Group #15777920 — subgroups

Subgroups — #15777920

Order 3328.

390 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 203 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	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112913	C₂	1	4	yes	maximal
4	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
9	C₂²×#6090	C₂²	1	2	yes	—
10	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₁₃×#346	C₄	1	4	yes	complemented
13	#11718053	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₈	1	4	yes	—
18	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₂²×#1055	C₂×C₄	1	2	yes	—
20	C₂×C₁₃×C₁₆	—	2	4	no	—
21	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₈×Dic₁₃	—	2	4	no	—
25	C₈×Dic₁₃	—	2	4	no	—
26	C₂×#6090	—	2	4	no	—
27	C₂×#6090	C₂×C₄	1	4	yes	—
28	C₄×#1055	C₈	1	4	yes	—
29	#12591	—	13	1	no	maximal, Sylow
30	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₁₃	C₄²	1	2	yes	—
34	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
35	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃	#41	1	4	yes	—
38	C₄²×C₁₃	C₄²	1	2	yes	—
39	C₄²×C₁₃	C₂×C₈	1	2	yes	—
40	C₄×Dic₁₃	—	2	2	no	—
41	C₄×Dic₁₃	—	2	2	no	—
42	C₄×Dic₁₃	—	2	8	no	—
43	C₂×#1055	—	2	2	no	—
44	C₂×#1055	—	2	2	no	—
45	C₂×#1055	C₂×C₈	1	4	yes	—
46	C₁₃×C₁₆	—	2	4	no	—
47	#6090	—	2	4	no	—
48	C₂×#346	—	13	1	no	—
49	C₂×#346	D₁₃	1	1	yes	characteristic
50	C₄²×C₈	—	13	1	no	—
51	#1374	—	13	4	no	—
52	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	—	2	4	no	—
54	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
55	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#117	1	2	yes	—
57	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₁₃	—	2	4	no	—
60	C₂×Dic₁₃	—	2	4	no	—
61	C₂×Dic₁₃	M₃₂	1	8	yes	—
62	C₈×C₁₃	#97	1	4	yes	—
63	C₈×C₁₃	#117	1	2	yes	—
64	C₈×C₁₃	#117	1	2	yes	—
65	#1055	—	2	4	no	—
66	C₂²×C₁₆	—	13	2	no	—
67	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
68	C₂×C₄×C₈	—	13	1	no	—
69	C₂×C₄×C₈	—	13	2	no	—
70	C₂×C₄×C₈	—	13	2	no	—
71	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
72	C₄³	—	13	1	no	—
73	#346	—	13	4	no	—
74	#346	Dic₁₃	1	4	yes	complemented
75	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₂²×C₁₃	#431	1	2	yes	—
77	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
78	C₄×C₁₃	—	2	4	no	—
79	C₄×C₁₃	#376	1	1	yes	characteristic
80	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
81	C₄×C₁₃	#431	1	2	yes	—
82	Dic₁₃	—	2	8	no	—
83	C₂²×C₈	—	13	2	no	—
84	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
86	C₂×C₄²	—	13	4	no	—
87	C₂×C₄²	—	13	2	no	—
88	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	13	4	no	—
91	C₂×C₁₆	—	26	4	no	—
92	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
93	C₄×C₈	—	13	4	no	—
94	C₄×C₈	—	26	4	no	—
95	C₄×C₈	—	26	4	no	—
96	C₄×C₈	C₄×D₁₃	1	2	yes	—
97	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
98	C₂×C₁₃	#1744	1	1	yes	characteristic
99	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
100	C₂×C₁₃	#2163	1	4	yes	—
101	C₂²×C₄	—	13	4	no	—
102	C₂²×C₄	C₈×D₁₃	1	2	yes	—
103	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
104	C₂×C₈	—	13	4	no	—
105	C₂×C₈	—	26	2	no	—
106	C₂×C₈	—	26	2	no	—
107	C₂×C₈	#6094	1	4	yes	—
108	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
109	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
110	C₂×C₈	#6127	1	4	yes	—
111	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
112	C₄²	—	26	8	no	—
113	C₄²	—	26	2	no	—
114	C₄²	—	26	2	no	—
115	C₄²	C₈×D₁₃	1	2	yes	—
116	C₄²	C₄×Dic₁₃	1	2	yes	—
117	C₁₆	—	2	4	no	—
118	C₁₆	—	26	4	no	—
119	C₁₃	#12591	1	1	yes	characteristic, complemented, Sylow
120	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	13	8	no	—
123	C₂×C₄	—	26	4	no	—
124	C₂×C₄	—	26	4	no	—
125	C₂×C₄	C₈×Dic₁₃	1	4	yes	—
126	C₂×C₄	#88126	1	2	yes	—
127	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
128	C₂×C₄	C₂×#6094	1	2	yes	—
129	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
130	C₈	—	26	4	no	—
131	C₈	#88055	1	4	yes	—
132	C₈	#88126	1	2	yes	—
133	C₈	#88180	1	2	yes	—
134	C₂²	#11718346	1	2	yes	—
135	C₂²	D₁₃×M₃₂	1	4	yes	—
136	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	26	8	no	—
139	C₄	#11718346	1	2	yes	—
140	C₄	#11718593	1	1	yes	characteristic
141	C₄	C₂×#88055	1	1	yes	characteristic
142	C₂	#13114581	1	1	yes	characteristic
143	C₂	M₃₂×Dic₁₃	1	2	yes	—
144	C₂	#13116637	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718053

