Group #15779279 — subgroups

Subgroups — #15779279

Order 3328.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#13116592	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112917	C₂	1	4	yes	maximal
4	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×M₃₂	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	C₂×#88052	C₂²	1	2	yes	—
10	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₁₃×C₄⋊C₁₆	—	2	2	no	—
13	C₄⋊#6090	—	2	2	no	—
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	—	2	2	no	—
18	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₂²×#1055	C₂×C₄	1	2	yes	—
20	C₁₃×M₃₂	—	2	4	no	—
21	C₂×C₁₃×C₁₆	—	2	2	no	—
22	C₄×C₈×C₁₃	D₄	1	2	yes	—
23	C₄×C₈×C₁₃	Q₈	1	2	yes	—
24	C₈×Dic₁₃	—	2	2	no	—
25	C₈×Dic₁₃	—	2	2	no	—
26	C₈×Dic₁₃	—	2	4	no	—
27	#88052	—	2	4	no	—
28	C₂×#6090	—	2	2	no	—
29	C₄×#1055	C₈	1	2	yes	—
30	C₄×#1055	C₈	1	2	yes	—
31	#12883	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	—	2	2	no	—
36	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	#41	1	2	yes	—
40	C₂×C₈×C₁₃	#41	1	2	yes	—
41	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₄×Dic₁₃	—	2	8	no	—
45	C₄×Dic₁₃	—	2	2	no	—
46	C₄×Dic₁₃	M₁₆	1	4	yes	—
47	C₂×#1055	—	2	2	no	—
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	C₂×C₈	1	4	yes	—
50	C₁₃×C₁₆	—	4	2	no	—
51	#6090	—	4	2	no	—
52	#2150	—	13	1	no	—
53	#2150	D₁₃	1	1	yes	characteristic
54	C₄²×C₈	—	13	1	no	—
55	#1376	—	13	4	no	—
56	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#117	1	2	yes	—
62	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
64	C₂×Dic₁₃	—	2	8	no	—
65	C₂×Dic₁₃	—	2	4	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	#124	1	1	yes	characteristic
68	C₈×C₁₃	#124	1	1	yes	characteristic
69	C₈×C₁₃	#125	1	2	yes	—
70	#1055	—	2	4	no	—
71	C₂×M₃₂	—	13	2	no	—
72	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
73	C₂×C₄×C₈	—	13	1	no	—
74	C₂×C₄×C₈	—	13	2	no	—
75	C₂×C₄×C₈	—	13	2	no	—
76	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
77	C₄³	—	13	1	no	—
78	C₄⋊C₁₆	—	2	2	no	—
79	C₄⋊C₁₆	—	26	2	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₂²×C₁₃	#432	1	2	yes	—
83	C₄×C₁₃	—	2	2	no	—
84	C₄×C₁₃	#364	1	4	yes	—
85	C₄×C₁₃	#382	1	1	yes	characteristic
86	C₄×C₁₃	#423	1	1	yes	characteristic
87	C₄×C₁₃	#431	1	2	yes	—
88	Dic₁₃	—	2	8	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
92	C₂×C₄²	—	13	2	no	—
93	C₂×C₄²	—	26	2	no	—
94	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	26	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	26	2	no	—
99	C₄×C₈	—	13	2	no	—
100	C₄×C₈	—	13	2	no	—
101	C₄×C₈	—	26	4	no	—
102	C₄×C₈	—	26	2	no	—
103	C₄×C₈	—	26	2	no	—
104	C₄×C₈	Dic₂₆	1	2	yes	—
105	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
106	C₂×C₁₃	—	2	2	no	—
107	C₂×C₁₃	#1831	1	1	yes	characteristic
108	C₂×C₁₃	C₂×#364	1	2	yes	—
109	C₂²×C₄	—	26	2	no	—
110	C₂²×C₄	#6094	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	13	4	no	—
114	C₂×C₈	—	26	2	no	—
115	C₂×C₈	—	26	2	no	—
116	C₂×C₈	#6101	1	2	yes	—
117	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
118	C₂×C₈	#6129	1	2	yes	—
119	C₂×C₈	#6131	1	2	yes	—
120	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
121	C₄²	—	13	4	no	—
122	C₄²	—	26	8	no	—
123	C₄²	—	26	2	no	—
124	C₄²	C₈×D₁₃	1	1	yes	characteristic
125	C₄²	C₈×D₁₃	1	1	yes	characteristic
126	C₄²	#6101	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	52	2	no	—
129	C₁₃	#12883	1	1	yes	characteristic, complemented, Sylow
130	C₂³	C₂×#6094	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	26	8	no	—
133	C₂×C₄	—	26	4	no	—
134	C₂×C₄	#88054	1	2	yes	—
135	C₂×C₄	#88054	1	2	yes	—
136	C₂×C₄	#88146	1	2	yes	—
137	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
138	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
139	C₂×C₄	C₂×#6101	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	26	4	no	—
142	C₈	#88144	1	2	yes	—
143	C₈	#88190	1	1	yes	characteristic
144	C₈	#88201	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	#11718496	1	2	yes	—
147	C₂²	C₂×#88054	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	26	8	no	—
150	C₄	#11718103	1	4	yes	—
151	C₄	#11718494	1	2	yes	—
152	C₄	#11718638	1	1	yes	characteristic
153	C₄	#11718779	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	#13114952	1	1	yes	characteristic
156	C₂	C₂×#11718103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116592

