Group #14293644 — subgroups

Subgroups — #14293644

Order 1920.

384 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 132 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	#11750800	C₂	1	1	yes	maximal, characteristic
2	C₃×#64988	C₂	1	1	yes	maximal, characteristic
3	#11746960	C₂	1	4	yes	maximal
4	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#10598858	—	3	1	no	maximal, Hall
6	C₂×Dic₆₀	—	2	2	no	—
7	#91758	C₂²	1	2	yes	—
8	C₂×C₃×Dic₂₀	—	2	2	no	—
9	C₃×#4015	C₂²	1	2	yes	—
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	#90990	—	2	8	no	—
13	#71298	—	5	1	no	maximal, Hall
14	#64988	—	3	1	no	—
15	#64988	D₃	1	1	yes	characteristic
16	#64971	—	3	4	no	—
17	C₅×C₈²	—	3	1	no	—
18	C₂×Dic₃₀	—	2	2	no	—
19	C₂×C₃×Dic₁₀	—	2	2	no	—
20	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
21	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
22	C₄⋊Dic₁₅	—	2	4	no	—
23	Dic₆₀	—	4	4	no	—
24	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
25	C₃×C₄⋊Dic₅	—	2	4	no	—
26	C₃×Dic₂₀	—	4	4	no	—
27	C₃×#494	—	5	1	no	—
28	#5078	—	5	4	no	—
29	#4699	—	5	1	no	—
30	C₈×C₃⋊C₈	D₅	1	1	yes	characteristic
31	C₂×Dic₂₀	—	2	2	no	—
32	C₂×Dic₂₀	—	6	2	no	—
33	#4015	—	3	2	no	—
34	#4015	C₂×D₃	1	2	yes	—
35	C₄×C₅×C₈	—	3	2	no	—
36	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
37	#3950	—	6	8	no	—
38	#1713	—	15	1	no	Sylow
39	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
41	C₂×Dic₁₅	—	4	2	no	—
42	Dic₃₀	—	4	4	no	—
43	C₂×C₃×Dic₅	—	4	2	no	—
44	C₃×Dic₁₀	—	4	4	no	—
45	C₃×C₅×C₈	D₈	1	4	yes	—
46	C₅×C₃⋊C₈	Q₁₆	1	8	yes	complemented
47	C₂×C₃×Q₁₆	—	10	2	no	—
48	C₃×C₄⋊Q₈	—	5	2	no	—
49	C₂×Dic₁₂	—	10	2	no	—
50	#869	—	5	2	no	—
51	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
52	#807	—	10	8	no	—
53	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
54	C₂×Dic₁₀	—	2	2	no	—
55	C₂×Dic₁₀	—	6	2	no	—
56	C₂×C₅×C₈	—	3	4	no	—
57	C₂×C₅×C₈	#67	1	2	yes	—
58	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
59	C₄⋊Dic₅	—	2	4	no	—
60	C₄⋊Dic₅	—	6	4	no	—
61	Dic₂₀	—	4	4	no	—
62	Dic₂₀	—	12	4	no	—
63	#500	—	15	4	no	—
64	#494	—	5	1	no	—
65	#494	—	15	1	no	—
66	C₈²	—	3	1	no	—
67	C₂²×C₃×C₅	#127	1	1	yes	characteristic
68	C₃×C₄×C₅	C₂×D₈	1	2	yes	—
69	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
70	Dic₁₅	—	8	2	no	—
71	C₃×Dic₅	—	8	2	no	—
