Group #23744881 — subgroups

Subgroups — C₁₃×#10624730 #23744881

Order 9477.

2072 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 2072 conjugacy classes of subgroups — too many to draw. Only its 116 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	C₃⁵×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	1	9	yes	maximal, complemented
3	C₁₃×#6891	C₃	1	2	yes	maximal, complemented
4	C₁₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃⁴×C₁₃	—	3	3	no	—
6	C₃⁴×C₁₃	—	3	9	no	—
7	C₃⁴×C₁₃	—	3	27	no	—
8	C₃⁴×C₁₃	C₃²	1	3	yes	—
9	C₃⁴×C₁₃	C₃²	1	1	yes	characteristic
10	C₃×C₁₃×3₋¹⁺²	C₃²	1	6	yes	complemented
11	C₃×C₁₃×H(3,𝔽₃)	—	3	3	no	—
12	C₃×C₁₃×H(3,𝔽₃)	C₃²	1	3	yes	complemented
13	C₁₃×AΣL(1,𝔽₂₇)	—	3	27	no	—
14	C₁₃×C₃²⋊C₉	—	3	6	no	—
15	#10624730	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₃³×C₁₃	—	3	27	no	—
17	C₃³×C₁₃	—	3	243	no	—
18	C₃³×C₁₃	—	3	27	no	—
19	C₃³×C₁₃	—	3	54	no	—
20	C₃³×C₁₃	—	3	27	no	—
21	C₃³×C₁₃	—	3	9	no	—
22	C₃³×C₁₃	—	3	6	no	—
23	C₃³×C₁₃	—	3	3	no	—
24	C₃³×C₁₃	—	3	3	no	—
25	C₃³×C₁₃	—	3	3	no	—
26	C₃³×C₁₃	H(3,𝔽₃)	1	3	yes	—
27	C₃³×C₁₃	H(3,𝔽₃)	1	9	yes	complemented
28	C₃³×C₁₃	C₃³	1	1	yes	characteristic
29	C₁₃×3₋¹⁺²	—	3	18	no	—
30	C₁₃×H(3,𝔽₃)	—	3	9	no	—
31	C₁₃×H(3,𝔽₃)	—	9	9	no	—
32	C₃×C₉×C₁₃	—	3	6	no	—
33	C₃⁵	C₃×C₁₃	1	1	yes	characteristic, complemented
34	C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	1	9	yes	complemented
35	#6891	C₃×C₁₃	1	2	yes	complemented
36	C₃²⋊H(3,𝔽₃)	C₃×C₁₃	1	1	yes	characteristic, complemented
37	C₃²×C₁₃	—	3	27	no	—
38	C₃²×C₁₃	—	3	243	no	—
39	C₃²×C₁₃	—	3	54	no	—
40	C₃²×C₁₃	—	3	27	no	—
41	C₃²×C₁₃	—	3	3	no	—
42	C₃²×C₁₃	—	3	9	no	—
43	C₃²×C₁₃	—	3	27	no	—
44	C₃²×C₁₃	—	3	6	no	—
45	C₃²×C₁₃	—	3	3	no	—
46	C₃²×C₁₃	—	9	3	no	—
47	C₃²×C₁₃	—	9	9	no	—
48	C₃²×C₁₃	AΣL(1,𝔽₂₇)	1	9	yes	complemented
49	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	3	yes	—
50	C₃²×C₁₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₉×C₁₃	—	9	6	no	—
52	C₃⁴	—	3	27	no	—
53	C₃⁴	—	3	3	no	—
54	C₃⁴	—	3	9	no	—
55	C₃⁴	C₃²×C₁₃	1	1	yes	characteristic
56	C₃⁴	C₃²×C₁₃	1	3	yes	—
57	C₃×3₋¹⁺²	C₃²×C₁₃	1	6	yes	complemented
58	C₃×H(3,𝔽₃)	—	3	3	no	—
59	C₃×H(3,𝔽₃)	C₃²×C₁₃	1	3	yes	complemented
60	AΣL(1,𝔽₂₇)	—	3	27	no	—
61	C₃²⋊C₉	—	3	6	no	—
62	C₃×C₁₃	—	3	27	no	—
63	C₃×C₁₃	—	3	9	no	—
64	C₃×C₁₃	—	3	3	no	—
65	C₃×C₁₃	—	27	3	no	—
66	C₃×C₁₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
67	C₃×C₁₃	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
68	C₃³	—	3	27	no	—
69	C₃³	—	3	243	no	—
70	C₃³	—	3	27	no	—
71	C₃³	—	3	54	no	—
72	C₃³	—	3	27	no	—
73	C₃³	—	3	9	no	—
74	C₃³	—	3	3	no	—
75	C₃³	—	3	6	no	—
76	C₃³	—	3	3	no	—
77	C₃³	—	3	3	no	—
78	C₃³	C₁₃×H(3,𝔽₃)	1	3	yes	—
79	C₃³	C₁₃×H(3,𝔽₃)	1	9	yes	complemented
80	C₃³	C₃³×C₁₃	1	1	yes	characteristic
81	3₋¹⁺²	—	3	18	no	—
82	H(3,𝔽₃)	—	3	9	no	—
83	H(3,𝔽₃)	—	9	9	no	—
84	C₃×C₉	—	3	6	no	—
85	C₁₃	#10624730	1	1	yes	characteristic, direct product factor, Sylow
86	C₃²	—	3	27	no	—
87	C₃²	—	3	27	no	—
88	C₃²	—	3	243	no	—
89	C₃²	—	3	54	no	—
90	C₃²	—	3	3	no	—
91	C₃²	—	3	9	no	—
92	C₃²	—	3	6	no	—
93	C₃²	—	3	3	no	—
94	C₃²	—	3	27	no	—
95	C₃²	—	9	3	no	—
96	C₃²	—	9	9	no	—
97	C₃²	C₁₃×AΣL(1,𝔽₂₇)	1	9	yes	complemented
98	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	3	yes	—
99	C₃²	C₃×C₁₃×H(3,𝔽₃)	1	1	yes	characteristic
100	C₉	—	9	6	no	—
101	C₃	—	3	9	no	—
102	C₃	—	3	3	no	—
103	C₃	—	3	27	no	—
104	C₃	—	27	3	no	—
105	C₃	C₁₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
106	C₃	C₃×C₁₃×AΣL(1,𝔽₂₇)	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃⁵×C₁₃

