Group #16777150 — subgroups

Subgroups — #16777150

Order 3328.

210 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₁₃⋊#364	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#364	C₂	1	1	yes	maximal, characteristic
4	D₁₃⋊M₃₂	—	2	1	no	—
5	C₂×D₁₃⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×D₁₃⋊C₁₆	C₄	1	1	yes	characteristic
7	D₁₃×M₃₂	—	2	1	no	—
8	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	#11718261	C₄	1	2	yes	—
10	#11718257	—	2	1	no	—
11	#11718233	—	2	1	no	—
12	C₄×#6092	—	2	1	no	—
13	C₁₃×#364	C₄	1	1	yes	characteristic
14	#11718103	—	2	1	no	—
15	#11718055	C₄	1	1	yes	characteristic
16	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
18	C₄²×D₁₃	Q₈	1	1	yes	characteristic
19	#88106	—	2	2	no	—
20	C₂×#6092	—	2	1	no	—
21	C₂×#6092	—	2	1	no	—
22	#88098	—	4	1	no	—
23	D₁₃⋊C₁₆	—	2	1	no	—
24	D₁₃⋊C₁₆	—	2	1	no	—
25	D₁₃⋊C₁₆	—	4	1	no	—
26	D₁₃⋊C₁₆	C₈	1	2	yes	—
27	D₁₃⋊C₁₆	C₈	1	2	yes	—
28	C₁₃×M₃₂	—	2	1	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₃	D₄	1	1	yes	characteristic
32	#88052	—	2	1	no	—
33	C₄×#1055	D₄	1	1	yes	characteristic
34	#88038	—	4	1	no	—
35	C₁₆×D₁₃	—	4	1	no	—
36	#7160	—	13	1	no	maximal, Sylow
37	C₂×C₄×D₁₃	—	2	1	no	—
38	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₃	—	2	1	no	—
43	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
46	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
47	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
48	C₈×D₁₃	M₁₆	1	1	yes	characteristic
49	C₈×D₁₃	M₁₆	1	1	yes	characteristic
50	C₈×D₁₃	D₈	1	1	yes	characteristic
51	C₈×D₁₃	SD₁₆	1	1	yes	characteristic
52	C₈×D₁₃	SD₁₆	1	1	yes	characteristic
53	C₈×D₁₃	Q₁₆	1	1	yes	characteristic
54	#6092	—	2	2	no	—
55	#6092	—	2	2	no	—
56	#6092	—	4	2	no	—
57	C₁₃×C₁₆	—	4	1	no	—
58	#6090	—	4	1	no	—
59	C₂×#364	—	13	1	no	—
60	C₂×#364	—	13	1	no	—
61	C₂×C₄×C₁₆	—	13	1	no	—
62	C₂²×D₁₃	#108	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
65	C₂×Dic₁₃	—	2	1	no	—
66	C₂×Dic₁₃	#108	1	1	yes	characteristic
67	C₄×D₁₃	—	2	4	no	—
68	C₄×D₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₄×D₁₃	C₄⋊C₈	1	1	yes	characteristic
70	C₄×D₁₃	#106	1	1	yes	characteristic
71	C₄×D₁₃	#107	1	1	yes	characteristic
72	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
73	C₈×C₁₃	#97	1	1	yes	characteristic
74	C₈×C₁₃	#102	1	1	yes	characteristic
75	C₈×C₁₃	#103	1	1	yes	characteristic
76	#1055	C₂²⋊C₈	1	1	yes	characteristic
77	#1055	C₂²⋊C₈	1	1	yes	characteristic
78	#1055	#102	1	1	yes	characteristic
79	#1055	#103	1	1	yes	characteristic
80	C₂×M₃₂	—	26	1	no	—
81	C₂×M₃₂	—	26	1	no	—
82	C₂²×C₁₆	—	13	1	no	—
83	C₂²×C₁₆	—	13	1	no	—
84	C₂×C₄×C₈	—	13	1	no	—
85	#364	—	13	2	no	—
