Subgroups — #44525110

Order 30861.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₁₂₇⋊C₃	C₃	1	2	yes	maximal, complemented
2	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779843	C₃	1	4	yes	maximal, complemented
5	#23779842	C₃	1	2	yes	maximal, complemented
6	#23779841	C₃	1	2	yes	maximal, complemented
7	AΣL(1,𝔽₂₇)×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	C₃²×C₁₂₇⋊C₃	C₃²	1	2	yes	complemented
11	#16812184	C₃²	1	4	yes	complemented
12	#16812183	—	3	2	no	—
13	#16812182	—	3	2	no	—
14	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
15	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
16	C₃²⋊(C₁₂₇⋊C₃)	—	3	1	no	—
17	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	2	yes	—
18	C₉×C₁₂₇⋊C₃	—	3	2	no	—
19	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic, complemented
20	3₋¹⁺²×C₁₂₇	C₃²	1	2	yes	complemented
21	H(3,𝔽₃)×C₁₂₇	C₃²	1	1	yes	characteristic
22	C₃²×C₁₂₇	—	3	1	no	—
23	C₃²×C₁₂₇	—	3	1	no	—
24	C₃²×C₁₂₇	—	9	1	no	—
25	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
26	C₃×C₁₂₇⋊C₃	—	3	2	no	—
27	C₃×C₁₂₇⋊C₃	—	3	2	no	—
28	C₃×C₁₂₇⋊C₃	—	9	1	no	—
29	C₃×C₁₂₇⋊C₃	—	9	3	no	—
30	C₃×C₁₂₇⋊C₃	—	9	2	no	—
31	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
32	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
33	C₉×C₁₂₇	—	3	2	no	—
34	#11765917	—	3	4	no	—
35	C₃×C₁₂₇	—	3	1	no	—
36	C₃×C₁₂₇	—	9	1	no	—
37	C₃×C₁₂₇	—	9	1	no	—
38	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₁₂₇⋊C₃	—	3	1	no	—
40	C₁₂₇⋊C₃	—	3	2	no	—
41	C₁₂₇⋊C₃	—	9	2	no	—
42	C₁₂₇⋊C₃	—	9	2	no	—
43	#6909	—	127	1	no	maximal, Sylow
44	C₁₂₇	#6909	1	1	yes	characteristic, complemented, Sylow
45	C₃×3₋¹⁺²	—	127	2	no	—
46	C₃×H(3,𝔽₃)	—	127	1	no	—
47	C₃×H(3,𝔽₃)	—	127	1	no	—
48	AΣL(1,𝔽₂₇)	—	127	2	no	—
49	AΣL(1,𝔽₂₇)	—	127	4	no	—
50	AΣL(1,𝔽₂₇)	—	127	2	no	—
51	AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
52	C₃³	—	381	1	no	—
53	C₃³	—	127	2	no	—
54	C₃³	—	127	1	no	—
55	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
56	3₋¹⁺²	—	381	2	no	—
57	3₋¹⁺²	—	381	2	no	—
58	3₋¹⁺²	—	127	4	no	—
59	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	2	yes	complemented
60	H(3,𝔽₃)	—	381	2	no	—
61	H(3,𝔽₃)	—	381	1	no	—
62	H(3,𝔽₃)	—	381	2	no	—
63	H(3,𝔽₃)	—	127	2	no	—
64	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
65	C₃×C₉	—	381	2	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	9	1	no	—
69	C₃²	—	1143	2	no	—
70	C₃²	—	1143	1	no	—
71	C₃²	—	1143	3	no	—
72	C₃²	—	381	2	no	—
73	C₃²	—	381	2	no	—
74	C₃²	—	127	2	no	—
75	C₃²	—	127	1	no	—
76	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
77	C₉	—	3	2	no	—
78	C₉	—	381	4	no	—
79	C₃	—	3	1	no	—
80	C₃	—	9	1	no	—
81	C₃	—	9	1	no	—
82	C₃	—	1143	2	no	—
83	C₃	—	1143	2	no	—
84	C₃	—	381	1	no	—
85	C₃	—	381	2	no	—
86	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	AΣL(1,𝔽₂₇)×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6909
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×C₁₂₇⋊C₃	C₃	semidirect
2. C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	semidirect
3. H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	semidirect
4. #23779843	C₃	semidirect
5. #23779842	C₃	semidirect
6. #23779841	C₃	semidirect
7. AΣL(1,𝔽₂₇)×C₁₂₇	C₃	semidirect
10. C₃²×C₁₂₇⋊C₃	C₃²	semidirect
11. #16812184	C₃²	semidirect
19. C₃³×C₁₂₇	C₃²	semidirect
20. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
44. C₁₂₇	#6909	semidirect
51. AΣL(1,𝔽₂₇)	C₁₂₇⋊C₃	semidirect
55. C₃³	C₃×C₁₂₇⋊C₃	semidirect
59. 3₋¹⁺²	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
3₋¹⁺²×C₁₂₇⋊C₃	3
C₃×C₃²⋊(C₁₂₇⋊C₃)	3
H(3,𝔽₃)×C₁₂₇⋊C₃	3
#23779843	3
#23779842	3
#23779841	3
AΣL(1,𝔽₂₇)×C₁₂₇	3
#6909	127