Group #14278507 — subgroups

Subgroups — #14278507

Order 1920.

160 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11747986	C₂	1	1	yes	maximal, characteristic
2	#11746759	C₂	1	1	yes	maximal, characteristic
3	C₃×#64883	C₂	1	1	yes	maximal, characteristic
4	#10598616	—	3	1	no	maximal, Hall
5	#91388	—	2	1	no	—
6	C₂×C₃⋊#674	—	2	1	no	—
7	C₃×C₄⋊#170	—	2	1	no	—
8	#91102	C₄	1	1	yes	characteristic
9	C₅×#818	C₄	1	1	yes	characteristic
10	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	#90985	—	2	1	no	—
12	#67468	—	5	1	no	maximal, Hall
13	#65167	—	3	1	no	—
14	#64888	—	3	1	no	—
15	#64883	D₃	1	1	yes	characteristic
16	C₂×C₃×#170	—	4	1	no	—
17	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₁₅	—	2	1	no	—
19	C₅×C₄⋊Dic₃	—	2	1	no	—
20	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
21	C₂×C₃×#168	D₄	1	1	yes	characteristic
22	Dic₃⋊Dic₅	—	4	1	no	—
23	#6669	—	4	1	no	—
24	C₃⋊#674	—	4	1	no	—
25	#4683	—	5	1	no	—
26	C₃×#334	—	5	1	no	—
27	C₄⋊(C₃⋊C₁₆)	—	5	1	no	—
28	#4036	—	6	1	no	—
29	C₄⋊#170	—	2	1	no	—
30	C₂×#674	—	6	1	no	—
31	C₅×#107	—	3	1	no	—
32	#3950	—	3	1	no	—
33	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
34	#3940	—	6	1	no	—
35	#1363	—	15	1	no	Sylow
36	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₁₅	—	4	1	no	—
38	C₂×C₅×Dic₃	—	4	1	no	—
39	C₂×C₃×Dic₅	D₈	1	1	yes	characteristic
40	C₂×C₃×Dic₅	SD₁₆	1	1	yes	characteristic
41	C₃×#170	—	8	1	no	—
42	C₃×C₅×C₈	—	2	1	no	—
43	C₃×#168	SD₁₆	1	1	yes	characteristic
44	C₃×#168	Q₁₆	1	1	yes	characteristic
45	#870	—	10	1	no	—
46	C₃×C₄⋊C₈	—	10	1	no	—
47	C₃×C₄×C₈	—	5	1	no	—
48	#818	—	5	1	no	—
49	#818	AGL(1,𝔽₅)	1	1	yes	characteristic
50	#817	—	10	1	no	—
51	C₂×C₃⋊C₁₆	—	10	1	no	—
52	C₂×#170	—	4	1	no	—
53	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
54	C₅×C₄⋊C₄	—	6	1	no	—
55	C₄⋊Dic₅	—	6	1	no	—
56	C₄⋊Dic₅	—	12	1	no	—
57	#683	—	12	1	no	—
58	C₄×Dic₅	#67	1	1	yes	characteristic
59	C₂×#168	D₁₂	1	1	yes	characteristic
60	#674	—	12	1	no	—
61	#499	—	15	1	no	—
62	C₄⋊C₁₆	—	15	1	no	—
63	#334	—	5	1	no	—
64	C₂²×C₃×C₅	#102	1	1	yes	characteristic
65	C₃×C₄×C₅	#101	1	1	yes	characteristic
66	C₃×C₄×C₅	#103	1	1	yes	characteristic
67	Dic₁₅	—	8	1	no	—
68	C₃×Dic₅	—	2	1	no	—
69	C₃×Dic₅	—	2	1	no	—
70	C₅×Dic₃	—	8	1	no	—
71	C₂×C₃×C₈	—	5	1	no	—
72	C₂×C₃×C₈	—	20	1	no	—
73	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₃×C₄²	—	5	1	no	—
75	C₄⋊Dic₃	—	2	1	no	—
76	C₄⋊Dic₃	—	10	1	no	—
77	Dic₃⋊C₄	—	20	1	no	—
78	Dic₃⋊C₄	—	20	1	no	—