72	C₂×C₃×Q₈	—	10	2	no	—
73	C₂×Dic₆	—	10	2	no	—
74	C₃×Q₁₆	—	20	4	no	—
75	C₂×C₃×C₈	D₂₀	1	2	yes	—
76	C₃×C₄⋊C₄	—	10	4	no	—
77	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
78	C₄⋊Dic₃	—	10	4	no	—
79	C₂×C₃⋊C₈	D₂₀	1	4	yes	—
80	Dic₁₂	—	20	4	no	—
81	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
82	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
83	C₂×Dic₅	—	4	2	no	—
84	C₂×Dic₅	—	12	2	no	—
85	Dic₁₀	—	4	4	no	—
86	Dic₁₀	—	12	4	no	—
87	C₅×C₈	—	3	8	no	—
88	C₅×C₈	#213	1	4	yes	—
89	C₂×Q₁₆	—	10	2	no	—
90	C₂×Q₁₆	—	30	2	no	—
91	C₄⋊Q₈	—	5	2	no	—
92	C₄⋊Q₈	—	15	2	no	—
93	#107	—	30	8	no	—
94	C₄×C₈	—	3	2	no	—
95	C₄×C₈	D₃×D₅	1	1	yes	characteristic
96	C₂×C₃×C₅	#493	1	1	yes	characteristic
97	C₂×C₃×C₅	#494	1	2	yes	—
98	C₃×Q₈	—	20	4	no	—
99	C₂×C₃×C₄	—	20	2	no	—
100	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
101	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
102	C₂×Dic₃	—	20	2	no	—
103	Dic₆	—	20	4	no	—
104	C₃×C₈	D₄₀	1	4	yes	—
105	C₃⋊C₈	Dic₂₀	1	8	yes	complemented
106	C₂²×C₅	#940	1	1	yes	characteristic
107	C₄×C₅	D₃×Q₁₆	1	4	yes	—
108	C₄×C₅	C₂×#213	1	2	yes	—
109	Dic₅	—	8	2	no	—
110	Dic₅	—	24	2	no	—
111	C₂×Q₈	—	10	2	no	—
112	C₂×Q₈	—	30	2	no	—
113	Q₁₆	—	20	4	no	—
114	Q₁₆	—	60	4	no	—
115	C₂×C₈	—	3	4	no	—
116	C₂×C₈	#1202	1	2	yes	—
117	C₄⋊C₄	—	10	4	no	—
118	C₄⋊C₄	—	30	4	no	—
119	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
120	C₃×C₅	#1713	1	1	yes	characteristic, complemented, Hall
121	C₂²×C₃	#4020	1	1	yes	characteristic
122	C₃×C₄	—	40	2	no	—
123	C₃×C₄	C₂×D₄₀	1	2	yes	—
124	C₃×C₄	C₂×Dic₂₀	1	4	yes	—
125	Dic₃	—	40	2	no	—
126	C₂×C₅	#5073	1	1	yes	characteristic
127	C₂×C₅	#5183	1	2	yes	—
128	Q₈	—	20	4	no	—
129	Q₈	—	60	4	no	—
130	C₂×C₄	—	20	2	no	—
131	C₂×C₄	—	60	2	no	—
132	C₂×C₄	D₃×D₂₀	1	2	yes	—
133	C₂×C₄	C₂×#1202	1	1	yes	characteristic
134	C₈	—	3	8	no	—
135	C₈	#6653	1	4	yes	—
136	C₂×C₃	#64984	1	1	yes	characteristic
137	C₂×C₃	#64988	1	2	yes	—
138	C₅	#71298	1	1	yes	characteristic, complemented, Sylow
139	C₂²	#91458	1	1	yes	characteristic
140	C₄	—	40	2	no	—
141	C₄	—	120	2	no	—
142	C₄	D₃×Dic₂₀	1	4	yes	—
143	C₄	C₂×#6653	1	2	yes	—
144	C₃	#10598858	1	1	yes	characteristic, complemented, Sylow
145	C₂	#11747035	1	1	yes	characteristic
146	C₂	#11748033	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11750800

