Group #14278418 — subgroups

Subgroups — #14278418

Order 1920.

204 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 53 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×#65374	C₂	1	1	yes	maximal, characteristic
2	#11746732	C₂	1	2	yes	maximal, complemented
3	#10598757	—	3	1	no	maximal, Hall
4	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
5	C₂×C₃×#688	—	2	2	no	—
6	C₃×#4020	C₄	1	1	yes	characteristic
7	#91227	—	2	2	no	—
8	C₃×C₄×#168	C₂²	1	1	yes	characteristic
9	#67787	—	5	1	no	maximal, Hall
10	#65374	D₃	1	1	yes	characteristic
11	#64861	—	3	2	no	—
12	C₂×C₃×C₅×Q₈	—	2	1	no	—
13	C₂×C₃×D₂₀	—	2	1	no	—
14	C₂×C₃×D₂₀	—	4	1	no	—
15	C₃×C₅×C₄⋊C₄	—	4	1	no	—
16	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₃×#688	—	4	2	no	—
18	C₃×#688	—	4	2	no	—
19	C₂×C₃×#168	D₄	1	2	yes	—
20	C₃⋊#674	—	4	2	no	—
21	C₃×#492	—	5	1	no	—
22	#4463	—	5	2	no	—
23	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
24	C₂×#688	—	2	2	no	—
25	#4020	Dic₃	1	1	yes	characteristic
26	#3998	—	6	2	no	—
27	C₄×#168	C₂×D₃	1	1	yes	characteristic
28	#1359	—	15	1	no	Sylow
29	C₂²×C₃×D₅	—	4	1	no	—
30	C₃×C₅×Q₈	—	4	2	no	—
31	C₂×C₃×C₄×C₅	—	2	1	no	—
32	C₂×C₃×C₄×C₅	—	4	1	no	—
33	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×D₂₀	—	4	2	no	—
35	C₃×D₂₀	—	8	2	no	—
36	C₃×#168	D₈	1	2	yes	—
37	C₃×#168	SD₁₆	1	2	yes	—
38	C₂×C₃×SD₁₆	—	10	2	no	—
39	C₃×C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
40	C₃×#127	—	5	1	no	—
41	C₃×C₄×C₈	—	5	1	no	—
42	#812	—	10	2	no	—
43	C₂×C₅×Q₈	—	2	1	no	—
44	C₂×D₂₀	—	2	1	no	—
45	C₂×D₂₀	—	4	1	no	—
46	C₅×C₄⋊C₄	—	4	1	no	—
47	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
48	#688	—	4	2	no	—
49	#688	—	4	2	no	—
50	C₂×#168	#67	1	2	yes	—
51	#674	—	12	2	no	—
52	#492	—	5	1	no	—
53	#364	—	15	2	no	—
54	C₂²×C₃×C₅	#100	1	1	yes	characteristic
55	C₂×C₃×D₅	—	8	2	no	—
56	C₂×C₃×D₅	—	8	1	no	—
57	C₃×C₄×C₅	—	2	2	no	—
58	C₃×C₄×C₅	—	8	1	no	—
59	C₃×C₄×C₅	#102	1	2	yes	—
60	C₂×C₃×Q₈	—	2	1	no	—
61	C₂×C₃×D₄	—	10	1	no	—
62	C₂×C₃×D₄	—	20	1	no	—
63	C₃×SD₁₆	—	20	2	no	—
64	C₃×SD₁₆	—	20	2	no	—
65	C₂×C₃×C₈	—	5	2	no	—
66	C₃×C₄⋊C₄	—	4	1	no	—
67	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₃⋊C₁₆	—	20	2	no	—
69	C₂²×D₅	—	4	1	no	—
70	C₅×Q₈	—	4	2	no	—
71	C₂×C₄×C₅	—	2	1	no	—
72	C₂×C₄×C₅	—	4	1	no	—
73	C₂×C₄×C₅	#217	1	1	yes	characteristic
74	D₂₀	—	4	2	no	—
75	D₂₀	—	8	2	no	—
76	#168	#213	1	2	yes	—
77	#168	#214	1	2	yes	—
78	C₂×SD₁₆	—	10	2	no	—
79	C₄⋊Q₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
80	#127	—	5	1	no	—
81	M₃₂	—	30	2	no	—
82	C₄×C₈	—	5	1	no	—
83	C₂×C₃×C₅	—	2	1	no	—
84	C₂×C₃×C₅	#331	1	1	yes	characteristic
85	C₃×D₅	—	16	1	no	—
86	C₂³×C₃	—	20	1	no	—
87	C₃×Q₈	—	4	2	no	—
88	C₃×D₄	—	20	2	no	—
89	C₃×D₄	—	40	2	no	—
90	C₂×C₃×C₄	—	2	1	no	—
91	C₂×C₃×C₄	—	4	1	no	—
92	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₃×C₈	—	5	2	no	—
94	C₃×C₈	—	5	2	no	—
95	C₂²×C₅	#833	1	1	yes	characteristic
96	C₂×D₅	—	8	2	no	—
97	C₂×D₅	—	8	1	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	8	1	no	—
100	C₄×C₅	#832	1	2	yes	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	10	1	no	—
103	C₂×D₄	—	20	1	no	—
104	SD₁₆	—	20	2	no	—
105	SD₁₆	—	20	2	no	—
106	C₂×C₈	—	5	2	no	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
109	C₁₆	—	60	2	no	—
110	C₃×C₅	#1359	1	1	yes	characteristic, complemented, Hall
111	C₂²×C₃	—	40	2	no	—
112	C₂²×C₃	—	40	1	no	—
113	C₂²×C₃	#4013	1	1	yes	characteristic
114	C₃×C₄	—	2	2	no	—
115	C₃×C₄	—	8	1	no	—
116	C₃×C₄	#4007	1	2	yes	—
117	C₂×C₅	—	2	1	no	—
118	C₂×C₅	#4544	1	1	yes	characteristic
119	D₅	—	16	1	no	—
120	C₂³	—	20	1	no	—
121	Q₈	—	4	2	no	—
122	D₄	—	20	2	no	—
123	D₄	—	40	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#6763	1	1	yes	characteristic
127	C₈	—	5	2	no	—
128	C₈	—	5	2	no	—
129	C₂×C₃	—	2	1	no	—
130	C₂×C₃	—	80	1	no	—
131	C₂×C₃	#64925	1	1	yes	characteristic
132	C₅	#67787	1	1	yes	characteristic, complemented, Sylow
133	C₂²	—	40	2	no	—
134	C₂²	—	40	1	no	—
135	C₂²	#91242	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	8	1	no	—
138	C₄	#91236	1	2	yes	—
139	C₃	#10598757	1	1	yes	characteristic, complemented, Sylow
140	C₂	—	2	1	no	—
141	C₂	—	80	1	no	—
142	C₂	#11746796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#65374

