Group #15767704 — subgroups

Subgroups — #15767704

Order 3328.

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

normal subgroups only

This group has 272 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	#13115817	C₂	1	1	yes	maximal, characteristic
2	#13114956	C₂	1	1	yes	maximal, characteristic
3	#13113491	C₂	1	2	yes	maximal
4	#13112732	C₂	1	2	yes	maximal
5	C₁₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#11719175	C₂²	1	1	yes	characteristic
7	C₁₃×#431	C₂²	1	2	yes	—
8	C₁₃×#376	C₂²	1	1	yes	characteristic
9	#11718585	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1055	C₂²	1	2	yes	—
11	#11718116	—	2	2	no	—
12	#11718115	—	2	2	no	—
13	#11718074	—	2	2	no	—
14	#11718073	—	2	2	no	—
15	C₂×C₄²×C₁₃	D₄	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
17	#88180	—	2	1	no	—
18	#88178	—	2	2	no	—
19	C₂×#6099	—	2	1	no	—
20	C₂²×#1055	—	2	2	no	—
21	#88124	—	2	1	no	—
22	#88123	D₄	1	2	yes	—
23	C₄×Dic₂₆	—	2	2	no	—
24	C₁₃×C₄⋊C₈	—	2	2	no	—
25	C₁₃×C₂²⋊C₈	—	2	2	no	—
26	C₄×C₈×C₁₃	—	2	2	no	—
27	C₁₃×#95	—	2	2	no	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	#10630	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	#41	1	2	yes	—
35	C₂×Dic₂₆	—	2	2	no	—
36	C₂×C₈×C₁₃	—	4	2	no	—
37	C₄²×C₁₃	—	2	2	no	—
38	C₄²×C₁₃	—	2	2	no	—
39	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₁₃	#41	1	2	yes	—
41	C₄²×C₁₃	#41	1	1	yes	characteristic
42	C₂²⋊Dic₁₃	—	4	1	no	—
43	C₄⋊Dic₁₃	—	2	2	no	—
44	C₄⋊Dic₁₃	—	2	2	no	—
45	#6101	—	4	2	no	—
46	C₄×Dic₁₃	—	4	1	no	—
47	#6099	—	4	2	no	—
48	C₂×#1055	—	4	2	no	—
49	C₂×#1055	—	4	1	no	—
50	C₂×#1055	—	4	2	no	—
51	C₂×#1055	—	4	2	no	—
52	#1995	—	13	1	no	—
53	#1832	—	13	1	no	—
54	#1542	—	13	2	no	—
55	#1301	—	13	2	no	—
56	#1274	D₁₃	1	1	yes	characteristic
57	C₂³×C₁₃	#123	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	4	2	no	—
59	C₂×C₄×C₁₃	—	4	4	no	—
60	C₂×C₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₁₃	#123	1	2	yes	—
63	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₁₃	#126	1	2	yes	—
65	C₂×C₄×C₁₃	#135	1	2	yes	—
66	C₂×C₄×C₁₃	#137	1	2	yes	—
67	C₂×Dic₁₃	—	4	2	no	—
68	Dic₂₆	—	8	2	no	—
69	C₈×C₁₃	—	8	2	no	—
70	#1055	—	8	2	no	—
71	#1055	—	8	1	no	—
72	#533	—	13	1	no	—
73	#431	C₂×D₁₃	1	2	yes	—
74	#423	—	13	1	no	—
75	C₂×C₄⋊C₈	—	13	2	no	—
76	#376	C₂×D₁₃	1	1	yes	characteristic
77	#339	—	26	2	no	—
78	#336	—	26	2	no	—
79	#333	—	26	2	no	—
80	#332	—	26	2	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	#397	1	1	yes	characteristic
83	C₂²×C₁₃	#485	1	2	yes	—
84	C₄×C₁₃	—	2	2	no	—
85	C₄×C₁₃	—	2	2	no	—
86	C₄×C₁₃	—	8	2	no	—
87	C₄×C₁₃	#464	1	2	yes	—
88	C₄×C₁₃	#488	1	1	yes	characteristic
89	C₄×C₁₃	#491	1	1	yes	characteristic
90	Dic₁₃	—	8	2	no	—
91	C₂×M₁₆	—	26	1	no	—
92	C₂²×C₈	—	26	2	no	—
93	C₄⋊Q₈	—	13	2	no	—
94	#125	—	26	1	no	—
95	C₂²⋊Q₈	—	26	2	no	—
96	C₄×Q₈	—	26	2	no	—
97	#117	—	26	1	no	—
98	C₂×C₄²	D₅₂	1	2	yes	—
99	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	26	2	no	—
102	C₄⋊C₈	—	26	2	no	—
103	C₄⋊C₈	—	26	2	no	—
104	C₂²⋊C₈	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	#95	—	2	2	no	—
107	C₂×C₁₃	—	2	2	no	—
108	C₂×C₁₃	#2074	1	1	yes	characteristic
109	C₂×C₁₃	#2080	1	2	yes	—
110	C₂×Q₈	—	26	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
113	C₂²×C₄	#6129	1	2	yes	—
114	M₁₆	—	52	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	52	1	no	—
117	C₂×C₈	—	52	2	no	—
118	C₂×C₈	—	52	2	no	—
119	C₂×C₈	—	52	2	no	—
120	C₄⋊C₄	—	26	2	no	—
121	C₄⋊C₄	—	26	2	no	—
122	C₄⋊C₄	—	52	2	no	—
123	C₂²⋊C₄	—	52	1	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	52	1	no	—
127	C₄²	#6127	1	2	yes	—
128	C₄²	D₄×D₁₃	1	1	yes	characteristic
129	C₄²	#6131	1	1	yes	characteristic
130	C₁₃	#10630	1	1	yes	characteristic, complemented, Sylow
131	C₂³	#88140	1	1	yes	characteristic
132	Q₈	—	104	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	52	2	no	—
137	C₂×C₄	#88129	1	1	yes	characteristic
138	C₂×C₄	#88137	1	2	yes	—
139	C₂×C₄	#88149	1	1	yes	characteristic
140	C₂×C₄	#88152	1	2	yes	—
141	C₂×C₄	#88158	1	2	yes	—
142	C₂×C₄	#88163	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	104	2	no	—
145	C₈	—	104	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#11718335	1	1	yes	characteristic
148	C₂²	#11718384	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	8	2	no	—
152	C₄	—	104	2	no	—
153	C₄	#11718358	1	1	yes	characteristic
154	C₄	#11718364	1	1	yes	characteristic
155	C₄	#11718506	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#13116256	1	1	yes	characteristic
158	C₂	#13116288	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115817

