Subgroups — C₃×#4625 #10590103

Order 576.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#120⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#4625	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×AΣL(1,𝔽₁₆)	—	3	1	no	maximal
4	C₃×C₂²⋊A₄	C₄	1	1	yes	characteristic, complemented
5	C₃×SL(2,ℤ/4)	—	4	1	no	maximal
6	C₃×#120	D₃	1	1	yes	characteristic
7	#120⋊C₃	—	2	1	no	—
8	#120⋊C₃	C₂×C₃	1	1	yes	characteristic
9	C₃×#100	—	3	1	no	—
10	C₃×#99	—	3	1	no	—
11	C₂×C₃×A₄	—	4	1	no	—
12	AΣL(1,𝔽₁₆)	—	3	1	no	Sylow
13	C₂⁴×C₃	Dic₃	1	1	yes	characteristic, complemented
14	C₂²⋊A₄	—	2	1	no	—
15	C₂²⋊A₄	C₃×C₄	1	1	yes	characteristic, complemented
16	C₂×C₃×D₄	—	3	1	no	—
17	SL(2,ℤ/4)	—	4	1	no	—
18	C₃×M₁₆	—	6	1	no	—
19	C₃×C₂²⋊C₄	—	3	1	no	—
20	C₃×C₂²⋊C₄	—	6	1	no	—
21	C₃×A₄	—	4	1	no	—
22	C₃×A₄	—	16	1	no	—
23	C₃×Dic₃	—	16	1	no	—
24	#120	C₃×D₃	1	1	yes	characteristic
25	#100	—	3	1	no	—
26	#99	—	3	1	no	—
27	C₂³×C₃	—	3	1	no	—
28	C₂³×C₃	—	12	1	no	—
29	C₂³×C₃	S₄	1	1	yes	characteristic
30	C₂×A₄	—	4	1	no	—
31	C₂×A₄	—	8	1	no	—
32	C₃×D₄	—	12	1	no	—
33	C₂×C₃×C₄	—	3	1	no	—
34	C₂×C₃×C₄	—	12	1	no	—
35	C₃×C₈	—	12	1	no	—
36	C₂×C₃²	—	16	1	no	—
37	C₂⁴	C₃×Dic₃	1	1	yes	characteristic, complemented
38	C₂×D₄	—	3	1	no	—
39	M₁₆	—	6	1	no	—
40	C₂²⋊C₄	—	3	1	no	—
41	C₂²⋊C₄	—	6	1	no	—
42	C₂²×C₃	—	4	1	no	—
43	C₂²×C₃	—	6	1	no	—
44	C₂²×C₃	—	6	1	no	—
45	C₂²×C₃	—	12	1	no	—
46	C₂²×C₃	—	12	1	no	—
47	C₂²×C₃	SL(2,ℤ/4)	1	1	yes	characteristic
48	A₄	—	4	1	no	—
49	A₄	—	8	1	no	—
50	A₄	—	16	1	no	—
51	A₄	—	16	2	no	—
52	C₃×C₄	—	6	1	no	—
53	C₃×C₄	—	24	1	no	—
54	Dic₃	—	16	1	no	—
55	C₃²	—	16	1	no	Sylow
56	C₂³	—	3	1	no	—
57	C₂³	—	12	1	no	—
58	C₂³	C₃×S₄	1	1	yes	characteristic
59	D₄	—	12	1	no	—
60	C₂×C₄	—	3	1	no	—
61	C₂×C₄	—	12	1	no	—
62	C₈	—	12	1	no	—
63	C₂×C₃	—	3	1	no	—
64	C₂×C₃	—	4	1	no	—
65	C₂×C₃	—	12	1	no	—
66	C₂×C₃	—	16	1	no	—
67	C₂×C₃	—	32	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	6	1	no	—
70	C₂²	—	6	1	no	—
71	C₂²	—	12	1	no	—
72	C₂²	—	12	1	no	—
73	C₂²	C₃×SL(2,ℤ/4)	1	1	yes	characteristic
74	C₄	—	6	1	no	—
75	C₄	—	24	1	no	—
76	C₃	—	16	1	no	—
77	C₃	—	32	1	no	—
78	C₃	#4625	1	1	yes	characteristic, direct product factor
79	C₂	—	3	1	no	—
80	C₂	—	4	1	no	—
81	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₃	#4625
commutator, hypocenter	C₂²⋊A₄	C₃×C₄
Frattini	C₂³	C₃×S₄
Fitting	C₃×#120	D₃
socle	C₂²×C₃	SL(2,ℤ/4)
prime radical	C₃×#120⋊C₃	C₂
cosocle	#120⋊C₃	C₂×C₃
LCS	G ▷ C₂²⋊A₄
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. #4625	C₃	direct (N × H)
4. C₃×C₂²⋊A₄	C₄	semidirect
13. C₂⁴×C₃	Dic₃	semidirect
15. C₂²⋊A₄	C₃×C₄	semidirect
37. C₂⁴	C₃×Dic₃	semidirect
78. C₃	#4625	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#120⋊C₃	2
#4625	3
C₃×AΣL(1,𝔽₁₆)	3
C₃×SL(2,ℤ/4)	4

Subgroups — C3×#4625 #10590103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#4625 #10590103