Group #13027422 — subgroups

Subgroups — #13027422

Order 1280.

342 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 132 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	#10615059	C₂	1	1	yes	maximal, characteristic
2	#10604150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic
4	#10598728	C₂	1	4	yes	maximal
5	C₅×#533	C₄	1	1	yes	characteristic
6	C₂×#3998	C₂²	1	2	yes	—
7	C₂²×#674	C₂²	1	2	yes	—
8	#65313	—	2	2	no	—
9	Q₈×#168	—	2	2	no	—
10	#65288	C₄	1	2	yes	—
11	#65286	C₂²	1	2	yes	—
12	#65213	C₄	1	1	yes	characteristic
13	C₂×C₄×#168	C₂²	1	1	yes	characteristic
14	C₄⋊#674	—	2	2	no	—
15	C₄×#674	—	2	2	no	—
16	#13266	—	5	1	no	maximal, Sylow
17	C₅×C₄⋊Q₈	C₈	1	2	yes	—
18	C₅×#125	C₈	1	2	yes	—
19	C₅×C₂²⋊Q₈	—	2	2	no	—
20	C₄×C₅×Q₈	—	2	2	no	—
21	C₅×#117	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂×#681	C₂×C₄	1	2	yes	—
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₂²×#168	C₂³	1	1	yes	characteristic
26	#3998	—	2	4	no	—
27	C₂×#674	—	2	2	no	—
28	C₂×#674	—	2	2	no	—
29	C₂×#674	—	2	2	no	—
30	C₂×#674	Q₈	1	4	yes	complemented
31	C₄⋊#168	—	2	2	no	—
32	C₄⋊#168	—	2	4	no	—
33	#3935	—	2	2	no	—
34	C₄×#168	—	2	2	no	—
35	C₄×#168	D₄	1	2	yes	—
36	C₄×#168	D₄	1	2	yes	—
37	#2960	—	5	1	no	—
38	#2150	—	5	1	no	—
39	C₂×C₄×C₁₆	—	5	1	no	—
40	#1379	—	5	4	no	—
41	C₂×C₅×Q₈	—	2	2	no	—
42	C₂²×C₄×C₅	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₅×C₄⋊C₄	—	2	4	no	—
45	C₅×C₄⋊C₄	—	2	2	no	—
46	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
47	C₅×C₂²⋊C₄	—	2	2	no	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
52	#681	—	2	4	no	—
53	C₂×#168	—	2	2	no	—
54	C₂×#168	—	2	2	no	—
55	C₂×#168	C₂²⋊C₄	1	2	yes	—
56	C₂×#168	C₂×D₄	1	1	yes	characteristic
57	C₂×#168	C₂×D₄	1	1	yes	characteristic
58	C₂×#168	C₂×Q₈	1	2	yes	—
59	C₂×#168	#41	1	2	yes	—
60	#674	—	2	4	no	—
61	#674	—	4	2	no	—
62	#533	AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂×M₃₂	—	5	2	no	—
64	C₂²×C₁₆	—	5	2	no	—
65	#446	—	10	2	no	—
66	C₈×Q₈	—	10	2	no	—
67	#424	—	5	2	no	—
68	#423	—	5	1	no	—
69	#405	—	5	2	no	—
70	C₂×C₄×C₈	—	5	1	no	—
71	C₄⋊C₁₆	—	10	2	no	—
72	C₄×C₁₆	—	10	2	no	—
73	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
74	C₅×Q₈	—	4	4	no	—
75	C₂×C₄×C₅	—	2	4	no	—
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅	#131	1	2	yes	—
83	#168	—	2	2	no	—
84	#168	—	4	2	no	—
85	#168	C₂²⋊Q₈	1	2	yes	—
86	#168	#124	1	1	yes	characteristic
87	#168	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×M₁₆	—	5	2	no	—
89	C₂²×C₈	—	5	1	no	—
90	C₂²×C₈	—	5	1	no	—
91	C₄⋊Q₈	#170	1	2	yes	—
92	#125	#170	1	2	yes	—
93	C₂²⋊Q₈	—	2	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	#117	C₂×AGL(1,𝔽₅)	1	2	yes	—
96	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
97	M₃₂	—	10	4	no	—
98	C₂×C₁₆	—	5	4	no	—
99	C₂×C₁₆	—	10	2	no	—
100	C₂×C₁₆	—	10	2	no	—
101	C₂×C₁₆	—	10	2	no	—
102	C₄⋊C₈	—	10	4	no	—
103	C₄⋊C₈	—	10	2	no	—
104	#97	—	10	2	no	—
105	C₄×C₈	—	5	2	no	—
106	C₄×C₈	—	5	2	no	—
107	C₄×C₈	—	10	2	no	—
108	C₂²×C₅	—	2	2	no	—
109	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
110	C₂²×C₅	#446	1	2	yes	—
111	C₄×C₅	—	2	2	no	—
112	C₄×C₅	—	4	4	no	—
113	C₄×C₅	#350	1	4	yes	—
114	C₄×C₅	#391	1	1	yes	characteristic
115	C₄×C₅	#408	1	1	yes	characteristic
116	C₄×C₅	C₈×Q₈	1	2	yes	—
117	C₂×Q₈	—	2	2	no	—
118	C₂²×C₄	#704	1	2	yes	—
119	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
120	M₁₆	—	10	4	no	—
121	C₂×C₈	—	5	2	no	—
122	C₂×C₈	—	5	2	no	—
123	C₂×C₈	—	5	1	no	—
124	C₂×C₈	—	5	1	no	—
125	C₂×C₈	—	5	2	no	—
126	C₂×C₈	—	10	2	no	—
127	C₂×C₈	—	10	2	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	C₂×#170	1	4	yes	—
131	C₂²⋊C₄	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	C₂×#170	1	1	yes	characteristic
134	C₄²	C₂×#170	1	1	yes	characteristic
135	C₄²	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
136	C₁₆	—	10	4	no	—
137	C₁₆	—	20	2	no	—
138	C₂×C₅	—	2	2	no	—
139	C₂×C₅	#1984	1	1	yes	characteristic
140	C₂×C₅	C₂×#350	1	2	yes	—
141	C₂³	C₂×#704	1	1	yes	characteristic
142	Q₈	—	4	4	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	C₂²⋊#170	1	2	yes	—
146	C₂×C₄	C₂²⋊#170	1	2	yes	—
147	C₂×C₄	#4133	1	2	yes	—
148	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
149	C₂×C₄	C₂²×#170	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₈	—	5	2	no	—
152	C₈	—	5	1	no	—
153	C₈	—	5	1	no	—
154	C₈	—	10	2	no	—
155	C₈	—	20	2	no	—
156	C₅	#13266	1	1	yes	characteristic, complemented, Sylow
157	C₂²	—	2	2	no	—
158	C₂²	#65788	1	2	yes	—
159	C₂²	C₂×C₂²⋊#170	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#64932	1	4	yes	—
163	C₄	Q₈×#170	1	2	yes	—
164	C₄	#65787	1	1	yes	characteristic
165	C₄	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	#10604984	1	1	yes	characteristic
168	C₂	C₂×#64932	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10615059

