Subgroups — C₃×#2380 #84244

Order 384.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×#386	C₂	1	12	yes	maximal, complemented
3	C₃×#383	C₂	1	2	yes	maximal, complemented
4	#2380	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×D₄	C₂²	1	6	yes	—
6	C₂³×C₃×C₄	C₂²	1	4	yes	—
7	C₃×#127	C₄	1	8	yes	complemented
8	C₂×C₃×C₂²⋊C₄	C₂²	1	12	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₃×#95	C₂²	1	12	yes	—
11	C₂×#127	C₂×C₃	1	1	yes	characteristic
12	#386	C₂×C₃	1	12	yes	complemented
13	#383	C₂×C₃	1	2	yes	complemented
14	C₂⁴×C₃	C₂³	1	4	yes	—
15	C₂×C₃×D₄	—	4	6	no	—
16	C₂×C₃×D₄	—	4	6	no	—
17	C₂×C₃×D₄	C₂×C₄	1	24	yes	complemented
18	C₂²×C₃×C₄	—	4	12	no	—
19	C₂²×C₃×C₄	C₂³	1	8	yes	—
20	C₂²×C₃×C₄	C₂³	1	3	yes	—
21	C₃×C₂²⋊C₄	—	4	12	no	—
22	C₃×C₄²	C₂×C₄	1	4	yes	complemented
23	C₂²×D₄	C₂²×C₃	1	6	yes	—
24	C₂³×C₄	C₂²×C₃	1	4	yes	—
25	#127	C₃×C₄	1	8	yes	complemented
26	C₂×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
27	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
28	#95	C₂²×C₃	1	12	yes	—
29	C₂³×C₃	—	2	12	no	—
30	C₂³×C₃	—	2	12	no	—
31	C₂³×C₃	C₂²×C₄	1	8	yes	—
32	C₂³×C₃	C₂⁴	1	1	yes	characteristic
33	C₃×D₄	—	4	24	no	—
34	C₂×C₃×C₄	—	2	6	no	—
35	C₂×C₃×C₄	—	2	24	no	—
36	C₂×C₃×C₄	—	4	16	no	—
37	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
38	C₂⁴	C₂³×C₃	1	4	yes	—
39	C₂×D₄	—	4	6	no	—
40	C₂×D₄	—	4	6	no	—
41	C₂×D₄	C₂×C₃×C₄	1	24	yes	complemented
42	C₂²×C₄	—	4	12	no	—
43	C₂²×C₄	C₂³×C₃	1	8	yes	—
44	C₂²×C₄	C₂³×C₃	1	3	yes	—
45	C₂²⋊C₄	—	4	12	no	—
46	C₄²	C₂×C₃×C₄	1	4	yes	complemented
47	C₂²×C₃	—	2	24	no	—
48	C₂²×C₃	—	4	4	no	—
49	C₂²×C₃	—	4	12	no	—
50	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
51	C₂²×C₃	H(5,𝔽₂)	1	6	yes	—
52	C₃×C₄	—	2	6	no	—
53	C₃×C₄	—	4	8	no	—
54	C₂³	—	2	12	no	—
55	C₂³	—	2	12	no	—
56	C₂³	C₂²×C₃×C₄	1	8	yes	—
57	C₂³	C₂⁴×C₃	1	1	yes	characteristic
58	D₄	—	4	24	no	—
59	C₂×C₄	—	2	6	no	—
60	C₂×C₄	—	2	24	no	—
61	C₂×C₄	—	4	16	no	—
62	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
63	C₂×C₃	—	4	8	no	—
64	C₂×C₃	#518	1	3	yes	—
65	C₂×C₃	#560	1	4	yes	—
66	C₂²	—	2	24	no	—
67	C₂²	—	4	4	no	—
68	C₂²	—	4	12	no	—
69	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
70	C₂²	C₃×H(5,𝔽₂)	1	6	yes	—
71	C₄	—	2	6	no	—
72	C₄	—	4	8	no	—
73	C₃	#2380	1	1	yes	characteristic, direct product factor, Sylow
74	C₂	—	4	8	no	—
75	C₂	C₃×#518	1	3	yes	—
76	C₂	C₃×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂×C₃×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#386	C₂	semidirect
3. C₃×#383	C₂	semidirect
4. #2380	C₃	direct (N × H)
7. C₃×#127	C₄	semidirect
12. #386	C₂×C₃	semidirect
13. #383	C₂×C₃	semidirect
17. C₂×C₃×D₄	C₂×C₄	semidirect
22. C₃×C₄²	C₂×C₄	semidirect
25. #127	C₃×C₄	semidirect
41. C₂×D₄	C₂×C₃×C₄	semidirect
46. C₄²	C₂×C₃×C₄	semidirect
73. C₃	#2380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#127	2
C₃×#386	2
C₃×#383	2
#2380	3

Subgroups — C3×#2380 #84244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2380 #84244