Group #15777165 — subgroups

Subgroups — #15777165

Order 3328.

214 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 59 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	#13117089	C₂	1	1	yes	maximal, characteristic, complemented
2	#13116918	C₂	1	2	yes	maximal
3	#13116674	C₂	1	1	yes	maximal, characteristic
4	#13112874	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718570	—	2	2	no	—
7	C₂×Dic₁₀₄	—	2	1	no	—
8	C₂×Dic₁₀₄	—	2	1	no	—
9	#11718562	—	2	1	no	—
10	#11718361	C₂²	1	1	yes	characteristic
11	#11718360	C₂²	1	2	yes	—
12	C₁₃×#364	C₂²	1	2	yes	—
13	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718109	—	2	2	no	—
15	#11718100	—	2	1	no	—
16	#11718055	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	#88158	D₄	1	1	yes	characteristic
20	#88155	—	2	1	no	—
21	#88123	—	2	2	no	—
22	C₁₃×M₃₂	—	2	2	no	—
23	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
24	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	#88059	D₄	1	1	yes	characteristic
27	#88052	—	2	1	no	—
28	#88045	—	2	1	no	—
29	Dic₁₀₄	—	4	2	no	—
30	Dic₁₀₄	—	4	2	no	—
31	Dic₁₀₄	—	4	1	no	—
32	Dic₁₀₄	—	4	1	no	—
33	#88040	—	4	2	no	—
34	#88038	—	4	1	no	—
35	C₁₆×D₁₃	—	4	1	no	—
36	#12437	—	13	1	no	maximal, Sylow
37	#6127	—	2	1	no	—
38	C₂×Dic₂₆	—	4	2	no	—
39	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₁₃	#41	1	1	yes	characteristic
42	C₄⋊Dic₁₃	—	8	2	no	—
43	#6099	—	2	1	no	—
44	Dic₅₂	—	4	2	no	—
45	Dic₅₂	—	4	2	no	—
46	Dic₅₂	—	4	2	no	—
47	Dic₅₂	—	4	2	no	—
48	Dic₅₂	D₈	1	1	yes	characteristic
49	D₁₀₄	D₈	1	1	yes	characteristic
50	#6095	—	2	1	no	—
51	#6094	—	4	1	no	—
52	C₈×D₁₃	—	4	1	no	—
53	C₁₃×C₁₆	—	2	1	no	—
54	C₁₃×C₁₆	—	4	2	no	—
55	C₁₃×C₁₆	D₈	1	1	yes	characteristic
56	C₁₃×C₁₆	D₈	1	1	yes	characteristic
57	#6090	—	4	1	no	—
58	#2247	—	13	1	no	—
59	#2214	—	13	2	no	—
60	#2170	—	13	1	no	—
61	#1373	D₁₃	1	1	yes	characteristic, complemented
62	#1358	—	13	2	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
65	C₂²⋊D₁₃	—	4	1	no	—
66	C₂×Dic₁₃	—	8	2	no	—
67	D₅₂	—	2	1	no	—
68	C₄×D₁₃	—	4	1	no	—
69	Dic₂₆	—	2	1	no	—
70	Dic₂₆	—	8	2	no	—
71	Dic₂₆	—	8	2	no	—
72	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
73	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₈×C₁₃	#135	1	1	yes	characteristic
75	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	#1055	—	4	1	no	—
77	#510	—	26	2	no	—
78	C₂×Q₃₂	—	26	1	no	—
79	C₂×Q₃₂	—	26	1	no	—
80	#504	—	26	1	no	—
81	#494	—	13	2	no	—
82	#443	—	13	1	no	—
83	#364	—	13	1	no	—
84	#364	C₂×D₁₃	1	2	yes	—
85	#362	—	26	2	no	—
86	#350	—	26	1	no	—
87	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
88	C₂²×C₁₃	#472	1	1	yes	characteristic
89	C₂×D₁₃	—	4	1	no	—
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	#459	1	1	yes	characteristic
92	C₄×C₁₃	#466	1	1	yes	characteristic
93	Dic₁₃	—	4	1	no	—
94	Dic₁₃	—	16	2	no	—
95	#135	—	13	1	no	—
96	C₂×Q₁₆	—	26	2	no	—
97	C₂×Q₁₆	—	26	2	no	—
98	#131	—	26	1	no	—
99	C₄⋊Q₈	—	26	2	no	—
100	Q₃₂	—	52	2	no	—
101	Q₃₂	—	52	2	no	—
102	Q₃₂	—	52	1	no	—
103	Q₃₂	—	52	1	no	—
104	SD₃₂	—	52	2	no	—
105	M₃₂	—	2	2	no	—
106	M₃₂	—	26	1	no	—
107	M₃₂	—	52	1	no	—
108	C₂×C₁₆	—	52	1	no	—
109	C₂×C₁₆	D₅₂	1	1	yes	characteristic
110	C₂×C₁₆	D₅₂	1	1	yes	characteristic
111	#108	—	13	1	no	—
112	#104	—	26	1	no	—
113	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₁₃	—	2	1	no	—
115	C₂×C₁₃	#1667	1	1	yes	characteristic
116	D₁₃	—	8	1	no	—
117	#41	—	26	1	no	—
118	C₂×Q₈	—	52	2	no	—
119	Q₁₆	—	13	1	no	—
120	Q₁₆	—	52	2	no	—
121	Q₁₆	—	52	2	no	—
122	Q₁₆	—	52	2	no	—
123	Q₁₆	—	52	2	no	—
124	SD₁₆	—	26	1	no	—
125	D₈	—	13	1	no	—
126	M₁₆	—	26	1	no	—
127	M₁₆	—	52	1	no	—
128	C₂×C₈	—	52	1	no	—
129	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
130	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
131	C₄⋊C₄	—	104	2	no	—
132	C₄²	#6131	1	1	yes	characteristic
133	C₁₆	—	2	1	no	—
134	C₁₆	—	4	2	no	—
135	C₁₆	—	52	1	no	—
136	C₁₆	D₁₀₄	1	1	yes	characteristic
137	C₁₆	D₁₀₄	1	1	yes	characteristic
138	C₁₃	#12437	1	1	yes	characteristic, complemented, Sylow
139	Q₈	—	26	1	no	—
140	Q₈	—	104	2	no	—
141	Q₈	—	104	2	no	—
142	D₄	—	26	1	no	—
143	D₄	—	52	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	52	1	no	—
146	C₂×C₄	—	104	2	no	—
147	C₂×C₄	#88149	1	1	yes	characteristic
148	C₈	—	52	1	no	—
149	C₈	C₂×D₁₀₄	1	1	yes	characteristic
150	C₈	#88162	1	1	yes	characteristic
151	C₈	D₈×D₁₃	1	1	yes	characteristic
152	C₈	#88173	1	1	yes	characteristic
153	C₂²	—	52	1	no	—
154	C₂²	#11718558	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	52	1	no	—
157	C₄	—	208	2	no	—
158	C₄	#11718504	1	1	yes	characteristic
159	C₄	#11718544	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	104	1	no	—
162	C₂	#13114192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic, complemented

