Group #16790118 — subgroups

Subgroups — #16790118

Order 3328.

478 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 132 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	#13131063	C₂	1	2	yes	maximal
2	#13129883	C₂	1	4	yes	maximal, complemented
3	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×#1055	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	4	yes	—
8	C₂×#88043	—	2	4	no	—
9	C₂×#88043	C₄	1	2	yes	—
10	C₂×#88043	C₂²	1	2	yes	—
11	C₂²⋊#6092	—	2	4	no	—
12	C₂³×C₄×C₁₃	C₈	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	—	2	2	no	—
15	C₂×C₄²×C₁₃	—	2	2	no	—
16	C₂×C₄²×C₁₃	C₈	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×#1055	—	2	2	no	—
19	C₂²×#1055	—	2	4	no	—
20	C₂²×#1055	—	2	4	no	—
21	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	2	yes	—
23	C₂²×#1055	D₄	1	4	yes	—
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	#88106	—	2	8	no	—
26	C₂×#6092	—	2	4	no	—
27	#88043	—	2	4	no	—
28	#88043	—	2	4	no	—
29	#12846	—	13	1	no	maximal, Sylow
30	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	—	2	2	no	—
34	C₂²×C₄×C₁₃	—	2	2	no	—
35	C₂²×C₄×C₁₃	—	4	4	no	—
36	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	—	2	4	no	—
42	C₄²×C₁₃	—	2	2	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₂×#1055	—	2	8	no	—
45	C₂×#1055	—	2	2	no	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	—	2	8	no	—
49	C₂×#1055	C₂²⋊C₄	1	2	yes	—
50	C₂×#1055	M₁₆	1	4	yes	—
51	C₂×#1055	C₂×D₄	1	2	yes	—
52	C₂×#1055	#41	1	4	yes	—
53	#6092	—	4	4	no	—
54	C₂²×#97	—	13	1	no	—
55	#2112	—	13	2	no	—
56	#1377	—	13	4	no	—
57	C₂³×C₁₃	—	2	4	no	—
58	C₂³×C₁₃	—	2	2	no	—
59	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
60	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	—	2	4	no	—
66	C₂×C₄×C₁₃	—	4	4	no	—
67	C₂×C₄×C₁₃	—	4	4	no	—
68	C₂×C₄×C₁₃	—	4	4	no	—
69	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#117	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
74	#1055	—	2	2	no	—
75	#1055	—	2	4	no	—
76	#1055	#123	1	4	yes	—
77	C₂³×C₈	—	13	1	no	—
78	C₂³×C₈	—	13	1	no	—
79	C₂²×C₄²	#261	1	1	yes	characteristic
80	C₂×M₃₂	—	13	4	no	—
81	C₂×#97	—	13	2	no	—
82	C₂×#97	—	13	2	no	—
83	C₂×#97	—	26	4	no	—
84	C₂²⋊C₁₆	—	26	4	no	—
85	C₂²×C₁₃	—	2	8	no	—
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	—	2	2	no	—
88	C₂²×C₁₃	—	2	2	no	—
89	C₂²×C₁₃	#349	1	4	yes	—
90	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₁₃	#431	1	2	yes	—
92	C₄×C₁₃	—	2	2	no	—
93	C₄×C₁₃	—	4	4	no	—
94	C₄×C₁₃	#381	1	1	yes	characteristic
95	C₄×C₁₃	#407	1	1	yes	characteristic
96	C₄×C₁₃	#432	1	2	yes	—
97	C₂³×C₄	#1057	1	2	yes	—
98	C₂³×C₄	C₂×#261	1	1	yes	characteristic
99	C₂²×C₈	—	13	1	no	—
100	C₂²×C₈	—	13	4	no	—
101	C₂²×C₈	—	13	2	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₂²×C₈	—	26	4	no	—
104	C₂²×C₈	—	26	4	no	—
105	C₂²×C₈	—	26	2	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	—	2	2	no	—
108	C₂×C₄²	#1057	1	2	yes	—
109	C₂×C₄²	C₂×#261	1	2	yes	—
110	M₃₂	—	26	8	no	—
111	C₂×C₁₆	—	26	4	no	—
112	#97	—	26	4	no	—
113	#97	—	26	4	no	—
114	C₂×C₁₃	—	2	4	no	—
115	C₂×C₁₃	—	2	2	no	—
116	C₂×C₁₃	#1817	1	1	yes	characteristic
117	C₂×C₁₃	C₂×#349	1	2	yes	—
118	C₂⁴	C₂×#1057	1	1	yes	characteristic
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	—	4	4	no	—
124	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
125	C₂²×C₄	C₂×#1057	1	4	yes	—
126	C₂²×C₄	#6122	1	2	yes	—
127	C₂²×C₄	C₂²⋊#261	1	2	yes	—
128	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
129	C₂×C₈	—	13	4	no	—
130	C₂×C₈	—	13	2	no	—
131	C₂×C₈	—	13	4	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	26	8	no	—
134	C₂×C₈	—	26	8	no	—
135	C₂×C₈	—	26	2	no	—
136	C₂×C₈	—	26	2	no	—
137	C₂×C₈	—	26	2	no	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₁₆	—	52	4	no	—
142	C₁₃	#12846	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	C₂²⋊#1057	1	2	yes	—
146	C₂³	C₂²×#1057	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	C₂²⋊#1057	1	2	yes	—
156	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
157	C₂×C₄	#88237	1	2	yes	—
158	C₂×C₄	C₂×#6122	1	1	yes	characteristic
159	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
160	C₈	—	13	4	no	—
161	C₈	—	26	4	no	—
162	C₈	—	26	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#11718284	1	4	yes	—
168	C₂²	#11719119	1	2	yes	—
169	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	4	4	no	—
172	C₄	#11719126	1	1	yes	characteristic
173	C₄	#11719128	1	2	yes	—
174	C₄	#11719135	1	1	yes	characteristic
175	C₂	—	2	4	no	—
176	C₂	—	2	2	no	—
177	C₂	#13130726	1	1	yes	characteristic
178	C₂	C₂×#11718284	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13131063

