Subgroups — #41008064

Order 26973.

596 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582605	C₃	1	8	yes	maximal, complemented
2	#22582602	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582590	C₃	1	8	yes	maximal, complemented
4	#22582589	C₃	1	16	yes	maximal, complemented
5	#22582586	C₃	1	2	yes	maximal, complemented
6	#22582553	C₃	1	4	yes	maximal
7	C₃₇×#6898	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758605	C₃²	1	4	yes	—
10	C₃×#11758605	C₃²	1	8	yes	—
11	#15688506	C₃²	1	8	yes	complemented
12	#15688505	C₃²	1	8	yes	complemented
13	#15688504	C₃²	1	16	yes	complemented
14	#15688504	C₃²	1	32	yes	complemented
15	C₃²⋊#66524	C₃²	1	8	yes	—
16	#15688502	C₃²	1	8	yes	—
17	#15688502	C₃²	1	16	yes	—
18	C₉×#66524	C₃²	1	8	yes	complemented
19	C₃₇×C₉⋊C₉	C₃²	1	8	yes	complemented
20	C₃₇×C₃²⋊C₉	C₃²	1	4	yes	—
21	C₉²×C₃₇	C₃²	1	1	yes	characteristic, complemented
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	—	9	4	no	—
24	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
25	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
26	C₃×C₉×C₃₇	C₃³	1	4	yes	—
27	C₃×C₉×C₃₇	C₃³	1	8	yes	—
28	C₃×#66524	C₃³	1	16	yes	—
29	C₃×#66524	C₃³	1	8	yes	—
30	#11758605	—	9	4	no	—
31	#11758605	—	9	8	no	—
32	#10624912	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	3	4	no	—
34	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
35	C₃×C₃₇⋊C₃	—	3	4	no	—
36	C₃×C₃₇⋊C₃	—	3	8	no	—
37	C₃×C₃₇⋊C₃	—	9	9	no	—
38	C₉×C₃₇	—	3	4	no	—
39	C₉×C₃₇	—	3	8	no	—
40	#66524	—	3	16	no	—
41	#66524	—	3	8	no	—
42	#6900	—	37	8	no	—
43	#6900	—	37	1	no	—
44	#6898	—	37	8	no	—
45	#6898	C₃₇⋊C₃	1	1	yes	characteristic, complemented
46	#6897	—	37	16	no	—
47	#6897	—	37	2	no	—
48	#6891	—	37	4	no	—
49	C₃×C₃₇	—	9	1	no	—
50	C₃×C₃₇	3₋¹⁺⁴	1	4	yes	—
51	C₃₇⋊C₃	—	9	1	no	—
52	C₃₇⋊C₃	—	9	2	no	—
53	C₃⁴	—	37	1	no	—
54	C₃×3₋¹⁺²	—	37	8	no	—
55	C₃×3₋¹⁺²	—	37	4	no	—
56	C₉⋊C₉	—	37	32	no	—
57	C₉⋊C₉	—	37	16	no	—
58	C₉⋊C₉	—	37	8	no	—
59	C₉⋊C₉	—	37	8	no	—
60	C₉⋊C₉	C₃×C₃₇⋊C₃	1	8	yes	complemented
61	C₃²⋊C₉	—	37	8	no	—
62	C₃²⋊C₉	—	37	16	no	—
63	C₃²⋊C₉	—	37	8	no	—
64	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	4	yes	—
65	C₉²	—	37	8	no	—
66	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
67	C₃₇	#10624912	1	1	yes	characteristic, complemented, Sylow
68	C₃³	—	37	2	no	—
69	C₃³	—	37	1	no	—
70	C₃³	—	333	4	no	—
71	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	333	8	no	—
73	3₋¹⁺²	—	333	4	no	—
74	C₃×C₉	—	37	16	no	—
75	C₃×C₉	—	37	8	no	—
76	C₃×C₉	C₃²×C₃₇⋊C₃	1	8	yes	—
77	C₃×C₉	C₃²×C₃₇⋊C₃	1	4	yes	—
78	C₃²	—	3	4	no	—
79	C₃²	—	333	9	no	—
80	C₃²	—	111	4	no	—
81	C₃²	—	111	8	no	—
82	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
83	C₉	—	3	8	no	—
84	C₉	—	3	4	no	—
85	C₉	—	111	16	no	—
86	C₉	—	111	8	no	—
87	C₃	—	9	1	no	—
88	C₃	—	333	1	no	—
89	C₃	—	333	2	no	—
90	C₃	#22582678	1	4	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6898	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6898	C₃₇⋊C₃
hypocenter	C₃₇	#10624912
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6898

Complements

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

N	H ≅ G/N	type
1. #22582605	C₃	semidirect
2. #22582602	C₃	semidirect
3. #22582590	C₃	semidirect
4. #22582589	C₃	semidirect
5. #22582586	C₃	semidirect
7. C₃₇×#6898	C₃	semidirect
11. #15688506	C₃²	semidirect
12. #15688505	C₃²	semidirect
13. #15688504	C₃²	semidirect
14. #15688504	C₃²	semidirect
18. C₉×#66524	C₃²	semidirect
19. C₃₇×C₉⋊C₉	C₃²	semidirect
21. C₉²×C₃₇	C₃²	semidirect
45. #6898	C₃₇⋊C₃	semidirect
60. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
66. C₉²	C₃×C₃₇⋊C₃	semidirect
67. C₃₇	#10624912	semidirect

Maximal subgroups

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

subgroup	index
#22582605	3
#22582602	3
#22582590	3
#22582589	3
#22582586	3
#22582553	3
C₃₇×#6898	3
#10624912	37