Group #15783715 — subgroups

Subgroups — #15783715

Order 3328.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₂×Dic₂₀₈	C₂	1	2	yes	maximal
2	#13117089	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
4	#13113018	C₂	1	2	yes	maximal
5	C₁₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	—	2	1	no	—
7	C₂×Dic₁₀₄	C₂²	1	2	yes	—
8	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
9	#11718360	—	2	1	no	—
10	C₄×Dic₅₂	—	2	1	no	—
11	C₂×C₁₃×C₃₂	C₂²	1	2	yes	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718096	C₂²	1	1	yes	characteristic
14	#11718095	—	2	1	no	—
15	Dic₂₀₈	—	2	4	no	—
16	C₂×Dic₅₂	—	2	1	no	—
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	4	1	no	—
19	#88123	—	4	1	no	—
20	C₄×Dic₂₆	—	4	1	no	—
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	#88058	—	2	1	no	—
25	#88056	—	4	1	no	—
26	Dic₁₀₄	—	2	2	no	—
27	Dic₁₀₄	—	2	1	no	—
28	Dic₁₀₄	—	2	2	no	—
29	Dic₁₀₄	—	4	2	no	—
30	Dic₁₀₄	D₄	1	2	yes	—
31	C₁₃×C₃₂	—	2	2	no	—
32	#13688	—	13	1	no	maximal, Sylow
33	C₂×Dic₂₆	—	4	1	no	—
34	C₂×Dic₂₆	—	4	2	no	—
35	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₁₃	D₈	1	1	yes	characteristic
38	C₄⋊Dic₁₃	—	4	1	no	—
39	C₄⋊Dic₁₃	—	8	1	no	—
40	C₄⋊Dic₁₃	—	8	1	no	—
41	#6101	—	8	1	no	—
42	C₄×Dic₁₃	—	8	1	no	—
43	Dic₅₂	—	2	2	no	—
44	Dic₅₂	—	4	2	no	—
45	Dic₅₂	—	4	1	no	—
46	Dic₅₂	—	4	2	no	—
47	Dic₅₂	—	8	2	no	—
48	C₁₃×C₁₆	—	2	1	no	—
49	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
50	C₁₃×C₁₆	#41	1	1	yes	characteristic
51	C₂×Q₆₄	—	13	2	no	—
52	#2247	—	13	1	no	—
53	C₄×Q₃₂	—	13	1	no	—
54	C₄⋊C₃₂	D₁₃	1	1	yes	characteristic
55	#1416	—	13	2	no	—
56	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₁₃	—	8	1	no	—
60	C₂×Dic₁₃	—	8	1	no	—
61	C₂×Dic₁₃	—	8	2	no	—
62	Dic₂₆	—	4	2	no	—
63	Dic₂₆	—	8	1	no	—
64	Dic₂₆	—	8	2	no	—
65	Dic₂₆	—	8	2	no	—
66	C₈×C₁₃	—	2	1	no	—
67	C₈×C₁₃	#121	1	1	yes	characteristic
68	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂×Q₃₂	—	13	1	no	—
70	C₂×Q₃₂	—	13	2	no	—
71	C₂×Q₃₂	—	26	1	no	—
72	#494	—	26	1	no	—
73	C₄×Q₁₆	—	26	1	no	—
74	Q₆₄	—	26	4	no	—
75	C₂×C₃₂	C₂×D₁₃	1	2	yes	—
76	#366	—	13	1	no	—
