Group #28353997 — subgroups

Subgroups — #28353997

Order 13851.

236 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 52 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	#17987425	C₃	1	1	yes	maximal, characteristic, complemented
2	C₉×C₃²⋊(C₁₉⋊C₃)	C₃	1	2	yes	maximal, complemented
3	#17987323	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987323	C₃	1	2	yes	maximal, complemented
5	#17987319	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987319	C₃	1	2	yes	maximal, complemented
7	#17987310	C₃	1	2	yes	maximal, complemented
8	#17987310	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#6895	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#10587022	—	3	1	no	—
11	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
12	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃×C₉×C₁₉⋊C₃	—	3	2	no	—
14	C₃²⋊#4294	C₃²	1	2	yes	—
15	C₃²⋊#4294	C₃²	1	2	yes	—
16	C₃²⋊#4294	C₃²	1	1	yes	characteristic
17	C₃²⋊#4294	C₃²	1	1	yes	characteristic
18	#13095303	—	3	2	no	—
19	#13095303	—	3	2	no	—
20	#13095303	—	3	2	no	—
21	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
22	C₁₉×C₉⋊C₉	—	3	2	no	—
23	C₁₉×C₉⋊C₉	—	3	1	no	—
24	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
26	#10624622	—	19	1	no	maximal, Sylow
27	C₃³×C₁₉	C₃³	1	1	yes	characteristic
28	C₃²×C₁₉⋊C₃	—	3	2	no	—
29	C₃²×C₁₉⋊C₃	—	3	1	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	2	no	—
32	C₃×C₉×C₁₉	—	9	1	no	—
33	C₃×C₉×C₁₉	H(3,𝔽₃)	1	2	yes	—
34	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
36	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
37	C₃×#4294	—	3	2	no	—
38	C₃×#4294	—	3	2	no	—
39	C₃×#4294	—	3	1	no	—
40	C₃×#4294	—	3	1	no	—
41	#10587022	—	9	3	no	—
42	C₉×C₁₉⋊C₃	—	9	6	no	—
43	#6895	C₁₉⋊C₃	1	1	yes	characteristic, complemented
44	C₉⋊H(3,𝔽₃)	—	19	1	no	—
45	C₉×H(3,𝔽₃)	—	19	2	no	—
46	#6861	—	19	1	no	—
47	#6861	—	19	2	no	—
48	#6859	—	19	1	no	—
49	#6859	—	19	2	no	—
50	#6856	—	19	2	no	—
51	#6856	—	19	1	no	—
52	C₃²×C₁₉	—	3	1	no	—
53	C₃²×C₁₉	—	3	2	no	—
54	C₃²×C₁₉	—	3	1	no	—
55	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃×C₁₉⋊C₃	—	3	6	no	—
57	C₃×C₁₉⋊C₃	—	9	2	no	—
58	C₃×C₁₉⋊C₃	—	9	2	no	—
59	C₃×C₁₉⋊C₃	—	9	2	no	—
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	2	no	—
64	C₉×C₁₉	—	3	1	no	—
65	C₉×C₁₉	—	9	1	no	—
66	C₉×C₁₉	—	9	2	no	—
67	#4294	—	9	2	no	—
68	#4294	—	9	2	no	—
69	#4294	—	9	1	no	—
70	#4294	—	9	1	no	—
71	C₃×3₋¹⁺²	—	57	1	no	—
72	C₃×H(3,𝔽₃)	—	19	2	no	—
73	C₃×H(3,𝔽₃)	—	19	1	no	—
74	C₃²×C₉	—	57	2	no	—
75	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
76	C₉⋊C₉	—	3	2	no	—
77	C₉⋊C₉	—	3	1	no	—
78	C₃²⋊C₉	—	19	2	no	—
79	C₃²⋊C₉	—	19	1	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	2	no	—
85	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
86	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
87	C₃×C₁₉	—	9	1	no	—
88	C₃×C₁₉	#6908	1	2	yes	—
89	C₃×C₁₉	#6911	1	1	yes	characteristic
90	C₃×C₁₉	#6913	1	1	yes	characteristic
91	C₁₉⋊C₃	—	9	6	no	—
92	C₁₉⋊C₃	—	27	1	no	—
93	C₃³	—	57	2	no	—
94	C₃³	—	57	1	no	—
95	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
96	3₋¹⁺²	—	171	3	no	—
97	H(3,𝔽₃)	—	57	6	no	—
98	H(3,𝔽₃)	—	171	1	no	—
99	C₃×C₉	—	3	2	no	—
100	C₃×C₉	—	3	1	no	—
101	C₃×C₉	—	9	1	no	—
102	C₃×C₉	—	57	2	no	—
103	C₃×C₉	—	57	2	no	—
104	C₃×C₉	—	57	1	no	—
105	C₃×C₉	—	57	1	no	—
106	C₃×C₉	—	171	6	no	—
107	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
108	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
109	C₁₉	#10624622	1	1	yes	characteristic, complemented, Sylow
110	C₃²	—	3	2	no	—
111	C₃²	—	3	1	no	—
112	C₃²	—	3	1	no	—
113	C₃²	—	57	6	no	—
114	C₃²	—	171	2	no	—
115	C₃²	—	171	1	no	—
116	C₃²	—	171	1	no	—
117	C₃²	—	171	2	no	—
118	C₃²	—	171	2	no	—
119	C₃²	—	171	2	no	—
120	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
121	C₉	—	3	2	no	—
122	C₉	—	3	1	no	—
123	C₉	—	9	2	no	—
124	C₉	—	9	1	no	—
125	C₉	—	171	2	no	—
126	C₉	—	171	2	no	—
127	C₉	—	171	1	no	—
128	C₉	—	171	1	no	—
129	C₃	—	513	1	no	—
130	C₃	—	9	1	no	—
131	C₃	—	171	6	no	—
132	C₃	#17987497	1	2	yes	—
133	C₃	#17987515	1	1	yes	characteristic
134	C₃	#17987522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17987425

