Group #15767729 — subgroups

Subgroups — #15767729

Order 3328.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13114977	C₂	1	1	yes	maximal, characteristic
2	#13113508	C₂	1	2	yes	maximal
3	#13112694	C₂	1	2	yes	maximal
4	#13112681	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1960	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#103	—	2	2	no	—
8	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
9	C₁₃×#382	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	2	yes	—
11	#11718585	C₂²	1	2	yes	—
12	#11718144	—	2	2	no	—
13	#11718139	—	2	4	no	—
14	C₂²×Q₈×C₁₃	—	2	1	no	—
15	C₂²×C₈×C₁₃	D₄	1	2	yes	—
16	C₁₃×C₄⋊Q₈	—	2	2	no	—
17	C₁₃×C₄⋊Q₈	—	2	1	no	—
18	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×#6099	—	2	2	no	—
23	C₁₃×#103	—	4	4	no	—
24	C₁₃×#97	—	2	2	no	—
25	C₁₃×#95	—	2	2	no	—
26	C₂²⋊#1055	—	2	2	no	—
27	C₄⋊#1055	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄×#1055	—	2	2	no	—
31	#10635	—	13	1	no	maximal, Sylow
32	C₂×Q₈×C₁₃	—	2	2	no	—
33	C₂×Q₈×C₁₃	—	4	1	no	—
34	C₂×Q₈×C₁₃	—	4	2	no	—
35	C₂×Q₈×C₁₃	—	4	2	no	—
36	C₂²×C₄×C₁₃	—	4	1	no	—
37	C₂²×C₄×C₁₃	—	4	1	no	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	#41	1	2	yes	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₁₃×C₄⋊C₄	—	4	2	no	—
46	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₁₃	#41	1	1	yes	characteristic
48	C₄²×C₁₃	#41	1	2	yes	—
49	#6099	—	4	2	no	—
50	#6099	—	4	2	no	—
51	C₂×#1055	—	4	2	no	—
52	C₂×#1055	—	4	2	no	—
53	#1960	D₁₃	1	1	yes	characteristic
54	#1838	—	13	1	no	—
55	#1546	—	13	2	no	—
56	#1286	—	13	2	no	—
57	#1281	—	13	1	no	—
58	C₂³×C₁₃	#124	1	1	yes	characteristic
59	Q₈×C₁₃	—	4	4	no	—
60	Q₈×C₁₃	—	8	2	no	—
61	C₂×C₄×C₁₃	—	4	2	no	—
62	C₂×C₄×C₁₃	—	4	2	no	—
63	C₂×C₄×C₁₃	—	8	1	no	—
64	C₂×C₄×C₁₃	—	8	1	no	—
65	C₂×C₄×C₁₃	—	8	2	no	—
66	C₂×C₄×C₁₃	—	8	1	no	—
67	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#123	1	2	yes	—
69	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#126	1	2	yes	—
71	C₂×C₄×C₁₃	#135	1	2	yes	—
72	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₈×C₁₃	—	4	2	no	—
74	#1055	—	8	2	no	—
75	#1055	—	8	2	no	—
76	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
77	#431	—	13	2	no	—
78	#423	—	13	2	no	—
79	C₂×#103	—	2	2	no	—
80	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
81	#382	C₂×D₁₃	1	1	yes	characteristic
82	#341	—	26	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	—	8	2	no	—
91	C₄×C₁₃	—	16	1	no	—
92	C₄×C₁₃	#463	1	2	yes	—
93	C₄×C₁₃	#482	1	2	yes	—
94	C₂²×Q₈	—	2	1	no	—
95	C₂×M₁₆	—	26	2	no	—
96	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
97	C₄⋊Q₈	—	2	1	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄⋊Q₈	D₅₂	1	2	yes	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	1	no	—
102	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
103	C₄⋊C₈	—	26	2	no	—
104	C₄⋊C₈	—	26	2	no	—
105	C₄⋊C₈	—	26	2	no	—
106	#103	—	4	4	no	—
107	C₂²⋊C₈	—	26	2	no	—
108	#97	—	2	2	no	—
109	C₄×C₈	—	26	2	no	—
110	#95	—	2	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	—	2	2	no	—
116	C₂×Q₈	—	4	1	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂²×C₄	—	4	1	no	—
120	C₂²×C₄	—	4	1	no	—
121	C₂²×C₄	#6129	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
123	M₁₆	—	52	2	no	—
124	M₁₆	—	52	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	52	2	no	—
129	C₂×C₈	—	52	2	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	4	2	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄²	C₂×D₅₂	1	1	yes	characteristic
134	C₄²	#6127	1	1	yes	characteristic
135	C₄²	#6129	1	2	yes	—
136	C₁₃	#10635	1	1	yes	characteristic, complemented, Sylow
137	C₂³	#88190	1	1	yes	characteristic
138	Q₈	—	4	4	no	—
139	Q₈	—	8	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	#88133	1	2	yes	—
147	C₂×C₄	#88140	1	2	yes	—
148	C₂×C₄	#88184	1	1	yes	characteristic
149	C₂×C₄	#88186	1	2	yes	—
150	C₂×C₄	#88189	1	1	yes	characteristic
151	C₂×C₄	#88204	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	104	2	no	—
154	C₈	—	104	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#11718614	1	1	yes	characteristic
157	C₂²	#11718717	1	1	yes	characteristic
158	C₂²	#11718732	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	#11718677	1	2	yes	—
164	C₄	#11718696	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#13116249	1	1	yes	characteristic
168	C₂	#13116300	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114977

