Subgroups — #44525025

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
2	C₃×#16812184	C₃	1	2	yes	maximal, complemented
3	#23779831	C₃	1	9	yes	maximal, complemented
4	C₃×3₋¹⁺²×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	#16812184	—	3	6	no	—
9	C₃×#11765917	C₃²	1	6	yes	complemented
10	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
11	3₋¹⁺²×C₁₂₇	—	3	3	no	—
12	C₃×C₉×C₁₂₇	C₃²	1	3	yes	complemented
13	C₃²×C₁₂₇	—	3	3	no	—
14	C₃²×C₁₂₇	H(3,𝔽₃)	1	3	yes	—
15	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
16	C₃×C₁₂₇⋊C₃	—	3	27	no	—
17	C₃×C₁₂₇⋊C₃	—	3	3	no	—
18	C₃×C₁₂₇⋊C₃	—	3	6	no	—
19	C₃×C₁₂₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
20	C₉×C₁₂₇	—	3	3	no	—
21	#11765917	—	3	6	no	—
22	C₃×C₁₂₇	—	3	3	no	—
23	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₁₂₇	C₃×3₋¹⁺²	1	3	yes	—
25	C₁₂₇⋊C₃	—	3	9	no	—
26	#6891	—	127	1	no	maximal, Sylow
27	C₁₂₇	#6891	1	1	yes	characteristic, complemented, Sylow
28	C₃⁴	—	127	1	no	—
29	C₃×3₋¹⁺²	—	127	2	no	—
30	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
31	C₃²⋊C₉	—	127	9	no	—
32	C₃³	—	381	9	no	—
33	C₃³	—	381	3	no	—
34	C₃³	—	127	3	no	—
35	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
36	3₋¹⁺²	—	3	3	no	—
37	3₋¹⁺²	—	381	6	no	—
38	C₃×C₉	—	127	6	no	—
39	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
40	C₃²	—	3	3	no	—
41	C₃²	—	381	27	no	—
42	C₃²	—	381	3	no	—
43	C₃²	—	381	6	no	—
44	C₃²	—	127	9	no	—
45	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	3	yes	—
46	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
47	C₉	—	3	3	no	—
48	C₉	—	381	6	no	—
49	C₃	—	3	3	no	—
50	C₃	—	381	9	no	—
51	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
52	C₃	3₋¹⁺²×C₁₂₇⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator, socle, cosocle	C₃²×C₁₂₇	C₃³
Fitting	C₃×3₋¹⁺²×C₁₂₇	C₃
prime radical	C₃³×C₁₂₇⋊C₃	C₃
hypercenter	C₃×3₋¹⁺²	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6891
LCS	G ▷ C₃²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₃×#16812184	C₃	semidirect
3. #23779831	C₃	semidirect
4. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
9. C₃×#11765917	C₃²	semidirect
12. C₃×C₉×C₁₂₇	C₃²	semidirect
19. C₃×C₁₂₇⋊C₃	3₋¹⁺²	semidirect
27. C₁₂₇	#6891	semidirect
30. C₃×3₋¹⁺²	C₁₂₇⋊C₃	semidirect
39. C₃×C₉	C₃×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×#16812184	3
#23779831	3
C₃×3₋¹⁺²×C₁₂₇	3
#6891	127