Subgroups — #13102931

Order 1600.

384 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717195	C₂	1	1	yes	maximal, characteristic
3	#11716677	C₂	1	2	yes	maximal, complemented
4	#11716676	C₂	1	2	yes	maximal
5	C₄×#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
7	C₂×#6045	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊C₄	—	2	2	no	—
9	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	#87808	—	2	2	no	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65357	—	5	6	no	maximal
14	C₂²×C₅⋊D₅	—	2	1	no	—
15	C₂×C₄×C₅²	—	2	2	no	—
16	C₂×C₄×C₅²	D₄	1	2	yes	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	—	2	1	no	—
19	C₄⋊(C₅⋊D₅)	—	2	2	no	—
20	#6045	—	2	2	no	—
21	#6029	—	2	2	no	—
22	C₅×#125	D₅	1	6	yes	complemented
23	#4021	—	5	6	no	—
24	#3942	—	5	12	no	—
25	#3941	—	5	12	no	—
26	C₄×#168	—	5	6	no	—
27	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
28	C₂×C₅⋊D₅	—	4	1	no	—
29	C₂×C₅⋊D₅	—	4	2	no	—
30	C₄×C₅²	—	2	2	no	—
31	C₄×C₅²	—	4	2	no	—
32	C₄×C₅²	#41	1	2	yes	—
33	#1039	—	4	1	no	—
34	C₂×D₂₀	—	5	6	no	—
35	C₂×Dic₁₀	—	5	6	no	—
36	C₅×C₄⋊C₄	—	2	12	no	—
37	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
38	C₄²×C₅	C₂×D₅	1	6	yes	—
39	#685	—	10	12	no	—
40	C₂×#168	—	5	12	no	—
41	#488	—	25	1	no	Sylow
42	C₂×C₅²	#124	1	1	yes	characteristic
43	C₂×C₅²	#135	1	2	yes	—
44	C₅⋊D₅	—	8	1	no	—
45	C₂²×D₅	—	10	6	no	—
46	C₂×C₄×C₅	—	2	12	no	—
47	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
49	C₂×Dic₅	—	10	6	no	—
50	D₂₀	—	10	12	no	—
51	Dic₁₀	—	10	12	no	—
52	#168	—	10	12	no	—
53	#125	C₅⋊D₅	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	C₅²	#488	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
60	C₂×D₅	—	20	12	no	—
61	C₂×D₅	—	20	6	no	—
62	C₄×C₅	—	2	12	no	—
63	C₄×C₅	—	4	12	no	—
64	C₄×C₅	#713	1	12	yes	—
65	Dic₅	—	20	6	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×C₅⋊D₅	1	2	yes	—
71	C₂²⋊C₄	—	50	2	no	—
72	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
73	C₂×C₅	#4093	1	6	yes	—
74	C₂×C₅	#4096	1	12	yes	—
75	D₅	—	40	6	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⋊(C₅⋊D₅)	1	2	yes	—
82	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
83	C₈	—	50	2	no	—
84	C₅	#65357	1	6	yes	complemented
85	C₂²	—	100	1	no	—
86	C₂²	—	100	2	no	—
87	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	2	no	—
90	C₄	—	100	1	no	—
91	C₄	#87904	1	2	yes	—
92	C₂	—	200	1	no	—
93	C₂	#11717267	1	1	yes	characteristic
94	C₂	#11717270	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#125	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#488
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅²×#125	C₂	semidirect
3. #11716677	C₂	semidirect
5. C₄×#6029	C₂	semidirect
22. C₅×#125	D₅	semidirect
53. #125	C₅⋊D₅	semidirect
58. C₅²	#488	semidirect
84. C₅	#65357	semidirect

Maximal subgroups

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

subgroup	index
C₅²×#125	2
#11717195	2
#11716677	2
#11716676	2
C₄×#6029	2
#65357	5