Group #14278381 — subgroups

Subgroups — #14278381

Order 1920.

156 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 42 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	#11746931	C₂	1	1	yes	maximal, characteristic
2	#11746731	C₂	1	1	yes	maximal, characteristic
3	C₃×#64860	C₂	1	1	yes	maximal, characteristic
4	#10598752	—	3	1	no	maximal, Hall
5	#91759	C₄	1	1	yes	characteristic
6	C₅×#869	C₄	1	1	yes	characteristic
7	#91227	—	2	1	no	—
8	C₃×#3998	—	2	1	no	—
9	C₃×#3935	C₂²	1	1	yes	characteristic
10	#90973	—	2	1	no	—
11	#90973	—	2	1	no	—
12	#67521	—	5	1	no	maximal, Hall
13	#65371	—	3	1	no	—
14	#64860	—	3	1	no	—
15	#64860	D₃	1	1	yes	characteristic
16	C₂×C₅×Dic₆	—	2	1	no	—
17	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₁₅	—	2	1	no	—
19	#6712	—	4	1	no	—
20	C₅×C₄⋊Dic₃	—	4	1	no	—
21	C₂×C₃×#168	D₄	1	1	yes	characteristic
22	C₂×C₃×#168	D₄	1	1	yes	characteristic
23	C₃⋊#674	—	2	2	no	—
24	C₃×#674	—	2	2	no	—
25	#4703	—	5	1	no	—
26	C₃×#347	—	5	1	no	—
27	#4466	—	5	1	no	—
28	C₅×C₄⋊Q₈	—	3	1	no	—
29	#4016	—	3	1	no	—
30	#3998	—	2	1	no	—
31	#3998	—	6	1	no	—
32	#3967	—	6	1	no	—
33	#3967	—	6	1	no	—
34	#3935	C₂×D₃	1	1	yes	characteristic
35	#1356	—	15	1	no	Sylow
36	C₂×C₃×C₄×C₅	—	2	1	no	—
37	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₅	—	4	1	no	—
39	C₂×C₅×Dic₃	—	4	1	no	—
40	C₅×Dic₆	—	4	1	no	—
41	C₅×Dic₆	—	4	1	no	—
42	C₃×#168	—	2	1	no	—
43	C₃×#168	—	2	1	no	—
44	#870	—	5	1	no	—
45	#869	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃×M₃₂	—	10	1	no	—
47	C₃×#97	—	5	1	no	—
48	#816	—	10	1	no	—
49	#816	—	10	1	no	—
50	#812	—	10	1	no	—
51	C₂×C₅×Q₈	—	6	1	no	—
52	C₅×C₄⋊C₄	—	12	1	no	—
53	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
54	C₄⋊Dic₅	—	6	1	no	—
55	#683	—	12	1	no	—
56	C₂×#168	D₁₂	1	1	yes	characteristic
57	C₂×#168	#67	1	1	yes	characteristic
58	#674	—	2	2	no	—
59	#674	—	6	2	no	—
60	#491	—	15	1	no	—
61	#347	—	5	1	no	—
62	#347	—	15	1	no	—
63	C₂²×C₃×C₅	#100	1	1	yes	characteristic
64	C₃×C₄×C₅	—	4	1	no	—
65	C₃×C₄×C₅	#104	1	1	yes	characteristic
66	C₃×C₄×C₅	#104	1	1	yes	characteristic
67	Dic₁₅	—	8	1	no	—
68	C₅×Dic₃	—	8	1	no	—
69	C₂×Dic₆	—	2	1	no	—
70	C₂×C₃×C₈	—	5	1	no	—
71	C₂×C₃×C₈	—	5	1	no	—
72	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₄⋊Dic₃	—	4	1	no	—
74	C₄⋊Dic₃	—	10	1	no	—
75	Dic₃⋊C₄	—	20	1	no	—
76	C₃×C₁₆	—	10	2	no	—
77	C₃⋊C₁₆	—	10	2	no	—
78	C₅×Q₈	—	12	1	no	—
79	C₅×Q₈	—	12	1	no	—
80	C₂×C₄×C₅	—	2	1	no	—
81	C₂×C₄×C₅	—	12	1	no	—
82	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
83	C₂×Dic₅	—	12	1	no	—
84	#168	—	2	1	no	—
85	#168	—	2	1	no	—
86	C₄⋊Q₈	—	3	1	no	—
87	#125	—	15	1	no	—
88	M₃₂	—	10	1	no	—
89	M₃₂	—	30	1	no	—
90	#103	—	30	1	no	—
91	#103	—	30	1	no	—
92	#97	—	5	1	no	—
93	C₂×C₃×C₅	—	2	1	no	—
94	C₂×C₃×C₅	#329	1	1	yes	characteristic
95	C₂×C₃×C₄	—	2	1	no	—
96	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₂×Dic₃	—	4	1	no	—
98	C₂×Dic₃	—	20	1	no	—
99	Dic₆	—	4	1	no	—
100	Dic₆	—	4	1	no	—
101	C₃×C₈	—	10	1	no	—
102	C₃×C₈	—	10	1	no	—
103	C₂²×C₅	#823	1	1	yes	characteristic
104	C₄×C₅	—	4	1	no	—
105	C₄×C₅	—	24	1	no	—
106	C₄×C₅	#805	1	1	yes	characteristic
107	C₄×C₅	#825	1	1	yes	characteristic
108	Dic₅	—	24	1	no	—
109	C₂×Q₈	—	6	1	no	—
110	C₂×C₈	—	5	1	no	—
111	C₂×C₈	—	5	1	no	—
112	C₄⋊C₄	—	12	1	no	—
113	C₄⋊C₄	—	30	1	no	—
114	C₄⋊C₄	—	60	1	no	—
115	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₁₆	—	10	2	no	—
117	C₁₆	—	30	2	no	—
118	C₃×C₅	#1356	1	1	yes	characteristic, complemented, Hall
119	C₂²×C₃	#4013	1	1	yes	characteristic
120	C₃×C₄	—	4	1	no	—
121	C₃×C₄	#4008	1	1	yes	characteristic
122	C₃×C₄	#4008	1	1	yes	characteristic
123	Dic₃	—	8	1	no	—
124	Dic₃	—	40	1	no	—
125	C₂×C₅	—	2	1	no	—
126	C₂×C₅	#4464	1	1	yes	characteristic
127	Q₈	—	12	1	no	—
128	Q₈	—	12	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	12	1	no	—
131	C₂×C₄	—	60	1	no	—
132	C₂×C₄	(C₂×D₃)⋊AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₈	—	10	1	no	—
134	C₈	—	10	1	no	—
135	C₂×C₃	—	2	1	no	—
136	C₂×C₃	#64921	1	1	yes	characteristic
137	C₅	#67521	1	1	yes	characteristic, complemented, Sylow
138	C₂²	#91174	1	1	yes	characteristic
139	C₄	—	4	1	no	—
140	C₄	—	24	1	no	—
141	C₄	—	120	1	no	—
142	C₄	#91155	1	1	yes	characteristic
143	C₄	#91156	1	1	yes	characteristic
144	C₃	#10598752	1	1	yes	characteristic, complemented, Sylow
145	C₂	—	2	1	no	—
146	C₂	#11746504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11746931

