Group #28354101 — subgroups

Subgroups — #28354101

Order 13851.

214 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17987429	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987428	C₃	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²×#4294	C₃	1	2	yes	maximal, complemented
4	#17987321	C₃	1	2	yes	maximal, complemented
5	#17987318	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987315	C₃	1	2	yes	maximal, complemented
7	#17987311	C₃	1	2	yes	maximal, complemented
8	#17987309	C₃	1	1	yes	maximal, characteristic
9	C₁₉×#6857	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#10587025	C₃²	1	2	yes	—
11	C₃×#10587022	—	3	1	no	—
12	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃²×#4294	C₃²	1	1	yes	characteristic
14	#13095307	—	3	2	no	—
15	#13095306	—	3	2	no	—
16	C₃²⋊#4294	—	3	2	no	—
17	C₃²⋊#4294	—	3	1	no	—
18	C₃²⋊#4294	C₃²	1	2	yes	—
19	C₃²⋊#4294	C₃²	1	2	yes	—
20	C₃²⋊#4294	C₃²	1	1	yes	characteristic
21	#13095303	—	3	2	no	—
22	C₉×#4294	—	3	2	no	—
23	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
24	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
25	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	#10624627	—	19	1	no	maximal, Sylow
27	C₃³×C₁₉	—	3	1	no	—
28	C₃³×C₁₉	C₃³	1	1	yes	characteristic
29	C₃²×C₁₉⋊C₃	—	3	1	no	—
30	C₁₉×3₋¹⁺²	—	3	6	no	—
31	C₁₉×H(3,𝔽₃)	—	9	1	no	—
32	C₃×C₉×C₁₉	—	3	2	no	—
33	C₃×C₉×C₁₉	—	3	1	no	—
34	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
35	#10587025	—	9	2	no	—
36	C₃×#4294	—	3	2	no	—
37	C₃×#4294	—	3	2	no	—
38	C₃×#4294	—	3	1	no	—
39	C₃×#4294	—	3	2	no	—
40	C₃×#4294	—	9	1	no	—
41	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
42	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
43	#10587022	—	9	3	no	—
44	#6893	—	19	1	no	—
45	C₉⋊H(3,𝔽₃)	—	19	1	no	—
46	C₉×3₋¹⁺²	—	19	2	no	—
47	#6860	—	19	2	no	—
48	#6859	—	19	1	no	—
49	#6858	—	19	2	no	—
50	#6857	—	19	2	no	—
51	#6857	C₁₉⋊C₃	1	1	yes	characteristic, complemented
52	#6856	—	19	1	no	—
53	C₃²×C₁₉	—	3	1	no	—
54	C₃²×C₁₉	—	3	2	no	—
55	C₃²×C₁₉	—	3	1	no	—
56	C₃²×C₁₉	—	9	1	no	—
57	C₃²×C₁₉	—	9	2	no	—
58	C₃²×C₁₉	—	9	1	no	—
59	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₃×C₁₉⋊C₃	—	9	1	no	—
61	C₃×C₁₉⋊C₃	—	9	2	no	—
62	C₃×C₁₉⋊C₃	—	9	1	no	—
63	C₉×C₁₉	—	3	6	no	—
64	C₉×C₁₉	—	9	1	no	—
65	#4294	—	3	1	no	—
66	#4294	—	3	2	no	—
67	#4294	—	9	2	no	—
68	#4294	—	9	2	no	—
69	#4294	—	9	1	no	—
70	#4294	—	9	2	no	—
71	C₃×3₋¹⁺²	—	19	2	no	—
72	C₃×3₋¹⁺²	—	57	1	no	—
73	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
74	C₃×H(3,𝔽₃)	—	19	1	no	—
75	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
76	C₃²×C₉	—	19	1	no	—
77	C₉⋊C₉	—	57	2	no	—
78	C₉⋊C₉	—	57	2	no	—
79	C₃²⋊C₉	—	19	2	no	—
80	C₃²⋊C₉	—	19	2	no	—
81	C₃²⋊C₉	—	19	1	no	—
82	C₃²⋊C₉	—	57	2	no	—
83	C₃²⋊C₉	—	57	2	no	—
84	C₃²⋊C₉	—	57	1	no	—
85	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
86	C₉²	—	57	2	no	—
87	C₃×C₁₉	—	9	1	no	—
88	C₃×C₁₉	—	27	1	no	—
89	C₃×C₁₉	#6908	1	2	yes	—
90	C₃×C₁₉	#6909	1	1	yes	characteristic
91	C₃×C₁₉	#6913	1	1	yes	characteristic
92	C₁₉⋊C₃	—	27	1	no	—
93	C₃³	—	3	1	no	—
94	C₃³	—	57	1	no	—
95	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
96	3₋¹⁺²	—	3	6	no	—
97	3₋¹⁺²	—	171	3	no	—
98	3₋¹⁺²	—	171	2	no	—
99	H(3,𝔽₃)	—	9	1	no	—
100	H(3,𝔽₃)	—	171	1	no	—
101	C₃×C₉	—	3	2	no	—
102	C₃×C₉	—	3	1	no	—
103	C₃×C₉	—	19	2	no	—
104	C₃×C₉	—	19	1	no	—
105	C₃×C₉	—	57	2	no	—
106	C₃×C₉	—	57	2	no	—
107	C₃×C₉	—	57	1	no	—
108	C₃×C₉	—	57	2	no	—
109	C₃×C₉	—	171	1	no	—
110	C₁₉	#10624627	1	1	yes	characteristic, complemented, Sylow
111	C₃²	—	3	2	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	3	1	no	—
114	C₃²	—	9	2	no	—
115	C₃²	—	9	1	no	—
116	C₃²	—	9	1	no	—
117	C₃²	—	171	2	no	—
118	C₃²	—	171	1	no	—
119	C₃²	—	171	1	no	—
120	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
121	C₉	—	3	6	no	—
122	C₉	—	9	1	no	—
123	C₉	—	57	1	no	—
124	C₉	—	57	2	no	—
125	C₉	—	171	2	no	—
126	C₉	—	171	1	no	—
127	C₉	—	171	2	no	—
128	C₉	—	171	2	no	—
129	C₃	—	513	1	no	—
130	C₃	—	9	1	no	—
131	C₃	—	27	1	no	—
132	C₃	#17987496	1	2	yes	—
133	C₃	#17987502	1	1	yes	characteristic
134	C₃	#17987527	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987429

