Group #13085494 — subgroups

Subgroups — #13085494

Order 1440.

238 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₂×#67140	C₂	1	1	yes	maximal, characteristic
2	C₃×#6762	C₂	1	2	yes	maximal, complemented
3	#91174	—	3	2	no	maximal
4	C₂²×C₃⋊D₁₅	C₄	1	1	yes	characteristic
5	C₂²×C₅×C₃⋊D₃	C₄	1	1	yes	characteristic
6	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	#67140	—	2	2	no	—
8	C₃×C₃⋊#170	—	2	2	no	—
9	#63559	—	5	1	no	maximal, Hall
10	C₂×#1203	—	3	2	no	—
11	C₂×#1203	—	6	1	no	—
12	#6762	D₃	1	2	yes	complemented
13	C₃×#704	—	3	2	no	—
14	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃⋊D₁₅	—	2	1	no	—
16	C₂×C₃⋊D₁₅	—	4	1	no	—
17	C₂×C₅×C₃⋊D₃	—	2	1	no	—
18	C₂×C₅×C₃⋊D₃	—	4	1	no	—
19	C₃²×Dic₅	D₄	1	2	yes	—
20	#4013	—	9	1	no	Hall
21	C₂×C₄⋊(C₃⋊D₃)	—	5	1	no	—
22	C₃×#208	—	5	2	no	—
23	C₂²×D₁₅	—	3	2	no	—
24	C₂²×D₁₅	—	6	1	no	—
25	C₂²×C₅×D₃	—	3	2	no	—
26	C₂²×C₅×D₃	—	6	1	no	—
27	C₂×C₃×Dic₅	—	2	1	no	—
28	C₂×C₃×Dic₅	C₂×D₃	1	2	yes	—
29	#1203	—	6	4	no	—
30	#1203	—	6	2	no	—
31	#1203	—	6	2	no	—
32	C₃⋊#170	—	2	2	no	—
33	C₃×#170	—	6	2	no	—
34	#823	—	15	2	no	—
35	C₂×C₃²×C₅	—	2	1	no	—
36	C₂×C₃²×C₅	C₂²⋊C₄	1	1	yes	characteristic
37	C₃⋊D₁₅	—	4	1	no	—
38	C₅×C₃⋊D₃	—	4	1	no	—
39	C₂×C₂²⋊D₅	—	9	1	no	—
40	#704	—	3	2	no	—
41	C₂²×C₃⋊D₃	—	5	1	no	—
42	C₂²×C₃⋊D₃	AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₂×C₃²×C₄	—	5	1	no	—
44	C₄⋊(C₃⋊D₃)	—	10	2	no	—
45	C₃×C₃⋊C₈	—	10	2	no	—
46	C₂²×C₃×C₅	—	2	1	no	—
47	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
48	C₂×D₁₅	—	6	2	no	—
49	C₂×D₁₅	—	6	2	no	—
50	C₂×D₁₅	—	12	2	no	—
51	C₂×D₁₅	—	12	2	no	—
52	C₂×C₅×D₃	—	6	2	no	—
53	C₂×C₅×D₃	—	6	2	no	—
54	C₂×C₅×D₃	—	12	2	no	—
55	C₂×C₅×D₃	—	12	2	no	—
56	C₃×Dic₅	—	2	2	no	—
57	C₃×Dic₅	D₁₂	1	2	yes	—
58	C₃×Dic₅	#67	1	2	yes	—
59	C₂×D₁₂	—	15	2	no	—
60	C₂×D₁₂	—	30	1	no	—
61	C₃×M₁₆	—	15	2	no	—
62	#208	—	5	2	no	—
63	C₃²×C₅	#100	1	1	yes	characteristic, complemented, Hall
64	C₂³×C₅	—	9	1	no	—
65	C₂²×D₅	—	9	1	no	—
66	C₂²⋊D₅	—	18	2	no	—
67	C₂×Dic₅	D₃²	1	1	yes	characteristic
68	#170	—	6	2	no	—
69	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂×C₃⋊D₃	—	2	1	no	—
