Group #13100870 — subgroups

Subgroups — #13100870

Order 1600.

336 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 104 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×C₈×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	#11716992	C₂	1	1	yes	maximal, characteristic
3	C₂×#87769	C₂	1	1	yes	maximal, characteristic
4	C₅×#3947	C₂	1	2	yes	maximal, complemented
5	Dic₅×#168	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	C₈×C₅⋊D₅	C₄	1	4	yes	complemented
10	C₄×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂×C₅×#168	C₂²	1	2	yes	—
12	#87781	C₄	1	2	yes	—
13	#87779	C₄	1	2	yes	—
14	#87769	C₄	1	4	yes	complemented
15	#64997	—	5	2	no	maximal
16	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
20	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
21	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
22	C₅²×C₈	C₂×C₄	1	2	yes	—
23	#6029	C₂×C₄	1	2	yes	—
24	C₅×#168	C₂×C₄	1	4	yes	—
25	C₂×#676	—	5	2	no	—
26	C₂×C₈×D₅	—	5	2	no	—
27	C₂×C₈×D₅	—	10	2	no	—
28	#4018	—	5	2	no	—
29	C₅×#97	—	5	2	no	—
30	#3947	D₅	1	2	yes	complemented
31	C₈×Dic₅	—	5	2	no	—
32	C₄×#168	—	5	2	no	—
33	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₅⋊D₅	—	2	2	no	—
35	C₂×C₅⋊D₅	C₄²	1	2	yes	—
36	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
38	#1039	C₄²	1	2	yes	—
39	C₅×Dic₅	—	2	4	no	—
40	C₂×C₄×D₅	—	5	2	no	—
41	C₂×C₄×D₅	—	10	2	no	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	—	5	2	no	—
44	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
45	C₄²×C₅	—	5	2	no	—
46	#685	—	5	4	no	—
47	#683	—	5	4	no	—
48	C₄×Dic₅	C₂×D₅	1	2	yes	—
49	C₂×#168	—	5	2	no	—
50	C₂×#168	—	10	2	no	—
51	C₂×#168	C₂×D₅	1	2	yes	—
52	#676	—	5	8	no	—
53	C₈×D₅	—	5	8	no	—
54	C₈×D₅	—	10	8	no	—
55	#405	—	25	1	no	Sylow
56	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₅²	#131	1	2	yes	—
58	C₅⋊D₅	—	2	2	no	—
59	C₂²×D₅	—	5	2	no	—
60	C₂²×D₅	—	10	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	5	4	no	—
63	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
64	C₂×Dic₅	—	5	2	no	—
65	C₂×Dic₅	—	10	2	no	—
66	C₂×Dic₅	C₄×D₅	1	4	yes	—
67	C₄×D₅	—	5	4	no	—
68	C₄×D₅	—	5	4	no	—
69	C₄×D₅	—	10	4	no	—
70	C₄×D₅	—	10	4	no	—
71	C₅×C₈	—	2	4	no	—
72	C₅×C₈	—	5	4	no	—
73	C₅×C₈	C₄×D₅	1	4	yes	—
74	#168	—	5	4	no	—
75	#168	—	10	4	no	—
76	#168	C₄×D₅	1	4	yes	—
77	C₂×M₁₆	—	25	1	no	—
78	C₂²×C₈	—	25	1	no	—
79	#117	—	25	1	no	—
80	#97	—	5	2	no	—
81	C₄×C₈	—	25	2	no	—
82	C₅²	#405	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₅	—	2	2	no	—
84	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
85	C₂×D₅	—	5	4	no	—
86	C₂×D₅	—	10	4	no	—
87	C₂×D₅	—	10	8	no	—
88	C₂×D₅	—	10	4	no	—
89	C₄×C₅	—	2	2	no	—
90	C₄×C₅	—	2	2	no	—
91	C₄×C₅	—	10	4	no	—
92	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
93	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
94	Dic₅	—	2	4	no	—
95	Dic₅	—	5	4	no	—
96	Dic₅	—	10	4	no	—
97	C₂²×C₄	—	25	1	no	—
98	M₁₆	—	25	4	no	—
99	C₂×C₈	—	5	2	no	—
100	C₂×C₈	—	25	1	no	—
101	C₂×C₈	—	25	4	no	—
102	C₂×C₈	D₅²	1	1	yes	characteristic
103	C₄⋊C₄	—	25	2	no	—
104	C₂²⋊C₄	—	25	2	no	—
105	C₄²	—	5	2	no	—
106	C₂×C₅	—	2	4	no	—
107	C₂×C₅	—	2	2	no	—
108	C₂×C₅	C₄²×D₅	1	2	yes	—
109	C₂×C₅	#4053	1	4	yes	—
110	D₅	—	10	8	no	—
111	D₅	—	10	4	no	—
112	C₂³	—	25	1	no	—
113	C₂×C₄	—	5	4	no	—
114	C₂×C₄	—	25	1	no	—
115	C₂×C₄	—	25	2	no	—
116	C₂×C₄	—	25	2	no	—
117	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
118	C₈	—	5	4	no	—
119	C₈	—	25	2	no	—
120	C₈	#6036	1	2	yes	—
121	C₅	—	2	2	no	—
122	C₅	#64997	1	2	yes	complemented
123	C₂²	—	25	2	no	—
124	C₂²	—	50	2	no	—
125	C₂²	C₄×D₅²	1	1	yes	characteristic
126	C₄	—	10	4	no	—
127	C₄	—	25	2	no	—
128	C₄	C₄×D₅²	1	1	yes	characteristic
129	C₄	C₂×#6036	1	1	yes	characteristic
130	C₂	—	50	2	no	—
131	C₂	#11716923	1	2	yes	—
132	C₂	C₄×#6036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×C₅⋊D₅

