Group #16777399 — subgroups

Subgroups — #16777399

Order 3328.

270 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13130716	C₂	1	2	yes	maximal
2	C₂×#11718099	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
4	#11719127	—	2	2	no	—
5	C₂×#88104	—	2	2	no	—
6	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×#6090	C₄	1	1	yes	characteristic
8	#11718268	C₄	1	4	yes	—
9	#11718099	—	2	2	no	—
10	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
11	C₂×#6118	—	2	4	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×#1055	Q₈	1	1	yes	characteristic
14	C₂×C₈×D₁₃	—	4	1	no	—
15	C₂×C₈×D₁₃	—	4	1	no	—
16	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₁₃	D₄	1	2	yes	—
18	C₂²⋊#1057	—	4	2	no	—
19	#88111	—	4	2	no	—
20	#88104	—	4	2	no	—
21	#88104	—	4	2	no	—
22	C₂×C₁₃×C₁₆	—	2	1	no	—
23	C₂×C₁₃×C₁₆	—	4	1	no	—
24	C₂×#6090	—	2	1	no	—
25	C₂×#6090	—	4	1	no	—
26	#7231	—	13	1	no	maximal, Sylow
27	C₂³×D₁₃	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
30	C₂×C₄×D₁₃	—	4	1	no	—
31	C₂×C₄×D₁₃	—	4	1	no	—
32	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
34	C₂×#1057	—	4	4	no	—
35	#6118	—	4	4	no	—
36	#6118	—	4	4	no	—
37	C₂×C₈×C₁₃	—	2	1	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#1055	—	4	1	no	—
42	C₂×#1055	C₄⋊C₄	1	2	yes	—
43	C₈×D₁₃	—	2	2	no	—
44	C₈×D₁₃	—	2	2	no	—
45	C₈×D₁₃	—	4	2	no	—
46	C₁₃×C₁₆	—	4	1	no	—
47	#6090	—	4	1	no	—
48	C₂×C₂²⋊C₁₆	—	13	1	no	—
49	#1810	—	13	2	no	—
50	C₂³×C₁₃	#107	1	1	yes	characteristic
51	C₂²×D₁₃	—	2	1	no	—
52	C₂²×D₁₃	—	2	1	no	—
53	C₂²×D₁₃	—	4	1	no	—
54	C₂²×D₁₃	—	4	1	no	—
55	C₂²×D₁₃	#103	1	2	yes	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
60	C₂×Dic₁₃	—	4	1	no	—
61	C₂×Dic₁₃	#102	1	2	yes	—
62	C₄×D₁₃	—	2	2	no	—
63	C₄×D₁₃	—	2	2	no	—
64	C₄×D₁₃	—	4	2	no	—
65	#1057	—	8	4	no	—
66	C₈×C₁₃	—	2	1	no	—
67	C₈×C₁₃	#99	1	1	yes	characteristic
68	C₈×C₁₃	#99	1	1	yes	characteristic
69	#1055	—	2	2	no	—
70	C₂³×C₈	—	13	1	no	—
71	C₂²×C₁₆	—	13	1	no	—
72	C₂²×C₁₆	#261	1	1	yes	characteristic
73	C₂×#108	—	26	2	no	—
74	#407	—	26	2	no	—
75	C₂²⋊C₁₆	—	26	2	no	—
76	#341	—	13	4	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	#340	1	1	yes	characteristic
80	C₂²×C₁₃	#368	1	1	yes	characteristic
81	C₂²×C₁₃	#368	1	1	yes	characteristic
82	C₂×D₁₃	—	2	2	no	—
83	C₂×D₁₃	—	2	2	no	—
84	C₂×D₁₃	—	2	2	no	—
85	C₂×D₁₃	—	4	2	no	—
86	C₂×D₁₃	—	4	2	no	—
87	C₄×C₁₃	—	2	1	no	—
88	C₄×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
89	C₄×C₁₃	#342	1	1	yes	characteristic
90	Dic₁₃	—	2	2	no	—
91	C₂³×C₄	—	13	1	no	—
92	C₂×M₁₆	—	26	4	no	—
93	C₂²×C₈	—	13	1	no	—
94	C₂²×C₈	—	13	2	no	—
95	C₂²×C₈	—	13	2	no	—
96	C₂²×C₈	—	52	1	no	—
97	C₂²×C₈	—	52	1	no	—
98	C₂²×C₈	C₂×#261	1	1	yes	characteristic
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	4	1	no	—
101	C₂×C₁₆	—	26	1	no	—
102	C₂×C₁₆	—	52	1	no	—
103	#108	—	52	2	no	—
104	#108	—	52	2	no	—
105	C₂²⋊C₈	—	52	2	no	—
106	C₂²⋊C₈	—	52	2	no	—
107	C₂×C₁₃	—	2	1	no	—
108	C₂×C₁₃	—	2	1	no	—
109	C₂×C₁₃	#1289	1	1	yes	characteristic
110	C₂×C₁₃	#1381	1	1	yes	characteristic
111	C₂×C₁₃	#1385	1	1	yes	characteristic
112	D₁₃	—	4	2	no	—
113	C₂⁴	—	13	1	no	—
114	C₂²×C₄	—	13	1	no	—
115	C₂²×C₄	—	13	2	no	—
116	C₂²×C₄	—	13	2	no	—
117	C₂²×C₄	—	52	1	no	—
118	C₂²×C₄	—	52	1	no	—
119	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
120	M₁₆	—	52	4	no	—
121	M₁₆	—	52	4	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	13	2	no	—
125	C₂×C₈	—	26	2	no	—
126	C₂×C₈	—	26	2	no	—
127	C₂×C₈	—	52	2	no	—
128	C₂×C₈	—	52	1	no	—
129	C₂×C₈	—	52	4	no	—
130	C₂×C₈	C₄×#261	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
132	C₁₆	—	4	1	no	—
133	C₁₆	—	52	1	no	—
134	C₁₃	#7231	1	1	yes	characteristic, complemented, Sylow
135	C₂³	—	13	2	no	—
136	C₂³	—	26	1	no	—
137	C₂³	—	26	1	no	—
138	C₂³	—	52	1	no	—
139	C₂³	—	52	1	no	—
140	C₂³	#88102	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	13	2	no	—
144	C₂×C₄	—	26	2	no	—
145	C₂×C₄	—	26	2	no	—
146	C₂×C₄	—	52	2	no	—
147	C₂×C₄	—	52	1	no	—
148	C₂×C₄	#88101	1	1	yes	characteristic
149	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	26	2	no	—
152	C₈	—	104	4	no	—
153	C₈	#88107	1	1	yes	characteristic
154	C₈	#88119	1	1	yes	characteristic
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	26	2	no	—
158	C₂²	—	26	2	no	—
159	C₂²	—	26	2	no	—
160	C₂²	—	52	2	no	—
161	C₂²	—	52	2	no	—
162	C₂²	#11718223	1	1	yes	characteristic
163	C₂²	#11718224	1	1	yes	characteristic
164	C₂²	#11718265	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	26	2	no	—
167	C₄	#11718267	1	1	yes	characteristic
168	C₄	#11718289	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	52	2	no	—
172	C₂	#13129699	1	1	yes	characteristic
173	C₂	#13129896	1	1	yes	characteristic
174	C₂	#13129906	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130716

