Group #16777745 — subgroups

Subgroups — #16777745

Order 3328.

268 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 40 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	#13129778	C₂	1	2	yes	maximal, complemented
2	#13115642	C₂	1	1	yes	maximal, characteristic
3	C₂×#88128	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×#88061	—	2	2	no	—
6	#11718596	C₂²	1	1	yes	characteristic
7	#11718285	—	2	2	no	—
8	C₂²×D₅₂	—	2	1	no	—
9	C₂²×D₅₂	—	4	1	no	—
10	C₂²×C₈×C₁₃	—	2	1	no	—
11	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	—	4	1	no	—
13	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
14	#88128	—	2	1	no	—
15	#88128	—	4	1	no	—
16	#88128	C₈	1	2	yes	complemented
17	C₂²⋊#1057	—	4	2	no	—
18	C₄×C₈×C₁₃	—	4	1	no	—
19	#88061	—	4	4	no	—
20	#7354	—	13	1	no	maximal, Sylow
21	C₂³×D₁₃	—	4	1	no	—
22	C₂²×C₄×C₁₃	—	2	1	no	—
23	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×Dic₁₃	—	2	2	no	—
25	C₂×D₅₂	—	2	2	no	—
26	C₂×D₅₂	—	4	2	no	—
27	C₂×D₅₂	—	4	2	no	—
28	C₂×D₅₂	—	4	2	no	—
29	C₂×D₅₂	—	4	2	no	—
30	C₂×D₅₂	—	4	1	no	—
31	C₂×D₅₂	—	8	2	no	—
32	C₂×D₅₂	—	8	2	no	—
33	C₂×#1057	—	8	2	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	4	1	no	—
36	C₂×C₈×C₁₃	—	4	1	no	—
37	C₄²×C₁₃	—	2	1	no	—
38	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
40	C₄⋊Dic₁₃	—	4	2	no	—
41	C₄⋊Dic₁₃	—	8	2	no	—
42	#1957	—	13	1	no	—
43	#1325	—	13	2	no	—
44	C₂³×C₁₃	#99	1	1	yes	characteristic
45	C₂²×D₁₃	—	4	2	no	—
46	C₂²×D₁₃	—	8	2	no	—
47	C₂²×D₁₃	—	8	2	no	—
48	C₂²×D₁₃	—	8	1	no	—
49	C₂²×D₁₃	—	8	1	no	—
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
58	C₂×Dic₁₃	—	4	2	no	—
59	C₂×Dic₁₃	—	8	2	no	—
60	D₅₂	—	4	4	no	—
61	D₅₂	—	8	2	no	—
62	D₅₂	—	8	4	no	—
63	D₅₂	—	8	4	no	—
64	#1057	—	16	2	no	—
65	C₈×C₁₃	—	4	2	no	—
66	C₂×#127	—	13	1	no	—
67	C₂×#102	—	26	2	no	—
68	C₂×C₄×C₈	#261	1	1	yes	characteristic
69	#384	—	13	1	no	—
70	#324	—	26	2	no	—
71	C₂²×C₁₃	—	2	1	no	—
72	C₂²×C₁₃	—	2	1	no	—
73	C₂²×C₁₃	#323	1	1	yes	characteristic
74	C₂²×C₁₃	#353	1	1	yes	characteristic
75	C₂²×C₁₃	#355	1	1	yes	characteristic
76	C₂×D₁₃	—	8	4	no	—
77	C₂×D₁₃	—	8	2	no	—
78	C₂×D₁₃	—	16	2	no	—
79	C₂×D₁₃	—	16	1	no	—
80	C₂×D₁₃	—	16	1	no	—
81	C₄×C₁₃	—	2	2	no	—
82	C₄×C₁₃	—	2	2	no	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	2	1	no	—
85	Dic₁₃	—	8	2	no	—
86	C₂²×D₄	—	26	1	no	—
87	C₂²×D₄	—	52	1	no	—
88	C₂²×C₈	—	2	1	no	—
89	#127	—	13	2	no	—
90	#127	—	26	1	no	—
91	#127	—	52	1	no	—
92	C₂×C₄⋊C₄	—	13	2	no	—
93	C₂×C₄⋊C₄	—	52	1	no	—
94	C₂×C₄²	C₂×#261	1	1	yes	characteristic
95	#102	—	52	4	no	—
96	C₂²⋊C₈	—	52	2	no	—
97	C₄×C₈	—	4	1	no	—
98	C₂×C₁₃	—	2	1	no	—
99	C₂×C₁₃	—	2	1	no	—
100	C₂×C₁₃	#1318	1	1	yes	characteristic
101	C₂×C₁₃	#1408	1	1	yes	characteristic
102	C₂×C₁₃	#1410	1	1	yes	characteristic
103	D₁₃	—	16	2	no	—
104	C₂⁴	—	52	1	no	—
105	C₂×D₄	—	26	2	no	—
106	C₂×D₄	—	52	2	no	—
107	C₂×D₄	—	52	2	no	—
108	C₂×D₄	—	52	2	no	—
109	C₂×D₄	—	52	2	no	—
110	C₂×D₄	—	52	1	no	—
111	C₂×D₄	—	104	2	no	—
112	C₂×D₄	—	104	2	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	26	2	no	—
115	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	104	2	no	—
120	C₄⋊C₄	—	52	2	no	—
121	C₄⋊C₄	—	104	2	no	—
122	C₄²	—	2	1	no	—
123	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
124	C₄²	#6118	1	1	yes	characteristic
125	C₁₃	#7354	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	52	2	no	—
127	C₂³	—	104	2	no	—
128	C₂³	—	104	2	no	—
129	C₂³	—	104	1	no	—
130	C₂³	—	104	1	no	—
131	C₂³	#88107	1	1	yes	characteristic
132	D₄	—	52	4	no	—
133	D₄	—	104	2	no	—
134	D₄	—	104	4	no	—
135	D₄	—	104	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	52	2	no	—
143	C₂×C₄	—	104	2	no	—
144	C₂×C₄	#88111	1	1	yes	characteristic
145	C₂×C₄	#88121	1	1	yes	characteristic
146	C₈	—	4	2	no	—
147	C₈	—	208	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	104	4	no	—
151	C₂²	—	104	2	no	—
152	C₂²	—	208	2	no	—
153	C₂²	—	208	1	no	—
154	C₂²	—	208	1	no	—
155	C₂²	#11718225	1	1	yes	characteristic
156	C₂²	#11718228	1	1	yes	characteristic
157	C₂²	#11718287	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	104	2	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	208	2	no	—
166	C₂	#13129765	1	1	yes	characteristic
167	C₂	#13129968	1	1	yes	characteristic
168	C₂	#13129972	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129778