#13116918

C₂

yes

maximal

#13116674

C₂

yes

maximal, characteristic

#13112874

C₂

yes

maximal, complemented

C₁₃×#1373

C₂

yes

maximal, characteristic, complemented

#11718570

—

C₂×Dic₁₀₄

—

C₂×Dic₁₀₄

—

#11718562

—

#11718361

C₂²

yes

characteristic

#11718360

C₂²

yes

—

C₁₃×#364

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88158

D₄

yes

characteristic

#88155

—

#88123

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88059

D₄

yes

characteristic

#88052

—

#88045

—

Dic₁₀₄

—

Dic₁₀₄

—

Dic₁₀₄

—

Dic₁₀₄

—

#88040

—

#88038

—

C₁₆×D₁₃

—

#12437

—

maximal, Sylow

#6127

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

#6099

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

D₈

yes

characteristic

D₁₀₄

D₈

yes

characteristic

#6095

—

#6094

—

C₈×D₁₃

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

D₈

yes

characteristic

C₁₃×C₁₆

D₈

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

#1358

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#121

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

C₂×D₈

yes

characteristic

C₈×C₁₃

C₂×D₈

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

—

—

C₂×Q₃₂

—

C₂×Q₃₂

—

—

—

—

—

C₂×D₁₃

yes

—

#362

—

#350

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#472

yes

characteristic

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

—

#131

—

C₄⋊Q₈

—

100

Q₃₂

—

101

Q₃₂

—

102

Q₃₂

—

103

Q₃₂

—

104

SD₃₂

—

105

M₃₂

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

D₅₂

yes

characteristic

110

C₂×C₁₆

D₅₂

yes

characteristic

111

#108

—

112

#104

—

113

C₄×C₈

C₂²×D₁₃

yes

characteristic

114

C₂×C₁₃

—

115

C₂×C₁₃

#1667

yes

characteristic

116

D₁₃

—

117

#41

—

118

C₂×Q₈

—

119

Q₁₆

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

SD₁₆

—

125

D₈

—

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

C₂×D₅₂

yes

characteristic

130

C₂×C₈

D₄×D₁₃

yes

characteristic

131

C₄⋊C₄

—

104

—

132

C₄²

#6131

yes

characteristic

133

C₁₆

—

134

C₁₆

—

135

C₁₆

—

136

C₁₆

D₁₀₄

yes

characteristic

137

C₁₆

D₁₀₄

yes

characteristic

138

C₁₃

#12437

yes

characteristic, complemented, Sylow

139

Q₈

—

140

Q₈

—

104

—

141

Q₈

—

104

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

104

—

147

C₂×C₄

#88149

yes

characteristic

148

C₈

—

149

C₈

C₂×D₁₀₄

yes

characteristic

150

C₈

#88162

yes

characteristic

151

C₈

D₈×D₁₃

yes

characteristic

152

C₈

#88173

yes

characteristic

153

C₂²

—

154

C₂²

#11718558

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

208

—

158

C₄

#11718504

yes

characteristic

159

C₄

#11718544

yes

characteristic

160

C₂

—

161

C₂

—

104

—

162

C₂

#13114192

yes

characteristic

	subgroup	G/H
center	C₂	#13114192
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1373	C₂
socle	C₂×C₁₃	#1667
prime radical	#88158	D₄
hypercenter	#1373	D₁₃
hypocenter	C₁₃	#12437
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₂

#13114192

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1373

C₂

C₂×C₁₃

D₄

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
1. #13117089	C₂	semidirect
4. #13112874	C₂	semidirect
5. C₁₃×#1373	C₂	semidirect
61. #1373	D₁₃	semidirect
138. C₁₃	#12437	semidirect

H ≅ G/N

type

1. #13117089

C₂

semidirect

4. #13112874

C₂

semidirect

5. C₁₃×#1373

C₂

semidirect

61. #1373

D₁₃

semidirect

138. C₁₃

#12437

semidirect

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

subgroup	index
#13117089	2
#13116918	2
#13116674	2
#13112874	2
C₁₃×#1373	2
#12437	13

subgroup

index

C₁₃×#1373

Subgroups — #15777165

Conjugacy classes of subgroups

Complements

Maximal subgroups