C₂

yes

maximal, characteristic

C₃×#64988

C₂

yes

maximal, characteristic

#11746960

C₂

yes

maximal

C₅×C₈×C₃⋊C₈

C₂

yes

maximal, characteristic

#10598858

—

maximal, Hall

C₂×Dic₆₀

—

#91758

C₂²

yes

—

C₂×C₃×Dic₂₀

—

C₃×#4015

C₂²

yes

—

C₃×C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₃⋊C₈

C₂²

yes

—

—

—

—

D₃

yes

characteristic

#64971

—

C₅×C₈²

—

C₂×Dic₃₀

—

C₂×C₃×Dic₁₀

—

C₂×C₃×C₅×C₈

D₄

yes

—

C₃×C₄²×C₅

C₂³

yes

characteristic

C₄⋊Dic₁₅

—

Dic₆₀

—

C₂×C₅×C₃⋊C₈

D₄

yes

—

C₃×C₄⋊Dic₅

—

C₃×Dic₂₀

—

C₃×#494

—

#5078

—

#4699

—

C₈×C₃⋊C₈

D₅

yes

characteristic

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

#4015

C₂×D₃

yes

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×D₃

yes

—

—

C₂×C₃×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₃×C₄×C₅

C₂×D₄

yes

—

C₂×Dic₁₅

—

Dic₃₀

—

C₂×C₃×Dic₅

—

C₃×Dic₁₀

—

C₃×C₅×C₈

D₈

yes

—

C₅×C₃⋊C₈

Q₁₆

yes

complemented

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

—

C₂×Dic₁₂

—

#869

—

C₃×C₄×C₈

C₂×D₅

yes

characteristic

#807

—

C₄×C₃⋊C₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

#67

yes

—

C₄²×C₅

C₂²×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

#500

—

#494

—

#494

—

C₈²

—

C₂²×C₃×C₅

#127

yes

characteristic

C₃×C₄×C₅

C₂×D₈

yes

—

C₃×C₄×C₅

C₂×Q₁₆

yes

—

Dic₁₅

—

C₃×Dic₅

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

D₂₀

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂²×D₅

yes

characteristic

C₄⋊Dic₃

—

C₂×C₃⋊C₈

D₂₀

yes

—

Dic₁₂

—

C₂×C₄×C₅

D₃×D₄

yes

—

C₂×C₄×C₅

C₂×#67

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#213

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#107

—

C₄×C₈

—

C₄×C₈

D₃×D₅

yes

C₂×C₃×C₅

yes

C₂×C₃×C₅

yes

—

C₃×Q₈

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

C₂×D₂₀

yes

—

101

C₂×C₃×C₄

C₂×D₂₀

yes

characteristic

102

C₂×Dic₃

—

103

Dic₆

—

104

C₃×C₈

D₄₀

yes

—

105

C₃⋊C₈

Dic₂₀

yes

complemented

106

C₂²×C₅

#940

yes

characteristic

107

C₄×C₅

D₃×Q₁₆

yes

—

108

C₄×C₅

C₂×#213

yes

—

109

Dic₅

—

110

Dic₅

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

Q₁₆

—

114

Q₁₆

—

115

C₂×C₈

—

116

C₂×C₈

#1202

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂×D₃×D₅

yes

characteristic

120

C₃×C₅

#1713

yes

characteristic, complemented, Hall

121

C₂²×C₃

#4020

yes

characteristic

122

C₃×C₄

—

123

C₃×C₄

C₂×D₄₀

yes

—

124

C₃×C₄

C₂×Dic₂₀

yes

—

125

Dic₃

—

126

C₂×C₅

#5073

yes

characteristic

127

C₂×C₅

#5183

yes

—

128

Q₈

—

129

Q₈

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

D₃×D₂₀

yes

—

133

C₂×C₄

C₂×#1202

yes

characteristic

134

C₈

—

135

C₈

#6653

yes

—

136

C₂×C₃

#64984

yes

characteristic

137

C₂×C₃

#64988

yes

—

138

C₅

#71298

yes

characteristic, complemented, Sylow

139

C₂²

#91458

yes

characteristic

140

C₄

—

141

C₄

—

120

—

142

C₄

D₃×Dic₂₀

yes

—

143

C₄

C₂×#6653

yes

—

144

C₃

#10598858

yes

characteristic, complemented, Sylow

145

C₂

#11747035

yes

characteristic

146

C₂

#11748033

yes

—

	subgroup	G/H
center	C₂²	#91458
commutator, cosocle	C₃×C₄²×C₅	C₂³
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle, prime radical	C₂²×C₃×C₅	#127
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1713
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

#91458

commutator, cosocle

C₃×C₄²×C₅

C₂³

Frattini

C₄²

C₂×D₃×D₅

Fitting

C₃×C₄×C₅×C₈

C₂²

socle, prime radical

C₂²×C₃×C₅

#127

hypercenter

C₄×C₈

D₃×D₅

hypocenter

C₃×C₅

#1713

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
46. C₅×C₃⋊C₈	Q₁₆	semidirect
105. C₃⋊C₈	Dic₂₀	semidirect
120. C₃×C₅	#1713	semidirect
138. C₅	#71298	semidirect
144. C₃	#10598858	semidirect

H ≅ G/N

type

46. C₅×C₃⋊C₈

Q₁₆

semidirect

105. C₃⋊C₈

Dic₂₀

semidirect

120. C₃×C₅

#1713

semidirect

138. C₅

#71298

semidirect

144. C₃

#10598858

semidirect

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

subgroup	index
#11750800	2
C₃×#64988	2
#11746960	2
C₅×C₈×C₃⋊C₈	2
#10598858	3
#71298	5

subgroup

index

C₅×C₈×C₃⋊C₈

Subgroups — #14293644

Conjugacy classes of subgroups

Complements

Maximal subgroups