	1	1	yes	characteristic
73	C₅	#64765	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	4	1	no	—
78	C₂²	C₂²⋊Dic₂₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	200	2	no	—
82	C₂	—	2	1	no	—
83	C₂	—	4	2	no	—
84	C₂	—	8	1	no	—
85	C₂	#11716382	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11716382
commutator	C₂×C₄×C₂₅	C₂×C₄
Frattini	C₂×C₅×D₄	C₂×D₅
Fitting	C₂₅×#127	C₂
socle	C₂×C₅	#3966
prime radical	C₅×#127	D₅
cosocle	C₂×D₄×C₂₅	C₂²
hypercenter	#127	D₂₅
hypocenter	C₂₅	#353
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂²×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
2. #11716382	C₂	semidirect
4. C₂×D₄×C₂₅	C₄	semidirect
6. C₄²×C₂₅	C₄	semidirect
47. C₂₅	#353	semidirect
57. C₂×D₄	Dic₂₅	semidirect
60. C₄²	Dic₂₅	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#127	2
#11716382	2
#64765	5