Group #15781516 — subgroups

Subgroups — #15781516

Order 3328.

366 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 163 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×#11718344	C₂	1	1	yes	maximal, characteristic
2	#13115000	C₂	1	1	yes	maximal, characteristic
3	#13112928	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#88124	—	2	1	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	C₂²	1	2	yes	—
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88057	—	2	1	no	—
11	C₂×#88057	—	2	1	no	—
12	#11718597	—	2	1	no	—
13	#11718344	—	2	2	no	—
14	#11718344	C₄	1	4	yes	complemented
15	C₄×C₁₃×C₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×#6101	—	4	2	no	—
22	#88124	—	2	4	no	—
23	#88124	—	4	2	no	—
24	C₂×C₁₃×C₁₆	—	2	2	no	—
25	C₂×C₁₃×C₁₆	—	2	2	no	—
26	C₂×C₁₃×C₁₆	Q₈	1	4	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	D₄	1	2	yes	—
29	#88071	—	4	2	no	—
30	#88058	—	2	2	no	—
31	#88058	—	2	4	no	—
32	#88058	C₂×C₄	1	4	yes	—
33	#88057	—	2	4	no	—
34	#88057	—	4	2	no	—
35	#13281	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	D₈	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	4	2	no	—
39	C₂²×Dic₁₃	—	4	2	no	—
40	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₁₃	#41	1	2	yes	—
47	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
48	C₄²×C₁₃	SD₁₆	1	2	yes	—
49	C₄⋊Dic₁₃	—	2	4	no	—
50	C₄⋊Dic₁₃	—	4	2	no	—
51	C₄⋊Dic₁₃	—	4	2	no	—
52	C₄⋊Dic₁₃	—	4	2	no	—
53	#6101	—	4	8	no	—
54	C₁₃×C₁₆	—	2	4	no	—
55	C₂×#499	—	13	1	no	—
56	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
57	#1846	—	13	1	no	—
58	#1380	—	13	4	no	—
59	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	#102	1	4	yes	—
62	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₁₃	—	4	4	no	—
69	C₂×Dic₁₃	—	8	4	no	—
70	C₂×Dic₁₃	—	8	2	no	—
71	C₂×Dic₁₃	—	8	4	no	—
72	C₈×C₁₃	#103	1	4	yes	—
73	C₈×C₁₃	C₄×Q₈	1	2	yes	—
74	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
75	C₂×#125	—	26	1	no	—
76	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
77	#499	—	13	4	no	—
78	#499	—	26	2	no	—
79	C₂×#107	—	13	2	no	—
80	C₂×#107	—	13	2	no	—
81	C₂×#106	—	26	1	no	—
82	C₂×#106	—	26	1	no	—
83	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
84	#382	—	26	1	no	—
85	C₄×C₁₆	—	2	2	no	—
86	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
87	C₂²×C₁₃	#486	1	1	yes	characteristic
88	C₂²×C₁₃	#500	1	1	yes	characteristic
89	C₂²×C₁₃	#508	1	4	yes	—
90	C₄×C₁₃	—	2	4	no	—
91	C₄×C₁₃	#391	1	1	yes	characteristic
92	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
93	C₄×C₁₃	#487	1	1	yes	characteristic
94	C₄×C₁₃	#498	1	1	yes	characteristic
95	Dic₁₃	—	8	4	no	—
96	Dic₁₃	—	16	2	no	—
97	C₂²×C₈	D₅₂	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
99	#125	—	26	4	no	—
100	#125	—	52	2	no	—
101	C₂×C₄⋊C₄	—	26	2	no	—
102	C₂×C₄⋊C₄	—	26	2	no	—
103	C₂×C₄⋊C₄	—	52	2	no	—
104	C₂×C₄²	D₅₂	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	Dic₂₆	1	4	yes	—
108	#107	—	13	4	no	—
109	#107	—	26	2	no	—
110	#107	—	26	4	no	—
111	#106	—	26	4	no	—
112	#106	—	52	2	no	—
113	C₄×C₈	C₄×D₁₃	1	2	yes	—
114	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
115	#95	—	52	2	no	—
116	C₂×C₁₃	#1990	1	1	yes	characteristic
117	C₂×C₁₃	#2138	1	2	yes	—
118	C₂×C₁₃	#2242	1	2	yes	—
119	C₂×C₁₃	#2253	1	2	yes	—
120	C₂²×C₄	—	52	2	no	—
121	C₂²×C₄	—	52	2	no	—
122	C₂²×C₄	D₁₀₄	1	2	yes	—
123	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
124	C₂×C₈	#6095	1	2	yes	—
125	C₂×C₈	Dic₅₂	1	2	yes	—
126	C₂×C₈	#6103	1	2	yes	—
127	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
129	C₂×C₈	#6127	1	2	yes	—
130	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	C₄⋊C₄	—	26	4	no	—
132	C₄⋊C₄	—	52	2	no	—
133	C₄⋊C₄	—	52	8	no	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄⋊C₄	—	52	2	no	—
136	C₄²	#6095	1	2	yes	—
137	C₄²	#6103	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₁₃	#13281	1	1	yes	characteristic, complemented, Sylow
140	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	52	4	no	—
143	C₂×C₄	—	104	4	no	—
144	C₂×C₄	—	104	4	no	—
145	C₂×C₄	—	104	2	no	—
146	C₂×C₄	#88061	1	4	yes	—
147	C₂×C₄	#88123	1	1	yes	characteristic
148	C₂×C₄	#88129	1	1	yes	characteristic
149	C₂×C₄	C₂×#6095	1	1	yes	characteristic
150	C₂×C₄	C₂×#6095	1	1	yes	characteristic
151	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
152	C₂×C₄	C₂×#6103	1	1	yes	characteristic
153	C₈	#88056	1	4	yes	—
154	C₈	C₄×Dic₂₆	1	2	yes	—
155	C₈	#88178	1	2	yes	—
156	C₂²	#11718343	1	1	yes	characteristic
157	C₂²	#11718355	1	1	yes	characteristic
158	C₂²	#11718565	1	4	yes	—
159	C₂²	C₂×#88061	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	104	4	no	—
162	C₄	—	208	2	no	—
163	C₄	#11718341	1	1	yes	characteristic
164	C₄	#11718342	1	1	yes	characteristic
165	C₄	#11718590	1	1	yes	characteristic
166	C₄	C₂×#88056	1	1	yes	characteristic
167	C₂	#13115797	1	1	yes	characteristic
168	C₂	#13116544	1	2	yes	—
169	C₂	#13117068	1	2	yes	—
170	C₂	#13117115	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718344

