Group #16791149 — subgroups

Subgroups — #16791149

Order 3328.

238 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₂×D₁₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#13131044	C₂	1	1	yes	maximal, characteristic
3	#13129942	C₂	1	2	yes	maximal, complemented
4	C₃₂×#261	C₂	1	2	yes	maximal, complemented
5	D₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic
6	#11719089	C₄	1	1	yes	characteristic
7	D₁₃⋊M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88036	C₄	1	2	yes	complemented
10	#11718216	C₄	1	2	yes	—
11	C₁₆×#261	C₂²	1	2	yes	—
12	D₁₃⋊C₃₂	C₄	1	2	yes	complemented
13	D₁₃⋊C₃₂	C₂²	1	1	yes	characteristic
14	D₁₃⋊C₃₂	C₂²	1	1	yes	characteristic
15	C₁₃×M₆₄	C₄	1	1	yes	characteristic, complemented
16	#11718091	C₄	1	1	yes	characteristic, complemented
17	#11718039	C₄	1	2	yes	complemented
18	D₁₃×C₃₂	C₂²	1	2	yes	—
19	#88237	C₈	1	1	yes	characteristic
20	D₁₃⋊M₁₆	C₈	1	1	yes	characteristic
21	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
22	#88106	C₈	1	2	yes	—
23	#88100	C₈	1	2	yes	—
24	C₈×#261	C₂×C₄	1	2	yes	—
25	#88098	C₈	1	2	yes	—
26	D₁₃⋊C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×#6090	C₂×C₄	1	1	yes	characteristic
29	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
33	#88036	C₂×C₄	1	2	yes	—
34	#88036	C₂×C₄	1	2	yes	—
35	C₁₃×C₃₂	C₂×C₄	1	2	yes	—
36	#88034	C₂×C₄	1	2	yes	—
37	#13549	—	13	1	no	maximal, Sylow
38	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²⋊#261	C₁₆	1	2	yes	—
40	#6122	C₁₆	1	2	yes	—
41	C₄⋊#261	C₁₆	1	2	yes	—
42	C₄×#261	C₂×C₈	1	2	yes	—
43	#6118	C₁₆	1	2	yes	—
44	D₁₃⋊C₈	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
46	C₂×#1055	C₂×C₈	1	1	yes	characteristic
47	C₈×D₁₃	C₄²	1	1	yes	characteristic
48	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
49	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
50	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
51	#6092	C₂×C₈	1	4	yes	—
52	C₁₃×C₁₆	C₄²	1	1	yes	characteristic
53	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
54	#6090	C₄²	1	1	yes	characteristic
55	#6090	C₂²×C₄	1	1	yes	characteristic
56	C₂×M₆₄	—	13	1	no	—
57	C₂²×C₃₂	—	13	1	no	—
58	#2108	—	13	1	no	—
59	#1397	—	13	2	no	—
60	C₄×C₃₂	—	13	2	no	—
61	C₂²×D₁₃	C₂×C₁₆	1	1	yes	characteristic
62	C₂×#261	C₂×C₁₆	1	4	yes	—
63	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
64	C₂×Dic₁₃	C₂×C₁₆	1	1	yes	characteristic
65	C₄×D₁₃	C₄×C₈	1	1	yes	characteristic
66	C₄×D₁₃	C₂×C₁₆	1	1	yes	characteristic
67	C₄×D₁₃	C₂×C₁₆	1	1	yes	characteristic
68	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
69	#1057	C₂×C₁₆	1	4	yes	—
70	C₈×C₁₃	C₄×C₈	1	1	yes	characteristic
