Subgroups — #41007344

Order 26973.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#732×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
2	#22582536	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582507	C₃	1	2	yes	maximal, complemented
4	#22582506	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6879	C₃	1	1	yes	maximal, characteristic, complemented
6	H(3,𝔽₃)×C₃₇⋊C₃	—	3	3	no	—
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
8	#15688519	C₃²	1	6	yes	complemented
9	#15688518	—	3	6	no	—
10	#15688504	—	3	2	no	—
11	#15688502	—	3	1	no	—
12	C₉×#66524	C₃²	1	2	yes	complemented
13	C₃₇×#732	C₃²	1	3	yes	complemented
14	C₉²×C₃₇	C₃²	1	1	yes	characteristic, complemented
15	C₃²×C₃₇⋊C₃	—	9	3	no	—
16	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
17	H(3,𝔽₃)×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
20	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	6	no	—
22	C₃×#66524	—	3	2	no	—
23	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
24	C₉×C₃₇⋊C₃	—	3	3	no	—
25	#10624839	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	9	3	no	—
27	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₃×C₃₇⋊C₃	—	9	1	no	—
29	C₃×C₃₇⋊C₃	—	9	6	no	—
30	C₃×C₃₇⋊C₃	—	27	9	no	—
31	C₃×C₃₇⋊C₃	#732	1	3	yes	—
32	C₉×C₃₇	—	3	1	no	—
33	C₉×C₃₇	—	9	1	no	—
34	#66524	—	3	2	no	—
35	#66524	—	9	2	no	—
36	C₃×#732	—	37	3	no	—
37	#6887	—	37	1	no	—
38	#6879	—	37	6	no	—
39	#6879	—	37	2	no	—
40	#6879	C₃₇⋊C₃	1	1	yes	characteristic, complemented
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	27	3	no	—
43	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
44	C₃₇⋊C₃	—	3	3	no	—
45	C₃₇⋊C₃	—	27	6	no	—
46	C₃×H(3,𝔽₃)	—	111	3	no	—
47	C₃²×C₉	—	37	1	no	—
48	#732	—	37	6	no	—
49	#732	—	111	6	no	—
50	#732	C₃×C₃₇⋊C₃	1	3	yes	complemented
51	C₉⋊C₉	—	111	2	no	—
52	C₃²⋊C₉	—	111	1	no	—
53	C₉²	—	37	2	no	—
54	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
55	C₃₇	#10624839	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	37	1	no	—
57	C₃³	—	333	3	no	—
58	H(3,𝔽₃)	—	3	3	no	—
59	H(3,𝔽₃)	—	333	6	no	—
60	H(3,𝔽₃)	—	111	6	no	—
61	C₃×C₉	—	3	1	no	—
62	C₃×C₉	—	37	2	no	—
63	C₃×C₉	—	111	3	no	—
64	C₃×C₉	—	111	2	no	—
65	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
66	C₃²	—	9	3	no	—
67	C₃²	—	37	3	no	—
68	C₃²	—	333	1	no	—
69	C₃²	—	333	6	no	—
70	C₃²	—	999	9	no	—
71	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
72	C₉	—	3	1	no	—
73	C₉	—	9	1	no	—
74	C₉	—	333	2	no	—
75	C₉	—	111	2	no	—
76	C₃	—	3	1	no	—
77	C₃	—	27	3	no	—
78	C₃	—	111	3	no	—
79	C₃	—	999	6	no	—
80	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	#10624839
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. #732×C₃₇⋊C₃	C₃	semidirect
2. #22582536	C₃	semidirect
3. #22582507	C₃	semidirect
4. #22582506	C₃	semidirect
5. C₃₇×#6879	C₃	semidirect
8. #15688519	C₃²	semidirect
12. C₉×#66524	C₃²	semidirect
13. C₃₇×#732	C₃²	semidirect
14. C₉²×C₃₇	C₃²	semidirect
40. #6879	C₃₇⋊C₃	semidirect
50. #732	C₃×C₃₇⋊C₃	semidirect
54. C₉²	C₃×C₃₇⋊C₃	semidirect
55. C₃₇	#10624839	semidirect

Maximal subgroups

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

subgroup	index
#732×C₃₇⋊C₃	3
#22582536	3
#22582507	3
#22582506	3
C₃₇×#6879	3
#10624839	37