Group #15781512 — subgroups

Subgroups — #15781512

Order 3328.

334 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 131 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	#13116592	C₂	1	1	yes	maximal, characteristic
3	#13112946	C₂	1	4	yes	maximal
4	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
5	C₂×#88124	—	2	1	no	—
6	C₂×C₁₃×#108	C₂²	1	2	yes	—
7	C₄×C₁₃×M₁₆	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88057	—	2	1	no	—
11	C₂×#88052	C₂²	1	2	yes	—
12	#11718344	—	2	2	no	—
13	#11718344	C₄	1	4	yes	—
14	C₁₃×#341	—	2	1	no	—
15	C₄⋊#6090	—	2	2	no	—
16	C₂×C₁₃×M₁₆	—	2	2	no	—
17	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×#6101	—	4	2	no	—
22	#88124	—	2	4	no	—
23	#88124	—	4	1	no	—
24	#88124	—	4	1	no	—
25	C₁₃×#108	—	2	2	no	—
26	C₁₃×#108	Q₈	1	4	yes	—
27	C₁₃×#97	—	4	1	no	—
28	C₄×C₈×C₁₃	—	4	1	no	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	#88058	—	2	2	no	—
32	#88058	C₂×C₄	1	4	yes	—
33	#88057	—	2	4	no	—
34	#88052	—	2	4	no	—
35	C₂×#6090	—	2	2	no	—
36	#13280	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	4	2	no	—
40	C₁₃×M₁₆	—	2	4	no	—
41	C₁₃×M₁₆	—	4	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₁₃	#41	1	2	yes	—
49	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₁₃	D₈	1	1	yes	characteristic
51	C₄²×C₁₃	D₈	1	1	yes	characteristic
52	C₄⋊Dic₁₃	—	2	4	no	—
53	C₄⋊Dic₁₃	—	4	2	no	—
54	#6101	—	4	4	no	—
55	#6101	—	4	4	no	—
56	#6090	—	4	2	no	—
57	C₂×#499	—	13	1	no	—
58	#2150	—	13	1	no	—
59	C₄×#108	D₁₃	1	1	yes	characteristic
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	#102	1	2	yes	—
64	C₂×C₄×C₁₃	#102	1	2	yes	—
65	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
67	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#137	1	2	yes	—
69	C₂×Dic₁₃	—	4	4	no	—
70	C₂×Dic₁₃	—	8	4	no	—
71	C₈×C₁₃	—	2	2	no	—
72	C₈×C₁₃	—	4	4	no	—
73	C₈×C₁₃	C₄×Q₈	1	2	yes	—
74	C₈×C₁₃	#124	1	1	yes	characteristic
75	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×#125	—	26	1	no	—
77	C₂×M₃₂	—	13	2	no	—
78	#499	—	13	4	no	—
79	#499	—	26	2	no	—
80	C₂×#108	C₂×D₁₃	1	2	yes	—
81	C₂×#107	—	13	2	no	—
82	C₂×#106	—	26	1	no	—
83	C₄×M₁₆	—	2	1	no	—
84	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
85	C₄⋊C₁₆	—	26	2	no	—
86	#341	—	2	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	#391	1	1	yes	characteristic
93	C₄×C₁₃	#419	1	1	yes	characteristic
94	C₄×C₁₃	#474	1	2	yes	—
95	Dic₁₃	—	8	4	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
99	#125	—	26	4	no	—
100	#125	—	52	1	no	—
101	#125	—	52	1	no	—
102	C₂×C₄⋊C₄	—	26	2	no	—
103	C₂×C₄⋊C₄	—	52	2	no	—
104	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
105	M₃₂	—	26	4	no	—
106	C₂×C₁₆	—	26	2	no	—
107	#108	—	2	2	no	—
108	#108	Dic₂₆	1	4	yes	—
109	#107	—	13	4	no	—
110	#107	—	26	2	no	—
111	#106	—	26	4	no	—
112	#97	—	4	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	C₄×D₁₃	1	2	yes	—
115	C₄×C₈	D₅₂	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	—	52	2	no	—
120	C₂²×C₄	#6106	1	2	yes	—
121	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
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	C₂×D₅₂	1	1	yes	characteristic
130	C₂×C₈	#6127	1	2	yes	—
131	C₄⋊C₄	—	26	4	no	—
132	C₄⋊C₄	—	52	2	no	—
133	C₄⋊C₄	—	52	4	no	—
134	C₄⋊C₄	—	52	4	no	—
135	C₄²	#6103	1	2	yes	—
136	C₄²	#6104	1	1	yes	characteristic
137	C₄²	#6104	1	1	yes	characteristic
138	C₁₆	—	52	2	no	—
139	C₁₃	#13280	1	1	yes	characteristic, complemented, Sylow
140	C₂³	C₂×#6106	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	52	4	no	—
143	C₂×C₄	—	104	4	no	—
144	C₂×C₄	#88049	1	2	yes	—
145	C₂×C₄	#88049	1	2	yes	—
146	C₂×C₄	#88178	1	2	yes	—
147	C₂×C₄	C₂×#6103	1	1	yes	characteristic
148	C₂×C₄	C₂×#6104	1	1	yes	characteristic
149	C₂×C₄	#88205	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	C₄×Dic₂₆	1	2	yes	—
153	C₈	#88123	1	1	yes	characteristic
154	C₈	#88129	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#88049	1	1	yes	characteristic
157	C₂²	#11718681	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	104	4	no	—
161	C₄	#11718590	1	1	yes	characteristic
162	C₄	#11718665	1	1	yes	characteristic
163	C₄	#11718668	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#13115788	1	1	yes	characteristic
166	C₂	#13116583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718344

