Subgroups — #44525086

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

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

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. C₂₇×C₁₂₇⋊C₃	C₃	semidirect
6. C₃×C₂₇×C₁₂₇	C₃	semidirect
27. C₁₂₇	#6903	semidirect
33. C₃×C₂₇	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
C₂₇×C₁₂₇⋊C₃	3
C₃×C₂₇×C₁₂₇	3
#6903	127