Group #14293575 — subgroups

Subgroups — #14293575

Order 1920.

424 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 116 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	#11750796	C₂	1	1	yes	maximal, characteristic
2	C₅×#4699	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746912	C₂	1	4	yes	maximal, complemented
4	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602090	—	3	1	no	maximal, Hall
6	C₂×D₁₂₀	—	2	2	no	—
7	#91763	C₂²	1	2	yes	—
8	C₂×C₅×Dic₁₂	—	2	2	no	—
9	C₅×#869	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×#168	C₂²	1	2	yes	—
12	#90969	—	2	8	no	—
13	#68060	—	5	1	no	maximal, Hall
14	C₅×#494	—	3	1	no	—
15	#65377	—	3	4	no	—
16	#64984	—	3	1	no	—
17	C₈×#168	D₃	1	1	yes	characteristic, complemented
18	C₂×D₆₀	—	2	4	no	—
19	C₂×D₆₀	—	2	2	no	—
20	C₂×C₅×Dic₆	—	2	2	no	—
21	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
22	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
23	D₁₂₀	—	4	4	no	—
24	C₅×C₄⋊Dic₃	—	2	4	no	—
25	C₅×Dic₁₂	—	4	4	no	—
26	C₂×C₃×#168	D₄	1	4	yes	—
27	#4699	D₅	1	1	yes	characteristic, complemented
28	#4695	—	5	1	no	—
29	#4694	—	5	4	no	—
30	C₃×C₈²	—	5	1	no	—
31	C₂×C₅×Q₁₆	—	6	2	no	—
32	C₅×C₄⋊Q₈	—	3	2	no	—
33	C₂×D₄₀	—	6	2	no	—
34	#4020	—	3	2	no	—
35	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
36	#3941	—	6	8	no	—
37	C₄×#168	C₂×D₃	1	2	yes	—
38	#1709	—	15	1	no	Sylow
39	C₂²×D₁₅	—	4	2	no	—
40	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
42	D₆₀	—	4	4	no	—
43	D₆₀	—	4	8	no	—
44	C₂×C₅×Dic₃	—	4	2	no	—
45	C₅×Dic₆	—	4	4	no	—
46	C₃×C₅×C₈	D₈	1	4	yes	—
47	C₃×#168	—	2	4	no	—
48	C₂×Dic₁₂	—	2	2	no	—
49	C₂×D₂₄	—	10	2	no	—
50	#874	—	5	2	no	—
51	#869	C₂×D₅	1	2	yes	—
52	C₃×C₄×C₈	—	5	2	no	—
53	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
54	#821	—	10	8	no	—
55	C₂×C₅×Q₈	—	6	2	no	—
56	C₂×D₂₀	—	6	4	no	—
57	C₂×D₂₀	—	6	2	no	—
58	C₅×Q₁₆	—	12	4	no	—
59	C₂×C₅×C₈	D₁₂	1	2	yes	—
60	C₅×C₄⋊C₄	—	6	4	no	—
61	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
62	C₂×#168	D₁₂	1	4	yes	—
63	D₄₀	—	12	4	no	—
64	#494	—	3	1	no	—
65	#493	—	15	1	no	—
66	#486	—	15	4	no	—
67	C₈²	—	5	1	no	—
68	C₂²×C₃×C₅	#127	1	1	yes	characteristic
69	C₂×D₁₅	—	8	2	no	—
70	C₂×D₁₅	—	8	4	no	—
71	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
72	C₃×C₄×C₅	#135	1	4	yes	—
73	C₅×Dic₃	—	8	2	no	—
74	C₂×D₁₂	—	10	4	no	—
75	C₂×D₁₂	—	10	2	no	—
76	C₂×Dic₆	—	2	2	no	—
77	C₂×C₃×C₈	—	5	4	no	—
