Group #28353603 — subgroups

Subgroups — #28353603

Order 13851.

470 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₃²×#10587022	C₃	1	1	yes	maximal, characteristic
2	C₃×#13095306	C₃	1	3	yes	maximal, complemented
3	C₃×#13095303	C₃	1	3	yes	maximal
4	#17987308	C₃	1	2	yes	maximal, complemented
5	#17987308	C₃	1	3	yes	maximal, complemented
6	C₁₉×#6855	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#10587022	—	3	3	no	—
9	C₃×#10587022	—	3	3	no	—
10	C₃×#10587022	—	3	3	no	—
11	C₃×#10587022	—	3	3	no	—
12	C₃²×#4294	C₃²	1	6	yes	—
13	C₃²×#4294	C₃²	1	1	yes	characteristic
14	C₃²×#4294	C₃²	1	1	yes	characteristic
15	#13095306	—	3	9	no	—
16	#13095303	—	3	9	no	—
17	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
18	C₃²×C₉×C₁₉	C₃²	1	3	yes	—
19	#10624565	—	19	1	no	maximal, Sylow
20	C₃³×C₁₉	C₃³	1	1	yes	characteristic
21	C₃²×C₁₉⋊C₃	—	3	1	no	—
22	C₃²×C₁₉⋊C₃	—	3	3	no	—
23	C₃²×C₁₉⋊C₃	—	3	1	no	—
24	C₃²×C₁₉⋊C₃	—	3	1	no	—
25	C₃²×C₁₉⋊C₃	—	3	6	no	—
26	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	3	yes	—
27	C₃×C₉×C₁₉	—	3	3	no	—
28	C₃×C₉×C₁₉	—	3	1	no	—
29	C₃×C₉×C₁₉	—	3	1	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	6	no	—
32	C₃×C₉×C₁₉	3₋¹⁺²	1	3	yes	—
33	C₃×C₉×C₁₉	3₋¹⁺²	1	9	yes	complemented
34	C₃×#4294	—	3	1	no	—
35	C₃×#4294	—	3	1	no	—
36	C₃×#4294	—	3	1	no	—
37	C₃×#4294	—	3	1	no	—
38	C₃×#4294	—	3	1	no	—
39	C₃×#4294	—	3	1	no	—
40	C₃×#4294	—	3	6	no	—
41	C₃×#4294	—	3	6	no	—
42	C₃×#4294	—	3	12	no	—
43	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
44	C₃×#4294	3₋¹⁺²	1	3	yes	—
45	#10587022	—	3	27	no	—
46	C₃²×3₋¹⁺²	—	19	1	no	—
47	C₃×C₉⋊C₉	—	19	3	no	—
48	C₃×C₃²⋊C₉	—	19	3	no	—
49	#6855	—	19	3	no	—
50	#6855	—	19	2	no	—
51	#6855	C₁₉⋊C₃	1	1	yes	characteristic, complemented
52	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
54	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
56	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
57	C₃²×C₁₉	#737	1	6	yes	—
58	C₃×C₁₉⋊C₃	—	3	3	no	—
59	C₃×C₁₉⋊C₃	—	3	9	no	—
60	C₃×C₁₉⋊C₃	—	9	3	no	—
61	C₃×C₁₉⋊C₃	—	9	3	no	—
62	C₃×C₁₉⋊C₃	—	9	2	no	—
63	C₃×C₁₉⋊C₃	—	9	1	no	—
64	C₉×C₁₉	—	3	3	no	—
65	C₉×C₁₉	—	3	9	no	—
66	#4294	—	3	3	no	—
67	#4294	—	3	3	no	—
68	#4294	—	9	6	no	—
69	C₃⁴	—	19	1	no	—
70	C₃×3₋¹⁺²	—	57	3	no	—
71	C₃×3₋¹⁺²	—	57	3	no	—
72	C₃×3₋¹⁺²	—	57	3	no	—
73	C₃×3₋¹⁺²	—	57	3	no	—
74	C₃²×C₉	—	19	1	no	—
75	C₃²×C₉	—	19	1	no	—
76	C₃²×C₉	—	19	6	no	—
77	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
78	C₃²×C₉	C₃×C₁₉⋊C₃	1	3	yes	—
79	C₉⋊C₉	—	57	9	no	—
80	C₃²⋊C₉	—	57	9	no	—
81	C₃×C₁₉	#6891	1	1	yes	characteristic
82	C₃×C₁₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
83	C₃×C₁₉	#6894	1	3	yes	—
84	C₃×C₁₉	#6896	1	3	yes	—
85	C₃×C₁₉	#6899	1	2	yes	—
86	C₃×C₁₉	#6900	1	1	yes	characteristic
87	C₁₉⋊C₃	—	9	3	no	—
88	C₃³	—	19	3	no	—
89	C₃³	—	57	1	no	—
90	C₃³	—	57	1	no	—
91	C₃³	—	57	3	no	—
92	C₃³	—	57	1	no	—
93	C₃³	—	57	6	no	—
94	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
95	3₋¹⁺²	—	57	27	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	3	6	no	—
101	C₃×C₉	—	19	3	no	—
102	C₃×C₉	—	19	3	no	—
103	C₃×C₉	—	57	1	no	—
104	C₃×C₉	—	57	1	no	—
105	C₃×C₉	—	57	1	no	—
106	C₃×C₉	—	57	1	no	—
107	C₃×C₉	—	57	1	no	—
108	C₃×C₉	—	57	1	no	—
109	C₃×C₉	—	57	6	no	—
110	C₃×C₉	—	57	6	no	—
111	C₃×C₉	—	57	12	no	—
112	C₃×C₉	#10587022	1	3	yes	—
113	C₃×C₉	#10587022	1	9	yes	complemented
114	C₁₉	#10624565	1	1	yes	characteristic, complemented, Sylow
115	C₃²	—	57	3	no	—
116	C₃²	—	57	9	no	—
117	C₃²	—	171	3	no	—
118	C₃²	—	171	3	no	—
119	C₃²	—	171	2	no	—
120	C₃²	—	171	1	no	—
121	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
122	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
123	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
124	C₃²	C₃×#10587022	1	1	yes	characteristic
125	C₃²	C₃×#10587022	1	3	yes	—
126	C₃²	#13095333	1	6	yes	—
127	C₉	—	3	3	no	—
128	C₉	—	3	9	no	—
129	C₉	—	57	3	no	—
130	C₉	—	57	3	no	—
131	C₉	—	171	6	no	—
132	C₃	—	171	3	no	—
133	C₃	#17987420	1	1	yes	characteristic
134	C₃	#17987428	1	3	yes	—
135	C₃	#17987434	1	3	yes	—
136	C₃	#17987450	1	3	yes	—
137	C₃	#17987466	1	2	yes	—
138	C₃	#17987469	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#10587022

