Subgroups — #44525153

Order 30861.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779861	C₃	1	18	yes	maximal, complemented
2	#23779860	C₃	1	9	yes	maximal, complemented
3	C₃×#16812184	C₃	1	2	yes	maximal, complemented
4	3₋¹⁺²×C₁₂₇⋊C₃	C₃	1	9	yes	maximal, complemented
5	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
6	C₃×3₋¹⁺²×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₁₂₇⋊C₃	C₃²	1	3	yes	—
8	#16812184	C₃²	1	6	yes	—
9	#16812184	C₃²	1	6	yes	complemented
10	#16812184	C₃²	1	6	yes	complemented
11	#16812183	C₃²	1	27	yes	complemented
12	#16812182	C₃²	1	27	yes	complemented
13	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	3	yes	—
14	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	6	yes	—
15	C₃×#11765917	C₃²	1	6	yes	complemented
16	C₉×C₁₂₇⋊C₃	C₃²	1	27	yes	complemented
17	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
18	3₋¹⁺²×C₁₂₇	C₃²	1	3	yes	—
19	3₋¹⁺²×C₁₂₇	C₃²	1	6	yes	complemented
20	C₃×C₉×C₁₂₇	C₃²	1	3	yes	complemented
21	C₃²×C₁₂₇	—	9	1	no	—
22	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
23	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
24	C₃²×C₁₂₇	C₃³	1	2	yes	—
25	C₃×C₁₂₇⋊C₃	—	9	3	no	—
26	C₃×C₁₂₇⋊C₃	C₃³	1	9	yes	—
27	C₉×C₁₂₇	C₃³	1	9	yes	—
28	#11765917	C₃³	1	18	yes	—
29	C₃×C₁₂₇	—	3	1	no	—
30	C₃×C₁₂₇	—	3	1	no	—
31	C₃×C₁₂₇	—	3	2	no	—
32	C₃×C₁₂₇	C₃⁴	1	1	yes	characteristic
33	C₁₂₇⋊C₃	—	3	9	no	—
34	3₋¹⁺⁴	—	127	1	no	maximal, Sylow
35	C₁₂₇	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
36	#737	—	127	18	no	—
37	#737	—	127	9	no	—
38	C₃×3₋¹⁺²	—	127	2	no	—
39	C₃×3₋¹⁺²	—	127	9	no	—
40	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
41	C₃×H(3,𝔽₃)	—	127	1	no	—
42	C₃³	—	127	3	no	—
43	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
44	3₋¹⁺²	—	127	6	no	—
45	3₋¹⁺²	—	127	6	no	—
46	3₋¹⁺²	—	127	6	no	—
47	3₋¹⁺²	—	127	27	no	—
48	3₋¹⁺²	—	127	27	no	—
49	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	3	yes	—
50	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	6	yes	complemented
51	H(3,𝔽₃)	—	127	6	no	—
52	H(3,𝔽₃)	—	127	3	no	—
53	C₃×C₉	—	127	6	no	—
54	C₃×C₉	—	127	27	no	—
55	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
56	C₃²	—	9	1	no	—
57	C₃²	—	1143	3	no	—
58	C₃²	—	127	9	no	—
59	C₃²	C₃²×C₁₂₇⋊C₃	1	2	yes	—
60	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
61	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
62	C₉	—	127	18	no	—
63	C₉	C₃²×C₁₂₇⋊C₃	1	9	yes	—
64	C₃	—	3	1	no	—
65	C₃	—	3	2	no	—
66	C₃	—	3	1	no	—
67	C₃	—	381	9	no	—
68	C₃	C₃³×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₁₂₇⋊C₃
commutator, socle, cosocle	C₃×C₁₂₇	C₃⁴
Fitting	C₃×3₋¹⁺²×C₁₂₇	C₃
prime radical	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₁₂₇⋊C₃
hypocenter	C₁₂₇	3₋¹⁺⁴
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #23779861	C₃	semidirect
2. #23779860	C₃	semidirect
3. C₃×#16812184	C₃	semidirect
4. 3₋¹⁺²×C₁₂₇⋊C₃	C₃	semidirect
6. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
9. #16812184	C₃²	semidirect
10. #16812184	C₃²	semidirect
11. #16812183	C₃²	semidirect
12. #16812182	C₃²	semidirect
15. C₃×#11765917	C₃²	semidirect
16. C₉×C₁₂₇⋊C₃	C₃²	semidirect
19. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
20. C₃×C₉×C₁₂₇	C₃²	semidirect
35. C₁₂₇	3₋¹⁺⁴	semidirect
40. C₃×3₋¹⁺²	C₁₂₇⋊C₃	semidirect
50. 3₋¹⁺²	C₃×C₁₂₇⋊C₃	semidirect
55. 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
#23779861	3
#23779860	3
C₃×#16812184	3
3₋¹⁺²×C₁₂₇⋊C₃	3
C₃×C₃²⋊(C₁₂₇⋊C₃)	3
C₃×3₋¹⁺²×C₁₂₇	3
3₋¹⁺⁴	127