Group #13115752 — subgroups

Subgroups — #13115752

Order 1664.

276 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#88076	C₂	1	2	yes	maximal
3	#11718886	C₂	1	2	yes	maximal
4	C₂×#88059	C₂	1	1	yes	maximal, characteristic
5	#11718762	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₁₃×Q₁₆	—	2	1	no	—
8	C₂×C₁₃×Q₁₆	—	2	1	no	—
9	C₂×C₁₃×Q₁₆	—	2	2	no	—
10	C₂×C₁₃×Q₁₆	C₄	1	4	yes	complemented
11	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#88180	C₂²	1	1	yes	characteristic
13	C₂×#6099	C₂²	1	2	yes	—
14	C₂×#6099	C₂²	1	1	yes	characteristic
15	#88076	—	2	4	no	—
16	#88075	—	2	4	no	—
17	#88059	—	2	2	no	—
18	#88055	—	2	1	no	—
19	C₈×Dic₁₃	—	2	1	no	—
20	C₂×Q₈×C₁₃	—	2	2	no	—
21	C₂×Q₈×C₁₃	—	2	2	no	—
22	C₂×Q₈×C₁₃	—	2	2	no	—
23	C₂×Q₈×C₁₃	—	2	2	no	—
24	C₂×Q₈×C₁₃	—	2	2	no	—
25	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₃	—	2	2	no	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₁₃×Q₁₆	—	2	4	no	—
29	C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²⋊Dic₁₃	—	2	1	no	—
35	C₄⋊Dic₁₃	D₄	1	2	yes	—
36	C₄×Dic₁₃	—	2	1	no	—
37	#6099	—	2	1	no	—
38	#6099	—	2	2	no	—
39	#6099	—	2	2	no	—
40	#6099	D₄	1	2	yes	—
41	C₂×#1055	—	2	1	no	—
42	C₂×#1055	—	2	2	no	—
43	#1979	—	13	1	no	maximal, Sylow
44	C₂³×C₁₃	#41	1	1	yes	characteristic
45	Q₈×C₁₃	—	2	4	no	—
46	Q₈×C₁₃	—	2	4	no	—
47	Q₈×C₁₃	—	4	2	no	—
48	Q₈×C₁₃	—	4	2	no	—
49	C₂×C₄×C₁₃	—	2	4	no	—
50	C₂×C₄×C₁₃	—	4	2	no	—
51	C₂×C₄×C₁₃	—	4	2	no	—
52	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₂×Dic₁₃	—	2	2	no	—
59	C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
60	C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	#1055	—	2	2	no	—
62	#1055	—	4	2	no	—
63	C₂²×Q₁₆	D₁₃	1	1	yes	characteristic
64	C₂×#108	—	13	1	no	—
65	#419	—	13	2	no	—
66	C₂×#101	—	13	2	no	—
67	#405	—	13	1	no	—
68	C₂²×C₁₃	—	2	2	no	—
69	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
70	C₂²×C₁₃	#121	1	1	yes	characteristic
71	C₂²×C₁₃	#124	1	1	yes	characteristic
72	C₄×C₁₃	—	4	4	no	—
73	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₄×C₁₃	#121	1	1	yes	characteristic
76	C₄×C₁₃	#127	1	1	yes	characteristic
77	Dic₁₃	—	4	2	no	—
78	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₂×Q₁₆	—	2	1	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	Dic₁₃	1	4	yes	complemented
83	C₂×M₁₆	—	13	1	no	—
84	C₂×M₁₆	—	13	2	no	—
85	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
86	#117	—	13	1	no	—
87	#108	—	26	2	no	—
88	#103	—	26	4	no	—
89	#101	—	26	4	no	—
90	#97	—	26	1	no	—
91	C₄×C₈	—	26	1	no	—
92	C₂×C₁₃	—	2	1	no	—
93	C₂×C₁₃	—	2	1	no	—
94	C₂×C₁₃	#390	1	1	yes	characteristic
95	C₂×C₁₃	#473	1	2	yes	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₁₃	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
104	Q₁₆	—	2	4	no	—
105	Q₁₆	—	2	4	no	—
106	M₁₆	—	13	2	no	—
107	M₁₆	—	26	1	no	—
108	M₁₆	—	26	2	no	—
109	M₁₆	—	26	2	no	—
110	C₂×C₈	—	26	1	no	—
111	C₂×C₈	—	26	2	no	—
112	C₂×C₈	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
116	C₄⋊C₄	—	13	2	no	—
117	C₂²⋊C₄	—	26	1	no	—
118	C₄²	—	26	1	no	—
119	C₁₃	#1979	1	1	yes	characteristic, complemented, Sylow
120	C₂³	#6129	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	26	2	no	—
129	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
131	C₂×C₄	#6129	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
135	C₈	—	26	2	no	—
136	C₈	—	52	2	no	—
137	C₈	C₂²⋊Dic₁₃	1	2	yes	—
138	C₈	C₂²⋊Dic₁₃	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	D₄×Dic₁₃	1	1	yes	characteristic
141	C₂²	#88190	1	1	yes	characteristic
142	C₂²	#88192	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	52	2	no	—
145	C₄	D₄×Dic₁₃	1	1	yes	characteristic
146	C₄	#88192	1	1	yes	characteristic
147	C₄	#88194	1	1	yes	characteristic
148	C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#11718853	1	2	yes	—
152	C₂	#11718882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₃×Q₁₆

