Subgroups — #13098345

Order 1600.

243 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, complemented
2	C₂×#87722	C₂	1	4	yes	maximal, complemented
3	#11716436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6014	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₂₅	—	2	2	no	—
6	C₂×D₄×C₂₅	C₂²	1	2	yes	—
7	C₂×D₁₀₀	—	2	2	no	—
8	C₂×D₁₀₀	C₂²	1	2	yes	—
9	C₄²×C₂₅	C₂²	1	1	yes	characteristic
10	#87722	—	2	8	no	—
11	C₂×#6014	C₂²	1	2	yes	—
12	#65362	—	5	1	no	maximal
13	C₂³×C₂₅	—	2	2	no	—
14	C₂²×D₂₅	—	2	2	no	—
15	D₄×C₂₅	—	2	4	no	—
16	D₄×C₂₅	—	4	4	no	—
17	C₂×C₄×C₂₅	D₄	1	2	yes	—
18	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
19	D₁₀₀	—	2	4	no	—
20	D₁₀₀	—	4	4	no	—
21	#6014	D₄	1	4	yes	complemented
22	C₅×#127	D₅	1	1	yes	characteristic
23	C₂×#686	—	5	4	no	—
24	#4020	—	5	1	no	—
25	C₄×#168	—	5	1	no	—
26	C₂²×C₂₅	—	4	2	no	—
27	C₂²×C₂₅	—	4	4	no	—
28	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
29	C₂×D₂₅	—	4	2	no	—
30	C₂×D₂₅	—	4	4	no	—
31	C₄×C₂₅	D₈	1	4	yes	complemented
32	C₄×C₂₅	C₂×D₄	1	2	yes	—
33	C₂×C₅×D₄	—	2	2	no	—
34	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
35	C₂×D₂₀	—	5	2	no	—
36	C₂×D₂₀	—	10	2	no	—
37	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
38	#686	—	10	8	no	—
39	C₂×#168	—	5	2	no	—
40	#493	—	25	1	no	Sylow
41	C₂×C₂₅	—	8	2	no	—
42	C₂×C₂₅	#127	1	1	yes	characteristic
43	C₂×C₂₅	C₂×D₈	1	2	yes	—
44	D₂₅	—	8	2	no	—
45	C₂³×C₅	—	2	2	no	—
46	C₂²×D₅	—	10	2	no	—
47	C₅×D₄	—	2	4	no	—
48	C₅×D₄	—	4	4	no	—
49	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
51	D₂₀	—	10	4	no	—
52	D₂₀	—	20	4	no	—
53	#168	—	5	4	no	—
54	C₂×D₈	—	25	4	no	—
55	#127	—	25	1	no	—
56	#127	D₂₅	1	1	yes	characteristic, complemented
57	C₄×C₈	—	25	1	no	—
58	C₂₅	#493	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	—	4	4	no	—
60	C₂²×C₅	—	4	2	no	—
61	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
62	C₂×D₅	—	20	4	no	—
63	C₂×D₅	—	20	2	no	—
64	C₄×C₅	#686	1	4	yes	—
65	C₄×C₅	D₄×D₅	1	2	yes	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	25	2	no	—
68	C₂×D₄	—	50	2	no	—
69	C₂×D₄	C₂×D₂₅	1	2	yes	—
70	D₈	—	50	8	no	—
71	C₂×C₈	—	25	2	no	—
72	C₄²	C₂×D₂₅	1	1	yes	characteristic
73	C₂×C₅	—	8	2	no	—
74	C₂×C₅	C₂×#686	1	2	yes	—
75	C₂×C₅	#4086	1	1	yes	characteristic
76	D₅	—	40	2	no	—
77	C₂³	—	2	2	no	—
78	C₂³	—	50	2	no	—
79	D₄	—	2	4	no	—
80	D₄	—	4	4	no	—
81	D₄	—	50	4	no	—
82	D₄	—	100	4	no	—
83	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
84	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
85	C₈	—	25	4	no	—
86	C₅	#65362	1	1	yes	characteristic
87	C₂²	—	4	4	no	—
88	C₂²	—	4	2	no	—
89	C₂²	—	100	4	no	—
90	C₂²	—	100	2	no	—
91	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
92	C₄	#87722	1	4	yes	complemented
93	C₄	D₄×D₂₅	1	2	yes	—
94	C₂	—	8	2	no	—
95	C₂	—	200	2	no	—
96	C₂	C₂×#87722	1	2	yes	—
97	C₂	#11716502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#127	C₂
socle	C₂²×C₅	C₂×C₂²⋊D₅
hypercenter	#127	D₂₅
hypocenter	C₂₅	#493
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂₅×#127	C₂	semidirect
2. C₂×#87722	C₂	semidirect
3. #11716436	C₂	semidirect
4. C₄×#6014	C₂	semidirect
21. #6014	D₄	semidirect
31. C₄×C₂₅	D₈	semidirect
56. #127	D₂₅	semidirect
58. C₂₅	#493	semidirect
92. C₄	#87722	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
C₂×#87722	2
#11716436	2
C₄×#6014	2
#65362	5