Group #15756387 — subgroups

Subgroups — #15756387

Order 3328.

418 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 115 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×#11718120	C₂	1	2	yes	maximal
2	#13112684	C₂	1	4	yes	maximal
3	C₁₃×#2313	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
5	C₁₃×#387	—	2	4	no	—
6	C₁₃×#381	C₄	1	2	yes	—
7	C₁₃×#380	C₄	1	2	yes	—
8	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
9	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
10	#11718120	—	2	4	no	—
11	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
16	C₂×C₁₃×C₄⋊C₄	D₄	1	4	yes	—
17	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
18	C₂×C₄²×C₁₃	—	2	2	no	—
19	C₂²×#1055	—	2	4	no	—
20	C₁₃×#95	—	2	2	no	—
21	C₁₃×#95	—	2	4	no	—
22	C₁₃×#95	—	2	2	no	—
23	C₂²⋊#1055	—	2	4	no	—
24	C₂²⋊#1055	—	2	4	no	—
25	#8737	—	13	1	no	maximal, Sylow
26	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	—	2	2	no	—
28	C₂²×C₄×C₁₃	—	2	4	no	—
29	C₂²×C₄×C₁₃	—	2	2	no	—
30	C₂²×C₄×C₁₃	—	2	4	no	—
31	C₂²×C₄×C₁₃	—	8	1	no	—
32	C₂²×C₄×C₁₃	—	8	1	no	—
33	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₁₃	#41	1	4	yes	—
37	C₁₃×C₄⋊C₄	—	2	8	no	—
38	C₁₃×C₄⋊C₄	—	8	2	no	—
39	C₁₃×C₂²⋊C₄	—	2	8	no	—
40	C₄²×C₁₃	—	8	2	no	—
41	C₂×#1055	—	4	4	no	—
42	C₂×#1055	—	8	4	no	—
43	#2313	D₁₃	1	1	yes	characteristic
44	C₂×#324	—	13	2	no	—
45	#1282	—	13	4	no	—
46	C₂³×C₁₃	—	4	1	no	—
47	C₂³×C₁₃	—	4	1	no	—
48	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂³×C₁₃	#117	1	2	yes	—
50	C₂³×C₁₃	#131	1	4	yes	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	—	4	4	no	—
55	C₂×C₄×C₁₃	—	4	2	no	—
56	C₂×C₄×C₁₃	—	4	4	no	—
57	C₂×C₄×C₁₃	—	4	2	no	—
58	C₂×C₄×C₁₃	—	4	4	no	—
59	C₂×C₄×C₁₃	—	4	4	no	—
60	C₂×C₄×C₁₃	—	4	4	no	—
61	C₂×C₄×C₁₃	—	8	2	no	—
62	C₂×C₄×C₁₃	#123	1	4	yes	—
63	#1055	—	8	4	no	—
64	C₂²×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
65	C₂×C₂²⋊C₈	—	13	4	no	—
66	#387	—	2	4	no	—
67	#381	Dic₁₃	1	2	yes	—
68	#380	Dic₁₃	1	2	yes	—
69	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
70	#324	—	26	4	no	—
71	C₂²×C₁₃	—	2	1	no	—
72	C₂²×C₁₃	—	2	2	no	—
73	C₂²×C₁₃	—	2	1	no	—
74	C₂²×C₁₃	—	2	2	no	—
75	C₂²×C₁₃	—	4	4	no	—
76	C₂²×C₁₃	#381	1	1	yes	characteristic
77	C₂²×C₁₃	#408	1	2	yes	—
78	C₂²×C₁₃	#433	1	4	yes	—
79	C₄×C₁₃	—	2	4	no	—
80	C₄×C₁₃	—	4	2	no	—
81	C₄×C₁₃	—	4	2	no	—
82	C₄×C₁₃	—	8	4	no	—
83	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
84	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
85	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
86	C₂²×C₈	—	26	4	no	—
87	C₂×C₄⋊C₄	—	2	4	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
90	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
91	C₂×C₄²	—	2	2	no	—
92	C₂²⋊C₈	—	26	4	no	—
93	C₂²⋊C₈	—	26	4	no	—
94	#95	—	2	2	no	—
95	#95	—	2	4	no	—
96	#95	—	2	2	no	—
97	C₂×C₁₃	—	2	2	no	—
98	C₂×C₁₃	—	2	4	no	—
99	C₂×C₁₃	#1818	1	1	yes	characteristic
100	C₂×C₁₃	#1828	1	1	yes	characteristic
101	C₂×C₁₃	#1830	1	1	yes	characteristic
102	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	8	1	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
110	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
111	C₂²×C₄	#6127	1	4	yes	—
112	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
113	C₂×C₈	—	52	4	no	—
114	C₂×C₈	—	104	4	no	—
115	C₄⋊C₄	—	2	8	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₂²⋊C₄	—	2	8	no	—
118	C₄²	—	8	2	no	—
119	C₁₃	#8737	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	4	1	no	—
121	C₂³	—	4	1	no	—
122	C₂³	#88180	1	2	yes	—
123	C₂³	#88202	1	4	yes	—
124	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	#88183	1	4	yes	—
137	C₈	—	104	4	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	#11718661	1	4	yes	—
144	C₂²	#11718872	1	1	yes	characteristic
145	C₂²	#11718895	1	2	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	4	no	—
150	C₂	—	2	2	no	—
151	C₂	—	2	4	no	—
152	C₂	#13114907	1	1	yes	characteristic
153	C₂	#13114944	1	1	yes	characteristic
154	C₂	#13114950	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718120

