Subgroups — #44533656

Order 31347.

596 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₄₃⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#16812506	C₃	1	9	yes	maximal, complemented
3	#23782798	C₃	1	2	yes	maximal, complemented
4	C₄₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₄₃⋊C₃	C₃²	1	3	yes	—
6	C₃×#11923958	C₃²	1	6	yes	—
7	C₃×C₃²⋊(C₄₃⋊C₃)	—	3	9	no	—
8	C₃×C₃²⋊(C₄₃⋊C₃)	—	3	3	no	—
9	#16812506	—	3	27	no	—
10	C₃²⋊(C₄₃⋊C₉)	—	3	6	no	—
11	C₃⁴×C₄₃	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₄₃	—	3	3	no	—
13	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	3	yes	—
14	C₃²×C₄₃⋊C₃	—	3	6	no	—
15	C₃²×C₄₃⋊C₃	—	3	3	no	—
16	C₃²×C₄₃⋊C₃	—	9	9	no	—
17	C₃²×C₄₃⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
18	#11923958	—	3	18	no	—
19	C₃²⋊(C₄₃⋊C₃)	—	3	27	no	—
20	C₃×C₄₃⋊C₉	—	3	6	no	—
21	C₃³×C₄₃	—	3	3	no	—
22	C₃³×C₄₃	—	3	3	no	—
23	C₃³×C₄₃	—	9	3	no	—
24	C₃³×C₄₃	H(3,𝔽₃)	1	3	yes	—
25	C₃³×C₄₃	C₃³	1	1	yes	characteristic
26	H(3,𝔽₃)×C₄₃	—	3	9	no	—
27	H(3,𝔽₃)×C₄₃	—	9	9	no	—
28	#10624735	—	43	1	no	maximal, Sylow
29	C₃²×C₄₃	—	3	3	no	—
30	C₃²×C₄₃	—	3	3	no	—
31	C₃²×C₄₃	—	3	3	no	—
32	C₃²×C₄₃	—	3	6	no	—
33	C₃²×C₄₃	—	9	9	no	—
34	C₃²×C₄₃	—	9	3	no	—
35	C₃²×C₄₃	—	9	9	no	—
36	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₃×C₄₃⋊C₃	—	3	9	no	—
39	C₃×C₄₃⋊C₃	—	9	18	no	—
40	C₃×C₄₃⋊C₃	—	9	9	no	—
41	C₃×C₄₃⋊C₃	—	9	9	no	—
42	C₄₃⋊C₉	—	9	6	no	—
43	C₃²×H(3,𝔽₃)	—	43	1	no	—
44	C₃×AΣL(1,𝔽₂₇)	—	43	9	no	—
45	#6891	—	43	2	no	—
46	C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic, complemented
47	C₃×C₄₃	—	3	3	no	—
48	C₃×C₄₃	—	9	3	no	—
49	C₃×C₄₃	—	27	3	no	—
50	C₃×C₄₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₄₃	#6909	1	3	yes	—
52	C₄₃⋊C₃	—	9	9	no	—
53	C₃⁴	—	43	3	no	—
54	C₃⁴	C₃×C₄₃⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	43	6	no	—
56	C₃×H(3,𝔽₃)	—	3	3	no	—
57	C₃×H(3,𝔽₃)	—	129	9	no	—
58	C₃×H(3,𝔽₃)	—	129	3	no	—
59	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	3	yes	—
60	AΣL(1,𝔽₂₇)	—	129	27	no	—
61	C₃²⋊C₉	—	129	6	no	—
62	C₄₃	#10624735	1	1	yes	characteristic, complemented, Sylow
63	C₃³	—	3	3	no	—
64	C₃³	—	3	3	no	—
65	C₃³	—	9	3	no	—
66	C₃³	—	43	9	no	—
67	C₃³	—	129	6	no	—
68	C₃³	—	129	3	no	—
69	C₃³	—	387	9	no	—
70	C₃³	C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
71	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	129	18	no	—
73	H(3,𝔽₃)	—	3	9	no	—
74	H(3,𝔽₃)	—	9	9	no	—
75	H(3,𝔽₃)	—	129	27	no	—
76	C₃×C₉	—	129	6	no	—
77	C₃²	—	3	3	no	—
78	C₃²	—	3	3	no	—
79	C₃²	—	3	3	no	—
80	C₃²	—	3	6	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	9	3	no	—
83	C₃²	—	9	9	no	—
84	C₃²	—	129	9	no	—
85	C₃²	—	387	9	no	—
86	C₃²	—	387	18	no	—
87	C₃²	—	387	9	no	—
88	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	3	yes	—
89	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
90	C₉	—	387	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	3	no	—
93	C₃	—	27	3	no	—
94	C₃	—	387	9	no	—
95	C₃	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic
96	C₃	#23782881	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)
commutator, cosocle	C₃³×C₄₃	C₃³
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting	C₄₃×C₃²⋊H(3,𝔽₃)	C₃
socle	C₃²×C₄₃	C₃×H(3,𝔽₃)
hypercenter	C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624735
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₄₃⋊C₃)	C₃	semidirect
2. C₃×#16812506	C₃	semidirect
3. #23782798	C₃	semidirect
4. C₄₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
17. C₃²×C₄₃⋊C₃	H(3,𝔽₃)	semidirect
46. C₃²⋊H(3,𝔽₃)	C₄₃⋊C₃	semidirect
62. C₄₃	#10624735	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃²⋊(C₄₃⋊C₃)	3
C₃×#16812506	3
#23782798	3
C₄₃×C₃²⋊H(3,𝔽₃)	3
#10624735	43