Subgroups — #41008313

Order 26973.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	3	yes	maximal
2	#22582536	C₃	1	1	yes	maximal, characteristic
3	#15688543	—	3	3	no	—
4	#15688542	—	3	3	no	—
5	#15688541	—	3	3	no	—
6	(C₃₇⋊C₃)⋊C₂₇	C₉	1	9	yes	complemented
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	#15688504	—	3	1	no	—
9	#15688504	—	3	1	no	—
10	#15688502	—	3	1	no	—
11	C₉×#66524	C₉	1	2	yes	—
12	C₉²×C₃₇	C₉	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
16	C₃×#66524	—	3	2	no	—
17	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×#66524	3₋¹⁺²	1	1	yes	characteristic
19	C₉×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
20	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
21	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
22	#11758603	—	3	9	no	—
23	#10624465	—	37	1	no	maximal, Sylow
24	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
25	C₃×C₃₇⋊C₃	—	9	1	no	—
26	C₃×C₃₇⋊C₃	C₉⋊C₉	1	3	yes	—
27	C₉×C₃₇	—	9	1	no	—
28	C₉×C₃₇	C₉²	1	1	yes	characteristic
29	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
30	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
31	#66524	—	9	2	no	—
32	#66524	C₃²⋊C₉	1	3	yes	—
33	#66524	C₉⋊C₉	1	3	yes	—
34	C₃×C₂₇⋊C₃	—	37	3	no	—
35	#6887	—	37	1	no	—
36	C₃×C₃₇	—	3	1	no	—
37	C₃×C₃₇	#6855	1	1	yes	characteristic
38	C₃₇⋊C₃	—	3	3	no	—
39	C₃²×C₉	—	37	1	no	—
40	C₂₇⋊C₃	—	111	3	no	—
41	C₂₇⋊C₃	—	111	3	no	—
42	C₂₇⋊C₃	—	111	3	no	—
43	C₃×C₂₇	—	37	9	no	—
44	C₉⋊C₉	—	111	1	no	—
45	C₉⋊C₉	—	111	1	no	—
46	C₃²⋊C₉	—	111	1	no	—
47	C₉²	—	37	2	no	—
48	C₉²	#66526	1	1	yes	characteristic
49	C₃₇	#10624465	1	1	yes	characteristic, complemented, Sylow
50	C₃³	—	37	1	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	37	1	no	—
53	C₃×C₉	—	37	1	no	—
54	C₃×C₉	—	37	3	no	—
55	C₃×C₉	—	37	3	no	—
56	C₃×C₉	—	37	3	no	—
57	C₃×C₉	—	111	2	no	—
58	C₃×C₉	C₃×#66526	1	1	yes	characteristic
59	C₂₇	—	111	9	no	—
60	C₃²	—	37	3	no	—
61	C₃²	—	333	1	no	—
62	C₃²	C₃²⋊#66526	1	1	yes	characteristic
63	C₉	—	9	1	no	—
64	C₉	—	37	3	no	—
65	C₉	—	37	3	no	—
66	C₉	—	333	2	no	—
67	C₉	C₉×#66526	1	1	yes	characteristic
68	C₉	#15688537	1	1	yes	characteristic
69	C₉	#15688538	1	1	yes	characteristic
70	C₃	—	3	1	no	—
71	C₃	—	111	3	no	—
72	C₃	(C₃×#66526)⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₉×#66526
commutator	C₉×C₃₇	C₉²
Frattini	C₃×C₉	C₃×#66526
Fitting	C₉²×C₃₇	C₉
socle	C₃×C₃₇	#6855
prime radical	C₃²×C₃₇⋊C₃	3₋¹⁺²
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₉²	#66526
hypocenter	C₃₇	#10624465
LCS	G ▷ C₉×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
6. (C₃₇⋊C₃)⋊C₂₇	C₉	semidirect
49. C₃₇	#10624465	semidirect

Maximal subgroups

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

subgroup	index
(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	3
#22582536	3
#10624465	37