Group #14298898 — subgroups

Subgroups — C₃×#10603857 #14298898

Order 1920.

376 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 140 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	C₃×#65682	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64896	C₂	1	4	yes	maximal
4	#10603857	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×#700	C₂²	1	4	yes	—
6	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
7	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
8	C₃×#4004	—	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	C₃×#1837	—	5	1	no	maximal, Hall
13	#65682	C₂×C₃	1	2	yes	—
14	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
15	#64896	C₂×C₃	1	4	yes	—
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	C₃×#700	—	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	C₃×#423	—	5	2	no	—
27	C₂×C₃×C₄×C₈	—	5	1	no	—
28	C₃×#341	—	5	4	no	—
29	C₂×#700	C₂²×C₃	1	4	yes	—
30	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
31	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
32	#4004	—	2	4	no	—
33	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
34	C₈×Dic₅	C₃×C₄	1	2	yes	—
35	C₄×#168	C₃×C₄	1	1	yes	characteristic
36	#1837	—	5	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	C₂×C₃×M₁₆	—	5	4	no	—
48	C₂²×C₃×C₈	—	5	2	no	—
49	C₂×C₃×C₄²	—	5	1	no	—
50	C₃×C₄⋊C₈	—	10	4	no	—
51	C₃×C₄×C₈	—	5	2	no	—
52	C₃×C₄×C₈	—	5	1	no	—
53	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂×C₄×D₅	C₃×Q₈	1	2	yes	—
55	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
56	C₂×#170	—	2	4	no	—
57	#700	—	2	8	no	—
58	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
59	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
60	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
61	C₄×Dic₅	C₃×D₄	1	2	yes	—
62	C₂×#168	C₂×C₃×C₄	1	2	yes	—
63	C₈×D₅	—	2	4	no	—
64	#423	—	5	2	no	—
65	C₂×C₄×C₈	—	5	1	no	—
66	#341	—	5	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	C₃×M₁₆	—	10	8	no	—
77	C₂×C₃×C₈	—	5	2	no	—
78	C₂×C₃×C₈	—	10	4	no	—
79	C₂×C₃×C₈	—	10	4	no	—
80	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
81	C₃×C₄²	—	5	2	no	—
82	C₃×C₄²	—	5	1	no	—
83	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₂²×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
87	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
88	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
89	C₄×D₅	—	2	4	no	—
90	C₄×D₅	C₃×#41	1	4	yes	—
91	#170	—	4	4	no	—
92	C₅×C₈	—	2	2	no	—
93	#168	—	2	2	no	—
94	C₂×M₁₆	—	5	4	no	—
95	C₂²×C₈	—	5	2	no	—
96	C₂×C₄²	—	5	1	no	—
97	C₄⋊C₈	—	10	4	no	—
98	C₄×C₈	—	5	2	no	—
99	C₄×C₈	—	5	1	no	—
100	C₄×C₈	C₃×AGL(1,𝔽₅)	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×C₃×C₄⋊C₄	1	1	yes	characteristic
115	C₂×D₅	—	2	2	no	—
116	C₂×D₅	C₃×#125	1	2	yes	—
117	C₄×C₅	—	2	2	no	—
118	C₄×C₅	C₃×#117	1	2	yes	—
119	Dic₅	C₃×#108	1	4	yes	—
120	Dic₅	C₃×#124	1	2	yes	—
121	C₂²×C₄	—	5	1	no	—
122	C₂²×C₄	—	5	2	no	—
123	M₁₆	—	10	8	no	—
124	C₂×C₈	—	5	2	no	—
125	C₂×C₈	—	10	4	no	—
126	C₂×C₈	—	10	4	no	—
127	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
128	C₄²	—	5	2	no	—
129	C₄²	—	5	1	no	—
130	C₄²	C₂×C₃×AGL(1,𝔽₅)	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	C₃×#382	1	1	yes	characteristic
140	C₂×C₅	C₂×C₃×#108	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	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
148	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	10	2	no	—
151	C₈	—	20	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	C₃×#1837	1	1	yes	characteristic, complemented, Sylow
156	C₂²	—	5	2	no	—
157	C₂²	—	10	2	no	—
158	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	5	2	no	—
161	C₄	—	5	4	no	—
162	C₄	C₃×#4130	1	2	yes	—
163	C₃	#10603857	1	1	yes	characteristic, direct product factor, Sylow
164	C₂	—	10	2	no	—
165	C₂	C₃×#65689	1	1	yes	characteristic
166	C₂	C₃×D₅⋊#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#65682