C₂

yes

maximal

#13112684

C₂

yes

maximal

C₁₃×#2313

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₁₃×#387

—

C₁₃×#381

C₄

yes

—

C₁₃×#380

C₄

yes

—

C₄×C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊#1055

C₂²

yes

—

#11718120

—

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₁₃×C₂²⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₁₃

—

C₂²×#1055

—

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

—

C₂²⋊#1055

—

C₂²⋊#1055

—

#8737

—

maximal, Sylow

C₂⁴×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

#41

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₂²⋊C₄

—

C₄²×C₁₃

—

C₂×#1055

—

C₂×#1055

—

#2313

D₁₃

yes

characteristic

C₂×#324

—

#1282

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₁₃

#117

yes

—

C₂³×C₁₃

#131

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#123

yes

—

#1055

—

C₂²×C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₂×C₂²⋊C₈

—

#387

—

#381

Dic₁₃

yes

—

#380

Dic₁₃

yes

—

C₄×C₂²⋊C₄

C₂×D₁₃

yes

—

#324

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#381

yes

characteristic

C₂²×C₁₃

#408

yes

—

C₂²×C₁₃

#433

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂³×C₄

C₂×Dic₁₃

yes

characteristic

C₂³×C₄

C₂×Dic₁₃

yes

characteristic

C₂³×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₂²⋊C₄

C₂×Dic₁₃

yes

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#95

—

#95

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#1818

yes

characteristic

100

C₂×C₁₃

#1828

yes

characteristic

101

C₂×C₁₃

#1830

yes

characteristic

102

C₂⁴

C₂²×Dic₁₃

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

110

C₂²×C₄

C₂²⋊Dic₁₃

yes

—

111

C₂²×C₄

#6127

yes

—

112

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

104

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₁₃

#8737

yes

characteristic, complemented, Sylow

120

C₂³

—

121

C₂³

—

122

C₂³

#88180

yes

—

123

C₂³

#88202

yes

—

124

C₂³

C₂×C₂²⋊Dic₁₃

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#88183

yes

—

137

C₈

—

104

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11718661

yes

—

144

C₂²

#11718872

yes

characteristic

145

C₂²

#11718895

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

#13114907

yes

characteristic

153

C₂

#13114944

yes

characteristic

154

C₂

#13114950

yes

characteristic

	subgroup	G/H
center	C₂²	#11718872
commutator, prime radical	C₂⁴×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#2313	C₂
socle	C₂²×C₁₃	#381
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#2313	D₁₃
hypocenter	C₁₃	#8737
LCS	G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2313

subgroup

G/H

center

C₂²

#11718872

commutator, prime radical

C₂⁴×C₁₃

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₁₃

Fitting

C₁₃×#2313

C₂

socle

C₂²×C₁₃

#381

cosocle

C₂³×C₄×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂⁴×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2313

N	H ≅ G/N	type
119. C₁₃	#8737	semidirect

H ≅ G/N

type

119. C₁₃

#8737

semidirect

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

subgroup	index
C₂×#11718120	2
#13112684	2
C₁₃×#2313	2
#8737	13

subgroup

index

C₂×#11718120

#13112684

C₁₃×#2313

#8737

Subgroups — #15756387

Conjugacy classes of subgroups

Complements

Maximal subgroups