Group #13130784 — subgroups

Subgroups — #13130784

Order 1664.

324 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 90 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×#6118	C₂	1	1	yes	maximal, characteristic
2	C₂×#88112	C₂	1	2	yes	maximal, complemented
3	C₂×#88111	C₂	1	1	yes	maximal, characteristic
4	#11718327	C₂	1	1	yes	maximal, characteristic
5	#11718290	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#1057	C₂²	1	1	yes	characteristic
8	C₂²×#1057	C₂²	1	1	yes	characteristic
9	C₂²×#1057	C₂²	1	2	yes	—
10	C₂×#6118	—	2	2	no	—
11	C₂×#6118	—	2	4	no	—
12	C₂×#6103	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
14	#88112	—	2	4	no	—
15	#88111	—	2	2	no	—
16	C₁₃×#95	C₄	1	1	yes	characteristic
17	#88071	C₄	1	1	yes	characteristic
18	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄×D₁₃	—	2	2	no	—
24	C₂×C₄×D₁₃	—	2	2	no	—
25	C₂×C₄×D₁₃	D₄	1	2	yes	—
26	C₂×C₄×D₁₃	Q₈	1	2	yes	—
27	C₂×#1057	—	2	2	no	—
28	C₂×#1057	—	2	2	no	—
29	C₂×#1057	—	2	1	no	—
30	C₂×#1057	—	2	1	no	—
31	C₂×#1057	—	2	2	no	—
32	C₂×#1057	—	2	2	no	—
33	C₂×#1057	D₄	1	4	yes	—
34	#6118	—	2	8	no	—
35	#6103	—	2	4	no	—
36	C₄×Dic₁₃	—	2	2	no	—
37	C₄×Dic₁₃	—	2	2	no	—
38	#1851	—	13	1	no	maximal, Sylow
39	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×D₁₃	—	2	2	no	—
41	C₂²×D₁₃	—	2	2	no	—
42	C₂²×D₁₃	—	2	1	no	—
43	C₂²×D₁₃	—	2	1	no	—
44	C₂²×D₁₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	2	no	—
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₁₃	—	2	2	no	—
51	C₂×Dic₁₃	—	2	2	no	—
52	C₂×Dic₁₃	M₁₆	1	4	yes	complemented
53	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
54	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×Dic₁₃	#41	1	2	yes	—
57	C₄×D₁₃	—	2	2	no	—
58	C₄×D₁₃	—	2	4	no	—
59	C₄×D₁₃	—	2	2	no	—
60	#1057	—	2	4	no	—
61	#1057	—	4	1	no	—
62	#1057	—	4	1	no	—
63	C₂²×M₁₆	—	13	1	no	—
64	C₂×C₄⋊C₈	—	13	2	no	—
65	C₂×C₂²⋊C₈	—	13	1	no	—
66	C₄×C₂²⋊C₄	—	13	1	no	—
67	#336	—	13	2	no	—
68	C₂²×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₃	C₄×D₄	1	2	yes	—
70	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
71	C₂²×C₁₃	#131	1	1	yes	characteristic
72	C₂²×C₁₃	#131	1	1	yes	characteristic
73	C₂×D₁₃	—	2	2	no	—
74	C₂×D₁₃	—	2	2	no	—
75	C₂×D₁₃	—	2	2	no	—
76	C₂×D₁₃	—	4	2	no	—
77	C₂×D₁₃	—	4	2	no	—
78	C₄×C₁₃	—	2	2	no	—
79	C₄×C₁₃	—	4	2	no	—
80	Dic₁₃	—	2	4	no	—
81	Dic₁₃	#120	1	1	yes	characteristic
82	Dic₁₃	C₂²⋊Q₈	1	2	yes	—
83	Dic₁₃	#123	1	1	yes	characteristic
84	C₂³×C₄	—	13	1	no	—
85	C₂×M₁₆	—	26	4	no	—
86	C₂×M₁₆	—	26	2	no	—
87	C₂²×C₈	—	13	2	no	—
88	C₂²×C₈	—	13	1	no	—
89	C₂²×C₈	—	13	1	no	—
90	C₂×C₂²⋊C₄	—	13	2	no	—
91	C₂×C₄²	—	13	2	no	—
92	C₄⋊C₈	—	26	4	no	—
93	C₂²⋊C₈	—	26	2	no	—
94	#95	—	13	1	no	—
95	#95	#261	1	1	yes	characteristic
96	C₂×C₁₃	#385	1	1	yes	characteristic
97	C₂×C₁₃	#423	1	1	yes	characteristic
98	C₂×C₁₃	#424	1	1	yes	characteristic
99	C₂×C₁₃	#435	1	2	yes	—
100	C₂×C₁₃	#436	1	2	yes	—
101	D₁₃	—	4	2	no	—
102	C₂⁴	—	13	1	no	—
103	C₂²×C₄	—	13	1	no	—
104	C₂²×C₄	—	13	2	no	—
105	C₂²×C₄	—	13	2	no	—
106	C₂²×C₄	—	13	2	no	—
107	C₂²×C₄	—	26	2	no	—
108	C₂²×C₄	—	26	2	no	—
109	C₂²×C₄	C₂×#261	1	2	yes	—
110	C₂²×C₄	C₂×#261	1	1	yes	characteristic
111	M₁₆	—	26	8	no	—
112	C₂×C₈	—	13	4	no	—
113	C₂×C₈	—	26	2	no	—
114	C₂×C₈	—	26	2	no	—
115	C₂×C₈	—	26	1	no	—
116	C₂×C₈	—	26	1	no	—
117	C₂×C₈	—	26	2	no	—
118	C₂×C₈	—	26	2	no	—
119	C₂²⋊C₄	—	26	4	no	—
120	C₄²	—	26	2	no	—
121	C₄²	—	26	2	no	—
122	C₁₃	#1851	1	1	yes	characteristic, complemented, Sylow
123	C₂³	—	13	2	no	—
124	C₂³	—	26	2	no	—
125	C₂³	—	26	1	no	—
126	C₂³	—	26	2	no	—
127	C₂³	—	26	1	no	—
128	C₂³	C₂²×#261	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	13	1	no	—
134	C₂×C₄	—	13	1	no	—
135	C₂×C₄	—	13	2	no	—
136	C₂×C₄	—	13	2	no	—
137	C₂×C₄	—	13	4	no	—
138	C₂×C₄	—	26	2	no	—
139	C₂×C₄	—	26	4	no	—
140	C₂×C₄	—	26	2	no	—
141	C₂×C₄	—	26	2	no	—
142	C₂×C₄	—	26	2	no	—
143	C₂×C₄	C₄⋊#261	1	2	yes	—
144	C₈	—	26	4	no	—
145	C₈	—	52	1	no	—
146	C₈	—	52	1	no	—
147	C₂²	—	26	2	no	—
148	C₂²	—	26	2	no	—
149	C₂²	—	26	2	no	—
150	C₂²	—	52	2	no	—
151	C₂²	—	52	2	no	—
152	C₂²	D₁₃⋊M₁₆	1	2	yes	—
153	C₂²	C₂×C₄⋊#261	1	1	yes	characteristic
154	C₂²	#88238	1	1	yes	characteristic
155	C₂²	D₄×#261	1	2	yes	—
156	C₂²	#88240	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	13	1	no	—
160	C₄	—	13	2	no	—
161	C₄	—	13	1	no	—
162	C₄	—	26	4	no	—
163	C₂	—	52	2	no	—
164	C₂	#11719053	1	1	yes	characteristic
165	C₂	#11719066	1	2	yes	—
166	C₂	#11719070	1	1	yes	characteristic
167	C₂	#11719075	1	2	yes	—
168	C₂	#11719079	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6118

