Subgroups — C₁₂₇×#6909 #44524902

Order 30861.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₂₇	C₃	1	2	yes	maximal, complemented
2	C₃×H(3,𝔽₃)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×H(3,𝔽₃)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	AΣL(1,𝔽₂₇)×C₁₂₇	C₃	1	9	yes	maximal, complemented
5	C₃³×C₁₂₇	—	3	1	no	—
6	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
7	C₃³×C₁₂₇	C₃²	1	3	yes	complemented
8	3₋¹⁺²×C₁₂₇	—	3	2	no	—
9	3₋¹⁺²×C₁₂₇	—	3	2	no	—
10	3₋¹⁺²×C₁₂₇	C₃²	1	6	yes	complemented
11	H(3,𝔽₃)×C₁₂₇	—	3	2	no	—
12	H(3,𝔽₃)×C₁₂₇	—	3	1	no	—
13	H(3,𝔽₃)×C₁₂₇	—	3	2	no	—
14	H(3,𝔽₃)×C₁₂₇	C₃²	1	3	yes	—
15	C₃×C₉×C₁₂₇	—	3	2	no	—
16	C₃²×C₁₂₇	—	3	3	no	—
17	C₃²×C₁₂₇	—	3	3	no	—
18	C₃²×C₁₂₇	—	9	1	no	—
19	C₃²×C₁₂₇	—	9	3	no	—
20	C₃²×C₁₂₇	—	9	3	no	—
21	C₃²×C₁₂₇	H(3,𝔽₃)	1	2	yes	—
22	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
24	C₉×C₁₂₇	—	3	6	no	—
25	C₃×C₁₂₇	—	3	1	no	—
26	C₃×C₁₂₇	—	3	1	no	—
27	C₃×C₁₂₇	—	3	2	no	—
28	C₃×C₁₂₇	—	9	3	no	—
29	C₃×C₁₂₇	—	9	3	no	—
30	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	#6909	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₁₂₇	#6909	1	1	yes	characteristic, direct product factor, Sylow
33	C₃×3₋¹⁺²	C₃×C₁₂₇	1	2	yes	complemented
34	C₃×H(3,𝔽₃)	C₃×C₁₂₇	1	1	yes	characteristic, complemented
35	C₃×H(3,𝔽₃)	C₃×C₁₂₇	1	1	yes	characteristic, complemented
36	AΣL(1,𝔽₂₇)	C₃×C₁₂₇	1	9	yes	complemented
37	C₃³	—	3	1	no	—
38	C₃³	C₃²×C₁₂₇	1	1	yes	characteristic
39	C₃³	C₃²×C₁₂₇	1	3	yes	complemented
40	3₋¹⁺²	—	3	2	no	—
41	3₋¹⁺²	—	3	2	no	—
42	3₋¹⁺²	C₃²×C₁₂₇	1	6	yes	complemented
43	H(3,𝔽₃)	—	3	2	no	—
44	H(3,𝔽₃)	—	3	1	no	—
45	H(3,𝔽₃)	—	3	2	no	—
46	H(3,𝔽₃)	C₃²×C₁₂₇	1	3	yes	—
47	C₃×C₉	—	3	2	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	3	3	no	—
50	C₃²	—	9	1	no	—
51	C₃²	—	9	3	no	—
52	C₃²	—	9	3	no	—
53	C₃²	H(3,𝔽₃)×C₁₂₇	1	2	yes	—
54	C₃²	H(3,𝔽₃)×C₁₂₇	1	1	yes	characteristic
55	C₃²	C₃³×C₁₂₇	1	1	yes	characteristic
56	C₉	—	3	6	no	—
57	C₃	—	3	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	3	2	no	—
60	C₃	—	9	3	no	—
61	C₃	—	9	3	no	—
62	C₃	C₃×H(3,𝔽₃)×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₁₂₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₂₇ ◁ C₃³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₁₂₇	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₁₂₇	C₃	semidirect
4. AΣL(1,𝔽₂₇)×C₁₂₇	C₃	semidirect
7. C₃³×C₁₂₇	C₃²	semidirect
10. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
31. #6909	C₁₂₇	direct (N × H)
32. C₁₂₇	#6909	direct (N × H)
33. C₃×3₋¹⁺²	C₃×C₁₂₇	semidirect
34. C₃×H(3,𝔽₃)	C₃×C₁₂₇	semidirect
35. C₃×H(3,𝔽₃)	C₃×C₁₂₇	semidirect
36. AΣL(1,𝔽₂₇)	C₃×C₁₂₇	semidirect
39. C₃³	C₃²×C₁₂₇	semidirect
42. 3₋¹⁺²	C₃²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₁₂₇	3
C₃×H(3,𝔽₃)×C₁₂₇	3
AΣL(1,𝔽₂₇)×C₁₂₇	3
#6909	127

Subgroups — C127×#6909 #44524902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#6909 #44524902