Subgroups — #41005303

Order 26973.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	3	yes	maximal, complemented
2	#22582456	C₃	1	3	yes	maximal, complemented
3	#22582452	C₃	1	3	yes	maximal, complemented
4	#22582443	C₃	1	3	yes	maximal, complemented
5	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
7	C₃×C₉×C₃₇⋊C₃	—	3	3	no	—
8	C₃²⋊#66524	C₃²	1	3	yes	—
9	C₃²⋊#66524	C₃²	1	3	yes	—
10	#15688502	—	3	6	no	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₉⋊C₉	—	3	2	no	—
13	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₉²×C₃₇	—	3	1	no	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃²×C₃₇⋊C₃	—	3	3	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	3	no	—
19	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
20	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
22	C₃×#66524	—	3	3	no	—
23	C₃×#66524	—	3	3	no	—
24	C₉×C₃₇⋊C₃	—	9	9	no	—
25	#10624585	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃×C₃₇⋊C₃	—	3	9	no	—
30	C₃×C₃₇⋊C₃	—	9	3	no	—
31	C₃×C₃₇⋊C₃	—	9	6	no	—
32	C₉×C₃₇	—	3	2	no	—
33	C₉×C₃₇	—	9	3	no	—
34	C₉×C₃₇	#732	1	3	yes	—
35	#66524	—	9	3	no	—
36	#66524	—	9	3	no	—
37	C₉×H(3,𝔽₃)	—	37	3	no	—
38	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
39	#6861	—	37	3	no	—
40	#6859	—	37	3	no	—
41	#6856	—	37	3	no	—
42	C₃×C₃₇	—	9	1	no	—
43	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
44	C₃×C₃₇	#6908	1	3	yes	—
45	C₃₇⋊C₃	—	9	9	no	—
46	C₃×H(3,𝔽₃)	—	37	3	no	—
47	C₃²×C₉	—	111	3	no	—
48	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
49	C₉⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	37	3	no	—
51	C₃²⋊C₉	—	37	3	no	—
52	C₃²⋊C₉	—	111	6	no	—
53	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
54	C₉²	—	3	1	no	—
55	C₃₇	#10624585	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	111	3	no	—
57	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
58	H(3,𝔽₃)	—	111	9	no	—
59	C₃×C₉	—	3	3	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	333	9	no	—
62	C₃×C₉	—	111	3	no	—
63	C₃×C₉	—	111	3	no	—
64	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
65	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
66	C₃²	—	3	1	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	333	3	no	—
69	C₃²	—	333	6	no	—
70	C₃²	—	111	9	no	—
71	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
72	C₉	—	3	2	no	—
73	C₉	—	9	3	no	—
74	C₉	—	333	3	no	—
75	C₉	—	333	3	no	—
76	C₉	#15688519	1	3	yes	—
77	C₃	—	9	1	no	—
78	C₃	—	333	9	no	—
79	C₃	C₃×#15688519	1	1	yes	characteristic
80	C₃	#22582630	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624585
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₉×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
2. #22582456	C₃	semidirect
3. #22582452	C₃	semidirect
4. #22582443	C₃	semidirect
5. C₃₇×#6887	C₃	semidirect
38. #6887	C₃₇⋊C₃	semidirect
55. C₃₇	#10624585	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582456	3
#22582452	3
#22582443	3
C₃₇×#6887	3
#10624585	37