Group #14512498 — subgroups

Subgroups — #14512498

Order 1920.

218 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 38 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	Dic₅×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	SL(2,𝔽₃)⋊#170	C₂	1	2	yes	maximal, complemented
3	#10605002	—	3	1	no	maximal, Hall
4	C₂×C₅×#41⋊C₃	C₄	1	1	yes	characteristic
5	#91231	—	4	1	no	maximal
6	Dic₅×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
7	Dic₅×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
8	#73069	—	5	1	no	maximal, Hall
9	#41×Dic₅	D₃	1	1	yes	characteristic
10	#65759	—	3	1	no	—
11	#65754	—	3	1	no	—
12	Q₈⋊#170	—	3	2	no	—
13	D₄⋊#170	—	3	2	no	—
14	C₅×#41⋊C₃	—	2	1	no	—
15	C₂×C₅×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃⋊#170	—	4	2	no	—
17	C₃×C₄×Dic₅	—	4	1	no	—
18	C₄×#41⋊C₃	—	5	1	no	—
19	SL(2,𝔽₃)⋊C₈	—	5	2	no	—
20	C₂×C₅×#41	Dic₃	1	1	yes	characteristic
21	Q₈×Dic₅	Dic₃	1	1	yes	characteristic
22	Q₈×Dic₅	C₂×D₃	1	1	yes	characteristic
23	D₄×Dic₅	—	3	1	no	—
24	D₄×Dic₅	—	3	1	no	—
25	#4072	—	3	1	no	—
26	C₂×C₄×Dic₅	—	3	1	no	—
27	C₂²⋊#170	—	6	1	no	—
28	C₂²⋊#170	—	6	1	no	—
29	#4004	—	6	1	no	—
30	#4002	—	6	1	no	—
31	C₄⋊#170	—	3	2	no	—
32	C₄×#170	—	3	2	no	—
33	#1488	—	15	1	no	Sylow
34	C₂×C₃×C₄×C₅	—	4	1	no	—
35	C₂×C₃×Dic₅	—	4	1	no	—
36	C₂×C₃×Dic₅	—	4	1	no	—
37	C₅×SL(2,𝔽₃)	M₁₆	1	1	yes	characteristic
38	C₃⋊#170	—	4	4	no	—
39	C₂×#41⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₄×SL(2,𝔽₃)	—	5	1	no	—
41	C₄×SL(2,𝔽₃)	—	5	1	no	—
42	#803	—	20	1	no	—
43	C₅×#41	—	2	1	no	—
44	C₅×#41	—	6	1	no	—
45	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
46	C₂×C₅×D₄	—	3	1	no	—
47	C₂²×C₄×C₅	—	3	1	no	—
48	C₂²×Dic₅	—	3	1	no	—
49	C₂²×Dic₅	—	3	1	no	—
50	C₂×#170	—	6	2	no	—
51	C₂×#170	—	6	2	no	—
52	C₂²⋊Dic₅	—	6	1	no	—
53	C₄⋊Dic₅	—	3	1	no	—
54	C₄⋊Dic₅	—	3	1	no	—
55	C₄×Dic₅	—	3	1	no	—
56	C₄×Dic₅	—	3	1	no	—
57	C₄×Dic₅	—	3	1	no	—
58	C₄×Dic₅	S₄	1	1	yes	characteristic
59	C₄×#41	—	5	1	no	—
60	#432	—	15	1	no	—
61	#431	—	15	1	no	—
62	Q₈⋊C₈	—	15	2	no	—
63	D₄⋊C₈	—	15	2	no	—
64	C₂²×C₃×C₅	—	4	1	no	—
65	C₃×C₄×C₅	—	8	1	no	—
66	C₃×Dic₅	—	4	2	no	—
67	C₃×Dic₅	—	8	1	no	—
68	#41⋊C₃	—	2	1	no	—
69	C₂×SL(2,𝔽₃)	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₃×C₄²	—	20	1	no	—
71	C₂×C₃⋊C₈	—	20	2	no	—
72	C₂³×C₅	—	3	1	no	—
73	C₅×Q₈	—	3	1	no	—
74	C₅×Q₈	#208	1	1	yes	characteristic
75	C₅×D₄	—	3	2	no	—
76	C₅×D₄	—	6	1	no	—
77	C₂×C₄×C₅	—	3	1	no	—
78	C₂×C₄×C₅	—	6	1	no	—
79	C₂×C₄×C₅	—	6	1	no	—
80	C₂×C₄×C₅	SL(2,ℤ/4)	1	1	yes	characteristic
81	C₂×Dic₅	—	3	1	no	—
82	C₂×Dic₅	—	3	1	no	—
83	C₂×Dic₅	—	6	2	no	—
