Group #16776910 — subgroups

Subgroups — #16776910

Order 3328.

304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 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	#13130622	C₂	1	2	yes	maximal
2	#13114952	C₂	1	1	yes	maximal, characteristic
3	C₂×#88110	—	2	2	no	—
4	C₂×C₄⋊#1057	—	2	2	no	—
5	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
8	#11718290	C₄	1	4	yes	—
9	#11718142	—	2	2	no	—
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	2	yes	—
15	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
17	C₂²×#1055	—	2	1	no	—
18	#88110	—	4	2	no	—
19	#88110	—	4	2	no	—
20	C₄⋊#1057	—	2	4	no	—
21	C₄⋊#1057	—	2	4	no	—
22	C₁₃×C₄⋊C₈	—	4	1	no	—
23	C₄⋊#1055	—	4	1	no	—
24	#7049	—	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	—	2	8	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	2	no	—
38	C₄×Dic₁₃	—	2	2	no	—
39	C₄×Dic₁₃	—	4	2	no	—
40	C₄×Dic₁₃	—	4	1	no	—
41	C₄×Dic₁₃	D₈	1	1	yes	characteristic
42	C₄×Dic₁₃	SD₁₆	1	2	yes	—
43	C₄×Dic₁₃	Q₁₆	1	1	yes	characteristic
44	C₂×#1055	—	4	1	no	—
45	C₂×#1055	—	4	2	no	—
46	#1831	—	13	1	no	—
47	#1767	—	13	2	no	—
48	C₂³×C₁₃	#95	1	1	yes	characteristic
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#101	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#104	1	2	yes	—
56	C₂×C₄×C₁₃	#104	1	2	yes	—
57	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
59	C₂×Dic₁₃	—	2	2	no	—
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	#1057	—	4	8	no	—
67	C₈×C₁₃	—	8	1	no	—
68	#1055	—	8	1	no	—
69	C₂×C₄⋊C₈	—	13	1	no	—
70	C₂×C₄⋊C₈	—	26	2	no	—
71	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
72	C₂×#97	—	26	2	no	—
73	C₄³	—	13	1	no	—
74	#336	—	13	4	no	—
75	#336	—	26	2	no	—
76	C₂²×C₁₃	#338	1	1	yes	characteristic
77	C₂²×C₁₃	#339	1	2	yes	—
78	C₂²×C₁₃	#340	1	1	yes	characteristic
79	C₂²×C₁₃	#343	1	1	yes	characteristic
80	C₂²×C₁₃	#344	1	2	yes	—
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	—	4	1	no	—
84	C₄×C₁₃	#329	1	1	yes	characteristic
85	C₄×C₁₃	#329	1	1	yes	characteristic
86	C₄×C₁₃	#330	1	2	yes	—
87	Dic₁₃	—	2	2	no	—
88	Dic₁₃	—	2	2	no	—
89	Dic₁₃	—	4	2	no	—
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	—	26	4	no	—
92	C₂²×C₈	—	26	1	no	—
93	C₂×C₄²	—	13	1	no	—
94	C₂×C₄²	—	13	2	no	—
95	C₂×C₄²	—	13	2	no	—
96	C₂×C₄²	—	13	1	no	—
97	C₂×C₄²	C₂×#261	1	1	yes	characteristic
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	26	4	no	—
100	C₄⋊C₈	—	26	4	no	—
101	C₄⋊C₈	—	52	1	no	—
102	#97	—	52	2	no	—
103	#97	—	52	2	no	—
104	C₂×C₁₃	#1291	1	2	yes	—
105	C₂×C₁₃	#1292	1	1	yes	characteristic
106	C₂×C₁₃	#1295	1	1	yes	characteristic
107	C₂×C₁₃	#1303	1	2	yes	—
108	C₂×C₁₃	#1307	1	1	yes	characteristic
109	C₂²×C₄	—	13	2	no	—
110	C₂²×C₄	—	13	2	no	—
111	C₂²×C₄	C₄×#261	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
113	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	26	8	no	—
117	C₂×C₈	—	52	4	no	—
118	C₂×C₈	—	52	4	no	—
119	C₂×C₈	—	52	2	no	—
120	C₂×C₈	—	52	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	13	2	no	—
124	C₄²	—	13	1	no	—
125	C₄²	—	13	1	no	—
126	C₄²	—	26	2	no	—
127	C₄²	—	26	2	no	—
128	C₄²	—	52	1	no	—
129	C₄²	—	52	2	no	—
130	C₁₃	#7049	1	1	yes	characteristic, complemented, Sylow
131	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	13	2	no	—
137	C₂×C₄	—	13	2	no	—
138	C₂×C₄	—	26	2	no	—
139	C₂×C₄	—	26	2	no	—
140	C₂×C₄	—	26	2	no	—
141	C₂×C₄	—	26	2	no	—
142	C₂×C₄	—	26	2	no	—
143	C₂×C₄	#88101	1	1	yes	characteristic
144	C₂×C₄	#88102	1	1	yes	characteristic
145	C₂×C₄	#88113	1	1	yes	characteristic
146	C₂×C₄	#88116	1	2	yes	—
147	C₂×C₄	#88118	1	2	yes	—
148	C₂×C₄	#88121	1	1	yes	characteristic
149	C₈	—	8	1	no	—
150	C₈	—	52	8	no	—
151	C₈	—	104	1	no	—
152	C₂²	#11718247	1	2	yes	—
153	C₂²	#11718265	1	1	yes	characteristic
154	C₂²	#11718269	1	1	yes	characteristic
155	C₂²	#11718274	1	2	yes	—
156	C₂²	#11718291	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	1	no	—
160	C₄	—	26	2	no	—
161	C₄	—	26	2	no	—
162	C₄	—	52	2	no	—
163	C₄	#11718242	1	2	yes	—
164	C₄	#11718293	1	1	yes	characteristic
165	C₄	#11718299	1	1	yes	characteristic
166	C₂	#13129702	1	2	yes	—
167	C₂	#13129706	1	1	yes	characteristic
168	C₂	#13129712	1	1	yes	characteristic
169	C₂	#13129733	1	2	yes	—
170	C₂	#13129742	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130622

