Group #15767730 — subgroups

Subgroups — #15767730

Order 3328.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13115656	C₂	1	1	yes	maximal, characteristic
2	#13113509	C₂	1	2	yes	maximal
3	#13112696	C₂	1	2	yes	maximal
4	#13112681	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1838	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	C₂²	1	1	yes	characteristic
7	C₁₃×#423	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
9	C₂×#88056	—	2	2	no	—
10	#11718597	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	2	yes	—
12	C₁₃×#341	—	2	2	no	—
13	#11718074	—	2	4	no	—
14	C₂²×Dic₂₆	—	2	1	no	—
15	C₂×C₁₃×M₁₆	—	2	2	no	—
16	C₂²×C₈×C₁₃	D₄	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	—	2	2	no	—
19	C₂×C₄⋊Dic₁₃	—	2	1	no	—
20	#88123	—	2	2	no	—
21	#88123	—	2	1	no	—
22	#88123	D₄	1	2	yes	—
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₁₃×#97	—	2	2	no	—
26	#88071	—	2	2	no	—
27	C₂²⋊#1055	—	2	2	no	—
28	#88056	—	4	4	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄×#1055	—	2	2	no	—
31	#10635	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	#41	1	2	yes	—
34	C₂²×Dic₁₃	—	4	1	no	—
35	C₂²×Dic₁₃	—	4	1	no	—
36	C₂×Dic₂₆	—	2	2	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₂×Dic₂₆	—	4	2	no	—
39	C₂×Dic₂₆	—	4	1	no	—
40	C₁₃×M₁₆	—	4	2	no	—
41	C₁₃×M₁₆	—	4	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₃	#41	1	2	yes	—
48	C₄²×C₁₃	#41	1	1	yes	characteristic
49	C₄⋊Dic₁₃	—	2	4	no	—
50	C₄⋊Dic₁₃	—	4	2	no	—
51	C₄⋊Dic₁₃	—	4	2	no	—
52	C₂×#1055	—	4	2	no	—
53	#1960	—	13	1	no	—
54	#1838	D₁₃	1	1	yes	characteristic
55	#1546	—	13	2	no	—
56	#1286	—	13	2	no	—
57	#1281	—	13	1	no	—
58	C₂³×C₁₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₁₃	—	4	2	no	—
60	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
61	C₂×C₄×C₁₃	#123	1	2	yes	—
62	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#126	1	2	yes	—
64	C₂×C₄×C₁₃	#135	1	2	yes	—
65	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
66	C₂×Dic₁₃	—	4	2	no	—
67	C₂×Dic₁₃	—	8	1	no	—
68	C₂×Dic₁₃	—	8	1	no	—
69	C₂×Dic₁₃	—	8	2	no	—
70	C₂×Dic₁₃	—	8	1	no	—
71	Dic₂₆	—	4	4	no	—
72	Dic₂₆	—	8	2	no	—
73	C₈×C₁₃	—	4	2	no	—
74	C₈×C₁₃	—	8	2	no	—
75	#1055	—	8	2	no	—
76	C₂×C₄⋊Q₈	—	13	1	no	—
77	#431	—	13	2	no	—
78	#423	C₂×D₁₃	1	2	yes	—
79	C₂×#103	—	26	2	no	—
80	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
81	#382	—	13	1	no	—
82	#341	—	2	2	no	—
83	#332	—	26	4	no	—
84	C₂²×C₁₃	—	2	2	no	—
85	C₂²×C₁₃	#397	1	1	yes	characteristic
86	C₂²×C₁₃	#488	1	1	yes	characteristic
87	C₂²×C₁₃	#490	1	1	yes	characteristic
88	C₄×C₁₃	—	2	2	no	—
89	C₄×C₁₃	—	2	2	no	—
90	C₄×C₁₃	#463	1	2	yes	—
91	C₄×C₁₃	#482	1	2	yes	—
92	Dic₁₃	—	8	2	no	—
93	Dic₁₃	—	16	1	no	—
94	C₂²×Q₈	—	26	1	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
97	C₄⋊Q₈	—	13	2	no	—
98	C₄⋊Q₈	—	26	2	no	—
99	C₄⋊Q₈	—	26	1	no	—
100	C₂×C₄⋊C₄	—	26	2	no	—
101	C₂×C₄⋊C₄	—	26	1	no	—
102	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	26	2	no	—
106	#103	—	52	4	no	—
107	C₂²⋊C₈	—	26	2	no	—
108	#97	—	2	2	no	—
109	C₄×C₈	—	26	2	no	—
110	#95	—	26	2	no	—
111	C₂×C₁₃	—	2	1	no	—
112	C₂×C₁₃	—	2	1	no	—
113	C₂×C₁₃	#2073	1	1	yes	characteristic
114	C₂×C₁₃	#2082	1	2	yes	—
115	C₂×Q₈	—	26	2	no	—
116	C₂×Q₈	—	52	1	no	—
117	C₂×Q₈	—	52	2	no	—
118	C₂×Q₈	—	52	2	no	—
119	C₂²×C₄	—	52	1	no	—
120	C₂²×C₄	—	52	1	no	—
121	C₂²×C₄	#6131	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	52	2	no	—
130	C₄⋊C₄	—	26	4	no	—
131	C₄⋊C₄	—	52	2	no	—
132	C₄⋊C₄	—	52	2	no	—
133	C₄²	#6127	1	2	yes	—
134	C₄²	D₄×D₁₃	1	1	yes	characteristic
135	C₄²	#6129	1	1	yes	characteristic
136	C₁₃	#10635	1	1	yes	characteristic, complemented, Sylow
137	C₂³	#88201	1	1	yes	characteristic
138	Q₈	—	52	4	no	—
139	Q₈	—	104	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	52	2	no	—
142	C₂×C₄	—	104	1	no	—
143	C₂×C₄	—	104	1	no	—
144	C₂×C₄	—	104	2	no	—
145	C₂×C₄	—	104	1	no	—
146	C₂×C₄	#88148	1	2	yes	—
147	C₂×C₄	#88152	1	2	yes	—
148	C₂×C₄	#88183	1	1	yes	characteristic
149	C₂×C₄	#88192	1	1	yes	characteristic
150	C₂×C₄	#88203	1	2	yes	—
151	C₂×C₄	#88204	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	104	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#11718654	1	1	yes	characteristic
157	C₂²	#11718728	1	1	yes	characteristic
158	C₂²	#11718729	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	104	2	no	—
162	C₄	—	208	1	no	—
163	C₄	#11718664	1	2	yes	—
164	C₄	#11718718	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#13116250	1	1	yes	characteristic
168	C₂	#13116301	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115656

