Group #13100511 — subgroups

Subgroups — #13100511

Order 1600.

304 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 80 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	#11717192	C₂	1	1	yes	maximal, characteristic
2	C₂×#87769	C₂	1	1	yes	maximal, characteristic
3	C₂×#87767	C₂	1	1	yes	maximal, characteristic
4	C₅×C₄⋊#168	C₂	1	2	yes	maximal, complemented
5	#11716613	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₄²×C₅²	C₂²	1	1	yes	characteristic
8	#87808	C₄	1	2	yes	—
9	#87806	C₄	1	2	yes	—
10	C₄×#1039	C₂²	1	1	yes	characteristic
11	C₂×C₅×#168	C₂²	1	2	yes	—
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	#87769	—	2	2	no	—
14	#87767	—	2	2	no	—
15	#65128	—	5	2	no	maximal
16	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×#1039	C₂×C₄	1	2	yes	—
20	C₂×#1039	C₂×C₄	1	1	yes	characteristic
21	C₄×C₅⋊D₅	D₄	1	2	yes	—
22	C₄×C₅⋊D₅	Q₈	1	2	yes	—
23	C₅×#168	—	2	2	no	—
24	C₅×#168	—	2	2	no	—
25	C₂×#676	—	5	2	no	—
26	C₂×C₈×D₅	—	5	2	no	—
27	#4018	—	5	2	no	—
28	#4018	—	10	2	no	—
29	C₅×C₄⋊C₈	—	5	2	no	—
30	#3946	—	5	2	no	—
31	#3946	—	5	2	no	—
32	C₄⋊#168	D₅	1	2	yes	complemented
33	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₅⋊D₅	—	2	2	no	—
35	C₂×C₅⋊D₅	C₄⋊C₄	1	2	yes	—
36	C₄×C₅²	—	2	2	no	—
37	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
38	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
39	#1039	—	2	2	no	—
40	#1039	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×D₅	—	5	2	no	—
42	C₂×C₄×D₅	—	10	2	no	—
43	C₂×C₅×C₈	—	5	2	no	—
44	C₂×C₅×C₈	—	5	2	no	—
45	C₄²×C₅	—	2	2	no	—
46	C₄²×C₅	C₂×D₅	1	2	yes	—
47	#685	—	5	2	no	—
48	#685	—	5	2	no	—
49	#685	—	10	4	no	—
50	#683	—	5	2	no	—
51	#683	—	5	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	C₂×#168	C₂×D₅	1	2	yes	—
57	#676	—	10	4	no	—
58	C₈×D₅	—	10	4	no	—
59	#424	—	25	1	no	Sylow
60	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₅²	#131	1	2	yes	—
62	C₅⋊D₅	—	2	2	no	—
63	C₂²×D₅	—	5	2	no	—
64	C₂²×D₅	—	10	2	no	—
65	C₂×C₄×C₅	—	2	4	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
68	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
69	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
70	C₂×Dic₅	—	5	2	no	—
71	C₂×Dic₅	—	5	2	no	—
72	C₂×Dic₅	—	5	2	no	—
73	C₂×Dic₅	—	10	2	no	—
74	C₂×Dic₅	—	10	4	no	—
75	C₄×D₅	—	5	4	no	—
76	C₄×D₅	—	5	4	no	—
77	C₄×D₅	—	10	4	no	—
78	C₄×D₅	—	10	4	no	—
79	C₅×C₈	—	10	2	no	—
80	C₅×C₈	—	10	2	no	—
81	#168	—	2	2	no	—
82	#168	—	2	2	no	—
83	C₂×M₁₆	—	25	1	no	—
84	C₂²×C₈	—	25	1	no	—
85	#117	—	25	1	no	—
86	C₄⋊C₈	—	5	2	no	—
87	C₄⋊C₈	—	25	2	no	—
88	C₅²	#424	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₅	—	2	2	no	—
90	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
91	C₂×D₅	—	5	4	no	—
92	C₂×D₅	—	10	4	no	—
93	C₂×D₅	—	10	2	no	—
94	C₂×D₅	—	10	2	no	—
95	C₂×D₅	—	20	4	no	—
96	C₄×C₅	—	2	2	no	—
97	C₄×C₅	—	2	2	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	2	2	no	—
100	C₄×C₅	—	4	4	no	—
101	C₄×C₅	D₄×D₅	1	2	yes	—
102	C₄×C₅	Q₈×D₅	1	2	yes	—
103	Dic₅	—	5	4	no	—
104	Dic₅	—	10	4	no	—
105	Dic₅	—	10	2	no	—
106	Dic₅	—	10	2	no	—
107	Dic₅	—	20	4	no	—
108	C₂²×C₄	—	25	1	no	—
109	M₁₆	—	50	2	no	—
110	C₂×C₈	—	5	2	no	—
111	C₂×C₈	—	5	2	no	—
112	C₂×C₈	—	50	2	no	—
113	C₄⋊C₄	—	25	2	no	—
114	C₂²⋊C₄	—	25	2	no	—
115	C₄²	—	25	1	no	—
116	C₄²	D₅²	1	1	yes	characteristic
117	C₂×C₅	—	2	4	no	—
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	D₅×C₄⋊C₄	1	2	yes	—
120	C₂×C₅	#4047	1	2	yes	—
121	C₂×C₅	#4053	1	2	yes	—
122	D₅	—	10	4	no	—
123	D₅	—	20	4	no	—
124	C₂³	—	25	1	no	—
125	C₂×C₄	—	25	1	no	—
126	C₂×C₄	—	25	2	no	—
127	C₂×C₄	—	25	2	no	—
128	C₂×C₄	—	25	2	no	—
129	C₂×C₄	#6036	1	2	yes	—
130	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
131	C₈	—	10	2	no	—
132	C₈	—	10	2	no	—
133	C₅	—	2	2	no	—
134	C₅	#65128	1	2	yes	complemented
135	C₂²	—	25	2	no	—
136	C₂²	—	50	2	no	—
137	C₂²	C₂×#6036	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	25	2	no	—
140	C₄	—	50	2	no	—
141	C₄	#87872	1	1	yes	characteristic
142	C₄	#87877	1	1	yes	characteristic
143	C₂	—	50	2	no	—
144	C₂	#11716935	1	2	yes	—
145	C₂	#11717018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717192

