Subgroups — #41007799

Order 26973.

1148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688530	C₃	1	3	yes	maximal, complemented
2	#22582562	C₃	1	24	yes	maximal, complemented
3	#22582554	C₃	1	8	yes	maximal, complemented
4	C₃×C₃²⋊#66524	C₃	1	4	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
6	#15688530	—	3	27	no	—
7	C₃×#11758608	C₃²	1	24	yes	complemented
8	C₃²×#66524	C₃²	1	12	yes	complemented
9	C₃²⋊#66524	C₃²	1	1	yes	characteristic
10	C₃²⋊#66524	C₃²	1	8	yes	—
11	C₃²⋊#66524	C₃²	1	24	yes	complemented
12	C₃²⋊#66524	C₃²	1	48	yes	complemented
13	C₃⁴×C₃₇	C₃²	1	4	yes	—
14	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
15	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	8	yes	—
16	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
17	C₃³×C₃₇	—	3	12	no	—
18	C₃³×C₃₇	—	3	12	no	—
19	C₃³×C₃₇	—	9	8	no	—
20	C₃³×C₃₇	C₃³	1	8	yes	—
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	H(3,𝔽₃)×C₃₇	—	3	24	no	—
24	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
25	#11758608	—	3	72	no	—
26	C₃×#66524	—	3	36	no	—
27	C₃×#66524	C₃³	1	3	yes	—
28	C₃×#66524	C₃³	1	24	yes	—
29	#10624894	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	4	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	—	3	8	no	—
34	C₃²×C₃₇	—	3	8	no	—
35	C₃²×C₃₇	—	3	8	no	—
36	C₃²×C₃₇	—	3	16	no	—
37	C₃²×C₃₇	—	9	36	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
39	C₃²×C₃₇	#737	1	3	yes	complemented
40	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
41	#66524	—	3	3	no	—
42	#66524	—	3	24	no	—
43	C₃×#737	—	37	3	no	—
44	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
45	#6893	—	37	24	no	—
46	#6891	—	37	8	no	—
47	C₃×C₃²⋊C₉	—	37	4	no	—
48	C₃×C₃₇	—	3	3	no	—
49	C₃×C₃₇	—	3	12	no	—
50	C₃×C₃₇	—	9	8	no	—
51	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
53	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
54	C₃⁴	C₃×C₃₇⋊C₃	1	4	yes	—
55	#737	—	111	27	no	—
56	C₃×3₋¹⁺²	—	37	24	no	—
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	8	yes	—
59	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
60	C₃²×C₉	—	37	12	no	—
61	C₃²⋊C₉	—	37	1	no	—
62	C₃²⋊C₉	—	37	8	no	—
63	C₃²⋊C₉	—	37	24	no	—
64	C₃²⋊C₉	—	37	48	no	—
65	C₃₇	#10624894	1	1	yes	characteristic, complemented, Sylow
66	C₃³	—	3	12	no	—
67	C₃³	—	3	12	no	—
68	C₃³	—	9	8	no	—
69	C₃³	C₃²×C₃₇⋊C₃	1	8	yes	—
70	C₃³	C₃²×C₃₇⋊C₃	1	4	yes	—
71	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	111	72	no	—
73	H(3,𝔽₃)	—	3	24	no	—
74	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
75	C₃×C₉	—	37	3	no	—
76	C₃×C₉	—	37	24	no	—
77	C₃×C₉	—	111	36	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	4	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	8	no	—
82	C₃²	—	3	8	no	—
83	C₃²	—	3	8	no	—
84	C₃²	—	3	16	no	—
85	C₃²	—	9	36	no	—
86	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
87	C₃²	#15688530	1	3	yes	complemented
88	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
89	C₉	—	111	3	no	—
90	C₉	—	111	24	no	—
91	C₃	—	3	3	no	—
92	C₃	—	3	12	no	—
93	C₃	—	9	8	no	—
94	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
95	C₃	C₃×#15688530	1	1	yes	characteristic
96	C₃	#22582680	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624894
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#15688530	C₃	semidirect
2. #22582562	C₃	semidirect
3. #22582554	C₃	semidirect
4. C₃×C₃²⋊#66524	C₃	semidirect
7. C₃×#11758608	C₃²	semidirect
8. C₃²×#66524	C₃²	semidirect
11. C₃²⋊#66524	C₃²	semidirect
12. C₃²⋊#66524	C₃²	semidirect
39. C₃²×C₃₇	#737	semidirect
65. C₃₇	#10624894	semidirect
87. C₃²	#15688530	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688530	3
#22582562	3
#22582554	3
C₃×C₃²⋊#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624894	37