84	C₂×Dic₅	—	12	1	no	—
85	C₂×Dic₅	SL(2,ℤ/4)	1	1	yes	characteristic
86	C₂×Dic₅	C₂×S₄	1	1	yes	characteristic
87	#170	—	12	2	no	—
88	#170	—	12	2	no	—
89	C₂×#41	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
90	C₄×Q₈	—	5	1	no	—
91	C₄×Q₈	—	5	1	no	—
92	C₄×D₄	—	15	1	no	—
93	C₄×D₄	—	15	1	no	—
94	#117	—	15	1	no	—
95	C₂×C₄²	—	15	1	no	—
96	C₄⋊C₈	—	15	2	no	—
97	C₄⋊C₈	—	30	1	no	—
98	C₂²⋊C₈	—	30	1	no	—
99	C₂²⋊C₈	—	30	1	no	—
100	#97	—	30	1	no	—
101	C₄×C₈	—	15	2	no	—
102	C₂×C₃×C₅	—	4	1	no	—
103	C₂×C₃×C₅	—	4	1	no	—
104	C₂×C₃×C₅	—	4	1	no	—
105	C₂×C₃×C₄	—	4	1	no	—
106	C₂×C₃×C₄	—	20	1	no	—
107	C₂×C₃×C₄	—	20	1	no	—
108	SL(2,𝔽₃)	#704	1	1	yes	characteristic
109	C₃⋊C₈	—	20	4	no	—
110	C₂²×C₅	—	6	1	no	—
111	C₂²×C₅	—	6	1	no	—
112	C₂²×C₅	—	6	1	no	—
113	C₂²×C₅	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	—	3	1	no	—
116	C₄×C₅	—	3	1	no	—
117	Dic₅	—	2	1	no	—
118	Dic₅	—	6	1	no	—
119	Dic₅	—	6	1	no	—
120	Dic₅	#985	1	2	yes	—
121	#41	—	2	1	no	—
122	#41	—	6	1	no	—
123	C₂×Q₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
124	C₂×D₄	—	3	1	no	—
125	C₂²×C₄	—	3	1	no	—
126	C₂²×C₄	—	15	1	no	—
127	C₂²×C₄	—	15	1	no	—
128	C₂×C₈	—	30	2	no	—
129	C₂×C₈	—	30	2	no	—
130	C₄⋊C₄	—	15	1	no	—
131	C₄⋊C₄	—	15	1	no	—
132	C₂²⋊C₄	—	30	1	no	—
133	C₄²	—	5	1	no	—
134	C₄²	—	15	1	no	—
135	C₄²	—	15	1	no	—
136	C₄²	—	15	1	no	—
137	C₃×C₅	—	4	1	no	Hall
138	C₂²×C₃	—	4	1	no	—
139	C₃×C₄	—	8	1	no	—
140	C₃×C₄	—	20	2	no	—
141	C₃×C₄	—	40	1	no	—
142	C₂×C₅	—	12	1	no	—
143	C₂×C₅	#5409	1	1	yes	characteristic
144	C₂×C₅	#5423	1	1	yes	characteristic
145	C₂×C₅	#5424	1	1	yes	characteristic
146	C₂³	—	3	1	no	—
147	Q₈	—	3	1	no	—
148	Q₈	#6762	1	1	yes	characteristic
149	D₄	—	3	2	no	—
150	D₄	—	6	1	no	—
151	C₂×C₄	—	3	1	no	—
152	C₂×C₄	—	5	1	no	—
153	C₂×C₄	—	5	1	no	—
154	C₂×C₄	—	6	1	no	—
155	C₂×C₄	—	6	1	no	—
156	C₂×C₄	—	15	1	no	—
157	C₂×C₄	—	15	1	no	—
158	C₂×C₄	—	30	2	no	—
159	C₂×C₄	—	60	1	no	—
160	C₂×C₄	A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
161	C₈	—	60	2	no	—
162	C₈	—	60	2	no	—
163	C₂×C₃	—	4	1	no	—
164	C₂×C₃	—	4	1	no	—
165	C₂×C₃	—	4	1	no	—
166	C₅	#73069	1	1	yes	characteristic, complemented, Sylow
167	C₂²	—	6	1	no	—
168	C₂²	—	6	1	no	—
169	C₂²	—	6	1	no	—
170	C₂²	C₂×A₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	3	1	no	—
173	C₄	—	3	1	no	—
174	C₄	—	5	2	no	—
175	C₄	—	10	1	no	—
176	C₄	—	30	1	no	—
177	C₄	—	30	1	no	—
178	C₃	—	4	1	no	Sylow
179	C₂	—	12	1	no	—
180	C₂	#11756724	1	1	yes	characteristic
181	C₂	#11756725	1	1	yes	characteristic
182	C₂	#11756728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Dic₅×#41⋊C₃

