Group #14298899 — subgroups

Subgroups — #14298899

Order 1920.

376 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 129 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	#11752341	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#11746767	C₂	1	4	yes	maximal
4	#10603857	—	3	1	no	maximal, Hall
5	C₂×#6758	C₂²	1	4	yes	—
6	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
7	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
8	#91233	—	2	4	no	—
9	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₃×C₈×Dic₅	C₄	1	2	yes	—
11	C₃×C₄×#168	C₄	1	1	yes	characteristic
12	#69791	—	5	1	no	maximal, Hall
13	#65682	—	3	2	no	—
14	C₄×C₈×D₅	D₃	1	1	yes	characteristic
15	#64896	—	3	4	no	—
16	C₂×C₃×C₄×D₅	Q₈	1	2	yes	—
17	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
18	C₂×C₃⋊#170	—	2	4	no	—
19	#6758	—	2	8	no	—
20	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
22	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×Dic₅	D₄	1	2	yes	—
24	C₂×C₃×#168	C₂×C₄	1	2	yes	—
25	C₃×C₈×D₅	—	2	4	no	—
26	C₂×C₃×C₄×C₈	—	5	1	no	—
27	#4924	—	5	2	no	—
28	#4552	—	5	4	no	—
29	C₂×#700	—	3	4	no	—
30	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
31	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
32	#4004	—	6	4	no	—
33	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic
34	C₈×Dic₅	Dic₃	1	2	yes	—
35	C₄×#168	Dic₃	1	1	yes	characteristic
36	#1837	—	15	1	no	Sylow
37	C₂²×C₃×D₅	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
41	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
42	C₃×C₄×D₅	—	2	4	no	—
43	C₃×C₄×D₅	#41	1	4	yes	—
44	C₃⋊#170	—	4	4	no	—
45	C₃×C₅×C₈	—	2	2	no	—
46	C₃×#168	—	2	2	no	—
47	C₂²×C₃×C₈	—	5	2	no	—
48	C₂×C₃×C₄²	—	5	1	no	—
49	C₂×#208	—	5	4	no	—
50	C₃×C₄×C₈	—	5	2	no	—
51	C₃×C₄×C₈	—	5	1	no	—
52	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄⋊(C₃⋊C₈)	—	10	4	no	—
54	C₂×C₄×D₅	Dic₆	1	2	yes	—
55	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
56	C₂×#170	—	6	4	no	—
57	#700	—	6	8	no	—
58	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
59	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
60	C₄×Dic₅	D₁₂	1	2	yes	—
61	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
62	C₂×#168	C₂×Dic₃	1	2	yes	—
63	C₈×D₅	—	2	4	no	—
64	#423	—	15	2	no	—
65	C₂×C₄×C₈	—	5	1	no	—
66	#341	—	15	4	no	—
67	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₃×D₅	—	2	2	no	—
69	C₂×C₃×D₅	#125	1	2	yes	—
70	C₃×C₄×C₅	—	2	2	no	—
71	C₃×C₄×C₅	#117	1	2	yes	—
72	C₃×Dic₅	#108	1	4	yes	—
73	C₃×Dic₅	#124	1	2	yes	—
74	C₂²×C₃×C₄	—	5	1	no	—
75	C₂²×C₃×C₄	—	5	2	no	—
76	C₂×C₃×C₈	—	5	2	no	—
77	C₂×C₃×C₈	—	10	4	no	—
78	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
79	C₃×C₄²	—	5	2	no	—
80	C₃×C₄²	—	5	1	no	—
81	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	#208	—	10	8	no	—
83	C₂×C₃⋊C₈	—	10	4	no	—
84	C₂²×D₅	C₂×Dic₆	1	1	yes	characteristic
85	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
86	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
87	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
88	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
89	C₄×D₅	—	2	4	no	—
90	C₄×D₅	#235	1	4	yes	—
91	#170	—	12	4	no	—
92	C₅×C₈	—	2	2	no	—
93	#168	—	2	2	no	—
94	C₂×M₁₆	—	15	4	no	—
95	C₂²×C₈	—	5	2	no	—
96	C₂×C₄²	—	5	1	no	—
97	C₄⋊C₈	—	30	4	no	—
98	C₄×C₈	—	5	2	no	—
99	C₄×C₈	—	5	1	no	—
100	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
101	C₂×C₃×C₅	#382	1	1	yes	characteristic
102	C₂×C₃×C₅	C₂×#108	1	2	yes	—
103	C₃×D₅	—	2	2	no	—
104	C₂³×C₃	—	5	1	no	—
105	C₂×C₃×C₄	—	5	2	no	—
106	C₂×C₃×C₄	—	5	1	no	—
107	C₂×C₃×C₄	—	5	4	no	—
108	C₂×C₃×C₄	—	10	4	no	—
109	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
110	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₃×C₈	—	2	2	no	—
112	C₃×C₈	—	10	2	no	—
113	C₃⋊C₈	—	20	4	no	—
114	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
115	C₂×D₅	—	2	2	no	—
116	C₂×D₅	#870	1	2	yes	—
117	C₄×C₅	—	2	2	no	—
118	C₄×C₅	#926	1	2	yes	—
119	Dic₅	#819	1	4	yes	—
120	Dic₅	#875	1	2	yes	—
121	C₂²×C₄	—	5	1	no	—
122	C₂²×C₄	—	5	2	no	—
123	M₁₆	—	30	8	no	—
124	C₂×C₈	—	5	2	no	—
125	C₂×C₈	—	10	4	no	—
126	C₂×C₈	—	30	4	no	—
127	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
128	C₄²	—	5	2	no	—
129	C₄²	—	5	1	no	—
130	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
131	C₃×C₅	#1837	1	1	yes	characteristic, complemented, Hall
132	C₂²×C₃	—	5	2	no	—
133	C₂²×C₃	—	10	2	no	—
134	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	5	2	no	—
137	C₃×C₄	—	5	4	no	—
138	C₃×C₄	#4130	1	2	yes	—
139	C₂×C₅	#4936	1	1	yes	characteristic
140	C₂×C₅	C₂×#819	1	2	yes	—
141	D₅	—	2	2	no	—
142	C₂³	—	5	1	no	—
143	C₂×C₄	—	5	1	no	—
144	C₂×C₄	—	5	2	no	—
145	C₂×C₄	—	5	4	no	—
146	C₂×C₄	—	10	4	no	—
147	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
148	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	10	2	no	—
151	C₈	—	60	4	no	—
152	C₂×C₃	—	10	2	no	—
153	C₂×C₃	#65689	1	1	yes	characteristic
154	C₂×C₃	D₅⋊#108	1	2	yes	—
155	C₅	#69791	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	5	2	no	—
157	C₂²	—	10	2	no	—
158	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	5	2	no	—
161	C₄	—	5	4	no	—
162	C₄	#91989	1	2	yes	—
163	C₃	#10603857	1	1	yes	characteristic, complemented, Sylow
164	C₂	—	10	2	no	—
165	C₂	#11752348	1	1	yes	characteristic
166	C₂	D₅⋊#819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11752341

