Group #15756596 — subgroups

Subgroups — #15756596

Order 3328.

342 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 155 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	#13114988	C₂	1	1	yes	maximal, characteristic
2	#13114953	C₂	1	1	yes	maximal, characteristic
3	#13112671	C₂	1	4	yes	maximal
4	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
6	C₄³×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊#1055	—	2	2	no	—
8	C₂×C₄⋊#1055	C₂²	1	2	yes	—
9	C₂×C₄⋊#1055	C₂²	1	2	yes	—
10	C₁₃×#336	C₄	1	2	yes	—
11	C₁₃×#336	C₄	1	2	yes	—
12	#11718116	—	2	2	no	—
13	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×#1055	—	2	2	no	—
20	C₂²×#1055	—	2	2	no	—
21	C₄×C₈×C₁₃	—	2	2	no	—
22	C₄×C₈×C₁₃	—	2	2	no	—
23	C₄⋊#1055	—	2	4	no	—
24	C₄⋊#1055	—	2	4	no	—
25	C₄⋊#1055	—	4	4	no	—
26	#8789	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₁₃	—	2	4	no	—
33	C₂×C₈×C₁₃	—	2	4	no	—
34	C₂×C₈×C₁₃	—	2	4	no	—
35	C₄²×C₁₃	—	2	4	no	—
36	C₄²×C₁₃	—	2	2	no	—
37	C₄²×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	—	2	2	no	—
39	C₄²×C₁₃	—	2	2	no	—
40	C₄²×C₁₃	#41	1	2	yes	—
41	C₄²×C₁₃	#41	1	2	yes	—
42	C₂×#1055	—	4	2	no	—
43	C₂×#1055	—	4	2	no	—
44	C₂×#1055	—	4	2	no	—
45	C₂×#1055	—	4	4	no	—
46	C₂×#1055	—	4	2	no	—
47	#1842	—	13	1	no	—
48	#1831	—	13	1	no	—
49	#1744	D₁₃	1	1	yes	characteristic
50	#1277	—	13	4	no	—
51	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	—	2	2	no	—
55	C₂×C₄×C₁₃	—	2	2	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	#108	1	4	yes	—
59	C₂×C₄×C₁₃	#117	1	2	yes	—
60	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#125	1	2	yes	—
63	C₂×C₄×C₁₃	#131	1	2	yes	—
64	C₂×C₄×C₁₃	#131	1	2	yes	—
65	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
66	C₂×C₄×C₁₃	#137	1	2	yes	—
67	C₈×C₁₃	—	4	4	no	—
68	#1055	—	8	2	no	—
69	#1055	—	8	2	no	—
70	C₂×C₄⋊C₈	—	13	2	no	—
71	C₂×C₄⋊C₈	—	13	2	no	—
72	C₂×C₄⋊C₈	—	26	2	no	—
73	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
74	C₄³	C₂×D₁₃	1	1	yes	characteristic
75	#336	—	26	2	no	—
76	#336	Dic₁₃	1	2	yes	—
77	#336	Dic₁₃	1	2	yes	—
78	C₂²×C₁₃	#382	1	1	yes	characteristic
79	C₂²×C₁₃	#424	1	1	yes	characteristic
80	C₂²×C₁₃	#424	1	1	yes	characteristic
81	C₂²×C₁₃	#428	1	2	yes	—
82	C₂²×C₁₃	C₂×#108	1	2	yes	—
83	C₄×C₁₃	—	2	4	no	—
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	4	2	no	—
86	C₄×C₁₃	#433	1	4	yes	—
87	C₄×C₁₃	#489	1	2	yes	—
88	C₄×C₁₃	#490	1	1	yes	characteristic
89	C₄×C₁₃	#491	1	1	yes	characteristic
90	C₂²×C₈	—	26	2	no	—
91	C₂²×C₈	—	26	2	no	—
92	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
93	C₂×C₄²	Dic₂₆	1	2	yes	—
94	C₂×C₄²	D₅₂	1	2	yes	—
95	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
97	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
98	C₄⋊C₈	—	26	4	no	—
99	C₄⋊C₈	—	26	4	no	—
100	C₄⋊C₈	—	52	4	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₂×C₁₃	#1573	1	2	yes	—
104	C₂×C₁₃	#1574	1	2	yes	—
105	C₂×C₁₃	#1832	1	1	yes	characteristic
106	C₂×C₁₃	#1836	1	1	yes	characteristic
107	C₂×C₁₃	#1837	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
109	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
110	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
112	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	52	2	no	—
117	C₂×C₈	—	52	2	no	—
118	C₂×C₈	—	52	4	no	—
119	C₂×C₈	—	52	2	no	—
120	C₂×C₈	—	52	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	4	no	—
126	C₄²	#6127	1	2	yes	—
127	C₄²	#6127	1	2	yes	—
128	C₁₃	#8789	1	1	yes	characteristic, complemented, Sylow
129	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#88059	1	4	yes	—
137	C₂×C₄	#88124	1	2	yes	—
138	C₂×C₄	#88129	1	1	yes	characteristic
139	C₂×C₄	#88129	1	1	yes	characteristic
140	C₂×C₄	#88162	1	2	yes	—
141	C₂×C₄	#88163	1	2	yes	—
142	C₂×C₄	#88180	1	2	yes	—
143	C₂×C₄	#88202	1	2	yes	—
144	C₂×C₄	#88202	1	2	yes	—
145	C₈	—	4	4	no	—
146	C₈	—	104	2	no	—
147	C₈	—	104	2	no	—
148	C₂²	#11718597	1	1	yes	characteristic
149	C₂²	#11718660	1	1	yes	characteristic
150	C₂²	#11718660	1	1	yes	characteristic
151	C₂²	C₂×#88059	1	2	yes	—
152	C₂²	#11718782	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#11718364	1	1	yes	characteristic
157	C₄	#11718374	1	1	yes	characteristic
158	C₄	#11718375	1	2	yes	—
159	C₄	#11718661	1	4	yes	—
160	C₂	#13113621	1	2	yes	—
161	C₂	#13113625	1	2	yes	—
162	C₂	#13114956	1	1	yes	characteristic
163	C₂	#13114970	1	1	yes	characteristic
164	C₂	#13114974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13114988

