Group #28356692 — subgroups

Subgroups — #28356692

Order 13851.

132 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 30 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	1	yes	maximal, characteristic, complemented
2	C₃²⋊#10587020	C₃	1	2	yes	maximal, complemented
3	#17987406	C₃	1	1	yes	maximal, characteristic
4	#13095342	—	3	3	no	—
5	C₃×#10587020	—	3	1	no	—
6	C₃×#10587020	—	3	2	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	C₉	1	1	yes	characteristic
8	C₃²×#4294	C₃²	1	1	yes	characteristic
9	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
10	C₃²⋊#4294	C₉	1	1	yes	characteristic
11	#13095303	—	3	2	no	—
12	C₁₉×C₃²⋊C₉	C₉	1	1	yes	characteristic
13	#10624453	—	19	1	no	maximal, Sylow
14	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
15	C₃²×C₁₉⋊C₃	—	3	1	no	—
16	C₃×C₉×C₁₉	—	3	1	no	—
17	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
18	C₃×#4294	—	3	1	no	—
19	C₃×#4294	—	3	1	no	—
20	C₃×#4294	—	3	1	no	—
21	C₃×#4294	—	3	1	no	—
22	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×#4294	3₋¹⁺²	1	2	yes	—
24	C₉×C₁₉⋊C₃	—	9	1	no	—
25	C₉×C₁₉⋊C₃	—	9	1	no	—
26	C₉×C₁₉⋊C₃	—	9	1	no	—
27	#10587020	—	3	3	no	—
28	#10587020	—	9	2	no	—
29	C₃×C₂₇⋊C₃	—	19	1	no	—
30	C₉×H(3,𝔽₃)	—	19	1	no	—
31	C₃²⋊C₂₇	—	19	2	no	—
32	C₃²×C₁₉	—	3	1	no	—
33	C₃²×C₁₉	—	3	1	no	—
34	C₃²×C₁₉	—	3	1	no	—
35	C₃²×C₁₉	—	3	1	no	—
36	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	—	3	3	no	—
38	C₃×C₁₉⋊C₃	—	9	1	no	—
39	C₃×C₁₉⋊C₃	—	9	1	no	—
40	C₃×C₁₉⋊C₃	—	9	1	no	—
41	C₉×C₁₉	—	9	1	no	—
42	#4294	—	3	2	no	—
43	#4294	—	9	1	no	—
44	#4294	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
45	#4294	#731	1	1	yes	characteristic
46	#4294	#732	1	1	yes	characteristic
47	C₃×H(3,𝔽₃)	—	19	1	no	—
48	C₃²×C₉	—	19	1	no	—
49	C₃²×C₉	—	57	1	no	—
50	C₂₇⋊C₃	—	57	3	no	—
51	C₃×C₂₇	—	57	1	no	—
52	C₃×C₂₇	—	57	2	no	—
53	C₃²⋊C₉	—	19	1	no	—
54	C₃²⋊C₉	—	57	2	no	—
55	C₃²⋊C₉	#4296	1	1	yes	characteristic
56	C₃×C₁₉	—	9	1	no	—
57	C₃×C₁₉	#6869	1	1	yes	characteristic
58	C₃×C₁₉	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
59	C₃×C₁₉	#6872	1	1	yes	characteristic
60	C₃×C₁₉	#6873	1	1	yes	characteristic
61	C₁₉⋊C₃	—	9	3	no	—
62	C₃³	—	57	1	no	—
63	C₃³	C₃×#4296	1	1	yes	characteristic
64	H(3,𝔽₃)	—	57	3	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	19	1	no	—
67	C₃×C₉	—	19	2	no	—
68	C₃×C₉	—	57	1	no	—
69	C₃×C₉	—	57	1	no	—
70	C₃×C₉	—	57	1	no	—
71	C₃×C₉	—	57	1	no	—
72	C₃×C₉	—	171	1	no	—
73	C₃×C₉	—	171	1	no	—
74	C₃×C₉	—	171	1	no	—
75	C₂₇	—	57	3	no	—
76	C₂₇	—	171	2	no	—
77	C₁₉	#10624453	1	1	yes	characteristic, complemented, Sylow
78	C₃²	—	3	1	no	—
79	C₃²	—	3	1	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	57	3	no	—
83	C₃²	—	171	1	no	—
84	C₃²	—	171	1	no	—
85	C₃²	—	171	1	no	—
86	C₃²	C₃²⋊#4296	1	1	yes	characteristic
87	C₉	—	9	1	no	—
88	C₉	—	19	1	no	—
89	C₉	—	19	1	no	—
90	C₉	—	19	1	no	—
91	C₉	—	57	2	no	—
92	C₉	—	171	1	no	—
93	C₃	—	9	1	no	—
94	C₃	—	171	3	no	—
95	C₃	#17987563	1	1	yes	characteristic
96	C₃	H(3,𝔽₃)⋊#4296	1	1	yes	characteristic
97	C₃	#17987567	1	1	yes	characteristic
98	C₃	#17987569	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#13095342

