Group #13107228 — subgroups

Subgroups — #13107228

Order 1600.

880 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717216	C₂	1	4	yes	maximal, complemented
3	#11717215	C₂	1	4	yes	maximal, complemented
4	#11717214	C₂	1	4	yes	maximal, complemented
5	#11717195	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	4	yes	—
8	C₂×C₄×C₅⋊D₅	C₂²	1	4	yes	—
9	C₂×#6045	C₂²	1	2	yes	—
10	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
11	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
12	C₄²×C₅²	C₂²	1	1	yes	characteristic
13	#87808	C₂²	1	4	yes	—
14	#87808	C₂²	1	4	yes	—
15	#87808	C₂²	1	4	yes	—
16	#87806	C₂²	1	4	yes	—
17	#66034	—	5	6	no	maximal
18	C₂²×C₅⋊D₅	C₂³	1	4	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂³	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	4	yes	—
22	C₂×#1039	C₂³	1	4	yes	—
23	C₄⋊(C₅⋊D₅)	—	4	2	no	—
24	C₄⋊(C₅⋊D₅)	—	4	4	no	—
25	C₄×C₅⋊D₅	—	4	4	no	—
26	#6045	—	4	2	no	—
27	C₅×#125	D₅	1	6	yes	complemented
28	#4042	—	5	24	no	—
29	#4041	—	5	24	no	—
30	#4040	—	5	24	no	—
31	#4021	—	5	12	no	—
32	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
33	C₂×C₅⋊D₅	—	2	4	no	—
34	C₂×C₅⋊D₅	—	2	4	no	—
35	C₂×C₅⋊D₅	—	2	4	no	—
36	C₄×C₅²	—	2	2	no	—
37	C₄×C₅²	—	2	4	no	—
38	C₄×C₅²	—	2	1	no	—
39	#1039	—	2	4	no	—
40	C₂×D₂₀	—	5	12	no	—
41	C₂×D₂₀	—	5	24	no	—
42	C₂×C₄×D₅	—	5	24	no	—
43	C₂×Dic₁₀	—	5	12	no	—
44	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
45	C₅×C₄⋊C₄	C₂×D₅	1	24	yes	—
46	C₄²×C₅	C₂×D₅	1	6	yes	—
47	#685	—	5	24	no	—
48	#685	—	5	24	no	—
49	#685	—	5	24	no	—
50	#683	—	5	24	no	—
51	#562	—	25	1	no	Sylow
52	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
53	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
54	C₅⋊D₅	—	4	4	no	—
55	C₂²×D₅	—	5	24	no	—
56	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
57	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
58	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
59	C₂×Dic₅	—	5	24	no	—
60	D₂₀	—	20	12	no	—
61	D₂₀	—	20	24	no	—
62	C₄×D₅	—	20	24	no	—
63	Dic₁₀	—	20	12	no	—
64	#125	C₅⋊D₅	1	1	yes	characteristic, complemented
65	#124	—	25	2	no	—
66	#123	—	25	4	no	—
67	C₂²⋊Q₈	—	25	4	no	—
68	#121	—	25	4	no	—
69	C₅²	#562	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₅	C₂³×D₅	1	6	yes	—
71	C₂×D₅	—	10	24	no	—
72	C₂×D₅	—	10	24	no	—
73	C₂×D₅	—	10	24	no	—
74	C₄×C₅	—	2	6	no	—
75	C₄×C₅	—	2	12	no	—
76	C₄×C₅	—	2	24	no	—
77	Dic₅	—	10	24	no	—
78	C₂×Q₈	—	25	2	no	—
79	C₂×D₄	—	25	4	no	—
80	C₂×D₄	—	25	2	no	—
81	C₂²×C₄	—	25	4	no	—
82	C₄⋊C₄	—	25	4	no	—
83	C₄⋊C₄	C₂×C₅⋊D₅	1	4	yes	—
84	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
85	C₂²⋊C₄	—	25	4	no	—
86	C₂²⋊C₄	—	25	4	no	—
87	C₂²⋊C₄	—	25	4	no	—
88	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
89	C₂×C₅	#4147	1	6	yes	—
90	C₂×C₅	#4149	1	12	yes	—
91	D₅	—	20	24	no	—
92	C₂³	—	25	4	no	—
93	Q₈	—	100	2	no	—
94	D₄	—	100	4	no	—
95	D₄	—	100	2	no	—
96	C₂×C₄	—	25	4	no	—
97	C₂×C₄	—	100	4	no	—
98	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
99	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
100	C₂×C₄	C₂²×C₅⋊D₅	1	4	yes	—
101	C₅	#66034	1	6	yes	complemented
102	C₂²	—	50	4	no	—
103	C₂²	—	50	4	no	—
104	C₂²	—	50	4	no	—
105	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	4	no	—
108	C₄	—	2	2	no	—
109	C₄	—	50	4	no	—
110	C₂	—	100	4	no	—
111	C₂	#11717520	1	1	yes	characteristic
112	C₂	#11717522	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#125

C₂

yes

maximal, characteristic, complemented

#11717216

C₂

yes

maximal, complemented

#11717215

C₂

yes

maximal, complemented

#11717214

C₂

yes

maximal, complemented

#11717195

C₂

yes

maximal, complemented

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×#6045

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

#87808

C₂²

yes

—

#87808

C₂²

yes

—

#87808

C₂²

yes

—

#87806

C₂²

yes

—

#66034

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×#1039

C₂³

yes

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄×C₅⋊D₅

—

#6045

—

C₅×#125

D₅

yes

—

—

—

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

#1039

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

—

C₂×C₅²

yes

—

C₂×C₅²

2₋¹⁺⁴

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

#125

C₅⋊D₅

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

C₅²

#562

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

#4147

yes

—

C₂×C₅

#4149

yes

—

D₅

—

C₂³

—

Q₈

—

100

—

D₄

—

100

—

D₄

—

100

—

C₂×C₄

—

C₂×C₄

—

100

—

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₂×C₄

C₂²×C₅⋊D₅

yes

—

100

C₂×C₄

C₂²×C₅⋊D₅

yes

—

101

C₅

#66034

yes

complemented

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₂³×C₅⋊D₅

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₂

—

100

—

111

C₂

#11717520

yes

characteristic

112

C₂

#11717522

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#125	C₂
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#562
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₅²×#125

C₂

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₅²×#125	C₂	semidirect
2. #11717216	C₂	semidirect
3. #11717215	C₂	semidirect
4. #11717214	C₂	semidirect
5. #11717195	C₂	semidirect
27. C₅×#125	D₅	semidirect
64. #125	C₅⋊D₅	semidirect
69. C₅²	#562	semidirect
101. C₅	#66034	semidirect

H ≅ G/N

type

1. C₅²×#125

C₂

semidirect

2. #11717216

C₂

semidirect

3. #11717215

C₂

semidirect

4. #11717214

C₂

semidirect

5. #11717195

C₂

semidirect

27. C₅×#125

D₅

semidirect

64. #125

C₅⋊D₅

semidirect

69. C₅²

#562

semidirect

101. C₅

#66034

semidirect

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

subgroup	index
C₅²×#125	2
#11717216	2
#11717215	2
#11717214	2
#11717195	2
#66034	5

subgroup

index

C₅²×#125

Subgroups — #13107228

Conjugacy classes of subgroups

Complements

Maximal subgroups