C₂

yes

maximal, characteristic

#11746732

C₂

yes

maximal, complemented

#10598757

—

maximal, Hall

C₃×C₅×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₃×#688

—

C₃×#4020

C₄

yes

characteristic

#91227

—

C₃×C₄×#168

C₂²

yes

—

D₃

yes

characteristic

#64861

—

C₂×C₃×C₅×Q₈

—

C₂×C₃×D₂₀

—

C₂×C₃×D₂₀

—

C₃×C₅×C₄⋊C₄

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₃×#688

—

C₃×#688

—

C₂×C₃×#168

D₄

yes

—

C₃⋊#674

—

C₃×#492

—

#4463

—

C₅×C₄⋊Q₈

Dic₃

yes

characteristic

C₂×#688

—

#4020

Dic₃

yes

characteristic

#3998

—

C₄×#168

C₂×D₃

yes

characteristic

#1359

—

Sylow

C₂²×C₃×D₅

—

C₃×C₅×Q₈

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₃×D₂₀

—

C₃×D₂₀

—

C₃×#168

D₈

yes

—

C₃×#168

SD₁₆

yes

—

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

AGL(1,𝔽₅)

yes

characteristic

C₃×#127

—

C₃×C₄×C₈

—

#812

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Dic₃

yes

—

—

yes

—

#674

—

#492

—

#364

—

C₂²×C₃×C₅

#100

yes

characteristic

C₂×C₃×D₅

—

C₂×C₃×D₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

#102

yes

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₃×SD₁₆

—

C₃×SD₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃⋊C₁₆

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#217

yes

characteristic

D₂₀

—

D₂₀

—

#168

#213

yes

—

#168

#214

yes

—

C₂×SD₁₆

—

C₄⋊Q₈

ΓL(1,𝔽₁₆)

yes

characteristic

#127

—

M₃₂

—

C₄×C₈

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#331

yes

characteristic

C₃×D₅

—

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₂²×C₅

#833

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

#832

yes

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

SD₁₆

—

105

SD₁₆

—

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄²

C₂×ΓL(1,𝔽₁₆)

yes

characteristic

109

C₁₆

—

110

C₃×C₅

#1359

yes

characteristic, complemented, Hall

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

#4013

yes

characteristic

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

#4007

yes

—

117

C₂×C₅

—

118

C₂×C₅

#4544

yes

characteristic

119

D₅

—

120

C₂³

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#6763

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

#64925

yes

characteristic

132

C₅

#67787

yes

characteristic, complemented, Sylow

133

C₂²

—

134

C₂²

—

135

C₂²

#91242

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#91236

yes

—

139

C₃

#10598757

yes

characteristic, complemented, Sylow

140

C₂

—

141

C₂

—

142

C₂

#11746796

yes

characteristic

	subgroup	G/H
center	C₂	#11746796
commutator	C₃×C₄²×C₅	C₂×C₄
Frattini	C₄²	C₂×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×C₄⋊Q₈	C₄
socle	C₂×C₃×C₅	#331
prime radical	C₃×#4020	C₄
cosocle	C₃×C₄×#168	C₂²
hypercenter	C₄⋊Q₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1359
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂

#11746796

commutator

C₃×C₄²×C₅

C₂×C₄

Frattini

C₄²

C₂×ΓL(1,𝔽₁₆)

Fitting

C₃×C₅×C₄⋊Q₈

C₄

C₂×C₃×C₅

C₄

C₃×C₄×#168

C₂²

hypercenter

C₄⋊Q₈

ΓL(1,𝔽₁₆)

hypocenter

C₃×C₅

#1359

LCS

G ▷ C₃×C₄²×C₅ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #11746732	C₂	semidirect
110. C₃×C₅	#1359	semidirect
132. C₅	#67787	semidirect
139. C₃	#10598757	semidirect

H ≅ G/N

type

2. #11746732

C₂

semidirect

110. C₃×C₅

#1359

semidirect

132. C₅

#67787

semidirect

139. C₃

#10598757

semidirect

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

subgroup	index
C₃×#65374	2
#11746732	2
#10598757	3
#67787	5

subgroup

index

Subgroups — #14278418

Conjugacy classes of subgroups

Complements

Maximal subgroups