Group #15781513 — subgroups

Subgroups — #15781513

Order 3328.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#13116592	C₂	1	1	yes	maximal, characteristic
2	C₄×#88059	C₂	1	1	yes	maximal, characteristic
3	#13112944	C₂	1	4	yes	maximal
4	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#125	—	2	1	no	—
6	C₁₃×#499	—	2	2	no	—
7	C₁₃×#499	C₄	1	4	yes	—
8	C₂×C₁₃×#107	C₂²	1	2	yes	—
9	C₂×C₁₃×#106	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88059	C₂²	1	2	yes	—
12	C₂×#88052	C₂²	1	2	yes	—
13	C₄×#6099	—	2	1	no	—
14	#11718144	—	2	1	no	—
15	C₄⋊#6090	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₁₃×#125	—	2	4	no	—
19	C₁₃×#125	—	4	1	no	—
20	C₁₃×#125	—	4	1	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×#6099	—	2	2	no	—
25	C₁₃×#107	—	2	2	no	—
26	C₁₃×#107	C₂×C₄	1	4	yes	—
27	C₁₃×#106	—	2	4	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	D₄	1	2	yes	—
30	#88059	—	2	2	no	—
31	#88059	Q₈	1	4	yes	—
32	#88052	—	2	4	no	—
33	C₂×#6090	—	2	2	no	—
34	#88043	—	4	1	no	—
35	C₄×#1055	—	4	1	no	—
36	#13280	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	—	4	2	no	—
38	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₁₃	#41	1	2	yes	—
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	4	4	no	—
48	C₁₃×C₄⋊C₄	—	4	4	no	—
49	C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
51	C₄²×C₁₃	D₈	1	1	yes	characteristic
52	C₄²×C₁₃	D₈	1	1	yes	characteristic
53	#6099	—	2	4	no	—
54	#6099	—	4	2	no	—
55	C₂×#1055	—	4	2	no	—
56	#6090	—	4	2	no	—
57	C₂×#499	D₁₃	1	1	yes	characteristic
58	#2150	—	13	1	no	—
59	C₄×#108	—	13	1	no	—
60	#1387	—	13	4	no	—
61	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	4	2	no	—
63	C₂×C₄×C₁₃	—	4	4	no	—
64	C₂×C₄×C₁₃	—	8	4	no	—
65	C₂×C₄×C₁₃	#102	1	2	yes	—
66	C₂×C₄×C₁₃	#102	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#137	1	2	yes	—
71	C₈×C₁₃	—	2	2	no	—
72	C₈×C₁₃	C₄×Q₈	1	2	yes	—
73	C₈×C₁₃	#124	1	1	yes	characteristic
74	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
75	#1055	—	4	4	no	—
76	C₂×#125	—	2	1	no	—
77	C₂×M₃₂	—	13	2	no	—
78	#499	—	2	2	no	—
79	#499	Dic₁₃	1	4	yes	—
80	C₂×#108	—	13	2	no	—
81	C₂×#107	C₂×D₁₃	1	2	yes	—
82	C₂×#106	—	2	1	no	—
83	C₄×M₁₆	—	26	1	no	—
84	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
85	C₄⋊C₁₆	—	26	2	no	—
86	#341	—	26	1	no	—
87	C₂²×C₁₃	—	2	2	no	—
88	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
89	C₂²×C₁₃	#475	1	2	yes	—
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	8	4	no	—
93	C₄×C₁₃	#391	1	1	yes	characteristic
94	C₄×C₁₃	#419	1	1	yes	characteristic
95	C₄×C₁₃	#474	1	2	yes	—
96	C₂×M₁₆	—	26	2	no	—
97	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
99	#125	—	2	4	no	—
100	#125	—	4	1	no	—
101	#125	—	4	1	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
105	M₃₂	—	26	4	no	—
106	C₂×C₁₆	—	26	2	no	—
107	#108	—	13	4	no	—
108	#108	—	26	2	no	—
109	#107	—	2	2	no	—
110	#107	C₂×Dic₁₃	1	4	yes	—
111	#106	—	2	4	no	—
112	#97	—	52	1	no	—
113	C₄×C₈	—	52	1	no	—
114	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
115	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
116	C₂×C₁₃	—	2	2	no	—
117	C₂×C₁₃	#1988	1	1	yes	characteristic
118	C₂×C₁₃	#2148	1	2	yes	—
119	C₂²×C₄	—	4	2	no	—
120	C₂²×C₄	#6105	1	2	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
122	M₁₆	—	26	4	no	—
123	M₁₆	—	52	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	52	2	no	—
126	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
127	C₂×C₈	Q₈×D₁₃	1	2	yes	—
128	C₂×C₈	#6131	1	2	yes	—
129	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
130	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	4	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄²	#6104	1	1	yes	characteristic
136	C₄²	#6104	1	1	yes	characteristic
137	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
138	C₁₆	—	52	2	no	—
139	C₁₃	#13280	1	1	yes	characteristic, complemented, Sylow
140	C₂³	C₂×#6105	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#88072	1	2	yes	—
145	C₂×C₄	#88072	1	2	yes	—
146	C₂×C₄	C₂×#6104	1	1	yes	characteristic
147	C₂×C₄	#88196	1	2	yes	—
148	C₂×C₄	#88200	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	52	4	no	—
152	C₈	#88198	1	1	yes	characteristic
153	C₈	Q₈×Dic₁₃	1	2	yes	—
154	C₈	#88201	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	#11718755	1	2	yes	—
157	C₂²	C₂×#88072	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#11718748	1	2	yes	—
162	C₄	#11718886	1	1	yes	characteristic
163	C₄	#11718891	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	#13115789	1	1	yes	characteristic
166	C₂	#13116585	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116592

