Group #13100725 — subgroups

Subgroups — #13100725

Order 1600.

236 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11717210	C₂	1	1	yes	maximal, characteristic
2	#11716985	C₂	1	1	yes	maximal, characteristic
3	C₅×#3942	C₂	1	2	yes	maximal
4	#11716607	C₂	1	2	yes	maximal
5	#11716596	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
7	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
8	C₄⋊#1039	—	2	1	no	—
9	C₄⋊#1039	C₂²	1	1	yes	characteristic
10	#87806	—	2	1	no	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	#87781	—	2	1	no	—
14	#87781	—	2	1	no	—
15	#65086	—	5	2	no	maximal
16	C₂×C₄×C₅²	—	2	1	no	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	—	2	1	no	—
19	C₂×#1039	D₄	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	C₂×C₅×Dic₅	—	2	2	no	—
22	C₅×Dic₁₀	—	2	2	no	—
23	C₅×Dic₁₀	—	2	2	no	—
24	C₅×#168	—	2	2	no	—
25	#4090	—	5	2	no	—
26	#4036	—	5	2	no	—
27	#4036	—	10	2	no	—
28	C₅×#103	—	5	2	no	—
29	#3967	—	5	2	no	—
30	#3948	—	5	2	no	—
31	#3947	—	5	2	no	—
32	#3942	D₅	1	2	yes	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	—	4	1	no	—
35	C₄×C₅²	#41	1	2	yes	—
36	#1039	—	2	1	no	—
37	#1039	—	2	1	no	—
38	#1039	—	4	1	no	—
39	C₅×Dic₅	—	4	2	no	—
40	C₂×C₅×Q₈	—	5	2	no	—
41	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
42	C₂×C₅×C₈	—	5	2	no	—
43	C₅×C₄⋊C₄	—	2	2	no	—
44	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
45	C₄⋊Dic₅	—	5	2	no	—
46	C₄⋊Dic₅	—	10	2	no	—
47	C₄⋊Dic₅	—	10	2	no	—
48	C₄⋊Dic₅	—	10	4	no	—
49	#683	—	10	2	no	—
50	#683	—	10	2	no	—
51	#683	—	10	2	no	—
52	#683	—	10	4	no	—
53	C₄×Dic₅	—	5	2	no	—
54	C₄×Dic₅	—	10	2	no	—
55	C₂×#168	C₂×D₅	1	2	yes	—
56	#491	—	25	1	no	Sylow
57	C₂×C₅²	#124	1	1	yes	characteristic
58	C₂×C₅²	#137	1	1	yes	characteristic
59	C₂×C₅²	#137	1	1	yes	characteristic
60	C₅×Q₈	—	10	2	no	—
61	C₅×Q₈	—	10	2	no	—
62	C₂×C₄×C₅	—	2	4	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	10	2	no	—
66	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
67	C₂×Dic₅	—	2	2	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	5	2	no	—
70	C₂×Dic₅	—	10	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	—	10	4	no	—
73	C₂×Dic₅	—	10	2	no	—
74	Dic₁₀	—	2	2	no	—
75	Dic₁₀	—	2	2	no	—
76	C₅×C₈	—	10	2	no	—
77	#168	—	2	2	no	—
78	C₄⋊Q₈	—	25	1	no	—
79	#125	—	25	1	no	—
80	#103	—	5	2	no	—
81	#103	—	25	2	no	—
82	#97	—	25	1	no	—
83	C₅²	#491	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	D₄×D₅	1	2	yes	—
86	C₄×C₅	—	2	4	no	—
87	C₄×C₅	—	4	4	no	—
88	C₄×C₅	—	4	2	no	—
89	C₄×C₅	—	20	2	no	—
90	C₄×C₅	#709	1	2	yes	—
91	C₄×C₅	#711	1	2	yes	—
92	Dic₅	—	4	2	no	—
93	Dic₅	—	10	2	no	—
94	Dic₅	—	10	2	no	—
95	Dic₅	—	20	2	no	—
96	Dic₅	—	20	2	no	—
97	Dic₅	—	20	4	no	—
98	Dic₅	—	20	2	no	—
99	C₂×Q₈	—	5	2	no	—
100	C₂×C₈	—	5	2	no	—
101	C₄⋊C₄	—	25	1	no	—
102	C₄⋊C₄	—	50	1	no	—
103	C₄⋊C₄	—	50	1	no	—
104	C₄⋊C₄	—	50	1	no	—
105	C₄⋊C₄	—	50	1	no	—
106	C₄⋊C₄	D₅²	1	1	yes	characteristic
107	C₄²	—	25	1	no	—
108	C₂×C₅	—	2	2	no	—
109	C₂×C₅	—	2	2	no	—
110	C₂×C₅	—	2	2	no	—
111	C₂×C₅	#4031	1	2	yes	—
112	C₂×C₅	#4061	1	2	yes	—
113	C₂×C₅	#4064	1	2	yes	—
114	Q₈	—	10	2	no	—
115	Q₈	—	10	2	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	10	2	no	—
118	C₂×C₄	—	25	1	no	—
119	C₂×C₄	—	25	1	no	—
120	C₂×C₄	—	50	1	no	—
121	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
122	C₈	—	10	2	no	—
123	C₅	—	2	2	no	—
124	C₅	#65086	1	2	yes	complemented
125	C₂²	#87886	1	1	yes	characteristic
126	C₄	—	4	1	no	—
127	C₄	—	20	2	no	—
128	C₄	—	50	1	no	—
129	C₄	—	50	1	no	—
130	C₄	—	100	1	no	—
131	C₄	#87870	1	2	yes	—
132	C₂	#11717031	1	1	yes	characteristic
133	C₂	#11717049	1	1	yes	characteristic
134	C₂	#11717063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717210

C₂

yes

maximal, characteristic

#11716985

C₂

yes

maximal, characteristic

C₅×#3942

C₂

yes

maximal

#11716607

C₂

yes

maximal

#11716596

C₂

yes

maximal, characteristic

C₂×C₅×Dic₁₀

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

C₄⋊#1039

—

C₄⋊#1039

C₂²

yes

characteristic

#87806

—

C₄×#1039

C₂²

yes

characteristic

C₂×C₅×#168

C₂²

yes

—

—

—

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

—

—

—

—

—

—

—

—

—

D₅

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

#1039

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

—

—

—

—

C₅²

#491

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#709

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₅²

yes

characteristic

107

C₄²

—

108

C₂×C₅

—

109

C₂×C₅

—

110

C₂×C₅

—

111

C₂×C₅

#4031

yes

—

112

C₂×C₅

#4061

yes

—

113

C₂×C₅

#4064

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×D₅²

yes

characteristic

122

C₈

—

123

C₅

—

124

C₅

#65086

yes

complemented

125

C₂²

#87886

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

100

—

131

C₄

#87870

yes

—

132

C₂

#11717031

yes

characteristic

133

C₂

#11717049

yes

characteristic

134

C₂

#11717063

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×C₄⋊C₄	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊C₄	D₅²
hypocenter	C₅²	#491
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×C₄⋊C₄

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₄⋊C₄

D₅²

hypocenter

C₅²

#491

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
83. C₅²	#491	semidirect
124. C₅	#65086	semidirect

H ≅ G/N

type

83. C₅²

#491

semidirect

124. C₅

#65086

semidirect

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

subgroup	index
#11717210	2
#11716985	2
C₅×#3942	2
#11716607	2
#11716596	2
#65086	5

subgroup

index

Subgroups — #13100725

Conjugacy classes of subgroups

Complements

Maximal subgroups