Group #16777021 — subgroups

Subgroups — #16777021

Order 3328.

268 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 56 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	#13129664	C₂	1	2	yes	maximal, complemented
2	#13115663	C₂	1	1	yes	maximal, characteristic
3	#11719182	C₄	1	1	yes	characteristic
4	#11719119	—	2	2	no	—
5	C₁₃×#423	C₄	1	1	yes	characteristic
6	C₂×#88065	—	2	2	no	—
7	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
8	C₂×#6127	—	2	1	no	—
9	C₂×C₁₃×M₁₆	—	2	1	no	—
10	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
11	#88184	—	4	1	no	—
12	C₂×C₄×Dic₁₃	—	4	1	no	—
13	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
14	#88129	—	2	1	no	—
15	#88128	C₈	1	1	yes	characteristic, complemented
16	C₄×D₅₂	—	4	1	no	—
17	#88123	C₈	1	1	yes	characteristic, complemented
18	C₂²⋊#1057	—	4	2	no	—
19	C₄⋊#1057	—	2	4	no	—
20	C₄×#1057	—	4	2	no	—
21	C₁₃×C₄⋊C₈	—	4	1	no	—
22	#88065	—	4	2	no	—
23	#88065	—	4	2	no	—
24	#7106	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	—	2	1	no	—
26	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₃	—	4	1	no	—
28	C₂²×Dic₁₃	—	2	2	no	—
29	#6127	—	4	2	no	—
30	#6127	—	4	2	no	—
31	C₂×D₅₂	—	2	1	no	—
32	C₂×D₅₂	—	4	1	no	—
33	C₂×C₄×D₁₃	—	4	1	no	—
34	C₂×Dic₂₆	—	2	1	no	—
35	C₂×#1057	—	4	4	no	—
36	C₁₃×M₁₆	—	4	2	no	—
37	C₂×C₈×C₁₃	—	4	1	no	—
38	C₄²×C₁₃	—	2	1	no	—
39	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
41	#6103	—	4	2	no	—
42	C₄⋊Dic₁₃	—	4	1	no	—
43	C₄×Dic₁₃	—	2	2	no	—
44	C₄×Dic₁₃	—	4	4	no	—
45	C₄×Dic₁₃	D₈	1	2	yes	—
46	C₄×Dic₁₃	SD₁₆	1	2	yes	—
47	#1962	—	13	1	no	—
48	#1276	—	13	2	no	—
49	C₂³×C₁₃	#99	1	1	yes	characteristic
50	C₂²×D₁₃	—	4	1	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	4	2	no	—
54	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#102	1	2	yes	—
57	C₂×C₄×C₁₃	#104	1	2	yes	—
58	C₂²⋊D₁₃	—	8	2	no	—
59	C₂×Dic₁₃	—	2	2	no	—
60	C₂×Dic₁₃	—	2	2	no	—
61	C₂×Dic₁₃	—	4	4	no	—
62	C₂×Dic₁₃	—	4	1	no	—
63	D₅₂	—	4	2	no	—
64	D₅₂	—	8	2	no	—
65	C₄×D₁₃	—	8	1	no	—
66	C₄×D₁₃	—	8	1	no	—
67	Dic₂₆	—	4	2	no	—
68	#1057	—	8	4	no	—
69	C₈×C₁₃	—	8	1	no	—
70	#532	—	13	1	no	—
71	#431	—	26	2	no	—
72	#423	#261	1	1	yes	characteristic
73	C₂×#104	—	26	2	no	—
74	C₄³	—	13	1	no	—
75	C₂²×C₁₃	—	4	1	no	—
76	C₂²×C₁₃	#323	1	1	yes	characteristic
77	C₂²×C₁₃	#327	1	2	yes	—
78	C₂×D₁₃	—	8	2	no	—
79	C₂×D₁₃	—	8	1	no	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	—	4	1	no	—
82	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
83	C₄×C₁₃	#329	1	1	yes	characteristic
84	C₄×C₁₃	#331	1	1	yes	characteristic
85	Dic₁₃	—	2	4	no	—
86	Dic₁₃	—	4	2	no	—
87	Dic₁₃	—	8	1	no	—
88	C₂×#41	—	26	1	no	—
89	C₂×M₁₆	—	2	1	no	—
90	C₄⋊Q₈	—	13	1	no	—
91	#127	—	13	1	no	—
92	#124	—	26	1	no	—
93	#121	—	52	1	no	—
94	C₄×D₄	—	52	1	no	—
95	C₂×C₄²	—	13	2	no	—
96	C₂×C₄²	—	52	1	no	—
97	C₂×C₄²	C₂×#261	1	1	yes	characteristic
98	C₄⋊C₈	—	4	1	no	—
99	C₄⋊C₈	—	26	4	no	—
100	#104	—	52	2	no	—
101	#104	—	52	2	no	—
102	C₂²⋊C₈	—	52	2	no	—
103	C₄×C₈	—	52	2	no	—
104	C₂×C₁₃	—	2	2	no	—
105	C₂×C₁₃	—	2	1	no	—
106	C₂×C₁₃	#1270	1	1	yes	characteristic
107	D₁₃	—	16	1	no	—
108	#41	—	52	2	no	—
109	#41	—	52	2	no	—
110	C₂×Q₈	—	26	1	no	—
111	C₂×D₄	—	26	1	no	—
112	C₂×D₄	—	52	1	no	—
113	C₂×D₄	—	52	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	26	2	no	—
116	C₂²×C₄	—	52	1	no	—
117	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
118	M₁₆	—	4	2	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	52	4	no	—
121	C₄⋊C₄	—	52	1	no	—
122	C₂²⋊C₄	—	52	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	13	2	no	—
125	C₄²	—	13	2	no	—
126	C₄²	—	26	2	no	—
127	C₄²	—	52	4	no	—
128	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
129	C₄²	#6118	1	1	yes	characteristic
130	C₁₃	#7106	1	1	yes	characteristic, complemented, Sylow
131	C₂³	—	52	1	no	—
132	C₂³	#88107	1	1	yes	characteristic
133	Q₈	—	52	2	no	—
134	D₄	—	52	2	no	—
135	D₄	—	104	2	no	—
136	D₄	—	104	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	26	2	no	—
141	C₂×C₄	—	26	2	no	—
142	C₂×C₄	—	52	4	no	—
143	C₂×C₄	—	52	1	no	—
144	C₂×C₄	—	104	1	no	—
145	C₂×C₄	—	104	1	no	—
146	C₂×C₄	#88111	1	1	yes	characteristic
147	C₂×C₄	#88115	1	2	yes	—
148	C₂×C₄	#88118	1	2	yes	—
149	C₂×C₄	#88121	1	1	yes	characteristic
150	C₈	—	8	1	no	—
151	C₈	—	104	4	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	104	2	no	—
154	C₂²	—	104	1	no	—
155	C₂²	#11718240	1	2	yes	—
156	C₂²	#11718287	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	1	no	—
159	C₄	—	26	4	no	—
160	C₄	—	52	2	no	—
161	C₄	—	104	1	no	—
162	C₄	#11718243	1	2	yes	—
163	C₄	#11718297	1	1	yes	characteristic
164	C₄	#11718299	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	2	no	—
167	C₂	—	208	1	no	—
168	C₂	#13129653	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129664