C₂

yes

maximal, characteristic

#13113508

C₂

yes

maximal

#13112694

C₂

yes

maximal

#13112681

C₂

yes

maximal, characteristic

C₁₃×#1960

C₂

yes

maximal, characteristic

C₂×C₁₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₁₃×#103

—

C₂×C₁₃×#97

C₂²

yes

characteristic

C₁₃×#382

C₂²

yes

characteristic

#11718586

C₂²

yes

—

#11718585

C₂²

yes

—

#11718144

—

#11718139

—

C₂²×Q₈×C₁₃

—

C₂²×C₈×C₁₃

D₄

yes

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×C₄⋊Q₈

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×#6099

—

C₁₃×#103

—

C₁₃×#97

—

C₁₃×#95

—

C₂²⋊#1055

—

—

—

—

—

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

C₄²×C₁₃

yes

—

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#124

yes

characteristic

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

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

—

C₈×C₁₃

—

#1055

—

#1055

—

C₂×C₄⋊Q₈

C₂×D₁₃

yes

—

—

—

C₂×#97

C₂×D₁₃

yes

characteristic

#382

C₂×D₁₃

yes

characteristic

#341

—

#332

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#463

yes

—

C₄×C₁₃

#482

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₅₂

yes

—

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

#6129

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

C₂×D₅₂

yes

characteristic

134

C₄²

#6127

yes

characteristic

135

C₄²

#6129

yes

—

136

C₁₃

#10635

yes

characteristic, complemented, Sylow

137

C₂³

#88190

yes

characteristic

138

Q₈

—

139

Q₈

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#88133

yes

—

147

C₂×C₄

#88140

yes

—

148

C₂×C₄

#88184

yes

characteristic

149

C₂×C₄

#88186

yes

—

150

C₂×C₄

#88189

yes

characteristic

151

C₂×C₄

#88204

yes

—

152

C₈

—

153

C₈

—

104

—

154

C₈

—

104

—

155

C₂²

—

156

C₂²

#11718614

yes

characteristic

157

C₂²

#11718717

yes

characteristic

158

C₂²

#11718732

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#11718677

yes

—

164

C₄

#11718696

yes

—

165

C₂

—

166

C₂

—

167

C₂

#13116249

yes

characteristic

168

C₂

#13116300

yes

—

	subgroup	G/H
center	C₂²	#11718614
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1960	C₂
socle	C₂²×C₁₃	#397
prime radical	C₂³×C₁₃	#124
hypercenter	#1960	D₁₃
hypocenter	C₁₃	#10635
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#11718614

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1960

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

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
#13114977	2
#13113508	2
#13112694	2
#13112681	2
C₁₃×#1960	2
#10635	13

subgroup

index

C₁₃×#1960

Subgroups — #15767729

Conjugacy classes of subgroups

Complements

Maximal subgroups