Group #28356928 — subgroups

Subgroups — #28356928

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	#17987568	C₃	1	9	yes	maximal, complemented
4	C₃×#13095331	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	#13095331	—	3	2	no	—
13	#13095331	—	3	4	no	—
14	#13095331	C₉	1	6	yes	—
15	#13095331	C₃²	1	3	yes	—
16	C₃×#10587025	—	3	2	no	—
17	C₃×#10587025	C₉	1	2	yes	—
18	C₃²×#4294	—	3	1	no	—
19	C₃²×#4294	C₃²	1	1	yes	characteristic
20	C₃×C₁₉×H(3,𝔽₃)	C₉	1	1	yes	characteristic
21	#10624704	—	19	1	no	maximal, Sylow
22	C₃³×C₁₉	—	3	1	no	—
23	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
24	C₁₉×H(3,𝔽₃)	—	3	2	no	—
25	C₁₉×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
26	#10587025	—	3	4	no	—
27	#10587025	—	3	12	no	—
28	#10587025	—	3	6	no	—
29	#10587025	C₃×C₉	1	6	yes	—
30	C₃×#4294	—	3	2	no	—
31	C₃×#4294	—	3	6	no	—
32	C₃×#4294	—	3	3	no	—
33	C₃×#4294	C₃×C₉	1	2	yes	—
34	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
35	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
36	C₃×#4294	3₋¹⁺²	1	2	yes	—
37	C₃×#4294	3₋¹⁺²	1	4	yes	—
38	C₃×#4294	C₃³	1	1	yes	characteristic
39	#10587020	—	3	6	no	—
40	#10587020	—	3	3	no	—
41	C₃×#737	—	19	1	no	—
42	C₃×C₂₇⋊C₃	—	19	1	no	—
43	C₃×C₂₇⋊C₃	—	19	2	no	—
44	#6873	—	19	9	no	—
45	C₃²×C₁₉	—	3	3	no	—
46	C₃²×C₁₉	—	9	1	no	—
47	C₃²×C₁₉	—	9	3	no	—
48	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
49	C₃²×C₁₉	C₃²⋊C₉	1	2	yes	—
50	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
51	#4294	—	3	6	no	—
52	#4294	C₃²⋊C₉	1	2	yes	—
53	#4294	C₃²⋊C₉	1	4	yes	—
54	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	#4294	C₃×3₋¹⁺²	1	2	yes	—
56	#737	—	19	6	no	—
57	#737	—	19	3	no	—
58	#737	—	57	2	no	—
59	#737	—	57	4	no	—
60	C₃×3₋¹⁺²	—	19	2	no	—
61	C₃×3₋¹⁺²	—	57	2	no	—
62	C₃×H(3,𝔽₃)	#4296	1	1	yes	characteristic
63	C₃²×C₉	—	19	1	no	—
64	C₃²×C₉	—	57	1	no	—
65	C₂₇⋊C₃	—	19	6	no	—
66	C₂₇⋊C₃	—	57	2	no	—
67	C₂₇⋊C₃	—	57	1	no	—
68	C₂₇⋊C₃	—	57	2	no	—
69	C₂₇⋊C₃	—	57	2	no	—
70	C₃×C₂₇	—	19	3	no	—
71	C₃×C₂₇	—	57	2	no	—
72	C₃×C₁₉	—	3	1	no	—
73	C₃×C₁₉	—	3	2	no	—
74	C₃×C₁₉	—	3	1	no	—
75	C₃×C₁₉	—	9	3	no	—
76	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
77	C₃³	—	3	1	no	—
78	C₃³	C₃×#4296	1	1	yes	characteristic
79	3₋¹⁺²	—	19	6	no	—
80	3₋¹⁺²	—	57	12	no	—
81	3₋¹⁺²	—	57	6	no	—
82	3₋¹⁺²	—	57	4	no	—
83	H(3,𝔽₃)	—	3	2	no	—
84	H(3,𝔽₃)	C₃×#4296	1	3	yes	—
85	C₃×C₉	—	19	2	no	—
86	C₃×C₉	—	19	1	no	—
87	C₃×C₉	—	19	2	no	—
88	C₃×C₉	—	19	4	no	—
89	C₃×C₉	—	19	2	no	—
90	C₃×C₉	—	19	1	no	—
91	C₃×C₉	—	57	3	no	—
92	C₃×C₉	—	57	6	no	—
93	C₃×C₉	—	57	2	no	—
94	C₂₇	—	57	3	no	—
95	C₂₇	—	57	6	no	—
96	C₁₉	#10624704	1	1	yes	characteristic, complemented, Sylow
97	C₃²	—	3	3	no	—
98	C₃²	—	9	1	no	—
99	C₃²	—	9	3	no	—
100	C₃²	C₃²⋊#4296	1	1	yes	characteristic
101	C₃²	C₃²⋊#4296	1	2	yes	—
102	C₃²	C₃²×#4296	1	1	yes	characteristic
103	C₉	—	19	1	no	—
104	C₉	—	19	2	no	—
105	C₉	—	19	4	no	—
106	C₉	—	19	2	no	—
107	C₉	—	57	6	no	—
108	C₃	—	3	1	no	—
109	C₃	—	3	2	no	—
110	C₃	—	3	1	no	—
111	C₃	—	9	3	no	—
112	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