C₂

yes

maximal

#13114952

C₂

yes

maximal, characteristic

C₂×#88110

—

C₂×C₄⋊#1057

—

C₂×C₁₃×C₄⋊C₈

C₄

yes

characteristic

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₄

yes

characteristic

#11718290

C₄

yes

—

#11718142

—

C₂²×#1057

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

characteristic

C₂×C₄×Dic₁₃

Q₈

yes

characteristic

C₂²×#1055

—

—

—

—

—

C₁₃×C₄⋊C₈

—

C₄⋊#1055

—

#7049

—

maximal, Sylow

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

characteristic

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

#104

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#102

yes

—

C₂×Dic₁₃

#103

yes

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#261

yes

characteristic

C₂×#97

—

C₄³

—

#336

—

#336

—

C₂²×C₁₃

#338

yes

characteristic

C₂²×C₁₃

#339

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#261

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

#97

—

104

C₂×C₁₃

#1291

yes

—

105

C₂×C₁₃

#1292

yes

characteristic

106

C₂×C₁₃

#1295

yes

characteristic

107

C₂×C₁₃

#1303

yes

—

108

C₂×C₁₃

#1307

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₄×#261

yes

characteristic

112

C₂²×C₄

C₄⋊#261

yes

characteristic

113

C₂²×C₄

C₂²⋊#261

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₁₃

#7049

yes

characteristic, complemented, Sylow

131

C₂³

(C₂×#261)⋊C₄

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#88101

yes

characteristic

144

C₂×C₄

#88102

yes

characteristic

145

C₂×C₄

#88113

yes

characteristic

146

C₂×C₄

#88116

yes

—

147

C₂×C₄

#88118

yes

—

148

C₂×C₄

#88121

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

104

—

152

C₂²

#11718247

yes

—

153

C₂²

#11718265

yes

characteristic

154

C₂²

#11718269

yes

characteristic

155

C₂²

#11718274

yes

—

156

C₂²

#11718291

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11718242

yes

—

164

C₄

#11718293

yes

characteristic

165

C₄

#11718299

yes

characteristic

166

C₂

#13129702

yes

—

167

C₂

#13129706

yes

characteristic

168

C₂

#13129712

yes

characteristic

169

C₂

#13129733

yes

—

170

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	#7049
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
130. C₁₃	#7049	semidirect

H ≅ G/N

type

130. C₁₃

#7049

semidirect

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

subgroup	index
#13130622	2
#13114952	2
#7049	13

subgroup

index

#13130622

#13114952

#7049

Subgroups — #16776910

Conjugacy classes of subgroups

Complements

Maximal subgroups