71	C₂×C₃⋊D₃	—	4	1	no	—
72	C₂×C₃⋊D₃	—	10	1	no	—
73	C₂×C₃⋊D₃	—	20	1	no	—
74	C₃²×C₄	—	5	2	no	—
75	#100	—	45	1	no	Sylow
76	C₂×C₃×C₅	—	2	2	no	—
77	C₂×C₃×C₅	—	2	2	no	—
78	C₂×C₃×C₅	—	2	1	no	—
79	C₂×C₃×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
80	D₁₅	—	12	2	no	—
81	D₁₅	—	12	2	no	—
82	C₅×D₃	—	12	2	no	—
83	C₅×D₃	—	12	2	no	—
84	C₂²×D₃	—	3	2	no	—
85	C₂²×D₃	—	6	1	no	—
86	C₂²×D₃	—	15	2	no	—
87	C₂²×D₃	—	30	1	no	—
88	C₂×C₃×C₄	—	5	2	no	—
89	C₂×C₃×C₄	—	10	1	no	—
90	D₁₂	—	30	2	no	—
91	D₁₂	—	30	2	no	—
92	D₁₂	—	30	4	no	—
93	C₃×C₈	—	30	2	no	—
94	C₃⋊C₈	—	10	2	no	—
95	C₂²×C₅	—	18	1	no	—
96	C₂²×C₅	—	36	1	no	—
97	C₂²×C₅	#624	1	1	yes	characteristic
98	C₂×D₅	—	18	1	no	—
99	C₂×D₅	—	36	1	no	—
100	Dic₅	#626	1	2	yes	—
101	C₂×C₃²	—	2	1	no	—
102	C₂×C₃²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₃⋊D₃	—	4	1	no	—
104	C₃⋊D₃	—	20	1	no	—
105	C₂×D₄	—	45	1	no	—
106	M₁₆	—	15	2	no	—
107	C₃×C₅	—	2	1	no	—
108	C₃×C₅	#823	1	2	yes	complemented
109	C₂²×C₃	—	2	1	no	—
110	C₂²×C₃	D₃×AGL(1,𝔽₅)	1	2	yes	—
111	C₂×D₃	—	6	2	no	—
112	C₂×D₃	—	6	2	no	—
113	C₂×D₃	—	12	2	no	—
114	C₂×D₃	—	12	2	no	—
115	C₂×D₃	—	30	2	no	—
116	C₂×D₃	—	30	2	no	—
117	C₂×D₃	—	60	2	no	—
118	C₂×D₃	—	60	2	no	—
119	C₃×C₄	—	5	2	no	—
120	C₃×C₄	—	5	2	no	—
121	C₃×C₄	—	10	2	no	—
122	C₂×C₅	—	2	1	no	—
123	C₂×C₅	—	36	1	no	—
124	C₂×C₅	#3722	1	1	yes	characteristic
125	D₅	—	36	1	no	—
126	C₃²	#4013	1	1	yes	characteristic, complemented, Sylow
127	C₂³	—	9	1	no	—
128	C₂³	—	45	1	no	—
129	D₄	—	90	2	no	—
130	C₂×C₄	—	5	1	no	—
131	C₈	—	30	2	no	—
132	C₂×C₃	—	2	2	no	—
133	C₂×C₃	—	2	2	no	—
134	C₂×C₃	—	2	1	no	—
135	C₂×C₃	(C₂×D₃)⋊AGL(1,𝔽₅)	1	2	yes	—
136	D₃	—	12	2	no	—
137	D₃	—	12	2	no	—
138	D₃	—	60	2	no	—
139	D₃	—	60	2	no	—
140	C₅	#63559	1	1	yes	characteristic, complemented, Sylow
141	C₂²	—	18	1	no	—
142	C₂²	—	36	1	no	—
143	C₂²	—	90	1	no	—
144	C₂²	—	180	1	no	—
145	C₂²	ΓL(1,𝔽₁₆)⋊D₃	1	1	yes	characteristic
146	C₄	—	5	2	no	—
147	C₃	—	2	1	no	—
148	C₃	#91174	1	2	yes	complemented
149	C₂	—	2	1	no	—
150	C₂	—	36	1	no	—
151	C₂	—	180	1	no	—
152	C₂	#10623978	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67140

