Group #16777608 — subgroups

Subgroups — #16777608

Order 3328.

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

normal subgroups only

This group has 212 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×#88104	C₂	1	1	yes	maximal, characteristic
3	#13113590	C₂	1	1	yes	maximal, characteristic
4	C₂×#88104	—	2	1	no	—
5	C₄×#6118	—	2	1	no	—
6	#11718498	—	2	1	no	—
7	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	#11718268	C₄	1	2	yes	—
9	#11718260	C₄	1	2	yes	—
10	#11718259	—	2	1	no	—
11	#11718231	—	2	1	no	—
12	C₁₃×#334	C₄	1	1	yes	characteristic
13	#11718072	—	2	1	no	—
14	#11718049	C₄	1	1	yes	characteristic
15	C₂×#6118	—	2	2	no	—
16	C₂×#6094	—	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	#88110	—	4	1	no	—
21	C₄×#1057	—	4	1	no	—
22	C₂×#6092	—	2	2	no	—
23	#88104	—	2	4	no	—
24	C₁₃×C₄⋊C₈	—	2	1	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	#88054	—	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	#7300	—	13	1	no	maximal, Sylow
32	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₃	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
35	C₂×#1057	—	4	2	no	—
36	#6118	—	2	4	no	—
37	#6118	—	4	2	no	—
38	C₂×C₈×C₁₃	—	4	1	no	—
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₃	D₈	1	1	yes	characteristic
44	C₄×Dic₁₃	SD₁₆	1	1	yes	characteristic
45	C₂×#1055	—	4	1	no	—
46	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
48	#6094	—	8	2	no	—
49	C₈×D₁₃	—	2	2	no	—
50	C₈×D₁₃	—	4	1	no	—
51	#6092	—	4	2	no	—
52	#2163	—	13	1	no	—
53	C₄×#108	—	13	1	no	—
54	#1566	—	13	1	no	—
55	C₂²×D₁₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
59	C₂×Dic₁₃	#102	1	1	yes	characteristic
60	C₂×Dic₁₃	#106	1	1	yes	characteristic
61	C₂×Dic₁₃	#107	1	1	yes	characteristic
62	C₄×D₁₃	—	2	1	no	—
63	C₄×D₁₃	—	2	1	no	—
64	C₄×D₁₃	—	2	2	no	—
65	C₄×D₁₃	—	4	1	no	—
66	#1057	—	4	4	no	—
67	C₈×C₁₃	—	2	1	no	—
68	C₈×C₁₃	—	2	1	no	—
69	C₈×C₁₃	—	8	1	no	—
70	#1055	—	2	1	no	—
71	#1055	—	8	1	no	—
72	#1055	#104	1	2	yes	—
73	C₂×#108	—	26	1	no	—
74	#423	—	26	1	no	—
75	C₄×M₁₆	—	26	1	no	—
76	C₂×C₄×C₈	—	13	1	no	—
77	C₄⋊C₁₆	—	13	2	no	—
78	C₂²⋊C₁₆	—	26	1	no	—
79	#346	—	26	1	no	—
80	#341	—	13	2	no	—
81	#334	—	13	1	no	—
82	#334	—	26	1	no	—
83	#334	#261	1	1	yes	characteristic
84	C₂²×C₁₃	#340	1	1	yes	characteristic
85	C₂×D₁₃	—	2	1	no	—
86	C₂×D₁₃	—	2	1	no	—
87	C₂×D₁₃	—	2	1	no	—
88	C₄×C₁₃	—	2	1	no	—
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	#339	1	1	yes	characteristic
91	C₄×C₁₃	#344	1	1	yes	characteristic
92	Dic₁₃	—	2	1	no	—
93	Dic₁₃	—	2	1	no	—
94	Dic₁₃	#365	1	2	yes	—
95	C₂×M₁₆	—	26	2	no	—
96	C₂×M₁₆	—	52	1	no	—
97	C₂²×C₈	—	13	1	no	—
98	C₂²×C₈	—	13	1	no	—
99	C₂×C₄²	—	13	1	no	—
100	C₂×C₁₆	—	26	2	no	—
101	#108	—	26	4	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	26	1	no	—
104	C₄⋊C₈	—	52	1	no	—
105	#97	—	52	1	no	—
106	C₄×C₈	—	13	1	no	—
107	C₄×C₈	—	13	1	no	—
108	C₄×C₈	—	13	1	no	—
109	C₄×C₈	—	52	1	no	—
110	C₄×C₈	C₂×#261	1	1	yes	characteristic
111	C₂×C₁₃	#1303	1	1	yes	characteristic
112	C₂×C₁₃	#1383	1	1	yes	characteristic
113	C₂×C₁₃	#1388	1	1	yes	characteristic
114	D₁₃	—	4	1	no	—
115	C₂²×C₄	—	13	1	no	—
116	C₂²×C₄	—	13	1	no	—
117	C₂²×C₄	—	13	1	no	—
118	M₁₆	—	26	4	no	—
119	M₁₆	—	52	2	no	—
120	M₁₆	—	104	2	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	13	1	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	26	2	no	—
125	C₂×C₈	—	52	1	no	—
126	C₂×C₈	—	52	2	no	—
127	C₂×C₈	—	52	1	no	—
128	C₂×C₈	C₄×#261	1	1	yes	characteristic
129	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
130	C₄²	—	13	1	no	—
131	C₄²	—	13	1	no	—
132	C₄²	—	13	1	no	—
133	C₄²	C₂²⋊#261	1	1	yes	characteristic
134	C₁₆	—	52	2	no	—
135	C₁₃	#7300	1	1	yes	characteristic, complemented, Sylow
136	C₂³	—	13	1	no	—
137	C₂×C₄	—	13	1	no	—
138	C₂×C₄	—	13	1	no	—
139	C₂×C₄	—	13	1	no	—
140	C₂×C₄	—	26	2	no	—
141	C₂×C₄	—	26	1	no	—
142	C₂×C₄	—	26	1	no	—
143	C₂×C₄	—	52	1	no	—
144	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
145	C₂×C₄	#88115	1	1	yes	characteristic
146	C₂×C₄	#88117	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	8	1	no	—
150	C₈	—	13	2	no	—
151	C₈	—	26	1	no	—
152	C₈	—	52	4	no	—
153	C₈	—	104	1	no	—
154	C₂²	—	26	1	no	—
155	C₂²	—	26	1	no	—
156	C₂²	—	26	1	no	—
157	C₂²	#11718246	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	13	2	no	—
161	C₄	—	26	1	no	—
162	C₄	—	26	1	no	—
163	C₄	#11718266	1	1	yes	characteristic
164	C₄	#11718272	1	1	yes	characteristic
165	C₂	—	52	1	no	—
166	C₂	#13129732	1	1	yes	characteristic
167	C₂	#13129900	1	1	yes	characteristic
168	C₂	#13129915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131199