86	#364	—	13	1	no	—
87	#364	—	26	1	no	—
88	#364	—	26	1	no	—
89	#364	#261	1	1	yes	characteristic
90	C₄×C₁₆	—	26	1	no	—
91	C₄×C₁₆	—	26	1	no	—
92	C₂²×C₁₃	#341	1	1	yes	characteristic
93	C₂×D₁₃	—	2	2	no	—
94	C₂×D₁₃	#334	1	1	yes	characteristic
95	C₂×D₁₃	#335	1	1	yes	characteristic
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	#336	1	1	yes	characteristic
98	C₄×C₁₃	#340	1	1	yes	characteristic
99	Dic₁₃	—	2	2	no	—
100	Dic₁₃	#334	1	1	yes	characteristic
101	Dic₁₃	#335	1	1	yes	characteristic
102	C₂²×C₈	—	13	1	no	—
103	C₂²×C₈	—	13	1	no	—
104	C₂×C₄²	—	13	1	no	—
105	M₃₂	—	2	1	no	—
106	M₃₂	—	26	2	no	—
107	M₃₂	—	26	1	no	—
108	M₃₂	—	52	1	no	—
109	M₃₂	—	52	1	no	—
110	C₂×C₁₆	—	13	2	no	—
111	C₂×C₁₆	—	13	2	no	—
112	C₂×C₁₆	—	26	1	no	—
113	C₂×C₁₆	—	26	1	no	—
114	C₂×C₁₆	—	26	1	no	—
115	C₂×C₁₆	—	26	1	no	—
116	C₂×C₁₆	—	52	1	no	—
117	C₂×C₁₆	—	52	1	no	—
118	C₄×C₈	—	13	1	no	—
119	C₄×C₈	—	13	1	no	—
120	C₄×C₈	—	13	1	no	—
121	C₄×C₈	C₂×#261	1	1	yes	characteristic
122	C₂×C₁₃	—	2	1	no	—
123	C₂×C₁₃	#1277	1	1	yes	characteristic
124	D₁₃	—	2	1	no	—
125	D₁₃	#1373	1	2	yes	—
126	C₂²×C₄	—	13	1	no	—
127	C₂²×C₄	—	26	1	no	—
128	C₂×C₈	—	13	1	no	—
129	C₂×C₈	—	13	1	no	—
130	C₂×C₈	—	13	1	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	13	1	no	—
133	C₂×C₈	—	13	1	no	—
134	C₂×C₈	—	13	1	no	—
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	1	no	—
137	C₂×C₈	—	13	1	no	—
138	C₂×C₈	C₄×#261	1	1	yes	characteristic
139	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
140	C₄²	—	13	1	no	—
141	C₄²	—	26	1	no	—
142	C₄²	C₄⋊#261	1	1	yes	characteristic
143	C₁₆	—	4	1	no	—
144	C₁₆	—	26	2	no	—
145	C₁₆	—	26	2	no	—
146	C₁₆	—	52	2	no	—
147	C₁₆	—	52	1	no	—
148	C₁₃	#7160	1	1	yes	characteristic, complemented, Sylow
149	C₂³	—	13	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	13	1	no	—
152	C₂×C₄	—	13	1	no	—
153	C₂×C₄	—	13	1	no	—
154	C₂×C₄	—	13	1	no	—
155	C₂×C₄	—	13	1	no	—
156	C₂×C₄	—	26	4	no	—
157	C₂×C₄	—	26	1	no	—
158	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
159	C₈	—	13	1	no	—
160	C₈	—	13	1	no	—
161	C₈	—	13	1	no	—
162	C₈	—	13	1	no	—
163	C₈	C₈×#261	1	1	yes	characteristic
164	C₈	#88100	1	1	yes	characteristic
165	C₈	#88115	1	1	yes	characteristic
166	C₈	#88117	1	1	yes	characteristic
167	C₂²	—	13	1	no	—
168	C₂²	—	13	1	no	—
169	C₂²	—	26	2	no	—
170	C₂²	#11718273	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	13	1	no	—
173	C₄	—	13	1	no	—
174	C₄	—	26	2	no	—
175	C₄	#11718246	1	1	yes	characteristic
176	C₄	(C₂×#261)⋊C₈	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	13	2	no	—
179	C₂	—	26	1	no	—
180	C₂	#13129665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₃⋊#364