C₂

yes

maximal, complemented

#13115642

C₂

yes

maximal, characteristic

C₂×#88128

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃

C₄

yes

characteristic

C₂×#88061

—

#11718596

C₂²

yes

characteristic

#11718285

—

C₂²×D₅₂

—

C₂²×D₅₂

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

D₄

yes

—

#88128

—

#88128

—

#88128

C₈

yes

complemented

C₂²⋊#1057

—

C₄×C₈×C₁₃

—

#88061

—

#7354

—

maximal, Sylow

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×#1057

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#1957

—

#1325

—

C₂³×C₁₃

#99

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃

#100

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

#1057

—

C₈×C₁₃

—

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

yes

—

—

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

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#261

yes

characteristic

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

100

C₂×C₁₃

#1318

yes

characteristic

101

C₂×C₁₃

#1408

yes

characteristic

102

C₂×C₁₃

#1410

yes

characteristic

103

D₁₃

—

104

C₂⁴

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

104

—

112

C₂×D₄

—

104

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²⋊#261

yes

characteristic

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

104

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

104

—

122

C₄²

—

123

C₄²

D₁₃⋊C₈

yes

characteristic

124

C₄²

#6118

yes

characteristic

125

C₁₃

#7354

yes

characteristic, complemented, Sylow

126

C₂³

—

127

C₂³

—

104

—

128

C₂³

—

104

—

129

C₂³

—

104

—

130

C₂³

—

104

—

131

C₂³

#88107

yes

characteristic

132

D₄

—

133

D₄

—

104

—

134

D₄

—

104

—

135

D₄

—

104

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

104

—

144

C₂×C₄

#88111

yes

characteristic

145

C₂×C₄

#88121

yes

characteristic

146

C₈

—

147

C₈

—

208

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

104

—

151

C₂²

—

104

—

152

C₂²

—

208

—

153

C₂²

—

208

—

154

C₂²

—

208

—

155

C₂²

#11718225

yes

characteristic

156

C₂²

#11718228

yes

characteristic

157

C₂²

#11718287

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

104

—

163

C₂

—

164

C₂

—

165

C₂

—

208

—

166

C₂

#13129765

yes

characteristic

167

C₂

#13129968

yes

characteristic

168

C₂

#13129972

yes

characteristic

	subgroup	G/H
center	C₂²	#11718225
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#353
prime radical	C₂×#88128	C₄
cosocle	#11718596	C₂²
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#7354
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²

#11718225

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₂×C₄×C₈×C₁₃

C₄

C₂²×C₁₃

C₄

C₂²

C₂×C₄×C₈

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
1. #13129778	C₂	semidirect
16. #88128	C₈	semidirect
125. C₁₃	#7354	semidirect

H ≅ G/N

type

1. #13129778

C₂

semidirect

16. #88128

C₈

semidirect

125. C₁₃

#7354

semidirect

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

subgroup	index
#13129778	2
#13115642	2
#7354	13

subgroup

index

#13129778

#13115642

#7354

Subgroups — #16777745

Conjugacy classes of subgroups

Complements

Maximal subgroups