Subgroups — #13102928

Order 1600.

344 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
2	#11717190	C₂	1	1	yes	maximal, characteristic
3	#11716675	C₂	1	2	yes	maximal
4	#11716674	C₂	1	2	yes	maximal
5	C₄×#6029	C₂	1	1	yes	maximal, characteristic
6	C₅²×C₄⋊C₄	—	2	2	no	—
7	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
8	C₄²×C₅²	C₂²	1	1	yes	characteristic
9	C₄⋊#1039	C₂²	1	2	yes	—
10	#87806	—	2	2	no	—
11	C₂×#6029	C₂²	1	1	yes	characteristic
12	C₂×#6029	C₂²	1	1	yes	characteristic
13	#65353	—	5	6	no	maximal
14	C₂×C₄×C₅²	—	2	2	no	—
15	C₂×C₄×C₅²	D₄	1	2	yes	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	—	2	2	no	—
18	#6029	Q₈	1	2	yes	—
19	#6029	Q₈	1	2	yes	—
20	C₅×#125	D₅	1	6	yes	—
21	#4016	—	5	6	no	—
22	#3940	—	5	12	no	—
23	#3939	—	5	12	no	—
24	C₄×#168	—	5	6	no	—
25	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
26	C₄×C₅²	—	2	2	no	—
27	C₄×C₅²	—	4	2	no	—
28	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
30	#1039	—	4	2	no	—
31	C₅×C₄⋊C₄	—	2	12	no	—
32	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
33	C₄²×C₅	C₂×D₅	1	6	yes	—
34	C₄⋊Dic₅	—	5	12	no	—
35	#683	—	10	12	no	—
36	C₂×#168	—	5	6	no	—
37	C₂×#168	—	5	6	no	—
38	#499	—	25	1	no	Sylow
39	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₅²	#135	1	2	yes	—
41	C₂×C₄×C₅	—	2	12	no	—
42	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
43	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
44	C₂×Dic₅	—	10	12	no	—
45	#168	—	5	12	no	—
46	#168	—	5	12	no	—
47	#125	—	25	1	no	—
48	#125	C₅⋊D₅	1	1	yes	characteristic
49	#107	—	25	2	no	—
50	#106	—	25	2	no	—
51	C₄×C₈	—	25	1	no	—
52	C₅²	#499	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
54	C₄×C₅	—	2	12	no	—
55	C₄×C₅	—	4	12	no	—
56	C₄×C₅	Q₈×D₅	1	6	yes	—
57	C₄×C₅	Q₈×D₅	1	6	yes	—
58	Dic₅	—	20	12	no	—
59	C₂×C₈	—	25	1	no	—
60	C₂×C₈	—	25	1	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	25	2	no	—
63	C₄⋊C₄	—	50	2	no	—
64	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
65	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
66	C₂×C₅	#4090	1	6	yes	—
67	C₂×C₅	#4096	1	12	yes	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	50	2	no	—
70	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
71	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
72	C₈	—	25	2	no	—
73	C₈	—	25	2	no	—
74	C₅	#65353	1	6	yes	complemented
75	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	2	no	—
78	C₄	—	100	2	no	—
79	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
80	C₄	Q₈×C₅⋊D₅	1	1	yes	characteristic
81	C₂	#11717264	1	1	yes	characteristic
82	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, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#499
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
52. C₅²	#499	semidirect
74. C₅	#65353	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
#11717190	2
#11716675	2
#11716674	2
C₄×#6029	2
#65353	5