Subgroups — #44525115

Order 30861.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812183	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×C₁₂₇⋊C₃	C₃	1	2	yes	maximal, complemented
3	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#23779846	C₃	1	4	yes	maximal, complemented
5	#23779845	C₃	1	2	yes	maximal, complemented
6	#23779844	C₃	1	2	yes	maximal, complemented
7	#731×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
9	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
10	#16812184	C₃²	1	4	yes	complemented
11	#16812183	—	3	2	no	—
12	#16812183	—	3	2	no	—
13	#16812183	—	3	1	no	—
14	#16812182	—	3	2	no	—
15	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
16	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	2	yes	—
17	C₃×#11765917	C₃²	1	2	yes	complemented
18	C₉×C₁₂₇⋊C₃	—	3	2	no	—
19	3₋¹⁺²×C₁₂₇	C₃²	1	2	yes	complemented
20	H(3,𝔽₃)×C₁₂₇	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic, complemented
22	C₃²×C₁₂₇	—	3	1	no	—
23	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
24	C₃×C₁₂₇⋊C₃	—	3	2	no	—
25	C₃×C₁₂₇⋊C₃	—	9	1	no	—
26	C₃×C₁₂₇⋊C₃	—	9	3	no	—
27	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
28	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
29	C₉×C₁₂₇	—	3	1	no	—
30	C₉×C₁₂₇	—	3	2	no	—
31	#11765917	—	3	4	no	—
32	#11765917	—	3	2	no	—
33	C₃×C₁₂₇	—	3	1	no	—
34	C₃×C₁₂₇	—	9	1	no	—
35	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₁₂₇⋊C₃	—	3	1	no	—
37	C₁₂₇⋊C₃	—	3	2	no	—
38	C₁₂₇⋊C₃	—	9	2	no	—
39	#6910	—	127	1	no	maximal, Sylow
40	C₁₂₇	#6910	1	1	yes	characteristic, complemented, Sylow
41	C₃×3₋¹⁺²	—	127	2	no	—
42	C₃×3₋¹⁺²	—	127	1	no	—
43	C₃×H(3,𝔽₃)	—	127	1	no	—
44	#731	—	127	4	no	—
45	#731	—	127	2	no	—
46	#731	—	127	2	no	—
47	#731	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
48	C₃³	—	381	1	no	—
49	C₃³	—	127	1	no	—
50	3₋¹⁺²	—	381	2	no	—
51	3₋¹⁺²	—	381	2	no	—
52	3₋¹⁺²	—	381	2	no	—
53	3₋¹⁺²	—	381	1	no	—
54	3₋¹⁺²	—	127	4	no	—
55	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	2	yes	complemented
56	H(3,𝔽₃)	—	381	2	no	—
57	H(3,𝔽₃)	—	127	2	no	—
58	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
59	C₃×C₉	—	381	2	no	—
60	C₃×C₉	—	127	2	no	—
61	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
62	C₃²	—	3	1	no	—
63	C₃²	—	1143	1	no	—
64	C₃²	—	1143	3	no	—
65	C₃²	—	381	2	no	—
66	C₃²	—	127	2	no	—
67	C₃²	—	127	1	no	—
68	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	3	1	no	—
71	C₉	—	381	4	no	—
72	C₉	—	381	2	no	—
73	C₃	—	3	1	no	—
74	C₃	—	9	1	no	—
75	C₃	—	1143	2	no	—
76	C₃	—	381	1	no	—
77	C₃	—	381	2	no	—
78	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	#731×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
prime radical	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃
hypercenter	#731	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6910
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

Complements

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

N	H ≅ G/N	type
1. C₃×#16812183	C₃	semidirect
2. 3₋¹⁺²×C₁₂₇⋊C₃	C₃	semidirect
4. #23779846	C₃	semidirect
5. #23779845	C₃	semidirect
6. #23779844	C₃	semidirect
7. #731×C₁₂₇	C₃	semidirect
10. #16812184	C₃²	semidirect
17. C₃×#11765917	C₃²	semidirect
19. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
21. C₃×C₉×C₁₂₇	C₃²	semidirect
40. C₁₂₇	#6910	semidirect
47. #731	C₁₂₇⋊C₃	semidirect
55. 3₋¹⁺²	C₃×C₁₂₇⋊C₃	semidirect
61. 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
C₃×#16812183	3
3₋¹⁺²×C₁₂₇⋊C₃	3
H(3,𝔽₃)×C₁₂₇⋊C₃	3
#23779846	3
#23779845	3
#23779844	3
#731×C₁₂₇	3
#6910	127