Group #13102746 — subgroups

Subgroups — #13102746

Order 1600.

536 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#87801	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717215	C₂	1	2	yes	maximal, complemented
4	C₅²×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716676	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	2	yes	—
8	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
11	#87808	—	2	2	no	—
12	C₂×#6029	C₂²	1	1	yes	characteristic
13	#87801	—	2	2	no	—
14	C₈×C₅⋊D₅	—	2	2	no	—
15	#65172	—	5	6	no	maximal
16	C₂²×C₅⋊D₅	—	2	2	no	—
17	C₂²×C₅⋊D₅	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	C₄⋊(C₅⋊D₅)	—	2	2	no	—
22	C₄⋊(C₅⋊D₅)	—	2	2	no	—
23	C₄⋊(C₅⋊D₅)	—	4	4	no	—
24	C₄×C₅⋊D₅	—	2	1	no	—
25	C₄×C₅⋊D₅	—	2	1	no	—
26	C₅²×C₈	D₄	1	2	yes	complemented
27	#6029	—	2	1	no	—
28	C₂×D₄₀	—	5	6	no	—
29	C₂×C₈×D₅	—	5	6	no	—
30	#4041	—	5	12	no	—
31	C₅×#107	D₅	1	6	yes	complemented
32	#3941	—	5	12	no	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₂×C₅⋊D₅	—	2	1	no	—
35	C₂×C₅⋊D₅	—	2	1	no	—
36	C₂×C₅⋊D₅	—	4	2	no	—
37	C₂×C₅⋊D₅	—	4	2	no	—
38	C₂×C₅⋊D₅	—	4	2	no	—
39	C₂×C₅⋊D₅	D₈	1	2	yes	complemented
40	C₄×C₅²	—	4	2	no	—
41	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅²	#41	1	1	yes	characteristic
43	#1039	—	2	1	no	—
44	C₂×D₂₀	—	5	12	no	—
45	C₂×D₂₀	—	10	12	no	—
46	C₂×C₄×D₅	—	5	6	no	—
47	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
48	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
49	#685	—	10	12	no	—
50	C₂×#168	—	5	6	no	—
51	D₄₀	—	10	12	no	—
52	C₈×D₅	—	10	12	no	—
53	#466	—	25	1	no	Sylow
54	C₂×C₅²	#121	1	1	yes	characteristic
55	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
56	C₂×C₅²	#135	1	1	yes	characteristic
57	C₅⋊D₅	—	2	2	no	—
58	C₅⋊D₅	—	8	2	no	—
59	C₂²×D₅	—	5	6	no	—
60	C₂²×D₅	—	10	12	no	—
61	C₂×C₄×C₅	—	2	12	no	—
62	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
63	C₂×Dic₅	—	5	6	no	—
64	D₂₀	—	10	12	no	—
65	D₂₀	—	10	12	no	—
66	D₂₀	—	20	24	no	—
67	C₄×D₅	—	10	6	no	—
68	C₄×D₅	—	10	6	no	—
69	C₅×C₈	D₂₀	1	12	yes	complemented
70	#168	—	10	6	no	—
71	C₂×D₈	—	25	1	no	—
72	C₂²×C₈	—	25	1	no	—
73	#121	—	25	2	no	—
74	#107	C₅⋊D₅	1	1	yes	characteristic, complemented
75	#102	—	25	2	no	—
76	C₅²	#466	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₅	D₄×D₅	1	6	yes	—
78	C₂×D₅	—	5	12	no	—
79	C₂×D₅	—	10	6	no	—
80	C₂×D₅	—	10	6	no	—
81	C₂×D₅	—	20	12	no	—
82	C₂×D₅	—	20	12	no	—
83	C₂×D₅	—	20	12	no	—
84	C₄×C₅	—	4	12	no	—
85	C₄×C₅	C₂×D₂₀	1	6	yes	—
86	C₄×C₅	#713	1	6	yes	—
87	Dic₅	—	10	6	no	—
88	C₂×D₄	—	25	2	no	—
89	C₂×D₄	—	50	2	no	—
90	C₂²×C₄	—	25	1	no	—
91	D₈	—	50	2	no	—
92	C₂×C₈	—	25	1	no	—
93	C₂×C₈	—	50	2	no	—
94	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
95	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
96	C₂²⋊C₄	—	50	2	no	—
97	C₂×C₅	#4041	1	6	yes	—
98	C₂×C₅	D₅×D₈	1	6	yes	—
99	C₂×C₅	#4065	1	6	yes	—
100	D₅	—	10	12	no	—
101	D₅	—	40	12	no	—
102	C₂³	—	25	1	no	—
103	C₂³	—	50	2	no	—
104	D₄	—	50	2	no	—
105	D₄	—	50	2	no	—
106	D₄	—	100	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	25	1	no	—
109	C₂×C₄	—	50	1	no	—
110	C₂×C₄	—	50	1	no	—
111	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
112	C₈	—	50	1	no	—
113	C₈	C₄⋊(C₅⋊D₅)	1	2	yes	complemented
114	C₅	#65172	1	6	yes	complemented
115	C₂²	—	25	2	no	—
116	C₂²	—	50	1	no	—
117	C₂²	—	50	1	no	—
118	C₂²	—	100	2	no	—
119	C₂²	—	100	2	no	—
120	C₂²	—	100	2	no	—
121	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
122	C₄	—	4	2	no	—
123	C₄	—	50	1	no	—
124	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
125	C₄	#87904	1	1	yes	characteristic
126	C₂	—	50	2	no	—
127	C₂	—	200	2	no	—
128	C₂	#11717215	1	1	yes	characteristic
129	C₂	D₈×C₅⋊D₅	1	1	yes	characteristic
130	C₂	#11717239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87801

