Group #28354867 — subgroups

Subgroups — #28354867

Order 13851.

196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 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×C₂₇×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₉×#10587023	C₃	1	1	yes	maximal, characteristic
3	#17987357	C₃	1	2	yes	maximal, complemented
4	#17987355	C₃	1	2	yes	maximal, complemented
5	#17987355	C₃	1	2	yes	maximal, complemented
6	#17987332	C₃	1	2	yes	maximal
7	#17987330	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987329	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#6864	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#10587023	C₉	1	1	yes	characteristic
11	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
12	C₃×#10587018	C₃²	1	2	yes	—
13	C₃×#10587018	C₃²	1	2	yes	—
14	C₃×#10587018	C₃²	1	2	yes	—
15	C₂₇×C₁₉⋊C₃	—	3	3	no	—
16	#13095307	C₉	1	2	yes	—
17	#13095305	C₉	1	2	yes	—
18	#13095305	C₉	1	2	yes	—
19	#13095303	C₉	1	2	yes	—
20	C₉×#4294	C₃²	1	1	yes	characteristic
21	C₉×#4294	C₃²	1	1	yes	characteristic
22	C₃×C₁₉×C₂₇	C₃²	1	2	yes	—
23	C₃×C₁₉×C₂₇	C₃²	1	1	yes	characteristic
24	C₉²×C₁₉	C₃²	1	1	yes	characteristic
25	#10624727	—	19	1	no	maximal, Sylow
26	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
27	C₃×C₉×C₁₉	C₃×C₉	1	2	yes	—
28	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
29	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
30	C₃×#4294	C₃×C₉	1	2	yes	—
31	C₃×#4294	C₃×C₉	1	2	yes	—
32	C₃×#4294	C₃×C₉	1	1	yes	characteristic
33	C₃×#4294	C₃×C₉	1	1	yes	characteristic
34	C₃×#4294	C₃×C₉	1	2	yes	—
35	#10587023	—	3	3	no	—
36	C₉×C₁₉⋊C₃	—	3	1	no	—
37	C₉×C₁₉⋊C₃	—	3	2	no	—
38	C₁₉×C₂₇	—	3	2	no	—
39	C₁₉×C₂₇	3₋¹⁺²	1	3	yes	complemented
40	#10587018	—	3	2	no	—
41	#10587018	—	3	2	no	—
42	#10587018	—	3	2	no	—
43	C₃²×C₂₇	—	19	1	no	—
44	C₉×3₋¹⁺²	—	19	1	no	—
45	C₉⋊C₂₇	—	19	2	no	—
46	C₉⋊C₂₇	—	19	2	no	—
47	C₉⋊C₂₇	—	19	2	no	—
48	C₃²⋊C₂₇	—	19	2	no	—
49	#6864	—	19	1	no	—
50	#6864	—	19	1	no	—
