Group #15766029 — subgroups

Subgroups — #15766029

Order 3328.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13115813	C₂	1	1	yes	maximal, characteristic
2	#13113399	C₂	1	2	yes	maximal
3	#13112694	C₂	1	2	yes	maximal
4	#13112669	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2864	C₂	1	1	yes	maximal, characteristic
6	#11719175	C₂²	1	1	yes	characteristic
7	C₁₃×#389	—	2	2	no	—
8	C₁₃×#387	—	2	4	no	—
9	C₁₃×#382	C₂²	1	2	yes	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	2	yes	—
12	#11718585	C₂²	1	1	yes	characteristic
13	#11718115	—	2	2	no	—
14	#11718044	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄²×C₁₃	—	2	2	no	—
18	C₂×C₄²×C₁₃	D₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
20	#88180	—	2	1	no	—
21	#88178	—	2	2	no	—
22	C₂×#6099	—	2	1	no	—
23	#88124	—	2	1	no	—
24	#88123	Q₈	1	2	yes	—
25	C₄×Dic₂₆	—	2	2	no	—
26	C₁₃×#95	—	2	4	no	—
27	C₁₃×#95	—	2	2	no	—
28	C₁₃×#95	D₄	1	4	yes	—
29	C₂²⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄×#1055	—	2	2	no	—
33	#10367	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	—	2	2	no	—
35	C₂²×C₄×C₁₃	—	2	4	no	—
36	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₂₆	—	2	2	no	—
39	C₁₃×C₄⋊C₄	—	4	8	no	—
40	C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₁₃×C₄⋊C₄	—	4	2	no	—
42	C₄²×C₁₃	—	2	2	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	4	4	no	—
45	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₁₃	#41	1	2	yes	—
47	C₄²×C₁₃	#41	1	1	yes	characteristic
48	C₂²⋊Dic₁₃	—	4	1	no	—
49	C₄⋊Dic₁₃	—	2	2	no	—
50	C₄⋊Dic₁₃	—	2	2	no	—
51	#6101	—	4	2	no	—
52	C₄×Dic₁₃	—	4	1	no	—
53	#6099	—	4	2	no	—
54	C₂×#1055	—	4	2	no	—
55	C₂×#1055	—	4	1	no	—
56	#2864	D₁₃	1	1	yes	characteristic
57	#1994	—	13	1	no	—
58	#1518	—	13	2	no	—
59	#1286	—	13	2	no	—
60	#1276	—	13	1	no	—
61	C₂³×C₁₃	#120	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	4	2	no	—
64	C₂×C₄×C₁₃	—	4	2	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	—	4	8	no	—
67	C₂×C₄×C₁₃	—	4	4	no	—
68	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#124	1	2	yes	—
71	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	2	yes	—
73	C₂×C₄×C₁₃	#137	1	2	yes	—
74	C₂×Dic₁₃	—	4	2	no	—
75	Dic₂₆	—	8	2	no	—
76	#1055	—	8	2	no	—
77	#1055	—	8	1	no	—
78	#533	—	13	1	no	—
79	#431	—	13	2	no	—
80	#423	—	13	1	no	—
81	#389	—	2	2	no	—
82	#387	—	2	4	no	—
83	#382	C₂×D₁₃	1	2	yes	—
84	C₄³	C₂×D₁₃	1	1	yes	characteristic
85	#339	—	26	2	no	—
86	#328	—	26	4	no	—
87	C₂²×C₁₃	—	2	2	no	—
88	C₂²×C₁₃	#393	1	1	yes	characteristic
89	C₂²×C₁₃	#452	1	2	yes	—
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	4	4	no	—
93	C₄×C₁₃	—	8	4	no	—
94	C₄×C₁₃	#479	1	2	yes	—
95	C₄×C₁₃	#488	1	1	yes	characteristic
96	C₄×C₁₃	#491	1	1	yes	characteristic
97	Dic₁₃	—	8	2	no	—
98	C₂×M₁₆	—	26	1	no	—
99	C₄⋊Q₈	—	13	2	no	—
100	#125	—	26	1	no	—
101	C₂²⋊Q₈	—	26	2	no	—
102	C₄×Q₈	—	26	2	no	—
103	#117	—	26	1	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
108	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
109	C₄⋊C₈	—	26	2	no	—
110	C₄⋊C₈	—	26	2	no	—
111	C₂²⋊C₈	—	26	2	no	—
112	C₄×C₈	—	26	2	no	—
113	#95	—	2	4	no	—
114	#95	—	2	2	no	—
115	#95	C₂²⋊D₁₃	1	4	yes	—
116	C₂×C₁₃	—	2	2	no	—
117	C₂×C₁₃	#2026	1	1	yes	characteristic
118	C₂×C₁₃	#2031	1	2	yes	—
119	C₂×Q₈	—	26	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	4	no	—
122	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
124	M₁₆	—	52	2	no	—
125	C₂×C₈	—	52	2	no	—
126	C₂×C₈	—	52	1	no	—
127	C₄⋊C₄	—	4	8	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	26	2	no	—
131	C₄⋊C₄	—	26	2	no	—
132	C₄⋊C₄	—	52	2	no	—
133	C₂²⋊C₄	—	52	1	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	4	4	no	—
137	C₄²	—	52	1	no	—
138	C₄²	Q₈×D₁₃	1	1	yes	characteristic
139	C₄²	#6131	1	1	yes	characteristic
140	C₄²	#6131	1	2	yes	—
141	C₁₃	#10367	1	1	yes	characteristic, complemented, Sylow
142	C₂³	#88207	1	1	yes	characteristic
143	Q₈	—	104	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	52	2	no	—
151	C₂×C₄	#88200	1	2	yes	—
152	C₂×C₄	#88200	1	1	yes	characteristic
153	C₂×C₄	#88201	1	2	yes	—
154	C₂×C₄	#88201	1	1	yes	characteristic
155	C₂×C₄	#88204	1	2	yes	—
156	C₂×C₄	#88205	1	2	yes	—
157	C₈	—	104	2	no	—
158	C₈	—	104	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#11718892	1	1	yes	characteristic
161	C₂²	#11718900	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	4	no	—
166	C₄	—	104	2	no	—
167	C₄	#11718724	1	2	yes	—
168	C₄	#11718729	1	1	yes	characteristic
169	C₄	#11718730	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#13115984	1	1	yes	characteristic
172	C₂	#13116005	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115813

