Group #16807371 — subgroups

Subgroups — #16807371

Order 3328.

494 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 232 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	#13131252	C₂	1	6	yes	maximal, complemented
2	Q₈×#6092	C₂	1	2	yes	maximal, complemented
3	#13131200	C₂	1	3	yes	maximal
4	#13131195	C₂	1	3	yes	maximal
5	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×D₁₃	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	Q₈×#1055	C₂²	1	1	yes	characteristic
9	D₁₃×C₄⋊C₈	C₄	1	3	yes	—
10	#11718491	C₂²	1	3	yes	—
11	C₄×C₈×D₁₃	C₂²	1	3	yes	—
12	C₈×Dic₂₆	C₄	1	3	yes	—
13	#11718260	C₂²	1	6	yes	—
14	#11718259	C₂²	1	1	yes	characteristic
15	#11718259	C₂²	1	3	yes	—
16	#11718258	C₂²	1	6	yes	—
17	C₄⋊#6092	C₂²	1	6	yes	—
18	C₄×#6092	C₂²	1	6	yes	—
19	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
22	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
23	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×D₁₃	C₈	1	3	yes	—
25	#88141	C₈	1	3	yes	—
26	C₄²×D₁₃	C₂×C₄	1	3	yes	—
27	C₄×Dic₂₆	C₂×C₄	1	3	yes	—
28	C₂×#6092	C₂³	1	2	yes	—
29	C₂×#6092	C₂³	1	6	yes	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
32	#88054	C₂×C₄	1	6	yes	—
33	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
34	C₈×Dic₁₃	C₂³	1	3	yes	—
35	C₄⋊#1055	C₂×C₄	1	3	yes	—
36	C₄×#1055	C₂³	1	3	yes	—
37	#33498	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
39	Q₈×D₁₃	—	2	4	no	—
40	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
41	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₂₆	C₂×C₈	1	3	yes	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
47	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
48	C₄⋊Dic₁₃	C₂×C₈	1	3	yes	—
49	#6101	C₂×C₈	1	6	yes	—
50	C₄×Dic₁₃	C₂×C₈	1	3	yes	—
51	C₄×Dic₁₃	C₂×C₈	1	3	yes	—
52	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
53	C₂×#1055	C₂²×C₄	1	3	yes	—
54	C₂×#1055	C₂⁴	1	1	yes	characteristic
55	C₈×D₁₃	—	2	2	no	—
56	C₈×D₁₃	—	4	3	no	—
57	#6092	—	2	2	no	—
58	#6092	—	2	6	no	—
59	C₂×C₈×Q₈	—	13	1	no	—
60	#2183	—	13	6	no	—
61	Q₈×C₁₆	—	13	2	no	—
62	#2163	—	13	3	no	—
63	#2162	—	13	3	no	—
64	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
65	Q₈×C₁₃	M₃₂	1	4	yes	—
66	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
69	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
71	C₂×Dic₁₃	C₂²×C₈	1	3	yes	—
72	C₄×D₁₃	—	2	6	no	—
73	C₄×D₁₃	—	2	1	no	—
74	C₄×D₁₃	—	2	1	no	—
75	C₄×D₁₃	—	2	6	no	—
76	Dic₂₆	—	2	6	no	—
77	C₈×C₁₃	—	2	1	no	—
78	C₈×C₁₃	—	2	3	no	—
79	#1055	—	2	3	no	—
80	#1055	2₋¹⁺⁴	1	2	yes	—
81	C₂×C₄×Q₈	—	13	1	no	—
82	C₈×Q₈	—	13	1	no	—
83	C₈×Q₈	—	13	3	no	—
84	C₈×Q₈	—	13	3	no	—
85	C₈×Q₈	#261	1	1	yes	characteristic
86	C₂×C₄⋊C₈	—	13	3	no	—
