Group #16777418 — subgroups

Subgroups — #16777418

Order 3328.

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

normal subgroups only

This group has 280 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	#13130702	C₂	1	2	yes	maximal
2	#13115772	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
4	C₂×#88113	—	2	2	no	—
5	#11719129	—	2	2	no	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
8	#11718291	C₄	1	4	yes	—
9	#11718142	—	2	2	no	—
10	C₂²×Dic₂₆	D₄	1	1	yes	characteristic
11	C₂×#6122	—	2	4	no	—
12	C₂²×C₈×C₁₃	—	2	1	no	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
15	C₂×#6101	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
17	C₂²×#1055	—	2	1	no	—
18	C₄×Dic₂₆	—	4	1	no	—
19	C₄×Dic₂₆	—	4	1	no	—
20	#88113	—	4	2	no	—
21	#88113	—	4	2	no	—
22	#88112	—	4	2	no	—
23	C₄⋊#1057	—	4	2	no	—
24	C₄×C₈×C₁₃	—	4	1	no	—
25	C₄⋊#1055	—	4	1	no	—
26	#7238	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
31	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₂₆	—	4	1	no	—
33	C₂×Dic₂₆	—	4	1	no	—
34	C₂×Dic₂₆	D₈	1	1	yes	characteristic
35	C₂×Dic₂₆	SD₁₆	1	2	yes	—
36	C₂×Dic₂₆	Q₁₆	1	1	yes	characteristic
37	#6122	—	4	4	no	—
38	#6122	—	4	4	no	—
39	C₂×#1057	—	4	4	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	4	1	no	—
42	C₂×C₈×C₁₃	—	4	1	no	—
43	C₄²×C₁₃	—	2	1	no	—
44	C₄²×C₁₃	—	2	1	no	—
45	C₄⋊Dic₁₃	—	4	1	no	—
46	#6101	—	4	2	no	—
47	C₄×Dic₁₃	—	2	2	no	—
48	C₄×Dic₁₃	—	2	2	no	—
49	C₂×#1055	—	4	2	no	—
50	C₂×#1055	—	4	1	no	—
51	#1984	—	13	1	no	—
52	#1801	—	13	2	no	—
53	C₂³×C₁₃	#95	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	—	2	1	no	—
58	C₂×C₄×C₁₃	—	2	1	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
64	C₂×Dic₁₃	—	2	2	no	—
65	C₂×Dic₁₃	—	2	2	no	—
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	4	2	no	—
68	C₂×Dic₁₃	#102	1	2	yes	—
69	C₂×Dic₁₃	#103	1	2	yes	—
70	Dic₂₆	—	4	1	no	—
71	Dic₂₆	—	4	2	no	—
72	Dic₂₆	—	4	1	no	—
73	#1057	—	8	4	no	—
74	C₈×C₁₃	—	4	2	no	—
75	#1055	—	8	1	no	—
76	C₂×C₄×Q₈	—	13	1	no	—
77	#423	—	26	2	no	—
78	C₂×C₄⋊C₈	—	13	1	no	—
79	C₂×#101	—	26	2	no	—
80	C₂×C₄×C₈	#261	1	1	yes	characteristic
81	#343	—	13	4	no	—
82	#336	—	26	2	no	—
83	C₂²×C₁₃	—	2	1	no	—
84	C₂²×C₁₃	—	2	1	no	—
85	C₂²×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
86	C₂²×C₁₃	#340	1	1	yes	characteristic
87	C₂²×C₁₃	#342	1	1	yes	characteristic
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	4	1	no	—
92	C₄×C₁₃	#355	1	1	yes	characteristic
93	C₄×C₁₃	#356	1	1	yes	characteristic
94	Dic₁₃	—	2	2	no	—
95	Dic₁₃	—	2	2	no	—
96	Dic₁₃	—	4	2	no	—
97	C₂²×Q₈	—	13	1	no	—
98	C₂×M₁₆	—	26	4	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	26	1	no	—
101	C₄×Q₈	—	52	1	no	—
102	C₄×Q₈	—	52	1	no	—
103	C₂×C₄⋊C₄	—	13	2	no	—
104	C₂×C₄⋊C₄	—	13	1	no	—
105	C₂×C₄²	—	13	2	no	—
106	C₂×C₄²	C₂×#261	1	1	yes	characteristic
107	C₄⋊C₈	—	52	1	no	—
108	C₄⋊C₈	—	52	2	no	—
109	C₄⋊C₈	—	52	2	no	—
110	#101	—	52	2	no	—
111	#101	—	52	2	no	—
112	C₄×C₈	—	4	1	no	—
113	C₂×C₁₃	—	2	1	no	—
114	C₂×C₁₃	—	2	1	no	—
115	C₂×C₁₃	#1289	1	1	yes	characteristic
116	C₂×C₁₃	#1392	1	1	yes	characteristic
117	C₂×C₁₃	#1394	1	1	yes	characteristic
118	C₂×Q₈	—	13	2	no	—
119	C₂×Q₈	—	13	1	no	—
120	C₂×Q₈	—	13	1	no	—
121	C₂×Q₈	—	52	1	no	—
122	C₂×Q₈	—	52	1	no	—
123	C₂²×C₄	—	13	2	no	—
124	C₂²×C₄	—	13	2	no	—
125	C₂²×C₄	C₄×#261	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
128	M₁₆	—	52	4	no	—
129	M₁₆	—	52	4	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	52	2	no	—
134	C₂×C₈	—	52	4	no	—
135	C₂×C₈	—	52	1	no	—
136	C₄⋊C₄	—	52	2	no	—
137	C₄⋊C₄	—	52	1	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	26	2	no	—
141	C₄²	—	26	2	no	—
142	C₁₃	#7238	1	1	yes	characteristic, complemented, Sylow
143	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
144	Q₈	—	52	2	no	—
145	Q₈	—	52	1	no	—
146	Q₈	—	52	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	13	2	no	—
154	C₂×C₄	—	13	2	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	—	26	2	no	—
157	C₂×C₄	—	26	2	no	—
158	C₂×C₄	—	52	2	no	—
159	C₂×C₄	#88101	1	1	yes	characteristic
160	C₂×C₄	#88102	1	1	yes	characteristic
161	C₂×C₄	#88107	1	1	yes	characteristic
162	C₂×C₄	#88119	1	1	yes	characteristic
163	C₈	—	4	2	no	—
164	C₈	—	104	4	no	—
165	C₈	—	104	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#11718265	1	1	yes	characteristic
169	C₂²	#11718267	1	1	yes	characteristic
170	C₂²	#11718289	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	26	2	no	—
176	C₄	—	26	2	no	—
177	C₄	—	52	2	no	—
178	C₄	#11718227	1	1	yes	characteristic
179	C₄	#11718228	1	1	yes	characteristic
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	#13129699	1	1	yes	characteristic
183	C₂	#13129927	1	1	yes	characteristic
184	C₂	#13129933	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130702

