Subgroups — #44533511

Order 31347.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812504	C₃	1	3	yes	maximal, complemented
2	C₃²×C₄₃⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₉×C₃²⋊(C₄₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊(C₄₃⋊C₂₇)	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₃×C₄₃⋊C₃)⋊C₂₇	C₃	1	6	yes	maximal, complemented
6	C₄₃×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₃²⋊(C₄₃⋊C₃)	C₉	1	1	yes	characteristic
8	C₃×C₉×C₄₃⋊C₃	C₃²	1	3	yes	—
9	#16812504	—	3	9	no	—
10	C₃×C₄₃⋊C₂₇	—	3	3	no	—
11	C₃×C₄₃⋊C₂₇	C₃²	1	2	yes	—
12	C₃×C₄₃⋊C₂₇	C₃²	1	3	yes	—
13	C₃×C₄₃⋊C₂₇	C₃²	1	1	yes	characteristic
14	C₃²⋊(C₄₃⋊C₉)	C₉	1	2	yes	—
15	(C₃×C₄₃⋊C₃)⋊C₉	C₉	1	6	yes	—
16	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
17	C₃×C₂₇×C₄₃	C₃²	1	3	yes	—
18	C₃²×C₄₃⋊C₃	C₃×C₉	1	3	yes	—
19	C₃²⋊(C₄₃⋊C₃)	—	3	3	no	—
20	C₃×C₄₃⋊C₉	C₃×C₉	1	6	yes	—
21	C₉×C₄₃⋊C₃	—	3	3	no	—
22	C₉×C₄₃⋊C₃	—	3	6	no	—
23	C₄₃⋊C₂₇	—	3	2	no	—
24	C₄₃⋊C₂₇	—	3	3	no	—
25	C₄₃⋊C₂₇	H(3,𝔽₃)	1	3	yes	complemented
26	C₃³×C₄₃	C₃×C₉	1	1	yes	characteristic
27	C₃×C₉×C₄₃	—	3	1	no	—
28	C₃×C₉×C₄₃	—	3	2	no	—
29	C₃×C₉×C₄₃	C₃×C₉	1	2	yes	—
30	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
31	C₂₇×C₄₃	—	3	3	no	—
32	C₂₇⋊H(3,𝔽₃)	—	43	1	no	maximal, Sylow
33	C₃²×C₄₃	—	3	1	no	—
34	C₃²×C₄₃	—	3	2	no	—
35	C₃²×C₄₃	C₂₇⋊C₃	1	3	yes	complemented
36	C₃²×C₄₃	C₃²×C₉	1	1	yes	characteristic
37	C₃×C₄₃⋊C₃	—	3	3	no	—
38	C₃×C₄₃⋊C₃	—	3	3	no	—
39	C₃×C₄₃⋊C₃	—	3	6	no	—
40	C₉×C₄₃	—	3	2	no	—
41	C₉×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₉×C₄₃	#737	1	2	yes	—
43	C₄₃⋊C₉	—	3	6	no	—
44	C₃×C₂₇⋊C₃	—	43	3	no	—
45	C₃²×C₂₇	—	43	1	no	—
46	C₉×H(3,𝔽₃)	—	43	1	no	—
47	C₃²⋊C₂₇	—	43	1	no	—
48	C₃²⋊C₂₇	—	43	6	no	—
49	C₃²⋊C₂₇	C₄₃⋊C₃	1	1	yes	characteristic, complemented
50	C₃×C₄₃	—	3	3	no	—
51	C₃×C₄₃	C₉×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₄₃	C₃×C₂₇⋊C₃	1	1	yes	characteristic
53	C₃×C₄₃	#6903	1	2	yes	—
54	C₄₃⋊C₃	—	9	3	no	—
55	C₃×H(3,𝔽₃)	—	43	1	no	—
56	C₃²×C₉	—	43	3	no	—
57	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
58	C₂₇⋊C₃	—	129	9	no	—
59	C₃×C₂₇	—	43	1	no	—
60	C₃×C₂₇	—	43	2	no	—
61	C₃×C₂₇	—	43	3	no	—
62	C₃×C₂₇	—	129	3	no	—
63	C₃×C₂₇	C₃×C₄₃⋊C₃	1	3	yes	—
64	C₃²⋊C₉	—	43	2	no	—
65	C₃²⋊C₉	—	43	6	no	—
66	C₄₃	C₂₇⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
67	C₃³	—	43	3	no	—
68	C₃³	C₉×C₄₃⋊C₃	1	1	yes	characteristic
69	H(3,𝔽₃)	—	129	3	no	—
70	C₃×C₉	—	3	1	no	—
71	C₃×C₉	—	3	2	no	—
72	C₃×C₉	—	43	6	no	—
73	C₃×C₉	—	129	3	no	—
74	C₃×C₉	—	129	6	no	—
75	C₃×C₉	C₉×C₄₃⋊C₃	1	2	yes	—
76	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
77	C₂₇	—	3	3	no	—
78	C₂₇	—	43	3	no	—
79	C₂₇	—	129	2	no	—
80	C₂₇	—	129	3	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	129	3	no	—
84	C₃²	—	129	3	no	—
85	C₃²	—	129	6	no	—
86	C₃²	#16812504	1	3	yes	complemented
87	C₃²	C₃×C₉×C₄₃⋊C₃	1	1	yes	characteristic
88	C₉	—	3	2	no	—
89	C₉	—	129	6	no	—
90	C₉	H(3,𝔽₃)×C₄₃⋊C₃	1	1	yes	characteristic
91	C₉	#16812526	1	2	yes	—
92	C₃	—	3	3	no	—
93	C₃	—	387	3	no	—
94	C₃	#23782783	1	1	yes	characteristic
95	C₃	C₃×#16812504	1	1	yes	characteristic
96	C₃	#23782858	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₄₃⋊C₃
commutator, socle	C₃²×C₄₃	C₃²×C₉
Fitting	C₄₃×C₃²⋊C₂₇	C₃
prime radical	C₃×C₃²⋊(C₄₃⋊C₃)	C₉
cosocle	C₃×C₉×C₄₃	C₃³
hypercenter	C₃²⋊C₂₇	C₄₃⋊C₃
hypocenter	C₄₃	C₂₇⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃×C₉ ◁ C₃²⋊C₂₇

Complements

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

N	H ≅ G/N	type
1. C₃×#16812504	C₃	semidirect
2. C₃²×C₄₃⋊C₂₇	C₃	semidirect
4. C₃²⋊(C₄₃⋊C₂₇)	C₃	semidirect
5. (C₃×C₄₃⋊C₃)⋊C₂₇	C₃	semidirect
6. C₄₃×C₃²⋊C₂₇	C₃	semidirect
25. C₄₃⋊C₂₇	H(3,𝔽₃)	semidirect
35. C₃²×C₄₃	C₂₇⋊C₃	semidirect
49. C₃²⋊C₂₇	C₄₃⋊C₃	semidirect
66. C₄₃	C₂₇⋊H(3,𝔽₃)	semidirect
86. C₃²	#16812504	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812504	3
C₃²×C₄₃⋊C₂₇	3
C₉×C₃²⋊(C₄₃⋊C₃)	3
C₃²⋊(C₄₃⋊C₂₇)	3
(C₃×C₄₃⋊C₃)⋊C₂₇	3
C₄₃×C₃²⋊C₂₇	3
C₂₇⋊H(3,𝔽₃)	43