C₂

yes

maximal

#13129883

C₂

yes

maximal, complemented

C₂²×#88043

C₂

yes

maximal, characteristic

C₂²×C₄²×C₁₃

C₄

yes

characteristic

C₂³×#1055

C₄

yes

characteristic

C₂³×#1055

C₂²

yes

characteristic

C₂×#88106

C₂²

yes

—

C₂×#88043

—

C₂×#88043

C₄

yes

—

C₂×#88043

C₂²

yes

—

C₂²⋊#6092

—

C₂³×C₄×C₁₃

C₈

yes

—

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₈

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂²×#1055

—

C₂²×#1055

—

C₂²×#1055

—

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

C₂²×#1055

D₄

yes

—

C₂²×#1055

C₂³

yes

—

—

—

—

—

C₂⁴×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#1055

M₁₆

yes

—

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

#41

yes

—

#6092

—

C₂²×#97

—

#2112

—

#1377

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂²⋊C₈

yes

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

—

yes

—

C₂³×C₈

—

C₂³×C₈

—

C₂²×C₄²

#261

yes

characteristic

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×#97

—

C₂²⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#349

yes

—

C₂²×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₁₃

#431

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂³×C₄

#1057

yes

—

C₂³×C₄

C₂×#261

yes

characteristic

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

#1057

yes

—

109

C₂×C₄²

C₂×#261

yes

—

110

M₃₂

—

111

C₂×C₁₆

—

112

#97

—

113

#97

—

114

C₂×C₁₃

—

115

C₂×C₁₃

—

116

C₂×C₁₃

#1817

yes

characteristic

117

C₂×C₁₃

C₂×#349

yes

—

118

C₂⁴

C₂×#1057

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×#1057

yes

characteristic

125

C₂²×C₄

C₂×#1057

yes

—

126

C₂²×C₄

#6122

yes

—

127

C₂²×C₄

C₂²⋊#261

yes

—

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

C₂×C₈

—

137

C₂×C₈

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₁₆

—

142

C₁₃

#12846

yes

characteristic, complemented, Sylow

143

C₂³

—

144

C₂³

—

145

C₂³

C₂²⋊#1057

yes

—

146

C₂³

C₂²×#1057

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

C₂²⋊#1057

yes

—

156

C₂×C₄

D₁₃⋊M₁₆

yes

—

157

C₂×C₄

#88237

yes

—

158

C₂×C₄

C₂×#6122

yes

characteristic

159

C₂×C₄

C₂×C₂²⋊#261

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#11718284

yes

—

168

C₂²

#11719119

yes

—

169

C₂²

C₂×C₂²⋊#1057

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

#11719126

yes

characteristic

173

C₄

#11719128

yes

—

174

C₄

#11719135

yes

characteristic

175

C₂

—

176

C₂

—

177

C₂

#13130726

yes

characteristic

178

C₂

C₂×#11718284

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#261
commutator	C₂³×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	#12846
LCS	G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#261

commutator

C₂³×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₂²×C₄²×C₁₃

C₄

socle

C₂²×C₁₃

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₁₃

C₂×C₈

cosocle

C₂²×#1055

C₂³

C₂²×C₄²

C₁₃

G ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #13129883	C₂	semidirect
142. C₁₃	#12846	semidirect

H ≅ G/N

type

2. #13129883

C₂

semidirect

142. C₁₃

#12846

semidirect

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

subgroup	index
#13131063	2
#13129883	2
C₂²×#88043	2
#12846	13

subgroup

index

#13131063

#13129883

C₂²×#88043

#12846

Subgroups — #16790118

Conjugacy classes of subgroups

Complements

Maximal subgroups