Group #28356924 — subgroups

Subgroups — #28356924

Order 13851.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₃×#13095342	C₃	1	2	yes	maximal, complemented
2	C₃×#13095342	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987567	C₃	1	9	yes	maximal, complemented
4	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
5	#13095342	—	3	2	no	—
6	#13095342	—	3	2	no	—
7	#13095342	—	3	1	no	—
8	#13095342	—	3	2	no	—
9	#13095342	C₃²	1	6	yes	complemented
10	C₃×#10587020	—	3	2	no	—
11	C₃×#10587020	C₃²	1	3	yes	complemented
12	#13095333	—	3	2	no	—
13	#13095333	—	3	2	no	—
14	#13095333	—	3	2	no	—
15	#13095333	C₉	1	6	yes	—
16	#13095333	C₃²	1	3	yes	—
17	C₃×#10587025	C₉	1	1	yes	characteristic
18	C₃×#10587022	—	3	1	no	—
19	C₃×#10587023	—	3	1	no	—
20	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	1	yes	characteristic
21	C₃²×#4294	C₃²	1	1	yes	characteristic
22	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
23	C₃×C₁₉×3₋¹⁺²	C₉	1	1	yes	characteristic
24	#10624702	—	19	1	no	maximal, Sylow
25	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
26	C₃²×C₁₉⋊C₃	—	3	1	no	—
27	C₁₉×3₋¹⁺²	—	3	2	no	—
28	C₁₉×3₋¹⁺²	C₃×C₉	1	3	yes	—
29	C₃×C₉×C₁₉	—	3	1	no	—
30	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
31	C₃²⋊(C₁₉⋊C₃)	C₃×C₉	1	3	yes	—
32	#10587025	—	3	2	no	—
33	#10587025	C₃×C₉	1	3	yes	—
34	C₃×#4294	—	3	1	no	—
35	C₃×#4294	C₃×C₉	1	2	yes	—
36	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
38	C₃×#4294	3₋¹⁺²	1	1	yes	characteristic
39	C₃×#4294	3₋¹⁺²	1	1	yes	characteristic
40	C₃×#4294	3₋¹⁺²	1	2	yes	—
41	C₃×#4294	3₋¹⁺²	1	2	yes	—
42	C₃×#4294	C₃³	1	1	yes	characteristic
43	#10587023	—	3	6	no	—
44	#10587023	—	3	3	no	—
45	#10587022	—	3	6	no	—
46	#10587022	—	3	3	no	—
47	C₉×C₁₉⋊C₃	—	3	6	no	—
48	C₉×C₁₉⋊C₃	—	3	3	no	—
49	#10587020	—	3	6	no	—
50	#10587020	—	3	3	no	—
51	C₃×#737	—	19	1	no	—
52	C₃×C₂₇⋊C₃	—	19	1	no	—
53	C₃×C₂₇⋊C₃	—	19	2	no	—
54	#6872	—	19	9	no	—
55	C₃²×C₁₉	—	9	1	no	—
56	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
57	C₃²×C₁₉	C₃²⋊C₉	1	2	yes	—
58	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
59	C₃×C₁₉⋊C₃	—	3	3	no	—
60	C₃×C₁₉⋊C₃	—	9	3	no	—
61	C₉×C₁₉	—	3	3	no	—
62	#4294	—	3	3	no	—
63	#4294	C₃²⋊C₉	1	2	yes	—
64	#4294	C₃²⋊C₉	1	2	yes	—
65	#4294	C₃²⋊C₉	1	2	yes	—
66	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
67	#4294	C₃×3₋¹⁺²	1	1	yes	characteristic
68	#4294	C₃×3₋¹⁺²	1	1	yes	characteristic
69	#737	—	19	6	no	—
70	#737	—	19	3	no	—
71	#737	—	57	2	no	—
72	#737	—	57	2	no	—
73	#737	—	57	2	no	—
74	C₃×3₋¹⁺²	—	19	1	no	—
75	C₃×3₋¹⁺²	—	57	1	no	—
76	C₃×3₋¹⁺²	—	57	1	no	—
77	C₃×3₋¹⁺²	#4296	1	1	yes	characteristic
78	C₃×H(3,𝔽₃)	—	19	1	no	—
79	C₃²×C₉	—	19	1	no	—
80	C₃²×C₉	—	57	1	no	—
81	C₂₇⋊C₃	—	19	6	no	—
82	C₂₇⋊C₃	—	57	2	no	—
83	C₂₇⋊C₃	—	57	1	no	—
84	C₂₇⋊C₃	—	57	2	no	—
85	C₂₇⋊C₃	—	57	2	no	—
86	C₃×C₂₇	—	19	3	no	—
87	C₃×C₂₇	—	57	2	no	—
88	C₃×C₁₉	—	3	1	no	—
89	C₃×C₁₉	—	3	2	no	—
90	C₃×C₁₉	—	3	1	no	—
91	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
92	C₁₉⋊C₃	—	9	3	no	—
93	C₃³	—	57	1	no	—
94	C₃³	C₃×#4296	1	1	yes	characteristic
95	3₋¹⁺²	—	3	2	no	—
96	3₋¹⁺²	—	19	3	no	—
97	3₋¹⁺²	—	57	6	no	—
98	3₋¹⁺²	—	57	3	no	—
99	3₋¹⁺²	—	57	6	no	—
100	3₋¹⁺²	—	57	3	no	—
101	3₋¹⁺²	—	57	2	no	—
102	3₋¹⁺²	C₃×#4296	1	3	yes	—
103	H(3,𝔽₃)	—	19	3	no	—
104	H(3,𝔽₃)	—	57	2	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	19	2	no	—
107	C₃×C₉	—	19	1	no	—
108	C₃×C₉	—	19	1	no	—
109	C₃×C₉	—	19	1	no	—
110	C₃×C₉	—	19	2	no	—
111	C₃×C₉	—	19	2	no	—
112	C₃×C₉	—	19	2	no	—
113	C₃×C₉	—	19	1	no	—
114	C₃×C₉	—	57	3	no	—
115	C₃×C₉	—	57	6	no	—
116	C₃×C₉	—	57	1	no	—
117	C₂₇	—	57	3	no	—
118	C₂₇	—	57	6	no	—
119	C₁₉	#10624702	1	1	yes	characteristic, complemented, Sylow
120	C₃²	—	9	1	no	—
121	C₃²	—	57	3	no	—
122	C₃²	—	171	3	no	—
123	C₃²	C₃²⋊#4296	1	1	yes	characteristic
124	C₃²	C₃²⋊#4296	1	2	yes	—
125	C₃²	C₃²×#4296	1	1	yes	characteristic
126	C₉	—	3	3	no	—
127	C₉	—	19	1	no	—
128	C₉	—	19	2	no	—
129	C₉	—	19	2	no	—
130	C₉	—	19	2	no	—
131	C₉	—	19	1	no	—
132	C₉	—	19	1	no	—
133	C₉	—	57	3	no	—
134	C₃	—	3	1	no	—
135	C₃	—	3	2	no	—
136	C₃	—	3	1	no	—
137	C₃	—	171	3	no	—
138	C₃	C₃×C₃²⋊#4296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095342

