Subgroups — #41004944

Order 26973.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688500	C₃	1	2	yes	maximal, complemented
3	C₃×C₃₇×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₃₇⋊C₃	C₉	1	1	yes	characteristic
5	C₃×#11758601	C₉	1	2	yes	—
6	C₂₇×C₃₇⋊C₃	—	3	1	no	—
7	C₂₇×C₃₇⋊C₃	—	3	1	no	—
8	C₂₇×C₃₇⋊C₃	—	3	1	no	—
9	#15688500	—	3	2	no	—
10	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₈₁	—	3	1	no	—
12	C₃²×C₃₇⋊C₃	C₂₇	1	1	yes	characteristic
13	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
14	C₃×#66524	C₂₇	1	2	yes	—
15	C₉×C₃₇⋊C₃	—	3	1	no	—
16	C₉×C₃₇⋊C₃	—	3	1	no	—
17	C₉×C₃₇⋊C₃	—	3	1	no	—
18	C₂₇×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
20	C₂₇×C₃₇	3₋¹⁺²	1	1	yes	characteristic
21	#11758601	—	3	2	no	—
22	C₃²⋊C₈₁	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₃×C₂₇	1	1	yes	characteristic
24	C₃×C₃₇⋊C₃	—	3	1	no	—
25	C₃×C₃₇⋊C₃	—	3	1	no	—
26	C₃×C₃₇⋊C₃	—	3	1	no	—
27	C₃×C₃₇⋊C₃	C₈₁	1	3	yes	complemented
28	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
29	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
30	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
31	#66524	—	3	2	no	—
32	C₃²×C₂₇	—	37	1	no	—
33	C₃×C₈₁	—	37	2	no	—
34	C₃×C₈₁	C₃₇⋊C₃	1	1	yes	characteristic, complemented
35	C₃×C₃₇	C₃²⋊C₂₇	1	1	yes	characteristic
36	C₃×C₃₇	C₃×C₈₁	1	1	yes	characteristic
37	C₃×C₃₇	C₈₁⋊C₃	1	1	yes	characteristic
38	C₃×C₃₇	C₈₁⋊C₃	1	1	yes	characteristic
39	C₃₇⋊C₃	—	3	3	no	—
40	C₃²×C₉	—	37	1	no	—
41	C₃×C₂₇	—	37	2	no	—
42	C₃×C₂₇	—	111	1	no	—
43	C₃×C₂₇	—	111	1	no	—
44	C₃×C₂₇	—	111	1	no	—
45	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
46	C₈₁	—	3	1	no	—
47	C₈₁	—	111	2	no	—
48	C₃₇	C₃²⋊C₈₁	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	37	1	no	—
50	C₃×C₉	—	37	2	no	—
51	C₃×C₉	—	111	1	no	—
52	C₃×C₉	—	111	1	no	—
53	C₃×C₉	—	111	1	no	—
54	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
55	C₂₇	—	111	2	no	—
56	C₂₇	#11758605	1	1	yes	characteristic
57	C₂₇	#11758606	1	1	yes	characteristic
58	C₂₇	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
59	C₃²	—	37	3	no	—
60	C₃²	—	111	1	no	—
61	C₃²	—	111	1	no	—
62	C₃²	—	111	1	no	—
63	C₃²	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
64	C₉	—	111	2	no	—
65	C₉	#15688502	1	1	yes	characteristic
66	C₉	#15688509	1	1	yes	characteristic
67	C₉	#15688510	1	1	yes	characteristic
68	C₃	—	111	3	no	—
69	C₃	#22582465	1	1	yes	characteristic
70	C₃	C₈₁×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃	#22582498	1	1	yes	characteristic
72	C₃	#22582499	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₂₇	C₃×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃×C₈₁
Fitting	C₃×C₃₇×C₈₁	C₃
socle	C₃²×C₃₇	C₃×C₂₇
prime radical	C₃²×C₃₇⋊C₃	C₂₇
cosocle	C₃×C₂₇×C₃₇	C₃²
hypercenter	C₃×C₈₁	C₃₇⋊C₃
hypocenter	C₃₇	C₃²⋊C₈₁
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₂₇ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. C₃×#15688500	C₃	semidirect
3. C₃×C₃₇×C₈₁	C₃	semidirect
27. C₃×C₃₇⋊C₃	C₈₁	semidirect
34. C₃×C₈₁	C₃₇⋊C₃	semidirect
48. C₃₇	C₃²⋊C₈₁	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂₇×C₃₇⋊C₃	3
C₃×#15688500	3
C₃×C₃₇×C₈₁	3
C₃²⋊C₈₁	37