Group #13116083 — subgroups

Subgroups — #13116083

Order 1664.

298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂×#88123	C₂	1	1	yes	maximal, characteristic
2	C₂×#88056	C₂	1	2	yes	maximal
3	C₂×#88056	C₂	1	2	yes	maximal
4	#11718589	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃	—	2	1	no	—
10	C₂×C₄⋊Dic₁₃	—	2	1	no	—
11	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6101	—	2	1	no	—
14	#88123	—	2	4	no	—
15	#88071	—	2	1	no	—
16	#88056	—	2	4	no	—
17	#88056	—	2	4	no	—
18	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₁₃	—	2	1	no	—
22	C₂²×Dic₁₃	—	2	2	no	—
23	C₂²×Dic₁₃	—	2	1	no	—
24	C₂×Dic₂₆	—	2	2	no	—
25	C₂×Dic₂₆	—	2	2	no	—
26	C₂×Dic₂₆	—	2	2	no	—
27	C₂×Dic₂₆	—	2	2	no	—
28	C₂×Dic₂₆	—	2	2	no	—
29	C₂×Dic₂₆	D₄	1	4	yes	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	—	2	2	no	—
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₄²×C₁₃	—	2	2	no	—
34	C₄⋊Dic₁₃	—	2	4	no	—
35	C₄⋊Dic₁₃	—	2	4	no	—
36	C₄⋊Dic₁₃	—	2	1	no	—
37	C₄⋊Dic₁₃	—	2	2	no	—
38	C₄⋊Dic₁₃	D₄	1	2	yes	—
39	#6101	—	4	2	no	—
40	#2043	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	2	1	no	—
46	C₂×C₄×C₁₃	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₃	Q₁₆	1	2	yes	—
48	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃	#41	1	2	yes	—
51	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
52	C₂×Dic₁₃	—	2	4	no	—
53	C₂×Dic₁₃	—	2	2	no	—
54	C₂×Dic₁₃	—	4	2	no	—
55	C₂×Dic₁₃	—	4	2	no	—
56	C₂×Dic₁₃	—	4	1	no	—
57	C₂×Dic₁₃	—	4	1	no	—
58	C₂×Dic₁₃	—	4	1	no	—
59	C₂×Dic₁₃	—	4	1	no	—
60	C₂×Dic₁₃	—	4	1	no	—
61	Dic₂₆	—	2	4	no	—
62	Dic₂₆	—	2	4	no	—
63	Dic₂₆	—	4	2	no	—
64	Dic₂₆	—	4	2	no	—
65	C₈×C₁₃	—	4	2	no	—
66	C₂×C₄⋊Q₈	—	13	1	no	—
67	C₂×#103	—	13	2	no	—
68	C₂×#103	—	13	2	no	—
69	#389	—	13	1	no	—
70	#336	D₁₃	1	1	yes	characteristic
71	C₂²×C₁₃	#120	1	1	yes	characteristic
72	C₂²×C₁₃	#121	1	1	yes	characteristic
73	C₂²×C₁₃	#124	1	1	yes	characteristic
74	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₁₃	#137	1	1	yes	characteristic
77	C₂²×C₁₃	#137	1	1	yes	characteristic
78	C₄×C₁₃	—	2	2	no	—
79	C₄×C₁₃	—	2	2	no	—
80	C₄×C₁₃	#121	1	2	yes	—
81	C₄×C₁₃	#123	1	2	yes	—
82	Dic₁₃	—	4	4	no	—
83	Dic₁₃	—	4	2	no	—
84	Dic₁₃	—	8	1	no	—
85	C₂²×Q₈	—	13	2	no	—
86	C₂²×C₈	C₂×D₁₃	1	2	yes	—
87	C₄⋊Q₈	—	26	4	no	—
88	C₂×C₄⋊C₄	—	13	1	no	—
89	C₂×C₄⋊C₄	—	13	1	no	—
90	C₂×C₄⋊C₄	—	26	1	no	—
91	C₂×C₄⋊C₄	—	26	1	no	—
92	C₂×C₄⋊C₄	—	26	1	no	—
93	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
94	#103	—	26	4	no	—
95	#103	—	26	4	no	—
96	#95	—	26	1	no	—
97	C₂×C₁₃	#394	1	1	yes	characteristic
98	C₂×C₁₃	#448	1	1	yes	characteristic
99	C₂×C₁₃	#451	1	1	yes	characteristic
100	C₂×C₁₃	#461	1	1	yes	characteristic
101	C₂×C₁₃	#462	1	1	yes	characteristic
102	C₂×C₁₃	#487	1	1	yes	characteristic
103	C₂×C₁₃	#491	1	1	yes	characteristic
104	C₂×Q₈	—	13	4	no	—
105	C₂×Q₈	—	26	2	no	—
106	C₂×Q₈	—	26	2	no	—
107	C₂×Q₈	—	26	2	no	—
108	C₂×Q₈	—	26	2	no	—
109	C₂×Q₈	—	26	2	no	—
110	C₂²×C₄	—	26	1	no	—
111	C₂²×C₄	—	26	2	no	—
112	C₂²×C₄	—	26	1	no	—
113	C₂²×C₄	D₅₂	1	1	yes	characteristic
114	C₂²×C₄	D₅₂	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₄⋊C₄	—	13	2	no	—
120	C₄⋊C₄	—	26	4	no	—
121	C₄⋊C₄	—	26	4	no	—
122	C₄⋊C₄	—	26	1	no	—
123	C₄⋊C₄	—	26	2	no	—
124	C₄⋊C₄	—	52	2	no	—
125	C₄²	—	2	2	no	—
126	C₁₃	#2043	1	1	yes	characteristic, complemented, Sylow
127	C₂³	C₂×D₅₂	1	1	yes	characteristic
128	Q₈	—	26	4	no	—
129	Q₈	—	26	4	no	—
130	Q₈	—	52	2	no	—
131	Q₈	—	52	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	26	4	no	—
137	C₂×C₄	—	26	2	no	—
138	C₂×C₄	—	52	2	no	—
139	C₂×C₄	—	52	2	no	—
140	C₂×C₄	—	52	1	no	—
141	C₂×C₄	—	52	1	no	—
142	C₂×C₄	—	52	1	no	—
143	C₂×C₄	—	52	1	no	—
144	C₂×C₄	—	52	1	no	—
145	C₂×C₄	#6095	1	2	yes	—
146	C₂×C₄	Dic₅₂	1	2	yes	—
147	C₂×C₄	#6127	1	2	yes	—
148	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
149	C₂×C₄	D₄×D₁₃	1	2	yes	—
150	C₂×C₄	#6131	1	1	yes	characteristic
151	C₈	—	4	2	no	—
152	C₂²	#88129	1	1	yes	characteristic
153	C₂²	#88136	1	1	yes	characteristic
154	C₂²	#88149	1	1	yes	characteristic
155	C₂²	C₂×#6095	1	1	yes	characteristic
156	C₂²	C₂×Dic₅₂	1	1	yes	characteristic
157	C₂²	#88163	1	1	yes	characteristic
158	C₂²	#88163	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	52	2	no	—
162	C₄	—	52	4	no	—
163	C₄	—	104	1	no	—
164	C₄	#88138	1	2	yes	—
165	C₄	#88148	1	2	yes	—
166	C₂	#11718333	1	1	yes	characteristic
167	C₂	#11718342	1	1	yes	characteristic
168	C₂	#11718364	1	1	yes	characteristic
169	C₂	#11718399	1	1	yes	characteristic
170	C₂	#11718400	1	1	yes	characteristic
171	C₂	#11718502	1	1	yes	characteristic
172	C₂	#11718510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88123

