Group #15783514 — subgroups

Subgroups — #15783514

Order 3328.

188 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 65 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×#11718114	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13116694	C₂	1	1	yes	maximal, characteristic
4	#13113037	C₂	1	1	yes	maximal, characteristic
5	#13113018	C₂	1	1	yes	maximal, characteristic
6	#13112969	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₁₃	—	2	1	no	—
10	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
11	#11718534	—	2	1	no	—
12	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#366	C₂²	1	1	yes	characteristic
14	#11718156	—	2	1	no	—
15	#11718114	C₄	1	4	yes	complemented
16	#11718096	C₂²	1	1	yes	characteristic
17	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88034	C₂²	1	1	yes	characteristic
19	#11718084	—	2	1	no	—
20	C₂×C₁₃×Q₁₆	—	2	1	no	—
21	Q₈×Dic₁₃	—	4	1	no	—
22	C₂×Dic₅₂	—	2	1	no	—
23	#88141	—	4	1	no	—
24	C₁₃×Q₃₂	—	2	1	no	—
25	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
27	C₁₃×#107	—	2	1	no	—
28	#88075	—	4	1	no	—
29	#88058	—	2	1	no	—
30	C₈×Dic₁₃	D₄	1	1	yes	characteristic
31	C₂×#6090	D₄	1	1	yes	characteristic
32	#88050	—	4	1	no	—
33	Dic₁₀₄	—	2	1	no	—
34	Dic₁₀₄	C₂×C₄	1	2	yes	—
35	C₁₃×C₃₂	—	2	1	no	—
36	#88034	C₂×C₄	1	2	yes	—
37	#13653	—	13	1	no	maximal, Sylow
38	C₂×Q₈×C₁₃	—	4	1	no	—
39	C₂×Dic₂₆	—	4	1	no	—
40	C₁₃×Q₁₆	—	2	2	no	—
41	C₁₃×Q₁₆	—	4	1	no	—
42	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	—	4	1	no	—
44	C₄⋊Dic₁₃	—	4	1	no	—
45	C₄⋊Dic₁₃	—	8	1	no	—
46	#6101	—	8	1	no	—
47	C₄×Dic₁₃	—	8	1	no	—
48	C₄×Dic₁₃	—	8	1	no	—
49	C₄×Dic₁₃	D₈	1	1	yes	characteristic
50	C₂×#1055	D₈	1	1	yes	characteristic
51	Dic₅₂	—	2	2	no	—
52	Dic₅₂	—	4	1	no	—
53	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
54	C₁₃×C₁₆	#41	1	1	yes	characteristic
55	#6090	—	2	1	no	—
56	C₂×Q₆₄	—	13	1	no	—
57	C₄×Q₃₂	—	13	1	no	—
58	C₄×Q₃₂	—	13	1	no	—
59	#1424	—	13	1	no	—
60	#1416	—	13	1	no	—
61	#1416	D₁₃	1	1	yes	characteristic
62	#1397	—	13	1	no	—
63	Q₈×C₁₃	—	4	2	no	—
64	Q₈×C₁₃	—	8	1	no	—
65	C₂×C₄×C₁₃	—	8	1	no	—
66	C₂×C₄×C₁₃	—	8	1	no	—
67	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
68	C₂×Dic₁₃	—	8	1	no	—
69	C₂×Dic₁₃	—	8	1	no	—
70	C₂×Dic₁₃	D₁₆	1	1	yes	characteristic
71	C₂×Dic₁₃	D₁₆	1	1	yes	characteristic
72	Dic₂₆	—	4	2	no	—
73	Dic₂₆	—	8	1	no	—
74	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₈×C₁₃	#135	1	1	yes	characteristic
