Group #15753370 — subgroups

Subgroups — #15753370

Order 3328.

330 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 139 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	#13114522	C₂	1	1	yes	maximal, characteristic
2	#13113128	C₂	1	1	yes	maximal, characteristic
3	#13112666	C₂	1	2	yes	maximal
4	#13112664	C₂	1	2	yes	maximal
5	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
10	#11718593	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	2	yes	—
12	C₂×#88043	C₂²	1	1	yes	characteristic
13	C₁₃×#322	C₄	1	4	yes	complemented
14	#11718046	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₂²×#1055	C₂×C₄	1	2	yes	—
20	C₁₃×C₄⋊C₈	—	2	2	no	—
21	C₁₃×C₂²⋊C₈	—	2	2	no	—
22	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	#88071	—	2	2	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	#88055	—	2	4	no	—
28	C₈×Dic₁₃	—	2	4	no	—
29	C₄⋊#1055	—	2	2	no	—
30	#88043	—	2	2	no	—
31	C₄×#1055	C₂×C₄	1	4	yes	—
32	#8140	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	#41	1	2	yes	—
35	C₂²×Dic₁₃	—	2	2	no	—
36	C₂×C₈×C₁₃	—	2	2	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	4	no	—
39	C₂×C₈×C₁₃	C₄²	1	2	yes	—
40	C₂×C₈×C₁₃	C₄²	1	2	yes	—
41	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₁₃	#41	1	2	yes	—
44	C₄×Dic₁₃	—	2	2	no	—
45	C₄×Dic₁₃	M₁₆	1	4	yes	—
46	C₂×#1055	—	2	2	no	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	—	2	4	no	—
49	C₂×#1055	M₁₆	1	4	yes	—
50	#1726	—	13	1	no	—
51	#1452	—	13	1	no	—
52	#1452	D₁₃	1	1	yes	characteristic
53	#1275	—	13	2	no	—
54	#1274	—	13	2	no	—
55	C₂³×C₁₃	#117	1	1	yes	characteristic
56	C₂×C₄×C₁₃	—	4	2	no	—
57	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#117	1	2	yes	—
59	C₂×C₄×C₁₃	#117	1	2	yes	—
60	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
63	C₂×Dic₁₃	—	2	4	no	—
64	C₂×Dic₁₃	—	4	4	no	—
65	C₈×C₁₃	—	2	2	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	—	4	4	no	—
68	#1055	—	2	4	no	—
69	#1055	—	4	4	no	—
70	#431	—	13	2	no	—
71	#431	C₂×D₁₃	1	2	yes	—
72	C₂×#97	—	13	2	no	—
73	C₂×#97	—	13	1	no	—
74	C₂×C₄×C₈	—	13	2	no	—
75	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
76	#376	—	13	1	no	—
77	#322	—	13	4	no	—
78	#322	Dic₁₃	1	4	yes	complemented
79	C₂²×C₁₃	—	2	2	no	—
80	C₂²×C₁₃	#376	1	1	yes	characteristic
81	C₂²×C₁₃	#432	1	2	yes	—
82	C₄×C₁₃	—	2	2	no	—
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
86	C₄×C₁₃	#432	1	2	yes	—
87	Dic₁₃	—	4	4	no	—
88	C₂²×C₈	—	13	2	no	—
89	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
90	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
91	C₂×C₄²	—	13	2	no	—
92	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	26	2	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₂²⋊C₈	—	26	2	no	—
97	#97	—	26	4	no	—
98	#97	—	26	2	no	—
99	C₄×C₈	—	13	4	no	—
100	C₄×C₈	—	26	4	no	—
101	C₄×C₈	C₄×D₁₃	1	2	yes	—
102	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
103	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
104	#95	—	26	2	no	—
105	C₂×C₁₃	—	2	2	no	—
106	C₂×C₁₃	#1745	1	1	yes	characteristic
107	C₂×C₁₃	#1747	1	2	yes	—
108	C₂²×C₄	—	26	2	no	—
109	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
110	C₂²×C₄	#6127	1	2	yes	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	13	4	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	—	26	2	no	—
117	C₂×C₈	—	26	4	no	—
118	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
119	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
120	C₄²	—	13	4	no	—
121	C₄²	—	26	2	no	—
122	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
123	C₄²	#6127	1	2	yes	—
124	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
125	C₁₃	#8140	1	1	yes	characteristic, complemented, Sylow
126	C₂³	#88126	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	26	4	no	—
129	C₂×C₄	—	52	4	no	—
130	C₂×C₄	#88126	1	2	yes	—
131	C₂×C₄	#88126	1	1	yes	characteristic
132	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
133	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
134	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
135	C₂×C₄	#88180	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	4	4	no	—
139	C₈	—	26	4	no	—
140	C₈	—	52	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#11718366	1	2	yes	—
143	C₂²	#11718593	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	52	4	no	—
147	C₄	#11718366	1	2	yes	—
148	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
149	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
150	C₂	—	2	2	no	—
151	C₂	#13114584	1	1	yes	characteristic
152	C₂	#13114591	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114522