C₂

yes

maximal

C₃×C₄×C₈×D₅

C₂

yes

maximal, characteristic

C₃×#64896

C₂

yes

maximal

#10603857

C₃

yes

maximal, characteristic, direct product factor, Hall

C₂×C₃×#700

C₂²

yes

—

C₂×C₃×C₈×D₅

C₂²

yes

—

C₃×C₄²×D₅

C₂²

yes

characteristic

C₃×#4004

—

C₃×C₄×C₅×C₈

C₄

yes

characteristic

C₃×C₈×Dic₅

C₄

yes

—

C₃×C₄×#168

C₄

yes

—

C₂×C₃

yes

—

C₄×C₈×D₅

C₂×C₃

yes

characteristic

#64896

C₂×C₃

yes

—

C₂×C₃×C₄×D₅

Q₈

yes

—

C₂×C₃×C₄×D₅

C₂³

yes

characteristic

C₂×C₃×#170

—

C₃×#700

—

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

—

C₃×#423

—

C₂×C₃×C₄×C₈

—

C₃×#341

—

C₂×#700

C₂²×C₃

yes

—

C₂×C₈×D₅

C₂²×C₃

yes

—

C₄²×D₅

C₂²×C₃

yes

characteristic

#4004

—

C₄×C₅×C₈

C₃×C₄

yes

characteristic

C₈×Dic₅

C₃×C₄

yes

—

C₄×#168

C₃×C₄

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

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

AGL(1,𝔽₅)

yes

characteristic

C₂×C₄×D₅

C₃×Q₈

yes

—

C₂×C₄×D₅

C₂³×C₃

yes

characteristic

C₂×#170

—

#700

—

C₂×C₅×C₈

C₂×C₃×C₄

yes

—

C₄²×C₅

C₂×C₃×C₄

yes

characteristic

C₄×Dic₅

C₂×C₃×C₄

yes

characteristic

C₄×Dic₅

C₃×D₄

yes

—

C₂×#168

C₂×C₃×C₄

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

—

C₃×M₁₆

—

C₂×C₃×C₈

—

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

C₂²×D₅

C₂×C₃×Q₈

yes

characteristic

C₂×C₄×C₅

C₃×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₃×C₄

yes

characteristic

C₂×Dic₅

C₃×C₄⋊C₄

yes

—

C₂×Dic₅

C₂×C₃×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

C₃×#41

yes

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₃×AGL(1,𝔽₅)

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×C₃×C₄⋊C₄

yes

characteristic

115

C₂×D₅

—

116

C₂×D₅

C₃×#125

yes

—

117

C₄×C₅

—

118

C₄×C₅

C₃×#117

yes

—

119

Dic₅

C₃×#108

yes

—

120

Dic₅

C₃×#124

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂×C₃×AGL(1,𝔽₅)

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂×C₃×AGL(1,𝔽₅)

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

C₃×#382

yes

characteristic

140

C₂×C₅

C₂×C₃×#108

yes

—

141

D₅

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₃×C₄⋊AGL(1,𝔽₅)

yes

—

148

C₂×C₄

C₂²×C₃×AGL(1,𝔽₅)

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

C₃×#1837

yes

characteristic, complemented, Sylow

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₃×C₄⋊AGL(1,𝔽₅)

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₃×#4130

yes

—

163

C₃

#10603857

yes

characteristic, direct product factor, Sylow

164

C₂

—

165

C₂

C₃×#65689

yes

characteristic

166

C₂

C₃×D₅⋊#108

yes

—

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×Q₈
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1837
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

subgroup

G/H

center

C₂²×C₃

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₃×C₄

Frattini

C₂×C₄

C₂²×C₃×AGL(1,𝔽₅)

Fitting

C₃×C₄×C₅×C₈

C₄

socle

C₂²×C₃×C₅

C₂×C₄⋊C₄

prime radical

C₂²×C₃×D₅

C₂×Q₈

cosocle

C₂×C₄×D₅

C₂³×C₃

C₃×C₄×C₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈

N	H ≅ G/N	type
4. #10603857	C₃	direct (N × H)
131. C₃×C₅	#1837	semidirect
155. C₅	C₃×#1837	semidirect
163. C₃	#10603857	direct (N × H)

H ≅ G/N

type

4. #10603857

C₃

direct (N × H)

131. C₃×C₅

#1837

semidirect

155. C₅

C₃×#1837

semidirect

163. C₃

#10603857

direct (N × H)

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

subgroup	index
C₃×#65682	2
C₃×C₄×C₈×D₅	2
C₃×#64896	2
#10603857	3
C₃×#1837	5

subgroup

index

C₃×C₄×C₈×D₅

Subgroups — C₃×#10603857 #14298898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#10603857 #14298898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603857 #14298898