C₂

yes

maximal, characteristic

#13113509

C₂

yes

maximal

#13112696

C₂

yes

maximal

#13112681

C₂

yes

maximal, characteristic

C₁₃×#1838

C₂

yes

maximal, characteristic

C₂×#88123

C₂²

yes

characteristic

C₁₃×#423

C₂²

yes

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₂×#88056

—

#11718597

C₂²

yes

characteristic

#11718586

C₂²

yes

—

C₁₃×#341

—

#11718074

—

C₂²×Dic₂₆

—

C₂×C₁₃×M₁₆

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

#88123

—

#88123

—

#88123

D₄

yes

—

C₁₃×C₄⋊C₈

—

C₁₃×C₄⋊C₈

—

C₁₃×#97

—

#88071

—

C₂²⋊#1055

—

—

—

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

#1960

—

#1838

D₁₃

yes

—

—

—

C₂³×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

#126

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

#1055

—

C₂×C₄⋊Q₈

—

#431

—

#423

C₂×D₁₃

yes

—

C₂×#103

—

C₂×#97

C₂×D₁₃

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#463

yes

—

C₄×C₁₃

#482

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₂²×D₁₃

yes

characteristic

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#103

—

107

C₂²⋊C₈

—

108

#97

—

109

C₄×C₈

—

110

#95

—

111

C₂×C₁₃

—

112

C₂×C₁₃

—

113

C₂×C₁₃

#2073

yes

characteristic

114

C₂×C₁₃

#2082

yes

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

—

118

C₂×Q₈

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

#6131

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄²

#6127

yes

—

134

C₄²

D₄×D₁₃

yes

characteristic

135

C₄²

#6129

yes

characteristic

136

C₁₃

#10635

yes

characteristic, complemented, Sylow

137

C₂³

#88201

yes

characteristic

138

Q₈

—

139

Q₈

—

104

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

104

—

143

C₂×C₄

—

104

—

144

C₂×C₄

—

104

—

145

C₂×C₄

—

104

—

146

C₂×C₄

#88148

yes

—

147

C₂×C₄

#88152

yes

—

148

C₂×C₄

#88183

yes

characteristic

149

C₂×C₄

#88192

yes

characteristic

150

C₂×C₄

#88203

yes

—

151

C₂×C₄

#88204

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

104

—

155

C₂²

—

156

C₂²

#11718654

yes

characteristic

157

C₂²

#11718728

yes

characteristic

158

C₂²

#11718729

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

104

—

162

C₄

—

208

—

163

C₄

#11718664

yes

—

164

C₄

#11718718

yes

—

165

C₂

—

166

C₂

—

167

C₂

#13116250

yes

characteristic

168

C₂

#13116301

yes

—

	subgroup	G/H
center	C₂²	#11718654
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1838	C₂
socle	C₂²×C₁₃	#397
prime radical	C₂³×C₁₃	#124
hypercenter	#1838	D₁₃
hypocenter	C₁₃	#10635
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂²

#11718654

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1838

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
136. C₁₃	#10635	semidirect

H ≅ G/N

type

136. C₁₃

#10635

semidirect

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

subgroup	index
#13115656	2
#13113509	2
#13112696	2
#13112681	2
C₁₃×#1838	2
#10635	13

subgroup

index

C₁₃×#1838

Subgroups — #15767730

Conjugacy classes of subgroups

Complements

Maximal subgroups