C₃

yes

maximal, characteristic, complemented

C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

yes

maximal, complemented

C₁₃×#6891

C₃

yes

maximal, complemented

C₁₃×C₃²⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic, complemented

C₃⁴×C₁₃

—

C₃⁴×C₁₃

—

C₃⁴×C₁₃

—

C₃⁴×C₁₃

C₃²

yes

—

C₃⁴×C₁₃

C₃²

yes

characteristic

C₃×C₁₃×3₋¹⁺²

C₃²

yes

complemented

C₃×C₁₃×H(3,𝔽₃)

—

C₃×C₁₃×H(3,𝔽₃)

C₃²

yes

complemented

C₁₃×AΣL(1,𝔽₂₇)

—

C₁₃×C₃²⋊C₉

—

#10624730

C₁₃

yes

maximal, characteristic, direct product factor, Sylow

C₃³×C₁₃

—

C₃³×C₁₃

—

243

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

—

C₃³×C₁₃

H(3,𝔽₃)

yes

—

C₃³×C₁₃

H(3,𝔽₃)

yes

complemented

C₃³×C₁₃

C₃³

yes

characteristic

C₁₃×3₋¹⁺²

—

C₁₃×H(3,𝔽₃)

—

C₁₃×H(3,𝔽₃)

—

C₃×C₉×C₁₃

—

C₃⁵

C₃×C₁₃

yes

characteristic, complemented

C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

yes

complemented

#6891

C₃×C₁₃

yes

complemented

C₃²⋊H(3,𝔽₃)

C₃×C₁₃

yes

characteristic, complemented

C₃²×C₁₃

—

C₃²×C₁₃

—

243

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

—

C₃²×C₁₃

AΣL(1,𝔽₂₇)

yes

complemented

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

—

C₃²×C₁₃

C₃×H(3,𝔽₃)

yes

characteristic

C₉×C₁₃

—

C₃⁴

—

C₃⁴

—

C₃⁴

—

C₃⁴

C₃²×C₁₃

yes

characteristic

C₃⁴

C₃²×C₁₃

yes

—

C₃×3₋¹⁺²

C₃²×C₁₃

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

C₃²×C₁₃

yes

complemented

AΣL(1,𝔽₂₇)

