Group #16777211 — subgroups

Subgroups — #16777211

Order 3328.

238 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₄×#88104	C₂	1	1	yes	maximal, characteristic
2	#13130384	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#88104	C₄	1	2	yes	—
5	#11719051	—	2	1	no	—
6	C₄×#6118	—	2	1	no	—
7	C₂×C₁₆×D₁₃	—	2	1	no	—
8	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	#11718268	—	2	1	no	—
10	#11718256	—	2	1	no	—
11	#11718253	C₄	1	2	yes	—
12	C₁₃×#346	C₄	1	1	yes	characteristic
13	#11718094	—	2	1	no	—
14	#11718053	C₄	1	1	yes	characteristic
15	C₂×#6118	—	2	2	no	—
16	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
18	C₄²×D₁₃	D₄	1	1	yes	characteristic
19	#88111	—	4	1	no	—
20	#88110	—	4	1	no	—
21	C₄⋊#1057	—	2	2	no	—
22	C₄×#1057	—	4	1	no	—
23	C₄×#1057	—	4	1	no	—
24	#88104	—	2	2	no	—
25	#88104	C₈	1	4	yes	complemented
26	C₂×C₁₃×C₁₆	—	2	1	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×#6090	—	2	1	no	—
31	C₄×#1055	Q₈	1	1	yes	characteristic
32	C₁₆×D₁₃	—	2	4	no	—
33	#7177	—	13	1	no	maximal, Sylow
34	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
37	C₂×#1057	—	4	2	no	—
38	C₂×#1057	—	4	2	no	—
39	#6118	—	2	4	no	—
40	#6118	—	4	2	no	—
41	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×Dic₁₃	D₈	1	1	yes	characteristic
46	C₄×Dic₁₃	SD₁₆	1	1	yes	characteristic
47	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
48	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₁₃	—	2	2	no	—
50	C₈×D₁₃	C₂×C₈	1	2	yes	—
51	C₈×D₁₃	M₁₆	1	2	yes	—
52	C₁₃×C₁₆	—	2	2	no	—
53	#6090	—	2	2	no	—
54	C₂×#346	—	13	1	no	—
55	C₄×#108	—	13	1	no	—
56	#1564	—	13	1	no	—
57	C₂²×D₁₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
61	C₂×Dic₁₃	#102	1	1	yes	characteristic
62	C₂×Dic₁₃	#102	1	1	yes	characteristic
63	C₂×Dic₁₃	#103	1	1	yes	characteristic
64	C₄×D₁₃	—	4	1	no	—
65	C₄×D₁₃	—	4	1	no	—
66	C₄×D₁₃	C₂²⋊C₈	1	2	yes	—
67	C₄×D₁₃	#104	1	2	yes	—
68	#1057	—	4	4	no	—
69	#1057	—	8	2	no	—
70	C₈×C₁₃	—	2	1	no	—
71	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
72	C₈×C₁₃	#97	1	1	yes	characteristic
73	#1055	—	2	1	no	—
74	#1055	C₄⋊C₈	1	2	yes	—
75	C₂²×C₁₆	—	26	1	no	—
76	#431	—	26	1	no	—
77	C₂×#108	—	13	2	no	—
78	C₄×M₁₆	—	26	1	no	—
79	C₂×C₄×C₈	—	13	1	no	—
80	#346	—	13	1	no	—
81	#346	—	26	1	no	—
82	#346	#261	1	1	yes	characteristic
83	#341	—	26	1	no	—
84	#335	—	26	1	no	—
85	#334	—	13	2	no	—
86	C₂²×C₁₃	#340	1	1	yes	characteristic
87	C₂×D₁₃	—	2	1	no	—
88	C₂×D₁₃	—	2	1	no	—
89	C₂×D₁₃	Q₈⋊C₈	1	2	yes	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	#336	1	1	yes	characteristic
93	C₄×C₁₃	#339	1	1	yes	characteristic
94	Dic₁₃	—	2	1	no	—
95	Dic₁₃	—	2	1	no	—
96	Dic₁₃	D₄⋊C₈	1	2	yes	—
97	C₂×M₁₆	—	26	2	no	—
98	C₂²×C₈	—	13	1	no	—
99	C₂²×C₈	—	13	1	no	—
100	C₂×C₄²	—	13	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	26	4	no	—
103	C₂×C₁₆	—	26	1	no	—
104	#108	—	13	4	no	—
105	#108	—	26	2	no	—
106	C₄⋊C₈	—	26	2	no	—
107	C₂²⋊C₈	—	52	1	no	—
108	#97	—	52	1	no	—
109	C₄×C₈	—	13	1	no	—
110	C₄×C₈	—	13	1	no	—
111	C₄×C₈	—	13	1	no	—
112	C₄×C₈	—	52	1	no	—
113	C₄×C₈	—	52	1	no	—
114	C₄×C₈	C₂×#261	1	1	yes	characteristic
115	C₂×C₁₃	#1279	1	1	yes	characteristic
116	C₂×C₁₃	#1388	1	1	yes	characteristic
117	C₂×C₁₃	#1389	1	1	yes	characteristic
118	D₁₃	—	2	2	no	—
119	C₂²×C₄	—	13	1	no	—
120	C₂²×C₄	—	13	1	no	—
121	C₂²×C₄	—	13	1	no	—
122	M₁₆	—	26	4	no	—
123	M₁₆	—	52	2	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	2	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	13	1	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	—	52	2	no	—
130	C₂×C₈	—	52	2	no	—
131	C₂×C₈	C₄×#261	1	1	yes	characteristic
132	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
133	C₄²	—	13	1	no	—
134	C₄²	—	13	1	no	—
135	C₄²	—	13	1	no	—
136	C₄²	C₄⋊#261	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	—	26	2	no	—
139	C₁₃	#7177	1	1	yes	characteristic, complemented, Sylow
140	C₂³	—	13	1	no	—
141	C₂×C₄	—	13	1	no	—
142	C₂×C₄	—	13	2	no	—
143	C₂×C₄	—	13	2	no	—
144	C₂×C₄	—	13	1	no	—
145	C₂×C₄	—	13	1	no	—
146	C₂×C₄	—	52	1	no	—
147	C₂×C₄	—	52	1	no	—
148	C₂×C₄	#88101	1	1	yes	characteristic
149	C₂×C₄	#88102	1	1	yes	characteristic
150	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	13	2	no	—
153	C₈	—	26	1	no	—
154	C₈	—	52	4	no	—
155	C₈	—	104	2	no	—
156	C₈	C₈×#261	1	1	yes	characteristic
157	C₈	#88100	1	1	yes	characteristic
158	C₂²	—	13	2	no	—
159	C₂²	—	26	1	no	—
160	C₂²	—	26	1	no	—
161	C₂²	#11718265	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	13	2	no	—
165	C₄	—	26	1	no	—
166	C₄	—	26	1	no	—
167	C₄	#11718234	1	1	yes	characteristic
168	C₄	(C₂×#261)⋊C₈	1	1	yes	characteristic
169	C₂	—	26	2	no	—
170	C₂	#13129671	1	1	yes	characteristic
171	C₂	#13129914	1	1	yes	characteristic
172	C₂	#13129917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#88104

