Subgroups — #44525151

Order 30861.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779861	C₃	1	16	yes	maximal, complemented
2	#23779860	C₃	1	8	yes	maximal, complemented
3	#23779859	C₃	1	2	yes	maximal, complemented
4	C₃×#16812182	C₃	1	4	yes	maximal, complemented
5	3₋¹⁺²×C₁₂₇⋊C₃	C₃	1	8	yes	maximal, complemented
6	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
7	#737×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₂₇⋊C₃	C₃²	1	4	yes	—
9	#16812184	C₃²	1	16	yes	complemented
10	#16812183	C₃²	1	24	yes	complemented
11	#16812182	C₃²	1	8	yes	—
12	#16812182	C₃²	1	1	yes	characteristic
13	#16812182	C₃²	1	8	yes	complemented
14	#16812182	C₃²	1	16	yes	complemented
15	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	8	yes	—
16	C₃×#11765917	C₃²	1	8	yes	complemented
17	C₉×C₁₂₇⋊C₃	C₃²	1	24	yes	complemented
18	3₋¹⁺²×C₁₂₇	C₃²	1	8	yes	complemented
19	H(3,𝔽₃)×C₁₂₇	C₃²	1	1	yes	characteristic
20	C₃×C₉×C₁₂₇	C₃²	1	4	yes	complemented
21	C₃²×C₁₂₇	C₃³	1	4	yes	—
22	C₃×C₁₂₇⋊C₃	—	9	4	no	—
23	C₃×C₁₂₇⋊C₃	C₃³	1	8	yes	—
24	C₃×C₁₂₇⋊C₃	C₃³	1	1	yes	characteristic
25	C₉×C₁₂₇	C₃³	1	1	yes	characteristic
26	C₉×C₁₂₇	C₃³	1	8	yes	—
27	#11765917	C₃³	1	16	yes	—
28	#11765917	C₃³	1	2	yes	—
29	C₃×C₁₂₇	—	3	4	no	—
30	C₃×C₁₂₇	C₃⁴	1	1	yes	characteristic
31	C₁₂₇⋊C₃	—	3	1	no	—
32	C₁₂₇⋊C₃	—	3	8	no	—
33	3₋¹⁺⁴	—	127	1	no	maximal, Sylow
34	C₁₂₇	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
35	#737	—	127	16	no	—
36	#737	—	127	2	no	—
37	#737	—	127	8	no	—
38	#737	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
39	C₃×3₋¹⁺²	—	127	8	no	—
40	C₃×3₋¹⁺²	—	127	4	no	—
41	C₃×H(3,𝔽₃)	—	127	1	no	—
42	C₃³	—	127	4	no	—
43	3₋¹⁺²	—	127	16	no	—
44	3₋¹⁺²	—	127	8	no	—
45	3₋¹⁺²	—	127	1	no	—
46	3₋¹⁺²	—	127	16	no	—
47	3₋¹⁺²	—	127	8	no	—
48	3₋¹⁺²	—	127	24	no	—
49	3₋¹⁺²	C₃×C₁₂₇⋊C₃	1	8	yes	complemented
50	H(3,𝔽₃)	—	127	8	no	—
51	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
52	C₃×C₉	—	127	8	no	—
53	C₃×C₉	—	127	24	no	—
54	C₃×C₉	C₃×C₁₂₇⋊C₃	1	4	yes	complemented
55	C₃²	—	1143	4	no	—
56	C₃²	—	127	8	no	—
57	C₃²	—	127	1	no	—
58	C₃²	C₃²×C₁₂₇⋊C₃	1	4	yes	—
59	C₉	—	127	2	no	—
60	C₉	—	127	16	no	—
61	C₉	C₃²×C₁₂₇⋊C₃	1	8	yes	—
62	C₉	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
63	C₃	—	3	4	no	—
64	C₃	—	381	1	no	—
65	C₃	—	381	8	no	—
66	C₃	C₃³×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃³×C₁₂₇⋊C₃
commutator, socle, cosocle	C₃×C₁₂₇	C₃⁴
Fitting	#737×C₁₂₇	C₃
prime radical	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃
hypercenter	#737	C₁₂₇⋊C₃
hypocenter	C₁₂₇	3₋¹⁺⁴
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ #737

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. #23779859	C₃	semidirect
4. C₃×#16812182	C₃	semidirect
5. 3₋¹⁺²×C₁₂₇⋊C₃	C₃	semidirect
7. #737×C₁₂₇	C₃	semidirect
9. #16812184	C₃²	semidirect
10. #16812183	C₃²	semidirect
13. #16812182	C₃²	semidirect
14. #16812182	C₃²	semidirect
16. C₃×#11765917	C₃²	semidirect
17. C₉×C₁₂₇⋊C₃	C₃²	semidirect
18. 3₋¹⁺²×C₁₂₇	C₃²	semidirect
20. C₃×C₉×C₁₂₇	C₃²	semidirect
34. C₁₂₇	3₋¹⁺⁴	semidirect
38. #737	C₁₂₇⋊C₃	semidirect
49. 3₋¹⁺²	C₃×C₁₂₇⋊C₃	semidirect
54. 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
#23779859	3
C₃×#16812182	3
3₋¹⁺²×C₁₂₇⋊C₃	3
H(3,𝔽₃)×C₁₂₇⋊C₃	3
#737×C₁₂₇	3
3₋¹⁺⁴	127