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

C₂×#88056

C₂

yes

maximal

#11718589

C₂

yes

maximal, characteristic

C₁₃×#336

C₂

yes

maximal, characteristic

C₂²×Dic₂₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

#6101

—

#2043

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

—

C₂×C₄×C₁₃

Q₁₆

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

—

—

—

—

D₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₃

C₂×Q₁₆

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#121

yes

—

C₄×C₁₃

#123

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

100

C₂×C₁₃

#461

yes

characteristic

101

C₂×C₁₃

#462

yes

characteristic

102

C₂×C₁₃

#487

yes

characteristic

103

C₂×C₁₃

#491

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

D₅₂

yes

characteristic

114

C₂²×C₄

D₅₂

yes

characteristic

115

C₂²×C₄

C₂²×D₁₃

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₁₃

#2043

yes

characteristic, complemented, Sylow

127

C₂³

C₂×D₅₂

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#6095

yes

—

146

C₂×C₄

Dic₅₂

yes

—

147

C₂×C₄

#6127

yes

—

148

C₂×C₄

D₄×D₁₃

yes

characteristic

149

C₂×C₄

D₄×D₁₃

yes

—

150

C₂×C₄

#6131

yes

characteristic

151

C₈

—

152

C₂²

#88129

yes

characteristic

153

C₂²

#88136

yes

characteristic

154

C₂²

#88149

yes

characteristic

155

C₂²

C₂×#6095

yes

characteristic

156

C₂²

C₂×Dic₅₂

yes

characteristic

157

C₂²

#88163

yes

characteristic

158

C₂²

#88163

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

104

—

164

C₄

#88138

yes

—

165

C₄

#88148

yes

—

166

C₂

#11718333

yes

characteristic

167

C₂

#11718342

yes

characteristic

168

C₂

#11718364

yes

characteristic

169

C₂

#11718399

yes

characteristic

170

C₂

#11718400

yes

characteristic

171

C₂

#11718502

yes

characteristic

172

C₂

#11718510

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#336	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
hypercenter	#336	D₁₃
hypocenter	C₁₃	#2043
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₁₃×#336

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
126. C₁₃	#2043	semidirect

H ≅ G/N

type

126. C₁₃

#2043

semidirect

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

subgroup	index
C₂×#88123	2
C₂×#88056	2
#11718589	2
C₁₃×#336	2
#2043	13

subgroup

index

C₁₃×#336

Subgroups — #13116083

Conjugacy classes of subgroups

Complements

Maximal subgroups