Group #14278927 — subgroups

Subgroups — #14278927

Order 1920.

248 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 89 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	D₅⋊#819	C₂	1	2	yes	maximal
2	C₂×C₃×#3930	C₂	1	1	yes	maximal, characteristic
3	#10598668	—	3	1	no	maximal, Hall
4	D₅⋊#208	—	2	2	no	—
5	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
6	#91225	—	2	2	no	—
7	#91224	C₄	1	4	yes	—
8	C₂×C₃×C₅×C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃×#672	C₄	1	1	yes	characteristic
10	C₃×#3930	—	2	2	no	—
11	#67820	—	5	1	no	maximal, Hall
12	D₅⋊#108	—	3	2	no	—
13	C₂×#3930	D₃	1	1	yes	characteristic
14	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
15	#6762	—	4	2	no	—
16	#6758	—	2	4	no	—
17	D₅⋊(C₃⋊C₈)	—	4	2	no	—
18	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×#168	Q₈	1	1	yes	characteristic
20	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
21	C₃×C₈×D₅	D₄	1	2	yes	—
22	C₃×C₅×C₁₆	—	2	1	no	—
23	C₃×#672	—	2	1	no	—
24	C₂×C₃×M₃₂	—	5	1	no	—
25	C₂×#819	—	5	2	no	—
26	D₅⋊M₁₆	—	6	2	no	—
27	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
28	#3996	—	6	2	no	—
29	#3995	—	3	4	no	—
30	C₂×C₅×C₁₆	Dic₃	1	1	yes	characteristic
31	C₂×#672	Dic₃	1	1	yes	characteristic
32	#3930	—	2	2	no	—
33	#1385	—	15	1	no	Sylow
34	C₂²×C₃×D₅	Q₁₆	1	1	yes	characteristic
35	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
37	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
38	C₃×C₄×D₅	SD₁₆	1	2	yes	—
39	C₃⋊#170	—	4	4	no	—
40	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
41	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₃×#168	C₄⋊C₄	1	2	yes	—
43	C₂²×C₃×C₈	—	5	1	no	—
44	C₂×#208	—	10	2	no	—
45	C₃×M₃₂	—	10	2	no	—
46	C₂×C₃×C₁₆	—	5	1	no	—
47	C₂×C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
48	#819	—	5	4	no	—
49	#819	—	10	2	no	—
50	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
51	#704	—	12	2	no	—
52	#700	—	6	4	no	—
53	D₅⋊C₈	—	12	2	no	—
54	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
55	C₂×#168	Dic₆	1	1	yes	characteristic
56	C₈×D₅	C₄×D₃	1	2	yes	—
57	C₈×D₅	#67	1	2	yes	—
58	C₅×C₁₆	—	2	1	no	—
59	#672	—	2	1	no	—
60	C₂×M₃₂	—	5	1	no	—
61	C₂×#108	—	15	2	no	—
62	C₂²×C₃×C₅	#107	1	1	yes	characteristic
63	C₂×C₃×D₅	—	4	1	no	—
64	C₂×C₃×D₅	#103	1	2	yes	—
65	C₃×C₄×C₅	#95	1	1	yes	characteristic
66	C₃×C₄×C₅	#106	1	1	yes	characteristic
67	C₃×Dic₅	#102	1	2	yes	—
68	C₂²×C₃×C₄	—	5	1	no	—
69	C₂×C₃×C₈	—	5	1	no	—
70	C₂×C₃×C₈	—	5	2	no	—
71	C₂×C₃×C₈	—	5	2	no	—
72	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	#208	—	10	4	no	—