77	#358	—	26	1	no	—
78	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
79	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	Q₆₄	1	2	yes	complemented
82	C₄×C₁₃	#459	1	1	yes	characteristic
83	C₄×C₁₃	#509	1	1	yes	characteristic
84	Dic₁₃	—	8	2	no	—
85	Dic₁₃	—	16	2	no	—
86	Dic₁₃	—	16	1	no	—
87	C₂×Q₁₆	—	26	1	no	—
88	C₂×Q₁₆	—	26	2	no	—
89	C₂×Q₁₆	—	52	1	no	—
90	C₄⋊Q₈	—	52	1	no	—
91	C₄×Q₈	—	52	1	no	—
92	Q₃₂	—	13	2	no	—
93	Q₃₂	—	26	2	no	—
94	Q₃₂	—	26	1	no	—
95	Q₃₂	—	26	2	no	—
96	Q₃₂	—	52	2	no	—
97	C₂×C₁₆	D₅₂	1	1	yes	characteristic
98	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
99	#107	—	26	1	no	—
100	#103	—	52	1	no	—
101	C₄×C₈	D₅₂	1	1	yes	characteristic
102	C₃₂	—	2	2	no	—
103	C₂×C₁₃	#2206	1	1	yes	characteristic
104	C₂×C₁₃	C₂×Q₆₄	1	1	yes	characteristic
105	C₂×C₁₃	#2264	1	1	yes	characteristic
106	C₂×Q₈	—	52	1	no	—
107	C₂×Q₈	—	52	2	no	—
108	Q₁₆	—	26	2	no	—
109	Q₁₆	—	52	2	no	—
110	Q₁₆	—	52	1	no	—
111	Q₁₆	—	52	2	no	—
112	Q₁₆	—	104	2	no	—
113	C₂×C₈	D₁₀₄	1	1	yes	characteristic
114	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
115	C₄⋊C₄	—	52	1	no	—
116	C₄⋊C₄	—	104	1	no	—
117	C₄⋊C₄	—	104	1	no	—
118	C₄⋊C₄	—	104	1	no	—
119	C₄²	—	104	1	no	—
120	C₄²	D₁₀₄	1	1	yes	characteristic
121	C₁₆	—	2	1	no	—
122	C₁₆	D₄×D₁₃	1	1	yes	characteristic
123	C₁₆	#6131	1	1	yes	characteristic
124	C₁₃	#13688	1	1	yes	characteristic, complemented, Sylow
125	Q₈	—	52	2	no	—
126	Q₈	—	104	1	no	—
127	Q₈	—	104	2	no	—
128	Q₈	—	104	2	no	—
129	C₂×C₄	—	104	1	no	—
130	C₂×C₄	—	104	1	no	—
131	C₂×C₄	—	104	2	no	—
132	C₂×C₄	D₂₀₈	1	1	yes	characteristic
133	C₂×C₄	D₂₀₈	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#88149	1	1	yes	characteristic
137	C₈	#88162	1	1	yes	characteristic
138	C₂²	C₂×D₂₀₈	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	104	2	no	—
141	C₄	—	208	2	no	—
142	C₄	—	208	1	no	—
143	C₄	Dic₂₀₈	1	2	yes	complemented
144	C₄	#11718504	1	1	yes	characteristic
145	C₄	#11718569	1	1	yes	characteristic
146	C₂	#13116872	1	1	yes	characteristic
147	C₂	C₂×Dic₂₀₈	1	1	yes	characteristic
148	C₂	#13117156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₀₈

