Subgroups — C₃×#550 #5880

Order 192.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	2	yes	maximal, complemented
2	C₃×#127	C₂	1	3	yes	maximal, complemented
3	C₃×#121	C₂	1	6	yes	maximal, complemented
4	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×D₄	C₂	1	3	yes	maximal, complemented
6	#550	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#41	—	2	8	no	—
8	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄	C₂²	1	3	yes	—
10	C₂×C₃×D₄	C₂²	1	6	yes	—
11	C₂×C₃×D₄	C₂²	1	6	yes	—
12	C₂²×C₃×C₄	C₂²	1	6	yes	—
13	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊C₄	C₂²	1	3	yes	—
15	C₃×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄²	C₂²	1	3	yes	—
17	C₂×#41	C₂×C₃	1	2	yes	complemented
18	#127	C₂×C₃	1	3	yes	complemented
19	#121	C₂×C₃	1	6	yes	complemented
20	C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₄×D₄	C₂×C₃	1	3	yes	complemented
22	C₂³×C₃	C₂³	1	6	yes	—
23	C₃×Q₈	D₄	1	4	yes	complemented
24	C₃×D₄	—	2	12	no	—
25	C₃×D₄	—	2	6	no	—
26	C₃×D₄	—	4	6	no	—
27	C₂×C₃×C₄	—	2	12	no	—
28	C₂×C₃×C₄	C₂³	1	3	yes	—
29	C₂×C₃×C₄	C₂³	1	2	yes	—
30	C₂×C₃×C₄	C₂³	1	3	yes	—
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	#41	—	2	8	no	—
33	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic
34	C₂×D₄	C₂²×C₃	1	3	yes	—
35	C₂×D₄	C₂²×C₃	1	6	yes	—
36	C₂×D₄	C₂²×C₃	1	6	yes	—
37	C₂²×C₄	C₂²×C₃	1	6	yes	—
38	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
39	C₄⋊C₄	C₂²×C₃	1	3	yes	—
40	C₂²⋊C₄	C₂²×C₃	1	6	yes	—
41	C₄²	C₂²×C₃	1	3	yes	—
42	C₂²×C₃	—	2	6	no	—
43	C₂²×C₃	—	2	6	no	—
44	C₂²×C₃	—	2	6	no	—
45	C₂²×C₃	C₂⁴	1	1	yes	characteristic
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	—	2	3	no	—
48	C₃×C₄	C₂×D₄	1	6	yes	—
49	C₃×C₄	#41	1	2	yes	complemented
50	C₂³	C₂³×C₃	1	6	yes	—
51	Q₈	C₃×D₄	1	4	yes	complemented
52	D₄	—	2	12	no	—
53	D₄	—	2	6	no	—
54	D₄	—	4	6	no	—
55	C₂×C₄	—	2	12	no	—
56	C₂×C₄	C₂³×C₃	1	3	yes	—
57	C₂×C₄	C₂³×C₃	1	2	yes	—
58	C₂×C₄	C₂³×C₃	1	3	yes	—
59	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
60	C₂×C₃	—	4	6	no	—
61	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₃	C₂×#41	1	1	yes	characteristic
63	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂²	—	2	6	no	—
65	C₂²	—	2	6	no	—
66	C₂²	—	2	6	no	—
67	C₂²	C₂⁴×C₃	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	2	3	no	—
70	C₄	C₂×C₃×D₄	1	6	yes	—
71	C₄	C₃×#41	1	2	yes	complemented
72	C₃	#550	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	—	4	6	no	—
74	C₂	C₂²×C₃×D₄	1	1	yes	characteristic
75	C₂	C₂×C₃×#41	1	1	yes	characteristic
76	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
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
1. C₂×C₃×#41	C₂	semidirect
2. C₃×#127	C₂	semidirect
3. C₃×#121	C₂	semidirect
4. C₃×C₄×Q₈	C₂	semidirect
5. C₃×C₄×D₄	C₂	semidirect
6. #550	C₃	direct (N × H)
17. C₂×#41	C₂×C₃	semidirect
18. #127	C₂×C₃	semidirect
19. #121	C₂×C₃	semidirect
20. C₄×Q₈	C₂×C₃	semidirect
21. C₄×D₄	C₂×C₃	semidirect
23. C₃×Q₈	D₄	semidirect
49. C₃×C₄	#41	semidirect
51. Q₈	C₃×D₄	semidirect
71. C₄	C₃×#41	semidirect
72. C₃	#550	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×#41	2
C₃×#127	2
C₃×#121	2
C₃×C₄×Q₈	2
C₃×C₄×D₄	2
#550	3

Subgroups — C3×#550 #5880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#550 #5880