74	#208	—	20	2	no	—
75	C₂×C₃⋊C₈	—	20	2	no	—
76	C₃×C₁₆	—	2	1	no	—
77	C₃×C₁₆	—	10	1	no	—
78	C₂²×D₅	Dic₁₂	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
80	C₂×Dic₅	D₂₄	1	1	yes	characteristic
81	C₄×D₅	#204	1	2	yes	—
82	C₄×D₅	(C₂×D₃)⋊C₄	1	2	yes	—
83	#170	—	12	4	no	—
84	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
85	C₅×C₈	#217	1	1	yes	characteristic
86	#168	Dic₃⋊C₄	1	2	yes	—
87	C₂×M₁₆	—	30	2	no	—
88	C₂²×C₈	—	5	1	no	—
89	M₃₂	—	10	2	no	—
90	C₂×C₁₆	—	5	1	no	—
91	C₂×C₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
92	#108	—	15	4	no	—
93	#108	—	30	2	no	—
94	C₂×C₃×C₅	—	2	1	no	—
95	C₂×C₃×C₅	#340	1	1	yes	characteristic
96	C₃×D₅	—	2	2	no	—
97	C₂³×C₃	—	5	1	no	—
98	C₂×C₃×C₄	—	5	1	no	—
99	C₂×C₃×C₄	—	5	2	no	—
100	C₂×C₃×C₄	—	5	2	no	—
101	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₃×C₈	—	5	2	no	—
103	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
104	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₃⋊C₈	—	20	4	no	—
106	C₂²×C₅	#818	1	1	yes	characteristic
107	C₂×D₅	—	4	1	no	—
108	C₂×D₅	#816	1	2	yes	—
109	C₄×C₅	#817	1	1	yes	characteristic
110	C₄×C₅	C₃⋊#95	1	1	yes	characteristic
111	Dic₅	#821	1	2	yes	—
112	C₂²×C₄	—	5	1	no	—
113	M₁₆	—	30	4	no	—
114	M₁₆	—	60	2	no	—
115	C₂×C₈	—	5	1	no	—
116	C₂×C₈	—	5	2	no	—
117	C₂×C₈	—	5	2	no	—
118	C₂×C₈	—	60	2	no	—
119	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
120	C₁₆	—	2	1	no	—
121	C₁₆	—	10	1	no	—
122	C₃×C₅	#1385	1	1	yes	characteristic, complemented, Hall
123	C₂²×C₃	—	5	2	no	—
124	C₂²×C₃	—	20	1	no	—
125	C₂²×C₃	#3994	1	1	yes	characteristic
126	C₃×C₄	—	5	2	no	—
127	C₃×C₄	#3993	1	1	yes	characteristic
128	C₃×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
129	C₂×C₅	—	2	1	no	—
130	C₂×C₅	#4551	1	1	yes	characteristic
131	D₅	—	2	2	no	—
132	C₂³	—	5	1	no	—
133	C₂×C₄	—	5	1	no	—
134	C₂×C₄	—	5	2	no	—
135	C₂×C₄	—	5	2	no	—
136	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
137	C₈	—	5	2	no	—
138	C₈	—	60	4	no	—
139	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
140	C₈	#6763	1	1	yes	characteristic
141	C₂×C₃	—	2	1	no	—
142	C₂×C₃	—	10	2	no	—
143	C₂×C₃	#64893	1	1	yes	characteristic
144	C₅	#67820	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	5	2	no	—
146	C₂²	—	20	1	no	—
147	C₂²	#91223	1	1	yes	characteristic
148	C₄	—	5	2	no	—
149	C₄	#91222	1	1	yes	characteristic
150	C₄	#91235	1	1	yes	characteristic
151	C₃	#10598668	1	1	yes	characteristic, complemented, Sylow
152	C₂	—	2	1	no	—
153	C₂	—	10	2	no	—
154	C₂	#11746764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅⋊#819