C₂

yes

maximal

C₃×C₄×C₈×D₅

C₂

yes

maximal, characteristic

#11746767

C₂

yes

—

C₂²

yes

—

C₂×C₃×C₈×D₅

C₂²

yes

—

C₃×C₄²×D₅

C₂²

yes

characteristic

#91233

—

C₃×C₄×C₅×C₈

C₄

yes

characteristic

C₃×C₈×Dic₅

C₄

yes

—

C₃×C₄×#168

C₄

yes

—

—

C₄×C₈×D₅

D₃

yes

characteristic

#64896

—

C₂×C₃×C₄×D₅

Q₈

yes

—

C₂×C₃×C₄×D₅

C₂³

yes

characteristic

C₂×C₃⋊#170

—

#6758

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

C₂×C₄

yes

characteristic

C₃×C₄×Dic₅

D₄

yes

—

C₂×C₃×#168

C₂×C₄

yes

—

C₃×C₈×D₅

—

C₂×C₃×C₄×C₈

—

#4924

—

#4552

—

C₂×#700

—

C₂×C₈×D₅

C₂×D₃

yes

—

C₄²×D₅

C₂×D₃

yes

characteristic

#4004

—

C₄×C₅×C₈

Dic₃

yes

characteristic

C₈×Dic₅

Dic₃

yes

—

C₄×#168

Dic₃

yes

characteristic

#1837

—

Sylow

C₂²×C₃×D₅

C₂×Q₈

yes

characteristic

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₃×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₃×Dic₅