C₂

yes

maximal, characteristic

C₄×C₈×Dic₁₃

C₂

yes

maximal, characteristic

#13112917

C₂

yes

maximal

C₁₃×#2150

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×C₈×Dic₁₃

C₄

yes

—

C₂×C₈×Dic₁₃

C₂²

yes

—

C₂×#88052

C₂²

yes

—

C₄²×Dic₁₃

C₄

yes

characteristic

C₂×C₄×#1055

C₄

yes

characteristic

C₁₃×C₄⋊C₁₆

—

C₄⋊#6090

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₃

—

C₂×C₄×Dic₁₃

C₂×C₄

yes

—

C₂²×#1055

C₂×C₄

yes

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

C₈×Dic₁₃

—

#88052

—

C₂×#6090

—

C₄×#1055

C₈

yes

—

C₄×#1055

C₈

yes

—

#12883

—

maximal, Sylow

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

#41

yes

—

C₂×C₈×C₁₃

#41

yes

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄×Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

M₁₆

yes

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

C₂×C₈

yes

—

C₁₃×C₁₆

—

#6090

—

#2150

—

#2150

D₁₃

yes

characteristic

C₄²×C₈

—

#1376

—

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

—

#1055

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₁₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₃

#432

yes

—

C₄×C₁₃

—

C₄×C₁₃

#364

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

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

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

Dic₂₆

yes

—

105

C₄×C₈

C₂²⋊D₁₃

yes

—

106

C₂×C₁₃

—

107

C₂×C₁₃

#1831

yes

characteristic

108

C₂×C₁₃

C₂×#364

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

#6094

yes

—

111

C₂²×C₄

C₂×C₄×D₁₃

yes

characteristic

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

#6101

yes

—

117

C₂×C₈

C₂×Dic₂₆

yes

characteristic

118

C₂×C₈

#6129

yes

—

119

C₂×C₈

#6131

yes

—

120

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₈×D₁₃

yes

characteristic

125

C₄²

C₈×D₁₃

yes

characteristic

126

C₄²

#6101

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₃

#12883

yes

characteristic, complemented, Sylow

130

C₂³

C₂×#6094

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#88054

yes

—

135

C₂×C₄

#88054

yes

—

136

C₂×C₄

#88146

yes

—

137

C₂×C₄

C₂×C₈×D₁₃

yes

characteristic

138

C₂×C₄

M₁₆×D₁₃

yes

—

139

C₂×C₄

C₂×#6101

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#88144

yes

—

143

C₈

#88190

yes

characteristic

144

C₈

#88201

yes

characteristic

145

C₂²

—

146

C₂²

#11718496

yes

—

147

C₂²

C₂×#88054

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

#11718103

yes

—

151

C₄

#11718494

yes

—

152

C₄

#11718638

yes

characteristic

153

C₄

#11718779

yes

characteristic

154

C₂

—

155

C₂

#13114952

yes

characteristic

156

C₂

C₂×#11718103

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2150	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2150	D₁₃
hypocenter	C₁₃	#12883
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₁₃

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₁₃

Fitting

C₁₃×#2150

C₂

socle

C₂²×C₁₃

C₂×C₄⋊C₈

prime radical

C₂³×C₁₃

C₂×M₁₆

cosocle

C₂²×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
129. C₁₃	#12883	semidirect

H ≅ G/N

type

129. C₁₃

#12883

semidirect

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

subgroup	index
#13116592	2
C₄×C₈×Dic₁₃	2
#13112917	2
C₁₃×#2150	2
#12883	13

subgroup

index

#13116592

C₄×C₈×Dic₁₃

#13112917

C₁₃×#2150

#12883

Subgroups — #15779279

Conjugacy classes of subgroups

Complements

Maximal subgroups