Subgroups — #41007355

Order 26973.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582631	C₃	1	3	yes	maximal, complemented
2	#22582536	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582510	C₃	1	2	yes	maximal, complemented
4	#22582501	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6879	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688530	—	3	3	no	—
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	#15688521	C₃²	1	6	yes	—
9	#15688515	—	3	3	no	—
10	#15688512	—	3	3	no	—
11	#15688504	—	3	1	no	—
12	#15688504	—	3	1	no	—
13	#15688502	—	3	1	no	—
14	C₉×#66524	C₃²	1	2	yes	—
15	C₃₇×#732	C₃²	1	3	yes	—
16	C₉²×C₃₇	C₃²	1	1	yes	characteristic
17	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	H(3,𝔽₃)×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
21	#11758608	—	3	6	no	—
22	#11758608	—	9	6	no	—
23	C₃×#66524	—	3	2	no	—
24	C₃×#66524	—	9	3	no	—
25	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
27	C₉×C₃₇⋊C₃	—	3	3	no	—
28	#10624841	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	9	3	no	—
30	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₃₇⋊C₃	—	3	1	no	—
32	C₃×C₃₇⋊C₃	—	9	1	no	—
33	C₉×C₃₇	—	3	1	no	—
34	C₉×C₃₇	—	9	1	no	—
35	#66524	—	3	1	no	—
36	#66524	—	9	2	no	—
37	#66524	—	9	6	no	—
38	#66524	#732	1	3	yes	—
39	#6908	—	37	3	no	—
40	#6887	—	37	1	no	—
41	#6880	—	37	2	no	—
42	#6879	C₃₇⋊C₃	1	1	yes	characteristic, complemented
43	#6878	—	37	6	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	27	3	no	—
46	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
47	C₃₇⋊C₃	—	9	1	no	—
48	#737	—	111	3	no	—
49	C₃²×C₉	—	37	1	no	—
50	#733	—	37	6	no	—
51	#732	C₃×C₃₇⋊C₃	1	3	yes	—
52	#731	—	111	3	no	—
53	AΣL(1,𝔽₂₇)	—	111	3	no	—
54	C₉⋊C₉	—	111	1	no	—
55	C₉⋊C₉	—	111	1	no	—
56	C₃²⋊C₉	—	111	1	no	—
57	C₉²	—	37	2	no	—
58	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃₇	#10624841	1	1	yes	characteristic, complemented, Sylow
60	C₃³	—	37	1	no	—
61	3₋¹⁺²	—	333	6	no	—
62	3₋¹⁺²	—	111	6	no	—
63	H(3,𝔽₃)	—	3	3	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	37	1	no	—
66	C₃×C₉	—	37	1	no	—
67	C₃×C₉	—	333	3	no	—
68	C₃×C₉	—	111	3	no	—
69	C₃×C₉	—	111	2	no	—
70	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
71	C₃²	—	9	3	no	—
72	C₃²	—	333	1	no	—
73	C₃²	—	111	1	no	—
74	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
75	C₉	—	3	1	no	—
76	C₉	—	9	1	no	—
77	C₉	—	37	3	no	—
78	C₉	—	333	2	no	—
79	C₉	—	333	6	no	—
80	C₉	—	111	1	no	—
81	C₃	—	3	1	no	—
82	C₃	—	27	3	no	—
83	C₃	—	333	1	no	—
84	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#732×C₃₇⋊C₃
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6879	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6879	C₃₇⋊C₃
hypocenter	C₃₇	#10624841
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6879

Complements

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

N	H ≅ G/N	type
1. #22582631	C₃	semidirect
2. #22582536	C₃	semidirect
3. #22582510	C₃	semidirect
4. #22582501	C₃	semidirect
5. C₃₇×#6879	C₃	semidirect
42. #6879	C₃₇⋊C₃	semidirect
59. C₃₇	#10624841	semidirect

Maximal subgroups

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

subgroup	index
#22582631	3
#22582536	3
#22582510	3
#22582501	3
C₃₇×#6879	3
#10624841	37