Group #13124487 — subgroups

Subgroups — #13124487

Order 1664.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂²×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719521	C₂	1	2	yes	maximal
3	C₂×#88155	C₂	1	1	yes	maximal, characteristic
4	#11718886	C₂	1	2	yes	maximal, complemented
5	#11718850	C₂	1	4	yes	maximal, complemented
6	#11718849	C₂	1	4	yes	maximal, complemented
7	#11718767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₂×#6127	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×Q₁₆	—	2	4	no	—
11	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#88201	—	2	2	no	—
15	#88200	C₂²	1	8	yes	—
16	#88198	—	2	2	no	—
17	#88183	—	2	4	no	—
18	#88180	C₂²	1	2	yes	—
19	C₂×#6099	C₂²	1	1	yes	characteristic
20	#88155	—	2	4	no	—
21	C₂×#6094	C₂²	1	2	yes	—
22	C₂×C₈×D₁₃	C₂²	1	2	yes	—
23	#88075	C₂²	1	8	yes	—
24	#88058	C₂²	1	2	yes	—
25	#88057	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₃	—	2	2	no	—
27	C₂×Q₈×C₁₃	—	2	4	no	—
28	C₂×Q₈×C₁₃	—	2	4	no	—
29	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₁₃	D₄	1	2	yes	—
33	#6127	—	2	2	no	—
34	#6127	—	2	2	no	—
35	C₂×D₅₂	D₄	1	1	yes	characteristic
36	C₂×C₄×D₁₃	C₂³	1	2	yes	—
37	C₂×Dic₂₆	D₄	1	1	yes	characteristic
38	C₁₃×Q₁₆	—	2	4	no	—
39	C₁₃×Q₁₆	—	2	4	no	—
40	C₂×C₈×C₁₃	D₄	1	4	yes	—
41	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₁₃	—	2	2	no	—
44	#6103	—	2	8	no	—
45	C₄⋊Dic₁₃	C₂³	1	4	yes	—
46	#6101	—	2	8	no	—
47	C₄×Dic₁₃	—	2	2	no	—
48	#6099	—	2	2	no	—
49	C₂×#1055	C₂³	1	2	yes	—
50	#6094	—	2	4	no	—
51	C₈×D₁₃	—	2	4	no	—
52	#3059	—	13	1	no	maximal, Sylow
53	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×D₁₃	C₂×D₄	1	2	yes	—
55	Q₈×C₁₃	—	2	4	no	—
56	Q₈×C₁₃	—	2	4	no	—
57	Q₈×C₁₃	—	4	4	no	—
58	C₂×C₄×C₁₃	—	2	4	no	—
59	C₂×C₄×C₁₃	—	4	4	no	—
60	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₃	#41	1	2	yes	—
63	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂²⋊D₁₃	—	2	4	no	—
65	C₂×Dic₁₃	—	2	4	no	—
66	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
67	D₅₂	—	2	2	no	—
68	C₄×D₁₃	—	2	2	no	—
69	C₄×D₁₃	—	2	2	no	—
70	Dic₂₆	—	2	2	no	—
71	C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₈×C₁₃	C₂×D₄	1	2	yes	—
73	#1055	—	2	2	no	—
74	C₂²×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
75	C₂×#131	—	13	1	no	—
76	#536	—	13	2	no	—
77	#470	—	13	4	no	—
78	#467	—	13	4	no	—
79	#427	—	13	1	no	—
80	#419	—	13	2	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	#121	1	2	yes	—
83	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×D₁₃	—	2	2	no	—
85	C₂×D₁₃	—	2	4	no	—