87	C₂×C₄×C₈	—	13	3	no	—
88	C₄⋊C₁₆	—	13	6	no	—
89	C₄⋊C₁₆	—	13	6	no	—
90	C₂²⋊C₁₆	—	13	1	no	—
91	C₂²⋊C₁₆	—	13	3	no	—
92	#346	—	13	6	no	—
93	C₄×C₁₆	—	13	6	no	—
94	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
95	C₂×D₁₃	—	2	1	no	—
96	C₂×D₁₃	—	2	1	no	—
97	C₂×D₁₃	#504	1	2	yes	—
98	C₄×C₁₃	—	2	3	no	—
99	C₄×C₁₃	C₂×M₃₂	1	6	yes	—
100	C₄×C₁₃	#519	1	1	yes	characteristic
101	C₄×C₁₃	#568	1	1	yes	characteristic
102	Dic₁₃	—	2	3	no	—
103	Dic₁₃	—	2	3	no	—
104	Dic₁₃	—	2	1	no	—
105	C₂²×Q₈	—	13	1	no	—
106	C₂²×C₈	—	13	1	no	—
107	C₂²×C₈	—	13	3	no	—
108	C₄×Q₈	—	13	1	no	—
109	C₄×Q₈	—	13	3	no	—
110	C₄×Q₈	—	13	3	no	—
111	C₄×Q₈	C₂×#261	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	13	3	no	—
113	C₂×C₄²	—	13	3	no	—
114	C₂×C₁₆	—	13	2	no	—
115	C₂×C₁₆	—	13	6	no	—
116	C₄⋊C₈	—	13	6	no	—
117	C₄⋊C₈	—	13	3	no	—
118	C₄⋊C₈	C₂×#261	1	3	yes	—
119	C₄×C₈	—	13	3	no	—
120	C₄×C₈	—	13	3	no	—
121	C₄×C₈	—	13	3	no	—
122	C₄×C₈	C₂×#261	1	3	yes	—
123	C₂×C₁₃	#2982	1	1	yes	characteristic
124	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
125	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
126	D₁₃	—	2	2	no	—
127	C₂×Q₈	—	13	3	no	—
128	C₂×Q₈	—	26	4	no	—
129	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
130	C₂²×C₄	—	13	3	no	—
131	C₂²×C₄	—	13	3	no	—
132	C₂²×C₄	—	13	1	no	—
133	C₂×C₈	—	13	1	no	—
134	C₂×C₈	—	13	3	no	—
135	C₂×C₈	—	26	2	no	—
136	C₂×C₈	—	52	3	no	—
137	C₂×C₈	C₂²×#261	1	1	yes	characteristic
138	C₂×C₈	C₂²×#261	1	3	yes	—
139	C₄⋊C₄	—	13	6	no	—
140	C₄⋊C₄	—	13	3	no	—
141	C₄⋊C₄	D₁₃⋊C₈	1	3	yes	—
142	C₄²	—	13	3	no	—
143	C₄²	—	13	3	no	—
144	C₄²	—	13	3	no	—
145	C₄²	C₂²×#261	1	3	yes	—
146	C₁₆	—	26	2	no	—
147	C₁₆	—	26	6	no	—
148	C₁₃	#33498	1	1	yes	characteristic, complemented, Sylow
149	C₂³	—	13	1	no	—
150	Q₈	—	26	6	no	—
151	Q₈	#88098	1	4	yes	—
152	C₂×C₄	—	13	1	no	—
153	C₂×C₄	—	13	3	no	—
154	C₂×C₄	—	13	3	no	—
155	C₂×C₄	—	26	6	no	—
156	C₂×C₄	—	26	6	no	—
157	C₂×C₄	—	26	1	no	—
158	C₂×C₄	—	26	1	no	—
159	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
160	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
161	C₂×C₄	C₂³×#261	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	3	no	—
164	C₈	—	13	2	no	—
165	C₈	—	26	3	no	—
166	C₂²	—	13	2	no	—
167	C₂²	—	26	1	no	—
168	C₂²	—	26	1	no	—
169	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
170	C₄	—	2	3	no	—
171	C₄	—	26	3	no	—
172	C₄	—	26	3	no	—
173	C₄	—	26	1	no	—
174	C₄	C₂×#88098	1	6	yes	—
175	C₄	#11719627	1	1	yes	characteristic
176	C₄	#11719629	1	1	yes	characteristic
177	C₂	—	26	2	no	—
178	C₂	#13131730	1	1	yes	characteristic
179	C₂	C₂²×#88098	1	1	yes	characteristic
180	C₂	D₁₃⋊#504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131252

