Group #16776891 — subgroups

Subgroups — #16776891

Order 3328.

292 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 102 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	#13130753	C₂	1	2	yes	maximal
2	#13114987	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#1057	—	2	2	no	—
4	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₂×#88054	—	2	2	no	—
6	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
8	#11718290	—	2	2	no	—
9	#11718290	C₄	1	4	yes	—
10	C₂²×#1057	—	2	4	no	—
11	C₂²×C₈×C₁₃	—	2	1	no	—
12	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
14	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
16	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
17	C₂²×#1055	—	2	1	no	—
18	C₄⋊#1057	—	4	2	no	—
19	C₄⋊#1057	—	4	2	no	—
20	C₁₃×C₄⋊C₈	—	4	1	no	—
21	#88054	—	4	2	no	—
22	#88054	—	4	2	no	—
23	C₄⋊#1055	—	4	1	no	—
24	#7040	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
29	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
30	C₂×#1057	—	4	4	no	—
31	C₂×#1057	—	4	4	no	—
32	C₂×#1057	—	4	4	no	—
33	C₂×C₈×C₁₃	—	4	1	no	—
34	C₂×C₈×C₁₃	—	4	2	no	—
35	C₄²×C₁₃	—	2	1	no	—
36	C₄²×C₁₃	—	2	1	no	—
37	C₄×Dic₁₃	—	2	1	no	—
38	C₄×Dic₁₃	—	2	1	no	—
39	C₄×Dic₁₃	—	4	2	no	—
40	C₄×Dic₁₃	—	4	1	no	—
41	C₄×Dic₁₃	D₈	1	2	yes	—
42	C₄×Dic₁₃	SD₁₆	1	2	yes	—
43	C₄×Dic₁₃	SD₁₆	1	2	yes	—
44	C₄×Dic₁₃	Q₁₆	1	2	yes	—
45	C₂×#1055	—	4	1	no	—
46	C₂×#1055	—	4	2	no	—
47	#1842	—	13	1	no	—
48	#1831	—	13	2	no	—
49	C₂³×C₁₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#101	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#102	1	2	yes	—
57	C₂×C₄×C₁₃	#103	1	2	yes	—
58	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
60	C₂×Dic₁₃	—	2	2	no	—
61	C₂×Dic₁₃	—	2	2	no	—
62	C₂×Dic₁₃	—	2	2	no	—
63	C₂×Dic₁₃	—	2	2	no	—
64	C₂×Dic₁₃	#102	1	2	yes	—
65	C₂×Dic₁₃	#103	1	2	yes	—
66	C₂×Dic₁₃	#106	1	2	yes	—
67	C₂×Dic₁₃	#107	1	2	yes	—
68	#1057	—	8	4	no	—
69	C₈×C₁₃	—	8	1	no	—
70	#1055	—	8	1	no	—
71	C₂×C₄⋊C₈	—	13	1	no	—
72	C₂×C₄⋊C₈	—	26	2	no	—
73	C₂×C₄⋊C₈	—	26	2	no	—
74	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
75	C₄³	—	13	1	no	—
76	#336	—	13	4	no	—
77	#336	—	26	2	no	—
78	C₂²×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
79	C₂²×C₁₃	#340	1	2	yes	—
80	C₂²×C₁₃	#341	1	1	yes	characteristic
81	C₂²×C₁₃	#343	1	1	yes	characteristic
82	C₂²×C₁₃	#344	1	2	yes	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	—	4	1	no	—
86	C₄×C₁₃	#331	1	1	yes	characteristic
87	C₄×C₁₃	#332	1	2	yes	—
88	C₄×C₁₃	#333	1	1	yes	characteristic
89	Dic₁₃	—	2	2	no	—
90	Dic₁₃	—	2	2	no	—
91	Dic₁₃	—	2	2	no	—
92	Dic₁₃	—	2	2	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	—	26	4	no	—
95	C₂²×C₈	—	26	1	no	—
96	C₂×C₄²	—	13	2	no	—
97	C₂×C₄²	—	13	1	no	—
98	C₂×C₄²	—	13	2	no	—
99	C₂×C₄²	—	13	1	no	—
100	C₂×C₄²	C₂×#261	1	1	yes	characteristic
101	C₄⋊C₈	—	4	1	no	—
102	C₄⋊C₈	—	52	1	no	—
103	C₄⋊C₈	—	52	2	no	—
104	C₄⋊C₈	—	52	2	no	—
105	C₄⋊C₈	—	52	2	no	—
106	C₄⋊C₈	—	52	2	no	—
107	C₂×C₁₃	#1286	1	2	yes	—
108	C₂×C₁₃	#1290	1	1	yes	characteristic
109	C₂×C₁₃	#1296	1	1	yes	characteristic
110	C₂×C₁₃	#1301	1	2	yes	—
111	C₂×C₁₃	#1307	1	1	yes	characteristic
112	C₂²×C₄	—	13	2	no	—
113	C₂²×C₄	—	13	2	no	—
114	C₂²×C₄	C₄×#261	1	1	yes	characteristic
115	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
116	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	52	4	no	—
120	C₂×C₈	—	52	4	no	—
121	C₂×C₈	—	52	4	no	—
122	C₂×C₈	—	52	2	no	—
123	C₂×C₈	—	52	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	13	2	no	—
127	C₄²	—	13	2	no	—
128	C₄²	—	13	2	no	—
129	C₄²	—	13	2	no	—
130	C₄²	—	26	1	no	—
131	C₄²	—	26	1	no	—
132	C₄²	—	52	1	no	—
133	C₄²	—	52	2	no	—
134	C₁₃	#7040	1	1	yes	characteristic, complemented, Sylow
135	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	13	2	no	—
141	C₂×C₄	—	13	2	no	—
142	C₂×C₄	—	13	2	no	—
143	C₂×C₄	—	13	2	no	—
144	C₂×C₄	—	26	2	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	—	26	2	no	—
147	C₂×C₄	—	26	2	no	—
148	C₂×C₄	#88103	1	1	yes	characteristic
149	C₂×C₄	#88104	1	1	yes	characteristic
150	C₂×C₄	#88113	1	1	yes	characteristic
151	C₂×C₄	#88115	1	2	yes	—
152	C₂×C₄	#88117	1	2	yes	—
153	C₂×C₄	#88121	1	1	yes	characteristic
154	C₈	—	8	1	no	—
155	C₈	—	104	4	no	—
156	C₈	—	104	1	no	—
157	C₂²	#11718246	1	2	yes	—
158	C₂²	#11718267	1	1	yes	characteristic
159	C₂²	#11718268	1	1	yes	characteristic
160	C₂²	#11718274	1	2	yes	—
161	C₂²	#11718291	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	26	2	no	—
166	C₄	—	26	2	no	—
167	C₄	—	26	2	no	—
168	C₄	—	26	2	no	—
169	C₄	#11718245	1	2	yes	—
170	C₄	#11718297	1	1	yes	characteristic
171	C₄	#11718303	1	1	yes	characteristic
172	C₂	#13129691	1	2	yes	—
173	C₂	#13129701	1	1	yes	characteristic
174	C₂	#13129714	1	1	yes	characteristic
175	C₂	#13129728	1	2	yes	—
176	C₂	#13129742	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130753