C₂

yes

maximal, characteristic

#13114953

C₂

yes

maximal, characteristic

#13112671

C₂

yes

maximal

C₁₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊#1055

—

C₂×C₄⋊#1055

C₂²

yes

—

C₂×C₄⋊#1055

C₂²

yes

—

C₁₃×#336

C₄

yes

—

C₁₃×#336

C₄

yes

—

#11718116

—

C₂²×C₈×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

Q₈

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

C₂²×#1055

—

C₄×C₈×C₁₃

—

C₄×C₈×C₁₃

—

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

#1277

—

C₂³×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#108

yes

—

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

#131

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₄³

C₂×D₁₃

yes

characteristic

#336

—

#336

Dic₁₃

yes

—

#336

Dic₁₃

yes

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

C₂×#108

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#433

yes

—

C₄×C₁₃

#489

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₁₃

yes

—

C₂×C₄²

Dic₂₆

yes

—

C₂×C₄²

D₅₂

yes

—

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

C₂×Dic₁₃

yes

characteristic

C₂×C₄²

C₂²×D₁₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₂×C₁₃

#1573

yes

—

104

C₂×C₁₃

#1574

yes

—

105

C₂×C₁₃

#1832

yes

characteristic

106

C₂×C₁₃

#1836

yes

characteristic

107

C₂×C₁₃

#1837

yes

characteristic

108

C₂²×C₄

C₄⋊Dic₁₃

yes

—

109

C₂²×C₄

C₄⋊Dic₁₃

yes

—

110

C₂²×C₄

C₂×Dic₂₆

yes

characteristic

111

C₂²×C₄

C₂×D₅₂

yes

characteristic

112

C₂²×C₄

C₂²×Dic₁₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

#6127

yes

—

127

C₄²

#6127

yes

—

128

C₁₃

#8789

yes

characteristic, complemented, Sylow

129

C₂³

C₂×C₄⋊Dic₁₃

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#88059

yes

—

137

C₂×C₄

#88124

yes

—

138

C₂×C₄

#88129

yes

characteristic

139

C₂×C₄

#88129

yes

characteristic

140

C₂×C₄

#88162

yes

—

141

C₂×C₄

#88163

yes

—

142

C₂×C₄

#88180

yes

—

143

C₂×C₄

#88202

yes

—

144

C₂×C₄

#88202

yes

—

145

C₈

—

146

C₈

—

104

—

147

C₈

—

104

—

148

C₂²

#11718597

yes

characteristic

149

C₂²

#11718660

yes

characteristic

150

C₂²

#11718660

yes

characteristic

151

C₂²

C₂×#88059

yes

—

152

C₂²

#11718782

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#11718364

yes

characteristic

157

C₄

#11718374

yes

characteristic

158

C₄

#11718375

yes

—

159

C₄

#11718661

yes

—

160

C₂

#13113621

yes

—

161

C₂

#13113625

yes

—

162

C₂

#13114956

yes

characteristic

163

C₂

#13114970

yes

characteristic

164

C₂

#13114974

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#8789
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₅₂

commutator

C₂²×C₄×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1744

C₂

socle, prime radical

C₂³×C₁₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
128. C₁₃	#8789	semidirect

H ≅ G/N

type

128. C₁₃

#8789

semidirect

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

subgroup	index
#13114988	2
#13114953	2
#13112671	2
C₁₃×#1744	2
#8789	13

subgroup

index

C₁₃×#1744

Subgroups — #15756596

Conjugacy classes of subgroups

Complements

Maximal subgroups