86	C₄×C₁₃	—	4	4	no	—
87	C₄×C₁₃	#121	1	2	yes	—
88	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
90	Dic₁₃	—	2	2	no	—
91	Dic₁₃	—	4	4	no	—
92	C₂×#41	—	13	1	no	—
93	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
94	C₂×Q₁₆	—	2	4	no	—
95	C₂×Q₁₆	C₂×D₁₃	1	2	yes	—
96	C₂×Q₁₆	C₂×D₁₃	1	2	yes	—
97	#131	—	26	4	no	—
98	C₂×M₁₆	—	13	1	no	—
99	C₂×M₁₆	—	13	2	no	—
100	C₂²×C₈	—	13	2	no	—
101	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
102	C₄⋊Q₈	—	26	2	no	—
103	#124	—	26	2	no	—
104	#123	—	26	4	no	—
105	C₂²⋊Q₈	—	13	8	no	—
106	#117	—	13	2	no	—
107	#107	—	13	2	no	—
108	#106	—	13	2	no	—
109	#103	—	13	8	no	—
110	C₂×C₁₃	—	2	2	no	—
111	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
112	C₂×C₁₃	#578	1	2	yes	—
113	D₁₃	—	4	2	no	—
114	#41	—	26	2	no	—
115	#41	—	26	2	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	—	2	4	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂×Q₈	—	13	1	no	—
120	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
121	C₂×D₄	—	13	2	no	—
122	C₂×D₄	—	13	1	no	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	Q₁₆	—	2	4	no	—
127	Q₁₆	—	2	4	no	—
128	M₁₆	—	26	2	no	—
129	M₁₆	—	26	4	no	—
130	C₂×C₈	—	13	2	no	—
131	C₂×C₈	—	26	4	no	—
132	C₂×C₈	C₂²⋊D₁₃	1	4	yes	—
133	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
134	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
135	C₄⋊C₄	—	13	4	no	—
136	C₄⋊C₄	—	26	8	no	—
137	C₂²⋊C₄	—	26	8	no	—
138	C₂²⋊C₄	—	26	2	no	—
139	C₄²	—	26	2	no	—
140	C₁₃	#3059	1	1	yes	characteristic, complemented, Sylow
141	C₂³	—	13	2	no	—
142	C₂³	D₄×D₁₃	1	1	yes	characteristic
143	Q₈	—	2	4	no	—
144	Q₈	—	2	4	no	—
145	Q₈	—	4	4	no	—
146	Q₈	—	26	2	no	—
147	D₄	—	26	4	no	—
148	D₄	—	26	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	13	2	no	—
152	C₂×C₄	—	26	2	no	—
153	C₂×C₄	—	26	2	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
156	C₂×C₄	#6129	1	2	yes	—
157	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
158	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
159	C₈	—	26	2	no	—
160	C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
161	C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	26	4	no	—
164	C₂²	—	26	2	no	—
165	C₂²	#88192	1	2	yes	—
166	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	26	2	no	—
169	C₄	—	52	4	no	—
170	C₄	#88192	1	2	yes	—
171	C₄	C₂×#6129	1	1	yes	characteristic
172	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	52	2	no	—
175	C₂	#11719471	1	2	yes	—
176	C₂	C₂×#88192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×Q₁₆