C₂

yes

maximal, characteristic

#11746731

C₂

yes

maximal, characteristic

C₃×#64860

C₂

yes

maximal, characteristic

#10598752

—

maximal, Hall

#91759

C₄

yes

characteristic

C₅×#869

C₄

yes

—

—

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

C₂×C₅×Dic₆

—

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₄⋊Dic₁₅

—

#6712

—

C₅×C₄⋊Dic₃

—

C₂×C₃×#168

D₄

yes

characteristic

C₂×C₃×#168

D₄

yes

—

—

—

—

—

C₅×C₄⋊Q₈

—

—

—

—

—

—

C₂×D₃

yes

characteristic

#1356

—

Sylow

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₅

—

C₂×C₅×Dic₃

—

C₅×Dic₆

—

C₅×Dic₆

—

—

—

—

yes

C₃×M₃₂

—

C₃×#97

—

#816

—

#816

—

#812

—

C₂×C₅×Q₈

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₄×D₃

yes

characteristic

C₄⋊Dic₅

—

#683

—

C₂×#168

D₁₂

yes

yes

—

—

—

—

—

C₂²×C₃×C₅

#100

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

Dic₁₅

—

C₅×Dic₃

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

(C₂×D₃)⋊C₄

yes

characteristic

C₂×Dic₅

—

#168

—

#168

—

C₄⋊Q₈

—

#125

—

M₃₂

—

M₃₂

—

#103

—

#103

—

#97

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#329

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

100

Dic₆

—

101

C₃×C₈

—

102

C₃×C₈

—

103

C₂²×C₅

#823

yes

characteristic

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

#805

yes

characteristic

107

C₄×C₅

#825

yes

characteristic

108

Dic₅

—

109

C₂×Q₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

D₃×AGL(1,𝔽₅)

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₃×C₅

#1356

yes

characteristic, complemented, Hall

119

C₂²×C₃

#4013

yes

characteristic

120

C₃×C₄

—

121

C₃×C₄

#4008

yes

characteristic

122

C₃×C₄

#4008

yes

characteristic

123

Dic₃

—

124

Dic₃

—

125

C₂×C₅

—

126

C₂×C₅

#4464

yes

characteristic

127

Q₈

—

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂×C₃

—

136

C₂×C₃

#64921

yes

characteristic

137

C₅

#67521

yes

characteristic, complemented, Sylow

138

C₂²

#91174

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

120

—

142

C₄

#91155

yes

characteristic

143

C₄

#91156

yes

characteristic

144

C₃

#10598752

yes

characteristic, complemented, Sylow

145

C₂

—

146

C₂

#11746504

yes

characteristic

	subgroup	G/H
center	C₂	#11746504
commutator, Fitting	C₃×C₄²×C₅	C₂×C₄
Frattini, hypercenter	C₄²	D₃×AGL(1,𝔽₅)
socle	C₂×C₃×C₅	#329
prime radical	C₂²×C₃×C₅	#100
cosocle	C₃×#3935	C₂²
hypocenter	C₃×C₅	#1356
LCS	G ▷ C₃×C₄²×C₅ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄²

subgroup

G/H

center

C₂

#11746504

commutator, Fitting

C₃×C₄²×C₅

C₂×C₄

Frattini, hypercenter

C₄²

C₂×C₃×C₅

C₂²×C₃×C₅

C₂²

C₃×C₅

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄²

N	H ≅ G/N	type
118. C₃×C₅	#1356	semidirect
137. C₅	#67521	semidirect
144. C₃	#10598752	semidirect

H ≅ G/N

type

118. C₃×C₅

#1356

semidirect

137. C₅

#67521

semidirect

144. C₃

#10598752

semidirect

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

subgroup	index
#11746931	2
#11746731	2
C₃×#64860	2
#10598752	3
#67521	5

subgroup

index

Subgroups — #14278381

Conjugacy classes of subgroups

Complements

Maximal subgroups