Group #15749961 — subgroups

Subgroups — #15749961

Order 3328.

196 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13116635	C₂	1	1	yes	maximal, characteristic
2	#13113592	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1837	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#423	—	2	1	no	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	#11718585	—	2	1	no	—
7	C₁₃×#341	—	2	1	no	—
8	C₁₃×#341	C₄	1	2	yes	—
9	C₂²⋊#6090	—	2	1	no	—
10	C₄⋊#6090	—	2	1	no	—
11	C₄×#6090	C₄	1	2	yes	—
12	#11718049	—	2	1	no	—
13	#11718049	C₄	1	2	yes	—
14	C₂×C₁₃×M₁₆	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×#6099	—	4	1	no	—
19	C₁₃×C₄⋊C₈	—	4	1	no	—
20	C₁₃×C₄⋊C₈	—	4	1	no	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
25	C₂×#6090	—	2	2	no	—
26	C₄⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	4	1	no	—
28	#7219	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
32	C₁₃×M₁₆	—	4	2	no	—
33	C₁₃×M₁₆	—	4	2	no	—
34	C₂×C₈×C₁₃	—	4	1	no	—
35	C₂×C₈×C₁₃	—	4	1	no	—
36	C₂×C₈×C₁₃	—	4	2	no	—
37	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₃	D₈	1	1	yes	characteristic
44	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
45	#6099	—	8	2	no	—
46	C₂×#1055	—	4	2	no	—
47	#6090	—	4	2	no	—
48	#2162	—	13	1	no	—
49	#1837	D₁₃	1	1	yes	characteristic
50	#1566	—	13	1	no	—
51	C₂³×C₁₃	#103	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	1	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
62	C₈×C₁₃	—	2	1	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	—	2	1	no	—
65	C₈×C₁₃	—	2	1	no	—
66	C₈×C₁₃	—	8	2	no	—
67	#1055	—	8	2	no	—
68	#423	—	2	1	no	—
69	#423	—	26	1	no	—
70	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
71	C₄⋊C₁₆	—	26	1	no	—
72	C₂²⋊C₁₆	—	26	1	no	—
73	C₄×C₁₆	—	13	2	no	—
74	#341	—	2	1	no	—
75	#341	Dic₁₃	1	2	yes	—
76	#334	—	13	2	no	—
77	#334	—	26	1	no	—
78	C₂²×C₁₃	—	2	1	no	—
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	—	2	1	no	—
81	C₂²×C₁₃	#340	1	1	yes	characteristic
82	C₄×C₁₃	—	2	1	no	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	2	1	no	—
85	C₄×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
86	C₄×C₁₃	#344	1	1	yes	characteristic
87	C₄×C₁₃	#365	1	2	yes	—
88	C₄×C₁₃	#368	1	2	yes	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	52	1	no	—
91	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
93	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
94	C₂×C₁₆	—	26	2	no	—
95	C₄⋊C₈	—	4	1	no	—
96	C₄⋊C₈	—	4	1	no	—
97	C₄⋊C₈	—	26	2	no	—
98	C₄⋊C₈	—	52	1	no	—
99	C₄×C₈	Dic₂₆	1	1	yes	characteristic
100	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
101	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
102	C₄×C₈	D₅₂	1	1	yes	characteristic
103	C₂×C₁₃	—	4	1	no	—
104	C₂×C₁₃	#1286	1	1	yes	characteristic
105	C₂×C₁₃	#1381	1	1	yes	characteristic
106	C₂×C₁₃	#1383	1	1	yes	characteristic
107	C₂²×C₄	#6106	1	1	yes	characteristic
108	C₂²×C₄	#6107	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
110	M₁₆	—	4	2	no	—
111	M₁₆	—	4	2	no	—
112	M₁₆	—	104	2	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	52	2	no	—
117	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
118	C₂×C₈	#6101	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
120	C₂×C₈	#6103	1	1	yes	characteristic
121	C₄²	#6101	1	1	yes	characteristic
122	C₄²	#6103	1	1	yes	characteristic
123	C₄²	#6104	1	1	yes	characteristic
124	C₄²	#6105	1	1	yes	characteristic
125	C₁₆	—	52	2	no	—
126	C₁₃	#7219	1	1	yes	characteristic, complemented, Sylow
127	C₂³	#88075	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	#88047	1	1	yes	characteristic
133	C₂×C₄	#88048	1	1	yes	characteristic
134	C₂×C₄	#88049	1	1	yes	characteristic
135	C₂×C₄	#88050	1	1	yes	characteristic
136	C₂×C₄	#88071	1	1	yes	characteristic
137	C₂×C₄	#88072	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	104	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#11718121	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#11718085	1	2	yes	—
152	C₄	#11718086	1	2	yes	—
153	C₄	#11718123	1	1	yes	characteristic
154	C₄	#11718125	1	1	yes	characteristic
155	C₂	—	4	1	no	—
156	C₂	#13112694	1	1	yes	characteristic
157	C₂	#13112929	1	1	yes	characteristic
158	C₂	#13112936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116635