C₃

yes

maximal, characteristic, complemented

C₉×C₃²⋊(C₁₉⋊C₃)

C₃

yes

maximal, complemented

#17987323

C₃

yes

maximal, characteristic, complemented

#17987323

C₃

yes

maximal, complemented

#17987319

C₃

yes

maximal, characteristic, complemented

#17987319

C₃

yes

maximal, complemented

#17987310

C₃

yes

maximal, complemented

#17987310

C₃

yes

maximal, characteristic, complemented

C₁₉×#6895

C₃

yes

maximal, characteristic, complemented

C₃×#10587022

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

—

C₃²⋊#4294

C₃²

yes

characteristic

C₃²⋊#4294

C₃²

yes

—

—

—

C₃²×C₉×C₁₉

C₃²

yes

characteristic

C₁₉×C₉⋊C₉

—

C₁₉×C₉⋊C₉

—

C₁₉×C₃²⋊C₉

C₃²

yes

characteristic

C₁₉×C₃²⋊C₉

C₃²

yes

—

#10624622

—

maximal, Sylow

C₃³×C₁₉

C₃³

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃²×C₁₉⋊C₃

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

—

C₃×C₉×C₁₉

H(3,𝔽₃)

yes

characteristic

C₃²⋊(C₁₉⋊C₃)

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

—

C₉×C₁₉⋊C₃

—

#6895

C₁₉⋊C₃

yes

characteristic, complemented

C₉⋊H(3,𝔽₃)

—

C₉×H(3,𝔽₃)

—

—

—

—

—

—

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃×H(3,𝔽₃)

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

—

C₃²⋊C₉

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

—

C₃×C₁₉

#6908

yes

—

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃³

—

C₃³

C₃²×C₁₉⋊C₃

yes

characteristic

3₋¹⁺²

—

171

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

171

—

C₃×C₉

—

100

C₃×C₉

—

101

C₃×C₉

—

102

C₃×C₉

—

103

C₃×C₉

—

104

C₃×C₉

—

105

C₃×C₉

—

106

C₃×C₉

—

171

—

107

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

—

108

C₃×C₉

C₃²⋊(C₁₉⋊C₃)

yes

characteristic

109

C₁₉

#10624622

yes

characteristic, complemented, Sylow

110

C₃²

—

111

C₃²

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

171

—

115

C₃²

—

171

—

116

C₃²

—

171

—

117

C₃²

—

171

—

118

C₃²

—

171

—

119

C₃²

—

171

—

120

C₃²

C₃×C₃²⋊(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

—

171

—

132

C₃

#17987497

yes

—

133

C₃

#17987515

yes

characteristic

134

C₃

#17987522

yes

characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6895	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
hypercenter	#6895	C₁₉⋊C₃
hypocenter	C₁₉	#10624622
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6895

subgroup

G/H

center

C₃²

C₃×C₃²⋊(C₁₉⋊C₃)

commutator, cosocle

C₃³×C₁₉

C₃³

Frattini

C₃³

C₃²×C₁₉⋊C₃

Fitting

C₁₉×#6895

C₃

socle

C₃²×C₁₉

C₃×H(3,𝔽₃)

C₁₉⋊C₃

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6895

N	H ≅ G/N	type
1. #17987425	C₃	semidirect
2. C₉×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
3. #17987323	C₃	semidirect
4. #17987323	C₃	semidirect
5. #17987319	C₃	semidirect
6. #17987319	C₃	semidirect
7. #17987310	C₃	semidirect
8. #17987310	C₃	semidirect
9. C₁₉×#6895	C₃	semidirect
43. #6895	C₁₉⋊C₃	semidirect
109. C₁₉	#10624622	semidirect

H ≅ G/N

type

1. #17987425

C₃

semidirect

2. C₉×C₃²⋊(C₁₉⋊C₃)

C₃

semidirect

3. #17987323

C₃

semidirect

4. #17987323

C₃

semidirect

5. #17987319

C₃

semidirect

6. #17987319

C₃

semidirect

7. #17987310

C₃

semidirect

8. #17987310

C₃

semidirect

9. C₁₉×#6895

C₃

semidirect

43. #6895

C₁₉⋊C₃

semidirect

109. C₁₉

#10624622

semidirect

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

subgroup	index
#17987425	3
C₉×C₃²⋊(C₁₉⋊C₃)	3
#17987323	3
#17987319	3
#17987310	3
C₁₉×#6895	3
#10624622	19

subgroup

index

C₉×C₃²⋊(C₁₉⋊C₃)

C₁₉×#6895

Subgroups — #28353997

Conjugacy classes of subgroups

Complements

Maximal subgroups