C₂

yes

maximal, characteristic

#11716992

C₂

yes

maximal, characteristic

C₂×#87769

C₂

yes

maximal, characteristic

C₅×#3947

C₂

yes

maximal, complemented

Dic₅×#168

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₈×C₅⋊D₅

C₄

yes

complemented

C₄×C₅×Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

C₄

yes

—

#87779

C₄

yes

—

#87769

C₄

yes

complemented

#64997

—

maximal

C₂²×C₅⋊D₅

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

—

C₄×C₅⋊D₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₅²×C₈

C₂×C₄

yes

—

#6029

C₂×C₄

yes

—

C₅×#168

C₂×C₄

yes

—

C₂×#676

—

C₂×C₈×D₅

—

C₂×C₈×D₅

—

#4018

—

C₅×#97

—

#3947

D₅

yes

complemented

C₈×Dic₅

—

C₄×#168

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄²

yes

—

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₄²

yes

—

C₅×Dic₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₄²×C₅

—

#685

—

#683

—

C₄×Dic₅

C₂×D₅

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₂×D₅

yes

—

#676

—

C₈×D₅

—

C₈×D₅

—

#405

—

Sylow

C₂×C₅²

C₂×C₄²

yes

characteristic

C₂×C₅²

#131

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₄×D₅

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₄×D₅

yes

—

#168

—

#168

—

#168

C₄×D₅

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

#97

—

C₄×C₈

—

C₅²

#405

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

C₂×C₄×D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

D₅²

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₂×C₅

—

107

C₂×C₅

—

108

C₂×C₅

C₄²×D₅

yes

—

109

C₂×C₅

#4053

yes

—

110

D₅

—

111

D₅

—

112

C₂³

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₅²

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

#6036

yes

—

121

C₅

—

122

C₅

#64997

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

C₄×D₅²

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₄×D₅²

yes

characteristic

129

C₄

C₂×#6036

yes

characteristic

130

C₂

—

131

C₂

#11716923

yes

—

132

C₂

C₄×#6036

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂×C₄²
Fitting	C₂×C₅²×C₈	C₂²
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#405
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂×D₅²

commutator

C₂×C₅²

C₂×C₄²

Fitting

C₂×C₅²×C₈

C₂²

socle

C₂²×C₅²

C₂²×C₄

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂×C₈

D₅²

C₅²

G ▷ C₂×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
4. C₅×#3947	C₂	semidirect
5. Dic₅×#168	C₂	semidirect
9. C₈×C₅⋊D₅	C₄	semidirect
14. #87769	C₄	semidirect
30. #3947	D₅	semidirect
82. C₅²	#405	semidirect
122. C₅	#64997	semidirect

H ≅ G/N

type

4. C₅×#3947

C₂

semidirect

5. Dic₅×#168

C₂

semidirect

9. C₈×C₅⋊D₅

C₄

semidirect

14. #87769

C₄

semidirect

30. #3947

D₅

semidirect

82. C₅²

#405

semidirect

122. C₅

#64997

semidirect

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

subgroup	index
C₂×C₈×C₅⋊D₅	2
#11716992	2
C₂×#87769	2
C₅×#3947	2
Dic₅×#168	2
#64997	5

subgroup

index

C₂×C₈×C₅⋊D₅

Subgroups — #13100870

Conjugacy classes of subgroups

Complements

Maximal subgroups