C₂

yes

maximal, characteristic

#13113592

C₂

yes

maximal, characteristic

C₁₃×#1837

C₂

yes

maximal, characteristic

C₁₃×#423

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

#11718585

—

C₁₃×#341

—

C₁₃×#341

C₄

yes

—

C₂²⋊#6090

—

C₄⋊#6090

—

C₄×#6090

C₄

yes

—

#11718049

—

#11718049

C₄

yes

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

characteristic

C₂×#6099

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

Q₈

yes

—

—

—

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

SD₁₆

yes

characteristic

C₂²×C₄×C₁₃

Q₁₆

yes

characteristic

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

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

C₄⋊C₄

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄²×C₁₃

SD₁₆

yes

—

—

—

—

D₁₃

yes

characteristic

#1566

—

C₂³×C₁₃

#103

yes

characteristic

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

—

C₈×C₁₃

—

#1055

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#341

—

#341

Dic₁₃

yes

—

#334

—

#334

—

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

#368

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₁₃

yes

characteristic

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

C₂×C₄²

C₂²⋊D₁₃

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

Dic₂₆

yes

characteristic

100

C₄×C₈

C₄×D₁₃

yes

characteristic

101

C₄×C₈

C₄×D₁₃

yes

characteristic

102

C₄×C₈

D₅₂

yes

characteristic

103

C₂×C₁₃

—

104

C₂×C₁₃

#1286

yes

characteristic

105

C₂×C₁₃

#1381

yes

characteristic

106

C₂×C₁₃

#1383

yes

characteristic

107

C₂²×C₄

#6106

yes

characteristic

108

C₂²×C₄

#6107

yes

characteristic

109

C₂²×C₄

C₂²⋊Dic₁₃

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

104

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₄×Dic₁₃

yes

characteristic

118

C₂×C₈

#6101

yes

characteristic

119

C₂×C₈

C₄⋊Dic₁₃

yes

characteristic

120

C₂×C₈

#6103

yes

characteristic

121

C₄²

#6101

yes

characteristic

122

C₄²

#6103

yes

characteristic

123

C₄²

#6104

yes

characteristic

124

C₄²

#6105

yes

characteristic

125

C₁₆

—

126

C₁₃

#7219

yes

characteristic, complemented, Sylow

127

C₂³

#88075

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#88047

yes

characteristic

133

C₂×C₄

#88048

yes

characteristic

134

C₂×C₄

#88049

yes

characteristic

135

C₂×C₄

#88050

yes

characteristic

136

C₂×C₄

#88071

yes

characteristic

137

C₂×C₄

#88072

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

104

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11718121

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#11718085

yes

—

152

C₄

#11718086

yes

—

153

C₄

#11718123

yes

characteristic

154

C₄

#11718125

yes

characteristic

155

C₂

—

156

C₂

#13112694

yes

characteristic

157

C₂

#13112929

yes

characteristic

158

C₂

#13112936

yes

characteristic

	subgroup	G/H
center	C₂²	#11718121
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₁₃
Fitting	C₁₃×#1837	C₂
socle	C₂²×C₁₃	#340
prime radical	C₂³×C₁₃	#103
cosocle	C₂×C₄×C₈×C₁₃	C₂²
hypercenter	#1837	D₁₃
hypocenter	C₁₃	#7219
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1837

subgroup

G/H

center

C₂²

#11718121

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₁₃

Fitting

C₁₃×#1837

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×C₈×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1837

N	H ≅ G/N	type
126. C₁₃	#7219	semidirect

H ≅ G/N

type

126. C₁₃

#7219

semidirect

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

subgroup	index
#13116635	2
#13113592	2
C₁₃×#1837	2
#7219	13

subgroup

index

#13116635

#13113592

C₁₃×#1837

#7219

Subgroups — #15749961

Conjugacy classes of subgroups

Complements

Maximal subgroups