C₂

yes

maximal, characteristic, complemented

C₂×C₈×C₅⋊D₅

C₂

yes

maximal, characteristic, complemented

#11717215

C₂

yes

maximal, complemented

C₅²×#107

C₂

yes

maximal, characteristic, complemented

#11716676

C₂

yes

maximal, complemented

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

—

#87808

—

C₂×#6029

C₂²

yes

characteristic

#87801

—

C₈×C₅⋊D₅

—

#65172

—

maximal

C₂²×C₅⋊D₅

—

C₂²×C₅⋊D₅

D₄

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

—

C₅²×C₈

D₄

yes

complemented

#6029

—

C₂×D₄₀

—

C₂×C₈×D₅

—

#4041

—

C₅×#107

D₅

yes

complemented

#3941

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

D₈

yes

complemented

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

#685

—

C₂×#168

—

D₄₀

—

C₈×D₅

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×D₈

yes

characteristic

C₂×C₅²

#135

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

D₂₀

yes

complemented

#168

—

C₂×D₈

—

C₂²×C₈

—

#121

—

#107

C₅⋊D₅

yes

characteristic, complemented

#102

—

C₅²

#466

yes

characteristic, complemented, Sylow

C₂²×C₅

D₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×D₂₀

yes

—

C₄×C₅

#713

yes

—

Dic₅

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₂²⋊C₄

—

C₂×C₅

#4041

yes

—

C₂×C₅

D₅×D₈

yes

—

C₂×C₅

#4065

yes

—

100

D₅

—

101

D₅

—

102

C₂³

—

103

C₂³

—

104

D₄

—

105

D₄

—

106

D₄

—

100

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

112

C₈

—

113

C₈

C₄⋊(C₅⋊D₅)

yes

complemented

114

C₅

#65172

yes

complemented

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

100

—

119

C₂²

—

100

—

120

C₂²

—

100

—

121

C₂²

D₄×C₅⋊D₅

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

125

C₄

#87904

yes

characteristic

126

C₂

—

127

C₂

—

200

—

128

C₂

#11717215

yes

characteristic

129

C₂

D₈×C₅⋊D₅

yes

characteristic

130

C₂

#11717239

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#107	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	#107	C₅⋊D₅
hypocenter	C₅²	#466
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×C₅⋊D₅

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#107

C₂

socle

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. C₂×#87801	C₂	semidirect
2. C₂×C₈×C₅⋊D₅	C₂	semidirect
3. #11717215	C₂	semidirect
4. C₅²×#107	C₂	semidirect
5. #11716676	C₂	semidirect
26. C₅²×C₈	D₄	semidirect
31. C₅×#107	D₅	semidirect
39. C₂×C₅⋊D₅	D₈	semidirect
69. C₅×C₈	D₂₀	semidirect
74. #107	C₅⋊D₅	semidirect
76. C₅²	#466	semidirect
113. C₈	C₄⋊(C₅⋊D₅)	semidirect
114. C₅	#65172	semidirect

H ≅ G/N

type

1. C₂×#87801

C₂

semidirect

2. C₂×C₈×C₅⋊D₅

C₂

semidirect

3. #11717215

C₂

semidirect

4. C₅²×#107

C₂

semidirect

5. #11716676

C₂

semidirect

26. C₅²×C₈

D₄

semidirect

31. C₅×#107

D₅

semidirect

39. C₂×C₅⋊D₅

D₈

semidirect

69. C₅×C₈

D₂₀

semidirect

74. #107

C₅⋊D₅

semidirect

76. C₅²

#466

semidirect

113. C₈

C₄⋊(C₅⋊D₅)

semidirect

114. C₅

#65172

semidirect

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

subgroup	index
C₂×#87801	2
C₂×C₈×C₅⋊D₅	2
#11717215	2
C₅²×#107	2
#11716676	2
#65172	5

subgroup

index

C₂×C₈×C₅⋊D₅

C₅²×#107

Subgroups — #13102746

Conjugacy classes of subgroups

Complements

Maximal subgroups