Group #28356958 — subgroups

Subgroups — #28356958

Order 13851.

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

normal subgroups only

This group has 198 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	(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	2	yes	maximal
2	#17987589	C₃	1	1	yes	maximal, characteristic
3	#17987578	C₃	1	3	yes	maximal, complemented
4	#17987576	C₃	1	3	yes	maximal, complemented
5	#17987574	C₃	1	3	yes	maximal, complemented
6	C₃×#13095311	C₃	1	1	yes	maximal, characteristic
7	#13095344	—	3	2	no	—
8	#13095343	C₃²	1	6	yes	—
9	#13095342	—	3	2	no	—
10	(C₁₉⋊C₃)⋊C₂₇	—	3	2	no	—
11	#13095339	—	3	1	no	—
12	#13095338	C₃²	1	3	yes	—
13	C₃²⋊#4296	—	3	1	no	—
14	C₉×#4296	—	3	1	no	—
15	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
16	#13095311	C₉	1	2	yes	complemented
17	#13095311	C₉	1	2	yes	complemented
18	#13095311	C₉	1	2	yes	complemented
19	#13095311	C₃²	1	1	yes	characteristic
20	#13095311	C₃²	1	1	yes	characteristic
21	#13095311	C₃²	1	1	yes	characteristic
22	C₃×#10587018	C₉	1	2	yes	complemented
23	C₃×C₁₉×C₂₇	C₉	1	1	yes	characteristic, complemented
24	#10624709	—	19	1	no	maximal, Sylow
25	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
26	C₃×#4296	—	3	3	no	—
27	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
28	C₃×#4294	3₋¹⁺²	1	2	yes	—
29	C₉×C₁₉⋊C₃	C₃×C₉	1	2	yes	—
30	C₉×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
31	C₉×C₁₉⋊C₃	3₋¹⁺²	1	1	yes	characteristic
32	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
33	C₉×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
34	C₉×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
35	#10587020	—	3	6	no	—
36	C₁₉×C₂₇	C₃×C₉	1	1	yes	characteristic
37	C₁₉×C₂₇	C₃×C₉	1	1	yes	characteristic
38	C₁₉×C₂₇	C₃×C₉	1	1	yes	characteristic
39	#10587018	C₃×C₉	1	2	yes	—
40	#10587018	C₃×C₉	1	2	yes	—
41	#10587018	C₃×C₉	1	2	yes	—
42	C₃×C₂₇⋊C₃	—	19	1	no	—
43	C₃×C₂₇⋊C₃	—	19	2	no	—
