Group #13115653 — subgroups

Subgroups — #13115653

Order 1664.

344 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×#88075	C₂	1	2	yes	maximal, complemented
3	C₂×#88072	C₂	1	2	yes	maximal
4	#11718596	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#124	—	2	2	no	—
9	C₁₃×#124	C₄	1	4	yes	complemented
10	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	—	2	2	no	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₂²×#1055	C₂²	1	2	yes	—
15	#88075	—	2	4	no	—
16	#88072	—	2	4	no	—
17	#88043	—	2	2	no	—
18	C₂⁴×C₁₃	—	2	1	no	—
19	C₂×Q₈×C₁₃	—	2	1	no	—
20	C₂×Q₈×C₁₃	—	2	2	no	—
21	C₂×Q₈×C₁₃	—	2	2	no	—
22	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₁₃	—	2	2	no	—
24	C₂×D₄×C₁₃	—	2	1	no	—
25	C₂×D₄×C₁₃	—	2	2	no	—
26	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₃	—	2	1	no	—
28	C₂²×C₄×C₁₃	D₄	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₃	—	2	2	no	—
31	C₁₃×C₂²⋊C₄	—	2	8	no	—
32	C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₄²×C₁₃	D₄	1	2	yes	—
34	C₄⋊Dic₁₃	—	2	2	no	—
35	C₄⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	—	4	4	no	—
37	C₂×#1055	—	2	2	no	—
38	C₂×#1055	—	2	2	no	—
39	C₂×#1055	D₄	1	4	yes	—
40	#1959	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	—	2	2	no	—
42	C₂³×C₁₃	—	4	1	no	—
43	C₂³×C₁₃	—	4	1	no	—
44	C₂³×C₁₃	—	4	2	no	—
45	C₂³×C₁₃	—	4	2	no	—
46	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₃	—	2	4	no	—
48	Q₈×C₁₃	—	4	2	no	—
49	D₄×C₁₃	—	2	4	no	—
50	D₄×C₁₃	—	4	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	4	1	no	—
55	C₂×C₄×C₁₃	—	4	1	no	—
56	C₂×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#41	1	2	yes	—
61	C₂×Dic₁₃	—	4	2	no	—
62	C₂×Dic₁₃	—	4	4	no	—
63	#1055	—	2	4	no	—
64	C₂×#124	D₁₃	1	1	yes	characteristic
65	C₂×#103	—	13	2	no	—
66	C₂×#102	—	13	2	no	—
67	C₂×#97	—	13	1	no	—
68	#384	—	13	1	no	—
69	C₂²×C₁₃	—	4	2	no	—
70	C₂²×C₁₃	—	4	4	no	—
71	C₂²×C₁₃	—	8	2	no	—
72	C₂²×C₁₃	—	8	1	no	—
73	C₂²×C₁₃	—	8	1	no	—
74	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₁₃	#124	1	1	yes	characteristic
76	C₂²×C₁₃	#127	1	1	yes	characteristic
77	C₂²×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₂²×C₁₃	#137	1	2	yes	—
79	C₄×C₁₃	—	2	4	no	—
80	C₄×C₁₃	—	4	2	no	—
81	C₄×C₁₃	C₄×D₄	1	2	yes	—
82	C₄×C₁₃	#121	1	2	yes	—
83	Dic₁₃	—	8	2	no	—
84	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
85	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
86	C₂²×C₈	—	13	2	no	—
87	#124	—	2	2	no	—
88	#124	Dic₁₃	1	4	yes	complemented
89	C₂×C₄⋊C₄	—	13	2	no	—
90	C₂×C₄⋊C₄	—	26	2	no	—
91	C₂×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
93	#103	—	26	4	no	—
94	#102	—	26	4	no	—
95	#97	—	26	2	no	—
96	C₂×C₁₃	—	8	2	no	—
97	C₂×C₁₃	#390	1	1	yes	characteristic
98	C₂×C₁₃	#417	1	2	yes	—
99	C₂×C₁₃	#489	1	2	yes	—
100	C₂×C₁₃	#496	1	1	yes	characteristic
101	C₂×C₁₃	#497	1	1	yes	characteristic
102	C₂⁴	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	26	2	no	—
113	C₂²×C₄	C₂²⋊D₁₃	1	2	yes	—
114	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×C₈	—	13	4	no	—
116	C₂×C₈	—	26	2	no	—
117	C₂×C₈	—	26	2	no	—
118	C₄⋊C₄	—	26	2	no	—
119	C₄⋊C₄	—	26	2	no	—
120	C₄⋊C₄	—	52	4	no	—
121	C₂²⋊C₄	—	2	8	no	—
122	C₄²	C₂×Dic₁₃	1	2	yes	—
123	C₄²	C₂²⋊D₁₃	1	2	yes	—
124	C₁₃	#1959	1	1	yes	characteristic, complemented, Sylow
125	C₂³	—	2	2	no	—
126	C₂³	—	4	1	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	Q₈	—	2	4	no	—
132	Q₈	—	4	2	no	—
133	D₄	—	2	4	no	—
134	D₄	—	4	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	52	2	no	—
141	C₂×C₄	—	52	4	no	—
142	C₂×C₄	C₂²⋊Dic₁₃	1	4	yes	—
143	C₂×C₄	D₄×D₁₃	1	2	yes	—
144	C₂×C₄	#6129	1	2	yes	—
145	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
147	C₈	—	26	4	no	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	4	no	—
150	C₂²	—	8	2	no	—
151	C₂²	—	8	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	#88190	1	1	yes	characteristic
154	C₂²	#88194	1	1	yes	characteristic
155	C₂²	#88203	1	2	yes	—
156	C₂²	#88205	1	2	yes	—
157	C₂²	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	—	4	2	no	—
160	C₄	—	104	2	no	—
161	C₄	D₄×Dic₁₃	1	2	yes	—
162	C₄	#88192	1	2	yes	—
163	C₂	—	8	2	no	—
164	C₂	#11718716	1	2	yes	—
165	C₂	#11718719	1	1	yes	characteristic
166	C₂	#11718721	1	1	yes	characteristic
167	C₂	#11718882	1	1	yes	characteristic
168	C₂	#11718893	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#124