C₂

yes

maximal

#13115772

C₂

yes

maximal, characteristic

C₂×C₄×Dic₂₆

C₂²

yes

characteristic

C₂×#88113

—

#11719129

—

C₂×C₄×C₈×C₁₃

C₄

yes

characteristic

C₂×C₄⋊#1055

C₄

yes

characteristic

#11718291

C₄

yes

—

#11718142

—

C₂²×Dic₂₆

D₄

yes

characteristic

C₂×#6122

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

Q₈

yes

characteristic

C₂×#6101

C₂×C₄

yes

—

C₂×C₄×Dic₁₃

D₄

yes

—

C₂²×#1055

—

C₄×Dic₂₆

—

C₄×Dic₂₆

—

—

—

—

—

C₄×C₈×C₁₃

—

C₄⋊#1055

—

#7238

—

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×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

D₈

yes

characteristic

C₂×Dic₂₆

SD₁₆

yes

—

C₂×Dic₂₆

Q₁₆

yes

—

—

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

C₂³×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

#102

yes

—

C₂×Dic₁₃

#103

yes

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×C₄×Q₈

—

#423

—

C₂×C₄⋊C₈

—

C₂×#101

—

C₂×C₄×C₈

yes

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₄×Q₈

—

102

C₄×Q₈

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×#261

yes

characteristic

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

#101

—

111

#101

—

112

C₄×C₈

—

113

C₂×C₁₃

—

114

C₂×C₁₃

—

115

C₂×C₁₃

#1289

yes

characteristic

116

C₂×C₁₃

#1392

yes

characteristic

117

C₂×C₁₃

#1394

yes

characteristic

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₄×#261

yes

characteristic

126

C₂²×C₄

C₄⋊#261

yes

characteristic

127

C₂²×C₄

C₂²⋊#261

yes

characteristic

128

M₁₆

—

129

M₁₆

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₃

#7238

yes

characteristic, complemented, Sylow

143

C₂³

(C₂×#261)⋊C₄

yes

characteristic

144

Q₈

—

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

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

—

159

C₂×C₄

#88101

yes

characteristic

160

C₂×C₄

#88102

yes

characteristic

161

C₂×C₄

#88107

yes

characteristic

162

C₂×C₄

#88119

yes

characteristic

163

C₈

—

164

C₈

—

104

—

165

C₈

—

104

—

166

C₂²

—

167

C₂²

—

168

C₂²

#11718265

yes

characteristic

169

C₂²

#11718267

yes

characteristic

170

C₂²

#11718289

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#11718227

yes

characteristic

179

C₄

#11718228

yes

characteristic

180

C₂

—

181

C₂

—

182

C₂

#13129699

yes

characteristic

183

C₂

#13129927

yes

characteristic

184

C₂

#13129933

yes

characteristic

	subgroup	G/H
center	C₂²	#11718289
commutator	C₂²×C₄×C₁₃	C₄²
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#342
prime radical	C₂³×C₁₃	#95
cosocle	C₂×C₄×Dic₂₆	C₂²
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#7238
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²

#11718289

commutator

C₂²×C₄×C₁₃

C₄²

Frattini

C₂×C₄²

C₂×#261

Fitting

C₂×C₄×C₈×C₁₃

C₄

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×Dic₂₆

C₂²

C₂×C₄×C₈

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
142. C₁₃	#7238	semidirect

H ≅ G/N

type

142. C₁₃

#7238

semidirect

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

subgroup	index
#13130702	2
#13115772	2
#7238	13

subgroup

index

#13130702

#13115772

#7238

Subgroups — #16777418

Conjugacy classes of subgroups

Complements

Maximal subgroups