Group #13102552 — subgroups

Subgroups — #13102552

Order 1600.

520 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 183 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×#87799	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
3	#11717192	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716682	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#1039	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716670	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
9	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	#87808	C₄	1	2	yes	—
13	#87806	C₄	1	2	yes	—
14	C₄×#1039	C₂²	1	1	yes	characteristic
15	C₂×#6029	C₂²	1	1	yes	characteristic
16	C₂×#6029	C₂²	1	1	yes	characteristic
17	#87799	C₄	1	4	yes	complemented
18	C₈×C₅⋊D₅	C₄	1	4	yes	complemented
19	#64978	—	5	6	no	maximal
20	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂×#1039	C₂×C₄	1	2	yes	—
24	C₂×#1039	C₂×C₄	1	1	yes	characteristic
25	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
26	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
27	C₅²×C₈	C₂×C₄	1	2	yes	—
28	C₅²×C₈	C₂×C₄	1	2	yes	—
29	#6029	C₂×C₄	1	2	yes	—
30	#6029	C₂×C₄	1	2	yes	—
31	C₂×#676	—	5	6	no	—
32	C₂×C₈×D₅	—	5	6	no	—
33	#4018	—	5	6	no	—
34	C₄×C₅×C₈	D₅	1	6	yes	complemented
35	#3947	—	5	6	no	—
36	C₈×Dic₅	—	5	6	no	—
37	#3935	—	5	6	no	—
38	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₅⋊D₅	—	2	2	no	—
40	C₂×C₅⋊D₅	C₄²	1	2	yes	—
41	C₄×C₅²	—	2	2	no	—
42	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
44	#1039	—	2	2	no	—
45	#1039	C₄²	1	2	yes	—
46	C₂×C₄×D₅	—	5	6	no	—
47	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
48	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
49	C₄²×C₅	C₂×D₅	1	6	yes	—
50	#685	—	5	12	no	—
51	#683	—	5	12	no	—
52	C₄×Dic₅	—	5	6	no	—
53	C₂×#168	—	5	6	no	—
54	C₂×#168	—	5	6	no	—
55	#676	—	5	24	no	—
56	C₈×D₅	—	5	24	no	—
57	#405	—	25	1	no	Sylow
58	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₅²	#131	1	2	yes	—
60	C₅⋊D₅	—	2	2	no	—
61	C₂²×D₅	—	5	6	no	—
62	C₂×C₄×C₅	C₄×D₅	1	12	yes	—
63	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
64	C₂×Dic₅	—	5	6	no	—
65	C₂×Dic₅	—	5	12	no	—
66	C₄×D₅	—	5	12	no	—
67	C₄×D₅	—	5	12	no	—
68	C₅×C₈	C₄×D₅	1	12	yes	—
69	C₅×C₈	C₄×D₅	1	12	yes	—
70	#168	—	5	12	no	—
71	#168	—	5	12	no	—
72	C₂×M₁₆	—	25	1	no	—
73	C₂²×C₈	—	25	1	no	—
74	#117	—	25	1	no	—
75	#97	—	25	1	no	—
76	#97	—	25	1	no	—
77	C₄×C₈	—	25	1	no	—
78	C₄×C₈	C₅⋊D₅	1	1	yes	characteristic, complemented
79	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₅	C₂×C₄×D₅	1	6	yes	—
81	C₂×D₅	—	5	12	no	—
82	C₂×D₅	—	10	12	no	—
83	C₄×C₅	—	2	12	no	—
84	C₄×C₅	C₂×C₄×D₅	1	6	yes	—
85	C₄×C₅	C₂×C₄×D₅	1	6	yes	—
86	Dic₅	—	5	12	no	—
87	Dic₅	—	10	12	no	—
88	C₂²×C₄	—	25	1	no	—
89	M₁₆	—	25	4	no	—
90	C₂×C₈	—	25	1	no	—
91	C₂×C₈	—	25	1	no	—
92	C₂×C₈	—	25	4	no	—
93	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
94	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
95	C₄⋊C₄	—	25	2	no	—
96	C₂²⋊C₄	—	25	2	no	—
97	C₄²	—	25	1	no	—
98	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
99	C₂×C₅	C₄²×D₅	1	6	yes	—
100	C₂×C₅	#4047	1	12	yes	—
101	D₅	—	10	12	no	—
102	C₂³	—	25	1	no	—
103	C₂×C₄	—	25	1	no	—
104	C₂×C₄	—	25	2	no	—
105	C₂×C₄	—	25	2	no	—
106	C₂×C₄	—	25	2	no	—
107	C₂×C₄	C₄×C₅⋊D₅	1	2	yes	—
108	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
109	C₈	—	25	2	no	—
110	C₈	—	25	2	no	—
111	C₈	C₄×C₅⋊D₅	1	2	yes	—
112	C₈	C₄×C₅⋊D₅	1	2	yes	—
113	C₅	#64978	1	6	yes	complemented
114	C₂²	—	25	2	no	—
115	C₂²	—	50	2	no	—
116	C₂²	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	25	2	no	—
119	C₄	—	50	2	no	—
120	C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
121	C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
122	C₂	—	50	2	no	—
123	C₂	C₄²×C₅⋊D₅	1	1	yes	characteristic
124	C₂	#11717221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87799

