Group #16786962 — subgroups

Subgroups — #16786962

Order 3328.

342 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 118 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	#13131193	C₂	1	2	yes	maximal
2	#13129773	C₂	1	4	yes	maximal
3	C₂×#11718491	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
5	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂×#88054	C₂²	1	2	yes	—
7	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
8	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
9	#11718491	—	2	2	no	—
10	#11718491	—	2	2	no	—
11	C₂²⋊#6092	C₂²	1	4	yes	—
12	C₄⋊#6092	—	2	2	no	—
13	C₄×#6092	—	2	2	no	—
14	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
18	C₂×#6101	C₈	1	2	yes	—
19	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
20	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
21	C₂²×#1055	C₂³	1	1	yes	characteristic
22	C₄×Dic₂₆	—	2	2	no	—
23	C₄×Dic₂₆	—	2	2	no	—
24	C₂×#6092	—	2	4	no	—
25	C₁₃×C₄⋊C₈	—	2	2	no	—
26	#88054	—	2	4	no	—
27	C₈×Dic₁₃	—	2	2	no	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₄×#1055	D₄	1	2	yes	—
30	C₄×#1055	D₄	1	2	yes	—
31	#11493	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
36	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
37	C₂×Dic₂₆	—	2	2	no	—
38	C₂×Dic₂₆	—	2	2	no	—
39	C₂×Dic₂₆	C₁₆	1	4	yes	complemented
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	4	2	no	—
42	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₄²×C₁₃	M₁₆	1	2	yes	—
44	C₄⋊Dic₁₃	—	2	2	no	—
45	#6101	—	2	4	no	—
46	C₄×Dic₁₃	—	2	2	no	—
47	C₄×Dic₁₃	—	2	2	no	—
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	—	4	1	no	—
50	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×#1055	C₂×D₄	1	2	yes	—
53	#6092	—	4	4	no	—
54	C₂×C₈×Q₈	—	13	1	no	—
55	#2162	—	13	2	no	—
56	#1322	—	13	4	no	—
57	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	4	1	no	—
61	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
67	C₂×Dic₁₃	—	2	2	no	—
68	C₂×Dic₁₃	—	2	2	no	—
69	C₂×Dic₁₃	—	2	2	no	—
70	C₂×Dic₁₃	—	4	2	no	—
71	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
72	Dic₂₆	—	2	8	no	—
73	C₈×C₁₃	—	4	2	no	—
74	#1055	—	2	2	no	—
75	#1055	—	2	1	no	—
76	#1055	#101	1	2	yes	—
77	C₂×C₄×Q₈	—	13	1	no	—
78	C₈×Q₈	—	26	2	no	—
79	C₈×Q₈	—	26	2	no	—
80	C₂×C₄⋊C₈	—	13	2	no	—
81	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
82	C₂×C₄×C₈	—	13	1	no	—
83	C₂×C₄×C₈	—	13	2	no	—
84	C₄⋊C₁₆	—	26	2	no	—
85	C₂²⋊C₁₆	—	13	4	no	—
86	C₄×C₁₆	—	26	2	no	—
87	C₂²×C₁₃	—	2	1	no	—
88	C₂²×C₁₃	—	2	1	no	—
89	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
90	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
91	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
92	C₄×C₁₃	—	2	2	no	—
93	C₄×C₁₃	—	2	1	no	—
94	C₄×C₁₃	C₂²⋊C₁₆	1	4	yes	—
95	C₄×C₁₃	#410	1	1	yes	characteristic
96	C₄×C₁₃	C₂×#101	1	1	yes	characteristic
97	Dic₁₃	—	2	4	no	—
98	Dic₁₃	—	4	2	no	—
99	C₂²×Q₈	—	13	1	no	—
100	C₂²×C₈	—	13	1	no	—
101	C₂²×C₈	—	13	1	no	—
102	C₂²×C₈	C₂×#261	1	2	yes	—
103	C₄×Q₈	—	26	2	no	—
104	C₄×Q₈	—	26	2	no	—
105	C₂×C₄⋊C₄	—	13	2	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄²	—	13	2	no	—
108	C₂×C₄²	C₂×#261	1	1	yes	characteristic
109	C₂×C₁₆	—	26	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	26	4	no	—
112	C₄×C₈	—	13	2	no	—
113	C₄×C₈	—	13	2	no	—
114	C₄×C₈	—	26	2	no	—
115	C₄×C₈	—	26	2	no	—
116	C₂×C₁₃	—	2	1	no	—
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	#1461	1	1	yes	characteristic
119	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
120	C₂×C₁₃	#2115	1	1	yes	characteristic
121	C₂×Q₈	—	13	4	no	—
122	C₂×Q₈	—	26	2	no	—
123	C₂×Q₈	—	26	2	no	—
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	—	13	2	no	—
126	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	13	1	no	—
133	C₂×C₈	—	13	2	no	—
134	C₂×C₈	—	26	2	no	—
135	C₂×C₈	—	52	1	no	—
136	C₄⋊C₄	—	26	4	no	—
137	C₄⋊C₄	—	26	2	no	—
138	C₄²	—	26	2	no	—
139	C₄²	—	26	2	no	—
140	C₄²	#6118	1	2	yes	—
141	C₄²	C₂²⋊#261	1	2	yes	—
142	C₁₆	—	52	4	no	—
143	C₁₃	#11493	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
145	Q₈	—	26	8	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	13	4	no	—
150	C₂×C₄	—	26	2	no	—
151	C₂×C₄	—	26	2	no	—
152	C₂×C₄	—	26	2	no	—
153	C₂×C₄	—	52	2	no	—
154	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
155	C₂×C₄	#88098	1	2	yes	—
156	C₂×C₄	#88111	1	2	yes	—
157	C₂×C₄	#88111	1	2	yes	—
158	C₂×C₄	C₂×#6118	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	13	2	no	—
162	C₈	—	26	2	no	—
163	C₈	—	26	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
167	C₂²	C₂×#88098	1	1	yes	characteristic
168	C₂²	C₂×#88111	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	26	4	no	—
172	C₄	—	52	2	no	—
173	C₄	#11718259	1	4	yes	—
174	C₄	#11719142	1	1	yes	characteristic
175	C₄	D₁₃⋊#101	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#13130080	1	1	yes	characteristic
179	C₂	C₂×#11718259	1	1	yes	characteristic
180	C₂	#13131074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131193

