Group #13101161 — subgroups

Subgroups — #13101161

Order 1600.

278 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 52 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	#11717227	C₂	1	1	yes	maximal, characteristic, complemented
2	#11716948	C₂	1	2	yes	maximal, complemented
3	C₂×#87770	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3955	C₂	1	2	yes	maximal, complemented
5	#11716619	C₂	1	1	yes	maximal, characteristic, complemented
6	#87900	C₂²	1	1	yes	characteristic
7	C₂×C₅×D₂₀	C₂²	1	2	yes	—
8	C₅²×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#6029	C₂²	1	1	yes	characteristic
10	#87799	—	2	1	no	—
11	C₈×C₅⋊D₅	—	2	1	no	—
12	C₅×#681	C₂²	1	2	yes	—
13	#87773	—	2	2	no	—
14	#87771	—	2	2	no	—
15	#87770	—	2	1	no	—
16	#87770	—	2	1	no	—
17	#65038	—	5	2	no	maximal
18	C₂²×C₅×D₅	—	2	2	no	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
21	C₄⋊(C₅⋊D₅)	D₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	—	2	1	no	—
23	#6045	D₄	1	1	yes	characteristic
24	C₅×D₂₀	—	2	2	no	—
25	C₅×D₂₀	—	2	2	no	—
26	C₅²×C₈	—	2	1	no	—
27	#6029	D₄	1	1	yes	characteristic
28	#6029	D₄	1	1	yes	characteristic
29	C₅×#168	—	2	2	no	—
30	#4078	—	5	2	no	—
31	C₂×#686	—	5	2	no	—
32	#4054	—	5	2	no	—
33	#4053	—	5	2	no	—
34	#4053	—	10	2	no	—
35	C₅×#100	—	5	2	no	—
36	#3955	D₅	1	2	yes	complemented
37	#3954	—	5	2	no	—
38	C₂²×C₅²	#41	1	1	yes	characteristic
39	C₂×C₅⋊D₅	—	2	1	no	—
40	C₂×C₅×D₅	—	4	2	no	—
41	C₂×C₅×D₅	—	8	2	no	—
42	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
43	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
44	#1039	—	2	1	no	—
45	C₂×C₅×D₄	—	5	2	no	—
46	#709	—	5	2	no	—
47	#709	—	10	2	no	—
48	C₂×D₂₀	C₂×D₅	1	2	yes	—
49	C₅×M₁₆	—	2	2	no	—
50	C₅×M₁₆	—	5	2	no	—
51	C₅×M₁₆	C₂×D₅	1	2	yes	—
52	#687	—	10	2	no	—
53	#686	—	10	2	no	—
54	#686	—	10	2	no	—
55	#686	—	10	2	no	—
56	#681	C₂×D₅	1	2	yes	—
57	C₂×#168	—	5	2	no	—
58	C₂×#168	—	10	2	no	—
59	D₄₀	—	10	2	no	—
60	#677	—	10	2	no	—
61	#676	—	10	2	no	—
62	#676	—	10	4	no	—
63	C₈×D₅	—	10	2	no	—
64	C₈×D₅	—	10	4	no	—
65	#472	—	25	1	no	Sylow
66	C₂×C₅²	—	2	1	no	—
67	C₂×C₅²	#121	1	1	yes	characteristic
68	C₅⋊D₅	—	4	1	no	—
69	C₅×D₅	—	8	2	no	—
70	C₂³×C₅	—	10	2	no	—
71	C₂²×D₅	—	2	2	no	—
72	C₅×D₄	—	10	2	no	—
73	C₅×D₄	—	10	2	no	—
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
76	C₂²⋊D₅	—	10	4	no	—
77	C₂²⋊D₅	—	10	2	no	—
78	D₂₀	—	2	2	no	—
79	D₂₀	—	2	2	no	—
80	D₂₀	—	5	2	no	—
81	D₂₀	—	10	2	no	—
82	C₄×D₅	—	10	4	no	—
83	C₄×D₅	—	10	2	no	—
84	Dic₁₀	—	5	2	no	—
85	Dic₁₀	—	10	2	no	—
86	C₅×C₈	—	2	4	no	—
87	C₅×C₈	—	2	2	no	—
88	C₅×C₈	—	10	2	no	—
89	#168	—	2	2	no	—
90	#168	—	5	2	no	—
91	#168	—	5	2	no	—
92	#168	—	10	2	no	—
93	#168	—	10	2	no	—
94	AGL(1,ℤ/8)	—	25	2	no	—
95	C₂×D₈	—	25	1	no	—
96	#131	—	25	1	no	—
97	#108	—	25	1	no	—
98	#100	—	5	2	no	—
99	C₅²	#472	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	—	20	2	no	—
102	C₂²×C₅	—	40	2	no	—
103	C₂²×C₅	#709	1	2	yes	—
104	C₂×D₅	—	4	2	no	—
105	C₂×D₅	—	8	2	no	—
106	C₂×D₅	—	10	4	no	—
107	C₂×D₅	—	10	2	no	—
108	C₄×C₅	—	2	2	no	—
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	D₄×D₅	1	2	yes	—
111	C₄×C₅	D₄×D₅	1	2	yes	—
112	Dic₅	—	10	4	no	—
113	Dic₅	—	10	2	no	—
114	#41	—	25	1	no	—
115	C₂×D₄	—	5	2	no	—
116	SD₁₆	—	50	2	no	—
117	D₈	—	50	2	no	—
118	D₈	—	50	1	no	—
119	D₈	—	50	1	no	—
120	M₁₆	—	5	2	no	—
121	M₁₆	—	50	1	no	—
122	M₁₆	D₅²	1	1	yes	characteristic
123	C₂×C₈	—	25	1	no	—
124	C₂×C₈	—	50	1	no	—
125	C₂×C₅	—	2	2	no	—
126	C₂×C₅	—	2	2	no	—
127	C₂×C₅	—	4	2	no	—
128	C₂×C₅	—	40	2	no	—
129	C₂×C₅	#4030	1	2	yes	—
130	D₅	—	8	2	no	—
131	D₅	—	20	4	no	—
132	D₅	—	20	2	no	—
133	C₂³	—	10	2	no	—
134	Q₈	—	25	1	no	—
135	D₄	—	10	2	no	—
136	D₄	—	10	2	no	—
137	D₄	—	25	1	no	—
138	D₄	—	50	1	no	—
139	C₂×C₄	—	50	1	no	—
140	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	10	2	no	—
143	C₈	—	25	1	no	—
144	C₈	—	25	1	no	—
145	C₅	—	2	2	no	—
146	C₅	#65038	1	2	yes	complemented
147	C₂²	—	20	2	no	—
148	C₂²	—	40	2	no	—
149	C₂²	—	50	1	no	—
150	C₂²	#87873	1	1	yes	characteristic
151	C₄	—	50	1	no	—
152	C₄	#87877	1	1	yes	characteristic
153	C₄	#87886	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	40	2	no	—
156	C₂	—	100	1	no	—
157	C₂	#11717030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717227

