Subgroups — C₃×#559 #5889

Order 192.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×#124	C₂	1	4	yes	maximal, complemented
3	C₃×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×D₄	C₂	1	4	yes	maximal, complemented
6	#559	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×D₄	C₂²	1	4	yes	—
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂²	1	4	yes	—
11	C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₂²⋊C₄	C₂²	1	8	yes	—
14	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₄²	C₂²	1	4	yes	—
16	C₃×C₄²	C₂²	1	1	yes	characteristic
17	#127	C₂×C₃	1	1	yes	characteristic
18	#124	C₂×C₃	1	4	yes	complemented
19	#123	C₂×C₃	1	4	yes	complemented
20	C₄×Q₈	C₂×C₃	1	2	yes	complemented
21	C₄×D₄	C₂×C₃	1	4	yes	complemented
22	C₂³×C₃	C₂³	1	4	yes	—
23	C₃×Q₈	—	2	4	no	—
24	C₃×D₄	—	2	8	no	—
25	C₃×D₄	—	4	2	no	—
26	C₂×C₃×C₄	—	4	4	no	—
27	C₂×C₃×C₄	C₂³	1	4	yes	—
28	C₂×C₃×C₄	C₂³	1	4	yes	—
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	2	yes	—
31	C₂×Q₈	C₂²×C₃	1	2	yes	—
32	C₂×D₄	C₂²×C₃	1	4	yes	—
33	C₂×D₄	C₂²×C₃	1	2	yes	—
34	C₂²×C₄	C₂²×C₃	1	4	yes	—
35	C₄⋊C₄	C₂²×C₃	1	2	yes	—
36	C₄⋊C₄	C₂²×C₃	1	4	yes	—
37	C₂²⋊C₄	C₂²×C₃	1	8	yes	—
38	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
39	C₄²	C₂²×C₃	1	4	yes	—
40	C₄²	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₃	—	2	8	no	—
42	C₂²×C₃	—	2	4	no	—
43	C₂²×C₃	C₂⁴	1	1	yes	characteristic
44	C₃×C₄	—	2	4	no	—
45	C₃×C₄	—	2	4	no	—
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	#41	1	4	yes	—
48	C₂³	C₂³×C₃	1	4	yes	—
49	Q₈	—	2	4	no	—
50	D₄	—	2	8	no	—
51	D₄	—	4	2	no	—
52	C₂×C₄	—	4	4	no	—
53	C₂×C₄	C₂³×C₃	1	4	yes	—
54	C₂×C₄	C₂³×C₃	1	4	yes	—
55	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
56	C₂×C₄	C₂³×C₃	1	2	yes	—
57	C₂×C₃	—	4	4	no	—
58	C₂×C₃	C₂×#41	1	2	yes	—
59	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂²	—	2	8	no	—
61	C₂²	—	2	4	no	—
62	C₂²	C₂⁴×C₃	1	1	yes	characteristic
63	C₄	—	2	4	no	—
64	C₄	—	2	4	no	—
65	C₄	—	2	1	no	—
66	C₄	C₃×#41	1	4	yes	—
67	C₃	#559	1	1	yes	characteristic, direct product factor, Sylow
68	C₂	—	4	4	no	—
69	C₂	C₂×C₃×#41	1	2	yes	—
70	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×#124	C₂	semidirect
3. C₃×#123	C₂	semidirect
4. C₃×C₄×Q₈	C₂	semidirect
5. C₃×C₄×D₄	C₂	semidirect
6. #559	C₃	direct (N × H)
18. #124	C₂×C₃	semidirect
19. #123	C₂×C₃	semidirect
20. C₄×Q₈	C₂×C₃	semidirect
21. C₄×D₄	C₂×C₃	semidirect
67. C₃	#559	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×#127	2
C₃×#124	2
C₃×#123	2
C₃×C₄×Q₈	2
C₃×C₄×D₄	2
#559	3

Subgroups — C3×#559 #5889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#559 #5889