C₂

yes

maximal

#13114987

C₂

yes

maximal, characteristic

C₂×C₄⋊#1057

—

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₂×#88054

—

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₄

yes

characteristic

#11718290

—

#11718290

C₄

yes

—

C₂²×#1057

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

Q₈

yes

characteristic

C₂²×#1055

—

C₄⋊#1057

—

C₄⋊#1057

—

C₁₃×C₄⋊C₈

—

—

—

—

—

C₂²×C₄×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₃

C₂²⋊C₄

yes

—

C₂²×Dic₁₃

C₄⋊C₄

yes

—

C₂×#1057

—

C₂×#1057

—

C₂×#1057

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

C₄×Dic₁₃

Q₁₆

yes

—

—

—

—

—

C₂³×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#103

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#102

yes

—

C₂×Dic₁₃

#103

yes

—

C₂×Dic₁₃

#106

yes

—

C₂×Dic₁₃

#107

yes

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#261

yes

characteristic

C₄³

—

#336

—

#336

—

C₂²×C₁₃

H(3,ℤ/4)

yes

characteristic

C₂²×C₁₃

#340

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#331

yes

characteristic

C₄×C₁₃

#332

yes

—

C₄×C₁₃

#333

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂×#261

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₂×C₁₃

#1286

yes

—

108

C₂×C₁₃

#1290

yes

characteristic

109

C₂×C₁₃

#1296

yes

characteristic

110

C₂×C₁₃

#1301

yes

—

111

C₂×C₁₃

#1307

yes

characteristic

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄×#261

yes

characteristic

115

C₂²×C₄

C₄⋊#261

yes

characteristic

116

C₂²×C₄

C₂²⋊#261

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₁₃

#7040

yes

characteristic, complemented, Sylow

135

C₂³

(C₂×#261)⋊C₄

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#88103

yes

characteristic

149

C₂×C₄

#88104

yes

characteristic

150

C₂×C₄

#88113

yes

characteristic

151

C₂×C₄

#88115

yes

—

152

C₂×C₄

#88117

yes

—

153

C₂×C₄

#88121

yes

characteristic

154

C₈

—

155

C₈

—

104

—

156

C₈

—

104

—

157

C₂²

#11718246

yes

—

158

C₂²

#11718267

yes

characteristic

159

C₂²

#11718268

yes

characteristic

160

C₂²

#11718274

yes

—

161

C₂²

#11718291

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#11718245

yes

—

170

C₄

#11718297

yes

characteristic

171

C₄

#11718303

yes

characteristic

172

C₂

#13129691

yes

—

173

C₂

#13129701

yes

characteristic

174

C₂

#13129714

yes

characteristic

175

C₂

#13129728

yes

—

176

C₂

#13129742

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×#261)⋊C₄
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle, prime radical	C₂³×C₁₃	#95
cosocle	C₄²×Dic₁₃	C₂²
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#7040
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

(C₂×#261)⋊C₄

commutator

C₂²×C₄×C₁₃

C₄²

Frattini

C₂×C₄²

C₂×#261

Fitting

C₂×C₁₃×C₄⋊C₈

C₄

socle, prime radical

C₂³×C₁₃

#95

cosocle

C₄²×Dic₁₃

C₂²

C₂×C₄⋊C₈

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
134. C₁₃	#7040	semidirect

H ≅ G/N

type

134. C₁₃

#7040

semidirect

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

subgroup	index
#13130753	2
#13114987	2
#7040	13

subgroup

index

#13130753

#13114987

#7040

Subgroups — #16776891

Conjugacy classes of subgroups

Complements

Maximal subgroups