Group #16786956 — subgroups

Subgroups — #16786956

Order 3328.

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

normal subgroups only

This group has 356 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	#13131197	C₂	1	2	yes	maximal, complemented
2	#13129759	C₂	1	4	yes	maximal, complemented
3	#13123547	C₂	1	1	yes	maximal, characteristic
4	C₂×#88161	C₄	1	2	yes	—
5	#11719182	C₄	1	1	yes	characteristic
6	C₁₃×#423	C₄	1	1	yes	characteristic
7	#11718790	C₂²	1	2	yes	—
8	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
9	#11718499	—	2	2	no	—
10	#11718495	—	2	2	no	—
11	C₂²⋊#6092	C₂²	1	4	yes	—
12	#11718231	—	2	2	no	—
13	C₄⋊#6092	—	2	2	no	—
14	C₂×#6127	C₂×C₄	1	2	yes	—
15	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	#88184	—	2	2	no	—
18	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	C₂³	1	1	yes	characteristic
20	#88161	—	4	4	no	—
21	C₂×#6094	—	2	2	no	—
22	C₂×C₈×D₁₃	—	2	2	no	—
23	#88129	C₈	1	2	yes	—
24	#88128	C₈	1	1	yes	characteristic
25	C₄×D₅₂	—	2	2	no	—
26	#88123	C₈	1	1	yes	characteristic
27	C₂×#6092	—	2	4	no	—
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	#88060	—	2	4	no	—
30	C₄×#1055	D₄	1	2	yes	—
31	C₄×#1055	D₄	1	2	yes	—
32	#11491	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₃	—	2	2	no	—
36	#6127	—	4	2	no	—
37	#6127	—	4	2	no	—
38	C₂×D₅₂	—	2	2	no	—
39	C₂×D₅₂	C₂×C₈	1	2	yes	—
40	C₂×C₄×D₁₃	—	2	2	no	—
41	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
42	C₁₃×M₁₆	—	4	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
45	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
47	#6103	—	2	4	no	—
48	C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×#1055	—	2	2	no	—
50	C₂×#1055	—	4	1	no	—
51	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×#1055	C₂×D₄	1	2	yes	—
54	#6094	—	4	4	no	—
55	C₈×D₁₃	—	4	4	no	—
56	#6092	—	4	4	no	—
57	#2931	—	13	1	no	—
58	#2163	—	13	2	no	—
59	#1315	—	13	4	no	—
60	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂²×D₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	4	2	no	—
65	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
66	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
69	C₂²⋊D₁₃	—	4	4	no	—
70	C₂×Dic₁₃	—	2	2	no	—
71	D₅₂	—	4	4	no	—
72	D₅₂	—	4	2	no	—
73	C₄×D₁₃	—	4	2	no	—
74	C₄×D₁₃	—	4	2	no	—
75	Dic₂₆	—	4	2	no	—
76	C₈×C₁₃	—	4	2	no	—
77	#1055	—	2	2	no	—
78	#1055	—	2	1	no	—
79	#1055	#100	1	2	yes	—
80	C₂×#131	—	13	2	no	—
81	#532	—	13	1	no	—
82	#436	—	26	2	no	—
83	C₈×D₄	—	26	2	no	—
84	#423	#261	1	1	yes	characteristic
85	#408	—	13	2	no	—
86	C₂×C₄×C₈	—	13	1	no	—
87	C₄⋊C₁₆	—	26	2	no	—
88	C₂²⋊C₁₆	—	13	4	no	—
89	#346	—	26	2	no	—
90	C₂²×C₁₃	—	2	2	no	—
91	C₂²×C₁₃	—	2	1	no	—
92	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₂×D₁₃	—	4	4	no	—
94	C₂×D₁₃	—	4	2	no	—
95	C₄×C₁₃	—	2	2	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	#410	1	1	yes	characteristic
99	C₄×C₁₃	C₂×#100	1	1	yes	characteristic
100	Dic₁₃	—	4	2	no	—
101	C₂×#41	—	13	2	no	—
102	#131	—	52	4	no	—
103	C₂×M₁₆	—	26	2	no	—
104	C₂×M₁₆	C₂×#261	1	2	yes	—
105	C₂²×C₈	—	13	1	no	—
106	C₂²×C₈	—	13	1	no	—
107	C₂²×C₈	—	26	2	no	—
108	C₄⋊Q₈	—	13	1	no	—
109	#127	—	13	1	no	—
110	#124	—	13	2	no	—
111	#121	—	26	2	no	—
112	C₄×D₄	—	26	2	no	—
113	C₂×C₄²	C₂×#261	1	1	yes	characteristic
114	C₂×C₁₆	—	26	4	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	26	4	no	—
117	C₄×C₈	—	13	2	no	—
118	C₄×C₈	—	13	2	no	—
119	C₂×C₁₃	—	4	1	no	—
120	C₂×C₁₃	#1460	1	1	yes	characteristic
121	C₂×C₁₃	#2117	1	2	yes	—
122	D₁₃	—	8	2	no	—
123	#41	—	52	2	no	—
124	#41	—	52	2	no	—
125	C₂×Q₈	—	13	2	no	—
126	C₂×D₄	—	13	2	no	—
127	C₂×D₄	—	26	2	no	—
128	C₂×D₄	—	26	2	no	—
129	C₂²×C₄	—	26	2	no	—
130	C₂²×C₄	#6118	1	2	yes	—
131	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
132	M₁₆	—	4	2	no	—
133	M₁₆	—	52	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	2	no	—
137	C₂×C₈	—	13	1	no	—
138	C₂×C₈	—	26	2	no	—
139	C₂×C₈	—	52	4	no	—
140	C₂×C₈	—	52	1	no	—
141	C₄⋊C₄	—	26	2	no	—
142	C₂²⋊C₄	—	26	4	no	—
143	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
144	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
145	C₄²	C₂²⋊#261	1	2	yes	—
146	C₁₆	—	52	4	no	—
147	C₁₃	#11491	1	1	yes	characteristic, complemented, Sylow
148	C₂³	—	26	2	no	—
149	C₂³	C₂×#6118	1	1	yes	characteristic
150	Q₈	—	52	2	no	—
151	D₄	—	52	2	no	—
152	D₄	—	52	4	no	—
153	D₄	—	52	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	26	2	no	—
158	C₂×C₄	—	52	2	no	—
159	C₂×C₄	—	52	2	no	—
160	C₂×C₄	#88111	1	2	yes	—
161	C₂×C₄	#88111	1	2	yes	—
162	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
163	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
164	C₈	—	4	2	no	—
165	C₈	—	13	2	no	—
166	C₈	—	26	1	no	—
167	C₈	—	26	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	52	4	no	—
171	C₂²	—	52	2	no	—
172	C₂²	C₂×#88111	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	52	2	no	—
177	C₄	D₁₃⋊#100	1	1	yes	characteristic
178	C₄	#11719156	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	—	104	2	no	—
181	C₂	#13130077	1	1	yes	characteristic
182	C₂	#13131080	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131197