78	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
79	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
80	C₄⋊Dic₃	—	2	4	no	—
81	Dic₁₂	—	4	4	no	—
82	D₂₄	—	20	4	no	—
83	C₂²×D₅	—	12	2	no	—
84	C₅×Q₈	—	12	4	no	—
85	C₂×C₄×C₅	—	12	2	no	—
86	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
87	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
88	D₂₀	—	12	4	no	—
89	D₂₀	—	12	8	no	—
90	C₅×C₈	D₂₄	1	4	yes	—
91	#168	—	2	4	no	—
92	C₂×Q₁₆	—	6	2	no	—
93	C₂×D₈	—	30	2	no	—
94	C₄⋊Q₈	—	3	2	no	—
95	#127	—	15	2	no	—
96	#102	—	30	8	no	—
97	C₄×C₈	—	5	2	no	—
98	C₄×C₈	D₃×D₅	1	1	yes	characteristic
99	C₂×C₃×C₅	#493	1	1	yes	characteristic
100	C₂×C₃×C₅	#495	1	2	yes	—
101	D₁₅	—	16	2	no	—
102	C₂²×D₃	—	20	2	no	—
103	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
104	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
105	C₂×Dic₃	—	4	2	no	—
106	D₁₂	—	20	8	no	—
107	D₁₂	—	20	4	no	—
108	Dic₆	—	4	4	no	—
109	C₃×C₈	—	10	4	no	—
110	C₃×C₈	#686	1	4	yes	—
111	C₂²×C₅	#874	1	1	yes	characteristic
112	C₂×D₅	—	24	2	no	—
113	C₂×D₅	—	24	4	no	—
114	C₄×C₅	—	24	2	no	—
115	C₄×C₅	C₂×D₂₄	1	2	yes	—
116	C₄×C₅	#904	1	4	yes	—
117	C₂×Q₈	—	6	2	no	—
118	C₂×D₄	—	30	4	no	—
119	C₂×D₄	—	30	2	no	—
120	Q₁₆	—	12	4	no	—
121	D₈	—	60	4	no	—
122	C₂×C₈	—	5	4	no	—
123	C₂×C₈	#1203	1	2	yes	—
124	C₄⋊C₄	—	6	4	no	—
125	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
126	C₃×C₅	#1709	1	1	yes	characteristic, complemented, Hall
127	C₂²×C₃	#4086	1	1	yes	characteristic
128	C₂×D₃	—	40	2	no	—
129	C₂×D₃	—	40	4	no	—
130	C₃×C₄	#4065	1	4	yes	—
131	C₃×C₄	C₂×#686	1	2	yes	—
132	Dic₃	—	8	2	no	—
133	C₂×C₅	#4695	1	1	yes	characteristic
134	C₂×C₅	#4696	1	2	yes	—
135	D₅	—	48	2	no	—
136	C₂³	—	60	2	no	—
137	Q₈	—	12	4	no	—
138	D₄	—	60	8	no	—
139	D₄	—	60	4	no	—
140	C₂×C₄	—	12	2	no	—
141	C₂×C₄	D₅×D₁₂	1	2	yes	—
142	C₂×C₄	C₂×#1203	1	1	yes	characteristic
143	C₈	—	10	4	no	—
144	C₈	#6654	1	4	yes	—
145	C₂×C₃	#65362	1	1	yes	characteristic
146	C₂×C₃	#65480	1	2	yes	—
147	D₃	—	80	2	no	—
148	C₅	#68060	1	1	yes	characteristic, complemented, Sylow
149	C₂²	—	120	4	no	—
150	C₂²	—	120	2	no	—
151	C₂²	#91451	1	1	yes	characteristic
152	C₄	—	24	2	no	—
153	C₄	#91271	1	4	yes	—
154	C₄	C₂×#6654	1	2	yes	—
155	C₃	#10602090	1	1	yes	characteristic, complemented, Sylow
156	C₂	—	240	2	no	—
157	C₂	#11747029	1	1	yes	characteristic
158	C₂	#11747965	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750796

