Group #28356997 — subgroups

Subgroups — #28356997

Order 13851.

250 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 69 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	3₋¹⁺²×#4296	C₃	1	3	yes	maximal
2	C₃×#13095339	C₃	1	1	yes	maximal, characteristic
3	3₋¹⁺²⋊#4296	C₃	1	6	yes	maximal
4	#17987560	C₃	1	2	yes	maximal
5	C₁₉⋊C₃×#733	C₃	1	1	yes	maximal, characteristic
6	C₃²×#4296	C₃²	1	2	yes	—
7	C₃²×#4296	C₃²	1	1	yes	characteristic
8	#13095339	—	3	3	no	—
9	#13095339	—	3	3	no	—
10	#13095338	—	3	3	no	—
11	C₃²⋊#4296	C₃²	1	6	yes	—
12	C₉×#4296	—	3	3	no	—
13	3₋¹⁺²×C₁₉⋊C₃	C₃²	1	3	yes	—
14	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
15	#13095322	C₉	1	2	yes	complemented
16	#13095321	C₉	1	6	yes	complemented
17	C₁₉×#733	C₉	1	1	yes	characteristic, complemented
18	#10624723	—	19	1	no	maximal, Sylow
19	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
20	C₃×#4296	—	3	6	no	—
21	C₃×#4296	—	3	2	no	—
22	C₃×#4296	—	9	1	no	—
23	C₃×#4296	—	9	2	no	—
24	C₃×#4296	H(3,𝔽₃)	1	3	yes	—
25	C₁₉×3₋¹⁺²	C₃×C₉	1	3	yes	—
26	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
27	#10587025	C₃×C₉	1	6	yes	—
28	C₃×#4294	C₃×C₉	1	2	yes	—
29	#10587023	—	3	3	no	—
30	#10587022	—	3	3	no	—
31	C₉×C₁₉⋊C₃	—	3	1	no	—
32	C₉×C₁₉⋊C₃	—	3	3	no	—
33	C₉×C₁₉⋊C₃	—	3	1	no	—
34	C₉×C₁₉⋊C₃	—	3	1	no	—
35	C₃×#733	—	19	1	no	—
36	C₉×3₋¹⁺²	—	19	3	no	—
37	C₃×C₉⋊C₉	—	19	1	no	—
38	3₋¹⁺²⋊C₉	—	19	6	no	—
39	#6868	—	19	2	no	—
40	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
41	C₃×C₁₉⋊C₃	—	3	1	no	—
42	C₃×C₁₉⋊C₃	—	3	2	no	—
43	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
44	C₃×C₁₉⋊C₃	#737	1	1	yes	characteristic
45	C₃×C₁₉⋊C₃	#737	1	1	yes	characteristic
46	#4296	—	3	3	no	—
47	#4296	—	9	6	no	—
48	#4296	—	9	2	no	—
49	C₉×C₁₉	—	3	1	no	—
50	C₉×C₁₉	—	3	3	no	—
51	#4294	—	3	2	no	—
52	#4294	—	3	6	no	—
53	C₃×3₋¹⁺²	—	19	3	no	—
54	C₃²×C₉	—	19	1	no	—
55	C₃²×C₉	—	19	1	no	—
56	C₃²×C₉	—	19	2	no	—
57	#733	—	19	6	no	—
58	#733	—	19	2	no	—
59	#733	#4296	1	1	yes	characteristic, complemented
60	C₉⋊C₉	—	57	3	no	—
61	C₉⋊C₉	—	57	3	no	—
62	C₉⋊C₉	—	57	3	no	—
63	C₃²⋊C₉	—	19	6	no	—
64	C₉²	—	57	3	no	—
65	C₃×C₁₉	—	3	1	no	—
66	C₃×C₁₉	C₉×H(3,𝔽₃)	1	1	yes	characteristic
67	C₁₉⋊C₃	—	3	1	no	—
68	C₁₉⋊C₃	—	3	1	no	—
69	C₁₉⋊C₃	#6912	1	1	yes	characteristic
70	C₁₉⋊C₃	#6913	1	2	yes	—
71	C₃³	—	19	1	no	—
72	3₋¹⁺²	—	19	6	no	—
73	3₋¹⁺²	—	57	3	no	—
74	3₋¹⁺²	—	57	3	no	—
75	3₋¹⁺²	C₃×#4296	1	3	yes	—
76	C₃×C₉	—	19	3	no	—
77	C₃×C₉	—	19	2	no	—
78	C₃×C₉	—	57	3	no	—
79	C₃×C₉	—	57	1	no	—
80	C₃×C₉	—	57	1	no	—
81	C₃×C₉	—	57	1	no	—
82	C₃×C₉	—	57	6	no	—
83	C₃×C₉	—	57	2	no	—
84	C₃×C₉	—	171	1	no	—
85	C₃×C₉	—	171	2	no	—
86	C₃×C₉	C₃×#4296	1	1	yes	characteristic
87	C₁₉	#10624723	1	1	yes	characteristic, complemented, Sylow
88	C₃²	—	19	1	no	—
89	C₃²	—	19	1	no	—
90	C₃²	—	19	1	no	—
91	C₃²	—	57	1	no	—
92	C₃²	—	57	2	no	—
93	C₃²	C₃²×#4296	1	1	yes	characteristic
94	C₉	—	3	1	no	—
95	C₉	—	3	3	no	—
96	C₉	—	57	3	no	—
97	C₉	—	57	2	no	—
98	C₉	—	57	6	no	—
99	C₉	—	171	2	no	—
100	C₉	—	171	6	no	—
101	C₃	—	3	1	no	—
102	C₃	—	19	1	no	—
103	C₃	—	19	2	no	—
104	C₃	—	57	1	no	—
105	C₃	—	57	1	no	—
106	C₃	H(3,𝔽₃)×#4296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#4296

