Subgroups — C₃×#23779841 #44525090

Order 30861.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#16812184	C₃	1	2	yes	maximal, complemented
3	#23779841	C₃	1	9	yes	maximal, direct product factor
4	C₃×H(3,𝔽₃)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₁₂₇⋊C₃	—	3	2	no	—
6	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
7	C₃²×C₁₂₇⋊C₃	—	3	9	no	—
8	C₃²×C₁₂₇⋊C₃	C₃²	1	3	yes	complemented
9	#16812184	—	3	4	no	—
10	#16812184	C₃²	1	6	yes	complemented
11	C₃×#11765917	—	3	2	no	—
12	C₃³×C₁₂₇	—	3	1	no	—
13	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
14	H(3,𝔽₃)×C₁₂₇	—	3	2	no	—
15	H(3,𝔽₃)×C₁₂₇	C₃²	1	3	yes	complemented
16	C₃²×C₁₂₇	—	3	3	no	—
17	C₃²×C₁₂₇	—	3	3	no	—
18	C₃²×C₁₂₇	—	9	3	no	—
19	C₃²×C₁₂₇	H(3,𝔽₃)	1	2	yes	—
20	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
22	C₃×C₁₂₇⋊C₃	—	3	18	no	—
23	C₃×C₁₂₇⋊C₃	—	3	9	no	—
24	C₃×C₁₂₇⋊C₃	—	3	3	no	—
25	C₃×C₁₂₇⋊C₃	—	3	9	no	—
26	#11765917	—	3	6	no	—
27	C₃×C₁₂₇	—	3	2	no	—
28	C₃×C₁₂₇	—	3	1	no	—
29	C₃×C₁₂₇	—	9	3	no	—
30	C₃×C₁₂₇	AΣL(1,𝔽₂₇)	1	3	yes	complemented
31	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₁₂₇⋊C₃	—	3	9	no	—
33	C₃×AΣL(1,𝔽₂₇)	—	127	1	no	maximal, Sylow
34	C₁₂₇	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
35	C₃⁴	—	127	1	no	—
36	C₃×3₋¹⁺²	—	127	2	no	—
37	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
38	AΣL(1,𝔽₂₇)	—	127	9	no	—
39	C₃³	—	3	1	no	—
40	C₃³	—	381	9	no	—
41	C₃³	—	381	2	no	—
42	C₃³	—	381	1	no	—
43	C₃³	—	127	3	no	—
44	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
45	3₋¹⁺²	—	381	4	no	—
46	3₋¹⁺²	—	127	6	no	—
47	H(3,𝔽₃)	—	3	2	no	—
48	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
49	C₃×C₉	—	381	2	no	—
50	C₃²	—	3	3	no	—
51	C₃²	—	3	3	no	—
52	C₃²	—	9	3	no	—
53	C₃²	—	381	9	no	—
54	C₃²	—	381	18	no	—
55	C₃²	—	381	9	no	—
56	C₃²	—	381	3	no	—
57	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	2	yes	—
58	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
59	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
60	C₉	—	381	6	no	—
61	C₃	—	3	2	no	—
62	C₃	—	3	1	no	—
63	C₃	—	9	3	no	—
64	C₃	—	381	9	no	—
65	C₃	#23779841	1	3	yes	direct product factor
66	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₂₇⋊C₃)
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	C₃×H(3,𝔽₃)×C₁₂₇	C₃
socle	C₃²×C₁₂₇	H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₃×AΣL(1,𝔽₂₇)
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃³×C₁₂₇⋊C₃	C₃	semidirect
2. C₃×#16812184	C₃	semidirect
3. #23779841	C₃	direct (N × H)
4. C₃×H(3,𝔽₃)×C₁₂₇	C₃	semidirect
8. C₃²×C₁₂₇⋊C₃	C₃²	semidirect
10. #16812184	C₃²	semidirect
15. H(3,𝔽₃)×C₁₂₇	C₃²	semidirect
30. C₃×C₁₂₇	AΣL(1,𝔽₂₇)	semidirect
34. C₁₂₇	C₃×AΣL(1,𝔽₂₇)	semidirect
37. C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	semidirect
48. H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	semidirect
65. C₃	#23779841	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₁₂₇⋊C₃	3
C₃×#16812184	3
#23779841	3
C₃×H(3,𝔽₃)×C₁₂₇	3
C₃×AΣL(1,𝔽₂₇)	127

Subgroups — C3×#23779841 #44525090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#23779841 #44525090