Subgroups — #44535002

Order 31347.

730 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812524	C₃	1	1	yes	maximal, characteristic
2	#23782820	C₃	1	8	yes	maximal, complemented
3	#23782818	C₃	1	16	yes	maximal, complemented
4	#23782815	C₃	1	8	yes	maximal, complemented
5	H(3,𝔽₃)×C₄₃⋊C₉	C₃	1	2	yes	maximal
6	C₃×#16812499	C₃	1	4	yes	maximal, complemented
7	C₉×H(3,𝔽₃)×C₄₃	C₃	1	1	yes	maximal, characteristic
8	#16812524	—	3	9	no	—
9	C₃×#11923958	C₃²	1	8	yes	—
10	C₃²×C₄₃⋊C₉	C₃²	1	4	yes	—
11	C₃²×C₄₃⋊C₉	C₃²	1	8	yes	—
12	#16812499	C₃²	1	1	yes	characteristic
13	#16812499	C₃²	1	8	yes	—
14	#16812499	C₃²	1	16	yes	—
15	#16812499	C₃²	1	8	yes	—
16	(C₄₃⋊C₉)⋊C₉	C₃²	1	48	yes	—
17	C₃²⋊(C₄₃⋊C₉)	C₃²	1	16	yes	—
18	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₄₃	C₃²	1	4	yes	—
20	C₄₃×C₃²⋊C₉	C₃²	1	8	yes	—
21	#11923958	—	3	24	no	—
22	C₃×C₄₃⋊C₉	—	3	12	no	—
23	C₃×C₄₃⋊C₉	—	9	8	no	—
24	C₃×C₄₃⋊C₉	C₃³	1	16	yes	—
25	C₃×C₄₃⋊C₉	C₃³	1	8	yes	—
26	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
27	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
28	C₃³×C₄₃	C₃³	1	4	yes	—
29	H(3,𝔽₃)×C₄₃	3₋¹⁺²	1	9	yes	—
30	C₃×C₉×C₄₃	—	3	12	no	—
31	C₃×C₉×C₄₃	C₃³	1	8	yes	—
32	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
33	#10624900	—	43	1	no	maximal, Sylow
34	C₃²×C₄₃	—	3	4	no	—
35	C₃²×C₄₃	—	3	8	no	—
36	C₃²×C₄₃	C₃×3₋¹⁺²	1	12	yes	—
37	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
38	C₉×C₄₃	—	3	8	no	—
39	C₉×C₄₃	#737	1	3	yes	complemented
40	C₄₃⋊C₉	—	3	1	no	—
41	C₄₃⋊C₉	—	3	8	no	—
42	C₄₃⋊C₉	—	3	2	no	—
43	C₄₃⋊C₉	—	3	16	no	—
44	C₃×#737	—	43	1	no	—
45	#6895	—	43	8	no	—
46	#6894	—	43	8	no	—
47	#6894	—	43	16	no	—
48	C₉×H(3,𝔽₃)	—	43	2	no	—
49	C₉×H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic
50	C₃×C₉⋊C₉	—	43	4	no	—
51	C₃×C₄₃	—	3	12	no	—
52	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
53	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
54	C₃×C₄₃	3₋¹⁺⁴	1	2	yes	—
55	#737	—	129	9	no	—
56	C₃×3₋¹⁺²	—	43	8	no	—
57	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
58	C₃²×C₉	—	43	4	no	—
59	C₃²×C₉	—	43	8	no	—
60	C₃²×C₉	C₃×C₄₃⋊C₃	1	4	yes	—
61	C₉⋊C₉	—	43	48	no	—
62	C₉⋊C₉	—	43	1	no	—
63	C₉⋊C₉	—	43	8	no	—
64	C₉⋊C₉	—	43	16	no	—
65	C₉⋊C₉	—	43	8	no	—
66	C₃²⋊C₉	—	43	16	no	—
67	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	8	yes	—
68	C₄₃	#10624900	1	1	yes	characteristic, complemented, Sylow
69	C₃³	C₃²×C₄₃⋊C₃	1	4	yes	—
70	3₋¹⁺²	—	129	24	no	—
71	H(3,𝔽₃)	#11923957	1	9	yes	—
72	C₃×C₉	—	3	12	no	—
73	C₃×C₉	—	43	16	no	—
74	C₃×C₉	—	43	1	no	—
75	C₃×C₉	—	43	2	no	—
76	C₃×C₉	—	43	8	no	—
77	C₃×C₉	—	129	12	no	—
78	C₃×C₉	—	387	8	no	—
79	C₃×C₉	C₃²×C₄₃⋊C₃	1	8	yes	—
80	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
81	C₃²	—	3	4	no	—
82	C₃²	—	3	8	no	—
83	C₃²	C₃×#11923957	1	12	yes	—
84	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
85	C₉	—	3	8	no	—
86	C₉	—	129	1	no	—
87	C₉	—	129	8	no	—
88	C₉	—	129	16	no	—
89	C₉	—	129	2	no	—
90	C₉	#16812524	1	3	yes	complemented
91	C₃	—	3	12	no	—
92	C₃	C₃²×#11923957	1	1	yes	characteristic
93	C₃	C₃×#16812524	1	1	yes	characteristic
94	C₃	#23782922	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₉×H(3,𝔽₃)×C₄₃	C₃
prime radical	C₃×H(3,𝔽₃)×C₄₃	C₃²
hypercenter	C₉×H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624900
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₉×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. #23782820	C₃	semidirect
3. #23782818	C₃	semidirect
4. #23782815	C₃	semidirect
6. C₃×#16812499	C₃	semidirect
39. C₉×C₄₃	#737	semidirect
68. C₄₃	#10624900	semidirect
90. C₉	#16812524	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812524	3
#23782820	3
#23782818	3
#23782815	3
H(3,𝔽₃)×C₄₃⋊C₉	3
C₃×#16812499	3
C₉×H(3,𝔽₃)×C₄₃	3
#10624900	43