Subgroups — #14518935

Order 1944.

714 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	#11758013	C₂	1	2	yes	maximal
3	C₃×#6229	C₃	1	4	yes	maximal, complemented
4	Dic₆×H(3,𝔽₃)	—	3	4	no	maximal
5	C₂×H(5,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃×C₄×H(3,𝔽₃)	—	2	16	no	—
7	C₃×C₄×H(3,𝔽₃)	D₃	1	4	yes	—
8	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	4	yes	—
9	C₃×#1084	C₂×C₃	1	8	yes	—
10	Dic₃×H(3,𝔽₃)	—	3	8	no	—
11	H(5,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
12	C₃²×Dic₆	—	3	16	no	—
13	Q₈×H(3,𝔽₃)	—	9	1	no	—
14	#6229	C₃²	1	1	yes	characteristic, complemented
15	C₂×C₃×H(3,𝔽₃)	—	2	16	no	—
16	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	4	yes	—
17	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	4	yes	—
18	C₃³×C₄	—	2	12	no	—
19	C₃³×C₄	C₃×D₃	1	16	yes	—
20	C₃²×Dic₃	—	3	32	no	—
21	C₄×H(3,𝔽₃)	—	2	12	no	—
22	C₄×H(3,𝔽₃)	—	2	16	no	—
23	C₄×H(3,𝔽₃)	—	2	16	no	—
24	C₄×H(3,𝔽₃)	—	9	2	no	—
25	C₄×H(3,𝔽₃)	C₃⋊D₃	1	1	yes	characteristic
26	C₄×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
27	#1084	C₂×C₃²	1	2	yes	—
28	C₃×H(3,𝔽₃)	—	2	16	no	—
29	C₃×H(3,𝔽₃)	Dic₆	1	4	yes	complemented
30	C₃×H(3,𝔽₃)	C₃×Q₈	1	4	yes	complemented
31	C₃²×Q₈	—	9	4	no	—
32	C₃×Dic₆	—	3	4	no	—
33	C₃×Dic₆	—	9	16	no	—
34	C₂×C₃³	—	2	12	no	—
35	C₂×C₃³	C₂×C₃×D₃	1	16	yes	—
36	C₂×H(3,𝔽₃)	—	2	12	no	—
37	C₂×H(3,𝔽₃)	—	2	16	no	—
38	C₂×H(3,𝔽₃)	—	2	16	no	—
39	C₂×H(3,𝔽₃)	C₂×C₃⋊D₃	1	1	yes	characteristic
40	C₂×H(3,𝔽₃)	C₂²×C₃²	1	1	yes	characteristic
41	C₃²×C₄	—	2	16	no	—
42	C₃²×C₄	—	9	16	no	—
43	C₃²×C₄	—	9	8	no	—
44	C₃²×C₄	—	18	12	no	—
45	C₃²×C₄	C₃²×D₃	1	4	yes	—
46	C₃²×C₄	C₃×C₃⋊D₃	1	4	yes	—
47	C₃×Dic₃	—	3	8	no	—
48	C₃×Dic₃	—	9	32	no	—
49	C₃³	—	2	12	no	—
50	C₃³	C₃×Dic₆	1	16	yes	complemented
51	H(3,𝔽₃)	—	2	12	no	—
52	H(3,𝔽₃)	—	2	16	no	—
53	H(3,𝔽₃)	—	2	16	no	—
54	H(3,𝔽₃)	#634	1	1	yes	characteristic
55	H(3,𝔽₃)	C₃²×Q₈	1	1	yes	characteristic
56	C₃×Q₈	—	9	1	no	—
57	C₃×Q₈	—	27	4	no	—
58	Dic₆	—	3	4	no	—
59	C₂×C₃²	—	2	16	no	—
60	C₂×C₃²	—	9	16	no	—
61	C₂×C₃²	—	18	12	no	—
62	C₂×C₃²	C₂×C₃²×D₃	1	4	yes	—
63	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	4	yes	—
64	C₃×C₄	—	3	4	no	—
65	C₃×C₄	—	3	4	no	—
66	C₃×C₄	—	6	16	no	—
67	C₃×C₄	—	9	2	no	—
68	C₃×C₄	—	27	8	no	—
69	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
70	Dic₃	—	3	8	no	—
71	C₃²	—	2	16	no	—
72	C₃²	—	9	16	no	—
73	C₃²	—	18	12	no	—
74	C₃²	C₃²×Dic₆	1	4	yes	—
75	C₃²	C₃×#634	1	4	yes	—
76	Q₈	—	9	1	no	Sylow
77	C₂×C₃	—	3	4	no	—
78	C₂×C₃	—	3	4	no	—
79	C₂×C₃	—	6	16	no	—
80	C₂×C₃	C₂×C₃²×C₃⋊D₃	1	1	yes	characteristic
81	C₄	—	9	2	no	—
82	C₄	#92392	1	1	yes	characteristic
83	C₃	—	3	4	no	—
84	C₃	—	3	4	no	—
85	C₃	—	6	16	no	—
86	C₃	C₃²×#634	1	1	yes	characteristic
87	C₂	C₂×#92392	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂×C₃²×C₃⋊D₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂²×C₃²
Fitting	C₄×H(5,𝔽₃)	C₂
prime radical	C₂×H(5,𝔽₃)	C₂²
hypercenter	C₄×H(3,𝔽₃)	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₃²×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₄×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
3. C₃×#6229	C₃	semidirect
11. H(5,𝔽₃)	Q₈	semidirect
14. #6229	C₃²	semidirect
29. C₃×H(3,𝔽₃)	Dic₆	semidirect
30. C₃×H(3,𝔽₃)	C₃×Q₈	semidirect
50. C₃³	C₃×Dic₆	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
#11758013	2
C₃×#6229	3
Dic₆×H(3,𝔽₃)	3