C₂

yes

maximal, characteristic

C₂×#88075

C₂

yes

maximal, complemented

C₂×#88072

C₂

yes

maximal

#11718596

C₂

yes

maximal, characteristic, complemented

C₂×#88043

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×D₄×C₁₃

C₂²

yes

characteristic

C₁₃×#124

—

C₁₃×#124

C₄

yes

complemented

C₂×C₁₃×C₂²⋊C₄

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂²×#1055

C₂²

yes

—

#88075

—

#88072

—

#88043

—

C₂⁴×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

C₂×C₄

yes

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₄

yes

—

#1959

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×D₄

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1055

—

C₂×#124

D₁₃

yes

characteristic

C₂×#103

—

C₂×#102

—

C₂×#97

—

#384

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#137

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₄

yes

—

C₄×C₁₃

#121

yes

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

characteristic

C₂²×D₄

C₂×D₁₃

yes

characteristic

C₂²×C₈

—

#124

—

#124

Dic₁₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#417

yes

—

C₂×C₁₃

#489

yes

—

100

C₂×C₁₃

#496

yes

characteristic

101

C₂×C₁₃

#497

yes

characteristic

102

C₂⁴

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂×Dic₁₃

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

C₂×Dic₁₃

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²⋊D₁₃

yes

—

114

C₂²×C₄

C₂²×D₁₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₂×Dic₁₃

yes

—

123

C₄²

C₂²⋊D₁₃

yes

—

124

C₁₃

#1959

yes

characteristic, complemented, Sylow

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂×C₂²⋊D₁₃

yes

characteristic

131

Q₈

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂²⋊Dic₁₃

yes

—

143

C₂×C₄

D₄×D₁₃

yes

—

144

C₂×C₄

#6129

yes

—

145

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#88190

yes

characteristic

154

C₂²

#88194

yes

characteristic

155

C₂²

#88203

yes

—

156

C₂²

#88205

yes

—

157

C₂²

C₂×C₂²⋊Dic₁₃

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

104

—

161

C₄

D₄×Dic₁₃

yes

—

162

C₄

#88192

yes

—

163

C₂

—

164

C₂

#11718716

yes

—

165

C₂

#11718719

yes

characteristic

166

C₂

#11718721

yes

characteristic

167

C₂

#11718882

yes

characteristic

168

C₂

#11718893

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#124	C₂
socle	C₂³×C₁₃	C₂×D₄
prime radical	C₂²×D₄×C₁₃	C₂²
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	#1959
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₁₃×#124

C₂

socle

C₂³×C₁₃

C₂×D₄

prime radical

C₂²×D₄×C₁₃

C₂²

cosocle

C₂²×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
2. C₂×#88075	C₂	semidirect
4. #11718596	C₂	semidirect
5. C₂×#88043	C₂	semidirect
9. C₁₃×#124	C₄	semidirect
88. #124	Dic₁₃	semidirect
124. C₁₃	#1959	semidirect

H ≅ G/N

type

2. C₂×#88075

C₂

semidirect

4. #11718596

C₂

semidirect

5. C₂×#88043

C₂

semidirect

9. C₁₃×#124

C₄

semidirect

88. #124

Dic₁₃

semidirect

124. C₁₃

#1959

semidirect

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

subgroup	index
C₂×C₁₃×#124	2
C₂×#88075	2
C₂×#88072	2
#11718596	2
C₂×#88043	2
#1959	13

subgroup

index

C₂×C₁₃×#124

Subgroups — #13115653

Conjugacy classes of subgroups

Complements

Maximal subgroups