C₂

yes

maximal, characteristic

#13116592

C₂

yes

maximal, characteristic

#13112946

C₂

yes

maximal

C₄×C₁₃×#108

C₂

yes

maximal, characteristic

C₂×#88124

—

C₂×C₁₃×#108

C₂²

yes

—

C₄×C₁₃×M₁₆

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88058

C₂²

yes

—

C₂×#88057

—

C₂×#88052

C₂²

yes

—

#11718344

—

#11718344

C₄

yes

—

C₁₃×#341

—

C₄⋊#6090

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

—

—

—

—

C₁₃×#108

—

C₁₃×#108

Q₈

yes

—

C₁₃×#97

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

#88058

—

#88058

C₂×C₄

yes

—

—

—

—

—

C₂²×C₄×C₁₃

SD₁₆

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

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

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

C₄²×C₁₃

D₈

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

—

D₁₃

yes

characteristic

#1387

—

C₂³×C₁₃

C₂×SD₁₆

yes

characteristic

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

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₄×Q₈

yes

—

C₈×C₁₃

#124

yes

characteristic

C₈×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

—

C₂×#108

C₂×D₁₃

yes

—

C₂×#107

—

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

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

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

Dic₂₆

yes

—

109

#107

—

110

#107

—

111

#106

—

112

#97

—

113

C₄×C₈

—

114

C₄×C₈

C₄×D₁₃

yes

—

115

C₄×C₈

D₅₂

yes

—

116

C₂×C₁₃

—

117

C₂×C₁₃

#1988

yes

characteristic

118

C₂×C₁₃

#2148

yes

—

119

C₂²×C₄

—

120

C₂²×C₄

#6106

yes

—

121

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

#6103

yes

—

127

C₂×C₈

C₂×Dic₂₆

yes

—

128

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

129

C₂×C₈

C₂×D₅₂

yes

characteristic

130

C₂×C₈

#6127

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

#6103

yes

—

136

C₄²

#6104

yes

characteristic

137

C₄²

#6104

yes

characteristic

138

C₁₆

—

139

C₁₃

#13280

yes

characteristic, complemented, Sylow

140

C₂³

C₂×#6106

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

104

—

144

C₂×C₄

#88049

yes

—

145

C₂×C₄

#88049

yes

—

146

C₂×C₄

#88178

yes

—

147

C₂×C₄

C₂×#6103

yes

characteristic

148

C₂×C₄

C₂×#6104

yes

characteristic

149

C₂×C₄

#88205

yes

—

150

C₈

—

151

C₈

—

152

C₈

C₄×Dic₂₆

yes

—

153

C₈

#88123

yes

characteristic

154

C₈

#88129

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#88049

yes

characteristic

157

C₂²

#11718681

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

104

—

161

C₄

#11718590

yes

characteristic

162

C₄

#11718665

yes

characteristic

163

C₄

#11718668

yes

—

164

C₂

—

165

C₂

#13115788

yes

characteristic

166

C₂

#13116583

yes

—

	subgroup	G/H
center	C₂²	C₂×#88049
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#13280
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#88049

commutator

C₂×C₈×C₁₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₄×C₁₃×#108

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 ◁ C₄×#108

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
C₂×#11718344	2
#13116592	2
#13112946	2
C₄×C₁₃×#108	2
#13280	13

subgroup

index

C₄×C₁₃×#108

Subgroups — #15781512

Conjugacy classes of subgroups

Complements

Maximal subgroups