C₂

yes

maximal, characteristic

#13114956

C₂

yes

maximal, characteristic

#13113491

C₂

yes

maximal

#13112732

C₂

yes

maximal

C₁₃×#1274

C₂

yes

maximal, characteristic

#11719175

C₂²

yes

characteristic

C₁₃×#431

C₂²

yes

—

C₁₃×#376

C₂²

yes

characteristic

#11718585

C₂²

yes

characteristic

C₂×C₄⋊#1055

C₂²

yes

—

—

—

—

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

—

—

—

C₂²×#1055

—

#88124

—

#88123

D₄

yes

—

C₄×Dic₂₆

—

C₁₃×C₄⋊C₈

—

C₁₃×C₂²⋊C₈

—

C₄×C₈×C₁₃

—

C₁₃×#95

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

#41

yes

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

—

—

—

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂³×C₁₃

#123

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

#124

yes

characteristic

C₂×C₄×C₁₃

#126

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

—

—

—

C₂×D₁₃

yes

—

#423

—

C₂×C₄⋊C₈

—

#376

C₂×D₁₃

yes

—

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

#397

yes

characteristic

C₂²×C₁₃

#485

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#464

yes

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

D₅₂

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

—

105

C₄×C₈

—

106

#95

—

107

C₂×C₁₃

—

108

C₂×C₁₃

#2074

yes

characteristic

109

C₂×C₁₃

#2080

yes

—

110

C₂×Q₈

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×D₅₂

yes

characteristic

113

C₂²×C₄

#6129

yes

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#6127

yes

—

128

C₄²

D₄×D₁₃

yes

characteristic

129

C₄²

#6131

yes

characteristic

130

C₁₃

#10630

yes

characteristic, complemented, Sylow

131

C₂³

#88140

yes

characteristic

132

Q₈

—

104

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#88129

yes

characteristic

138

C₂×C₄

#88137

yes

—

139

C₂×C₄

#88149

yes

characteristic

140

C₂×C₄

#88152

yes

—

141

C₂×C₄

#88158

yes

—

142

C₂×C₄

#88163

yes

—

143

C₈

—

144

C₈

—

104

—

145

C₈

—

104

—

146

C₂²

—

147

C₂²

#11718335

yes

characteristic

148

C₂²

#11718384

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

104

—

153

C₄

#11718358

yes

characteristic

154

C₄

#11718364

yes

characteristic

155

C₄

#11718506

yes

—

156

C₂

—

157

C₂

#13116256

yes

characteristic

158

C₂

#13116288

yes

—

	subgroup	G/H
center	C₂×C₄	#88129
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1274	C₂
socle	C₂²×C₁₃	#397
prime radical	C₂³×C₁₃	#123
hypercenter	#1274	D₁₃
hypocenter	C₁₃	#10630
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#88129

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1274

C₂

C₂²×C₁₃

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
130. C₁₃	#10630	semidirect

H ≅ G/N

type

130. C₁₃

#10630

semidirect

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

subgroup	index
#13115817	2
#13114956	2
#13113491	2
#13112732	2
C₁₃×#1274	2
#10630	13

subgroup

index

C₁₃×#1274

Subgroups — #15767704

Conjugacy classes of subgroups

Complements

Maximal subgroups