C₂

yes

maximal, characteristic

C₂×#88112

C₂

yes

maximal, complemented

C₂×#88111

C₂

yes

maximal, characteristic

#11718327

C₂

yes

maximal, characteristic

#11718290

C₂

yes

maximal, complemented

C₂²×C₄×D₁₃

C₂²

yes

characteristic

C₂²×#1057

C₂²

yes

characteristic

C₂²×#1057

C₂²

yes

characteristic

C₂²×#1057

C₂²

yes

—

C₂×#6118

—

C₂×#6118

—

C₂×#6103

C₄

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88112

—

#88111

—

C₁₃×#95

C₄

yes

characteristic

#88071

C₄

yes

characteristic

C₂³×D₁₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₄

yes

—

C₂²×Dic₁₃

C₂³

yes

characteristic

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

—

C₂×C₄×D₁₃

Q₈

yes

—

—

—

—

—

—

—

D₄

yes

—

#6118

—

#6103

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

#1851

—

maximal, Sylow

C₂³×C₁₃

C₂²×C₄

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

M₁₆

yes

complemented

C₂×Dic₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

C₂×Q₈

yes

characteristic

C₂×Dic₁₃

#41

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

#1057

—

#1057

—

C₂²×M₁₆

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₄×C₂²⋊C₄

—

#336

—

C₂²×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₃

C₄×D₄

yes

—

C₂²×C₁₃

C₂×M₁₆

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

#120

yes

characteristic

Dic₁₃

C₂²⋊Q₈

yes

—

Dic₁₃

#123

yes

characteristic

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

100

C₂×C₁₃

#436

yes

—

101

D₁₃

—

102

C₂⁴

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×#261

yes

—

110

C₂²×C₄

C₂×#261

yes

characteristic

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₁₃

#1851

yes

characteristic, complemented, Sylow

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₂²×#261

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

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

C₄⋊#261

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

D₁₃⋊M₁₆

yes

—

153

C₂²

C₂×C₄⋊#261

yes

characteristic

154

C₂²

#88238

yes

characteristic

155

C₂²

D₄×#261

yes

—

156

C₂²

#88240

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

#11719053

yes

characteristic

165

C₂

#11719066

yes

—

166

C₂

#11719070

yes

characteristic

167

C₂

#11719075

yes

—

168

C₂

#11719079

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#261
commutator, socle	C₂³×C₁₃	C₂²×C₄
Fitting	C₁₃×#95	C₄
prime radical	C₂³×D₁₃	C₂×C₄
cosocle	C₂²×Dic₁₃	C₂³
hypercenter	#95	#261
hypocenter	C₁₃	#1851
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#261

commutator, socle

C₂³×C₁₃

C₂²×C₄

Fitting

C₁₃×#95

C₄

prime radical

C₂³×D₁₃

C₂×C₄

cosocle

C₂²×Dic₁₃

C₂³

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
2. C₂×#88112	C₂	semidirect
5. #11718290	C₂	semidirect
52. C₂×Dic₁₃	M₁₆	semidirect
122. C₁₃	#1851	semidirect

H ≅ G/N

type

2. C₂×#88112

C₂

semidirect

5. #11718290

C₂

semidirect

52. C₂×Dic₁₃

M₁₆

semidirect

122. C₁₃

#1851

semidirect

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

subgroup	index
C₂²×#6118	2
C₂×#88112	2
C₂×#88111	2
#11718327	2
#11718290	2
#1851	13

subgroup

index

C₂²×#6118

Subgroups — #13130784

Conjugacy classes of subgroups

Complements

Maximal subgroups