C₂

yes

maximal

#13117089

C₂

yes

maximal, characteristic

C₄×Dic₁₀₄

C₂

yes

maximal, characteristic

#13113018

C₂

yes

maximal

C₁₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

—

C₂×Dic₁₀₄

C₂²

yes

—

C₂×Dic₁₀₄

C₂²

yes

characteristic

#11718360

—

C₄×Dic₅₂

—

C₂×C₁₃×C₃₂

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718096

C₂²

yes

characteristic

#11718095

—

Dic₂₀₈

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₄×Dic₂₆

—

C₂×C₁₃×C₁₆

D₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂³

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

#88058

—

#88056

—

Dic₁₀₄

—

Dic₁₀₄

—

Dic₁₀₄

—

Dic₁₀₄

—

Dic₁₀₄

D₄

yes

—

C₁₃×C₃₂

—

#13688

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₄²×C₁₃

D₈

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×D₄

yes

characteristic

C₁₃×C₁₆

#41

yes

characteristic

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

—

C₄⋊C₃₂

D₁₃

yes

characteristic

#1416

—

C₂×C₄×C₁₃

D₁₆

yes

characteristic

C₂×C₄×C₁₃

D₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

C₂×D₁₃

yes

—

#366

—

#358

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₂×D₁₆

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

Q₆₄

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

D₅₂

yes

characteristic

C₂×C₁₆

C₂²×D₁₃

yes

characteristic

#107

—

100

#103

—

101

C₄×C₈

D₅₂

yes

characteristic

102

C₃₂

—

103

C₂×C₁₃

#2206

yes

characteristic

104

C₂×C₁₃

C₂×Q₆₄

yes

characteristic

105

C₂×C₁₃

#2264

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

112

Q₁₆

—

104

—

113

C₂×C₈

D₁₀₄

yes

characteristic

114

C₂×C₈

C₂×D₅₂

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

104

—

117

C₄⋊C₄

—

104

—

118

C₄⋊C₄

—

104

—

119

C₄²

—

104

—

120

C₄²

D₁₀₄

yes

characteristic

121

C₁₆

—

122

C₁₆

D₄×D₁₃

yes

characteristic

123

C₁₆

#6131

yes

characteristic

124

C₁₃

#13688

yes

characteristic, complemented, Sylow

125

Q₈

—

126

Q₈

—

104

—

127

Q₈

—

104

—

128

Q₈

—

104

—

129

C₂×C₄

—

104

—

130

C₂×C₄

—

104

—

131

C₂×C₄

—

104

—

132

C₂×C₄

D₂₀₈

yes

characteristic

133

C₂×C₄

D₂₀₈

yes

characteristic

134

C₂×C₄

C₂×D₁₀₄

yes

characteristic

135

C₈

—

136

C₈

#88149

yes

characteristic

137

C₈

#88162

yes

characteristic

138

C₂²

C₂×D₂₀₈

yes

characteristic

139

C₄

—

140

C₄

—

104

—

141

C₄

—

208

—

142

C₄

—

208

—

143

C₄

Dic₂₀₈

yes

complemented

144

C₄

#11718504

yes

characteristic

145

C₄

#11718569

yes

characteristic

146

C₂

#13116872

yes

characteristic

147

C₂

C₂×Dic₂₀₈

yes

characteristic

148

C₂

#13117156

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₀₈
commutator, cosocle	C₂×C₁₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₁₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₁₃	C₂×D₁₆
hypercenter	C₄⋊C₃₂	D₁₃
hypocenter	C₁₃	#13688
LCS	G ▷ C₂×C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center

C₂²

C₂×D₂₀₈

commutator, cosocle

C₂×C₁₃×C₁₆

C₂³

Frattini

C₂×C₁₆

C₂²×D₁₃

Fitting

C₁₃×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₁₃

C₂×D₁₆

hypercenter

C₄⋊C₃₂

D₁₃

hypocenter

C₁₃

#13688

LCS

G ▷ C₂×C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
81. C₄×C₁₃	Q₆₄	semidirect
124. C₁₃	#13688	semidirect
143. C₄	Dic₂₀₈	semidirect

H ≅ G/N

type

81. C₄×C₁₃

Q₆₄

semidirect

124. C₁₃

#13688

semidirect

143. C₄

Dic₂₀₈

semidirect

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

subgroup	index
C₂×Dic₂₀₈	2
#13117089	2
C₄×Dic₁₀₄	2
#13113018	2
C₁₃×C₄⋊C₃₂	2
#13688	13

subgroup

index

C₂×Dic₂₀₈

#13117089

C₄×Dic₁₀₄

#13113018

C₁₃×C₄⋊C₃₂

#13688

Subgroups — #15783715

Conjugacy classes of subgroups

Complements

Maximal subgroups