C₂

yes

maximal

C₂×#11718099

C₂

yes

maximal, characteristic

C₂²×C₈×D₁₃

C₂²

yes

characteristic

#11719127

—

C₂×#88104

—

C₂²×C₁₃×C₁₆

C₄

yes

characteristic

C₂²×#6090

C₄

yes

characteristic

#11718268

C₄

yes

—

#11718099

—

C₂²×C₄×D₁₃

D₄

yes

characteristic

C₂×#6118

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

Q₈

yes

characteristic

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

C₂×C₄

yes

—

C₂×C₈×D₁₃

D₄

yes

—

C₂²⋊#1057

—

#88111

—

#88104

—

#88104

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

—

—

—

C₂³×D₁₃

Q₁₆

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₃

D₈

yes

characteristic

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₂²⋊C₄

yes

—

C₂×C₄×D₁₃

SD₁₆

yes

—

C₂×#1057

—

#6118

—

#6118

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

#6090

—

C₂×C₂²⋊C₁₆

—

#1810

—

C₂³×C₁₃

#107

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

#103

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

#102

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

yes

—

—

C₂²⋊C₁₆

—

#341

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#261

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

#108

—

104

#108

—

105

C₂²⋊C₈

—

106

C₂²⋊C₈

—

107

C₂×C₁₃

—

108

C₂×C₁₃

—

109

C₂×C₁₃

#1289

yes

characteristic

110

C₂×C₁₃

#1381

yes

characteristic

111

C₂×C₁₃

#1385

yes

characteristic

112

D₁₃

—

113

C₂⁴

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₄⋊#261

yes

characteristic

120

M₁₆

—

121

M₁₆

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄×#261

yes

characteristic

131

C₂×C₈

C₂²⋊#261

yes

characteristic

132

C₁₆

—

133

C₁₆

—

134

C₁₃

#7231

yes

characteristic, complemented, Sylow

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

#88102

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#88101

yes

characteristic

149

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

104

—

153

C₈

#88107

yes

characteristic

154

C₈

#88119

yes

characteristic

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

#11718223

yes

characteristic

163

C₂²

#11718224

yes

characteristic

164

C₂²

#11718265

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

#11718267

yes

characteristic

168

C₄

#11718289

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

—

172

C₂

#13129699

yes

characteristic

173

C₂

#13129896

yes

characteristic

174

C₂

#13129906

yes

characteristic

	subgroup	G/H
center	C₂²	#11718265
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂²×C₈	C₂×#261
Fitting	C₂²×C₁₃×C₁₆	C₄
socle	C₂²×C₁₃	#340
prime radical	C₂³×D₁₃	Q₁₆
cosocle	C₂²×C₈×D₁₃	C₂²
hypercenter	C₂²×C₁₆	#261
hypocenter	C₁₃	#7231
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#11718265

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂²×C₈

C₂×#261

Fitting

C₂²×C₁₃×C₁₆

C₄

socle

C₂²×C₁₃

#340

prime radical

C₂³×D₁₃

Q₁₆

cosocle

C₂²×C₈×D₁₃

C₂²

C₂²×C₁₆

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
134. C₁₃	#7231	semidirect

H ≅ G/N

type

134. C₁₃

#7231

semidirect

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

subgroup	index
#13130716	2
C₂×#11718099	2
#7231	13

subgroup

index

#13130716

C₂×#11718099

#7231

Subgroups — #16777399

Conjugacy classes of subgroups

Complements

Maximal subgroups