Subgroups — #44525105

Order 30861.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779859	C₃	1	3	yes	maximal, complemented
2	C₃×C₉×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#23779850	C₃	1	2	yes	maximal, complemented
4	#23779844	C₃	1	3	yes	maximal, complemented
5	#23779841	C₃	1	3	yes	maximal, complemented
6	#732×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
8	#16812184	—	3	6	no	—
9	#16812184	C₃²	1	6	yes	—
10	C₃×#11765917	—	3	3	no	—
11	C₃×#11765917	C₃²	1	1	yes	characteristic
12	C₃×#11765917	C₃²	1	1	yes	characteristic
13	C₉×C₁₂₇⋊C₃	—	3	1	no	—
14	C₉×C₁₂₇⋊C₃	—	3	2	no	—
15	H(3,𝔽₃)×C₁₂₇	C₃²	1	3	yes	—
16	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
17	C₃²×C₁₂₇	—	3	3	no	—
18	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
19	C₃×C₁₂₇⋊C₃	—	3	1	no	—
20	C₃×C₁₂₇⋊C₃	—	3	3	no	—
21	C₉×C₁₂₇	—	3	1	no	—
22	#11765917	—	3	1	no	—
23	#11765917	—	3	6	no	—
24	#11765917	H(3,𝔽₃)	1	1	yes	characteristic
25	#11765917	H(3,𝔽₃)	1	2	yes	—
26	C₃×C₁₂₇	—	3	1	no	—
27	C₃×C₁₂₇	—	9	3	no	—
28	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₁₂₇⋊C₃	—	3	3	no	—
30	#6908	—	127	1	no	maximal, Sylow
31	C₁₂₇	#6908	1	1	yes	characteristic, complemented, Sylow
32	#737	—	127	3	no	—
33	C₃²×C₉	—	127	1	no	—
34	#733	—	127	2	no	—
35	#732	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
36	#731	—	127	3	no	—
37	AΣL(1,𝔽₂₇)	—	127	3	no	—
38	C₃³	—	127	1	no	—
39	3₋¹⁺²	—	381	6	no	—
40	3₋¹⁺²	—	127	6	no	—
41	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	3	yes	—
42	C₃×C₉	—	381	1	no	—
43	C₃×C₉	—	381	2	no	—
44	C₃×C₉	—	381	3	no	—
45	C₃×C₉	—	127	1	no	—
46	C₃×C₉	—	127	1	no	—
47	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
48	C₃²	—	3	3	no	—
49	C₃²	—	381	3	no	—
50	C₃²	—	381	1	no	—
51	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
52	C₉	—	3	1	no	—
53	C₉	—	381	1	no	—
54	C₉	—	381	6	no	—
55	C₉	—	127	1	no	—
56	C₉	—	127	2	no	—
57	C₃	—	3	1	no	—
58	C₃	—	9	3	no	—
59	C₃	—	381	3	no	—
60	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	#732×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	#732	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6908
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. #23779859	C₃	semidirect
2. C₃×C₉×C₁₂₇⋊C₃	C₃	semidirect
3. #23779850	C₃	semidirect
4. #23779844	C₃	semidirect
5. #23779841	C₃	semidirect
6. #732×C₁₂₇	C₃	semidirect
31. C₁₂₇	#6908	semidirect
35. #732	C₁₂₇⋊C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23779859	3
C₃×C₉×C₁₂₇⋊C₃	3
#23779850	3
#23779844	3
#23779841	3
#732×C₁₂₇	3
#6908	127