C₂

yes

maximal, complemented

Q₈×#6092

C₂

yes

maximal, complemented

#13131200

C₂

yes

maximal

#13131195

C₂

yes

maximal

C₈×Q₈×D₁₃

C₂

yes

maximal, characteristic

C₄×Q₈×D₁₃

C₄

yes

characteristic

C₈×Q₈×C₁₃

C₄

yes

characteristic

Q₈×#1055

C₂²

yes

characteristic

D₁₃×C₄⋊C₈

C₄

yes

—

#11718491

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

—

C₈×Dic₂₆

C₄

yes

—

#11718260

C₂²

yes

—

#11718259

C₂²

yes

characteristic

#11718259

C₂²

yes

—

#11718258

C₂²

yes

—

C₄⋊#6092

C₂²

yes

—

C₄×#6092

C₂²

yes

—

C₂×Q₈×D₁₃

C₈

yes

characteristic

C₄×Q₈×C₁₃

C₂×C₄

yes

characteristic

Q₈×Dic₁₃

C₈

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

—

C₂×C₈×D₁₃

C₂³

yes

characteristic

C₄⋊C₄×D₁₃

C₈

yes

—

#88141

C₈

yes

—

C₄²×D₁₃

C₂×C₄

yes

—

C₄×Dic₂₆

C₂×C₄

yes

—

C₂×#6092

C₂³

yes

—

C₂×#6092

C₂³

yes

—

C₁₃×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

#88054

C₂×C₄

yes

—

C₈×Dic₁₃

C₂×C₄

yes

—

C₈×Dic₁₃

C₂³

yes

—

C₄⋊#1055

C₂×C₄

yes

—

C₄×#1055

C₂³

yes

—

#33498

—

maximal, Sylow

C₂×Q₈×C₁₃

C₂×C₈

yes

characteristic

Q₈×D₁₃

—

C₂×C₄×D₁₃

C₂×C₈

yes

—

C₂×C₄×D₁₃

C₂×C₈

yes

—

C₂×C₄×D₁₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

—

C₁₃×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₃

C₂²×C₄

yes

—

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

C₂×C₈

yes

—

C₄×Dic₁₃

C₂×C₈

yes

—

C₄×Dic₁₃

C₂×C₈

yes

—

C₄×Dic₁₃

C₂²×C₄

yes

—

C₂×#1055

C₂²×C₄

yes

—

C₂×#1055

C₂⁴

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

#6092

—

#6092

—

C₂×C₈×Q₈

—

#2183

—

Q₈×C₁₆

—

#2163

—

#2162

—

C₂²×D₁₃

C₂²×C₈

yes

characteristic

Q₈×C₁₃

M₃₂

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

—

C₂×C₄×C₁₃

C₂³×C₄

yes

characteristic

C₂×Dic₁₃

C₂²×C₈

yes

characteristic

C₂×Dic₁₃

C₂²×C₈

yes

—

C₂×Dic₁₃

C₂²×C₈

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

#1055

2₋¹⁺⁴

yes

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

#261

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂²×C₁₃

C₂³×C₈

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

#504

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×M₃₂

yes

—

100

C₄×C₁₃

#519

yes

characteristic

101

C₄×C₁₃

#568

yes

characteristic

102

Dic₁₃

—

103

Dic₁₃

—

104

Dic₁₃

—

105

C₂²×Q₈

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₄×Q₈

—

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₄×Q₈

C₂×#261

yes

characteristic

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

C₂×#261

yes

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#261

yes

—

123

C₂×C₁₃

#2982

yes

characteristic

124

C₂×C₁₃

C₂²×M₃₂

yes

characteristic

125

C₂×C₁₃

C₂×#504

yes

characteristic

126

D₁₃

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

D₁₃⋊C₈

yes

characteristic

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

C₂²×#261

yes

characteristic

138

C₂×C₈

C₂²×#261

yes

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

D₁₃⋊C₈

yes

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

145

C₄²

C₂²×#261

yes

—

146

C₁₆

—

147

C₁₆

—

148

C₁₃

#33498

yes

characteristic, complemented, Sylow

149

C₂³

—

150

Q₈

—

151

Q₈

#88098

yes

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

160

C₂×C₄

C₂×D₁₃⋊C₈

yes

—

161

C₂×C₄

C₂³×#261

yes

characteristic

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂²×D₁₃⋊C₈

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

C₂×#88098

yes

—

175

C₄

#11719627

yes

characteristic

176

C₄

#11719629

yes

characteristic

177

C₂

—

178

C₂

#13131730

yes

characteristic

179

C₂

C₂²×#88098

yes

characteristic

180

C₂

D₁₃⋊#504

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₈×Q₈×C₁₃	C₄
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	#33498
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#261

commutator, socle

C₂²×C₁₃

C₂³×C₈

Fitting

C₈×Q₈×C₁₃

C₄

prime radical

C₂²×D₁₃

C₂²×C₈

C₂⁴

C₈×Q₈

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #13131252	C₂	semidirect
2. Q₈×#6092	C₂	semidirect
148. C₁₃	#33498	semidirect

H ≅ G/N

type

1. #13131252

C₂

semidirect

2. Q₈×#6092

C₂

semidirect

148. C₁₃

#33498

semidirect

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

subgroup	index
#13131252	2
Q₈×#6092	2
#13131200	2
#13131195	2
C₈×Q₈×D₁₃	2
#33498	13

subgroup

index

C₈×Q₈×D₁₃

Subgroups — #16807371

Conjugacy classes of subgroups

Complements

Maximal subgroups