Subgroups — #44534947

Order 31347.

1148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812524	C₃	1	3	yes	maximal, complemented
2	#23782806	C₃	1	24	yes	maximal, complemented
3	#23782798	C₃	1	8	yes	maximal, complemented
4	C₃×C₃²⋊(C₄₃⋊C₉)	C₃	1	4	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)×C₄₃	C₃	1	1	yes	maximal, characteristic
6	#16812524	—	3	27	no	—
7	C₃×#11923958	C₃²	1	24	yes	complemented
8	C₃²×C₄₃⋊C₉	C₃²	1	12	yes	complemented
9	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
10	C₃²⋊(C₄₃⋊C₉)	C₃²	1	8	yes	—
11	C₃²⋊(C₄₃⋊C₉)	C₃²	1	24	yes	complemented
12	C₃²⋊(C₄₃⋊C₉)	C₃²	1	48	yes	complemented
13	C₃⁴×C₄₃	C₃²	1	4	yes	—
14	C₃×H(3,𝔽₃)×C₄₃	—	3	9	no	—
15	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	8	yes	—
16	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
17	#11923958	—	3	72	no	—
18	C₃×C₄₃⋊C₉	—	3	36	no	—
19	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
20	C₃×C₄₃⋊C₉	C₃³	1	24	yes	—
21	C₃³×C₄₃	—	3	12	no	—
22	C₃³×C₄₃	—	3	12	no	—
23	C₃³×C₄₃	—	9	8	no	—
24	C₃³×C₄₃	C₃³	1	8	yes	—
25	C₃³×C₄₃	C₃³	1	4	yes	—
26	C₃³×C₄₃	C₃³	1	1	yes	characteristic
27	H(3,𝔽₃)×C₄₃	—	3	24	no	—
28	H(3,𝔽₃)×C₄₃	H(3,𝔽₃)	1	9	yes	—
29	#10624894	—	43	1	no	maximal, Sylow
30	C₃²×C₄₃	—	3	1	no	—
31	C₃²×C₄₃	—	3	4	no	—
32	C₃²×C₄₃	—	3	2	no	—
33	C₃²×C₄₃	—	3	8	no	—
34	C₃²×C₄₃	—	3	8	no	—
35	C₃²×C₄₃	—	3	8	no	—
36	C₃²×C₄₃	—	3	16	no	—
37	C₃²×C₄₃	—	9	36	no	—
38	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	12	yes	—
39	C₃²×C₄₃	#737	1	3	yes	complemented
40	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
41	C₄₃⋊C₉	—	3	3	no	—
42	C₄₃⋊C₉	—	3	24	no	—
43	C₃×#737	—	43	3	no	—
44	C₃²×H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic
45	#6893	—	43	24	no	—
46	#6891	—	43	8	no	—
47	C₃×C₃²⋊C₉	—	43	4	no	—
48	C₃×C₄₃	—	3	3	no	—
49	C₃×C₄₃	—	3	12	no	—
50	C₃×C₄₃	—	9	8	no	—
51	C₃×C₄₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
53	C₃×C₄₃	3₋¹⁺⁴	1	2	yes	—
54	C₃⁴	C₃×C₄₃⋊C₃	1	4	yes	—
55	#737	—	129	27	no	—
56	C₃×3₋¹⁺²	—	43	24	no	—
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	8	yes	—
59	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
60	C₃²×C₉	—	43	12	no	—
61	C₃²⋊C₉	—	43	1	no	—
62	C₃²⋊C₉	—	43	8	no	—
63	C₃²⋊C₉	—	43	24	no	—
64	C₃²⋊C₉	—	43	48	no	—
65	C₄₃	#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×C₄₃⋊C₃	1	8	yes	—
70	C₃³	C₃²×C₄₃⋊C₃	1	4	yes	—
71	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
72	3₋¹⁺²	—	129	72	no	—
73	H(3,𝔽₃)	—	3	24	no	—
74	H(3,𝔽₃)	C₃²⋊(C₄₃⋊C₃)	1	9	yes	—
75	C₃×C₉	—	43	3	no	—
76	C₃×C₉	—	43	24	no	—
77	C₃×C₉	—	129	36	no	—
78	C₃²	—	3	4	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	8	no	—
81	C₃²	—	3	1	no	—
82	C₃²	—	3	8	no	—
83	C₃²	—	3	16	no	—
84	C₃²	—	3	8	no	—
85	C₃²	—	9	36	no	—
86	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	12	yes	—
87	C₃²	#16812524	1	3	yes	complemented
88	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
89	C₉	—	129	3	no	—
90	C₉	—	129	24	no	—
91	C₃	—	3	3	no	—
92	C₃	—	3	12	no	—
93	C₃	—	9	8	no	—
94	C₃	C₃²×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
95	C₃	C₃×#16812524	1	1	yes	characteristic
96	C₃	#23782924	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₄₃	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624894
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
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×#16812524	C₃	semidirect
2. #23782806	C₃	semidirect
3. #23782798	C₃	semidirect
4. C₃×C₃²⋊(C₄₃⋊C₉)	C₃	semidirect
7. C₃×#11923958	C₃²	semidirect
8. C₃²×C₄₃⋊C₉	C₃²	semidirect
11. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
12. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
39. C₃²×C₄₃	#737	semidirect
65. C₄₃	#10624894	semidirect
87. C₃²	#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
#23782806	3
#23782798	3
C₃×C₃²⋊(C₄₃⋊C₉)	3
C₃²×H(3,𝔽₃)×C₄₃	3
#10624894	43