76	#1055	—	2	1	no	—
77	C₂×Q₃₂	—	13	1	no	—
78	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
79	C₄×Q₁₆	—	26	1	no	—
80	C₄×Q₁₆	—	26	1	no	—
81	Q₆₄	—	13	4	no	—
82	C₂×C₃₂	—	13	1	no	—
83	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
84	#366	—	13	1	no	—
85	#366	C₂×D₁₃	1	1	yes	characteristic
86	#358	—	26	1	no	—
87	#358	—	26	1	no	—
88	C₄×C₁₆	—	13	1	no	—
89	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
90	C₄×C₁₃	—	8	2	no	—
91	C₄×C₁₃	—	16	1	no	—
92	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
93	C₄×C₁₃	#508	1	1	yes	characteristic
94	Dic₁₃	—	2	1	no	—
95	Dic₁₃	—	2	1	no	—
96	Dic₁₃	—	8	2	no	—
97	Dic₁₃	—	16	1	no	—
98	C₂×Q₁₆	—	2	1	no	—
99	C₂×Q₁₆	—	26	1	no	—
100	C₄×Q₈	—	52	1	no	—
101	C₄×Q₈	—	52	1	no	—
102	Q₃₂	—	2	1	no	—
103	Q₃₂	—	13	2	no	—
104	Q₃₂	—	26	1	no	—
105	Q₃₂	C₄×D₁₃	1	2	yes	—
106	C₂×C₁₆	—	13	1	no	—
107	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
108	#107	—	2	1	no	—
109	#107	—	26	1	no	—
110	#103	—	52	1	no	—
111	#103	—	52	1	no	—
112	C₄×C₈	—	13	1	no	—
113	C₃₂	—	2	1	no	—
114	C₃₂	—	13	2	no	—
115	C₂×C₁₃	C₄×D₁₆	1	1	yes	characteristic
116	C₂×C₁₃	#2264	1	1	yes	characteristic
117	C₂×C₁₃	#2264	1	1	yes	characteristic
118	C₂×Q₈	—	4	1	no	—
119	C₂×Q₈	—	52	1	no	—
120	Q₁₆	—	2	2	no	—
121	Q₁₆	—	4	1	no	—
122	Q₁₆	—	26	2	no	—
123	Q₁₆	—	52	1	no	—
124	C₂×C₈	—	13	1	no	—
125	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	52	1	no	—
128	C₄⋊C₄	—	104	1	no	—
129	C₄⋊C₄	—	104	1	no	—
130	C₄²	—	13	1	no	—
131	C₄²	—	104	1	no	—
132	C₄²	—	104	1	no	—
133	C₁₆	—	26	1	no	—
134	C₁₆	C₂×C₄×D₁₃	1	1	yes	characteristic
135	C₁₆	#6129	1	1	yes	characteristic
136	C₁₃	#13653	1	1	yes	characteristic, complemented, Sylow
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	52	2	no	—
140	Q₈	—	104	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	13	1	no	—
144	C₂×C₄	—	13	1	no	—
145	C₂×C₄	—	104	1	no	—
146	C₂×C₄	—	104	1	no	—
147	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
148	C₈	—	26	1	no	—
149	C₈	#88135	1	1	yes	characteristic
150	C₈	#88170	1	1	yes	characteristic
151	C₂²	D₁₃×D₁₆	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	26	1	no	—
155	C₄	—	26	1	no	—
156	C₄	—	104	2	no	—
157	C₄	—	208	1	no	—
158	C₄	#11718417	1	1	yes	characteristic
159	C₄	#11718577	1	1	yes	characteristic
160	C₂	#13116680	1	1	yes	characteristic
161	C₂	#13117152	1	1	yes	characteristic
162	C₂	#13117155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718114

