Group #16777611 — subgroups

Subgroups — #16777611

Order 3328.

230 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13131199	C₂	1	1	yes	maximal, characteristic
2	C₄×#88102	C₂	1	1	yes	maximal, characteristic
3	#13113596	C₂	1	1	yes	maximal, characteristic
4	C₂×#88102	—	2	1	no	—
5	C₄×C₄⋊#261	—	2	1	no	—
6	#11718496	—	2	1	no	—
7	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	#11718265	C₄	1	2	yes	—
9	#11718260	C₄	1	2	yes	complemented
10	#11718259	—	2	1	no	—
11	#11718231	—	2	1	no	—
12	C₁₃×#334	C₄	1	1	yes	characteristic, complemented
13	#11718071	—	2	1	no	—
14	#11718049	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#261	—	2	2	no	—
16	C₂×C₄×#261	—	4	1	no	—
17	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
19	C₄²×D₁₃	D₄	1	1	yes	characteristic
20	(C₂×#261)⋊C₄	—	4	1	no	—
21	C₂×#6092	—	2	2	no	—
22	#88102	—	2	4	no	—
23	C₁₃×C₄⋊C₈	—	2	1	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	#88060	—	4	1	no	—
26	#88055	—	4	1	no	—
27	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₁₃	Q₈	1	1	yes	characteristic
29	C₄⋊#1055	—	2	1	no	—
30	C₄×#1055	D₄	1	1	yes	characteristic
31	#7301	—	13	1	no	maximal, Sylow
32	C₂²×#261	—	4	2	no	—
33	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
35	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
36	C₄⋊#261	—	2	4	no	—
37	C₄⋊#261	—	4	2	no	—
38	C₄×#261	—	8	2	no	—
39	C₂×C₈×C₁₃	—	4	1	no	—
40	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₁₃	SD₁₆	1	1	yes	characteristic
45	C₄×Dic₁₃	Q₁₆	1	1	yes	characteristic
46	C₂×#1055	—	4	1	no	—
47	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₁₃	—	2	2	no	—
50	C₈×D₁₃	—	4	1	no	—
51	#6092	—	4	2	no	—
52	#2163	—	13	1	no	—
53	C₄×#107	—	13	1	no	—
54	#1568	—	13	1	no	—
55	C₂²×D₁₃	#102	1	1	yes	characteristic
56	C₂×#261	—	4	4	no	—
57	C₂×#261	—	8	2	no	—
58	C₂×#261	—	8	2	no	—
59	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
62	C₂×Dic₁₃	#103	1	1	yes	characteristic
63	C₂×Dic₁₃	#106	1	1	yes	characteristic
64	C₂×Dic₁₃	#107	1	1	yes	characteristic
65	C₄×D₁₃	—	2	1	no	—
66	C₄×D₁₃	—	2	1	no	—
67	C₄×D₁₃	—	2	1	no	—
68	C₄×D₁₃	—	2	1	no	—
69	C₄×D₁₃	—	4	1	no	—
70	C₈×C₁₃	—	2	1	no	—
71	C₈×C₁₃	—	2	1	no	—
72	C₈×C₁₃	—	8	1	no	—
73	#1055	—	2	1	no	—
74	#1055	—	8	1	no	—
75	#1055	#104	1	2	yes	—
76	#432	—	26	1	no	—
77	C₂×#107	—	26	1	no	—
78	C₂×C₄×C₈	—	13	1	no	—
79	C₄×C₄⋊C₄	—	26	1	no	—
80	C₄⋊C₁₆	—	13	2	no	—
81	C₂²⋊C₁₆	—	26	1	no	—
82	#346	—	26	1	no	—
83	#340	—	13	2	no	—
84	#335	—	26	1	no	—
85	#334	—	13	1	no	—
86	#334	#261	1	1	yes	characteristic, complemented
87	C₂²×C₁₃	#340	1	1	yes	characteristic
88	C₂×D₁₃	—	2	1	no	—
89	C₂×D₁₃	—	2	1	no	—
90	C₂×D₁₃	#361	1	1	yes	characteristic
91	C₂×D₁₃	#362	1	1	yes	characteristic
92	#261	—	8	4	no	—
93	C₄×C₁₃	—	2	1	no	—
94	C₄×C₁₃	—	2	1	no	—
95	C₄×C₁₃	#339	1	1	yes	characteristic
96	C₄×C₁₃	#344	1	1	yes	characteristic
97	Dic₁₃	—	2	1	no	—
98	Dic₁₃	—	2	1	no	—
99	Dic₁₃	—	2	1	no	—
100	C₂²×C₈	—	13	1	no	—
101	C₂²×C₈	—	13	1	no	—
102	C₂×C₄⋊C₄	—	26	2	no	—
103	C₂×C₄²	—	13	1	no	—
104	C₂×C₄²	—	52	1	no	—
105	C₂×C₁₆	—	26	2	no	—
106	#107	—	26	4	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	26	1	no	—
109	C₂²⋊C₈	—	52	1	no	—
110	#97	—	52	1	no	—
111	C₄×C₈	—	13	1	no	—
112	C₄×C₈	—	13	1	no	—
113	C₄×C₈	—	13	1	no	—
114	C₄×C₈	C₂×#261	1	1	yes	characteristic
115	#95	—	52	1	no	—
116	C₂×C₁₃	#1303	1	1	yes	characteristic
117	C₂×C₁₃	#1384	1	1	yes	characteristic
118	C₂×C₁₃	#1387	1	1	yes	characteristic
119	D₁₃	—	2	1	no	—
120	D₁₃	—	2	1	no	—
121	C₂²×C₄	—	13	1	no	—
122	C₂²×C₄	—	13	1	no	—
123	C₂²×C₄	—	13	1	no	—
124	C₂²×C₄	—	52	2	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	—	13	1	no	—
127	C₂×C₈	—	13	1	no	—
128	C₂×C₈	—	26	2	no	—
129	C₂×C₈	—	52	1	no	—
130	C₂×C₈	—	52	1	no	—
131	C₂×C₈	C₄×#261	1	1	yes	characteristic
132	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
133	C₄⋊C₄	—	26	4	no	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄²	—	13	1	no	—
136	C₄²	—	13	1	no	—
137	C₄²	—	13	1	no	—
138	C₄²	—	104	2	no	—
139	C₄²	C₂²⋊#261	1	1	yes	characteristic
140	C₁₆	—	52	2	no	—
141	C₁₃	#7301	1	1	yes	characteristic, complemented, Sylow
142	C₂³	—	13	1	no	—
143	C₂×C₄	—	13	1	no	—
144	C₂×C₄	—	13	1	no	—
145	C₂×C₄	—	13	1	no	—
146	C₂×C₄	—	26	1	no	—
147	C₂×C₄	—	26	1	no	—
148	C₂×C₄	—	26	1	no	—
149	C₂×C₄	—	26	1	no	—
150	C₂×C₄	—	52	1	no	—
151	C₂×C₄	—	52	4	no	—
152	C₂×C₄	—	104	2	no	—
153	C₂×C₄	—	104	2	no	—
154	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#88115	1	1	yes	characteristic
156	C₂×C₄	#88117	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	1	no	—
160	C₈	—	13	2	no	—
161	C₈	—	26	1	no	—
162	C₈	—	104	1	no	—
163	C₂²	—	13	1	no	—
164	C₂²	—	13	1	no	—
165	C₂²	—	26	1	no	—
166	C₂²	—	26	1	no	—
167	C₂²	#11718246	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	26	1	no	—
171	C₄	—	26	1	no	—
172	C₄	—	26	1	no	—
173	C₄	—	104	4	no	—
174	C₄	#11718266	1	1	yes	characteristic
175	C₄	#11718272	1	1	yes	characteristic
176	C₂	—	26	1	no	—
177	C₂	—	26	1	no	—
178	C₂	#13129732	1	1	yes	characteristic
179	C₂	#13129903	1	1	yes	characteristic
180	C₂	#13129912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131199