#17987568

C₃

yes

maximal, complemented

C₃×#13095331

C₃

yes

maximal, characteristic

—

—

—

—

C₃²

yes

complemented

C₃×#10587020

—

C₃×#10587020

C₃²

yes

—

—

C₉

yes

—

#13095331

C₃²

yes

—

C₃×#10587025

—

C₃×#10587025

C₉

yes

—

C₃²×#4294

—

C₃²×#4294

C₃²

yes

characteristic

C₃×C₁₉×H(3,𝔽₃)

C₉

yes

characteristic

#10624704

—

maximal, Sylow

C₃³×C₁₉

—

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₁₉×H(3,𝔽₃)

—

C₁₉×H(3,𝔽₃)

C₃×C₉

yes

—

—

—

—

C₃×C₉

yes

—

—

—

—

C₃×C₉

yes

—

yes

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

3₋¹⁺²

yes

—

C₃×#4294

C₃³

yes

—

—

—

C₃×C₂₇⋊C₃

—

C₃×C₂₇⋊C₃

—

#6873

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃²×C₁₉

C₃²⋊C₉

yes

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

#4294

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃²⋊C₉

yes

—

#4294

C₃×H(3,𝔽₃)

yes

characteristic

#4294

C₃×3₋¹⁺²

yes

—

—

—

—

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

#4296

yes

characteristic

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×C₁₉

C₃×C₃²⋊C₉

yes

characteristic

C₃³

—

C₃³

C₃×#4296

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×#4296

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₂₇

—

C₂₇

—

C₁₉

#10624704

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₃²⋊#4296

yes

characteristic

101

C₃²

C₃²⋊#4296

yes

—

102

C₃²

C₃²×#4296

yes

characteristic

103

C₉

—

104

C₉

—

105

C₉

—

106

C₉

—

107

C₉

—

108

C₃

—

109

C₃

—

110

C₃

—

111

C₃

—

112

C₃

C₃×C₃²⋊#4296

yes

characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#4296
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃²	C₃²×#4296
Fitting, prime radical	C₃×C₁₉×H(3,𝔽₃)	C₉
socle	C₃×C₁₉	C₃×C₃²⋊C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃×H(3,𝔽₃)	#4296
hypocenter	C₁₉	#10624704
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

subgroup

G/H

center

C₃

C₃×C₃²⋊#4296

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃²

C₃²×#4296

Fitting, prime radical

C₃×C₁₉×H(3,𝔽₃)

C₉

socle

C₃×C₁₉

C₃×C₃²⋊C₉

C₃³

C₃×H(3,𝔽₃)

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

N	H ≅ G/N	type
1. C₃×#13095342	C₃	semidirect
2. C₃×#13095342	C₃	semidirect
3. #17987568	C₃	semidirect
9. #13095342	C₃²	semidirect
11. C₃×#10587020	C₃²	semidirect
96. C₁₉	#10624704	semidirect

H ≅ G/N

type

1. C₃×#13095342

C₃

semidirect

2. C₃×#13095342

C₃

semidirect

3. #17987568

C₃

semidirect

9. #13095342

C₃²

semidirect

11. C₃×#10587020

C₃²

semidirect

96. C₁₉

#10624704

semidirect

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

subgroup	index
C₃×#13095342	3
#17987568	3
C₃×#13095331	3
#10624704	19

subgroup

index

Subgroups — #28356928

Conjugacy classes of subgroups

Complements

Maximal subgroups