Group #16791169 — subgroups

Subgroups — #16791169

Order 3328.

298 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 112 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×#11718249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#88036	C₂	1	1	yes	maximal, characteristic, complemented
3	#13129950	C₂	1	4	yes	maximal, complemented
4	C₂×#11718568	C₂	1	1	yes	maximal, characteristic
5	C₂×#88155	C₄	1	1	yes	characteristic
6	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×#6090	C₂²	1	1	yes	characteristic
8	#11718568	—	2	2	no	—
9	#11718568	—	2	2	no	—
10	C₂×#88038	C₄	1	2	yes	—
11	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
12	#11718249	—	2	2	no	—
13	C₂×#88036	—	2	2	no	—
14	C₂×#88036	C₂²	1	2	yes	—
15	C₂×#88036	C₂²	1	2	yes	—
16	C₂×#6127	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×#6099	C₈	1	1	yes	characteristic
19	#88155	—	2	2	no	—
20	#88155	—	2	2	no	—
21	C₂×#6094	C₈	1	2	yes	—
22	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
23	C₁₃×M₃₂	—	2	2	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×#6090	C₂×C₄	1	1	yes	characteristic
26	C₂×#6090	D₄	1	2	yes	—
27	C₂×#6090	D₄	1	2	yes	—
28	C₂×#6090	C₂³	1	1	yes	characteristic
29	#88038	—	2	4	no	—
30	C₁₆×D₁₃	—	2	2	no	—
31	C₁₆×D₁₃	—	2	2	no	—
32	#88036	—	2	2	no	—
33	#88036	—	2	2	no	—
34	#13554	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₂²⋊D₁₃	C₁₆	1	2	yes	—
37	#6127	—	2	2	no	—
38	#6127	—	2	2	no	—
39	C₂×D₅₂	C₁₆	1	1	yes	characteristic
40	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
41	C₂×Dic₂₆	C₁₆	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	#6099	—	2	2	no	—
47	C₂×#1055	C₂×C₈	1	2	yes	—
48	#6094	—	2	4	no	—
49	C₈×D₁₃	—	2	2	no	—
50	C₈×D₁₃	—	2	2	no	—
51	C₁₃×C₁₆	—	2	2	no	—
52	#6090	C₂²⋊C₄	1	2	yes	—
53	#6090	C₂×D₄	1	1	yes	characteristic
54	#6090	C₂×D₄	1	1	yes	characteristic
55	C₂×#504	—	13	1	no	—
56	C₂×M₆₄	—	13	1	no	—
57	C₂²×C₃₂	—	13	1	no	—
58	C₂²⋊C₃₂	—	13	4	no	—
59	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
60	C₂²×D₁₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
64	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
65	C₂²⋊D₁₃	—	2	4	no	—
66	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
67	D₅₂	—	2	2	no	—
68	C₄×D₁₃	—	2	2	no	—
69	C₄×D₁₃	—	2	2	no	—
70	Dic₂₆	—	2	2	no	—
71	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
72	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
74	#1055	—	2	2	no	—
75	C₂×#131	—	13	1	no	—
76	#504	—	26	2	no	—
77	#504	—	26	2	no	—
78	C₂×M₃₂	—	13	2	no	—
79	C₂×M₃₂	#261	1	1	yes	characteristic
80	C₂²×C₁₆	—	13	1	no	—