C₂

yes

maximal, characteristic

C₄×#88102

C₂

yes

maximal, characteristic

#13113596

C₂

yes

maximal, characteristic

C₂×#88102

—

C₄×C₄⋊#261

—

#11718496

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718265

C₄

yes

—

#11718260

C₄

yes

complemented

#11718259

—

#11718231

—

C₁₃×#334

C₄

yes

characteristic, complemented

#11718071

—

#11718049

C₄

yes

characteristic, complemented

C₂×C₄⋊#261

—

C₂×C₄×#261

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

D₄

yes

characteristic

(C₂×#261)⋊C₄

—

C₂×#6092

—

#88102

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88060

—

#88055

—

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₄⋊#1055

—

C₄×#1055

D₄

yes

characteristic

#7301

—

maximal, Sylow

C₂²×#261

—

C₂×C₄×D₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₃

D₈

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

—

—

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₃

C₄⋊C₄

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

characteristic

C₄×Dic₁₃

Q₁₆

yes

characteristic

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

C₈×D₁₃

—

C₈×D₁₃

—

—

—

—

—

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

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

yes

—

#432

—

C₂×#107

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

yes

characteristic, complemented

C₂²×C₁₃

#340

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

yes

C₂×D₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

#107

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#261

yes

characteristic

115

#95

—

116

C₂×C₁₃

#1303

yes

characteristic

117

C₂×C₁₃

#1384

yes

characteristic

118

C₂×C₁₃

#1387

yes

characteristic

119

D₁₃

—

120

D₁₃

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

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

C₄⋊#261

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

104

—

139

C₄²

C₂²⋊#261

yes

characteristic

140

C₁₆

—

141

C₁₃

#7301

yes

characteristic, complemented, Sylow

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

104

—

153

C₂×C₄

—

104

—

154

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

155

C₂×C₄

#88115

yes

characteristic

156

C₂×C₄

#88117

yes

characteristic

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

104

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

#11718246

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

104

—

174

C₄

#11718266

yes

characteristic

175

C₄

#11718272

yes

characteristic

176

C₂

—

177

C₂

—

178

C₂

#13129732

yes

characteristic

179

C₂

#13129903

yes

characteristic

180

C₂

#13129912

yes

characteristic

	subgroup	G/H
center	C₂²	#11718246
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#334	C₄
socle	C₂²×C₁₃	#340
prime radical	C₂²×D₁₃	#102
cosocle	C₄×C₈×D₁₃	C₂²
hypercenter	#334	#261
hypocenter	C₁₃	#7301
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#11718246

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#334

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄×C₈×D₁₃

C₂²

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
9. #11718260	C₄	semidirect
12. C₁₃×#334	C₄	semidirect
14. #11718049	C₄	semidirect
86. #334	#261	semidirect
141. C₁₃	#7301	semidirect

H ≅ G/N

type

9. #11718260

C₄

semidirect

12. C₁₃×#334

C₄

semidirect

14. #11718049

C₄

semidirect

86. #334

#261

semidirect

141. C₁₃

#7301

semidirect

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

subgroup	index
#13131199	2
C₄×#88102	2
#13113596	2
#7301	13

subgroup

index

Subgroups — #16777611

Conjugacy classes of subgroups

Complements

Maximal subgroups