Group #14278567 — subgroups

Subgroups — #14278567

Order 1920.

148 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 50 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	#11746956	C₂	1	1	yes	maximal, characteristic
2	#11746732	C₂	1	1	yes	maximal, characteristic
3	C₃×#64861	C₂	1	1	yes	maximal, characteristic
4	#10598744	—	3	1	no	maximal, Hall
5	#91759	C₄	1	1	yes	characteristic
6	C₅×#870	C₄	1	1	yes	characteristic
7	#91227	—	2	1	no	—
8	C₃×#3998	—	2	1	no	—
9	C₃×C₄×#168	C₂²	1	1	yes	characteristic
10	#90988	—	2	1	no	—
11	#90985	—	2	1	no	—
12	#67574	—	5	1	no	maximal, Hall
13	#65353	—	3	1	no	—
14	#64861	—	3	1	no	—
15	#64861	D₃	1	1	yes	characteristic
16	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₄⋊Dic₁₅	—	2	1	no	—
18	#6712	—	4	1	no	—
19	C₅×C₄⋊Dic₃	—	2	1	no	—
20	C₅×Dic₃⋊C₄	—	4	1	no	—
21	C₂×C₃×#168	D₄	1	1	yes	characteristic
22	C₂×C₃×#168	D₄	1	1	yes	characteristic
23	C₃⋊#674	—	4	1	no	—
24	C₃×#674	—	4	1	no	—
25	#4683	—	5	1	no	—
26	C₃×#364	—	5	1	no	—
27	#4463	—	5	1	no	—
28	C₅×#125	—	3	1	no	—
29	#4016	—	3	1	no	—
30	#3998	—	2	1	no	—
31	#3998	—	6	1	no	—
32	#3940	—	6	1	no	—
33	#3939	—	6	1	no	—
34	C₄×#168	C₂×D₃	1	1	yes	characteristic
35	#1366	—	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	C₃×#168	SD₁₆	1	1	yes	characteristic
41	C₃×#168	SD₁₆	1	1	yes	characteristic
42	C₃×#168	Q₁₆	1	1	yes	characteristic
43	C₃×#168	Q₁₆	1	1	yes	characteristic
44	#870	—	5	1	no	—
45	#870	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃×M₃₂	—	10	1	no	—
47	C₃×C₄×C₈	—	5	1	no	—
48	#818	—	10	1	no	—
49	#817	—	10	1	no	—
50	#812	—	10	1	no	—
51	C₅×C₄⋊C₄	—	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	—	4	1	no	—
59	#674	—	12	1	no	—
60	#499	—	15	1	no	—
61	#364	—	5	1	no	—
62	#364	—	15	1	no	—
63	C₂²×C₃×C₅	#100	1	1	yes	characteristic
64	C₃×C₄×C₅	—	4	1	no	—
65	C₃×C₄×C₅	#103	1	1	yes	characteristic
66	C₃×C₄×C₅	#103	1	1	yes	characteristic
67	Dic₁₅	—	8	1	no	—
68	C₅×Dic₃	—	8	1	no	—
69	C₂×C₃×C₈	—	5	1	no	—
70	C₂×C₃×C₈	—	5	1	no	—
71	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄⋊Dic₃	—	2	1	no	—
73	C₄⋊Dic₃	—	10	1	no	—
74	Dic₃⋊C₄	—	4	1	no	—
75	Dic₃⋊C₄	—	20	1	no	—
76	C₃×C₁₆	—	20	1	no	—
77	C₃⋊C₁₆	—	20	1	no	—
78	C₂×C₄×C₅	—	2	1	no	—
79	C₂×C₄×C₅	—	12	1	no	—
80	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
81	C₂×Dic₅	—	12	1	no	—
82	#168	#204	1	1	yes	characteristic
83	#168	Dic₁₂	1	1	yes	characteristic
84	#168	#214	1	1	yes	characteristic
85	#168	#216	1	1	yes	characteristic
86	#125	—	3	1	no	—
87	#125	—	15	1	no	—
88	M₃₂	—	10	1	no	—
89	M₃₂	—	30	1	no	—
90	#107	—	30	1	no	—
91	#106	—	30	1	no	—
92	C₄×C₈	—	5	1	no	—
93	C₂×C₃×C₅	—	2	1	no	—
94	C₂×C₃×C₅	#333	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	C₃×C₈	—	5	1	no	—
100	C₃×C₈	—	5	1	no	—
101	C₃×C₈	—	5	1	no	—
102	C₃×C₈	—	5	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	#810	1	1	yes	characteristic
107	C₄×C₅	#816	1	1	yes	characteristic
108	Dic₅	—	24	1	no	—
109	C₂×C₈	—	5	1	no	—
110	C₂×C₈	—	5	1	no	—
111	C₄⋊C₄	—	6	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	—	20	1	no	—
117	C₁₆	—	60	1	no	—
118	C₃×C₅	#1366	1	1	yes	characteristic, complemented, Hall
119	C₂²×C₃	#4013	1	1	yes	characteristic
120	C₃×C₄	—	4	1	no	—
121	C₃×C₄	#4009	1	1	yes	characteristic
122	C₃×C₄	#4009	1	1	yes	characteristic
123	Dic₃	—	8	1	no	—
124	Dic₃	—	40	1	no	—
125	C₂×C₅	—	2	1	no	—
126	C₂×C₅	#4481	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	12	1	no	—
129	C₂×C₄	—	60	1	no	—
130	C₂×C₄	(C₂×D₃)⋊AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₈	—	5	1	no	—
132	C₈	—	5	1	no	—
133	C₈	—	5	1	no	—
134	C₈	—	5	1	no	—
135	C₂×C₃	—	2	1	no	—
136	C₂×C₃	#64931	1	1	yes	characteristic
137	C₅	#67574	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	#91153	1	1	yes	characteristic
143	C₄	#91154	1	1	yes	characteristic
144	C₃	#10598744	1	1	yes	characteristic, complemented, Sylow
145	C₂	—	2	1	no	—
146	C₂	#11746511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11746956