—

C₃²⋊C₉

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

—

C₃×C₁₃

C₃²⋊H(3,𝔽₃)

yes

characteristic

C₃×C₁₃

C₃×AΣL(1,𝔽₂₇)

yes

—

C₃³

—

C₃³

—

243

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₁₃×H(3,𝔽₃)

yes

—

C₃³

C₁₃×H(3,𝔽₃)

yes

complemented

C₃³

C₃³×C₁₃

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₁₃

#10624730

yes

characteristic, direct product factor, Sylow

C₃²

—

C₃²

—

C₃²

—

243

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₁₃×AΣL(1,𝔽₂₇)

yes

complemented

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

—

C₃²

C₃×C₁₃×H(3,𝔽₃)

yes

characteristic

100

C₉

—

101

C₃

—

102

C₃

—

103

C₃

—

104

C₃

—

105

C₃

C₁₃×C₃²⋊H(3,𝔽₃)

yes

characteristic

106

C₃

C₃×C₁₃×AΣL(1,𝔽₂₇)

yes

—

	subgroup	G/H
center, socle	C₃²×C₁₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₁₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

subgroup

G/H

center, socle

C₃²×C₁₃

C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₃³

C₃³×C₁₃

LCS

G ▷ C₃³ ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₃²×C₁₃ ◁ C₃⁴×C₁₃ ◁ G

N	H ≅ G/N	type
1. C₃⁵×C₁₃	C₃	semidirect
2. C₃×C₁₃×AΣL(1,𝔽₂₇)	C₃	semidirect
3. C₁₃×#6891	C₃	semidirect
4. C₁₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
10. C₃×C₁₃×3₋¹⁺²	C₃²	semidirect
12. C₃×C₁₃×H(3,𝔽₃)	C₃²	semidirect
15. #10624730	C₁₃	direct (N × H)
27. C₃³×C₁₃	H(3,𝔽₃)	semidirect
33. C₃⁵	C₃×C₁₃	semidirect
34. C₃×AΣL(1,𝔽₂₇)	C₃×C₁₃	semidirect
35. #6891	C₃×C₁₃	semidirect
36. C₃²⋊H(3,𝔽₃)	C₃×C₁₃	semidirect
48. C₃²×C₁₃	AΣL(1,𝔽₂₇)	semidirect
57. C₃×3₋¹⁺²	C₃²×C₁₃	semidirect
59. C₃×H(3,𝔽₃)	C₃²×C₁₃	semidirect
79. C₃³	C₁₃×H(3,𝔽₃)	semidirect
85. C₁₃	#10624730	direct (N × H)
97. C₃²	C₁₃×AΣL(1,𝔽₂₇)	semidirect

H ≅ G/N

type

1. C₃⁵×C₁₃

C₃

semidirect

2. C₃×C₁₃×AΣL(1,𝔽₂₇)

C₃

semidirect

3. C₁₃×#6891

C₃

semidirect

4. C₁₃×C₃²⋊H(3,𝔽₃)

C₃

semidirect

10. C₃×C₁₃×3₋¹⁺²

C₃²

semidirect

12. C₃×C₁₃×H(3,𝔽₃)

C₃²

semidirect

15. #10624730

C₁₃

direct (N × H)

27. C₃³×C₁₃

H(3,𝔽₃)

semidirect

33. C₃⁵

C₃×C₁₃

semidirect

34. C₃×AΣL(1,𝔽₂₇)

C₃×C₁₃

semidirect

35. #6891

C₃×C₁₃

semidirect

36. C₃²⋊H(3,𝔽₃)

C₃×C₁₃

semidirect

48. C₃²×C₁₃

AΣL(1,𝔽₂₇)

semidirect

57. C₃×3₋¹⁺²

C₃²×C₁₃

semidirect

59. C₃×H(3,𝔽₃)

C₃²×C₁₃

semidirect

79. C₃³

C₁₃×H(3,𝔽₃)

semidirect

85. C₁₃

#10624730

direct (N × H)

97. C₃²

C₁₃×AΣL(1,𝔽₂₇)

semidirect

Subgroups — C₁₃×#10624730 #23744881

Conjugacy classes of subgroups

Complements

Subgroups — C13×#10624730 #23744881

Conjugacy classes of subgroups

Complements

Subgroups — C₁₃×#10624730 #23744881