Subgroups — #44525088

Order 30861.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779860	C₃	1	1	yes	maximal, characteristic
2	#23779840	C₃	1	1	yes	maximal, characteristic, complemented
3	#23779839	C₃	1	3	yes	maximal, complemented
4	#23779838	C₃	1	3	yes	maximal, complemented
5	C₃×#16812178	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂₇×C₁₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
7	C₂₇⋊C₃×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	#16812184	C₉	1	2	yes	—
9	#16812183	C₉	1	3	yes	—
10	#16812182	C₉	1	3	yes	—
11	C₃²⋊(C₁₂₇⋊C₃)	C₉	1	1	yes	characteristic
12	C₉×C₁₂₇⋊C₃	C₃²	1	3	yes	—
13	#16812178	C₃²	1	3	yes	—
14	#16812178	C₃²	1	1	yes	characteristic
15	#16812178	C₃²	1	1	yes	characteristic
16	#16812178	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
18	C₂₇×C₁₂₇	C₃²	1	3	yes	—
19	C₃²×C₁₂₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₁₂₇⋊C₃	C₃×C₉	1	3	yes	—
21	C₉×C₁₂₇	C₃×C₉	1	1	yes	characteristic
22	C₉×C₁₂₇	C₃×C₉	1	1	yes	characteristic
23	C₉×C₁₂₇	C₃³	1	1	yes	characteristic
24	#11765917	C₃×C₉	1	6	yes	—
25	C₃×C₁₂₇	—	3	1	no	—
26	C₃×C₁₂₇	C₃²×C₉	1	1	yes	characteristic
27	C₁₂₇⋊C₃	—	3	3	no	—
28	#6903	—	127	1	no	maximal, Sylow
29	C₁₂₇	#6903	1	1	yes	characteristic, complemented, Sylow
30	#737	—	127	1	no	—
31	C₂₇⋊C₃	—	127	1	no	—
32	C₂₇⋊C₃	—	127	3	no	—
33	C₂₇⋊C₃	—	127	3	no	—
34	C₂₇⋊C₃	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
35	C₃×C₂₇	—	127	1	no	—
36	C₃×C₂₇	—	127	3	no	—
37	3₋¹⁺²	—	127	2	no	—
38	3₋¹⁺²	—	127	3	no	—
39	3₋¹⁺²	—	127	3	no	—
40	H(3,𝔽₃)	—	127	1	no	—
41	C₃×C₉	—	127	3	no	—
42	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
43	C₂₇	—	127	1	no	—
44	C₂₇	—	127	3	no	—
45	C₂₇	—	127	1	no	—
46	C₂₇	—	127	1	no	—
47	C₂₇	C₃×C₁₂₇⋊C₃	1	3	yes	—
48	C₃²	—	127	3	no	—
49	C₃²	C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
50	C₉	—	127	6	no	—
51	C₉	C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
52	C₉	C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
53	C₉	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
54	C₃	—	3	1	no	—
55	C₃	—	381	3	no	—
56	C₃	C₃×C₉×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₉	C₃²×C₁₂₇⋊C₃
commutator, socle	C₃×C₁₂₇	C₃²×C₉
Fitting	C₂₇⋊C₃×C₁₂₇	C₃
prime radical	C₃²⋊(C₁₂₇⋊C₃)	C₉
cosocle	C₉×C₁₂₇	C₃³
hypercenter	C₂₇⋊C₃	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6903
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₉ ◁ C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
2. #23779840	C₃	semidirect
3. #23779839	C₃	semidirect
4. #23779838	C₃	semidirect
5. C₃×#16812178	C₃	semidirect
6. C₂₇×C₁₂₇⋊C₃	C₃	semidirect
7. C₂₇⋊C₃×C₁₂₇	C₃	semidirect
29. C₁₂₇	#6903	semidirect
34. C₂₇⋊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
#23779860	3
#23779840	3
#23779839	3
#23779838	3
C₃×#16812178	3
C₂₇×C₁₂₇⋊C₃	3
C₂₇⋊C₃×C₁₂₇	3
#6903	127