Subgroups — #41008322

Order 26973.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	2	yes	maximal
2	#22582722	C₃	1	1	yes	maximal, characteristic
3	C₃×C₂₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#15688543	—	3	2	no	—
5	#15688542	—	3	2	no	—
6	#15688541	C₉	1	6	yes	complemented
7	(C₃₇⋊C₃)⋊C₂₇	—	3	2	no	—
8	#15688538	—	3	1	no	—
9	#15688537	—	3	1	no	—
10	C₃²⋊#66526	C₉	1	3	yes	—
11	C₉×#66526	—	3	1	no	—
12	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
13	C₃×#11758601	C₉	1	2	yes	complemented
14	C₂₇×C₃₇⋊C₃	—	3	1	no	—
15	C₂₇×C₃₇⋊C₃	—	3	2	no	—
16	C₃×C₂₇×C₃₇	C₉	1	1	yes	characteristic, complemented
17	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
18	C₃×#66526	—	3	3	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×#66524	C₃×C₉	1	2	yes	—
21	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
23	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
24	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
25	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
26	C₉×C₃₇⋊C₃	3₋¹⁺²	1	1	yes	characteristic
27	#11758603	—	3	6	no	—
28	C₂₇×C₃₇	—	3	1	no	—
29	#11758601	—	3	2	no	—
30	#10624469	—	37	1	no	maximal, Sylow
31	C₃²×C₃₇	C₉²	1	1	yes	characteristic
32	C₃×C₃₇⋊C₃	—	3	3	no	—
33	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	C₉⋊C₉	1	2	yes	—
35	#66526	—	9	3	no	—
36	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
37	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
38	C₉×C₃₇	C₉⋊C₉	1	1	yes	characteristic
39	#66524	C₃²⋊C₉	1	2	yes	—
40	#66524	C₉⋊C₉	1	2	yes	—
41	#66524	C₉⋊C₉	1	2	yes	—
42	C₃×C₂₇⋊C₃	—	37	2	no	—
43	C₃²×C₂₇	—	37	1	no	—
44	#6887	—	37	1	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	#6855	1	1	yes	characteristic
47	C₃₇⋊C₃	—	3	2	no	—
48	C₃₇⋊C₃	#6872	1	3	yes	—
49	C₃²×C₉	—	37	1	no	—
50	C₂₇⋊C₃	—	37	6	no	—
51	C₂₇⋊C₃	—	111	2	no	—
52	C₂₇⋊C₃	—	111	2	no	—
53	C₃×C₂₇	—	37	2	no	—
54	C₃×C₂₇	—	111	1	no	—
55	C₃×C₂₇	—	111	2	no	—
56	C₃×C₂₇	—	111	2	no	—
57	C₃×C₂₇	#66526	1	1	yes	characteristic, complemented
58	C₉⋊C₉	—	111	1	no	—
59	C₉⋊C₉	—	111	1	no	—
60	C₃²⋊C₉	—	37	3	no	—
61	C₉²	—	111	1	no	—
62	C₃₇	#10624469	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	37	1	no	—
64	C₃×C₉	—	37	2	no	—
65	C₃×C₉	—	37	1	no	—
66	C₃×C₉	—	37	2	no	—
67	C₃×C₉	—	37	1	no	—
68	C₃×C₉	—	37	2	no	—
69	C₃×C₉	—	37	1	no	—
70	C₃×C₉	—	37	2	no	—
71	C₃×C₉	—	111	3	no	—
72	C₃×C₉	C₃×#66526	1	1	yes	characteristic
73	C₂₇	—	3	1	no	—
74	C₂₇	—	111	6	no	—
75	C₂₇	—	111	2	no	—
76	C₃²	—	37	1	no	—
77	C₃²	—	37	2	no	—
78	C₃²	—	111	3	no	—
79	C₃²	C₉×#66526	1	1	yes	characteristic
80	C₉	—	37	2	no	—
81	C₉	—	37	2	no	—
82	C₉	—	37	2	no	—
83	C₉	—	333	3	no	—
84	C₉	C₃²⋊#66526	1	1	yes	characteristic
85	C₉	#15688537	1	1	yes	characteristic
86	C₉	#15688538	1	1	yes	characteristic
87	C₃	—	3	1	no	—
88	C₃	—	37	3	no	—
89	C₃	—	111	2	no	—
90	C₃	(C₃×#66526)⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²⋊#66526
commutator	C₃²×C₃₇	C₉²
Frattini	C₃×C₉	C₃×#66526
Fitting	C₃×C₂₇×C₃₇	C₉
socle	C₃×C₃₇	#6855
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇⋊C₃	C₃²
hypercenter	C₃×C₂₇	#66526
hypocenter	C₃₇	#10624469
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
6. #15688541	C₉	semidirect
13. C₃×#11758601	C₉	semidirect
16. C₃×C₂₇×C₃₇	C₉	semidirect
57. C₃×C₂₇	#66526	semidirect
62. C₃₇	#10624469	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
#22582722	3
C₃×C₂₇×C₃₇⋊C₃	3
#10624469	37