C₃

yes

maximal, characteristic, complemented

C₃²⋊#10587020

C₃

yes

maximal, complemented

#17987406

C₃

yes

maximal, characteristic

#13095342

—

C₃×#10587020

—

C₃×#10587020

—

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

yes

characteristic

C₃²×#4294

C₃²

yes

characteristic

C₃×C₉×C₁₉⋊C₃

—

C₃²⋊#4294

C₉

yes

characteristic

#13095303

—

C₁₉×C₃²⋊C₉

C₉

yes

characteristic

#10624453

—

maximal, Sylow

C₃³×C₁₉

C₃×C₉

yes

characteristic

C₃²×C₁₉⋊C₃

—

C₃×C₉×C₁₉

—

C₃²⋊(C₁₉⋊C₃)

—

—

—

—

—

yes

3₋¹⁺²

yes

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

C₉×C₁₉⋊C₃

—

#10587020

—

#10587020

—

C₃×C₂₇⋊C₃

—

C₉×H(3,𝔽₃)

—

C₃²⋊C₂₇

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

—

C₃²×C₁₉

C₃²⋊C₉

yes

characteristic

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₃×C₁₉⋊C₃

—

C₉×C₁₉

—

#4294

—

#4294

—

#4294

AΣL(1,𝔽₂₇)

yes

yes

yes

C₃×H(3,𝔽₃)

—

C₃²×C₉

—

C₃²×C₉

—

C₂₇⋊C₃

—

C₃×C₂₇

—

C₃×C₂₇

—

C₃²⋊C₉

—

C₃²⋊C₉

—

C₃²⋊C₉

#4296

yes

characteristic

C₃×C₁₉

—

C₃×C₁₉

#6869

yes

characteristic

C₃×C₁₉

H(3,𝔽₃)⋊C₉

yes

C₃×C₁₉

yes

C₃×C₁₉

yes

C₁₉⋊C₃

—

C₃³

—

C₃³

C₃×#4296

yes

characteristic

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₃×C₉

—

171

—

C₂₇

—

C₂₇

—

171

—

C₁₉

#10624453

yes

characteristic, complemented, Sylow

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

—

171

—

C₃²

C₃²⋊#4296

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

171

—

C₃

—

C₃

—

171

—

C₃

yes

C₃

yes

C₃

yes

C₃

yes

	subgroup	G/H
center	C₃²	C₃²⋊#4296
commutator	C₃³×C₁₉	C₃×C₉
Frattini	C₃³	C₃×#4296
Fitting	C₁₉×C₃²⋊C₉	C₉
socle	C₃²×C₁₉	C₃²⋊C₉
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₉
cosocle	C₃²×#4294	C₃²
hypercenter	C₃²⋊C₉	#4296
hypocenter	C₁₉	#10624453
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

subgroup

G/H

center

C₃²

C₃²⋊#4296

commutator

C₃³×C₁₉

C₃×C₉

Frattini

C₃³

C₃×#4296

Fitting

C₁₉×C₃²⋊C₉

C₉

socle

C₃²×C₁₉

C₃²⋊C₉

prime radical

C₃×C₃²⋊(C₁₉⋊C₃)

C₉

cosocle

C₃²×#4294

C₃²

C₃²⋊C₉

C₁₉

G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉

UCS

C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

N	H ≅ G/N	type
1. C₃×#13095342	C₃	semidirect
2. C₃²⋊#10587020	C₃	semidirect
77. C₁₉	#10624453	semidirect

H ≅ G/N

type

1. C₃×#13095342

C₃

semidirect

2. C₃²⋊#10587020

C₃

semidirect

77. C₁₉

#10624453

semidirect

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

subgroup	index
C₃×#13095342	3
C₃²⋊#10587020	3
#17987406	3
#10624453	19

subgroup

index

Subgroups — #28356692

Conjugacy classes of subgroups

Complements

Maximal subgroups