Group #15777107 — subgroups

Subgroups — #15777107

Order 3328.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#13116887	C₂	1	2	yes	maximal
2	#13114454	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic
4	#13112886	C₂	1	2	yes	maximal
5	C₁₃×#1370	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	—	2	2	no	—
7	#11718360	C₂²	1	2	yes	—
8	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
9	#11718343	—	2	1	no	—
10	#11718343	—	2	1	no	—
11	C₈×Dic₂₆	—	2	1	no	—
12	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	#11718095	—	2	2	no	—
15	#11718050	C₂²	1	1	yes	characteristic
16	#11718048	—	2	1	no	—
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	C₂×Dic₅₂	D₄	1	1	yes	characteristic
20	#88123	—	2	2	no	—
21	C₄×Dic₂₆	—	2	1	no	—
22	C₂×C₁₃×C₁₆	—	2	2	no	—
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88058	—	4	2	no	—
27	#88058	—	4	1	no	—
28	#88058	—	4	1	no	—
29	#88058	D₄	1	1	yes	characteristic
30	#88056	—	2	1	no	—
31	#88054	—	4	1	no	—
32	C₈×Dic₁₃	—	4	1	no	—
33	C₄⋊#1055	—	2	1	no	—
34	Dic₁₀₄	—	4	4	no	—
35	#12427	—	13	1	no	maximal, Sylow
36	C₂×Dic₂₆	—	2	1	no	—
37	C₂×Dic₂₆	—	4	2	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
43	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊Dic₁₃	—	2	1	no	—
45	C₄⋊Dic₁₃	—	4	2	no	—
46	C₄⋊Dic₁₃	—	4	2	no	—
47	#6101	—	4	1	no	—
48	C₄×Dic₁₃	—	4	1	no	—
49	C₂×#1055	—	4	1	no	—
50	Dic₅₂	—	2	1	no	—
51	Dic₅₂	—	4	2	no	—
52	Dic₅₂	—	4	2	no	—
53	Dic₅₂	—	8	2	no	—
54	Dic₅₂	D₈	1	2	yes	—
55	C₁₃×C₁₆	—	4	2	no	—
56	#2209	—	13	2	no	—
57	#1713	—	13	1	no	—
58	C₈×Q₁₆	—	13	1	no	—
59	#1370	D₁₃	1	1	yes	characteristic
60	#1362	—	13	2	no	—
61	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×Dic₁₃	—	4	1	no	—
65	C₂×Dic₁₃	—	4	1	no	—
66	C₂×Dic₁₃	—	8	2	no	—
67	Dic₂₆	—	2	2	no	—
68	Dic₂₆	—	4	1	no	—
69	Dic₂₆	—	8	2	no	—
70	Dic₂₆	—	8	2	no	—
71	C₈×C₁₃	—	2	2	no	—
72	C₈×C₁₃	—	2	1	no	—
73	C₈×C₁₃	—	2	1	no	—
74	C₈×C₁₃	Q₃₂	1	2	yes	complemented
75	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
76	C₈×C₁₃	#135	1	1	yes	characteristic
77	#1055	—	8	1	no	—
78	C₂×Q₃₂	—	26	2	no	—
79	#500	—	26	1	no	—
80	#500	—	26	1	no	—
81	#494	—	13	2	no	—
82	C₈×Q₈	—	26	1	no	—
83	C₄×Q₁₆	—	13	1	no	—
84	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
85	#358	—	26	2	no	—
86	#335	—	13	1	no	—
87	Q₈⋊C₈	—	26	1	no	—
88	C₈²	C₂×D₁₃	1	1	yes	characteristic
89	C₂²×C₁₃	#459	1	1	yes	characteristic
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	#466	1	1	yes	characteristic
92	C₄×C₁₃	#472	1	1	yes	characteristic
93	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
94	C₄×C₁₃	#510	1	1	yes	characteristic
95	Dic₁₃	—	4	2	no	—
96	Dic₁₃	—	8	1	no	—
97	Dic₁₃	—	16	2	no	—
98	C₂×Q₁₆	—	13	1	no	—
99	C₂×Q₁₆	—	26	2	no	—
100	C₂×Q₁₆	—	26	2	no	—
101	C₄⋊Q₈	—	26	2	no	—
102	C₄×Q₈	—	26	1	no	—
103	Q₃₂	—	52	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	#107	—	13	1	no	—
106	#107	—	52	2	no	—
107	#107	—	52	1	no	—
108	#107	—	52	1	no	—
109	C₄⋊C₈	—	26	1	no	—
110	C₄⋊C₈	—	52	1	no	—
111	#103	—	26	1	no	—
112	C₄×C₈	—	52	1	no	—
113	C₄×C₈	D₅₂	1	1	yes	characteristic
114	C₄×C₈	D₅₂	1	1	yes	characteristic
115	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
116	C₂×C₁₃	#1667	1	1	yes	characteristic
117	C₂×C₁₃	#2209	1	1	yes	characteristic
118	C₂×C₁₃	#2214	1	1	yes	characteristic
119	C₂×Q₈	—	26	1	no	—
120	C₂×Q₈	—	52	2	no	—
121	Q₁₆	—	13	2	no	—
122	Q₁₆	—	26	1	no	—
123	Q₁₆	—	52	2	no	—
124	Q₁₆	—	52	2	no	—
125	Q₁₆	—	104	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	52	1	no	—
128	C₂×C₈	D₁₀₄	1	1	yes	characteristic
129	C₂×C₈	D₁₀₄	1	1	yes	characteristic
130	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
131	C₂×C₈	#6131	1	1	yes	characteristic
132	C₄⋊C₄	—	26	1	no	—
133	C₄⋊C₄	—	52	1	no	—
134	C₄⋊C₄	—	52	2	no	—
135	C₄⋊C₄	—	52	2	no	—
136	C₄²	—	52	1	no	—
137	C₄²	C₂×D₅₂	1	1	yes	characteristic
138	C₁₆	—	4	2	no	—
139	C₁₃	#12427	1	1	yes	characteristic, complemented, Sylow
140	Q₈	—	26	2	no	—
141	Q₈	—	52	1	no	—
142	Q₈	—	104	2	no	—
143	Q₈	—	104	2	no	—
144	C₂×C₄	—	52	1	no	—
145	C₂×C₄	—	52	1	no	—
146	C₂×C₄	—	104	2	no	—
147	C₂×C₄	#88149	1	1	yes	characteristic
148	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
149	C₂×C₄	#88162	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	104	1	no	—
154	C₈	Dic₁₀₄	1	2	yes	complemented
155	C₈	D₈×D₁₃	1	1	yes	characteristic
156	C₈	#88173	1	1	yes	characteristic
157	C₂²	#11718504	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	52	2	no	—
160	C₄	—	104	1	no	—
161	C₄	—	208	2	no	—
162	C₄	#11718544	1	1	yes	characteristic
163	C₄	#11718558	1	1	yes	characteristic
164	C₄	C₂×Dic₁₀₄	1	1	yes	characteristic
165	C₄	#11718570	1	1	yes	characteristic
166	C₂	#13114192	1	1	yes	characteristic
167	C₂	#13116887	1	1	yes	characteristic
168	C₂	#13116918	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116887