79	C₃⋊C₁₆	—	20	1	no	—
80	C₂×C₄×C₅	—	12	1	no	—
81	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
82	C₂×Dic₅	—	12	1	no	—
83	C₂×Dic₅	#213	1	1	yes	characteristic
84	C₂×Dic₅	#215	1	1	yes	characteristic
85	#170	—	8	1	no	—
86	C₅×C₈	—	2	1	no	—
87	#168	#204	1	1	yes	characteristic
88	#168	Dic₁₂	1	1	yes	characteristic
89	#125	—	30	1	no	—
90	C₂×C₁₆	—	30	1	no	—
91	#107	—	3	1	no	—
92	#107	—	15	1	no	—
93	#106	—	30	1	no	—
94	C₄⋊C₈	—	10	1	no	—
95	C₄×C₈	—	5	1	no	—
96	C₂×C₃×C₅	#332	1	1	yes	characteristic
97	C₂×C₃×C₅	#359	1	1	yes	characteristic
98	C₂×C₃×C₅	#360	1	1	yes	characteristic
99	C₂×C₃×C₄	—	5	1	no	—
100	C₂×C₃×C₄	—	5	1	no	—
101	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×Dic₃	—	4	1	no	—
103	C₂×Dic₃	—	20	1	no	—
104	C₃×C₈	—	2	1	no	—
105	C₃×C₈	—	5	1	no	—
106	C₃×C₈	—	5	1	no	—
107	C₃×C₈	—	40	1	no	—
108	C₂²×C₅	#809	1	1	yes	characteristic
109	C₄×C₅	—	24	1	no	—
110	C₄×C₅	#816	1	1	yes	characteristic
111	C₄×C₅	#824	1	1	yes	characteristic
112	Dic₅	—	2	1	no	—
113	Dic₅	—	2	1	no	—
114	Dic₅	—	24	1	no	—
115	C₂×C₈	—	5	1	no	—
116	C₂×C₈	—	20	1	no	—
117	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₄⋊C₄	—	6	1	no	—
119	C₄⋊C₄	—	30	1	no	—
120	C₄⋊C₄	—	60	1	no	—
121	C₄⋊C₄	—	60	1	no	—
122	C₄²	—	5	1	no	—
123	C₁₆	—	60	1	no	—
124	C₃×C₅	#1363	1	1	yes	characteristic, complemented, Hall
125	C₂²×C₃	#4007	1	1	yes	characteristic
126	C₃×C₄	—	10	1	no	—
127	C₃×C₄	—	10	1	no	—
128	C₃×C₄	#4005	1	1	yes	characteristic
129	C₃×C₄	#4009	1	1	yes	characteristic
130	Dic₃	—	8	1	no	—
131	Dic₃	—	40	1	no	—
132	C₂×C₅	#4482	1	1	yes	characteristic
133	C₂×C₅	#4488	1	1	yes	characteristic
134	C₂×C₅	#4497	1	1	yes	characteristic
135	C₂×C₄	—	5	1	no	—
136	C₂×C₄	—	5	1	no	—
137	C₂×C₄	—	12	1	no	—
138	C₂×C₄	—	60	1	no	—
139	C₂×C₄	(C₂×D₃)⋊AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	5	1	no	—
142	C₈	—	5	1	no	—
143	C₈	—	40	1	no	—
144	C₂×C₃	#64873	1	1	yes	characteristic
145	C₂×C₃	#64903	1	1	yes	characteristic
146	C₂×C₃	#64909	1	1	yes	characteristic
147	C₅	#67468	1	1	yes	characteristic, complemented, Sylow
148	C₂²	#91151	1	1	yes	characteristic
149	C₄	—	10	1	no	—
150	C₄	—	10	1	no	—
151	C₄	—	24	1	no	—
152	C₄	—	120	1	no	—
153	C₄	#91154	1	1	yes	characteristic
154	C₄	#91175	1	1	yes	characteristic
155	C₃	#10598616	1	1	yes	characteristic, complemented, Sylow
156	C₂	#11746461	1	1	yes	characteristic
157	C₂	#11746464	1	1	yes	characteristic
158	C₂	#11746509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11747986