C₂

yes

maximal, complemented

#13129759

C₂

yes

maximal, complemented

#13123547

C₂

yes

maximal, characteristic

C₂×#88161

C₄

yes

—

#11719182

C₄

yes

characteristic

C₁₃×#423

C₄

yes

characteristic

#11718790

C₂²

yes

—

C₂×C₄×#1055

C₂²

yes

characteristic

#11718499

—

#11718495

—

C₂²⋊#6092

C₂²

yes

—

#11718231

—

C₄⋊#6092

—

C₂×#6127

C₂×C₄

yes

—

C₂×C₁₃×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88184

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂³

yes

characteristic

#88161

—

C₂×#6094

—

C₂×C₈×D₁₃

—

#88129

C₈

yes

—

#88128

C₈

yes

characteristic

C₄×D₅₂

—

#88123

C₈

yes

characteristic

C₂×#6092

—

C₁₃×C₄⋊C₈

—

#88060

—

C₄×#1055

D₄

yes

—

C₄×#1055

D₄

yes

—

#11491

—

maximal, Sylow

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₁₃

—

#6127

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

C₂×C₈

yes

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

C₂×C₈

yes

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

#6094

—

C₈×D₁₃

—

—

—

—

—

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

C₂²×D₁₃

—

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

yes

—

C₂×#131

—

#532

—

#436

—

C₈×D₄

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

100

Dic₁₃

—

101

C₂×#41

—

102

#131

—

103

C₂×M₁₆

—

104

C₂×M₁₆

C₂×#261

yes

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₄⋊Q₈

—

109

#127

—

110

#124

—

111

#121

—

112

C₄×D₄

—

113

C₂×C₄²

C₂×#261

yes

characteristic

114

C₂×C₁₆

—

115

C₄⋊C₈

—

116

C₂²⋊C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₂×C₁₃

—

120

C₂×C₁₃

#1460

yes

characteristic

121

C₂×C₁₃

#2117

yes

—

122

D₁₃

—

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

#6118

yes

—

131

C₂²×C₄

C₂²×#261

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

D₁₃⋊C₈

yes

characteristic

144

C₄²

D₁₃⋊C₈

yes

characteristic

145

C₄²

C₂²⋊#261

yes

—

146

C₁₆

—

147

C₁₃

#11491

yes

characteristic, complemented, Sylow

148

C₂³

—

149

C₂³

C₂×#6118

yes

characteristic

150

Q₈

—

151

D₄

—

152

D₄

—

153

D₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#88111

yes

—

161

C₂×C₄

#88111

yes

—

162

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

163

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂×#88111

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

D₁₃⋊#100

yes

characteristic

178

C₄

#11719156

yes

characteristic

179

C₂

—

180

C₂

—

104

—

181

C₂

#13130077

yes

characteristic

182

C₂

#13131080

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#423	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	#11719182	C₄
cosocle	C₂²×#1055	C₂³
hypercenter	#423	#261
hypocenter	C₁₃	#11491
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#423

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

#11719182

C₄

cosocle

C₂²×#1055

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #13131197	C₂	semidirect
2. #13129759	C₂	semidirect
147. C₁₃	#11491	semidirect

H ≅ G/N

type

1. #13131197

C₂

semidirect

2. #13129759

C₂

semidirect

147. C₁₃

#11491

semidirect

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

subgroup	index
#13131197	2
#13129759	2
#13123547	2
#11491	13

subgroup

index

Subgroups — #16786956

Conjugacy classes of subgroups

Complements

Maximal subgroups