C₃

yes

maximal

C₃×#13095339

C₃

yes

maximal, characteristic

3₋¹⁺²⋊#4296

C₃

yes

maximal

#17987560

C₃

yes

maximal

C₁₉⋊C₃×#733

C₃

yes

maximal, characteristic

C₃²×#4296

C₃²

yes

—

C₃²×#4296

C₃²

yes

—

—

—

C₃²⋊#4296

C₃²

yes

—

C₉×#4296

—

3₋¹⁺²×C₁₉⋊C₃

C₃²

yes

—

C₃×C₉×C₁₉⋊C₃

C₃²

yes

characteristic

#13095322

C₉

yes

complemented

#13095321

C₉

yes

complemented

C₁₉×#733

C₉

yes

characteristic, complemented

#10624723

—

maximal, Sylow

C₃²×C₁₉⋊C₃

C₃³

yes

—

—

—

—

yes

—

C₁₉×3₋¹⁺²

C₃×C₉

yes

—

C₃×C₉×C₁₉

C₃×C₉

yes

characteristic

#10587025

C₃×C₉

yes

—

C₃×#4294

C₃×C₉

yes

—

#10587023

—

#10587022

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₃×#733

—

C₉×3₋¹⁺²

—

C₃×C₉⋊C₉

—

3₋¹⁺²⋊C₉

—

#6868

—

C₃²×C₁₉

C₃²×C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

C₃×H(3,𝔽₃)

yes

C₃×C₁₉⋊C₃

yes

C₃×C₁₉⋊C₃

yes

—

—

—

C₉×C₁₉

—

C₉×C₁₉

—

#4294

—

#4294

—

C₃×3₋¹⁺²

—

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

—

—

—

yes

characteristic, complemented

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃×C₁₉

—

C₃×C₁₉

C₉×H(3,𝔽₃)

yes

characteristic

C₁₉⋊C₃

—

C₁₉⋊C₃

—

C₁₉⋊C₃

#6912

yes

characteristic

C₁₉⋊C₃

#6913

yes

—

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×#4296

yes

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₃×C₉

C₃×#4296

yes

characteristic

C₁₉

#10624723

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×#4296

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

171

—

100

C₉

—

171

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

H(3,𝔽₃)×#4296

yes

characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×#4296
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃²	C₃²×#4296
Fitting	C₁₉×#733	C₉
socle	C₃×C₁₉	C₉×H(3,𝔽₃)
prime radical, cosocle	C₃²×C₁₉⋊C₃	C₃³
hypercenter	#733	#4296
hypocenter	C₁₉	#10624723
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #733

subgroup

G/H

center

C₃

H(3,𝔽₃)×#4296

commutator

C₃²×C₁₉

C₃²×C₉

Frattini

C₃²

C₃²×#4296

Fitting

C₁₉×#733

C₉

socle

C₃×C₁₉

C₉×H(3,𝔽₃)

prime radical, cosocle

C₃²×C₁₉⋊C₃

C₃³

C₁₉

G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃ ◁ C₃² ◁ #733

N	H ≅ G/N	type
15. #13095322	C₉	semidirect
16. #13095321	C₉	semidirect
17. C₁₉×#733	C₉	semidirect
59. #733	#4296	semidirect
87. C₁₉	#10624723	semidirect

H ≅ G/N

type

15. #13095322

C₉

semidirect

16. #13095321

C₉

semidirect

17. C₁₉×#733

C₉

semidirect

59. #733

#4296

semidirect

87. C₁₉

#10624723

semidirect

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

subgroup	index
3₋¹⁺²×#4296	3
C₃×#13095339	3
3₋¹⁺²⋊#4296	3
#17987560	3
C₁₉⋊C₃×#733	3
#10624723	19

subgroup

index

3₋¹⁺²×#4296

C₃×#13095339

3₋¹⁺²⋊#4296

#17987560

C₁₉⋊C₃×#733

#10624723

Subgroups — #28356997

Conjugacy classes of subgroups

Complements

Maximal subgroups