C₂

yes

maximal, characteristic

#11746759

C₂

yes

maximal, characteristic

C₃×#64883

C₂

yes

maximal, characteristic

#10598616

—

maximal, Hall

#91388

—

C₂×C₃⋊#674

—

C₃×C₄⋊#170

—

#91102

C₄

yes

characteristic

C₅×#818

C₄

yes

characteristic

C₃×C₈×Dic₅

C₂²

yes

—

—

—

—

D₃

yes

characteristic

C₂×C₃×#170

—

C₂×C₃×C₅×C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₁₅

—

C₅×C₄⋊Dic₃

—

C₃×C₄×Dic₅

D₄

yes

characteristic

C₂×C₃×#168

D₄

yes

characteristic

Dic₃⋊Dic₅

—

—

—

—

—

C₄⋊(C₃⋊C₁₆)

—

—

—

—

—

—

C₈×Dic₅

C₂×D₃

yes

—

—

C₂×C₃×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₅

—

C₂×C₅×Dic₃

—

C₂×C₃×Dic₅

D₈

yes

characteristic

C₂×C₃×Dic₅

SD₁₆

yes

characteristic

C₃×#170

—

C₃×C₅×C₈

—

C₃×#168

SD₁₆

yes

characteristic

C₃×#168

Q₁₆

yes

characteristic

#870

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

—

yes

—

C₂×C₃⋊C₁₆

—

C₂×#170

—

C₂×C₅×C₈

C₄×D₃

yes

characteristic

C₅×C₄⋊C₄

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

#67

yes

characteristic

C₂×#168

D₁₂

yes

characteristic

#674

—

#499

—

C₄⋊C₁₆

—

#334

—

C₂²×C₃×C₅

yes

C₃×C₄×C₅

yes

C₃×C₄×C₅

yes

Dic₁₅

—

C₃×Dic₅

—

C₃×Dic₅

—

C₅×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₃⋊C₁₆

—

C₂×C₄×C₅

—

C₂×C₄×C₅

(C₂×D₃)⋊C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

yes

C₂×Dic₅

yes

—

C₅×C₈

—

yes

Dic₁₂

yes

characteristic

#125

—

C₂×C₁₆

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₄×C₈

—

C₂×C₃×C₅

yes

C₂×C₃×C₅

yes

C₂×C₃×C₅

yes

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

102

C₂×Dic₃

—

103

C₂×Dic₃

—

104

C₃×C₈

—

105

C₃×C₈

—

106

C₃×C₈

—

107

C₃×C₈

—

108

C₂²×C₅

#809

yes

characteristic

109

C₄×C₅

—

110

C₄×C₅

#816

yes

characteristic

111

C₄×C₅

#824

yes

characteristic

112

Dic₅

—

113

Dic₅

—

114

Dic₅

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₃×AGL(1,𝔽₅)

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₁₆

—

124

C₃×C₅

#1363

yes

characteristic, complemented, Hall

125

C₂²×C₃

#4007

yes

characteristic

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

#4005

yes

characteristic

129

C₃×C₄

#4009

yes

characteristic

130

Dic₃

—

131

Dic₃

—

132

C₂×C₅

#4482

yes

characteristic

133

C₂×C₅

#4488

yes

characteristic

134

C₂×C₅

#4497

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂×C₃

#64873

yes

characteristic

145

C₂×C₃

#64903

yes

characteristic

146

C₂×C₃

#64909

yes

characteristic

147

C₅

#67468

yes

characteristic, complemented, Sylow

148

C₂²

#91151

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

120

—

153

C₄

#91154

yes

characteristic

154

C₄

#91175

yes

characteristic

155

C₃

#10598616

yes

characteristic, complemented, Sylow

156

C₂

#11746461

yes

characteristic

157

C₂

#11746464

yes

characteristic

158

C₂

#11746509

yes

characteristic

	subgroup	G/H
center	C₂²	#91151
commutator, Fitting	C₂×C₃×C₅×C₈	C₂×C₄
Frattini, hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
socle, prime radical	C₂²×C₃×C₅	#102
cosocle	C₃×C₈×Dic₅	C₂²
hypocenter	C₃×C₅	#1363
LCS	G ▷ C₂×C₃×C₅×C₈ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₂²

#91151

commutator, Fitting

C₂×C₃×C₅×C₈

C₂×C₄

Frattini, hypercenter

C₂×C₈

C₂²×C₃×C₅

C₃×C₈×Dic₅

C₂²

hypocenter

C₃×C₅

#1363

LCS

G ▷ C₂×C₃×C₅×C₈ ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
124. C₃×C₅	#1363	semidirect
147. C₅	#67468	semidirect
155. C₃	#10598616	semidirect

H ≅ G/N

type

124. C₃×C₅

#1363

semidirect

147. C₅

#67468

semidirect

155. C₃

#10598616

semidirect

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

subgroup	index
#11747986	2
#11746759	2
C₃×#64883	2
#10598616	3
#67468	5

subgroup

index

Subgroups — #14278507

Conjugacy classes of subgroups

Complements

Maximal subgroups