71	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	#1055	C₄×C₈	1	1	yes	characteristic
73	#1055	C₂²×C₈	1	1	yes	characteristic
74	C₂×M₃₂	—	13	1	no	—
75	C₂²×C₁₆	—	13	1	no	—
76	#405	—	13	1	no	—
77	M₆₄	—	13	1	no	—
78	M₆₄	—	13	2	no	—
79	M₆₄	#261	1	1	yes	characteristic, complemented
80	C₂×C₃₂	—	13	2	no	—
81	C₂×C₃₂	—	13	1	no	—
82	C₂×C₃₂	—	13	1	no	—
83	C₂×C₃₂	—	13	2	no	—
84	C₂×C₃₂	—	13	2	no	—
85	#346	—	13	2	no	—
86	C₄×C₁₆	—	13	2	no	—
87	C₂²×C₁₃	C₄×C₁₆	1	1	yes	characteristic
88	C₂×D₁₃	—	2	2	no	—
89	C₂×D₁₃	C₄×C₁₆	1	1	yes	characteristic
90	C₂×D₁₃	C₂²×C₁₆	1	1	yes	characteristic
91	#261	—	2	4	no	—
92	C₄×C₁₃	C₄×C₁₆	1	1	yes	characteristic
93	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
94	Dic₁₃	C₄×C₁₆	1	1	yes	characteristic
95	Dic₁₃	C₂²×C₁₆	1	1	yes	characteristic
96	C₂×M₁₆	—	13	1	no	—
97	C₂²×C₈	—	13	1	no	—
98	#117	—	13	1	no	—
99	M₃₂	—	13	2	no	—
100	M₃₂	—	13	2	no	—
101	C₂×C₁₆	—	13	2	no	—
102	C₂×C₁₆	—	13	1	no	—
103	C₂×C₁₆	—	13	1	no	—
104	C₂×C₁₆	—	13	1	no	—
105	C₂×C₁₆	—	13	1	no	—
106	C₂×C₁₆	—	13	1	no	—
107	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
108	#97	—	13	2	no	—
109	C₄×C₈	—	13	2	no	—
110	C₃₂	—	13	2	no	—
111	C₃₂	—	13	2	no	—
112	C₃₂	—	13	2	no	—
113	C₃₂	C₂×#261	1	2	yes	—
114	C₂×C₁₃	—	2	1	no	—
115	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
116	D₁₃	—	2	1	no	—
117	D₁₃	#2258	1	2	yes	—
118	C₂²×C₄	—	13	1	no	—
119	M₁₆	—	13	2	no	—
120	M₁₆	—	13	2	no	—
121	C₂×C₈	—	13	2	no	—
122	C₂×C₈	—	13	1	no	—
123	C₂×C₈	—	13	1	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	—	13	1	no	—
126	C₂×C₈	—	13	1	no	—
127	C₂×C₈	C₄×#261	1	1	yes	characteristic
128	C₄⋊C₄	—	13	2	no	—
129	C₂²⋊C₄	—	13	2	no	—
130	C₄²	—	13	2	no	—
131	C₁₆	—	13	4	no	—
132	C₁₆	—	13	1	no	—
133	C₁₆	—	13	1	no	—
134	C₁₆	C₄×#261	1	1	yes	characteristic
135	C₁₆	C₂²×#261	1	1	yes	characteristic
136	C₁₃	#13549	1	1	yes	characteristic, complemented, Sylow
137	C₂³	—	13	1	no	—
138	C₂×C₄	—	13	1	no	—
139	C₂×C₄	—	13	1	no	—
140	C₂×C₄	—	13	1	no	—
141	C₂×C₄	—	13	1	no	—
142	C₂×C₄	—	13	1	no	—
143	C₂×C₄	—	13	4	no	—
144	C₂×C₄	C₈×#261	1	1	yes	characteristic
145	C₈	—	13	4	no	—
146	C₈	—	13	1	no	—
147	C₈	—	13	1	no	—
148	C₈	C₈×#261	1	1	yes	characteristic
149	C₈	C₂×C₄×#261	1	1	yes	characteristic
150	C₂²	—	13	1	no	—
151	C₂²	—	13	1	no	—
152	C₂²	—	26	2	no	—
153	C₂²	C₁₆×#261	1	1	yes	characteristic
154	C₄	—	13	1	no	—
155	C₄	—	13	1	no	—
156	C₄	—	26	4	no	—
157	C₄	C₁₆×#261	1	1	yes	characteristic
158	C₄	C₂×C₈×#261	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	13	2	no	—
161	C₂	—	26	1	no	—
162	C₂	C₂×C₁₆×#261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₃⋊C₃₂