C₂

yes

maximal, complemented

#13115663

C₂

yes

maximal, characteristic

#11719182

C₄

yes

characteristic

#11719119

—

C₁₃×#423

C₄

yes

characteristic

C₂×#88065

—

C₄²×Dic₁₃

C₂²

yes

characteristic

C₂×#6127

—

C₂×C₁₃×M₁₆

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

#88184

—

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

D₄

yes

—

#88129

—

#88128

C₈

yes

characteristic, complemented

C₄×D₅₂

—

#88123

C₈

yes

characteristic, complemented

C₂²⋊#1057

—

C₄⋊#1057

—

C₄×#1057

—

C₁₃×C₄⋊C₈

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

—

#6127

—

#6127

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×Dic₂₆

—

C₂×#1057

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

—

C₄×Dic₁₃

SD₁₆

yes

—

#1962

—

#1276

—

C₂³×C₁₃

#99

yes

characteristic

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#104

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

#1057

—

C₈×C₁₃

—

—

—

yes

—

C₄³

—

C₂²×C₁₃

—

C₂²×C₁₃

#323

yes

characteristic

C₂²×C₁₃

#327

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

D₄⋊C₈

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#261

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

#104

—

101

#104

—

102

C₂²⋊C₈

—

103

C₄×C₈

—

104

C₂×C₁₃

—

105

C₂×C₁₃

—

106

C₂×C₁₃

#1270

yes

characteristic

107

D₁₃

—

108

#41

—

109

#41

—

110

C₂×Q₈

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊#261

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

D₁₃⋊C₈

yes

characteristic

129

C₄²

#6118

yes

characteristic

130

C₁₃

#7106

yes

characteristic, complemented, Sylow

131

C₂³

—

132

C₂³

#88107

yes

characteristic

133

Q₈

—

134

D₄

—

135

D₄

—

104

—

136

D₄

—

104

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

104

—

145

C₂×C₄

—

104

—

146

C₂×C₄

#88111

yes

characteristic

147

C₂×C₄

#88115

yes

—

148

C₂×C₄

#88118

yes

—

149

C₂×C₄

#88121

yes

characteristic

150

C₈

—

151

C₈

—

104

—

152

C₂²

—

153

C₂²

—

104

—

154

C₂²

—

104

—

155

C₂²

#11718240

yes

—

156

C₂²

#11718287

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

104

—

162

C₄

#11718243

yes

—

163

C₄

#11718297

yes

characteristic

164

C₄

#11718299

yes

characteristic

165

C₂

—

166

C₂

—

167

C₂

—

208

—

168

C₂

#13129653

yes

characteristic

	subgroup	G/H
center	C₄	#11718297
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₁₃×#423	C₄
socle	C₂×C₁₃	#1270
prime radical	#11719182	C₄
cosocle	C₄²×Dic₁₃	C₂²
hypercenter	#423	#261
hypocenter	C₁₃	#7106
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₄

#11718297

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₁₃×#423

C₄

C₂×C₁₃

C₄

C₄²×Dic₁₃

C₂²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #13129664	C₂	semidirect
15. #88128	C₈	semidirect
17. #88123	C₈	semidirect
130. C₁₃	#7106	semidirect

H ≅ G/N

type

1. #13129664

C₂

semidirect

15. #88128

C₈

semidirect

17. #88123

C₈

semidirect

130. C₁₃

#7106

semidirect

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

subgroup	index
#13129664	2
#13115663	2
#7106	13

subgroup

index

#13129664

#13115663

#7106

Subgroups — #16777021

Conjugacy classes of subgroups

Complements

Maximal subgroups