Subgroups — #41007404

Order 26973.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582633	C₃	1	3	yes	maximal, complemented
2	#22582523	C₃	1	3	yes	maximal, complemented
3	#22582518	C₃	1	3	yes	maximal, complemented
4	#22582513	C₃	1	3	yes	maximal, complemented
5	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
6	#15688531	—	3	3	no	—
7	#15688521	C₃²	1	3	yes	—
8	#15688521	C₃²	1	3	yes	—
9	#15688519	C₃²	1	3	yes	—
10	#15688516	—	3	3	no	—
11	#15688513	—	3	3	no	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃₇×C₉⋊C₉	—	3	1	no	—
14	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
15	C₃₇×C₃²⋊C₉	—	3	1	no	—
16	C₉²×C₃₇	—	3	1	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
24	#11758608	—	3	3	no	—
25	#11758608	—	3	3	no	—
26	#11758606	—	9	3	no	—
27	#11758605	—	9	3	no	—
28	C₉×C₃₇⋊C₃	—	9	3	no	—
29	#10624849	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	9	1	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	—	9	3	no	—
34	C₉×C₃₇	—	3	1	no	—
35	C₉×C₃₇	—	3	1	no	—
36	C₉×C₃₇	—	9	3	no	—
37	C₉×C₃₇	#732	1	3	yes	—
38	#66524	—	9	3	no	—
39	#66524	—	9	3	no	—
40	#6908	—	37	3	no	—
41	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
42	#6883	—	37	3	no	—
43	#6882	—	37	3	no	—
44	#6881	—	37	3	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	9	1	no	—
47	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
48	C₃₇⋊C₃	—	27	3	no	—
49	#737	—	111	3	no	—
50	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
51	#733	—	37	3	no	—
52	#733	—	37	3	no	—
53	#732	—	37	3	no	—
54	#731	—	111	3	no	—
55	AΣL(1,𝔽₂₇)	—	111	3	no	—
56	C₉⋊C₉	—	3	1	no	—
57	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
58	C₃²⋊C₉	—	3	1	no	—
59	C₉²	—	3	1	no	—
60	C₃₇	#10624849	1	1	yes	characteristic, complemented, Sylow
61	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
62	3₋¹⁺²	—	333	3	no	—
63	3₋¹⁺²	—	333	3	no	—
64	3₋¹⁺²	—	111	3	no	—
65	3₋¹⁺²	—	111	3	no	—
66	H(3,𝔽₃)	—	111	3	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	3	3	no	—
69	C₃×C₉	—	333	3	no	—
70	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
71	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
72	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
73	C₃²	—	3	1	no	—
74	C₃²	—	9	1	no	—
75	C₃²	—	333	3	no	—
76	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
77	C₉	—	3	1	no	—
78	C₉	—	3	1	no	—
79	C₉	—	9	3	no	—
80	C₉	—	333	3	no	—
81	C₉	—	333	3	no	—
82	C₉	#15688519	1	3	yes	—
83	C₃	—	3	1	no	—
84	C₃	—	9	1	no	—
85	C₃	—	999	3	no	—
86	C₃	C₃×#15688519	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#15688519
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624849
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #22582633	C₃	semidirect
2. #22582523	C₃	semidirect
3. #22582518	C₃	semidirect
4. #22582513	C₃	semidirect
5. C₃₇×#6887	C₃	semidirect
41. #6887	C₃₇⋊C₃	semidirect
60. C₃₇	#10624849	semidirect

Maximal subgroups

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

subgroup	index
#22582633	3
#22582523	3
#22582518	3
#22582513	3
C₃₇×#6887	3
#10624849	37