C₂

yes

maximal, characteristic

#13131044

C₂

yes

maximal, characteristic

#13129942

C₂

yes

maximal, complemented

C₃₂×#261

C₂

yes

maximal, complemented

D₁₃×M₆₄

C₂

yes

maximal, characteristic

#11719089

C₄

yes

characteristic

D₁₃⋊M₃₂

C₄

yes

characteristic

C₂×C₁₆×D₁₃

C₂²

yes

characteristic

C₂×#88036

C₄

yes

complemented

#11718216

C₄

yes

—

C₁₆×#261

C₂²

yes

—

D₁₃⋊C₃₂

C₄

yes

complemented

D₁₃⋊C₃₂

C₂²

yes

characteristic

D₁₃⋊C₃₂

C₂²

yes

characteristic

C₁₃×M₆₄

C₄

yes

characteristic, complemented

#11718091

C₄

yes

characteristic, complemented

#11718039

C₄

yes

complemented

D₁₃×C₃₂

C₂²

yes

—

#88237

C₈

yes

characteristic

D₁₃⋊M₁₆

C₈

yes

characteristic

C₂×C₈×D₁₃

C₂×C₄

yes

characteristic

#88106

C₈

yes

—

#88100

C₈

yes

—

C₈×#261

C₂×C₄

yes

—

#88098

C₈

yes

—

D₁₃⋊C₁₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×#6090

C₂×C₄

yes

characteristic

C₁₆×D₁₃

C₂×C₄

yes

characteristic

C₁₆×D₁₃

C₂×C₄

yes

characteristic

C₁₆×D₁₃

C₂×C₄

yes

characteristic

C₁₆×D₁₃

C₂³

yes

characteristic

#88036

C₂×C₄

yes

—

#88036

C₂×C₄

yes

—

C₁₃×C₃₂

C₂×C₄

yes

—

#88034

C₂×C₄

yes

—

#13549

—

maximal, Sylow

C₂×C₄×D₁₃

C₂×C₈

yes

characteristic

C₂²⋊#261

C₁₆

yes

—

#6122

C₁₆

yes

—

C₄⋊#261

C₁₆

yes

—

C₄×#261

C₂×C₈

yes

—

#6118

C₁₆

yes

—

D₁₃⋊C₈

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×#1055

C₂×C₈

yes

characteristic

C₈×D₁₃

C₄²

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂×C₈

yes

characteristic

C₈×D₁₃

C₂²×C₄

yes

characteristic

#6092

C₂×C₈

yes

—

C₁₃×C₁₆

C₄²

yes

characteristic

C₁₃×C₁₆

C₂²×C₄

yes

characteristic

#6090

C₄²

yes

characteristic

#6090

C₂²×C₄

yes

characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

#1397

—

C₄×C₃₂

—

C₂²×D₁₃

C₂×C₁₆

yes

characteristic

C₂×#261

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

C₄×C₈

yes

characteristic

C₂×Dic₁₃

C₂×C₁₆

yes

characteristic

C₄×D₁₃

C₄×C₈

yes

characteristic

C₄×D₁₃

C₂×C₁₆

yes

characteristic

C₄×D₁₃

C₂×C₁₆

yes

characteristic

C₄×D₁₃

C₂²×C₈

yes

characteristic

#1057

C₂×C₁₆

yes

—

C₈×C₁₃

C₄×C₈

yes

characteristic

C₈×C₁₃

C₂×C₄²

yes

characteristic

#1055

C₄×C₈

yes

characteristic

#1055

C₂²×C₈

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

M₆₄

#261

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂²×C₁₃

C₄×C₁₆

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

C₄×C₁₆

yes

characteristic

C₂×D₁₃

C₂²×C₁₆

yes

characteristic

#261

—

C₄×C₁₃

C₄×C₁₆

yes

characteristic

C₄×C₁₃

C₂×C₄×C₈

yes

characteristic

Dic₁₃

C₄×C₁₆

yes

characteristic

Dic₁₃

C₂²×C₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

C₂×#261

yes

characteristic

108

#97

—

109

C₄×C₈

—

110

C₃₂

—

111

C₃₂

—

112

C₃₂

—

113

C₃₂

C₂×#261

yes

—

114

C₂×C₁₃

—

115

C₂×C₁₃

C₂×C₄×C₁₆

yes

characteristic

116

D₁₃

—

117

D₁₃

#2258

yes

—

118

C₂²×C₄

—

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₄×#261

yes

characteristic

128

C₄⋊C₄

—

129

C₂²⋊C₄

—

130

C₄²

—

131

C₁₆

—

132

C₁₆

—

133

C₁₆

—

134

C₁₆

C₄×#261

yes

characteristic

135

C₁₆

C₂²×#261

yes

characteristic

136

C₁₃

#13549

yes

characteristic, complemented, Sylow

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₈×#261

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

C₈×#261

yes

characteristic

149

C₈

C₂×C₄×#261

yes

characteristic

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₁₆×#261

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₁₆×#261

yes

characteristic

158

C₄

C₂×C₈×#261

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

C₂×C₁₆×#261

yes

characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×#261
commutator, socle	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₁₃×M₆₄	C₄
prime radical	C₂²×D₁₃	C₂×C₁₆
cosocle	C₁₆×D₁₃	C₂³
hypercenter	M₆₄	#261
hypocenter	C₁₃	#13549
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₁₆ ◁ M₆₄

subgroup

G/H

center, Frattini

C₁₆

C₂²×#261

commutator, socle

C₂×C₁₃

C₂×C₄×C₁₆

Fitting

C₁₃×M₆₄

C₄

prime radical

C₂²×D₁₃

C₂×C₁₆

cosocle

C₁₆×D₁₃

C₂³

M₆₄

C₁₃

G ▷ C₂×C₁₃ ▷ C₁₃

C₁ ◁ C₁₆ ◁ M₆₄

N	H ≅ G/N	type
3. #13129942	C₂	semidirect
4. C₃₂×#261	C₂	semidirect
9. C₂×#88036	C₄	semidirect
12. D₁₃⋊C₃₂	C₄	semidirect
15. C₁₃×M₆₄	C₄	semidirect
16. #11718091	C₄	semidirect
17. #11718039	C₄	semidirect
79. M₆₄	#261	semidirect
136. C₁₃	#13549	semidirect

H ≅ G/N

type

3. #13129942

C₂

semidirect

4. C₃₂×#261

C₂

semidirect

9. C₂×#88036

C₄

semidirect

12. D₁₃⋊C₃₂

C₄

semidirect

15. C₁₃×M₆₄

C₄

semidirect

16. #11718091

C₄

semidirect

17. #11718039

C₄

semidirect

79. M₆₄

#261

semidirect

136. C₁₃

#13549

semidirect

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

subgroup	index
C₂×D₁₃⋊C₃₂	2
#13131044	2
#13129942	2
C₃₂×#261	2
D₁₃×M₆₄	2
#13549	13

subgroup

index

C₂×D₁₃⋊C₃₂

#13131044

#13129942

C₃₂×#261

D₁₃×M₆₄

#13549

Subgroups — #16791149

Conjugacy classes of subgroups

Complements

Maximal subgroups