Group #15750152 — subgroups

Subgroups — #15750152

Order 3328.

192 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13113595	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
5	#11718586	—	2	1	no	—
6	#11718585	—	2	1	no	—
7	C₁₃×C₄⋊C₁₆	—	2	1	no	—
8	C₁₃×C₂²⋊C₁₆	C₄	1	2	yes	—
9	C₄×C₁₃×C₁₆	—	2	1	no	—
10	#11718050	—	2	1	no	—
11	#11718050	C₄	1	2	yes	—
12	#11718050	C₄	1	2	yes	—
13	#11718049	—	2	1	no	—
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
17	C₂×#6099	—	4	1	no	—
18	C₂×C₁₃×C₁₆	—	2	2	no	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²⋊#1055	—	4	1	no	—
24	C₄⋊#1055	—	2	2	no	—
25	C₄⋊#1055	—	2	2	no	—
26	C₄⋊#1055	—	4	1	no	—
27	C₄×#1055	—	4	1	no	—
28	#7291	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	—	4	1	no	—
34	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₁₃	D₈	1	1	yes	characteristic
41	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
42	#6099	—	8	2	no	—
43	C₂×#1055	—	4	2	no	—
44	C₂×#1055	—	4	2	no	—
45	C₁₃×C₁₆	—	4	2	no	—
46	#2162	D₁₃	1	1	yes	characteristic
47	#1567	—	13	1	no	—
48	#1564	—	13	1	no	—
49	C₂³×C₁₃	#106	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
61	C₈×C₁₃	—	2	1	no	—
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	#108	1	2	yes	—
65	#1055	—	8	2	no	—
66	#1055	—	8	2	no	—
67	#431	—	26	1	no	—
68	#423	—	26	1	no	—
69	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
70	C₄⋊C₁₆	—	2	1	no	—
71	C₂²⋊C₁₆	Dic₁₃	1	2	yes	—
72	C₄×C₁₆	—	2	1	no	—
73	#335	—	13	2	no	—
74	#335	—	13	2	no	—
75	#335	—	26	1	no	—
76	#334	—	26	1	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	#340	1	1	yes	characteristic
81	C₄×C₁₃	—	2	1	no	—
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	#341	1	1	yes	characteristic
85	C₄×C₁₃	#344	1	1	yes	characteristic
86	C₄×C₁₃	#359	1	2	yes	—
87	C₄×C₁₃	#361	1	1	yes	characteristic
88	C₄×C₁₃	#362	1	1	yes	characteristic
89	C₂×M₁₆	—	52	1	no	—
90	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
91	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
92	C₂×C₄²	D₅₂	1	1	yes	characteristic
93	C₂×C₁₆	—	2	2	no	—
94	C₄⋊C₈	—	26	2	no	—
95	C₄⋊C₈	—	26	2	no	—
96	C₄⋊C₈	—	52	1	no	—
97	C₂²⋊C₈	—	52	1	no	—
98	C₄×C₈	—	52	1	no	—
99	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
100	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
101	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
103	C₂×C₁₃	—	4	1	no	—
104	C₂×C₁₃	#1301	1	1	yes	characteristic
105	C₂×C₁₃	#1382	1	1	yes	characteristic
106	C₂×C₁₃	#1387	1	1	yes	characteristic
107	C₂²×C₄	#6095	1	1	yes	characteristic
108	C₂²×C₄	#6095	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
110	M₁₆	—	104	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	52	2	no	—
114	C₂×C₈	—	52	2	no	—
115	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
116	C₂×C₈	#6101	1	1	yes	characteristic
117	C₂×C₈	#6101	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
119	C₄²	D₁₀₄	1	1	yes	characteristic
120	C₄²	Dic₅₂	1	1	yes	characteristic
121	C₄²	#6103	1	1	yes	characteristic
122	C₄²	#6103	1	1	yes	characteristic
123	C₁₆	—	4	2	no	—
124	C₁₃	#7291	1	1	yes	characteristic, complemented, Sylow
125	C₂³	#88057	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	#88056	1	1	yes	characteristic
132	C₂×C₄	#88056	1	1	yes	characteristic
133	C₂×C₄	#88058	1	1	yes	characteristic
134	C₂×C₄	#88061	1	1	yes	characteristic
135	C₂×C₄	#88061	1	1	yes	characteristic
136	C₂×C₄	#88071	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	104	2	no	—
141	C₈	—	104	2	no	—
142	C₈	#88062	1	2	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#11718143	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#11718102	1	2	yes	—
151	C₄	#11718108	1	1	yes	characteristic
152	C₄	#11718109	1	1	yes	characteristic
153	C₄	#11718125	1	1	yes	characteristic
154	C₄	#11718150	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	#13112732	1	1	yes	characteristic
157	C₂	#13112932	1	1	yes	characteristic
158	C₂	#13112945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13113595