44	#6887	—	19	1	no	—
45	#6875	—	19	3	no	—
46	#6875	—	19	3	no	—
47	#6875	—	19	3	no	—
48	C₃²×C₁₉	C₉⋊C₉	1	1	yes	characteristic
49	C₃×C₁₉⋊C₃	—	9	1	no	—
50	C₃×C₁₉⋊C₃	C₉⋊C₉	1	2	yes	—
51	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
52	#4296	—	9	3	no	—
53	C₉×C₁₉	C₉⋊C₉	1	1	yes	characteristic
54	C₉×C₁₉	C₉⋊C₉	1	1	yes	characteristic
55	C₉×C₁₉	C₃²×C₉	1	1	yes	characteristic
56	#4294	C₉⋊C₉	1	2	yes	—
57	#4294	C₉⋊C₉	1	2	yes	—
58	#4294	C₃×3₋¹⁺²	1	2	yes	—
59	C₃²×C₉	—	19	1	no	—
60	C₂₇⋊C₃	—	19	2	no	—
61	C₂₇⋊C₃	—	19	2	no	—
62	C₂₇⋊C₃	—	19	2	no	—
63	C₂₇⋊C₃	—	19	1	no	—
64	C₂₇⋊C₃	—	19	1	no	—
65	C₂₇⋊C₃	—	19	1	no	—
66	C₂₇⋊C₃	—	19	6	no	—
67	C₂₇⋊C₃	—	57	2	no	—
68	C₂₇⋊C₃	—	57	2	no	—
69	C₃×C₂₇	—	19	2	no	—
70	C₃×C₂₇	—	57	2	no	—
71	C₃×C₂₇	#4296	1	1	yes	characteristic, complemented
72	C₉⋊C₉	—	19	3	no	—
73	C₉⋊C₉	—	57	1	no	—
74	C₃²⋊C₉	—	57	1	no	—
75	C₉²	—	57	1	no	—
76	C₃×C₁₉	—	3	1	no	—
77	C₃×C₁₉	C₃×C₉⋊C₉	1	1	yes	characteristic
78	C₁₉⋊C₃	—	3	1	no	—
79	C₁₉⋊C₃	—	3	2	no	—
80	C₃³	—	19	1	no	—
81	C₃×C₉	—	19	2	no	—
82	C₃×C₉	—	19	1	no	—
83	C₃×C₉	—	19	2	no	—
84	C₃×C₉	—	19	1	no	—
85	C₃×C₉	—	19	2	no	—
86	C₃×C₉	—	19	2	no	—
87	C₃×C₉	—	19	1	no	—
88	C₃×C₉	—	57	3	no	—
89	C₃×C₉	C₃×#4296	1	1	yes	characteristic
90	C₂₇	—	19	2	no	—
91	C₂₇	—	19	2	no	—
92	C₂₇	—	19	2	no	—
93	C₂₇	—	57	6	no	—
94	C₂₇	C₃×#4296	1	1	yes	characteristic
95	C₂₇	C₃×#4296	1	1	yes	characteristic
96	C₂₇	C₃×#4296	1	1	yes	characteristic
97	C₁₉	#10624709	1	1	yes	characteristic, complemented, Sylow
98	C₃²	—	19	1	no	—
99	C₃²	—	19	2	no	—
100	C₃²	—	171	1	no	—
101	C₃²	#13095338	1	1	yes	characteristic
102	C₉	—	19	2	no	—
103	C₉	—	19	2	no	—
104	C₉	—	19	2	no	—
105	C₉	—	171	3	no	—
106	C₉	#13095338	1	1	yes	characteristic
107	C₉	#13095338	1	1	yes	characteristic
108	C₉	C₃²×#4296	1	1	yes	characteristic
109	C₃	—	3	1	no	—
110	C₃	—	57	1	no	—
111	C₃	—	57	2	no	—
112	C₃	C₃×#13095338	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