C₂

yes

maximal

C₂×C₃×#3930

C₂

yes

maximal, characteristic

#10598668

—

maximal, Hall

D₅⋊#208

—

C₂×C₃×C₈×D₅

C₂²

yes

characteristic

#91225

—

#91224

C₄

yes

—

C₂×C₃×C₅×C₁₆

C₄

yes

characteristic

C₂×C₃×#672

C₄

yes

—

—

—

D₃

yes

characteristic

C₂×C₃×C₄×D₅

D₄

yes

characteristic

#6762

—

#6758

—

D₅⋊(C₃⋊C₈)

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₃×#168

Q₈

yes

characteristic

C₃×C₈×D₅

C₂×C₄

yes

—

C₃×C₈×D₅

D₄

yes

—

C₃×C₅×C₁₆

—

C₃×#672

—

C₂×C₃×M₃₂

—

C₂×#819

—

D₅⋊M₁₆

—

C₂×C₈×D₅

C₂×D₃

yes

characteristic

#3996

—

#3995

—

C₂×C₅×C₁₆

Dic₃

yes

characteristic

C₂×#672

Dic₃

yes

—

—

C₂²×C₃×D₅

Q₁₆

yes

characteristic

C₂×C₃×C₄×C₅

C₄⋊C₄

yes

characteristic

C₂×C₃×Dic₅

D₈

yes

characteristic

C₃×C₄×D₅

C₂²⋊C₄

yes

—

C₃×C₄×D₅

SD₁₆

yes

—

C₃⋊#170

—

C₃×C₅×C₈

C₄²

yes

characteristic

C₃×C₅×C₈

C₂²⋊C₄

yes

characteristic

C₃×#168

C₄⋊C₄

yes

—

C₂²×C₃×C₈

—

C₂×#208

—

C₃×M₃₂

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

yes

—

—

C₂×C₄×D₅

D₁₂

yes

characteristic

#704

—

#700

—

D₅⋊C₈

—

C₂×C₅×C₈

C₂×Dic₃

yes

characteristic

C₂×#168

Dic₆

yes

characteristic

C₈×D₅

C₄×D₃

yes

—

C₈×D₅

#67

yes

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

C₂×#108

—

C₂²×C₃×C₅

#107

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

#103

yes

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

C₃×Dic₅

yes

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

yes

—

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₂²×D₅

Dic₁₂

yes

characteristic

C₂×C₄×C₅

C₄⋊Dic₃

yes

characteristic

C₂×Dic₅

D₂₄

yes

characteristic

C₄×D₅

#204

yes

—

C₄×D₅

(C₂×D₃)⋊C₄

yes

—

#170

—

C₅×C₈

C₄×Dic₃

yes

C₅×C₈

yes

Dic₃⋊C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

ΓL(1,𝔽₁₆)

yes

characteristic

#108

—

#108

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#340

yes

characteristic

C₃×D₅

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

C₄⋊AGL(1,𝔽₅)

yes

characteristic

102

C₃×C₈

—

103

C₃×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

104

C₃×C₈

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

105

C₃⋊C₈

—

106

C₂²×C₅

#818

yes

characteristic

107

C₂×D₅

—

108

C₂×D₅

#816

yes

—

109

C₄×C₅

#817

yes

characteristic

110

C₄×C₅

C₃⋊#95

yes

characteristic

111

Dic₅

#821

yes

—

112

C₂²×C₄

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

120

C₁₆

—

121

C₁₆

—

122

C₃×C₅

#1385

yes

characteristic, complemented, Hall

123

C₂²×C₃

—

124

C₂²×C₃

—

125

C₂²×C₃

#3994

yes

characteristic

126

C₃×C₄

—

127

C₃×C₄

#3993

yes

characteristic

128

C₃×C₄

(C₂×AGL(1,𝔽₅))⋊C₄

yes

characteristic

129

C₂×C₅

—

130

C₂×C₅

#4551

yes

characteristic

131

D₅

—

132

C₂³

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₄⋊ΓL(1,𝔽₁₆)

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

C₄×ΓL(1,𝔽₁₆)

yes

characteristic

140

C₈

#6763

yes

characteristic

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

#64893

yes

characteristic

144

C₅

#67820

yes

characteristic, complemented, Sylow

145

C₂²

—

146

C₂²

—

147

C₂²

#91223

yes

characteristic

148

C₄

—

149

C₄

#91222

yes

characteristic

150

C₄

#91235

yes

characteristic

151

C₃

#10598668

yes

characteristic, complemented, Sylow

152

C₂

—

153

C₂

—

154

C₂

#11746764

yes

characteristic

	subgroup	G/H
center	C₄	#91222
commutator	C₃×C₅×C₈	C₄²
Frattini	C₂×C₈	C₂×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×C₁₆	C₄
socle	C₂×C₃×C₅	#340
prime radical	C₂²×C₃×D₅	Q₁₆
cosocle	C₂×C₃×C₈×D₅	C₂²
hypercenter	C₂×C₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1385
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#91222

commutator

C₃×C₅×C₈

C₄²

Frattini

C₂×C₈

C₂×ΓL(1,𝔽₁₆)

Fitting

C₂×C₃×C₅×C₁₆

C₄

socle

C₂×C₃×C₅

#340

prime radical

C₂²×C₃×D₅

Q₁₆

cosocle

C₂×C₃×C₈×D₅

C₂²

hypercenter

C₂×C₁₆

ΓL(1,𝔽₁₆)

hypocenter

C₃×C₅

#1385

LCS

G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
122. C₃×C₅	#1385	semidirect
144. C₅	#67820	semidirect
151. C₃	#10598668	semidirect

H ≅ G/N

type

122. C₃×C₅

#1385

semidirect

144. C₅

#67820

semidirect

151. C₃

#10598668

semidirect

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

subgroup	index
D₅⋊#819	2
C₂×C₃×#3930	2
#10598668	3
#67820	5

subgroup

index

D₅⋊#819

C₂×C₃×#3930

#10598668

#67820

Subgroups — #14278927

Conjugacy classes of subgroups

Complements

Maximal subgroups