C₂

yes

maximal, characteristic

#13113399

C₂

yes

maximal

#13112694

C₂

yes

maximal

#13112669

C₂

yes

maximal, characteristic

C₁₃×#2864

C₂

yes

maximal, characteristic

#11719175

C₂²

yes

characteristic

C₁₃×#389

—

C₁₃×#387

—

C₁₃×#382

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

#11718586

C₂²

yes

—

#11718585

C₂²

yes

characteristic

#11718115

—

#11718044

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

—

—

—

—

Q₈

yes

—

C₄×Dic₂₆

—

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

D₄

yes

—

C₂²⋊#1055

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

—

—

—

D₁₃

yes

—

—

—

—

C₂³×C₁₃

#120

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

#124

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

—

—

—

—

—

—

—

C₂×D₁₃

yes

—

C₄³

C₂×D₁₃

yes

characteristic

#339

—

#328

—

C₂²×C₁₃

—

C₂²×C₁₃

#393

yes

characteristic

C₂²×C₁₃

#452

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#479

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×M₁₆

—

C₄⋊Q₈

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂²⋊D₁₃

yes

—

108

C₂×C₄²

C₂²×D₁₃

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₂²⋊C₈

—

112

C₄×C₈

—

113

#95

—

114

#95

—

115

#95

C₂²⋊D₁₃

yes

—

116

C₂×C₁₃

—

117

C₂×C₁₃

#2026

yes

characteristic

118

C₂×C₁₃

#2031

yes

—

119

C₂×Q₈

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

characteristic

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₂²⋊C₄

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

Q₈×D₁₃

yes

characteristic

139

C₄²

#6131

yes

characteristic

140

C₄²

#6131

yes

—

141

C₁₃

#10367

yes

characteristic, complemented, Sylow

142

C₂³

#88207

yes

characteristic

143

Q₈

—

104

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#88200

yes

—

152

C₂×C₄

#88200

yes

characteristic

153

C₂×C₄

#88201

yes

—

154

C₂×C₄

#88201

yes

characteristic

155

C₂×C₄

#88204

yes

—

156

C₂×C₄

#88205

yes

—

157

C₈

—

104

—

158

C₈

—

104

—

159

C₂²

—

160

C₂²

#11718892

yes

characteristic

161

C₂²

#11718900

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

104

—

167

C₄

#11718724

yes

—

168

C₄

#11718729

yes

characteristic

169

C₄

#11718730

yes

characteristic

170

C₂

—

171

C₂

#13115984

yes

characteristic

172

C₂

#13116005

yes

—

	subgroup	G/H
center	C₂²	#11718892
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2864	C₂
socle	C₂²×C₁₃	#393
prime radical	C₂³×C₁₃	#120
hypercenter	#2864	D₁₃
hypocenter	C₁₃	#10367
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

subgroup

G/H

center

C₂²

#11718892

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#2864

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

N	H ≅ G/N	type
141. C₁₃	#10367	semidirect

H ≅ G/N

type

141. C₁₃

#10367

semidirect

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

subgroup	index
#13115813	2
#13113399	2
#13112694	2
#13112669	2
C₁₃×#2864	2
#10367	13

subgroup

index

C₁₃×#2864

Subgroups — #15766029

Conjugacy classes of subgroups

Complements

Maximal subgroups