C₂

yes

maximal, characteristic

C₄×D₁₃⋊C₁₆

C₂

yes

maximal, characteristic

D₁₃×#364

C₂

yes

maximal, characteristic

D₁₃⋊M₃₂

—

C₂×D₁₃⋊C₁₆

C₄

yes

characteristic

C₂×D₁₃⋊C₁₆

C₄

yes

characteristic

D₁₃×M₃₂

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718261

C₄

yes

—

#11718257

—

#11718233

—

C₄×#6092

—

C₁₃×#364

C₄

yes

characteristic

#11718103

—

#11718055

C₄

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

Q₈

yes

—

—

—

—

D₁₃⋊C₁₆

—

D₁₃⋊C₁₆

—

D₁₃⋊C₁₆

—

D₁₃⋊C₁₆

C₈

yes

—

D₁₃⋊C₁₆

C₈

yes

—

C₁₃×M₃₂

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

#88052

—

C₄×#1055

D₄

yes

characteristic

#88038

—

C₁₆×D₁₃

—

#7160

—

maximal, Sylow

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

—

C₄×Dic₁₃

C₄⋊C₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

M₁₆

yes

characteristic

C₈×D₁₃

M₁₆

yes

characteristic

C₈×D₁₃

D₈

yes

characteristic

C₈×D₁₃

SD₁₆

yes

characteristic

C₈×D₁₃

SD₁₆

yes

characteristic

C₈×D₁₃

Q₁₆

yes

—

—

—

C₁₃×C₁₆

—

#6090

—

C₂×#364

—

C₂×#364

—

C₂×C₄×C₁₆

—

C₂²×D₁₃

#108

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#95

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

#108

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

C₄⋊C₈

yes

characteristic

C₄×D₁₃

C₄⋊C₈

yes

C₄×D₁₃

yes

C₄×D₁₃

yes

C₈×C₁₃

C₄×C₈

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

C₂²⋊C₈

yes

characteristic

#1055

C₂²⋊C₈

yes

yes

yes

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

—

—

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₁₃

#341

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

yes

C₂×D₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

100

Dic₁₃

#334

yes

characteristic

101

Dic₁₃

#335

yes

characteristic

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

M₃₂

—

106

M₃₂

—

107

M₃₂

—

108

M₃₂

—

109

M₃₂

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₂×#261

yes

characteristic

122

C₂×C₁₃

—

123

C₂×C₁₃

#1277

yes

characteristic

124

D₁₃

—

125

D₁₃

#1373

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂×C₈

—

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

C₂×C₈

C₄×#261

yes

characteristic

139

C₂×C₈

C₂²⋊#261

yes

characteristic

140

C₄²

—

141

C₄²

—

142

C₄²

C₄⋊#261

yes

characteristic

143

C₁₆

—

144

C₁₆

—

145

C₁₆

—

146

C₁₆

—

147

C₁₆

—

148

C₁₃

#7160

yes

characteristic, complemented, Sylow

149

C₂³

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

C₈×#261

yes

characteristic

164

C₈

#88100

yes

characteristic

165

C₈

#88115

yes

characteristic

166

C₈

#88117

yes

characteristic

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#11718273

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#11718246

yes

characteristic

176

C₄

(C₂×#261)⋊C₈

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

—

180

C₂

#13129665

yes

characteristic

	subgroup	G/H
center	C₈	#88115
commutator	C₈×C₁₃	C₄×C₈
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#364	C₄
socle	C₂×C₁₃	#1277
prime radical	C₂²×D₁₃	#108
cosocle	C₄×C₈×D₁₃	C₂²
hypercenter	#364	#261
hypocenter	C₁₃	#7160
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₈

#88115

commutator

C₈×C₁₃

C₄×C₈

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#364

C₄

C₂×C₁₃

C₂²×D₁₃

C₄×C₈×D₁₃

C₂²

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
148. C₁₃	#7160	semidirect

H ≅ G/N

type

148. C₁₃

#7160

semidirect

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

subgroup	index
D₁₃⋊#364	2
C₄×D₁₃⋊C₁₆	2
D₁₃×#364	2
#7160	13

subgroup

index

D₁₃⋊#364

C₄×D₁₃⋊C₁₆

D₁₃×#364

#7160

Subgroups — #16777150

Conjugacy classes of subgroups

Complements

Maximal subgroups