C₂

yes

maximal

#13129773

C₂

yes

maximal

C₂×#11718491

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₄

yes

characteristic

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×#88054

C₂²

yes

—

C₂×C₈×Dic₁₃

C₄

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718491

—

#11718491

—

C₂²⋊#6092

C₂²

yes

—

C₄⋊#6092

—

C₄×#6092

—

C₂²×Dic₂₆

C₈

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

C₈

yes

characteristic

C₂×#6101

C₈

yes

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

characteristic

C₄×Dic₂₆

—

C₄×Dic₂₆

—

C₂×#6092

—

C₁₃×C₄⋊C₈

—

#88054

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

C₄×#1055

D₄

yes

—

C₄×#1055

D₄

yes

—

#11493

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₁₆

yes

complemented

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

M₁₆

yes

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

#6092

—

C₂×C₈×Q₈

—

#2162

—

#1322

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

M₃₂

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×C₁₆

yes

—

Dic₂₆

—

C₈×C₁₃

—

—

—

yes

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#261

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₃

C₂²×C₁₆

yes

characteristic

C₂²×C₁₃

C₂×M₃₂

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂²⋊C₁₆

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×#261

yes

—

103

C₄×Q₈

—

104

C₄×Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

—

108

C₂×C₄²

C₂×#261

yes

characteristic

109

C₂×C₁₆

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₂×C₁₃

—

117

C₂×C₁₃

—

118

C₂×C₁₃

#1461

yes

characteristic

119

C₂×C₁₃

C₂×C₂²⋊C₁₆

yes

characteristic

120

C₂×C₁₃

#2115

yes

characteristic

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

127

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

128

C₂²×C₄

C₂²×#261

yes

characteristic

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

#6118

yes

—

141

C₄²

C₂²⋊#261

yes

—

142

C₁₆

—

143

C₁₃

#11493

yes

characteristic, complemented, Sylow

144

C₂³

C₂×D₁₃⋊C₈

yes

characteristic

145

Q₈

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

D₁₃⋊C₁₆

yes

—

155

C₂×C₄

#88098

yes

—

156

C₂×C₄

#88111

yes

—

157

C₂×C₄

#88111

yes

—

158

C₂×C₄

C₂×#6118

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₂²

—

165

C₂²

—

166

C₂²

C₂×D₁₃⋊C₁₆

yes

characteristic

167

C₂²

C₂×#88098

yes

characteristic

168

C₂²

C₂×#88111

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#11718259

yes

—

174

C₄

#11719142

yes

characteristic

175

C₄

D₁₃⋊#101

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#13130080

yes

characteristic

179

C₂

C₂×#11718259

yes

characteristic

180

C₂

#13131074

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#11493
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#261

commutator

C₂²×C₁₃

C₂²×C₁₆

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₂×C₁₃×C₄⋊C₈

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

C₂³×C₁₃

C₂²×C₈

cosocle

C₂²×#1055

C₂³

C₂×C₄⋊C₈

C₁₃

G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
39. C₂×Dic₂₆	C₁₆	semidirect
143. C₁₃	#11493	semidirect

H ≅ G/N

type

39. C₂×Dic₂₆

C₁₆

semidirect

143. C₁₃

#11493

semidirect

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

subgroup	index
#13131193	2
#13129773	2
C₂×#11718491	2
#11493	13

subgroup

index

Subgroups — #16786962

Conjugacy classes of subgroups

Complements

Maximal subgroups