C₂

yes

maximal, characteristic, complemented

#11719521

C₂

yes

maximal

C₂×#88155

C₂

yes

maximal, characteristic

#11718886

C₂

yes

maximal, complemented

#11718850

C₂

yes

maximal, complemented

#11718849

C₂

yes

maximal, complemented

#11718767

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂×#6127

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

C₂²

yes

—

C₂×C₁₃×Q₁₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

#88201

—

#88200

C₂²

yes

—

#88198

—

#88183

—

#88180

C₂²

yes

—

C₂×#6099

C₂²

yes

characteristic

#88155

—

C₂×#6094

C₂²

yes

—

C₂×C₈×D₁₃

C₂²

yes

—

#88075

C₂²

yes

—

#88058

C₂²

yes

—

#88057

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₂²⋊D₁₃

D₄

yes

—

#6127

—

#6127

—

C₂×D₅₂

D₄

yes

characteristic

C₂×C₄×D₁₃

C₂³

yes

—

C₂×Dic₂₆

D₄

yes

characteristic

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

#6103

—

C₄⋊Dic₁₃

C₂³

yes

—

#6101

—

C₄×Dic₁₃

—

#6099

—

C₂×#1055

C₂³

yes

—

#6094

—

C₈×D₁₃

—

#3059

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂²×D₁₃

C₂×D₄

yes

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂×D₄

yes

—

C₈×C₁₃

C₂×D₄

yes

—

#1055

—

C₂²×Q₁₆

D₁₃

yes

characteristic, complemented

—

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

#121

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

—

C₄×C₁₃

C₂²×D₄

yes

characteristic

C₄×C₁₃

C₂×#41

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×#41

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₁₃

yes

—

C₂×Q₁₆

C₂×D₁₃

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×D₁₃

yes

characteristic

102

C₄⋊Q₈

—

103

#124

—

104

#123

—

105

C₂²⋊Q₈

—

106

#117

—

107

#107

—

108

#106

—

109

#103

—

110

C₂×C₁₃

—

111

C₂×C₁₃

C₂×#121

yes

characteristic

112

C₂×C₁₃

#578

yes

—

113

D₁₃

—

114

#41

—

115

#41

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

C₂²×D₁₃

yes

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂²×D₁₃

yes

characteristic

126

Q₁₆

—

127

Q₁₆

—

128

M₁₆

—

129

M₁₆

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₂²⋊D₁₃

yes

—

133

C₂×C₈

C₂²×D₁₃

yes

characteristic

134

C₂×C₈

C₂²×D₁₃

yes

characteristic

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₂²⋊C₄

—

139

C₄²

—

140

C₁₃

#3059

yes

characteristic, complemented, Sylow

141

C₂³

—

142

C₂³

D₄×D₁₃

yes

characteristic

143

Q₈

—

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

D₄×D₁₃

yes

characteristic

156

C₂×C₄

#6129

yes

—

157

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

158

C₂×C₄

C₂³×D₁₃

yes

characteristic

159

C₈

—

160

C₈

C₂×C₂²⋊D₁₃

yes

—

161

C₈

C₂×C₂²⋊D₁₃

yes

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

#88192

yes

—

166

C₂²

C₂×D₄×D₁₃

yes

characteristic

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#88192

yes

—

171

C₄

C₂×#6129

yes

characteristic

172

C₄

C₂²×C₂²⋊D₁₃

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#11719471

yes

—

176

C₂

C₂×#88192

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#6127	C₂²
hypercenter	C₂²×Q₁₆	D₁₃
hypocenter	C₁₃	#3059
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃

commutator, cosocle

C₂×C₄×C₁₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃

Fitting

C₂²×C₁₃×Q₁₆

C₂

socle

C₂²×C₁₃

C₂²×D₄

prime radical

C₂×#6127

C₂²

hypercenter

C₂²×Q₁₆

D₁₃

hypocenter

C₁₃

#3059

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
1. C₂²×C₁₃×Q₁₆	C₂	semidirect
4. #11718886	C₂	semidirect
5. #11718850	C₂	semidirect
6. #11718849	C₂	semidirect
7. #11718767	C₂	semidirect
74. C₂²×Q₁₆	D₁₃	semidirect
140. C₁₃	#3059	semidirect

H ≅ G/N

type

1. C₂²×C₁₃×Q₁₆

C₂

semidirect

4. #11718886

C₂

semidirect

5. #11718850

C₂

semidirect

6. #11718849

C₂

semidirect

7. #11718767

C₂

semidirect

74. C₂²×Q₁₆

D₁₃

semidirect

140. C₁₃

#3059

semidirect

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

subgroup	index
C₂²×C₁₃×Q₁₆	2
#11719521	2
C₂×#88155	2
#11718886	2
#11718850	2
#11718849	2
#11718767	2
#3059	13

subgroup

index

C₂²×C₁₃×Q₁₆

Subgroups — #13124487

Conjugacy classes of subgroups

Complements

Maximal subgroups