C₂

yes

maximal, characteristic

Q₃₂×Dic₁₃

C₂

yes

maximal, characteristic

#13116694

C₂

yes

maximal, characteristic

#13113037

C₂

yes

maximal, characteristic

#13113018

C₂

yes

maximal, characteristic

#13112969

C₂

yes

maximal, characteristic

C₁₃×#1416

C₂

yes

maximal, characteristic

C₂×C₁₃×Q₃₂

C₂²

yes

characteristic

Q₁₆×Dic₁₃

—

C₂×Dic₁₀₄

C₂²

yes

characteristic

#11718534

—

C₂×C₁₃×C₃₂

C₂²

yes

characteristic

C₁₃×#366

C₂²

yes

characteristic

#11718156

—

#11718114

C₄

yes

complemented

#11718096

C₂²

yes

characteristic

C₁₆×Dic₁₃

C₂²

yes

characteristic

C₂×#88034

C₂²

yes

characteristic

#11718084

—

C₂×C₁₃×Q₁₆

—

Q₈×Dic₁₃

—

C₂×Dic₅₂

—

#88141

—

C₁₃×Q₃₂

—

C₁₃×Q₃₂

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₁₃×#107

—

#88075

—

#88058

—

C₈×Dic₁₃

D₄

yes

characteristic

C₂×#6090

D₄

yes

characteristic

#88050

—

Dic₁₀₄

—

Dic₁₀₄

C₂×C₄

yes

—

C₁₃×C₃₂

—

#88034

C₂×C₄

yes

—

#13653

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₈

yes

characteristic

C₂×#1055

D₈

yes

characteristic

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

C₂²×C₄

yes

C₁₃×C₁₆

yes

—

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1416

D₁₃

yes

characteristic

#1397

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

D₁₆

yes

characteristic

C₂×Dic₁₃

D₁₆

yes

characteristic

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

C₄×D₄

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₁₃

yes

characteristic

C₄×Q₁₆

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₁₃

yes

characteristic

#366

—

#366

C₂×D₁₃

yes

characteristic

#358

—

#358

—

C₄×C₁₆

—

C₂²×C₁₃

C₂×D₁₆

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×D₈

yes

characteristic

C₄×C₁₃

#508

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₄×Q₈

—

101

C₄×Q₈

—

102

Q₃₂

—

103

Q₃₂

—

104

Q₃₂

—

105

Q₃₂

C₄×D₁₃

yes

—

106

C₂×C₁₆

—

107

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

108

#107

—

109

#107

—

110

#103

—

111

#103

—

112

C₄×C₈

—

113

C₃₂

—

114

C₃₂

—

115

C₂×C₁₃

C₄×D₁₆

yes

characteristic

116

C₂×C₁₃

#2264

yes

characteristic

117

C₂×C₁₃

#2264

yes

characteristic

118

C₂×Q₈

—

119

C₂×Q₈

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

C₂×C₈

—

125

C₂×C₈

D₄×D₁₃

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

104

—

129

C₄⋊C₄

—

104

—

130

C₄²

—

131

C₄²

—

104

—

132

C₄²

—

104

—

133

C₁₆

—

134

C₁₆

C₂×C₄×D₁₃

yes

characteristic

135

C₁₆

#6129

yes

characteristic

136

C₁₃

#13653

yes

characteristic, complemented, Sylow

137

Q₈

—

138

Q₈

—

139

Q₈

—

140

Q₈

—

104

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

104

—

146

C₂×C₄

—

104

—

147

C₂×C₄

D₈×D₁₃

yes

characteristic

148

C₈

—

149

C₈

#88135

yes

characteristic

150

C₈

#88170

yes

characteristic

151

C₂²

D₁₃×D₁₆

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

104

—

157

C₄

—

208

—

158

C₄

#11718417

yes

characteristic

159

C₄

#11718577

yes

characteristic

160

C₂

#13116680

yes

characteristic

161

C₂

#13117152

yes

characteristic

162

C₂

#13117155

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₃×D₁₆
commutator	C₁₃×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₁₃×#1416	C₂
socle, prime radical	C₂²×C₁₃	C₂×D₁₆
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	#1416	D₁₃
hypocenter	C₁₃	#13653
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₃×D₁₆

commutator

C₁₃×C₁₆

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₁₃

Fitting

C₁₃×#1416

C₂

socle, prime radical

C₂²×C₁₃

C₂×D₁₆

cosocle

C₂×C₁₃×C₁₆

C₂³

D₁₃

C₁₃

G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
15. #11718114	C₄	semidirect
136. C₁₃	#13653	semidirect

H ≅ G/N

type

15. #11718114

C₄

semidirect

136. C₁₃

#13653

semidirect

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

subgroup	index
C₂×#11718114	2
Q₃₂×Dic₁₃	2
#13116694	2
#13113037	2
#13113018	2
#13112969	2
C₁₃×#1416	2
#13653	13

subgroup

index

Q₃₂×Dic₁₃

C₁₃×#1416

Subgroups — #15783514

Conjugacy classes of subgroups

Complements

Maximal subgroups