C₂

yes

maximal, characteristic

C₂×C₈×C₅⋊D₅

C₂

yes

maximal, characteristic

#11717192

C₂

yes

maximal, characteristic

C₄×C₅²×C₈

C₂

yes

maximal, characteristic, complemented

#11716682

C₂

yes

maximal, characteristic, complemented

C₈×#1039

C₂

yes

maximal, characteristic, complemented

#11716670

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

#87808

C₄

yes

—

#87806

C₄

yes

—

C₄×#1039

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#87799

C₄

yes

complemented

C₈×C₅⋊D₅

C₄

yes

complemented

#64978

—

maximal

C₂²×C₅⋊D₅

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

—

C₄×C₅⋊D₅

C₂×C₄

yes

—

C₅²×C₈

C₂×C₄

yes

—

C₅²×C₈

C₂×C₄

yes

—

#6029

C₂×C₄

yes

—

#6029

C₂×C₄

yes

—

C₂×#676

—

C₂×C₈×D₅

—

#4018

—

C₄×C₅×C₈

D₅

yes

complemented

#3947

—

C₈×Dic₅

—

#3935

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄²

yes

—

C₄×C₅²

—

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

—

#1039

C₄²

yes

—

C₂×C₄×D₅

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

#685

—

#683

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

#676

—

C₈×D₅

—

#405

—

Sylow

C₂×C₅²

C₂×C₄²

yes

characteristic

C₂×C₅²

#131

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

C₄×D₅

yes

—

C₅×C₈

C₄×D₅

yes

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₅⋊D₅

yes

characteristic, complemented

C₅²

#405

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

C₂×C₄×D₅

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

C₄²×D₅

yes

—

100

C₂×C₅

#4047

yes

—

101

D₅

—

102

C₂³

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄×C₅⋊D₅

yes

—

108

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

C₄×C₅⋊D₅

yes

—

112

C₈

C₄×C₅⋊D₅

yes

—

113

C₅

#64978

yes

complemented

114

C₂²

—

115

C₂²

—

116

C₂²

C₂×C₄×C₅⋊D₅

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×C₄×C₅⋊D₅

yes

characteristic

121

C₄

C₂×C₄×C₅⋊D₅

yes

characteristic

122

C₂

—

123

C₂

C₄²×C₅⋊D₅

yes

characteristic

124

C₂

#11717221

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₅⋊D₅
commutator	C₂×C₅²	C₂×C₄²
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₄×C₅²×C₈	C₂
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₄×C₈	C₅⋊D₅
hypocenter	C₅²	#405
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×C₅⋊D₅

commutator

C₂×C₅²

C₂×C₄²

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₄×C₅²×C₈

C₂

socle

C₂²×C₅²

C₂²×C₄

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₄×C₈

C₅⋊D₅

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
4. C₄×C₅²×C₈	C₂	semidirect
5. #11716682	C₂	semidirect
6. C₈×#1039	C₂	semidirect
7. #11716670	C₂	semidirect
17. #87799	C₄	semidirect
18. C₈×C₅⋊D₅	C₄	semidirect
34. C₄×C₅×C₈	D₅	semidirect
78. C₄×C₈	C₅⋊D₅	semidirect
79. C₅²	#405	semidirect
113. C₅	#64978	semidirect

H ≅ G/N

type

4. C₄×C₅²×C₈

C₂

semidirect

5. #11716682

C₂

semidirect

6. C₈×#1039

C₂

semidirect

7. #11716670

C₂

semidirect

17. #87799

C₄

semidirect

18. C₈×C₅⋊D₅

C₄

semidirect

34. C₄×C₅×C₈

D₅

semidirect

78. C₄×C₈

C₅⋊D₅

semidirect

79. C₅²

#405

semidirect

113. C₅

#64978

semidirect

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

subgroup	index
C₂×#87799	2
C₂×C₈×C₅⋊D₅	2
#11717192	2
C₄×C₅²×C₈	2
#11716682	2
C₈×#1039	2
#11716670	2
#64978	5

subgroup

index

C₂×C₈×C₅⋊D₅

C₄×C₅²×C₈

Subgroups — #13102552

Conjugacy classes of subgroups

Complements

Maximal subgroups