Subgroups — (C₃×C₁₂₇⋊C₃)⋊H(3,𝔽₃) #44525023

Order 30861.

378 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×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	2	yes	maximal, complemented
3	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₁₂₇⋊C₃	—	3	3	no	—
6	C₃²×C₁₂₇⋊C₃	—	3	9	no	—
7	C₃²×C₁₂₇⋊C₃	C₃²	1	3	yes	—
8	C₃²×C₁₂₇⋊C₃	C₃²	1	6	yes	complemented
9	C₃²⋊(C₁₂₇⋊C₃)	—	3	6	no	—
10	C₃²⋊(C₁₂₇⋊C₃)	—	3	27	no	—
11	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
12	C₃³×C₁₂₇	C₃²	1	3	yes	complemented
13	H(3,𝔽₃)×C₁₂₇	—	3	3	no	—
14	C₃²×C₁₂₇	—	3	3	no	—
15	C₃²×C₁₂₇	—	3	3	no	—
16	C₃²×C₁₂₇	—	3	9	no	—
17	C₃²×C₁₂₇	H(3,𝔽₃)	1	3	yes	complemented
18	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
19	C₃×C₁₂₇⋊C₃	—	3	27	no	—
20	C₃×C₁₂₇⋊C₃	—	3	6	no	—
21	C₃×C₁₂₇⋊C₃	—	3	3	no	—
22	C₃×C₁₂₇⋊C₃	—	3	18	no	—
23	C₃×C₁₂₇⋊C₃	—	3	6	no	—
24	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
25	C₃×C₁₂₇	—	3	3	no	—
26	C₃×C₁₂₇	—	9	3	no	—
27	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	3	yes	—
28	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₁₂₇⋊C₃	—	3	9	no	—
30	C₁₂₇⋊C₃	—	9	6	no	—
31	C₃²⋊H(3,𝔽₃)	—	127	1	no	maximal, Sylow
32	C₁₂₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
33	C₃⁴	—	127	1	no	—
34	C₃×H(3,𝔽₃)	—	127	2	no	—
35	C₃×H(3,𝔽₃)	—	127	9	no	—
36	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
37	C₃³	—	381	9	no	—
38	C₃³	—	381	3	no	—
39	C₃³	—	127	6	no	—
40	C₃³	—	127	3	no	—
41	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
42	C₃³	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
43	H(3,𝔽₃)	—	3	3	no	—
44	H(3,𝔽₃)	—	381	27	no	—
45	H(3,𝔽₃)	—	381	6	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	9	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	381	27	no	—
50	C₃²	—	381	18	no	—
51	C₃²	—	381	6	no	—
52	C₃²	—	381	6	no	—
53	C₃²	—	381	3	no	—
54	C₃²	—	127	9	no	—
55	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	3	yes	complemented
56	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
57	C₃	—	3	3	no	—
58	C₃	—	9	3	no	—
59	C₃	—	1143	6	no	—
60	C₃	—	381	9	no	—
61	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	3	yes	—
62	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator, socle, cosocle	C₃²×C₁₂₇	C₃³
Fitting	C₃×H(3,𝔽₃)×C₁₂₇	C₃
hypercenter	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₃²⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ 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×C₃²⋊(C₁₂₇⋊C₃)	C₃	semidirect
3. C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	semidirect
4. C₃×H(3,𝔽₃)×C₁₂₇	C₃	semidirect
8. C₃²×C₁₂₇⋊C₃	C₃²	semidirect
12. C₃³×C₁₂₇	C₃²	semidirect
17. C₃²×C₁₂₇	H(3,𝔽₃)	semidirect
24. C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	semidirect
32. C₁₂₇	C₃²⋊H(3,𝔽₃)	semidirect
36. C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	semidirect
42. C₃³	C₃×C₁₂₇⋊C₃	semidirect
55. C₃²	C₃²⋊(C₁₂₇⋊C₃)	semidirect

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×C₃²⋊(C₁₂₇⋊C₃)	3
C₃×H(3,𝔽₃)×C₁₂₇	3
C₃²⋊H(3,𝔽₃)	127

Subgroups — (C3×C127⋊C3)⋊H(3,𝔽3) #44525023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃×C₁₂₇⋊C₃)⋊H(3,𝔽₃) #44525023