C₂

yes

maximal, characteristic

C₂×#87769

C₂

yes

maximal, characteristic

C₂×#87767

C₂

yes

maximal, characteristic

C₅×C₄⋊#168

C₂

yes

maximal, complemented

#11716613

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₄²×C₅²

C₂²

yes

characteristic

#87808

C₄

yes

—

#87806

C₄

yes

—

C₄×#1039

C₂²

yes

characteristic

C₂×C₅×#168

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

—

—

—

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

D₄

yes

—

C₄×C₅⋊D₅

Q₈

yes

—

C₅×#168

—

C₅×#168

—

C₂×#676

—

C₂×C₈×D₅

—

#4018

—

#4018

—

C₅×C₄⋊C₈

—

#3946

—

#3946

—

C₄⋊#168

D₅

yes

complemented

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

#1039

—

#1039

C₄⋊C₄

yes

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

—

—

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

C₂×#168

C₂×D₅

yes

—

#676

—

C₈×D₅

—

#424

—

Sylow

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#131

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₄⋊C₈

—

C₄⋊C₈

—

C₅²

#424

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

D₄×D₅

yes

—

102

C₄×C₅

Q₈×D₅

yes

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

—

106

Dic₅

—

107

Dic₅

—

108

C₂²×C₄

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

D₅²

yes

characteristic

117

C₂×C₅

—

118

C₂×C₅

—

119

C₂×C₅

D₅×C₄⋊C₄

yes

—

120

C₂×C₅

#4047

yes

—

121

C₂×C₅

#4053

yes

—

122

D₅

—

123

D₅

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#6036

yes

—

130

C₂×C₄

C₂×D₅²

yes

characteristic

131

C₈

—

132

C₈

—

133

C₅

—

134

C₅

#65128

yes

complemented

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×#6036

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#87872

yes

characteristic

142

C₄

#87877

yes

characteristic

143

C₂

—

144

C₂

#11716935

yes

—

145

C₂

#11717018

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₅²
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₂²
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	#424
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂×D₅²

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₄²×C₅²

C₂²

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₄²

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₄²

N	H ≅ G/N	type
4. C₅×C₄⋊#168	C₂	semidirect
5. #11716613	C₂	semidirect
32. C₄⋊#168	D₅	semidirect
88. C₅²	#424	semidirect
134. C₅	#65128	semidirect

H ≅ G/N

type

4. C₅×C₄⋊#168

C₂

semidirect

5. #11716613

C₂

semidirect

32. C₄⋊#168

D₅

semidirect

88. C₅²

#424

semidirect

134. C₅

#65128

semidirect

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

subgroup	index
#11717192	2
C₂×#87769	2
C₂×#87767	2
C₅×C₄⋊#168	2
#11716613	2
#65128	5

subgroup

index

C₅×C₄⋊#168

Subgroups — #13100511

Conjugacy classes of subgroups

Complements

Maximal subgroups