Subgroups — #41007386

Order 26973.

210 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688519	C₃	1	2	yes	maximal, complemented
3	#22582513	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11758608	—	3	1	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
7	#15688521	—	3	1	no	—
8	#15688521	—	3	1	no	—
9	#15688521	C₃²	1	3	yes	complemented
10	#15688519	—	3	2	no	—
11	#15688519	—	3	2	no	—
12	#15688519	C₃²	1	6	yes	complemented
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₉⋊C₉	—	3	1	no	—
15	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
16	C₃₇×C₃²⋊C₉	—	3	1	no	—
17	C₉²×C₃₇	—	3	1	no	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₃₇⋊C₃	—	9	2	no	—
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×C₉×C₃₇	—	3	3	no	—
23	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
26	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
27	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
28	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
29	#11758608	—	3	3	no	—
30	#11758608	—	9	2	no	—
31	C₃×#66524	—	9	1	no	—
32	#10624846	—	37	1	no	maximal, Sylow
33	C₃²×C₃₇	—	9	1	no	—
34	C₃²×C₃₇	#732	1	1	yes	characteristic
35	C₃²×C₃₇	#732	1	2	yes	—
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₃₇⋊C₃	—	9	6	no	—
38	C₃×C₃₇⋊C₃	—	27	6	no	—
39	C₉×C₃₇	—	3	1	no	—
40	C₉×C₃₇	—	3	1	no	—
41	C₉×C₃₇	—	3	1	no	—
42	C₉×C₃₇	—	9	3	no	—
43	#66524	—	9	3	no	—
44	C₃×#733	—	37	1	no	—
45	C₃×#732	—	37	2	no	—
46	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
47	#6881	—	37	9	no	—
48	C₃×C₃₇	—	3	1	no	—
49	C₃×C₃₇	—	3	2	no	—
50	C₃×C₃₇	—	3	1	no	—
51	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
52	C₃₇⋊C₃	—	27	6	no	—
53	C₃×3₋¹⁺²	—	111	1	no	—
54	C₃×H(3,𝔽₃)	—	111	2	no	—
55	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
56	#733	—	37	3	no	—
57	#733	—	111	1	no	—
58	#733	—	111	1	no	—
59	#732	—	37	6	no	—
60	#732	—	111	2	no	—
61	#732	—	111	2	no	—
62	C₉⋊C₉	—	3	1	no	—
63	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
64	C₃²⋊C₉	—	3	1	no	—
65	C₉²	—	3	1	no	—
66	C₃₇	#10624846	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	333	2	no	—
68	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
69	3₋¹⁺²	—	333	2	no	—
70	3₋¹⁺²	—	111	3	no	—
71	H(3,𝔽₃)	—	333	2	no	—
72	H(3,𝔽₃)	—	333	2	no	—
73	H(3,𝔽₃)	—	111	6	no	—
74	C₃×C₉	—	3	1	no	—
75	C₃×C₉	—	3	2	no	—
76	C₃×C₉	—	3	3	no	—
77	C₃×C₉	—	333	1	no	—
78	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
79	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
80	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
81	C₃²	—	9	1	no	—
82	C₃²	—	333	6	no	—
83	C₃²	—	999	6	no	—
84	C₃²	#15688519	1	1	yes	characteristic
85	C₃²	#15688519	1	2	yes	—
86	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
87	C₉	—	3	1	no	—
88	C₉	—	3	1	no	—
89	C₉	—	3	1	no	—
90	C₉	—	9	3	no	—
91	C₉	—	333	3	no	—
92	C₃	—	3	1	no	—
93	C₃	—	3	2	no	—
94	C₃	—	3	1	no	—
95	C₃	—	999	6	no	—
96	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	#10624846
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
2. C₃×#15688519	C₃	semidirect
3. #22582513	C₃	semidirect
4. C₃₇×#6887	C₃	semidirect
9. #15688521	C₃²	semidirect
12. #15688519	C₃²	semidirect
15. C₃₇×C₉⋊C₉	C₃²	semidirect
46. #6887	C₃₇⋊C₃	semidirect
63. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
66. C₃₇	#10624846	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
C₃×#15688519	3
#22582513	3
C₃₇×#6887	3
#10624846	37