C₂

yes

maximal, characteristic

C₅×#4699

C₂

yes

maximal, characteristic, complemented

#11746912

C₂

yes

maximal, complemented

C₃×C₈×#168

C₂

yes

maximal, characteristic, complemented

#10602090

—

maximal, Hall

C₂×D₁₂₀

—

#91763

C₂²

yes

—

C₂×C₅×Dic₁₂

—

C₅×#869

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

C₃×C₄×#168

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×D₆₀

—

C₂×D₆₀

—

C₂×C₅×Dic₆

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

D₁₂₀

—

C₅×C₄⋊Dic₃

—

C₅×Dic₁₂

—

C₂×C₃×#168

D₄

yes

—

#4699

D₅

yes

characteristic, complemented

#4695

—

#4694

—

C₃×C₈²

—

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₂×D₄₀

—

#4020

—

C₄×C₅×C₈

C₂×D₃

yes

characteristic

#3941

—

C₄×#168

C₂×D₃

yes

—

#1709

—

Sylow

C₂²×D₁₅

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

D₆₀

—

D₆₀

—

C₂×C₅×Dic₃

—

C₅×Dic₆

—

C₃×C₅×C₈

D₈

yes

—

C₃×#168

—

C₂×Dic₁₂

—

C₂×D₂₄

—

#874

—

#869

C₂×D₅

yes

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#821

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×Q₁₆

—

C₂×C₅×C₈

D₁₂

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₂×#168

D₁₂

yes

—

D₄₀

—

#494

—

#493

—

#486

—

C₈²

—

C₂²×C₃×C₅

#127

yes

characteristic

C₂×D₁₅

—

C₂×D₁₅

—

C₃×C₄×C₅

C₂×D₈

yes

—

C₃×C₄×C₅

#135

yes

—

C₅×Dic₃

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊D₅

yes

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄⋊Dic₃

—

Dic₁₂

—

D₂₄

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₁₂

yes

—

C₂×C₄×C₅

C₂×D₁₂

yes

characteristic

D₂₀

—

D₂₀

—

C₅×C₈

D₂₄

yes

—

#168

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#102

—

C₄×C₈

—

C₄×C₈

D₃×D₅

yes

characteristic

C₂×C₃×C₅

#493

yes

characteristic

100

C₂×C₃×C₅

#495

yes

—

101

D₁₅

—

102

C₂²×D₃

—

103

C₂×C₃×C₄

D₄×D₅

yes

—

104

C₂×C₃×C₄

C₂×C₂²⋊D₅

yes

characteristic

105

C₂×Dic₃

—

106

D₁₂

—

107

D₁₂

—

108

Dic₆

—

109

C₃×C₈

—

110

C₃×C₈

#686

yes

—

111

C₂²×C₅

#874

yes

characteristic

112

C₂×D₅

—

113

C₂×D₅

—

114

C₄×C₅

—

115

C₄×C₅

C₂×D₂₄

yes

—

116

C₄×C₅

#904

yes

—

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

Q₁₆

—

121

D₈

—

122

C₂×C₈

—

123

C₂×C₈

#1203

yes

—

124

C₄⋊C₄

—

125

C₄²

C₂×D₃×D₅

yes

characteristic

126

C₃×C₅

#1709

yes

characteristic, complemented, Hall

127

C₂²×C₃

#4086

yes

characteristic

128

C₂×D₃

—

129

C₂×D₃

—

130

C₃×C₄

#4065

yes

—

131

C₃×C₄

C₂×#686

yes

—

132

Dic₃

—

133

C₂×C₅

#4695

yes

characteristic

134

C₂×C₅

#4696

yes

—

135

D₅

—

136

C₂³

—

137

Q₈

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

D₅×D₁₂

yes

—

142

C₂×C₄

C₂×#1203

yes

characteristic

143

C₈

—

144

C₈

#6654

yes

—

145

C₂×C₃

#65362

yes

characteristic

146

C₂×C₃

#65480

yes

—

147

D₃

—

148

C₅

#68060

yes

characteristic, complemented, Sylow

149

C₂²

—

120

—

150

C₂²

—

120

—

151

C₂²

#91451

yes

characteristic

152

C₄

—

153

C₄

#91271

yes

—

154

C₄

C₂×#6654

yes

—

155

C₃

#10602090

yes

characteristic, complemented, Sylow

156

C₂

—

240

—

157

C₂

#11747029

yes

characteristic

158

C₂

#11747965

yes

—

	subgroup	G/H
center	C₂²	#91451
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	#127
prime radical	#11750796	C₂
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1709
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91451

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

C₂²×C₃×C₅

C₂

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1709

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
2. C₅×#4699	C₂	semidirect
3. #11746912	C₂	semidirect
4. C₃×C₈×#168	C₂	semidirect
17. C₈×#168	D₃	semidirect
27. #4699	D₅	semidirect
126. C₃×C₅	#1709	semidirect
148. C₅	#68060	semidirect
155. C₃	#10602090	semidirect

H ≅ G/N

type

2. C₅×#4699

C₂

semidirect

3. #11746912

C₂

semidirect

4. C₃×C₈×#168

C₂

semidirect

17. C₈×#168

D₃

semidirect

27. #4699

D₅

semidirect

126. C₃×C₅

#1709

semidirect

148. C₅

#68060

semidirect

155. C₃

#10602090

semidirect

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

subgroup	index
#11750796	2
C₅×#4699	2
#11746912	2
C₃×C₈×#168	2
#10602090	3
#68060	5

subgroup

index

C₃×C₈×#168

Subgroups — #14293575

Conjugacy classes of subgroups

Complements

Maximal subgroups