C₂

yes

maximal, characteristic

C₄×#88104

C₂

yes

maximal, characteristic

#13113590

C₂

yes

maximal, characteristic

C₂×#88104

—

C₄×#6118

—

#11718498

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718268

C₄

yes

—

#11718260

C₄

yes

—

#11718259

—

#11718231

—

C₁₃×#334

C₄

yes

characteristic

#11718072

—

#11718049

C₄

yes

characteristic

C₂×#6118

—

C₂×#6094

—

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

C₂×C₈×D₁₃

D₄

yes

characteristic

C₄²×D₁₃

D₄

yes

—

—

—

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88054

—

C₈×Dic₁₃

C₂×C₄

yes

characteristic

C₈×Dic₁₃

Q₈

yes

characteristic

C₄⋊#1055

—

C₄×#1055

D₄

yes

characteristic

#7300

—

maximal, Sylow

C₂×C₄×D₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₃

SD₁₆

yes

characteristic

C₂×C₄×D₁₃

Q₁₆

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

D₈

yes

characteristic

C₄×Dic₁₃

SD₁₆

yes

characteristic

C₂×#1055

—

C₂×#1055

C₂²⋊C₄

yes

characteristic

C₂×#1055

C₄⋊C₄

yes

characteristic

#6094

—

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

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

yes

—

C₂×#108

—

#423

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

yes

C₂²×C₁₃

yes

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

#365

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

100

C₂×C₁₆

—

101

#108

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₂×#261

yes

characteristic

111

C₂×C₁₃

#1303

yes

characteristic

112

C₂×C₁₃

#1383

yes

characteristic

113

C₂×C₁₃

#1388

yes

characteristic

114

D₁₃

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

104

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₄×#261

yes

characteristic

129

C₂×C₈

C₄⋊#261

yes

characteristic

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²⋊#261

yes

characteristic

134

C₁₆

—

135

C₁₃

#7300

yes

characteristic, complemented, Sylow

136

C₂³

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

145

C₂×C₄

#88115

yes

characteristic

146

C₂×C₄

#88117

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

104

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#11718246

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11718266

yes

characteristic

164

C₄

#11718272

yes

characteristic

165

C₂

—

166

C₂

#13129732

yes

characteristic

167

C₂

#13129900

yes

characteristic

168

C₂

#13129915

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	#103
cosocle	C₄×C₈×D₁₃	C₂²
hypercenter	#334	#261
hypocenter	C₁₃	#7300
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
135. C₁₃	#7300	semidirect

H ≅ G/N

type

135. C₁₃

#7300

semidirect

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

subgroup	index
#13131199	2
C₄×#88104	2
#13113590	2
#7300	13

subgroup

index

Subgroups — #16777608

Conjugacy classes of subgroups

Complements

Maximal subgroups