Group #13115697 — subgroups

Subgroups — #13115697

Order 1664.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#88058	C₂	1	1	yes	maximal, characteristic
4	C₂×#88056	C₂	1	2	yes	maximal
5	#11718590	C₂	1	2	yes	maximal
6	C₂²×Dic₂₆	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×Dic₅₂	—	2	2	no	—
12	C₂×Dic₅₂	—	2	2	no	—
13	C₂×Dic₅₂	C₄	1	4	yes	complemented
14	C₄×C₈×C₁₃	—	2	2	no	—
15	#88071	—	2	4	no	—
16	#88058	—	2	2	no	—
17	#88056	—	2	4	no	—
18	C₂²×C₄×C₁₃	D₄	1	2	yes	—
19	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×Dic₁₃	—	2	2	no	—
21	C₂²×Dic₁₃	—	2	2	no	—
22	C₂×Dic₂₆	—	2	2	no	—
23	C₂×Dic₂₆	—	2	2	no	—
24	C₂×Dic₂₆	—	2	2	no	—
25	C₂×Dic₂₆	—	2	2	no	—
26	C₂×Dic₂₆	—	2	2	no	—
27	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₃	—	2	1	no	—
29	C₂×C₈×C₁₃	—	2	1	no	—
30	C₂×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₃	D₄	1	2	yes	—
32	C₂×C₈×C₁₃	D₄	1	2	yes	—
33	C₂×C₈×C₁₃	D₄	1	2	yes	—
34	C₄²×C₁₃	—	2	1	no	—
35	C₄²×C₁₃	—	2	1	no	—
36	C₄⋊Dic₁₃	—	2	2	no	—
37	C₄⋊Dic₁₃	—	2	2	no	—
38	Dic₅₂	—	2	8	no	—
39	#1969	—	13	1	no	maximal, Sylow
40	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃	—	2	2	no	—
42	C₂×C₄×C₁₃	—	2	2	no	—
43	C₂×C₄×C₁₃	Q₁₆	1	4	yes	complemented
44	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
50	C₂×Dic₁₃	—	2	4	no	—
51	C₂×Dic₁₃	—	4	2	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	4	2	no	—
54	C₂×Dic₁₃	—	4	4	no	—
55	Dic₂₆	—	2	4	no	—
56	Dic₂₆	—	2	4	no	—
57	Dic₂₆	—	4	2	no	—
58	Dic₂₆	—	4	2	no	—
59	C₈×C₁₃	—	2	2	no	—
60	C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
61	C₂²×Q₁₆	—	13	1	no	—
62	C₂×#107	—	13	1	no	—
63	C₂×#103	—	13	2	no	—
64	C₂×C₄×C₈	D₁₃	1	1	yes	characteristic
65	#391	—	13	2	no	—
66	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
67	C₂²×C₁₃	#121	1	1	yes	characteristic
68	C₂²×C₁₃	#127	1	1	yes	characteristic
69	C₂²×C₁₃	C₂×Q₁₆	1	2	yes	—
70	C₂²×C₁₃	#135	1	2	yes	—
71	C₄×C₁₃	—	2	4	no	—
72	C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
74	C₄×C₁₃	#121	1	1	yes	characteristic
75	C₄×C₁₃	#124	1	1	yes	characteristic
76	Dic₁₃	—	4	4	no	—
77	Dic₁₃	—	8	2	no	—
78	C₂²×Q₈	—	13	2	no	—
79	C₂×Q₁₆	—	13	4	no	—
80	C₂×Q₁₆	—	26	2	no	—
81	C₂×Q₁₆	—	26	2	no	—
82	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
84	C₂×C₄⋊C₄	—	13	2	no	—
85	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
86	#107	—	26	2	no	—
87	#103	—	26	4	no	—
88	C₄×C₈	—	2	2	no	—
89	#95	—	26	4	no	—
90	C₂×C₁₃	#390	1	1	yes	characteristic
91	C₂×C₁₃	C₄×Q₁₆	1	2	yes	—
92	C₂×C₁₃	#467	1	2	yes	—
93	C₂×C₁₃	#494	1	1	yes	characteristic
94	C₂×C₁₃	#495	1	1	yes	characteristic
95	C₂×Q₈	—	13	4	no	—
96	C₂×Q₈	—	26	2	no	—
97	C₂×Q₈	—	26	2	no	—
98	C₂×Q₈	—	26	2	no	—
99	C₂×Q₈	—	26	2	no	—
100	C₂×Q₈	—	26	2	no	—
101	C₂²×C₄	—	26	2	no	—
102	C₂²×C₄	—	26	2	no	—
103	C₂²×C₄	D₅₂	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
105	Q₁₆	—	26	8	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₄×D₁₃	1	2	yes	—
109	C₂×C₈	D₅₂	1	2	yes	—
110	C₂×C₈	D₅₂	1	2	yes	—
111	C₂×C₈	C₂²⋊D₁₃	1	2	yes	—
112	C₄⋊C₄	—	26	2	no	—
113	C₄⋊C₄	—	26	2	no	—
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₁₃	#1969	1	1	yes	characteristic, complemented, Sylow
117	C₂³	C₂×D₅₂	1	1	yes	characteristic
118	Q₈	—	26	4	no	—
119	Q₈	—	26	4	no	—
120	Q₈	—	52	2	no	—
121	Q₈	—	52	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	26	4	no	—
125	C₂×C₄	—	52	2	no	—
126	C₂×C₄	—	52	2	no	—
127	C₂×C₄	—	52	2	no	—
128	C₂×C₄	—	52	4	no	—
129	C₂×C₄	Dic₅₂	1	4	yes	complemented
130	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
133	C₂×C₄	#6127	1	1	yes	characteristic
134	C₂×C₄	#6127	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#6103	1	4	yes	—
138	C₂²	C₄×D₅₂	1	1	yes	characteristic
139	C₂²	#88128	1	1	yes	characteristic
140	C₂²	#88158	1	2	yes	—
141	C₂²	C₂×Dic₅₂	1	2	yes	—
142	C₂²	#88184	1	1	yes	characteristic
143	C₄	—	2	4	no	—
144	C₄	—	52	4	no	—
145	C₄	—	104	2	no	—
146	C₄	C₄×D₅₂	1	1	yes	characteristic
147	C₄	#88129	1	1	yes	characteristic
148	C₄	C₂×#6103	1	1	yes	characteristic
149	C₄	#88184	1	1	yes	characteristic
150	C₂	#11718357	1	1	yes	characteristic
151	C₂	C₄×Dic₅₂	1	2	yes	—
152	C₂	#11718360	1	1	yes	characteristic
153	C₂	#11718600	1	1	yes	characteristic
154	C₂	#11718777	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₂