C₂

yes

maximal, characteristic

C₄×C₈×Dic₁₃

C₂

yes

maximal, characteristic

#13112913

C₂

yes

maximal

C₂×C₁₃×#346

C₂

yes

maximal, characteristic

C₂²×C₁₃×C₁₆

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₈×Dic₁₃

C₄

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

—

C₂²×#6090

C₂²

yes

—

C₄²×Dic₁₃

C₄

yes

characteristic

C₂×C₄×#1055

C₄

yes

characteristic

C₁₃×#346

C₄

yes

complemented

#11718053

C₄

yes

complemented

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

C₈

yes

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

C₂×#6090

—

C₂×#6090

C₂×C₄

yes

—

C₄×#1055

C₈

yes

—

#12591

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×C₈

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

C₂×C₈

yes

—

C₂×C₈×C₁₃

C₄²

yes

—

C₂×C₈×C₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₄²×C₁₃

C₄²

yes

—

C₄²×C₁₃

C₂×C₈

yes

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂×C₈

yes

—

C₁₃×C₁₆

—

#6090

—

C₂×#346

—

C₂×#346

D₁₃

yes

characteristic

C₄²×C₈

—

#1374

—

C₂³×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×C₈

yes

—

C₂×C₄×C₁₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

M₃₂

yes

—

C₈×C₁₃

#97

yes

—

C₈×C₁₃

#117

yes

—

C₈×C₁₃

#117

yes

—

#1055

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄³

—

#346

—

#346

Dic₁₃

yes

complemented

C₂²×C₁₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₁₃

#431

yes

—

C₂²×C₁₃

C₂×M₃₂

yes

—

C₄×C₁₃

—

C₄×C₁₃

#376

yes

characteristic

C₄×C₁₃

C₂×#97

yes

characteristic

C₄×C₁₃

#431

yes

—

Dic₁₃

—

C₂²×C₈

—

C₂²×C₈

C₄×D₁₃

yes

characteristic

C₂²×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₁₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₁₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₂×C₁₃

#1744

yes

characteristic

C₂×C₁₃

C₄×M₃₂

yes

—

100

C₂×C₁₃

#2163

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₈×D₁₃

yes

—

103

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

#6094

yes

—

108

C₂×C₈

C₄×Dic₁₃

yes

—

109

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

110

C₂×C₈

#6127

yes

—

111

C₂×C₈

C₂²×Dic₁₃

yes

characteristic

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₈×D₁₃

yes

—

116

C₄²

C₄×Dic₁₃

yes

—

117

C₁₆

—

118

C₁₆

—

119

C₁₃

#12591

yes

characteristic, complemented, Sylow

120

C₂³

C₂×C₈×D₁₃

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₈×Dic₁₃

yes

—

126

C₂×C₄

#88126

yes

—

127

C₂×C₄

C₂×C₈×D₁₃

yes

characteristic

128

C₂×C₄

C₂×#6094

yes

—

129

C₂×C₄

C₂×C₄×Dic₁₃

yes

characteristic

130

C₈

—

131

C₈

#88055

yes

—

132

C₈

#88126

yes

—

133

C₈

#88180

yes

—

134

C₂²

#11718346

yes

—

135

C₂²

D₁₃×M₃₂

yes

—

136

C₂²

C₂×C₈×Dic₁₃

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#11718346

yes

—

140

C₄

#11718593

yes

characteristic

141

C₄

C₂×#88055

yes

characteristic

142

C₂

#13114581

yes

characteristic

143

C₂

M₃₂×Dic₁₃

yes

—

144

C₂

#13116637

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12591
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₁₃

commutator

C₂²×C₁₃

C₂×C₄×C₈

Fitting

C₂×C₁₃×#346

C₂

socle, prime radical

C₂³×C₁₃

C₂²×C₈

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂²×C₈ ◁ C₂×#346

N	H ≅ G/N	type
12. C₁₃×#346	C₄	semidirect
13. #11718053	C₄	semidirect
74. #346	Dic₁₃	semidirect
119. C₁₃	#12591	semidirect

H ≅ G/N

type

12. C₁₃×#346

C₄

semidirect

13. #11718053

C₄

semidirect

74. #346

Dic₁₃

semidirect

119. C₁₃

#12591

semidirect

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

subgroup	index
C₂×#11718053	2
C₄×C₈×Dic₁₃	2
#13112913	2
C₂×C₁₃×#346	2
#12591	13

subgroup

index

C₂×#11718053

C₄×C₈×Dic₁₃

#13112913

C₂×C₁₃×#346

#12591

Subgroups — #15777920

Conjugacy classes of subgroups

Complements

Maximal subgroups