C₂

yes

maximal, characteristic

#13113128

C₂

yes

maximal, characteristic

#13112666

C₂

yes

maximal

#13112664

C₂

yes

maximal

C₁₃×#1452

C₂

yes

maximal, characteristic

C₁₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88055

C₄

yes

—

C₂×C₈×Dic₁₃

C₄

yes

—

#11718593

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₂×#88043

C₂²

yes

characteristic

C₁₃×#322

C₄

yes

complemented

#11718046

C₄

yes

complemented

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

#88071

—

C₂²⋊#1055

—

#88055

—

C₈×Dic₁₃

—

C₄⋊#1055

—

#88043

—

C₄×#1055

C₂×C₄

yes

—

#8140

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

—

C₂×C₈×C₁₃

C₄²

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

M₁₆

yes

—

—

—

—

M₁₆

yes

—

#1726

—

#1452

—

#1452

D₁₃

yes

characteristic

#1275

—

#1274

—

C₂³×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#117

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

C₂×D₁₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

Dic₁₃

yes

complemented

C₂²×C₁₃

—

C₂²×C₁₃

#376

yes

characteristic

C₂²×C₁₃

#432

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄×M₁₆

yes

characteristic

C₄×C₁₃

C₄×M₁₆

yes

characteristic

C₄×C₁₃

#432

yes

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₄×D₁₃

yes

characteristic

C₂²×C₈

C₄×D₁₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

C₄×D₁₃

yes

—

102

C₄×C₈

C₂×Dic₁₃

yes

—

103

C₄×C₈

C₂×Dic₁₃

yes

—

104

#95

—

105

C₂×C₁₃

—

106

C₂×C₁₃

#1745

yes

characteristic

107

C₂×C₁₃

#1747

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

110

C₂²×C₄

#6127

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₄×Dic₁₃

yes

—

119

C₂×C₈

C₄×Dic₁₃

yes

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×C₄×D₁₃

yes

characteristic

123

C₄²

#6127

yes

—

124

C₄²

C₂²×Dic₁₃

yes

characteristic

125

C₁₃

#8140

yes

characteristic, complemented, Sylow

126

C₂³

#88126

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#88126

yes

—

131

C₂×C₄

#88126

yes

characteristic

132

C₂×C₄

M₁₆×D₁₃

yes

—

133

C₂×C₄

M₁₆×D₁₃

yes

—

134

C₂×C₄

C₂×C₄×Dic₁₃

yes

characteristic

135

C₂×C₄

#88180

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

#11718366

yes

—

143

C₂²

#11718593

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#11718366

yes

—

148

C₄

M₁₆×Dic₁₃

yes

characteristic

149

C₄

M₁₆×Dic₁₃

yes

characteristic

150

C₂

—

151

C₂

#13114584

yes

characteristic

152

C₂

#13114591

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1452	C₂
socle	C₂²×C₁₃	#376
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1452	D₁₃
hypocenter	C₁₃	#8140
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #1452

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₁₃

commutator

C₂×C₄×C₁₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1452

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₁₃

C₁ ◁ C₂×C₄ ◁ #1452

N	H ≅ G/N	type
13. C₁₃×#322	C₄	semidirect
14. #11718046	C₄	semidirect
78. #322	Dic₁₃	semidirect
125. C₁₃	#8140	semidirect

H ≅ G/N

type

13. C₁₃×#322

C₄

semidirect

14. #11718046

C₄

semidirect

78. #322

Dic₁₃

semidirect

125. C₁₃

#8140

semidirect

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

subgroup	index
#13114522	2
#13113128	2
#13112666	2
#13112664	2
C₁₃×#1452	2
#8140	13

subgroup

index

C₁₃×#1452

Subgroups — #15753370

Conjugacy classes of subgroups

Complements

Maximal subgroups