C₂

yes

maximal, characteristic

SL(2,𝔽₃)⋊#170

C₂

yes

maximal, complemented

#10605002

—

maximal, Hall

C₂×C₅×#41⋊C₃

C₄

yes

—

C₄

yes

characteristic

Dic₅×SL(2,𝔽₃)

C₂²

yes

—

D₃

yes

—

—

—

—

C₅×#41⋊C₃

—

C₂×C₅×SL(2,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₃⋊#170

—

C₃×C₄×Dic₅

—

C₄×#41⋊C₃

—

SL(2,𝔽₃)⋊C₈

—

C₂×C₅×#41

Dic₃

yes

characteristic

Q₈×Dic₅

Dic₃

yes

characteristic

Q₈×Dic₅

C₂×D₃

yes

characteristic

D₄×Dic₅

—

D₄×Dic₅

—

#4072

—

C₂×C₄×Dic₅

—

C₂²⋊#170

—

C₂²⋊#170

—

—

—

—

—

—

C₂×C₃×C₄×C₅

—

C₂×C₃×Dic₅

—

C₂×C₃×Dic₅

—

C₅×SL(2,𝔽₃)

M₁₆

yes

characteristic

C₃⋊#170

—

C₂×#41⋊C₃

AGL(1,𝔽₅)

yes

characteristic

C₄×SL(2,𝔽₃)

—

C₄×SL(2,𝔽₃)

—

#803

—

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

C₂×Dic₃

yes

characteristic

C₂×C₅×D₄

—

C₂²×C₄×C₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×#170

—

C₂×#170

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

S₄

yes

characteristic

C₄×#41

—

#432

—

#431

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₂²×C₃×C₅

—

C₃×C₄×C₅

—

C₃×Dic₅

—

C₃×Dic₅

—

#41⋊C₃

—

C₂×SL(2,𝔽₃)

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃×C₄²

—

C₂×C₃⋊C₈

—

C₂³×C₅

—

C₅×Q₈

—

C₅×Q₈

#208

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

SL(2,ℤ/4)

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

SL(2,ℤ/4)

yes

characteristic

C₂×Dic₅

C₂×S₄

yes

characteristic

#170

—

#170

—

C₂×#41

ΓL(1,𝔽₁₆)

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#97

—

101

C₄×C₈

—

102

C₂×C₃×C₅

—

103

C₂×C₃×C₅

—

104

C₂×C₃×C₅

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

SL(2,𝔽₃)

#704

yes

characteristic

109

C₃⋊C₈

—

110

C₂²×C₅

—

111

C₂²×C₅

—

112

C₂²×C₅

—

113

C₂²×C₅

C₂×SL(2,ℤ/4)

yes

characteristic

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

—

117

Dic₅

—

118

Dic₅

—

119

Dic₅

—

120

Dic₅

#985

yes

—

121

#41

—

122

#41

—

123

C₂×Q₈

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

124

C₂×D₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₃×C₅

—

Hall

138

C₂²×C₃

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₂×C₅

—

143

C₂×C₅

#5409

yes

characteristic

144

C₂×C₅

#5423

yes

characteristic

145

C₂×C₅

#5424

yes

characteristic

146

C₂³

—

147

Q₈

—

148

Q₈

#6762

yes

characteristic

149

D₄

—

150

D₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

A₄⋊AGL(1,𝔽₅)

yes

characteristic

161

C₈

—

162

C₈

—

163

C₂×C₃

—

164

C₂×C₃

—

165

C₂×C₃

—

166

C₅

#73069

yes

characteristic, complemented, Sylow

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂×A₄⋊AGL(1,𝔽₅)

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₃

—

Sylow

179

C₂

—

180

C₂

#11756724

yes

characteristic

181

C₂

#11756725

yes

characteristic

182

C₂

#11756728

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×A₄⋊AGL(1,𝔽₅)
commutator	C₂×C₅×SL(2,𝔽₃)	C₂×C₄
Fitting	C₂×C₅×#41	Dic₃
socle	C₂²×C₅	C₂×SL(2,ℤ/4)
prime radical	C₂×C₅×#41⋊C₃	C₄
cosocle	Dic₅×SL(2,𝔽₃)	C₂²
hypercenter	C₂×C₄	A₄⋊AGL(1,𝔽₅)
hypocenter	C₅×SL(2,𝔽₃)	M₁₆
LCS	G ▷ C₂×C₅×SL(2,𝔽₃) ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₂²

C₂×A₄⋊AGL(1,𝔽₅)

commutator

C₂×C₅×SL(2,𝔽₃)

C₂×C₄

Fitting

C₂×C₅×#41

Dic₃

socle

C₂²×C₅

C₂×SL(2,ℤ/4)

prime radical

C₂×C₅×#41⋊C₃

C₄

C₂²

C₂×C₄

C₅×SL(2,𝔽₃)

M₁₆

LCS

G ▷ C₂×C₅×SL(2,𝔽₃) ▷ C₅×SL(2,𝔽₃)

UCS

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
2. SL(2,𝔽₃)⋊#170	C₂	semidirect
166. C₅	#73069	semidirect

H ≅ G/N

type

2. SL(2,𝔽₃)⋊#170

C₂

semidirect

166. C₅

#73069

semidirect

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

subgroup	index
Dic₅×#41⋊C₃	2
SL(2,𝔽₃)⋊#170	2
#10605002	3
#91231	4
#73069	5

subgroup

index

Dic₅×#41⋊C₃

Subgroups — #14512498

Conjugacy classes of subgroups

Complements

Maximal subgroups