C₂

yes

maximal, characteristic, complemented

#11716948

C₂

yes

maximal, complemented

C₂×#87770

C₂

yes

maximal, characteristic, complemented

C₅×#3955

C₂

yes

maximal, complemented

#11716619

C₂

yes

maximal, characteristic, complemented

#87900

C₂²

yes

characteristic

C₂×C₅×D₂₀

C₂²

yes

—

C₅²×M₁₆

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#87799

—

C₈×C₅⋊D₅

—

C₅×#681

C₂²

yes

—

—

—

—

—

—

C₂²×C₅×D₅

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

characteristic

C₄×C₅⋊D₅

—

#6045

D₄

yes

characteristic

C₅×D₂₀

—

C₅×D₂₀

—

C₅²×C₈

—

#6029

D₄

yes

characteristic

#6029

D₄

yes

—

—

—

—

—

—

—

D₅

yes

complemented

#3954

—

C₂²×C₅²

#41

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

C₂×C₅×D₄

—

#709

—

#709

—

C₂×D₂₀

C₂×D₅

yes

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

C₂×D₅

yes

—

—

—

—

—

C₂×D₅

yes

—

C₂×#168

—

C₂×#168

—

D₄₀

—

#677

—

#676

—

#676

—

C₈×D₅

—

C₈×D₅

—

#472

—

Sylow

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₅⋊D₅

—

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂²⋊D₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

—

—

—

—

—

—

C₂×D₈

—

#131

—

#108

—

#100

—

C₅²

#472

yes

characteristic, complemented, Sylow

100

C₂²×C₅

—

101

C₂²×C₅

—

102

C₂²×C₅

—

103

C₂²×C₅

#709

yes

—

104

C₂×D₅

—

105

C₂×D₅

—

106

C₂×D₅

—

107

C₂×D₅

—

108

C₄×C₅

—

109

C₄×C₅

—

110

C₄×C₅

D₄×D₅

yes

—

111

C₄×C₅

D₄×D₅

yes

—

112

Dic₅

—

113

Dic₅

—

114

#41

—

115

C₂×D₄

—

116

SD₁₆

—

117

D₈

—

118

D₈

—

119

D₈

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

D₅²

yes

characteristic

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₅

—

126

C₂×C₅

—

127

C₂×C₅

—

128

C₂×C₅

—

129

C₂×C₅

#4030

yes

—

130

D₅

—

131

D₅

—

132

D₅

—

133

C₂³

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×D₅²

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₅

—

146

C₅

#65038

yes

complemented

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#87873

yes

characteristic

151

C₄

—

152

C₄

#87877

yes

characteristic

153

C₄

#87886

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

—

100

—

157

C₂

#11717030

yes

characteristic

	subgroup	G/H
center	C₂	#11717030
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×M₁₆	C₂²
socle	C₂×C₅²	#121
hypercenter	M₁₆	D₅²
hypocenter	C₅²	#472
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

subgroup

G/H

center

C₂

#11717030

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×M₁₆

C₂²

socle

C₂×C₅²

#121

hypercenter

M₁₆

D₅²

hypocenter

C₅²

#472

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

N	H ≅ G/N	type
1. #11717227	C₂	semidirect
2. #11716948	C₂	semidirect
3. C₂×#87770	C₂	semidirect
4. C₅×#3955	C₂	semidirect
5. #11716619	C₂	semidirect
36. #3955	D₅	semidirect
99. C₅²	#472	semidirect
146. C₅	#65038	semidirect

H ≅ G/N

type

1. #11717227

C₂

semidirect

2. #11716948

C₂

semidirect

3. C₂×#87770

C₂

semidirect

4. C₅×#3955

C₂

semidirect

5. #11716619

C₂

semidirect

36. #3955

D₅

semidirect

99. C₅²

#472

semidirect

146. C₅

#65038

semidirect

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

subgroup	index
#11717227	2
#11716948	2
C₂×#87770	2
C₅×#3955	2
#11716619	2
#65038	5

subgroup

index

Subgroups — #13101161

Conjugacy classes of subgroups

Complements

Maximal subgroups