Subgroups — #41004786

Order 26973.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582490	C₃	1	1	yes	maximal, characteristic
2	#22582489	C₃	1	1	yes	maximal, characteristic
3	#22582464	C₃	1	1	yes	maximal, characteristic
4	C₃₇×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758601	—	3	1	no	—
6	C₃×#11758601	—	3	1	no	—
7	C₃×#11758601	C₉	1	1	yes	characteristic
8	C₃×#11758601	C₉	1	1	yes	characteristic
9	C₃×#11758601	C₉	1	1	yes	characteristic
10	C₃×C₂₇×C₃₇	—	3	1	no	—
11	C₉²×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
14	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
15	C₃×C₉×C₃₇	3₋¹⁺²	1	1	yes	characteristic
16	C₂₇×C₃₇	—	9	1	no	—
17	#11758601	—	3	1	no	—
18	#11758601	—	3	1	no	—
19	#11758601	—	9	1	no	—
20	#11758601	—	9	1	no	—
21	#11758601	C₂₇	1	3	yes	complemented
22	#10624462	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	C₉⋊C₉	1	1	yes	characteristic
24	C₉×C₃₇	—	3	1	no	—
25	C₉×C₃₇	—	3	1	no	—
26	C₉×C₃₇	—	3	1	no	—
27	C₉×C₃₇	C₃×C₂₇	1	1	yes	characteristic
28	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
29	C₉×C₃₇	C₂₇⋊C₃	1	1	yes	characteristic
30	C₉⋊C₂₇	—	37	1	no	—
31	C₉⋊C₂₇	—	37	1	no	—
32	C₉⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic
33	#6864	—	37	1	no	—
34	C₃×C₃₇	C₉⋊C₂₇	1	1	yes	characteristic
35	C₃×C₃₇	#6875	1	1	yes	characteristic
36	C₃×C₃₇	#6875	1	1	yes	characteristic
37	C₃×C₃₇	#6875	1	1	yes	characteristic
38	C₃×C₂₇	—	3	1	no	—
39	C₃×C₂₇	—	37	1	no	—
40	C₃×C₂₇	—	37	1	no	—
41	C₃×C₂₇	—	37	1	no	—
42	C₃×C₂₇	—	111	1	no	—
43	C₃×C₂₇	—	111	1	no	—
44	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
45	C₃₇	#10624462	1	1	yes	characteristic, complemented, Sylow
46	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
47	C₃×C₉	#11758605	1	1	yes	characteristic
48	C₃×C₉	#11758606	1	1	yes	characteristic
49	C₃×C₉	#11758608	1	1	yes	characteristic
50	C₂₇	—	9	1	no	—
51	C₂₇	—	37	3	no	—
52	C₂₇	—	333	1	no	—
53	C₂₇	—	333	1	no	—
54	C₂₇	—	111	1	no	—
55	C₂₇	—	111	1	no	—
56	C₃²	#15688504	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	3	1	no	—
59	C₉	—	3	1	no	—
60	C₉	C₂₇×C₃₇⋊C₃	1	1	yes	characteristic
61	C₉	#15688509	1	1	yes	characteristic
62	C₉	#15688510	1	1	yes	characteristic
63	C₃	#22582488	1	1	yes	characteristic
64	C₃	#22582491	1	1	yes	characteristic
65	C₃	#22582494	1	1	yes	characteristic
66	C₃	#22582495	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688504
commutator	C₉×C₃₇	C₃×C₂₇
Frattini	C₉²	C₃×C₃₇⋊C₃
Fitting	C₃₇×C₉⋊C₂₇	C₃
socle, prime radical	C₃²×C₃₇	C₉⋊C₉
cosocle	C₉²×C₃₇	C₃²
hypercenter	C₉⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624462
LCS	G ▷ C₉×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉² ◁ C₉⋊C₂₇

Complements

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

N	H ≅ G/N	type
21. #11758601	C₂₇	semidirect
45. C₃₇	#10624462	semidirect

Maximal subgroups

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

subgroup	index
#22582490	3
#22582489	3
#22582464	3
C₃₇×C₉⋊C₂₇	3
#10624462	37