81	C₂²×C₁₆	—	13	2	no	—
82	M₆₄	—	26	2	no	—
83	C₂×C₃₂	—	13	2	no	—
84	C₂×C₃₂	—	13	2	no	—
85	C₂×C₃₂	—	26	2	no	—
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	C₂²⋊C₁₆	1	2	yes	—
88	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
89	C₂×D₁₃	—	2	2	no	—
90	C₂×D₁₃	—	2	4	no	—
91	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
92	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
93	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
94	Dic₁₃	—	2	2	no	—
95	C₂×#41	—	13	1	no	—
96	#131	—	26	2	no	—
97	#131	—	26	2	no	—
98	C₂×M₁₆	—	13	1	no	—
99	C₂×M₁₆	—	13	2	no	—
100	C₂²×C₈	—	13	2	no	—
101	C₂²×C₈	C₂×#261	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	—	26	4	no	—
104	C₂×C₁₆	—	13	2	no	—
105	C₂×C₁₆	—	13	2	no	—
106	C₂×C₁₆	—	13	1	no	—
107	C₂×C₁₆	—	13	1	no	—
108	C₂×C₁₆	—	26	2	no	—
109	C₂×C₁₆	—	26	2	no	—
110	C₂×C₁₆	C₂×#261	1	2	yes	—
111	C₃₂	—	26	2	no	—
112	C₃₂	—	26	2	no	—
113	C₂×C₁₃	—	2	2	no	—
114	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
115	C₂×C₁₃	#2258	1	2	yes	—
116	D₁₃	—	4	2	no	—
117	#41	—	26	2	no	—
118	#41	—	26	2	no	—
119	C₂×Q₈	—	13	1	no	—
120	C₂×D₄	—	13	2	no	—
121	C₂×D₄	—	13	1	no	—
122	C₂²×C₄	—	13	2	no	—
123	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
124	M₁₆	—	26	2	no	—
125	M₁₆	—	26	4	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	26	2	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
130	C₂×C₈	#6118	1	2	yes	—
131	C₂×C₈	C₂²⋊#261	1	2	yes	—
132	C₂×C₈	C₂²×#261	1	1	yes	characteristic
133	C₁₆	—	2	2	no	—
134	C₁₆	—	13	1	no	—
135	C₁₆	—	13	1	no	—
136	C₁₆	—	13	2	no	—
137	C₁₃	#13554	1	1	yes	characteristic, complemented, Sylow
138	C₂³	—	13	2	no	—
139	C₂³	D₁₃⋊C₁₆	1	1	yes	characteristic
140	Q₈	—	26	2	no	—
141	D₄	—	26	4	no	—
142	D₄	—	26	2	no	—
143	C₂×C₄	—	13	2	no	—
144	C₂×C₄	—	26	2	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	D₁₃⋊C₁₆	1	1	yes	characteristic
147	C₂×C₄	#88098	1	2	yes	—
148	C₂×C₄	#88111	1	2	yes	—
149	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
150	C₈	—	26	2	no	—
151	C₈	#88111	1	2	yes	—
152	C₈	C₂×#6118	1	1	yes	characteristic
153	C₈	C₂×C₂²⋊#261	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	26	2	no	—
156	C₂²	—	26	4	no	—
157	C₂²	#11718259	1	2	yes	—
158	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
159	C₄	—	26	2	no	—
160	C₄	#11718259	1	2	yes	—
161	C₄	C₂×#88098	1	1	yes	characteristic
162	C₄	C₂×#88111	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	52	2	no	—
165	C₂	C₂×#11718259	1	1	yes	characteristic
166	C₂	#13131260	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718249

