Group #13085438 — subgroups

Subgroups — #13085438

Order 1440.

276 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 66 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	#10624097	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#704	C₂	1	1	yes	maximal, characteristic, complemented
4	#91174	—	3	4	no	maximal
5	C₂²×C₃⋊D₁₅	C₄	1	1	yes	characteristic
6	C₂²×C₅×C₃⋊D₃	C₄	1	1	yes	characteristic
7	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
8	#67140	—	2	1	no	—
9	#67140	—	2	1	no	—
10	C₃⋊(C₃⋊#170)	—	2	1	no	—
11	C₃²×#170	—	2	1	no	—
12	#63650	—	5	1	no	maximal, Hall
13	C₂×#1203	—	3	4	no	—
14	#6762	—	3	4	no	—
15	C₃×#704	D₃	1	4	yes	complemented
16	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃⋊D₁₅	—	2	1	no	—
18	C₂×C₃⋊D₁₅	—	4	1	no	—
19	C₂×C₅×C₃⋊D₃	—	2	1	no	—
20	C₂×C₅×C₃⋊D₃	—	4	1	no	—
21	C₃²×Dic₅	D₄	1	1	yes	characteristic
22	C₃²×Dic₅	D₄	1	1	yes	characteristic
23	#4013	—	9	1	no	Hall
24	C₂×C₄⋊(C₃⋊D₃)	—	5	1	no	—
25	C₃²×M₁₆	—	5	1	no	—
26	#3748	—	5	1	no	—
27	C₂²×D₁₅	—	3	4	no	—
28	C₂²×C₅×D₃	—	3	4	no	—
29	C₂×C₃×Dic₅	C₂×D₃	1	4	yes	—
30	#1203	—	6	4	no	—
31	#1203	—	6	4	no	—
32	C₃⋊#170	—	6	4	no	—
33	C₃×#170	—	2	4	no	—
34	#823	—	15	4	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	—	9	1	no	—
41	#704	C₃⋊D₃	1	1	yes	characteristic, complemented
42	C₂²×C₃⋊D₃	—	5	1	no	—
43	C₂²×C₃⋊D₃	AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×C₃²×C₄	—	5	1	no	—
45	C₄⋊(C₃⋊D₃)	—	10	1	no	—
46	C₄⋊(C₃⋊D₃)	—	10	1	no	—
47	C₃²×C₈	—	10	1	no	—
48	C₃⋊(C₃⋊C₈)	—	10	1	no	—
49	C₂²×C₃×C₅	C₄×D₃	1	4	yes	—
50	C₂×D₁₅	—	6	4	no	—
51	C₂×D₁₅	—	12	4	no	—
52	C₂×C₅×D₃	—	6	4	no	—
53	C₂×C₅×D₃	—	12	4	no	—
54	C₃×Dic₅	D₁₂	1	4	yes	—
55	C₃×Dic₅	#67	1	4	yes	—
56	C₂×D₁₂	—	15	4	no	—
57	C₃×M₁₆	—	5	4	no	—
58	#208	—	15	4	no	—
59	C₃²×C₅	#100	1	1	yes	characteristic, complemented, Hall
60	C₂³×C₅	—	9	1	no	—
61	C₂²×D₅	—	9	1	no	—
62	C₂²⋊D₅	—	18	1	no	—
63	C₂²⋊D₅	—	18	1	no	—
64	C₂×Dic₅	C₂×C₃⋊D₃	1	1	yes	characteristic
65	#170	—	2	1	no	—