C₂

yes

maximal, characteristic

C₃×#6762

C₂

yes

maximal, complemented

#91174

—

maximal

C₂²×C₃⋊D₁₅

C₄

yes

characteristic

C₂²×C₅×C₃⋊D₃

C₄

yes

characteristic

C₂×C₃²×Dic₅

C₂²

yes

characteristic

#67140

—

C₃×C₃⋊#170

—

—

—

—

D₃

yes

complemented

C₃×#704

—

C₂²×C₃²×C₅

C₂×C₄

yes

characteristic

C₂×C₃⋊D₁₅

—

C₂×C₃⋊D₁₅

—

C₂×C₅×C₃⋊D₃

—

C₂×C₅×C₃⋊D₃

—

C₃²×Dic₅

D₄

yes

—

#4013

—

Hall

C₂×C₄⋊(C₃⋊D₃)

—

C₃×#208

—

C₂²×D₁₅

—

C₂²×D₁₅

—

C₂²×C₅×D₃

—

C₂²×C₅×D₃

—

C₂×C₃×Dic₅

—

C₂×C₃×Dic₅

C₂×D₃

yes

—

—

—

—

—

—

—

C₂×C₃²×C₅

—

C₂×C₃²×C₅

C₂²⋊C₄

yes

characteristic

C₃⋊D₁₅

—

C₅×C₃⋊D₃

—

C₂×C₂²⋊D₅

—

#704

—

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

AGL(1,𝔽₅)

yes

characteristic

C₂×C₃²×C₄

—

C₄⋊(C₃⋊D₃)

—

C₃×C₃⋊C₈

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₄×D₃

yes

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×C₅×D₃

—

C₂×C₅×D₃

—

C₂×C₅×D₃

—

C₂×C₅×D₃

—

C₃×Dic₅

—

C₃×Dic₅

D₁₂

yes

—

C₃×Dic₅

#67

yes

—

C₂×D₁₂

—

C₂×D₁₂

—

C₃×M₁₆

—

#208

—

C₃²×C₅

#100

yes

characteristic, complemented, Hall

C₂³×C₅

—

C₂²×D₅

—

C₂²⋊D₅

—

C₂×Dic₅

D₃²

yes

characteristic

#170

—

C₂²×C₃²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

#100

—

Sylow

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₂×C₃×C₅

(C₂×D₃)⋊C₄

yes

—

D₁₅

—

D₁₅

—

C₅×D₃

—

C₅×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#624

yes

characteristic

C₂×D₅

—

C₂×D₅

—

100

Dic₅

#626

yes

—

101

C₂×C₃²

—

102

C₂×C₃²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

103

C₃⋊D₃

—

104

C₃⋊D₃

—

105

C₂×D₄

—

106

M₁₆

—

107

C₃×C₅

—

108

C₃×C₅

#823

yes

complemented

109

C₂²×C₃

—

110

C₂²×C₃

D₃×AGL(1,𝔽₅)

yes

—

111

C₂×D₃

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₂×D₃

—

115

C₂×D₃

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₂×D₃

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

#3722

yes

characteristic

125

D₅

—

126

C₃²

#4013

yes

characteristic, complemented, Sylow

127

C₂³

—

128

C₂³

—

129

D₄

—

130

C₂×C₄

—

131

C₈

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

—

136

D₃

—

137

D₃

—

138

D₃

—

139

D₃

—

140

C₅

#63559

yes

characteristic, complemented, Sylow

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

180

—

145

C₂²

ΓL(1,𝔽₁₆)⋊D₃

yes

characteristic

146

C₄

—

147

C₃

—

148

C₃

#91174

yes

complemented

149

C₂

—

150

C₂

—

151

C₂

—

180

—

152

C₂

#10623978

yes

characteristic

	subgroup	G/H
center	C₂	#10623978
commutator, Fitting	C₂²×C₃²×C₅	C₂×C₄
Frattini, hypercenter	C₂²	ΓL(1,𝔽₁₆)⋊D₃
socle	C₂×C₃²×C₅	C₂²⋊C₄
prime radical	C₂×#67140	C₂
cosocle	C₂×C₃²×Dic₅	C₂²
hypocenter	C₃²×C₅	#100
LCS	G ▷ C₂²×C₃²×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₂²

subgroup

G/H

center

C₂

#10623978

commutator, Fitting

C₂²×C₃²×C₅

C₂×C₄

Frattini, hypercenter

C₂²

ΓL(1,𝔽₁₆)⋊D₃

socle

C₂×C₃²×C₅

C₂²⋊C₄

prime radical

C₂×#67140

C₂

cosocle

C₂×C₃²×Dic₅

C₂²

hypocenter

C₃²×C₅

#100

LCS

G ▷ C₂²×C₃²×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅

UCS

C₁ ◁ C₂ ◁ C₂²

N	H ≅ G/N	type
2. C₃×#6762	C₂	semidirect
12. #6762	D₃	semidirect
63. C₃²×C₅	#100	semidirect
108. C₃×C₅	#823	semidirect
126. C₃²	#4013	semidirect
140. C₅	#63559	semidirect
148. C₃	#91174	semidirect

H ≅ G/N

type

2. C₃×#6762

C₂

semidirect

12. #6762

D₃

semidirect

63. C₃²×C₅

#100

semidirect

108. C₃×C₅

#823

semidirect

126. C₃²

#4013

semidirect

140. C₅

#63559

semidirect

148. C₃

#91174

semidirect

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

subgroup	index
C₂×#67140	2
C₃×#6762	2
#91174	3
#63559	5

subgroup

index

Subgroups — #13085494

Conjugacy classes of subgroups

Complements

Maximal subgroups