Group #28353740 — subgroups

Subgroups — #28353740

Order 13851.

464 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 64 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	#17987421	C₃	1	2	yes	maximal, complemented
2	C₃×C₃²⋊#4294	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987312	C₃	1	9	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	C₃²	1	6	yes	complemented
6	C₃²×#4294	—	3	1	no	—
7	C₃²⋊#4294	—	3	1	no	—
8	C₃²⋊#4294	—	3	1	no	—
9	C₃²⋊#4294	—	3	2	no	—
10	C₃²⋊#4294	—	3	2	no	—
11	C₃²⋊#4294	—	3	2	no	—
12	C₃²⋊#4294	C₃²	1	3	yes	complemented
13	C₃⁴×C₁₉	—	3	1	no	—
14	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
15	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
16	C₃×C₁₉×H(3,𝔽₃)	—	3	1	no	—
17	C₃×C₁₉×H(3,𝔽₃)	—	3	9	no	—
18	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
19	#10624591	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	—	3	9	no	—
21	C₃³×C₁₉	—	3	3	no	—
22	C₃³×C₁₉	—	3	3	no	—
23	C₃³×C₁₉	—	3	3	no	—
24	C₃³×C₁₉	—	9	2	no	—
25	C₃³×C₁₉	—	9	6	no	—
26	C₃³×C₁₉	—	9	3	no	—
27	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
29	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
30	C₃³×C₁₉	C₃³	1	1	yes	characteristic
31	C₁₉×H(3,𝔽₃)	—	3	9	no	—
32	C₁₉×H(3,𝔽₃)	—	3	9	no	—
33	C₁₉×H(3,𝔽₃)	—	3	9	no	—
34	#10587025	—	9	6	no	—
35	C₃×#4294	—	3	6	no	—
36	C₃×#4294	—	3	3	no	—
37	C₃×#4294	—	9	3	no	—
38	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
39	#6891	—	19	2	no	—
40	C₃×C₃²⋊C₉	—	19	1	no	—
41	#6857	—	19	9	no	—
42	C₃²×C₁₉	—	3	1	no	—
43	C₃²×C₁₉	—	3	1	no	—
44	C₃²×C₁₉	—	3	2	no	—
45	C₃²×C₁₉	—	3	2	no	—
46	C₃²×C₁₉	—	3	1	no	—
47	C₃²×C₁₉	—	3	1	no	—
48	C₃²×C₁₉	—	3	9	no	—
49	C₃²×C₁₉	—	3	1	no	—
50	C₃²×C₁₉	—	3	1	no	—
51	C₃²×C₁₉	—	3	2	no	—
52	C₃²×C₁₉	—	3	2	no	—
53	C₃²×C₁₉	—	9	9	no	—
54	C₃²×C₁₉	—	9	27	no	—
55	C₃²×C₁₉	—	9	3	no	—
56	C₃²×C₁₉	—	9	6	no	—
57	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	3	yes	—
58	C₃²×C₁₉	#731	1	3	yes	—
59	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	#4294	—	9	3	no	—
61	#4294	—	9	6	no	—
62	C₃⁴	—	3	1	no	—
63	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
64	C₃×3₋¹⁺²	—	19	6	no	—
65	C₃×H(3,𝔽₃)	—	3	1	no	—
66	C₃×H(3,𝔽₃)	—	3	1	no	—
67	C₃×H(3,𝔽₃)	—	3	9	no	—
68	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
69	C₃²×C₉	—	57	1	no	—
70	C₃²⋊C₉	—	19	3	no	—
71	C₃²⋊C₉	—	57	1	no	—
72	C₃²⋊C₉	—	57	1	no	—
73	C₃²⋊C₉	—	57	2	no	—
74	C₃²⋊C₉	—	57	2	no	—
75	C₃²⋊C₉	—	57	2	no	—
76	C₃×C₁₉	—	3	3	no	—
77	C₃×C₁₉	—	3	3	no	—
78	C₃×C₁₉	—	9	1	no	—
79	C₃×C₁₉	—	9	1	no	—
80	C₃×C₁₉	—	9	9	no	—
81	C₃×C₁₉	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
82	C₃×C₁₉	C₃×#731	1	1	yes	characteristic
83	C₃×C₁₉	#6909	1	2	yes	—
84	C₃³	—	3	9	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	3	3	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	9	2	no	—
89	C₃³	—	9	6	no	—
90	C₃³	—	9	3	no	—
91	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
92	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
93	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
94	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
95	3₋¹⁺²	—	171	6	no	—
96	H(3,𝔽₃)	—	3	9	no	—
97	H(3,𝔽₃)	—	3	9	no	—
98	H(3,𝔽₃)	—	3	9	no	—
99	C₃×C₉	—	57	3	no	—
100	C₃×C₉	—	57	6	no	—
101	C₃×C₉	—	171	3	no	—
102	C₁₉	#10624591	1	1	yes	characteristic, complemented, Sylow
103	C₃²	—	3	1	no	—
104	C₃²	—	3	1	no	—
105	C₃²	—	3	2	no	—
106	C₃²	—	3	2	no	—
107	C₃²	—	3	1	no	—
108	C₃²	—	3	1	no	—
109	C₃²	—	3	9	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	2	no	—
113	C₃²	—	3	2	no	—
114	C₃²	—	9	9	no	—
115	C₃²	—	9	27	no	—
116	C₃²	—	9	3	no	—
117	C₃²	—	9	6	no	—
118	C₃²	#13095314	1	3	yes	—
119	C₃²	#13095316	1	3	yes	—
120	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
121	C₉	—	171	3	no	—
122	C₉	—	171	6	no	—
123	C₃	—	3	3	no	—
124	C₃	—	3	3	no	—
125	C₃	—	9	1	no	—
126	C₃	—	9	1	no	—
127	C₃	—	9	9	no	—
128	C₃	C₃×#13095314	1	1	yes	characteristic
129	C₃	C₃×#13095316	1	1	yes	characteristic
130	C₃	#17987504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17987421

