Subgroups — #44524939

Order 30861.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#16812178	C₃	1	2	yes	maximal, complemented
3	C₃×C₂₇×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₂₇⋊C₃	C₉	1	1	yes	characteristic
5	C₃×#11765917	C₉	1	2	yes	—
6	C₉×C₁₂₇⋊C₃	—	3	1	no	—
7	C₉×C₁₂₇⋊C₃	—	3	1	no	—
8	C₉×C₁₂₇⋊C₃	—	3	1	no	—
9	#16812178	—	3	2	no	—
10	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
11	C₂₇×C₁₂₇	—	3	1	no	—
12	C₃²×C₁₂₇	C₃×C₉	1	1	yes	characteristic
13	C₃×C₁₂₇⋊C₃	—	3	1	no	—
14	C₃×C₁₂₇⋊C₃	—	3	1	no	—
15	C₃×C₁₂₇⋊C₃	—	3	1	no	—
16	C₃×C₁₂₇⋊C₃	C₂₇	1	3	yes	complemented
17	C₉×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₉×C₁₂₇	3₋¹⁺²	1	1	yes	characteristic
19	C₉×C₁₂₇	3₋¹⁺²	1	1	yes	characteristic
20	#11765917	—	3	2	no	—
21	C₃×C₁₂₇	C₃²⋊C₉	1	1	yes	characteristic
22	C₃×C₁₂₇	C₃×C₂₇	1	1	yes	characteristic
23	C₃×C₁₂₇	C₂₇⋊C₃	1	1	yes	characteristic
24	C₃×C₁₂₇	C₂₇⋊C₃	1	1	yes	characteristic
25	C₁₂₇⋊C₃	—	3	3	no	—
26	C₃²⋊C₂₇	—	127	1	no	maximal, Sylow
27	C₁₂₇	C₃²⋊C₂₇	1	1	yes	characteristic, complemented, Sylow
28	C₃²×C₉	—	127	1	no	—
29	C₃×C₂₇	—	127	2	no	—
30	C₃×C₂₇	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
31	C₃³	—	127	1	no	—
32	C₃×C₉	—	381	1	no	—
33	C₃×C₉	—	381	1	no	—
34	C₃×C₉	—	381	1	no	—
35	C₃×C₉	—	127	2	no	—
36	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
37	C₂₇	—	3	1	no	—
38	C₂₇	—	381	2	no	—
39	C₃²	—	381	1	no	—
40	C₃²	—	381	1	no	—
41	C₃²	—	381	1	no	—
42	C₃²	—	127	3	no	—
43	C₃²	C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
44	C₉	—	381	2	no	—
45	C₉	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
46	C₉	#16812182	1	1	yes	characteristic
47	C₉	#16812183	1	1	yes	characteristic
48	C₃	—	381	3	no	—
49	C₃	#23779831	1	1	yes	characteristic
50	C₃	C₂₇×C₁₂₇⋊C₃	1	1	yes	characteristic
51	C₃	#23779838	1	1	yes	characteristic
52	C₃	#23779839	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×#16812178	C₃	semidirect
3. C₃×C₂₇×C₁₂₇	C₃	semidirect
16. C₃×C₁₂₇⋊C₃	C₂₇	semidirect
27. C₁₂₇	C₃²⋊C₂₇	semidirect
30. 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
C₃×C₉×C₁₂₇⋊C₃	3
C₃×#16812178	3
C₃×C₂₇×C₁₂₇	3
C₃²⋊C₂₇	127