C₂

yes

maximal, characteristic

#13113584

C₂

yes

maximal, characteristic

C₁₃×#2162

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718586

—

#11718585

—

C₁₃×C₄⋊C₁₆

—

C₁₃×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₃×C₁₆

—

#11718050

—

#11718050

C₄

yes

—

#11718050

C₄

yes

—

#11718049

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

Q₈

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×#6099

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₂²⋊#1055

—

—

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

SD₁₆

yes

characteristic

C₂²×C₄×C₁₃

SD₁₆

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

C₂²⋊C₄

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄²×C₁₃

Q₁₆

yes

—

—

—

C₁₃×C₁₆

—

#2162

D₁₃

yes

characteristic

#1567

—

#1564

—

C₂³×C₁₃

#106

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#108

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₁₃

yes

—

C₄×C₁₆

—

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#340

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×M₁₆

—

C₂²×C₈

Dic₂₆

yes

characteristic

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

D₅₂

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₁₃

yes

characteristic

100

C₄×C₈

C₄×D₁₃

yes

characteristic

101

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

102

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

103

C₂×C₁₃

—

104

C₂×C₁₃

#1301

yes

characteristic

105

C₂×C₁₃

#1382

yes

characteristic

106

C₂×C₁₃

#1387

yes

characteristic

107

C₂²×C₄

#6095

yes

characteristic

108

C₂²×C₄

#6095

yes

characteristic

109

C₂²×C₄

C₄⋊Dic₁₃

yes

characteristic

110

M₁₆

—

104

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₄×Dic₁₃

yes

characteristic

116

C₂×C₈

#6101

yes

characteristic

117

C₂×C₈

#6101

yes

characteristic

118

C₂×C₈

C₂²⋊Dic₁₃

yes

characteristic

119

C₄²

D₁₀₄

yes

characteristic

120

C₄²

Dic₅₂

yes

characteristic

121

C₄²

#6103

yes

characteristic

122

C₄²

#6103

yes

characteristic

123

C₁₆

—

124

C₁₃

#7291

yes

characteristic, complemented, Sylow

125

C₂³

#88057

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#88056

yes

characteristic

132

C₂×C₄

#88056

yes

characteristic

133

C₂×C₄

#88058

yes

characteristic

134

C₂×C₄

#88061

yes

characteristic

135

C₂×C₄

#88061

yes

characteristic

136

C₂×C₄

#88071

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

104

—

141

C₈

—

104

—

142

C₈

#88062

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#11718143

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#11718102

yes

—

151

C₄

#11718108

yes

characteristic

152

C₄

#11718109

yes

characteristic

153

C₄

#11718125

yes

characteristic

154

C₄

#11718150

yes

characteristic

155

C₂

—

156

C₂

#13112732

yes

characteristic

157

C₂

#13112932

yes

characteristic

158

C₂

#13112945

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#88061
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	#340
prime radical	C₂³×C₁₃	#106
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	#2162	D₁₃
hypocenter	C₁₃	#7291
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

subgroup

G/H

center

C₂×C₄

#88061

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₁₃

Fitting

C₁₃×#2162

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×C₈×C₁₃

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

N	H ≅ G/N	type
124. C₁₃	#7291	semidirect

H ≅ G/N

type

124. C₁₃

#7291

semidirect

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

subgroup	index
#13113595	2
#13113584	2
C₁₃×#2162	2
#7291	13

subgroup

index

#13113595

#13113584

C₁₃×#2162

#7291

Subgroups — #15750152

Conjugacy classes of subgroups

Complements

Maximal subgroups