C₄⋊C₄

yes

—

C₂×C₃×Dic₅

C₂×D₄

yes

characteristic

C₃×C₄×D₅

—

C₃×C₄×D₅

#41

yes

—

C₃⋊#170

—

C₃×C₅×C₈

—

C₃×#168

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

—

C₂×#208

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₄⋊(C₃⋊C₈)

—

C₂×C₄×D₅

Dic₆

yes

—

C₂×C₄×D₅

C₂²×D₃

yes

characteristic

C₂×#170

—

#700

—

C₂×C₅×C₈

C₂×Dic₃

yes

—

C₄²×C₅

C₂×Dic₃

yes

characteristic

C₄×Dic₅

D₁₂

yes

—

C₄×Dic₅

C₂×Dic₃

yes

characteristic

C₂×#168

C₂×Dic₃

yes

—

C₈×D₅

—

#423

—

C₂×C₄×C₈

—

#341

—

C₂²×C₃×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

#125

yes

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#117

yes

—

C₃×Dic₅

#108

yes

—

C₃×Dic₅

#124

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂²×D₅

C₂×Dic₆

yes

characteristic

C₂×C₄×C₅

C₄⋊Dic₃

yes

—

C₂×C₄×C₅

C₂²×Dic₃

yes

characteristic

C₂×Dic₅

C₄⋊Dic₃

yes

—

C₂×Dic₅

C₂×D₁₂

yes

characteristic

C₄×D₅

—

C₄×D₅

#235

yes

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

ΓL(1,𝔽₁₆)

yes

characteristic

101

C₂×C₃×C₅

#382

yes

characteristic

102

C₂×C₃×C₅

C₂×#108

yes

—

103

C₃×D₅

—

104

C₂³×C₃

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

110

C₂×C₃×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

111

C₃×C₈

—

112

C₃×C₈

—

113

C₃⋊C₈

—

114

C₂²×C₅

C₂×C₄⋊Dic₃

yes

characteristic

115

C₂×D₅

—

116

C₂×D₅

#870

yes

—

117

C₄×C₅

—

118

C₄×C₅

#926

yes

—

119

Dic₅

#819

yes

—

120

Dic₅

#875

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

131

C₃×C₅

#1837

yes

characteristic, complemented, Hall

132

C₂²×C₃

—

133

C₂²×C₃

—

134

C₂²×C₃

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

#4130

yes

—

139

C₂×C₅

#4936

yes

characteristic

140

C₂×C₅

C₂×#819

yes

—

141

D₅

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₄⋊ΓL(1,𝔽₁₆)

yes

—

148

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

152

C₂×C₃

—

153

C₂×C₃

#65689

yes

characteristic

154

C₂×C₃

D₅⋊#108

yes

—

155

C₅

#69791

yes

characteristic, complemented, Sylow

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₄⋊ΓL(1,𝔽₁₆)

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#91989

yes

—

163

C₃

#10603857

yes

characteristic, complemented, Sylow

164

C₂

—

165

C₂

#11752348

yes

characteristic

166

C₂

D₅⋊#819

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×Q₈
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1837
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊ΓL(1,𝔽₁₆)

commutator

C₂×C₃×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×ΓL(1,𝔽₁₆)

Fitting

C₃×C₄×C₅×C₈

C₄

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂²×C₃×D₅

C₂×Q₈

cosocle

C₂×C₃×C₄×D₅

C₂³

hypercenter

C₄×C₈

ΓL(1,𝔽₁₆)

hypocenter

C₃×C₅

#1837

LCS

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
131. C₃×C₅	#1837	semidirect
155. C₅	#69791	semidirect
163. C₃	#10603857	semidirect

H ≅ G/N

type

131. C₃×C₅

#1837

semidirect

155. C₅

#69791

semidirect

163. C₃

#10603857

semidirect

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

subgroup	index
#11752341	2
C₃×C₄×C₈×D₅	2
#11746767	2
#10603857	3
#69791	5

subgroup

index

C₃×C₄×C₈×D₅

Subgroups — #14298899

Conjugacy classes of subgroups

Complements

Maximal subgroups