Group #15756623 — subgroups

Subgroups — #15756623

Order 3328.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#13115000	C₂	1	1	yes	maximal, characteristic
2	#13113128	C₂	1	1	yes	maximal, characteristic
3	#13112696	C₂	1	2	yes	maximal
4	#13112696	C₂	1	2	yes	maximal
5	C₁₃×#1837	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#423	C₂²	1	1	yes	characteristic
7	C₁₃×#423	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	—	2	1	no	—
10	C₂×#88057	—	2	1	no	—
11	#11718597	C₂²	1	2	yes	—
12	#11718586	C₂²	1	2	yes	—
13	C₁₃×#341	—	2	1	no	—
14	C₁₃×#341	—	2	1	no	—
15	#11718046	C₄	1	4	yes	complemented
16	C₂×C₁₃×M₁₆	—	2	1	no	—
17	C₂×C₁₃×M₁₆	—	2	1	no	—
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	—	2	2	no	—
22	C₁₃×C₄⋊C₈	—	2	2	no	—
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	Q₈	1	2	yes	—
26	#88071	—	2	4	no	—
27	C₂²⋊#1055	—	2	2	no	—
28	#88058	—	4	2	no	—
29	#88057	—	4	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄×#1055	C₂×C₄	1	4	yes	—
32	#8795	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	#41	1	2	yes	—
35	C₂²×Dic₁₃	—	4	2	no	—
36	C₁₃×M₁₆	—	4	1	no	—
37	C₁₃×M₁₆	—	4	1	no	—
38	C₁₃×M₁₆	—	4	1	no	—
39	C₁₃×M₁₆	—	4	1	no	—
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	4	1	no	—
45	C₂×C₈×C₁₃	—	4	1	no	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₃	#41	1	2	yes	—
51	C₄⋊Dic₁₃	—	4	2	no	—
52	C₄⋊Dic₁₃	—	4	2	no	—
53	C₂×#1055	—	2	4	no	—
54	#1846	—	13	1	no	—
55	#1837	D₁₃	1	1	yes	characteristic
56	#1452	—	13	1	no	—
57	#1286	—	13	2	no	—
58	#1286	—	13	2	no	—
59	C₂³×C₁₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#117	1	2	yes	—
63	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#125	1	2	yes	—
65	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
69	C₂×Dic₁₃	—	8	4	no	—
70	C₂×Dic₁₃	—	8	2	no	—
71	C₈×C₁₃	—	2	2	no	—
72	C₈×C₁₃	—	2	2	no	—
73	C₈×C₁₃	—	8	1	no	—
74	C₈×C₁₃	—	8	1	no	—
75	#1055	—	4	4	no	—
76	#431	—	13	2	no	—
77	C₂×#107	—	26	1	no	—
78	C₂×#106	—	26	1	no	—
79	#423	C₂×D₁₃	1	1	yes	characteristic
80	#423	C₂×D₁₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
82	#382	—	13	2	no	—
83	#341	—	2	1	no	—
84	#341	—	2	1	no	—
85	#322	—	13	4	no	—
86	C₂²×C₁₃	—	2	2	no	—
87	C₂²×C₁₃	#382	1	1	yes	characteristic
88	C₂²×C₁₃	#490	1	1	yes	characteristic
89	C₂²×C₁₃	#491	1	1	yes	characteristic
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	#428	1	1	yes	characteristic
93	C₄×C₁₃	#428	1	1	yes	characteristic
94	C₄×C₁₃	#489	1	1	yes	characteristic
95	C₄×C₁₃	#489	1	1	yes	characteristic
96	Dic₁₃	—	16	2	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	26	2	no	—
102	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
103	#107	—	52	2	no	—
104	#106	—	52	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	26	2	no	—
108	C₂²⋊C₈	—	26	2	no	—
109	C₄×C₈	—	13	4	no	—
110	C₄×C₈	Dic₂₆	1	2	yes	—
111	C₄×C₈	C₄×D₁₃	1	2	yes	—
112	#95	—	26	4	no	—
113	C₂×C₁₃	—	2	1	no	—
114	C₂×C₁₃	—	2	1	no	—
115	C₂×C₁₃	#1833	1	2	yes	—
116	C₂×C₁₃	#1836	1	1	yes	characteristic
117	C₂²×C₄	—	52	2	no	—
118	C₂²×C₄	#6131	1	2	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	M₁₆	—	4	1	no	—
121	M₁₆	—	4	1	no	—
122	M₁₆	—	4	1	no	—
123	M₁₆	—	4	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	26	4	no	—
131	C₂×C₈	#6101	1	2	yes	—
132	C₂×C₈	#6101	1	2	yes	—
133	C₄⋊C₄	—	52	2	no	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
136	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
137	C₄²	#6129	1	2	yes	—
138	C₁₃	#8795	1	1	yes	characteristic, complemented, Sylow
139	C₂³	#88201	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	104	4	no	—
142	C₂×C₄	—	104	2	no	—
143	C₂×C₄	#88144	1	2	yes	—
144	C₂×C₄	#88146	1	2	yes	—
145	C₂×C₄	C₂×#6101	1	1	yes	characteristic
146	C₂×C₄	#88190	1	1	yes	characteristic
147	C₂×C₄	#88203	1	1	yes	characteristic
148	C₂×C₄	#88203	1	1	yes	characteristic
149	C₂×C₄	#88205	1	1	yes	characteristic
150	C₂×C₄	#88205	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	8	1	no	—
154	C₈	—	8	1	no	—
155	C₈	—	52	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#11718638	1	1	yes	characteristic
158	C₂²	#11718728	1	1	yes	characteristic
159	C₂²	#11718730	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	208	2	no	—
163	C₄	#11718656	1	1	yes	characteristic
164	C₄	#11718656	1	1	yes	characteristic
165	C₄	#11718716	1	1	yes	characteristic
166	C₄	#11718716	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#13114961	1	2	yes	—
170	C₂	#13114969	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13115000