C₂

yes

maximal, characteristic

#10604150

C₂

yes

maximal, characteristic

C₂×C₄×#674

C₂

yes

maximal, characteristic

#10598728

C₂

yes

maximal

C₅×#533

C₄

yes

characteristic

C₂×#3998

C₂²

yes

—

C₂²×#674

C₂²

yes

—

#65313

—

Q₈×#168

—

#65288

C₄

yes

—

#65286

C₂²

yes

—

#65213

C₄

yes

characteristic

C₂×C₄×#168

C₂²

yes

—

—

—

C₅×C₄⋊Q₈

C₈

yes

—

C₅×#125

C₈

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×#681

C₂×C₄

yes

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂³

yes

—

—

—

—

Q₈

yes

—

—

—

—

D₄

yes

—

C₄×#168

D₄

yes

—

#2960

—

#2150

—

C₂×C₄×C₁₆

—

#1379

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

M₁₆

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₈

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

—

—

C₂²⋊C₄

yes

—

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×D₄

yes

characteristic

C₂×#168

C₂×Q₈

yes

—

C₂×#168

#41

yes

—

—

—

yes

C₂×M₃₂

—

C₂²×C₁₆

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₂³×C₅

C₂×M₁₆

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×Q₈

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×C₄×C₅

#131

yes

—

#168

—

#168

—

#168

C₂²⋊Q₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

#170

yes

—

#125

#170

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

#97

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₂²×C₅

—

109

C₂²×C₅

C₂×C₂²⋊C₈

yes

characteristic

110

C₂²×C₅

#446

yes

—

111

C₄×C₅

—

112

C₄×C₅

—

113

C₄×C₅

#350

yes

—

114

C₄×C₅

#391

yes

characteristic

115

C₄×C₅

#408

yes

characteristic

116

C₄×C₅

C₈×Q₈

yes

—

117

C₂×Q₈

—

118

C₂²×C₄

#704

yes

—

119

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

C₂×#170

yes

—

131

C₂²⋊C₄

—

132

C₄²

—

133

C₄²

C₂×#170

yes

characteristic

134

C₄²

C₂×#170

yes

characteristic

135

C₄²

C₂²⋊AGL(1,𝔽₅)

yes

—

136

C₁₆

—

137

C₁₆

—

138

C₂×C₅

—

139

C₂×C₅

#1984

yes

characteristic

140

C₂×C₅

C₂×#350

yes

—

141

C₂³

C₂×#704

yes

characteristic

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₂²⋊#170

yes

—

146

C₂×C₄

C₂²⋊#170

yes

—

147

C₂×C₄

#4133

yes

—

148

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

—

149

C₂×C₄

C₂²×#170

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₅

#13266

yes

characteristic, complemented, Sylow

157

C₂²

—

158

C₂²

#65788

yes

—

159

C₂²

C₂×C₂²⋊#170

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

#64932

yes

—

163

C₄

Q₈×#170

yes

—

164

C₄

#65787

yes

characteristic

165

C₄

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

166

C₂

—

167

C₂

#10604984

yes

characteristic

168

C₂

C₂×#64932

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#533	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×#168	C₂³
hypercenter	#533	AGL(1,𝔽₅)
hypocenter	C₅	#13266
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#533

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₈

prime radical

C₂³×C₅

C₂×M₁₆

cosocle

C₂²×#168

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
30. C₂×#674	Q₈	semidirect
156. C₅	#13266	semidirect

H ≅ G/N

type

30. C₂×#674

Q₈

semidirect

156. C₅

#13266

semidirect

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

subgroup	index
#10615059	2
#10604150	2
C₂×C₄×#674	2
#10598728	2
#13266	5

subgroup

index

C₂×C₄×#674

Subgroups — #13027422

Conjugacy classes of subgroups

Complements

Maximal subgroups