C₃

yes

maximal, complemented

C₃×C₃²⋊#4294

C₃

yes

maximal, characteristic, complemented

#17987312

C₃

yes

maximal, complemented

C₃²×C₁₉×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#10587025

C₃²

yes

complemented

C₃²×#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

complemented

C₃⁴×C₁₉

—

C₃⁴×C₁₉

C₃²

yes

characteristic

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

—

#10624591

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

—

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

yes

C₃³×C₁₉

C₃³

yes

characteristic

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

—

—

—

—

—

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

yes

characteristic

#6891

—

C₃×C₃²⋊C₉

—

#6857

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

AΣL(1,𝔽₂₇)

yes

—

C₃²×C₁₉

#731

yes

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

#4294

—

#4294

—

C₃⁴

—

C₃⁴

C₃×C₁₉⋊C₃

yes

characteristic

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

—

C₃²×C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×AΣL(1,𝔽₂₇)

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

171

—

102

C₁₉

#10624591

yes

characteristic, complemented, Sylow

103

C₃²

—

104

C₃²

—

105

C₃²

—

106

C₃²

—

107

C₃²

—

108

C₃²

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

#13095314

yes

—

119

C₃²

#13095316

yes

—

120

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

yes

characteristic

121

C₉

—

171

—

122

C₉

—

171

—

123

C₃

—

124

C₃

—

125

C₃

—

126

C₃

—

127

C₃

—

128

C₃

C₃×#13095314

yes

characteristic

129

C₃

C₃×#13095316

yes

characteristic

130

C₃

#17987504

yes

—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624591
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting, prime radical

C₃²×C₁₉×H(3,𝔽₃)

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

hypercenter

C₃²×H(3,𝔽₃)

C₁₉⋊C₃

hypocenter

C₁₉

#10624591

LCS

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

N	H ≅ G/N	type
1. #17987421	C₃	semidirect
2. C₃×C₃²⋊#4294	C₃	semidirect
3. #17987312	C₃	semidirect
5. C₃×#10587025	C₃²	semidirect
12. C₃²⋊#4294	C₃²	semidirect
102. C₁₉	#10624591	semidirect

H ≅ G/N

type

1. #17987421

C₃

semidirect

2. C₃×C₃²⋊#4294

C₃

semidirect

3. #17987312

C₃

semidirect

5. C₃×#10587025

C₃²

semidirect

12. C₃²⋊#4294

C₃²

semidirect

102. C₁₉

#10624591

semidirect

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

subgroup	index
#17987421	3
C₃×C₃²⋊#4294	3
#17987312	3
C₃²×C₁₉×H(3,𝔽₃)	3
#10624591	19

subgroup

index

#17987421

C₃×C₃²⋊#4294

#17987312

C₃²×C₁₉×H(3,𝔽₃)

#10624591

Subgroups — #28353740

Conjugacy classes of subgroups

Complements

Maximal subgroups