Subgroups — #13107215

Order 1600.

688 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	#11717210	C₂	1	4	yes	maximal
3	#11717209	C₂	1	4	yes	maximal
4	#11717209	C₂	1	4	yes	maximal
5	#11717190	C₂	1	2	yes	maximal
6	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
7	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
8	C₄²×C₅²	C₂²	1	1	yes	characteristic
9	C₄⋊#1039	C₂²	1	2	yes	—
10	C₄⋊#1039	C₂²	1	2	yes	—
11	C₄⋊#1039	C₂²	1	4	yes	—
12	#87806	C₂²	1	4	yes	—
13	#87806	C₂²	1	4	yes	—
14	#87806	C₂²	1	4	yes	—
15	#87806	C₂²	1	4	yes	—
16	C₄×#1039	C₂²	1	4	yes	—
17	C₅⋊#564	—	5	6	no	maximal
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	2	yes	—
20	C₂×C₄×C₅²	C₂³	1	4	yes	—
21	C₂×#1039	C₂³	1	4	yes	—
22	C₂×#1039	C₂³	1	4	yes	—
23	C₅×#125	D₅	1	6	yes	—
24	#4036	—	5	24	no	—
25	#4035	—	5	24	no	—
26	#4035	—	5	24	no	—
27	#4016	—	5	12	no	—
28	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
29	C₄×C₅²	—	2	1	no	—
30	C₄×C₅²	—	2	2	no	—
31	C₄×C₅²	—	2	4	no	—
32	#1039	—	2	4	no	—
33	#1039	—	2	4	no	—
34	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
35	C₅×C₄⋊C₄	C₂×D₅	1	24	yes	—
36	C₄²×C₅	C₂×D₅	1	6	yes	—
37	C₄⋊Dic₅	—	5	12	no	—
38	C₄⋊Dic₅	—	5	12	no	—
39	C₄⋊Dic₅	—	5	24	no	—
40	#683	—	5	24	no	—
41	#683	—	5	24	no	—
42	#683	—	5	24	no	—
43	#683	—	5	24	no	—
44	C₄×Dic₅	—	5	24	no	—
45	#564	—	25	1	no	Sylow
46	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂×C₅²	2₋¹⁺⁴	1	2	yes	—
48	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
49	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
50	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
51	C₂×Dic₅	—	5	24	no	—
52	C₂×Dic₅	—	5	24	no	—
53	#125	—	25	4	no	—
54	#125	—	25	4	no	—
55	#125	—	25	4	no	—
56	#125	—	25	2	no	—
57	#125	C₅⋊D₅	1	1	yes	characteristic
58	C₅²	#564	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₅	C₂³×D₅	1	6	yes	—
60	C₄×C₅	—	2	6	no	—
61	C₄×C₅	—	2	24	no	—
62	C₄×C₅	—	2	12	no	—
63	Dic₅	—	10	24	no	—
64	Dic₅	—	10	24	no	—
65	C₄⋊C₄	—	25	4	no	—
66	C₄⋊C₄	—	25	4	no	—
67	C₄⋊C₄	—	25	4	no	—
68	C₄⋊C₄	—	25	4	no	—
69	C₄⋊C₄	—	25	4	no	—
70	C₄⋊C₄	—	25	2	no	—
71	C₄⋊C₄	—	25	2	no	—
72	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
73	C₄⋊C₄	C₂×C₅⋊D₅	1	4	yes	—
74	C₄²	—	25	4	no	—
75	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
76	C₂×C₅	#4147	1	6	yes	—
77	C₂×C₅	#4150	1	12	yes	—
78	C₂×C₄	—	25	4	no	—
79	C₂×C₄	—	25	4	no	—
80	C₂×C₄	C₂²×C₅⋊D₅	1	4	yes	—
81	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
82	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
83	C₅	C₅⋊#564	1	6	yes	complemented
84	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	4	no	—
87	C₄	—	2	2	no	—
88	C₄	—	50	4	no	—
89	C₄	—	50	4	no	—
90	C₂	#11717520	1	1	yes	characteristic
91	C₂	#11717523	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#125	C₂
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#564
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
58. C₅²	#564	semidirect
83. C₅	C₅⋊#564	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
#11717210	2
#11717209	2
#11717190	2
C₅⋊#564	5