C₂

yes

maximal, characteristic, complemented

C₂²×#88036

C₂

yes

maximal, characteristic, complemented

#13129950

C₂

yes

maximal, complemented

C₂×#11718568

C₂

yes

maximal, characteristic

C₂×#88155

C₄

yes

characteristic

C₂×C₁₃×M₃₂

C₄

yes

characteristic

C₂²×#6090

C₂²

yes

—

—

C₄

yes

—

C₂×C₁₆×D₁₃

C₂²

yes

—

#11718249

—

C₂×#88036

—

C₂×#88036

C₂²

yes

—

C₂×#88036

C₂²

yes

—

C₂×#6127

C₈

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×#6099

C₈

yes

—

—

C₈

yes

—

C₂×C₈×D₁₃

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

D₄

yes

—

C₂×#6090

D₄

yes

—

C₂×#6090

C₂³

yes

characteristic

#88038

—

C₁₆×D₁₃

—

C₁₆×D₁₃

—

—

—

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂×C₂²⋊D₁₃

C₁₆

yes

—

#6127

—

#6127

—

C₂×D₅₂

C₁₆

yes

characteristic

C₂×C₄×D₁₃

C₂×C₈

yes

—

C₂×Dic₂₆

C₁₆

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

#6099

—

C₂×#1055

C₂×C₈

yes

—

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

#6090

C₂²⋊C₄

yes

—

#6090

C₂×D₄

yes

characteristic

#6090

C₂×D₄

yes

characteristic

C₂×#504

—

C₂×M₆₄

—

C₂²×C₃₂

—

C₂²⋊C₃₂

—

C₂³×C₁₃

C₂×C₁₆

yes

characteristic

C₂²×D₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₃

M₃₂

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂×C₁₆

yes

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂²⋊C₈

yes

—

C₈×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

—

—

—

—

C₂×M₃₂

—

C₂×M₃₂

#261

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²⋊C₁₆

yes

—

C₂²×C₁₃

C₂²×C₁₆

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

C₂²⋊C₁₆

yes

—

C₄×C₁₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₁₃

C₂×M₃₂

yes

characteristic

Dic₁₃

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×#261

yes

characteristic

102

M₃₂

—

103

M₃₂

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

C₂×#261

yes

—

111

C₃₂

—

112

C₃₂

—

113

C₂×C₁₃

—

114

C₂×C₁₃

C₂×C₂²⋊C₁₆

yes

characteristic

115

C₂×C₁₃

#2258

yes

—

116

D₁₃

—

117

#41

—

118

#41

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

D₁₃⋊C₈

yes

characteristic

130

C₂×C₈

#6118

yes

—

131

C₂×C₈

C₂²⋊#261

yes

—

132

C₂×C₈

C₂²×#261

yes

characteristic

133

C₁₆

—

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₃

#13554

yes

characteristic, complemented, Sylow

138

C₂³

—

139

C₂³

D₁₃⋊C₁₆

yes

characteristic

140

Q₈

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

D₁₃⋊C₁₆

yes

characteristic

147

C₂×C₄

#88098

yes

—

148

C₂×C₄

#88111

yes

—

149

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

150

C₈

—

151

C₈

#88111

yes

—

152

C₈

C₂×#6118

yes

characteristic

153

C₈

C₂×C₂²⋊#261

yes

characteristic

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#11718259

yes

—

158

C₂²

C₂×D₁₃⋊C₁₆

yes

characteristic

159

C₄

—

160

C₄

#11718259

yes

—

161

C₄

C₂×#88098

yes

characteristic

162

C₄

C₂×#88111

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

C₂×#11718259

yes

characteristic

166

C₂

#13131260

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂×C₁₃×M₃₂	C₄
prime radical	C₂×#6127	C₈
cosocle	C₂×#6090	C₂³
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	#13554
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#261

commutator, socle

C₂²×C₁₃

C₂²×C₁₆

Fitting

C₂×C₁₃×M₃₂

C₄

C₈

C₂³

C₂×M₃₂

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#11718249	C₂	semidirect
2. C₂²×#88036	C₂	semidirect
3. #13129950	C₂	semidirect
137. C₁₃	#13554	semidirect

H ≅ G/N

type

1. C₂×#11718249

C₂

semidirect

2. C₂²×#88036

C₂

semidirect

3. #13129950

C₂

semidirect

137. C₁₃

#13554

semidirect

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

subgroup	index
C₂×#11718249	2
C₂²×#88036	2
#13129950	2
C₂×#11718568	2
#13554	13

subgroup

index

C₂²×#88036

Subgroups — #16791169

Conjugacy classes of subgroups

Complements

Maximal subgroups