Subgroups — #41006358

Order 26973.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582536	C₃	1	1	yes	maximal, characteristic
2	#22582488	C₃	1	6	yes	maximal, complemented
3	#22582465	C₃	1	3	yes	maximal
4	#22582464	C₃	1	1	yes	maximal, characteristic, complemented
5	C₉×#11758601	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃₇×#6864	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×#11758601	C₃²	1	3	yes	—
9	C₃×#11758601	C₃²	1	3	yes	—
10	#15688504	C₉	1	6	yes	—
11	#15688502	C₉	1	3	yes	—
12	C₉×#66524	C₃²	1	2	yes	—
13	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
14	C₉²×C₃₇	C₃²	1	1	yes	characteristic
15	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
17	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
18	C₃×#66524	C₃×C₉	1	6	yes	—
19	C₃×#66524	C₃×C₉	1	2	yes	—
20	C₉×C₃₇⋊C₃	—	3	3	no	—
21	C₂₇×C₃₇	—	3	3	no	—
22	#11758601	—	3	3	no	—
23	#11758601	—	3	3	no	—
24	#10624713	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
26	C₃×C₃₇⋊C₃	—	3	1	no	—
27	C₃×C₃₇⋊C₃	—	3	3	no	—
28	C₉×C₃₇	—	3	3	no	—
29	C₉×C₃₇	#737	1	3	yes	—
30	#66524	—	3	2	no	—
31	#66524	—	3	6	no	—
32	#6887	—	37	1	no	—
33	C₉⋊C₂₇	—	37	6	no	—
34	C₃²⋊C₂₇	—	37	3	no	—
35	#6864	—	37	1	no	—
36	#6864	C₃₇⋊C₃	1	1	yes	characteristic, complemented
37	C₉×C₂₇	—	37	1	no	—
38	C₃×C₃₇	#6887	1	1	yes	characteristic
39	C₃×C₃₇	#6903	1	3	yes	—
40	C₃₇⋊C₃	—	9	1	no	—
41	C₃²×C₉	—	37	1	no	—
42	C₃×C₂₇	—	37	3	no	—
43	C₃×C₂₇	—	37	3	no	—
44	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
45	C₉⋊C₉	—	37	6	no	—
46	C₃²⋊C₉	—	37	3	no	—
47	C₉²	—	37	2	no	—
48	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	C₃₇	#10624713	1	1	yes	characteristic, complemented, Sylow
50	C₃³	—	37	1	no	—
51	C₃×C₉	—	37	6	no	—
52	C₃×C₉	—	37	2	no	—
53	C₃×C₉	—	111	3	no	—
54	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
55	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
56	C₂₇	—	3	3	no	—
57	C₂₇	—	111	3	no	—
58	C₂₇	—	111	3	no	—
59	C₃²	—	111	1	no	—
60	C₃²	—	111	3	no	—
61	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
62	C₉	—	3	3	no	—
63	C₉	—	111	2	no	—
64	C₉	—	111	6	no	—
65	C₉	#15688531	1	3	yes	—
66	C₃	—	333	1	no	—
67	C₃	#22582536	1	1	yes	characteristic
68	C₃	#22582614	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×#6864	C₃
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6864	C₃₇⋊C₃
hypocenter	C₃₇	#10624713
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ #6864

Complements

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

N	H ≅ G/N	type
2. #22582488	C₃	semidirect
4. #22582464	C₃	semidirect
5. C₉×#11758601	C₃	semidirect
6. C₃₇×#6864	C₃	semidirect
36. #6864	C₃₇⋊C₃	semidirect
49. C₃₇	#10624713	semidirect

Maximal subgroups

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

subgroup	index
#22582536	3
#22582488	3
#22582465	3
#22582464	3
C₉×#11758601	3
C₃₇×#6864	3
#10624713	37