Group #16777406 — subgroups

Subgroups — #16777406

Order 3328.

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

normal subgroups only

This group has 264 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	#13130715	C₂	1	2	yes	maximal
2	#13116446	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
4	#11719127	—	2	2	no	—
5	C₂×#88103	—	2	2	no	—
6	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂×#88052	C₄	1	1	yes	characteristic
8	#11718268	C₄	1	4	yes	—
9	#11718099	—	2	2	no	—
10	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
11	C₂×#6118	—	2	4	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×#1055	Q₈	1	1	yes	characteristic
14	C₂×C₈×D₁₃	—	4	1	no	—
15	C₂×C₈×D₁₃	—	4	1	no	—
16	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₁₃	D₄	1	2	yes	—
18	C₂²⋊#1057	—	4	2	no	—
19	#88111	—	4	2	no	—
20	#88103	—	4	2	no	—
21	#88103	—	4	2	no	—
22	C₁₃×M₃₂	—	4	1	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	#88052	—	4	1	no	—
25	C₂×#6090	—	2	1	no	—
26	#7234	—	13	1	no	maximal, Sylow
27	C₂³×D₁₃	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
30	C₂×C₄×D₁₃	—	4	1	no	—
31	C₂×C₄×D₁₃	—	4	1	no	—
32	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×D₁₃	SD₁₆	1	2	yes	—
34	C₂×#1057	—	4	4	no	—
35	#6118	—	4	4	no	—
36	#6118	—	4	4	no	—
37	C₂×C₈×C₁₃	—	2	1	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#1055	—	4	1	no	—
42	C₂×#1055	C₄⋊C₄	1	2	yes	—
43	C₈×D₁₃	—	2	2	no	—
44	C₈×D₁₃	—	2	2	no	—
45	C₈×D₁₃	—	4	2	no	—
46	C₁₃×C₁₆	—	4	1	no	—
47	#6090	—	4	1	no	—
48	#2112	—	13	1	no	—
49	#1810	—	13	2	no	—
50	C₂³×C₁₃	#107	1	1	yes	characteristic
51	C₂²×D₁₃	—	4	1	no	—
52	C₂²×D₁₃	—	4	2	no	—
53	C₂²×D₁₃	#103	1	2	yes	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	—	2	1	no	—
56	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
58	C₂×Dic₁₃	—	4	1	no	—
59	C₂×Dic₁₃	#102	1	2	yes	—
60	C₄×D₁₃	—	2	2	no	—
61	C₄×D₁₃	—	2	2	no	—
62	C₄×D₁₃	—	4	2	no	—
63	#1057	—	8	4	no	—
64	C₈×C₁₃	—	2	1	no	—
65	C₈×C₁₃	#99	1	1	yes	characteristic
66	C₈×C₁₃	#99	1	1	yes	characteristic
67	#1055	—	2	2	no	—
68	C₂³×C₈	—	13	1	no	—
69	C₂×M₃₂	—	13	1	no	—
70	C₂×M₃₂	#261	1	1	yes	characteristic
71	C₂×#108	—	26	2	no	—
72	#407	—	26	2	no	—
73	C₂²⋊C₁₆	—	26	2	no	—
74	#341	—	13	4	no	—
75	C₂²×C₁₃	—	2	1	no	—
76	C₂²×C₁₃	—	2	2	no	—
77	C₂²×C₁₃	#340	1	1	yes	characteristic
78	C₂×D₁₃	—	2	2	no	—
79	C₂×D₁₃	—	2	4	no	—
80	C₂×D₁₃	—	4	4	no	—
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
83	C₄×C₁₃	#342	1	1	yes	characteristic
84	Dic₁₃	—	2	2	no	—
85	C₂³×C₄	—	13	1	no	—
86	C₂×M₁₆	—	26	4	no	—
87	C₂²×C₈	—	13	1	no	—
88	C₂²×C₈	—	13	2	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	—	52	1	no	—
91	C₂²×C₈	—	52	1	no	—
92	C₂²×C₈	C₂×#261	1	1	yes	characteristic
93	M₃₂	—	4	1	no	—
94	M₃₂	—	52	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	26	1	no	—
97	#108	—	52	2	no	—
98	#108	—	52	2	no	—
99	C₂²⋊C₈	—	52	2	no	—
100	C₂²⋊C₈	—	52	2	no	—
101	C₂×C₁₃	—	4	1	no	—
102	C₂×C₁₃	#1289	1	1	yes	characteristic
103	C₂×C₁₃	#1388	1	2	yes	—
104	D₁₃	—	4	2	no	—
105	C₂⁴	—	13	1	no	—
106	C₂²×C₄	—	13	1	no	—
107	C₂²×C₄	—	13	2	no	—
108	C₂²×C₄	—	13	2	no	—
109	C₂²×C₄	—	52	1	no	—
110	C₂²×C₄	—	52	1	no	—
111	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
112	M₁₆	—	52	4	no	—
113	M₁₆	—	52	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	13	2	no	—
117	C₂×C₈	—	26	2	no	—
118	C₂×C₈	—	26	2	no	—
119	C₂×C₈	—	52	2	no	—
120	C₂×C₈	—	52	1	no	—
121	C₂×C₈	—	52	4	no	—
122	C₂×C₈	C₄×#261	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
124	C₁₆	—	4	1	no	—
125	C₁₆	—	52	1	no	—
126	C₁₃	#7234	1	1	yes	characteristic, complemented, Sylow
127	C₂³	—	13	2	no	—
128	C₂³	—	52	1	no	—
129	C₂³	—	52	2	no	—
130	C₂³	#88102	1	1	yes	characteristic
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	13	2	no	—
134	C₂×C₄	—	26	2	no	—
135	C₂×C₄	—	26	2	no	—
136	C₂×C₄	—	52	2	no	—
137	C₂×C₄	—	52	1	no	—
138	C₂×C₄	#88101	1	1	yes	characteristic
139	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	26	2	no	—
142	C₈	—	104	4	no	—
143	C₈	#88107	1	1	yes	characteristic
144	C₈	#88119	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	26	4	no	—
148	C₂²	—	26	2	no	—
149	C₂²	—	52	4	no	—
150	C₂²	#11718265	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	26	2	no	—
153	C₄	#11718267	1	1	yes	characteristic
154	C₄	#11718289	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	—	52	2	no	—
157	C₂	#13129699	1	1	yes	characteristic
158	C₂	#13129914	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13130715

