Subgroups — #13098344

Order 1600.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716437	C₂	1	1	yes	maximal, characteristic
3	#11716358	C₂	1	2	yes	maximal
4	#11716357	C₂	1	2	yes	maximal, complemented
5	C₄×#6014	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₂₅	—	2	2	no	—
9	C₄⋊C₄×C₂₅	C₂²	1	2	yes	—
10	C₄²×C₂₅	C₂²	1	1	yes	characteristic
11	#87720	—	2	2	no	—
12	C₂×#6014	C₂²	1	2	yes	—
13	#65357	—	5	1	no	maximal
14	C₂²×D₂₅	—	2	1	no	—
15	C₂×C₄×C₂₅	—	2	2	no	—
16	C₂×C₄×C₂₅	D₄	1	2	yes	—
17	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₅	—	2	1	no	—
19	D₁₀₀	—	2	2	no	—
20	Dic₅₀	—	2	2	no	—
21	#6014	—	2	2	no	—
22	C₅×#125	D₅	1	1	yes	characteristic
23	#4021	—	5	1	no	—
24	#3942	—	5	2	no	—
25	#3941	—	5	2	no	—
26	C₄×#168	—	5	1	no	—
27	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
28	C₂×D₂₅	—	4	1	no	—
29	C₂×D₂₅	—	4	2	no	—
30	C₄×C₂₅	—	2	2	no	—
31	C₄×C₂₅	—	4	2	no	—
32	C₄×C₂₅	#41	1	2	yes	—
33	Dic₂₅	—	4	1	no	—
34	C₂×D₂₀	—	5	1	no	—
35	C₂×Dic₁₀	—	5	1	no	—
36	C₅×C₄⋊C₄	—	2	2	no	—
37	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
38	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
39	#685	—	10	2	no	—
40	C₂×#168	—	5	2	no	—
41	#488	—	25	1	no	Sylow
42	C₂×C₂₅	#124	1	1	yes	characteristic
43	C₂×C₂₅	#135	1	2	yes	—
44	D₂₅	—	8	1	no	—
45	C₂²×D₅	—	10	1	no	—
46	C₂×C₄×C₅	—	2	2	no	—
47	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
49	C₂×Dic₅	—	10	1	no	—
50	D₂₀	—	10	2	no	—
51	Dic₁₀	—	10	2	no	—
52	#168	—	10	2	no	—
53	#125	D₂₅	1	1	yes	characteristic, complemented
54	#124	—	25	1	no	—
55	#103	—	25	2	no	—
56	#102	—	25	2	no	—
57	C₄×C₈	—	25	1	no	—
58	C₂₅	#488	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
60	C₂×D₅	—	20	2	no	—
61	C₂×D₅	—	20	1	no	—
62	C₄×C₅	—	2	2	no	—
63	C₄×C₅	—	4	2	no	—
64	C₄×C₅	#713	1	2	yes	—
65	Dic₅	—	20	1	no	—
66	C₂×Q₈	—	25	1	no	—
67	C₂×D₄	—	25	1	no	—
68	C₂×C₈	—	25	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	C₂×D₂₅	1	2	yes	—
71	C₂²⋊C₄	—	50	2	no	—
72	C₄²	C₂×D₂₅	1	1	yes	characteristic
73	C₂×C₅	#4093	1	1	yes	characteristic
74	C₂×C₅	#4096	1	2	yes	—
75	D₅	—	40	1	no	—
76	C₂³	—	50	1	no	—
77	Q₈	—	50	2	no	—
78	D₄	—	50	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	50	1	no	—
81	C₂×C₄	C₂²⋊D₂₅	1	2	yes	—
82	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
83	C₈	—	50	2	no	—
84	C₅	#65357	1	1	yes	characteristic
85	C₂²	—	100	1	no	—
86	C₂²	—	100	2	no	—
87	C₂²	C₂×C₂²⋊D₂₅	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	2	no	—
90	C₄	—	100	1	no	—
91	C₄	#87748	1	2	yes	—
92	C₂	—	200	1	no	—
93	C₂	#11716509	1	1	yes	characteristic
94	C₂	#11716513	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₅
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#125	C₂
socle	C₂²×C₅	C₂×C₂²⋊D₅
prime radical	C₂×D₁₀₀	C₂²
hypercenter	#125	D₂₅
hypocenter	C₂₅	#488
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₂₅×#125	C₂	semidirect
4. #11716357	C₂	semidirect
5. C₄×#6014	C₂	semidirect
53. #125	D₂₅	semidirect
58. C₂₅	#488	semidirect

Maximal subgroups

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

subgroup	index
C₂₅×#125	2
#11716437	2
#11716358	2
#11716357	2
C₄×#6014	2
#65357	5