C₂

yes

maximal, characteristic

C₂×#88076

C₂

yes

maximal

#11718886

C₂

yes

maximal

C₂×#88059

C₂

yes

maximal, characteristic

#11718762

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

C₄

yes

complemented

C₂²×C₈×C₁₃

C₂²

yes

characteristic

#88180

C₂²

yes

characteristic

C₂×#6099

C₂²

yes

—

C₂×#6099

C₂²

yes

—

—

—

—

C₈×Dic₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

C₂×C₄

yes

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

D₄

yes

characteristic

C₂×C₈×C₁₃

D₄

yes

characteristic

C₂×C₈×C₁₃

D₄

yes

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

D₄

yes

—

C₄×Dic₁₃

—

—

—

—

D₄

yes

—

—

—

—

C₂³×C₁₃

yes

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

#41

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

C₂²⋊C₄

yes

—

C₈×C₁₃

C₂²⋊C₄

yes

—

#1055

—

#1055

—

C₂²×Q₁₆

D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄×D₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

C₄×D₄

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₁₃

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₁₃

yes

—

—

—

—

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

#473

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₁₃

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₁₃

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×Dic₁₃

yes

—

113

C₂×C₈

C₂²⋊D₁₃

yes

characteristic

114

C₂×C₈

C₂²⋊D₁₃

yes

characteristic

115

C₂×C₈

C₂²⋊D₁₃

yes

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₁₃

#1979

yes

characteristic, complemented, Sylow

120

C₂³

#6129

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₁₃

yes

characteristic

130

C₂×C₄

D₄×D₁₃

yes

characteristic

131

C₂×C₄

#6129

yes

characteristic

132

C₂×C₄

C₂²×Dic₁₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

C₂²⋊Dic₁₃

yes

—

138

C₈

C₂²⋊Dic₁₃

yes

—

139

C₂²

—

140

C₂²

D₄×Dic₁₃

yes

characteristic

141

C₂²

#88190

yes

characteristic

142

C₂²

#88192

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

D₄×Dic₁₃

yes

characteristic

146

C₄

#88192

yes

characteristic

147

C₄

#88194

yes

characteristic

148

C₄

C₂×C₂²⋊Dic₁₃

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#11718853

yes

—

152

C₂

#11718882

yes

characteristic

	subgroup	G/H
center	C₂²	#88192
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂²×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	#121
prime radical	C₂³×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂²×Q₁₆	D₁₃
hypocenter	C₁₃	#1979
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#88192

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂²×C₁₃×Q₁₆

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂²×Q₁₆

D₁₃

hypocenter

C₁₃

#1979

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
10. C₂×C₁₃×Q₁₆	C₄	semidirect
82. C₂×Q₁₆	Dic₁₃	semidirect
119. C₁₃	#1979	semidirect

H ≅ G/N

type

10. C₂×C₁₃×Q₁₆

C₄

semidirect

82. C₂×Q₁₆

Dic₁₃

semidirect

119. C₁₃

#1979

semidirect

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

subgroup	index
C₂²×C₁₃×Q₁₆	2
C₂×#88076	2
#11718886	2
C₂×#88059	2
#11718762	2
#1979	13

subgroup

index

C₂²×C₁₃×Q₁₆

Subgroups — #13115752

Conjugacy classes of subgroups

Complements

Maximal subgroups