Subgroups — #14517876

Order 1944.

592 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×#6180	C₃	1	4	yes	maximal, complemented
3	C₃⋊C₈×H(3,𝔽₃)	—	3	4	no	maximal
4	C₂×H(5,𝔽₃)	C₄	1	1	yes	characteristic
5	C₃×C₄×H(3,𝔽₃)	—	2	16	no	—
6	C₃×C₄×H(3,𝔽₃)	D₃	1	4	yes	—
7	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	4	yes	—
8	H(5,𝔽₃)	C₈	1	1	yes	characteristic, complemented, Sylow
9	C₃²×C₃⋊C₈	—	3	16	no	—
10	C₈×H(3,𝔽₃)	—	9	1	no	—
11	#6180	C₃²	1	1	yes	characteristic, complemented
12	C₂×C₃×H(3,𝔽₃)	—	2	16	no	—
13	C₂×C₃×H(3,𝔽₃)	Dic₃	1	4	yes	—
14	C₂×C₃×H(3,𝔽₃)	C₃×C₄	1	4	yes	—
15	C₃³×C₄	—	2	12	no	—
16	C₃³×C₄	C₃×D₃	1	16	yes	—
17	C₄×H(3,𝔽₃)	—	2	12	no	—
18	C₄×H(3,𝔽₃)	—	2	16	no	—
19	C₄×H(3,𝔽₃)	—	2	16	no	—
20	C₄×H(3,𝔽₃)	C₃⋊D₃	1	1	yes	characteristic
21	C₄×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
22	C₃×H(3,𝔽₃)	—	2	16	no	—
23	C₃×H(3,𝔽₃)	C₃⋊C₈	1	4	yes	complemented
24	C₃×H(3,𝔽₃)	C₃×C₈	1	4	yes	complemented
25	C₃²×C₈	—	9	4	no	—
26	C₃×C₃⋊C₈	—	3	4	no	—
27	C₃×C₃⋊C₈	—	9	16	no	—
28	C₂×C₃³	—	2	12	no	—
29	C₂×C₃³	C₃×Dic₃	1	16	yes	—
30	C₂×H(3,𝔽₃)	—	2	12	no	—
31	C₂×H(3,𝔽₃)	—	2	16	no	—
32	C₂×H(3,𝔽₃)	—	2	16	no	—
33	C₂×H(3,𝔽₃)	C₃⋊Dic₃	1	1	yes	characteristic
34	C₂×H(3,𝔽₃)	C₃²×C₄	1	1	yes	characteristic
35	C₃²×C₄	—	2	16	no	—
36	C₃²×C₄	—	9	16	no	—
37	C₃²×C₄	—	18	12	no	—
38	C₃²×C₄	C₃²×D₃	1	4	yes	—
39	C₃²×C₄	C₃×C₃⋊D₃	1	4	yes	—
40	C₃³	—	2	12	no	—
41	C₃³	C₃×C₃⋊C₈	1	16	yes	complemented
42	H(3,𝔽₃)	—	2	12	no	—
43	H(3,𝔽₃)	—	2	16	no	—
44	H(3,𝔽₃)	—	2	16	no	—
45	H(3,𝔽₃)	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
46	H(3,𝔽₃)	C₃²×C₈	1	1	yes	characteristic
47	C₃×C₈	—	9	1	no	—
48	C₃×C₈	—	27	4	no	—
49	C₃⋊C₈	—	3	4	no	—
50	C₂×C₃²	—	2	16	no	—
51	C₂×C₃²	—	9	16	no	—
52	C₂×C₃²	—	18	12	no	—
53	C₂×C₃²	C₃²×Dic₃	1	4	yes	—
54	C₂×C₃²	C₃×C₃⋊Dic₃	1	4	yes	—
55	C₃×C₄	—	3	4	no	—
56	C₃×C₄	—	3	4	no	—
57	C₃×C₄	—	6	16	no	—
58	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
59	C₃²	—	2	16	no	—
60	C₃²	—	9	16	no	—
61	C₃²	—	18	12	no	—
62	C₃²	C₃²×C₃⋊C₈	1	4	yes	—
63	C₃²	C₃×C₃⋊(C₃⋊C₈)	1	4	yes	—
64	C₈	—	9	1	no	Sylow
65	C₂×C₃	—	3	4	no	—
66	C₂×C₃	—	3	4	no	—
67	C₂×C₃	—	6	16	no	—
68	C₂×C₃	C₃²×C₃⋊Dic₃	1	1	yes	characteristic
69	C₄	#92392	1	1	yes	characteristic
70	C₃	—	3	4	no	—
71	C₃	—	3	4	no	—
72	C₃	—	6	16	no	—
73	C₃	C₃²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
74	C₂	#11758013	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₄	C₃²×C₃⋊D₃
commutator, hypocenter	H(3,𝔽₃)	C₃²×C₈
Fitting	C₄×H(5,𝔽₃)	C₂
socle	C₂×C₃	C₃²×C₃⋊Dic₃
prime radical	C₂×H(5,𝔽₃)	C₄
cosocle	C₄×H(3,𝔽₃)	C₂×C₃²
hypercenter	C₄×H(3,𝔽₃)	C₃⋊D₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₄ ◁ C₄×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃×#6180	C₃	semidirect
8. H(5,𝔽₃)	C₈	semidirect
11. #6180	C₃²	semidirect
23. C₃×H(3,𝔽₃)	C₃⋊C₈	semidirect
24. C₃×H(3,𝔽₃)	C₃×C₈	semidirect
41. C₃³	C₃×C₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×H(5,𝔽₃)	2
C₃×#6180	3
C₃⋊C₈×H(3,𝔽₃)	3