C₃

yes

maximal, characteristic

C₃×#13095306

C₃

yes

maximal, complemented

C₃×#13095303

C₃

yes

maximal

#17987308

C₃

yes

maximal, complemented

#17987308

C₃

yes

maximal, complemented

C₁₉×#6855

C₃

yes

maximal, characteristic, complemented

C₃³×C₁₉⋊C₃

C₃²

yes

—

—

—

—

C₃²×#4294

C₃²

yes

—

C₃²×#4294

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

#13095306

—

#13095303

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₃²×C₉×C₁₉

C₃²

yes

—

#10624565

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

C₃×C₉×C₁₉

3₋¹⁺²

yes

—

—

—

—

—

—

—

—

—

yes

—

C₃×#4294

3₋¹⁺²

yes

—

#10587022

—

C₃²×3₋¹⁺²

—

C₃×C₉⋊C₉

—

C₃×C₃²⋊C₉

—

#6855

—

#6855

—

#6855

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₃²×C₁₉

C₃×3₋¹⁺²

yes

—

C₃²×C₁₉

#737

yes

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

—

C₃⁴

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

C₃×C₁₉⋊C₃

yes

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃×C₁₉

#6891

yes

characteristic

C₃×C₁₉

C₉⋊H(3,𝔽₃)

yes

—

C₃×C₁₉

#6894

yes

—

C₃×C₁₉

#6896

yes

—

C₃×C₁₉

#6899

yes

—

C₃×C₁₉

#6900

yes

characteristic

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

#10587022

yes

—

113

C₃×C₉

#10587022

yes

complemented

114

C₁₉

#10624565

yes

characteristic, complemented, Sylow

115

C₃²

—

116

C₃²

—

117

C₃²

—

171

—

118

C₃²

—

171

—

119

C₃²

—

171

—

120

C₃²

—

171

—

121

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

122

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

123

C₃²

3₋¹⁺²×C₁₉⋊C₃

yes

characteristic

124

C₃²

C₃×#10587022

yes

characteristic

125

C₃²

C₃×#10587022

yes

—

126

C₃²

#13095333

yes

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

171

—

132

C₃

—

171

—

133

C₃

#17987420

yes

characteristic

134

C₃

#17987428

yes

—

135

C₃

#17987434

yes

—

136

C₃

#17987450

yes

—

137

C₃

#17987466

yes

—

138

C₃

#17987469

yes

characteristic

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator, socle, cosocle	C₃³×C₁₉	C₃³
Fitting	C₁₉×#6855	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6855	C₁₉⋊C₃
hypocenter	C₁₉	#10624565
LCS	G ▷ C₃³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ #6855

subgroup

G/H

center, Frattini

C₃³

C₃²×C₁₉⋊C₃

commutator, socle, cosocle

C₃³×C₁₉

C₃³

Fitting

C₁₉×#6855

C₃

prime radical

C₃³×C₁₉⋊C₃

C₃²

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₁₉

C₁ ◁ C₃³ ◁ #6855

N	H ≅ G/N	type
2. C₃×#13095306	C₃	semidirect
4. #17987308	C₃	semidirect
5. #17987308	C₃	semidirect
6. C₁₉×#6855	C₃	semidirect
33. C₃×C₉×C₁₉	3₋¹⁺²	semidirect
51. #6855	C₁₉⋊C₃	semidirect
113. C₃×C₉	#10587022	semidirect
114. C₁₉	#10624565	semidirect

H ≅ G/N

type

2. C₃×#13095306

C₃

semidirect

4. #17987308

C₃

semidirect

5. #17987308

C₃

semidirect

6. C₁₉×#6855

C₃

semidirect

33. C₃×C₉×C₁₉

3₋¹⁺²

semidirect

51. #6855

C₁₉⋊C₃

semidirect

113. C₃×C₉

#10587022

semidirect

114. C₁₉

#10624565

semidirect

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

subgroup	index
C₃²×#10587022	3
C₃×#13095306	3
C₃×#13095303	3
#17987308	3
C₁₉×#6855	3
#10624565	19

subgroup

index

C₁₉×#6855

Subgroups — #28353603

Conjugacy classes of subgroups

Complements

Maximal subgroups