C₂

yes

maximal, characteristic

#13113128

C₂

yes

maximal, characteristic

#13112696

C₂

yes

maximal

#13112696

C₂

yes

maximal

C₁₃×#1837

C₂

yes

maximal, characteristic

C₁₃×#423

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

—

C₂²

yes

—

#11718586

C₂²

yes

—

C₁₃×#341

—

C₁₃×#341

—

#11718046

C₄

yes

complemented

C₂×C₁₃×M₁₆

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂²×C₈×C₁₃

D₄

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#88071

—

C₂²⋊#1055

—

—

—

—

C₂×C₄

yes

—

#8795

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

#1846

—

#1837

D₁₃

yes

—

—

—

C₂³×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#125

yes

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

C₂×D₁₃

yes

characteristic

#423

C₂×D₁₃

yes

characteristic

C₂×C₄×C₈

C₂×D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

100

C₂²×C₈

C₂²⋊D₁₃

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₂²×D₁₃

yes

characteristic

103

#107

—

104

#106

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₂²⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

Dic₂₆

yes

—

111

C₄×C₈

C₄×D₁₃

yes

—

112

#95

—

113

C₂×C₁₃

—

114

C₂×C₁₃

—

115

C₂×C₁₃

#1833

yes

—

116

C₂×C₁₃

#1836

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

#6131

yes

—

119

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

#6101

yes

—

132

C₂×C₈

#6101

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

C₂×Dic₂₆

yes

characteristic

136

C₄²

C₂×C₄×D₁₃

yes

characteristic

137

C₄²

#6129

yes

—

138

C₁₃

#8795

yes

characteristic, complemented, Sylow

139

C₂³

#88201

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

104

—

142

C₂×C₄

—

104

—

143

C₂×C₄

#88144

yes

—

144

C₂×C₄

#88146

yes

—

145

C₂×C₄

C₂×#6101

yes

characteristic

146

C₂×C₄

#88190

yes

characteristic

147

C₂×C₄

#88203

yes

characteristic

148

C₂×C₄

#88203

yes

characteristic

149

C₂×C₄

#88205

yes

characteristic

150

C₂×C₄

#88205

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

#11718638

yes

characteristic

158

C₂²

#11718728

yes

characteristic

159

C₂²

#11718730

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

208

—

163

C₄

#11718656

yes

characteristic

164

C₄

#11718656

yes

characteristic

165

C₄

#11718716

yes

characteristic

166

C₄

#11718716

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#13114961

yes

—

170

C₂

#13114969

yes

characteristic

	subgroup	G/H
center	C₂²	#11718638
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1837	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#124
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1837	D₁₃
hypocenter	C₁₃	#8795
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#11718638

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1837

C₂

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄²×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
15. #11718046	C₄	semidirect
138. C₁₃	#8795	semidirect

H ≅ G/N

type

15. #11718046

C₄

semidirect

138. C₁₃

#8795

semidirect

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

subgroup	index
#13115000	2
#13113128	2
#13112696	2
C₁₃×#1837	2
#8795	13

subgroup

index

C₁₃×#1837

Subgroups — #15756623

Conjugacy classes of subgroups

Complements

Maximal subgroups