66	#170	—	18	1	no	—
67	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₂×C₃⋊D₃	—	2	1	no	—
69	C₂×C₃⋊D₃	—	4	1	no	—
70	C₂×C₃⋊D₃	—	10	1	no	—
71	C₂×C₃⋊D₃	—	20	1	no	—
72	C₃²×C₄	—	5	1	no	—
73	C₃²×C₄	—	5	1	no	—
74	#100	—	45	1	no	Sylow
75	C₂×C₃×C₅	—	2	4	no	—
76	C₂×C₃×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
77	D₁₅	—	12	4	no	—
78	C₅×D₃	—	12	4	no	—
79	C₂²×D₃	—	3	4	no	—
80	C₂²×D₃	—	15	4	no	—
81	C₂×C₃×C₄	—	5	4	no	—
82	D₁₂	—	30	4	no	—
83	D₁₂	—	30	4	no	—
84	C₃×C₈	—	10	4	no	—
85	C₃⋊C₈	—	30	4	no	—
86	C₂²×C₅	—	18	1	no	—
87	C₂²×C₅	—	36	1	no	—
88	C₂²×C₅	C₄×C₃⋊D₃	1	1	yes	characteristic
89	C₂×D₅	—	18	1	no	—
90	C₂×D₅	—	36	1	no	—
91	Dic₅	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
92	Dic₅	#638	1	1	yes	characteristic
93	C₂×C₃²	—	2	1	no	—
94	C₂×C₃²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
95	C₃⋊D₃	—	4	1	no	—
96	C₃⋊D₃	—	20	1	no	—
97	C₂×D₄	—	45	1	no	—
98	M₁₆	—	5	1	no	—
99	M₁₆	—	45	1	no	—
100	C₃×C₅	#823	1	4	yes	complemented
101	C₂²×C₃	D₃×AGL(1,𝔽₅)	1	4	yes	—
102	C₂×D₃	—	6	4	no	—
103	C₂×D₃	—	12	4	no	—
104	C₂×D₃	—	30	4	no	—
105	C₂×D₃	—	60	4	no	—
106	C₃×C₄	—	5	4	no	—
107	C₃×C₄	—	5	4	no	—
108	C₂×C₅	—	2	1	no	—
109	C₂×C₅	—	36	1	no	—
110	C₂×C₅	#3752	1	1	yes	characteristic
111	D₅	—	36	1	no	—
112	C₃²	#4013	1	1	yes	characteristic, complemented, Sylow
113	C₂³	—	9	1	no	—
114	C₂³	—	45	1	no	—
115	D₄	—	90	1	no	—
116	D₄	—	90	1	no	—
117	C₂×C₄	—	5	1	no	—
118	C₈	—	10	1	no	—
119	C₈	—	90	1	no	—
120	C₂×C₃	—	2	4	no	—
121	C₂×C₃	(C₂×D₃)⋊AGL(1,𝔽₅)	1	4	yes	—
122	D₃	—	12	4	no	—
123	D₃	—	60	4	no	—
124	C₅	#63650	1	1	yes	characteristic, complemented, Sylow
125	C₂²	—	18	1	no	—
126	C₂²	—	36	1	no	—
127	C₂²	—	90	1	no	—
128	C₂²	—	180	1	no	—
129	C₂²	C₃⋊D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₄	—	5	1	no	—
131	C₄	—	5	1	no	—
132	C₃	#91174	1	4	yes	complemented
133	C₂	—	2	1	no	—
134	C₂	—	36	1	no	—
135	C₂	—	180	1	no	—
136	C₂	(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67140