51	#6864	C₁₉⋊C₃	1	1	yes	characteristic, complemented
52	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
53	C₃×C₁₉⋊C₃	—	3	1	no	—
54	C₃×C₁₉⋊C₃	—	3	2	no	—
55	C₃×C₁₉⋊C₃	C₂₇⋊C₃	1	3	yes	—
56	C₉×C₁₉	—	3	1	no	—
57	C₉×C₁₉	—	3	2	no	—
58	C₉×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
59	C₉×C₁₉	#737	1	2	yes	—
60	#4294	—	3	2	no	—
61	#4294	—	3	1	no	—
62	#4294	—	3	1	no	—
63	#4294	—	3	2	no	—
64	#4294	—	3	2	no	—
65	C₃×3₋¹⁺²	—	19	1	no	—
66	C₃²×C₉	—	19	1	no	—
67	C₃×C₂₇	—	19	2	no	—
68	C₃×C₂₇	—	19	2	no	—
69	C₃×C₂₇	—	19	2	no	—
70	C₃×C₂₇	—	57	3	no	—
71	C₃×C₂₇	C₃×C₁₉⋊C₃	1	2	yes	—
72	C₃×C₂₇	C₃×C₁₉⋊C₃	1	1	yes	characteristic
73	C₉⋊C₉	—	19	2	no	—
74	C₉⋊C₉	—	19	2	no	—
75	C₉⋊C₉	—	19	2	no	—
76	C₃²⋊C₉	—	19	2	no	—
77	C₉²	—	19	1	no	—
78	C₉²	—	19	1	no	—
79	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
80	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
81	C₃×C₁₉	C₃×C₂₇⋊C₃	1	1	yes	characteristic
82	C₃×C₁₉	#6903	1	2	yes	—
83	C₁₉⋊C₃	—	3	3	no	—
84	C₃³	—	19	1	no	—
85	3₋¹⁺²	—	57	3	no	—
86	C₃×C₉	—	19	2	no	—
87	C₃×C₉	—	19	2	no	—
88	C₃×C₉	—	19	1	no	—
89	C₃×C₉	—	19	1	no	—
90	C₃×C₉	—	19	2	no	—
91	C₃×C₉	—	57	1	no	—
92	C₃×C₉	—	57	2	no	—
93	C₃×C₉	C₉×C₁₉⋊C₃	1	2	yes	—
94	C₃×C₉	C₉×C₁₉⋊C₃	1	1	yes	characteristic
95	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
96	C₂₇	—	3	2	no	—
97	C₂₇	—	57	2	no	—
98	C₂₇	—	57	2	no	—
99	C₂₇	—	57	2	no	—
100	C₂₇	#10587023	1	3	yes	complemented
101	C₁₉	#10624727	1	1	yes	characteristic, complemented, Sylow
102	C₃²	—	19	3	no	—
103	C₃²	—	57	1	no	—
104	C₃²	—	57	2	no	—
105	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
106	C₉	—	3	1	no	—
107	C₉	—	3	2	no	—
108	C₉	—	57	2	no	—
109	C₉	—	57	1	no	—
110	C₉	—	57	1	no	—
111	C₉	—	57	2	no	—
112	C₉	—	57	2	no	—
113	C₉	C₃×#10587023	1	1	yes	characteristic
114	C₉	#13095332	1	2	yes	—
115	C₃	—	57	3	no	—
116	C₃	C₉×#10587023	1	1	yes	characteristic
117	C₃	C₁₉⋊C₃×C₂₇⋊C₃	1	1	yes	characteristic
118	C₃	#17987481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₂₇×C₁₉⋊C₃