C₂

yes

maximal, characteristic

#13115000

C₂

yes

maximal, characteristic

#13112928

C₂

yes

maximal

C₂×C₄×C₁₃×C₁₆

C₂

yes

maximal, characteristic

C₂×#88124

—

C₂²×C₁₃×C₁₆

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88058

C₂²

yes

—

C₂×#88058

C₂²

yes

—

—

—

—

—

C₄

yes

complemented

C₄×C₁₃×C₁₆

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×#6101

—

#88124

—

#88124

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

Q₈

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

—

—

—

C₂×C₄

yes

—

—

—

—

C₂²×C₄×C₁₃

D₈

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₂×C₈×C₁₃

Q₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₁₃×C₁₆

—

C₂×#499

—

C₂×C₄×C₁₆

D₁₃

yes

characteristic

#1846

—

#1380

—

C₂³×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

#103

yes

—

C₈×C₁₃

C₄×Q₈

yes

—

C₈×C₁₃

C₂²⋊Q₈

yes

—

C₂×#125

—

C₂²×C₁₆

C₂×D₁₃

yes

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#382

—

C₄×C₁₆

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×C₈

D₅₂

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

#125

—

100

#125

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

D₅₂

yes

characteristic

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

Dic₂₆

yes

—

108

#107

—

109

#107

—

110

#107

—

111

#106

—

112

#106

—

113

C₄×C₈

C₄×D₁₃

yes

—

114

C₄×C₈

C₂²⋊D₁₃

yes

—

115

#95

—

116

C₂×C₁₃

#1990

yes

characteristic

117

C₂×C₁₃

#2138

yes

—

118

C₂×C₁₃

#2242

yes

—

119

C₂×C₁₃

#2253

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

D₁₀₄

yes

—

123

C₂²×C₄

C₂×D₅₂

yes

characteristic

124

C₂×C₈

#6095

yes

—

125

C₂×C₈

Dic₅₂

yes

—

126

C₂×C₈

#6103

yes

—

127

C₂×C₈

C₂×Dic₂₆

yes

—

128

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

129

C₂×C₈

#6127

yes

—

130

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

#6095

yes

—

137

C₄²

#6103

yes

—

138

C₁₆

—

139

C₁₃

#13281

yes

characteristic, complemented, Sylow

140

C₂³

C₂×D₁₀₄

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

104

—

144

C₂×C₄

—

104

—

145

C₂×C₄

—

104

—

146

C₂×C₄

#88061

yes

—

147

C₂×C₄

#88123

yes

characteristic

148

C₂×C₄

#88129

yes

characteristic

149

C₂×C₄

C₂×#6095

yes

characteristic

150

C₂×C₄

C₂×#6095

yes

characteristic

151

C₂×C₄

C₂×Dic₅₂

yes

characteristic

152

C₂×C₄

C₂×#6103

yes

characteristic

153

C₈

#88056

yes

—

154

C₈

C₄×Dic₂₆

yes

—

155

C₈

#88178

yes

—

156

C₂²

#11718343

yes

characteristic

157

C₂²

#11718355

yes

characteristic

158

C₂²

#11718565

yes

—

159

C₂²

C₂×#88061

yes

characteristic

160

C₄

—

161

C₄

—

104

—

162

C₄

—

208

—

163

C₄

#11718341

yes

characteristic

164

C₄

#11718342

yes

characteristic

165

C₄

#11718590

yes

characteristic

166

C₄

C₂×#88056

yes

characteristic

167

C₂

#13115797

yes

characteristic

168

C₂

#13116544

yes

—

169

C₂

#13117068

yes

—

170

C₂

#13117115

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#13281
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂³

C₂×D₁₀₄

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₂×C₄×C₁₃×C₁₆

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₈

cosocle

C₂²×C₈×C₁₃

C₂³

hypercenter

C₂×C₄×C₁₆

D₁₃

hypocenter

C₁₃

#13281

LCS

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
14. #11718344	C₄	semidirect
139. C₁₃	#13281	semidirect

H ≅ G/N

type

14. #11718344

C₄

semidirect

139. C₁₃

#13281

semidirect

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

subgroup	index
C₂×#11718344	2
#13115000	2
#13112928	2
C₂×C₄×C₁₃×C₁₆	2
#13281	13

subgroup

index

C₂×C₄×C₁₃×C₁₆

Subgroups — #15781516

Conjugacy classes of subgroups

Complements

Maximal subgroups