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic

C₈×Dic₅₂

C₂

yes

maximal, characteristic

#13112886

C₂

yes

maximal

C₁₃×#1370

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

—

#11718360

C₂²

yes

—

C₄×Dic₅₂

C₂²

yes

characteristic

#11718343

—

#11718343

—

C₈×Dic₂₆

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718095

—

#11718050

C₂²

yes

characteristic

#11718048

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

D₄

yes

characteristic

#88123

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

characteristic

#88056

—

#88054

—

C₈×Dic₁₃

—

C₄⋊#1055

—

Dic₁₀₄

—

#12427

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

D₈

yes

—

C₁₃×C₁₆

—

#2209

—

#1713

—

C₈×Q₁₆

—

#1370

D₁₃

yes

characteristic

#1362

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

Q₃₂

yes

complemented

C₈×C₁₃

C₂×D₈

yes

C₈×C₁₃

yes

—

C₂×Q₃₂

—

#500

—

#500

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

C₂×D₁₃

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#459

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

characteristic

C₄×C₁₃

#510

yes

characteristic

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₂×Q₁₆

—

101

C₄⋊Q₈

—

102

C₄×Q₈

—

103

Q₃₂

—

104

C₂×C₁₆

—

105

#107

—

106

#107

—

107

#107

—

108

#107

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

112

C₄×C₈

—

113

C₄×C₈

D₅₂

yes

characteristic

114

C₄×C₈

D₅₂

yes

characteristic

115

C₄×C₈

C₂²×D₁₃

yes

characteristic

116

C₂×C₁₃

#1667

yes

characteristic

117

C₂×C₁₃

#2209

yes

characteristic

118

C₂×C₁₃

#2214

yes

characteristic

119

C₂×Q₈

—

120

C₂×Q₈

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

125

Q₁₆

—

104

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

D₁₀₄

yes

characteristic

129

C₂×C₈

D₁₀₄

yes

characteristic

130

C₂×C₈

D₄×D₁₃

yes

characteristic

131

C₂×C₈

#6131

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

—

137

C₄²

C₂×D₅₂

yes

characteristic

138

C₁₆

—

139

C₁₃

#12427

yes

characteristic, complemented, Sylow

140

Q₈

—

141

Q₈

—

142

Q₈

—

104

—

143

Q₈

—

104

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

104

—

147

C₂×C₄

#88149

yes

characteristic

148

C₂×C₄

C₂×D₁₀₄

yes

characteristic

149

C₂×C₄

#88162

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

104

—

154

C₈

Dic₁₀₄

yes

complemented

155

C₈

D₈×D₁₃

yes

characteristic

156

C₈

#88173

yes

characteristic

157

C₂²

#11718504

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

104

—

161

C₄

—

208

—

162

C₄

#11718544

yes

characteristic

163

C₄

#11718558

yes

characteristic

164

C₄

C₂×Dic₁₀₄

yes

characteristic

165

C₄

#11718570

yes

characteristic

166

C₂

#13114192

yes

characteristic

167

C₂

#13116887

yes

characteristic

168

C₂

#13116918

yes

characteristic

	subgroup	G/H
center	C₂²	#11718504
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1370	C₂
socle, prime radical	C₂²×C₁₃	#459
hypercenter	#1370	D₁₃
hypocenter	C₁₃	#12427
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#11718504

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1370

C₂

C₂²×C₁₃

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
74. C₈×C₁₃	Q₃₂	semidirect
139. C₁₃	#12427	semidirect
154. C₈	Dic₁₀₄	semidirect

H ≅ G/N

type

74. C₈×C₁₃

Q₃₂

semidirect

139. C₁₃

#12427

semidirect

154. C₈

Dic₁₀₄

semidirect

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

subgroup	index
#13116887	2
#13114454	2
C₈×Dic₅₂	2
#13112886	2
C₁₃×#1370	2
#12427	13

subgroup

index

C₈×Dic₅₂

C₁₃×#1370

Subgroups — #15777107

Conjugacy classes of subgroups

Complements

Maximal subgroups