C₃

yes

maximal, characteristic, complemented

#17987428

C₃

yes

maximal, characteristic, complemented

3₋¹⁺²×#4294

C₃

yes

maximal, complemented

#17987321

C₃

yes

maximal, complemented

#17987318

C₃

yes

maximal, characteristic, complemented

#17987315

C₃

yes

maximal, complemented

#17987311

C₃

yes

maximal, complemented

#17987309

C₃

yes

maximal, characteristic

C₁₉×#6857

C₃

yes

maximal, characteristic, complemented

C₃×#10587025

C₃²

yes

—

C₃×#10587022

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

#13095307

—

#13095306

—

C₃²⋊#4294

—

C₃²⋊#4294

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

characteristic

#13095303

—

C₉×#4294

—

C₃×C₁₉×3₋¹⁺²

C₃²

yes

—

C₃×C₁₉×H(3,𝔽₃)

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

#10624627

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×H(3,𝔽₃)

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

—

yes

—

yes

—

—

C₉⋊H(3,𝔽₃)

—

C₉×3₋¹⁺²

—

—

—

—

—

C₁₉⋊C₃

yes

characteristic, complemented

#6856

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×C₁₉⋊C₃

yes

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃×C₁₉⋊C₃

yes

characteristic

C₃²×C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

#6908

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

171

—

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

100

H(3,𝔽₃)

—

171

—

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

—

171

—

110

C₁₉

#10624627

yes

characteristic, complemented, Sylow

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

171

—

118

C₃²

—

171

—

119

C₃²

—

171

—

120

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

yes

characteristic

121

C₉

—

122

C₉

—

123

C₉

—

124

C₉

—

125

C₉

—

171

—

126

C₉

—

171

—

127

C₉

—

171

—

128

C₉

—

171

—

129

C₃

—

513

—

130

C₃

—

131

C₃

—

132

C₃

#17987496

yes

—

133

C₃

#17987502

yes

characteristic

134

C₃

#17987527

yes

characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6857	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	#17987309	C₃
hypercenter	#6857	C₁₉⋊C₃
hypocenter	C₁₉	#10624627
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6857

subgroup

G/H

center

C₃²

H(3,𝔽₃)×C₁₉⋊C₃

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6857

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

C₃

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ #6857

N	H ≅ G/N	type
1. #17987429	C₃	semidirect
2. #17987428	C₃	semidirect
3. 3₋¹⁺²×#4294	C₃	semidirect
4. #17987321	C₃	semidirect
5. #17987318	C₃	semidirect
6. #17987315	C₃	semidirect
7. #17987311	C₃	semidirect
9. C₁₉×#6857	C₃	semidirect
51. #6857	C₁₉⋊C₃	semidirect
110. C₁₉	#10624627	semidirect

H ≅ G/N

type

1. #17987429

C₃

semidirect

2. #17987428

C₃

semidirect

3. 3₋¹⁺²×#4294

C₃

semidirect

4. #17987321

C₃

semidirect

5. #17987318

C₃

semidirect

6. #17987315

C₃

semidirect

7. #17987311

C₃

semidirect

9. C₁₉×#6857

C₃

semidirect

51. #6857

C₁₉⋊C₃

semidirect

110. C₁₉

#10624627

semidirect

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

subgroup	index
#17987429	3
#17987428	3
3₋¹⁺²×#4294	3
#17987321	3
#17987318	3
#17987315	3
#17987311	3
#17987309	3
C₁₉×#6857	3
#10624627	19

subgroup

index

3₋¹⁺²×#4294

C₁₉×#6857

Subgroups — #28354101

Conjugacy classes of subgroups

Complements

Maximal subgroups