(C₁₉⋊C₃)⋊(C₂₇⋊C₃)

C₃

yes

maximal

#17987589

C₃

yes

maximal, characteristic

#17987578

C₃

yes

maximal, complemented

#17987576

C₃

yes

maximal, complemented

#17987574

C₃

yes

maximal, complemented

C₃×#13095311

C₃

yes

maximal, characteristic

#13095344

—

#13095343

C₃²

yes

—

#13095342

—

(C₁₉⋊C₃)⋊C₂₇

—

#13095339

—

#13095338

C₃²

yes

—

C₃²⋊#4296

—

C₉×#4296

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

#13095311

C₉

yes

complemented

#13095311

C₉

yes

complemented

#13095311

C₉

yes

complemented

#13095311

C₃²

yes

characteristic

#13095311

C₃²

yes

characteristic

#13095311

C₃²

yes

characteristic

C₃×#10587018

C₉

yes

complemented

C₃×C₁₉×C₂₇

C₉

yes

characteristic, complemented

#10624709

—

maximal, Sylow

C₃²×C₁₉⋊C₃

3₋¹⁺²

yes

characteristic

C₃×#4296

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

C₃×#4294

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

C₃×C₉

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

characteristic

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

C₃³

yes

characteristic

#10587020

—

C₁₉×C₂₇

C₃×C₉

yes

characteristic

C₁₉×C₂₇

C₃×C₉

yes

characteristic

C₁₉×C₂₇

C₃×C₉

yes

characteristic

#10587018

C₃×C₉

yes

—

#10587018

C₃×C₉

yes

—

#10587018

C₃×C₉

yes

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

—

—

—

—

C₃²×C₁₉

C₉⋊C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₉⋊C₉

yes

—

C₃×C₁₉⋊C₃

C₃×3₋¹⁺²

yes

characteristic

#4296

—

C₉×C₁₉

C₉⋊C₉

yes

characteristic

C₉×C₁₉

C₉⋊C₉

yes

characteristic

C₉×C₁₉

C₃²×C₉

yes

characteristic

#4294

C₉⋊C₉

yes

—

#4294

C₉⋊C₉

yes

—

#4294

C₃×3₋¹⁺²

yes

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃×C₂₇

#4296

yes

characteristic, complemented

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

C₃×C₉⋊C₉

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₃³

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×#4296

yes

characteristic

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

—

C₂₇

yes

C₂₇

yes

C₂₇

yes

C₁₉

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

100

C₃²

—

171

—

101

C₃²

#13095338

yes

characteristic

102

C₉

—

103

C₉

—

104

C₉

—

105

C₉

—

171

—

106

C₉

#13095338

yes

characteristic

107

C₉

#13095338

yes

characteristic

108

C₉

C₃²×#4296

yes

characteristic

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

C₃×#13095338

yes

characteristic

	subgroup	G/H
center	C₉	#13095338
commutator	C₉×C₁₉	C₃²×C₉
Frattini	C₉	C₃²×#4296
Fitting	C₃×C₁₉×C₂₇	C₉
socle	C₃×C₁₉	C₃×C₉⋊C₉
prime radical	C₃²×C₁₉⋊C₃	3₋¹⁺²
cosocle	C₉×C₁₉⋊C₃	C₃³
hypercenter	C₃×C₂₇	#4296
hypocenter	C₁₉	#10624709
LCS	G ▷ C₉×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃×C₂₇

subgroup

G/H

center

C₉

#13095338

commutator

C₉×C₁₉

C₃²×C₉

Frattini

C₉

C₃²×#4296

Fitting

C₃×C₁₉×C₂₇

C₉

socle

C₃×C₁₉

C₃×C₉⋊C₉

prime radical

C₃²×C₁₉⋊C₃

3₋¹⁺²

cosocle

C₉×C₁₉⋊C₃

C₃³

C₃×C₂₇

C₁₉

G ▷ C₉×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃×C₂₇

N	H ≅ G/N	type
3. #17987578	C₃	semidirect
4. #17987576	C₃	semidirect
5. #17987574	C₃	semidirect
16. #13095311	C₉	semidirect
17. #13095311	C₉	semidirect
18. #13095311	C₉	semidirect
22. C₃×#10587018	C₉	semidirect
23. C₃×C₁₉×C₂₇	C₉	semidirect
71. C₃×C₂₇	#4296	semidirect
97. C₁₉	#10624709	semidirect

H ≅ G/N

type

3. #17987578

C₃

semidirect

4. #17987576

C₃

semidirect

5. #17987574

C₃

semidirect

16. #13095311

C₉

semidirect

17. #13095311

C₉

semidirect

18. #13095311

C₉

semidirect

22. C₃×#10587018

C₉

semidirect

23. C₃×C₁₉×C₂₇

C₉

semidirect

71. C₃×C₂₇

#4296

semidirect

97. C₁₉

#10624709

semidirect

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

subgroup	index
(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	3
#17987589	3
#17987578	3
#17987576	3
#17987574	3
C₃×#13095311	3
#10624709	19

subgroup

index

(C₁₉⋊C₃)⋊(C₂₇⋊C₃)

Subgroups — #28356958

Conjugacy classes of subgroups

Complements

Maximal subgroups