C₂

yes

maximal, characteristic

#11746732

C₂

yes

maximal, characteristic

C₃×#64861

C₂

yes

maximal, characteristic

#10598744

—

maximal, Hall

#91759

C₄

yes

characteristic

C₅×#870

C₄

yes

characteristic

#91227

—

C₃×#3998

—

C₃×C₄×#168

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

C₃×C₄²×C₅

C₂×C₄

yes

characteristic

C₄⋊Dic₁₅

—

#6712

—

C₅×C₄⋊Dic₃

—

C₅×Dic₃⋊C₄

—

C₂×C₃×#168

D₄

yes

characteristic

C₂×C₃×#168

D₄

yes

—

—

—

—

—

—

—

—

—

—

—

C₂×D₃

yes

characteristic

#1366

—

Sylow

C₂×C₃×C₄×C₅

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₅

—

C₂×C₅×Dic₃

—

C₃×#168

SD₁₆

yes

characteristic

C₃×#168

SD₁₆

yes

characteristic

C₃×#168

Q₁₆

yes

characteristic

C₃×#168

Q₁₆

yes

—

yes

C₃×M₃₂

—

C₃×C₄×C₈

—

#818

—

#817

—

#812

—

C₅×C₄⋊C₄

—

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×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

(C₂×D₃)⋊C₄

yes

characteristic

C₂×Dic₅

—

yes

Dic₁₂

yes

yes

yes

—

—

M₃₂

—

M₃₂

—

#107

—

#106

—

C₄×C₈

—

C₂×C₃×C₅

—

C₂×C₃×C₅

#333

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃×C₈

—

103

C₂²×C₅

#823

yes

characteristic

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

#810

yes

characteristic

107

C₄×C₅

#816

yes

characteristic

108

Dic₅

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

D₃×AGL(1,𝔽₅)

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₃×C₅

#1366

yes

characteristic, complemented, Hall

119

C₂²×C₃

#4013

yes

characteristic

120

C₃×C₄

—

121

C₃×C₄

#4009

yes

characteristic

122

C₃×C₄

#4009

yes

characteristic

123

Dic₃

—

124

Dic₃

—

125

C₂×C₅

—

126

C₂×C₅

#4481

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂×C₃

—

136

C₂×C₃

#64931

yes

characteristic

137

C₅

#67574

yes

characteristic, complemented, Sylow

138

C₂²

#91174

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

120

—

142

C₄

#91153

yes

characteristic

143

C₄

#91154

yes

characteristic

144

C₃

#10598744

yes

characteristic, complemented, Sylow

145

C₂

—

146

C₂

#11746511

yes

characteristic

	subgroup	G/H
center	C₂	#11746511
commutator, Fitting	C₃×C₄²×C₅	C₂×C₄
Frattini, hypercenter	C₄²	D₃×AGL(1,𝔽₅)
socle	C₂×C₃×C₅	#333
prime radical	C₂²×C₃×C₅	#100
cosocle	C₃×C₄×#168	C₂²
hypocenter	C₃×C₅	#1366
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

#11746511

commutator, Fitting

C₃×C₄²×C₅

C₂×C₄

Frattini, hypercenter

C₄²

C₂×C₃×C₅

C₂²×C₃×C₅

C₃×C₄×#168

C₂²

hypocenter

C₃×C₅

#1366

LCS

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	#1366	semidirect
137. C₅	#67574	semidirect
144. C₃	#10598744	semidirect

H ≅ G/N

type

118. C₃×C₅

#1366

semidirect

137. C₅

#67574

semidirect

144. C₃

#10598744

semidirect

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

subgroup	index
#11746956	2
#11746732	2
C₃×#64861	2
#10598744	3
#67574	5

subgroup

index

Subgroups — #14278567

Conjugacy classes of subgroups

Complements

Maximal subgroups