C₂

yes

maximal, characteristic

#13130384

C₂

yes

maximal, characteristic

D₁₃×#346

C₂

yes

maximal, characteristic

C₂×#88104

C₄

yes

—

#11719051

—

C₄×#6118

—

C₂×C₁₆×D₁₃

—

C₄×C₈×D₁₃

C₂²

yes

—

—

C₄

yes

—

C₁₃×#346

C₄

yes

characteristic

#11718094

—

#11718053

C₄

yes

characteristic

C₂×#6118

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

D₄

yes

—

—

—

—

—

—

C₈

yes

complemented

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

D₄

yes

characteristic

C₂×#6090

—

C₄×#1055

Q₈

yes

characteristic

C₁₆×D₁₃

—

#7177

—

maximal, Sylow

C₂×C₄×D₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

characteristic

C₂×C₄×D₁₃

Q₁₆

yes

—

—

—

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₃

D₈

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

C₂×C₈

yes

—

C₈×D₁₃

M₁₆

yes

—

C₁₃×C₁₆

—

—

—

—

—

C₂²×D₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₂×Dic₁₃

yes

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂²⋊C₈

yes

—

C₄×D₁₃

#104

yes

—

#1057

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

C₄×C₈

yes

C₈×C₁₃

yes

—

C₄⋊C₈

yes

—

C₂²×C₁₆

—

#431

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

—

—

yes

—

—

—

C₂²×C₁₃

#340

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

Q₈⋊C₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

D₄⋊C₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

#108

—

105

#108

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#261

yes

characteristic

115

C₂×C₁₃

#1279

yes

characteristic

116

C₂×C₁₃

#1388

yes

characteristic

117

C₂×C₁₃

#1389

yes

characteristic

118

D₁₃

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

C₄×#261

yes

characteristic

132

C₂×C₈

C₂²⋊#261

yes

characteristic

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₄⋊#261

yes

characteristic

137

C₁₆

—

138

C₁₆

—

139

C₁₃

#7177

yes

characteristic, complemented, Sylow

140

C₂³

—

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

#88102

yes

characteristic

150

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

104

—

156

C₈

C₈×#261

yes

characteristic

157

C₈

#88100

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

#11718265

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#11718234

yes

characteristic

168

C₄

(C₂×#261)⋊C₈

yes

characteristic

169

C₂

—

170

C₂

#13129671

yes

characteristic

171

C₂

#13129914

yes

characteristic

172

C₂

#13129917

yes

characteristic

	subgroup	G/H
center	C₂²	#11718265
commutator	C₈×C₁₃	C₄×C₈
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#346	C₄
socle	C₂²×C₁₃	#340
prime radical	C₂²×D₁₃	#103
cosocle	C₄×C₈×D₁₃	C₂²
hypercenter	#346	#261
hypocenter	C₁₃	#7177
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

#11718265

commutator

C₈×C₁₃

C₄×C₈

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#346

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄×C₈×D₁₃

C₂²

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
25. #88104	C₈	semidirect
139. C₁₃	#7177	semidirect

H ≅ G/N

type

25. #88104

C₈

semidirect

139. C₁₃

#7177

semidirect

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

subgroup	index
C₄×#88104	2
#13130384	2
D₁₃×#346	2
#7177	13

subgroup

index

C₄×#88104

#13130384

D₁₃×#346

#7177

Subgroups — #16777211

Conjugacy classes of subgroups

Complements

Maximal subgroups