C₂

yes

maximal, characteristic

#10624097

C₂

yes

maximal, characteristic, complemented

C₃²×#704

C₂

yes

maximal, characteristic, complemented

#91174

—

maximal

C₂²×C₃⋊D₁₅

C₄

yes

characteristic

C₂²×C₅×C₃⋊D₃

C₄

yes

characteristic

C₂×C₃²×Dic₅

C₂²

yes

—

—

—

C₃²×#170

—

—

—

—

D₃

yes

complemented

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

characteristic

C₃²×Dic₅

D₄

yes

characteristic

#4013

—

Hall

C₂×C₄⋊(C₃⋊D₃)

—

C₃²×M₁₆

—

#3748

—

C₂²×D₁₅

—

C₂²×C₅×D₃

—

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

—

#704

C₃⋊D₃

yes

characteristic, complemented

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

AGL(1,𝔽₅)

yes

characteristic

C₂×C₃²×C₄

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₂²×C₃×C₅

C₄×D₃

yes

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×C₅×D₃

—

C₂×C₅×D₃

—

C₃×Dic₅

D₁₂

yes

—

C₃×Dic₅

#67

yes

—

C₂×D₁₂

—

C₃×M₁₆

—

#208

—

C₃²×C₅

#100

yes

characteristic, complemented, Hall

C₂³×C₅

—

C₂²×D₅

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

C₂×C₃⋊D₃

yes

characteristic

#170

—

#170

—

C₂²×C₃²

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

—

#100

—

Sylow

C₂×C₃×C₅

—

C₂×C₃×C₅

(C₂×D₃)⋊C₄

yes

—

D₁₅

—

C₅×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₄×C₃⋊D₃

yes

characteristic

C₂×D₅

—

C₂×D₅

—

Dic₅

C₄⋊(C₃⋊D₃)

yes

characteristic

Dic₅

#638

yes

characteristic

C₂×C₃²

—

C₂×C₃²

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

C₃⋊D₃

—

C₃⋊D₃

—

C₂×D₄

—

M₁₆

—

M₁₆

—

100

C₃×C₅

#823

yes

complemented

101

C₂²×C₃

D₃×AGL(1,𝔽₅)

yes

—

102

C₂×D₃

—

103

C₂×D₃

—

104

C₂×D₃

—

105

C₂×D₃

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₂×C₅

—

109

C₂×C₅

—

110

C₂×C₅

#3752

yes

characteristic

111

D₅

—

112

C₃²

#4013

yes

characteristic, complemented, Sylow

113

C₂³

—

114

C₂³

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₈

—

119

C₈

—

120

C₂×C₃

—

121

C₂×C₃

(C₂×D₃)⋊AGL(1,𝔽₅)

yes

—

122

D₃

—

123

D₃

—

124

C₅

#63650

yes

characteristic, complemented, Sylow

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

180

—

129

C₂²

C₃⋊D₃×AGL(1,𝔽₅)

yes

characteristic

130

C₄

—

131

C₄

—

132

C₃

#91174

yes

complemented

133

C₂

—

134

C₂

—

135

C₂

—

180

—

136

C₂

(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)

yes

characteristic

	subgroup	G/H
center	C₂	(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)
commutator, Fitting	C₂²×C₃²×C₅	C₂×C₄
Frattini, hypercenter	C₂²	C₃⋊D₃×AGL(1,𝔽₅)
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

(C₂×AGL(1,𝔽₅))⋊(C₃⋊D₃)

commutator, Fitting

C₂²×C₃²×C₅

C₂×C₄

Frattini, hypercenter

C₂²

C₃⋊D₃×AGL(1,𝔽₅)

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. #10624097	C₂	semidirect
3. C₃²×#704	C₂	semidirect
15. C₃×#704	D₃	semidirect
41. #704	C₃⋊D₃	semidirect
59. C₃²×C₅	#100	semidirect
100. C₃×C₅	#823	semidirect
112. C₃²	#4013	semidirect
124. C₅	#63650	semidirect
132. C₃	#91174	semidirect

H ≅ G/N

type

2. #10624097

C₂

semidirect

3. C₃²×#704

C₂

semidirect

15. C₃×#704

D₃

semidirect

41. #704

C₃⋊D₃

semidirect

59. C₃²×C₅

#100

semidirect

100. C₃×C₅

#823

semidirect

112. C₃²

#4013

semidirect

124. C₅

#63650

semidirect

132. 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
#10624097	2
C₃²×#704	2
#91174	3
#63650	5

subgroup

index

C₃²×#704

Subgroups — #13085438

Conjugacy classes of subgroups

Complements

Maximal subgroups