C₂

yes

maximal

#13116446

C₂

yes

maximal, characteristic

C₂²×C₈×D₁₃

C₂²

yes

characteristic

#11719127

—

C₂×#88103

—

C₂×C₁₃×M₃₂

C₄

yes

characteristic

C₂×#88052

C₄

yes

characteristic

#11718268

C₄

yes

—

#11718099

—

C₂²×C₄×D₁₃

D₄

yes

characteristic

C₂×#6118

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

Q₈

yes

characteristic

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

—

C₂×C₈×D₁₃

C₂×C₄

yes

—

C₂×C₈×D₁₃

D₄

yes

—

C₂²⋊#1057

—

#88111

—

#88103

—

#88103

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

—

—

—

C₂³×D₁₃

Q₁₆

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₃

D₈

yes

characteristic

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

C₂²⋊C₄

yes

—

C₂×C₄×D₁₃

SD₁₆

yes

—

C₂×#1057

—

#6118

—

#6118

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

—

C₈×D₁₃

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

#6090

—

#2112

—

#1810

—

C₂³×C₁₃

#107

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

#103

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

#102

yes

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

yes

—

—

C₂²⋊C₁₆

—

#341

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#340

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#261

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

101

C₂×C₁₃

—

102

C₂×C₁₃

#1289

yes

characteristic

103

C₂×C₁₃

#1388

yes

—

104

D₁₃

—

105

C₂⁴

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₄⋊#261

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄×#261

yes

characteristic

123

C₂×C₈

C₂²⋊#261

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₁₃

#7234

yes

characteristic, complemented, Sylow

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

#88102

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#88101

yes

characteristic

139

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

104

—

143

C₈

#88107

yes

characteristic

144

C₈

#88119

yes

characteristic

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

#11718265

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

#11718267

yes

characteristic

154

C₄

#11718289

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#13129699

yes

characteristic

158

C₂

#13129914

yes

—

	subgroup	G/H
center	C₂²	#11718265
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂²×C₈	C₂×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂²×C₁₃	#340
prime radical	C₂³×D₁₃	Q₁₆
cosocle	C₂²×C₈×D₁₃	C₂²
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	#7234
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#11718265

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂²×C₈

C₂×#261

Fitting

C₂×C₁₃×M₃₂

C₄

socle

C₂²×C₁₃

#340

prime radical

C₂³×D₁₃

Q₁₆

cosocle

C₂²×C₈×D₁₃

C₂²

C₂×M₃₂

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
126. C₁₃	#7234	semidirect

H ≅ G/N

type

126. C₁₃

#7234

semidirect

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

subgroup	index
#13130715	2
#13116446	2
#7234	13

subgroup

index

#13130715

#13116446

#7234

Subgroups — #16777406

Conjugacy classes of subgroups

Complements

Maximal subgroups