C₃

yes

maximal, characteristic

C₉×#10587023

C₃

yes

maximal, characteristic

#17987357

C₃

yes

maximal, complemented

#17987355

C₃

yes

maximal, complemented

#17987355

C₃

yes

maximal, complemented

#17987332

C₃

yes

maximal

#17987330

C₃

yes

maximal, characteristic, complemented

#17987329

C₃

yes

maximal, characteristic, complemented

C₁₉×#6864

C₃

yes

maximal, characteristic, complemented

C₃×#10587023

C₉

yes

characteristic

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

C₃×#10587018

C₃²

yes

—

C₃×#10587018

C₃²

yes

—

C₃×#10587018

C₃²

yes

—

C₂₇×C₁₉⋊C₃

—

#13095307

C₉

yes

—

#13095305

C₉

yes

—

#13095305

C₉

yes

—

#13095303

C₉

yes

—

C₉×#4294

C₃²

yes

characteristic

C₉×#4294

C₃²

yes

characteristic

C₃×C₁₉×C₂₇

C₃²

yes

—

C₃×C₁₉×C₂₇

C₃²

yes

characteristic

C₉²×C₁₉

C₃²

yes

characteristic

#10624727

—

maximal, Sylow

C₃²×C₁₉⋊C₃

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×C₉×C₁₉

C₃³

yes

characteristic

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

characteristic

C₃×#4294

C₃×C₉

yes

—

#10587023

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₁₉×C₂₇

—

C₁₉×C₂₇

3₋¹⁺²

yes

—

—

—

C₃²×C₂₇

—

C₉×3₋¹⁺²

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₉⋊C₂₇

—

C₃²⋊C₂₇

—

#6864

—

#6864

—

#6864

C₁₉⋊C₃

yes

characteristic, complemented

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₂₇⋊C₃

yes

—

C₉×C₁₉

—

C₉×C₁₉

—

C₉×C₁₉

C₃×3₋¹⁺²

yes

characteristic

C₉×C₁₉

#737

yes

—

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

C₃×C₁₉⋊C₃

yes

—

C₃×C₂₇

C₃×C₁₉⋊C₃

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₉²

—

C₉²

C₃×C₁₉⋊C₃

yes

characteristic

C₃×C₁₉

C₉×3₋¹⁺²

yes

characteristic

C₃×C₁₉

C₃×C₂₇⋊C₃

yes

characteristic

C₃×C₁₉

#6903

yes

—

C₁₉⋊C₃

—

C₃³

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₉×C₁₉⋊C₃

yes

—

C₃×C₉

C₉×C₁₉⋊C₃

yes

characteristic

C₃×C₉

C₃²×C₁₉⋊C₃

yes

characteristic

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

—

100

C₂₇

#10587023

yes

complemented

101

C₁₉

#10624727

yes

characteristic, complemented, Sylow

102

C₃²

—

103

C₃²

—

104

C₃²

—

105

C₃²

C₃×C₉×C₁₉⋊C₃

yes

characteristic

106

C₉

—

107

C₉

—

108

C₉

—

109

C₉

—

110

C₉

—

111

C₉

—

112

C₉

—

113

C₉

C₃×#10587023

yes

characteristic

114

C₉

#13095332

yes

—

115

C₃

—

116

C₃

C₉×#10587023

yes

characteristic

117

C₃

C₁₉⋊C₃×C₂₇⋊C₃

yes

characteristic

118

C₃

#17987481

yes

—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
commutator, socle	C₃²×C₁₉	C₃²×C₉
Fitting	C₁₉×#6864	C₃
prime radical	C₃²×C₁₉⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₁₉	C₃³
hypercenter	#6864	C₁₉⋊C₃
hypocenter	C₁₉	#10624727
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ #6864

subgroup

G/H

center, Frattini

C₃×C₉

C₃²×C₁₉⋊C₃

commutator, socle

C₃²×C₁₉

C₃²×C₉

Fitting

C₁₉×#6864

C₃

prime radical

C₃²×C₁₉⋊C₃

C₃×C₉

cosocle

C₃×C₉×C₁₉

C₃³

C₁₉⋊C₃

C₁₉

G ▷ C₃²×C₁₉ ▷ C₁₉

C₁ ◁ C₃×C₉ ◁ #6864

N	H ≅ G/N	type
3. #17987357	C₃	semidirect
4. #17987355	C₃	semidirect
5. #17987355	C₃	semidirect
7. #17987330	C₃	semidirect
8. #17987329	C₃	semidirect
9. C₁₉×#6864	C₃	semidirect
39. C₁₉×C₂₇	3₋¹⁺²	semidirect
51. #6864	C₁₉⋊C₃	semidirect
100. C₂₇	#10587023	semidirect
101. C₁₉	#10624727	semidirect

H ≅ G/N

type

3. #17987357

C₃

semidirect

4. #17987355

C₃

semidirect

5. #17987355

C₃

semidirect

7. #17987330

C₃

semidirect

8. #17987329

C₃

semidirect

9. C₁₉×#6864

C₃

semidirect

39. C₁₉×C₂₇

3₋¹⁺²

semidirect

51. #6864

C₁₉⋊C₃

semidirect

100. C₂₇

#10587023

semidirect

101. C₁₉

#10624727

semidirect

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

subgroup	index
C₃×C₂₇×C₁₉⋊C₃	3
C₉×#10587023	3
#17987357	3
#17987355	3
#17987332	3
#17987330	3
#17987329	3
C₁₉×#6864	3
#10624727	19

subgroup

index

C₃×C₂₇×C₁₉⋊C₃

C₁₉×#6864

Subgroups — #28354867

Conjugacy classes of subgroups

Complements

Maximal subgroups