Subgroups — #44533602

Order 31347.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄₃⋊C₂₇	C₃	1	1	yes	maximal, characteristic
2	C₉×#11923958	C₃	1	1	yes	maximal, characteristic
3	(C₄₃⋊C₉)⋊C₂₇	C₃	1	6	yes	maximal, complemented
4	C₃²⋊(C₄₃⋊C₂₇)	C₃	1	1	yes	maximal, characteristic
5	C₂₇×C₄₃⋊C₉	C₃	1	3	yes	maximal, complemented
6	C₄₃×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
7	C₃×#11923958	C₉	1	1	yes	characteristic
8	C₃×C₄₃⋊C₂₇	—	3	3	no	—
9	C₃×C₄₃⋊C₂₇	C₃²	1	3	yes	—
10	C₃×C₄₃⋊C₂₇	C₃²	1	2	yes	—
11	C₃×C₄₃⋊C₂₇	C₃²	1	1	yes	characteristic
12	(C₄₃⋊C₉)⋊C₉	C₉	1	6	yes	—
13	C₃²⋊(C₄₃⋊C₉)	C₉	1	2	yes	—
14	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
15	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
16	C₃×C₂₇×C₄₃	C₃²	1	3	yes	—
17	#11923958	C₂₇	1	9	yes	complemented
18	C₃×C₄₃⋊C₉	C₃×C₉	1	6	yes	—
19	C₃×C₄₃⋊C₉	C₃×C₉	1	3	yes	—
20	C₄₃⋊C₂₇	—	3	2	no	—
21	C₄₃⋊C₂₇	—	3	3	no	—
22	C₄₃⋊C₂₇	3₋¹⁺²	1	3	yes	complemented
23	C₃³×C₄₃	C₃×C₉	1	1	yes	characteristic
24	C₃×C₉×C₄₃	—	3	1	no	—
25	C₃×C₉×C₄₃	—	3	1	no	—
26	C₃×C₉×C₄₃	—	3	1	no	—
27	C₃×C₉×C₄₃	C₃×C₉	1	2	yes	—
28	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
29	C₂₇×C₄₃	—	3	3	no	—
30	C₂₇×3₋¹⁺²	—	43	1	no	maximal, Sylow
31	C₃²×C₄₃	—	3	1	no	—
32	C₃²×C₄₃	—	3	2	no	—
33	C₃²×C₄₃	C₃×C₂₇	1	3	yes	—
34	C₃²×C₄₃	C₃²×C₉	1	1	yes	characteristic
35	C₉×C₄₃	—	3	2	no	—
36	C₉×C₄₃	C₃×3₋¹⁺²	1	1	yes	characteristic
37	C₉×C₄₃	#737	1	1	yes	characteristic
38	C₉×C₄₃	#737	1	1	yes	characteristic
39	C₄₃⋊C₉	—	3	6	no	—
40	C₄₃⋊C₉	C₃×C₂₇	1	9	yes	complemented
41	C₃²×C₂₇	—	43	1	no	—
42	C₉×3₋¹⁺²	—	43	1	no	—
43	C₉⋊C₂₇	—	43	6	no	—
44	C₃²⋊C₂₇	—	43	1	no	—
45	C₃²⋊C₂₇	C₄₃⋊C₃	1	1	yes	characteristic
46	C₉×C₂₇	—	43	3	no	—
47	C₃×C₄₃	—	3	3	no	—
48	C₃×C₄₃	C₉×3₋¹⁺²	1	1	yes	characteristic
49	C₃×C₄₃	C₃²×C₂₇	1	1	yes	characteristic
50	C₃×C₄₃	#6903	1	2	yes	—
51	C₃×3₋¹⁺²	—	43	1	no	—
52	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
53	C₃×C₂₇	—	43	1	no	—
54	C₃×C₂₇	—	43	3	no	—
55	C₃×C₂₇	—	43	2	no	—
56	C₃×C₂₇	—	129	3	no	—
57	C₃×C₂₇	C₃×C₄₃⋊C₃	1	3	yes	—
58	C₉⋊C₉	—	43	6	no	—
59	C₃²⋊C₉	—	43	2	no	—
60	C₉²	—	43	3	no	—
61	C₄₃	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
62	C₃³	C₉×C₄₃⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	43	9	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	43	3	no	—
68	C₃×C₉	—	43	6	no	—
69	C₃×C₉	C₉×C₄₃⋊C₃	1	2	yes	—
70	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
71	C₂₇	—	3	3	no	—
72	C₂₇	—	43	3	no	—
73	C₂₇	—	129	2	no	—
74	C₂₇	—	129	3	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	C₂₇×C₄₃⋊C₃	1	3	yes	—
78	C₃²	C₃×C₉×C₄₃⋊C₃	1	1	yes	characteristic
79	C₉	—	3	2	no	—
80	C₉	—	43	9	no	—
81	C₉	—	129	6	no	—
82	C₉	3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
83	C₉	#16812524	1	1	yes	characteristic
84	C₉	#16812526	1	1	yes	characteristic
85	C₃	—	3	3	no	—
86	C₃	#23782792	1	1	yes	characteristic
87	C₃	C₃×C₂₇×C₄₃⋊C₃	1	1	yes	characteristic
88	C₃	#23782858	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₄₃⋊C₃
commutator	C₃×C₄₃	C₃²×C₂₇
Fitting	C₄₃×C₃²⋊C₂₇	C₃
socle	C₃²×C₄₃	C₃²×C₉
prime radical	C₃³×C₄₃	C₃×C₉
cosocle	C₃×C₉×C₄₃	C₃³
hypercenter	C₃²⋊C₂₇	C₄₃⋊C₃
hypocenter	C₄₃	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃×C₉ ◁ C₃²⋊C₂₇

Complements

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

N	H ≅ G/N	type
3. (C₄₃⋊C₉)⋊C₂₇	C₃	semidirect
5. C₂₇×C₄₃⋊C₉	C₃	semidirect
17. #11923958	C₂₇	semidirect
22. C₄₃⋊C₂₇	3₋¹⁺²	semidirect
40. C₄₃⋊C₉	C₃×C₂₇	semidirect
61. C₄₃	C₂₇×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄₃⋊C₂₇	3
C₉×#11923958	3
(C₄₃⋊C₉)⋊C₂₇	3
C₃²⋊(C₄₃⋊C₂₇)	3
C₂₇×C₄₃⋊C₉	3
C₄₃×C₃²⋊C₂₇	3
C₂₇×3₋¹⁺²	43