C₃

yes

maximal, complemented

C₃×#13095342

C₃

yes

maximal, characteristic, complemented

#17987567

C₃

yes

maximal, complemented

C₃×#13095333

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

complemented

C₃×#10587020

—

C₃×#10587020

C₃²

yes

—

—

—

C₉

yes

—

#13095333

C₃²

yes

—

C₃×#10587025

C₉

yes

characteristic

C₃×#10587022

—

C₃×#10587023

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

C₃×C₁₉×3₋¹⁺²

C₉

yes

characteristic

#10624702

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₁₉×3₋¹⁺²

—

C₁₉×3₋¹⁺²

C₃×C₉

yes

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

C₃×C₉

yes

—

#10587025

—

#10587025

C₃×C₉

yes

—

C₃×#4294

—

C₃×#4294

C₃×C₉

yes

—

yes

yes

—

C₃×#4294

3₋¹⁺²

yes

characteristic

C₃×#4294

3₋¹⁺²

yes

characteristic

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

C₃³

yes

—

—

—

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

#10587020

—

#10587020

—

C₃×#737

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

#6872

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃×H(3,𝔽₃)

yes

characteristic

#4294

C₃×3₋¹⁺²

yes

characteristic

#4294

C₃×3₋¹⁺²

yes

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

#4296

yes

characteristic

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₁₉⋊C₃

—

C₃³

—

C₃³

C₃×#4296

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

100

3₋¹⁺²

—

101

3₋¹⁺²

—

102

3₋¹⁺²

C₃×#4296

yes

—

103

H(3,𝔽₃)

—

104

H(3,𝔽₃)

—

105

C₃×C₉

—

106

C₃×C₉

—

107

C₃×C₉

—

108

C₃×C₉

—

109

C₃×C₉

—

110

C₃×C₉

—

111

C₃×C₉

—

112

C₃×C₉

—

113

C₃×C₉

—

114

C₃×C₉

—

115

C₃×C₉

—

116

C₃×C₉

—

117

C₂₇

—

118

C₂₇

—

119

C₁₉

#10624702

yes

characteristic, complemented, Sylow

120

C₃²

—

121

C₃²

—

122

C₃²

—

171

—

123

C₃²

C₃²⋊#4296

yes

characteristic

124

C₃²

C₃²⋊#4296

yes

—

125

C₃²

C₃²×#4296

yes

characteristic

126

C₉

—

127

C₉

—

128

C₉

—

129

C₉

—

130

C₉

—

131

C₉

—

132

C₉

—

133

C₉

—

134

C₃

—

135

C₃

—

136

C₃

—

137

C₃

—

171

—

138

C₃

C₃×C₃²⋊#4296

yes

characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#4296
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃²	C₃²×#4296
Fitting	C₃×C₁₉×3₋¹⁺²	C₉
socle	C₃×C₁₉	C₃×C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃×3₋¹⁺²	#4296
hypocenter	C₁₉	#10624702
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

subgroup

G/H

center

C₃

C₃×C₃²⋊#4296

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃²

C₃²×#4296

Fitting

C₃×C₁₉×3₋¹⁺²

C₉

socle

C₃×C₁₉

C₃×C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

C₃³

C₃×3₋¹⁺²

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×#13095342	C₃	semidirect
2. C₃×#13095342	C₃	semidirect
3. #17987567	C₃	semidirect
9. #13095342	C₃²	semidirect
11. C₃×#10587020	C₃²	semidirect
119. C₁₉	#10624702	semidirect

H ≅ G/N

type

1. C₃×#13095342

C₃

semidirect

2. C₃×#13095342

C₃

semidirect

3. #17987567

C₃

semidirect

9. #13095342

C₃²

semidirect

11. C₃×#10587020

C₃²

semidirect

119. C₁₉

#10624702

semidirect

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

subgroup	index
C₃×#13095342	3
#17987567	3
C₃×#13095333	3
#10624702	19

subgroup

index

Subgroups — #28356924

Conjugacy classes of subgroups

Complements

Maximal subgroups