C₂

yes

maximal, characteristic

C₄×#88059

C₂

yes

maximal, characteristic

#13112944

C₂

yes

maximal

C₂×C₁₃×#499

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

—

C₁₃×#499

—

C₁₃×#499

C₄

yes

—

C₂×C₁₃×#107

C₂²

yes

—

C₂×C₁₃×#106

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88059

C₂²

yes

—

C₂×#88052

C₂²

yes

—

C₄×#6099

—

#11718144

—

C₄⋊#6090

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₁₃×#125

—

C₁₃×#125

—

C₁₃×#125

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×#6099

—

C₁₃×#107

—

C₁₃×#107

C₂×C₄

yes

—

C₁₃×#106

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#88059

—

#88059

Q₈

yes

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

SD₁₆

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²⋊C₄

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

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

D₈

yes

characteristic

C₄²×C₁₃

D₈

yes

—

—

—

—

D₁₃

yes

—

—

—

C₂³×C₁₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

#102

yes

—

C₂×C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

C₈×C₁₃

C₄×Q₈

yes

—

C₈×C₁₃

#124

yes

characteristic

C₈×C₁₃

C₄⋊Q₈

yes

characteristic

#1055

—

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

Dic₁₃

yes

—

C₂×#108

—

C₂×#107

C₂×D₁₃

yes

—

C₂×#106

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄⋊C₁₆

—

#341

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×#102

yes

characteristic

C₂²×C₁₃

#475

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

#125

—

100

#125

—

101

#125

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

108

#108

—

109

#107

—

110

#107

C₂×Dic₁₃

yes

—

111

#106

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

C₂×Dic₁₃

yes

—

115

C₄×C₈

C₂²⋊D₁₃

yes

—

116

C₂×C₁₃

—

117

C₂×C₁₃

#1988

yes

characteristic

118

C₂×C₁₃

#2148

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

#6105

yes

—

121

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

C₂²⋊Dic₁₃

yes

—

127

C₂×C₈

Q₈×D₁₃

yes

—

128

C₂×C₈

#6131

yes

—

129

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

130

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

#6104

yes

characteristic

136

C₄²

#6104

yes

characteristic

137

C₄²

C₂²⋊Dic₁₃

yes

—

138

C₁₆

—

139

C₁₃

#13280

yes

characteristic, complemented, Sylow

140

C₂³

C₂×#6105

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#88072

yes

—

145

C₂×C₄

#88072

yes

—

146

C₂×C₄

C₂×#6104

yes

characteristic

147

C₂×C₄

#88196

yes

—

148

C₂×C₄

#88200

yes

—

149

C₂×C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

#88198

yes

characteristic

153

C₈

Q₈×Dic₁₃

yes

—

154

C₈

#88201

yes

characteristic

155

C₂²

—

156

C₂²

#11718755

yes

—

157

C₂²

C₂×#88072

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#11718748

yes

—

162

C₄

#11718886

yes

characteristic

163

C₄

#11718891

yes

characteristic

164

C₂

—

165

C₂

#13115789

yes

characteristic

166

C₂

#13116585

yes

—

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#499	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#499	D₁₃
hypocenter	C₁₃	#13280
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

C₂×#88072

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₂×C₁₃×#499

C₂

socle

C₂²×C₁₃

C₂×#102

prime radical

C₂³×C₁₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
139. C₁₃	#13280	semidirect

H ≅ G/N

type

139. C₁₃

#13280

semidirect

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

subgroup	index
#13116592	2
C₄×#88059	2
#13112944	2
C₂×C₁₃×#499	2
#13280	13

subgroup

index

C₂×C₁₃×#499

Subgroups — #15781513

Conjugacy classes of subgroups

Complements

Maximal subgroups