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂

yes

maximal, characteristic

C₂×#88058

C₂

yes

maximal, characteristic

C₂×#88056

C₂

yes

maximal

#11718590

C₂

yes

maximal

C₂²×Dic₂₆

C₂²

yes

—

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂²×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

C₄

yes

complemented

C₄×C₈×C₁₃

—

#88071

—

#88058

—

#88056

—

C₂²×C₄×C₁₃

D₄

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

C₂×C₄

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₂×C₈×C₁₃

D₄

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

#1969

—

maximal, Sylow

C₂³×C₁₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

Q₁₆

yes

complemented

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

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂²⋊C₄

yes

—

C₂²×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

D₁₃

yes

characteristic

#391

—

C₂²×C₁₃

C₄×D₄

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

C₂×Q₁₆

yes

—

C₂²×C₁₃

#135

yes

—

C₄×C₁₃

—

C₄×C₁₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₃

C₄×D₄

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂²×C₈

C₂×D₁₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#107

—

#103

—

C₄×C₈

—

#95

—

C₂×C₁₃

#390

yes

characteristic

C₂×C₁₃

C₄×Q₁₆

yes

—

C₂×C₁₃

#467

yes

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

D₅₂

yes

—

104

C₂²×C₄

C₂²×D₁₃

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₄×D₁₃

yes

—

109

C₂×C₈

D₅₂

yes

—

110

C₂×C₈

D₅₂

yes

—

111

C₂×C₈

C₂²⋊D₁₃

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₁₃

#1969

yes

characteristic, complemented, Sylow

117

C₂³

C₂×D₅₂

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

Dic₅₂

yes

complemented

130

C₂×C₄

C₂×C₄×D₁₃

yes

characteristic

131

C₂×C₄

C₂×D₅₂

yes

characteristic

132

C₂×C₄

C₂×D₅₂

yes

characteristic

133

C₂×C₄

#6127

yes

characteristic

134

C₂×C₄

#6127

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

136

C₈

—

137

C₈

#6103

yes

—

138

C₂²

C₄×D₅₂

yes

characteristic

139

C₂²

#88128

yes

characteristic

140

C₂²

#88158

yes

—

141

C₂²

C₂×Dic₅₂

yes

—

142

C₂²

#88184

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

104

—

146

C₄

C₄×D₅₂

yes

characteristic

147

C₄

#88129

yes

characteristic

148

C₄

C₂×#6103

yes

characteristic

149

C₄

#88184

yes

characteristic

150

C₂

#11718357

yes

characteristic

151

C₂

C₄×Dic₅₂

yes

—

152

C₂

#11718360

yes

characteristic

153

C₂

#11718600

yes

characteristic

154

C₂

#11718777

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₂×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	C₂×C₄×C₈	D₁₃
hypocenter	C₁₃	#1969
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₅₂

commutator

C₂×C₄×C₁₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃

Fitting

C₂×C₄×C₈×C₁₃

C₂

socle, prime radical

C₂³×C₁₃

C₂×D₄

cosocle

C₂²×C₄×C₁₃

C₂³

hypercenter

C₂×C₄×C₈

D₁₃

hypocenter

C₁₃

#1969

LCS

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
13. C₂×Dic₅₂	C₄	semidirect
43. C₂×C₄×C₁₃	Q₁₆	semidirect
116. C₁₃	#1969	semidirect
129. C₂×C₄	Dic₅₂	semidirect

H ≅ G/N

type

13. C₂×Dic₅₂

C₄

semidirect

43. C₂×C₄×C₁₃

Q₁₆

semidirect

116. C₁₃

#1969

semidirect

129. C₂×C₄

Dic₅₂

semidirect

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

subgroup	index
C₂²×Dic₅₂	2
C₂×C₄×C₈×C₁₃	2
C₂×#88058	2
C₂×#88056	2
#11718590	2
#1969	13

subgroup

index

C₂²×Dic₅₂

C₂×C₄×C₈×C₁₃

Subgroups — #13115697

Conjugacy classes of subgroups

Complements

Maximal subgroups