Subgroups — C₃×#520 #5850

Order 192.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Q₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×D₄	C₂	1	6	yes	maximal, complemented
4	C₃×#117	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	#520	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#41	C₄	1	8	yes	complemented
8	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₄	C₂²	1	6	yes	—
11	C₂²×C₃×C₄	C₂²	1	3	yes	—
12	C₃×C₄⋊C₄	C₂²	1	3	yes	—
13	C₃×C₄⋊C₄	C₂²	1	6	yes	—
14	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₂²⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄²	C₂²	1	6	yes	—
17	C₂×#41	C₂×C₃	1	1	yes	characteristic
18	C₄×Q₈	C₂×C₃	1	2	yes	complemented
19	C₄×D₄	C₂×C₃	1	6	yes	complemented
20	#117	C₂×C₃	1	3	yes	complemented
21	C₂×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
22	C₂³×C₃	C₂³	1	3	yes	—
23	C₃×Q₈	C₂×C₄	1	4	yes	complemented
24	C₃×D₄	C₂×C₄	1	12	yes	complemented
25	C₂×C₃×C₄	—	4	6	no	—
26	C₂×C₃×C₄	C₂×C₄	1	12	yes	complemented
27	C₂×C₃×C₄	C₂³	1	2	yes	—
28	C₂×C₃×C₄	C₂³	1	6	yes	—
29	C₂×C₃×C₄	C₂³	1	3	yes	—
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	#41	C₃×C₄	1	8	yes	complemented
32	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic
33	C₂×D₄	C₂²×C₃	1	3	yes	—
34	C₂²×C₄	C₂²×C₃	1	6	yes	—
35	C₂²×C₄	C₂²×C₃	1	3	yes	—
36	C₄⋊C₄	C₂²×C₃	1	3	yes	—
37	C₄⋊C₄	C₂²×C₃	1	6	yes	—
38	C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
39	C₂²⋊C₄	C₂²×C₃	1	6	yes	—
40	C₄²	C₂²×C₃	1	6	yes	—
41	C₂²×C₃	—	2	3	no	—
42	C₂²×C₃	—	2	3	no	—
43	C₂²×C₃	C₂²×C₄	1	6	yes	—
44	C₂²×C₃	C₂⁴	1	1	yes	characteristic
45	C₃×C₄	—	2	6	no	—
46	C₃×C₄	—	2	2	no	—
47	C₃×C₄	C₂²×C₄	1	2	yes	—
48	C₃×C₄	C₂²×C₄	1	6	yes	—
49	C₂³	C₂³×C₃	1	3	yes	—
50	Q₈	C₂×C₃×C₄	1	4	yes	complemented
51	D₄	C₂×C₃×C₄	1	12	yes	complemented
52	C₂×C₄	—	4	6	no	—
53	C₂×C₄	C₂×C₃×C₄	1	12	yes	complemented
54	C₂×C₄	C₂³×C₃	1	2	yes	—
55	C₂×C₄	C₂³×C₃	1	6	yes	—
56	C₂×C₄	C₂³×C₃	1	3	yes	—
57	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
58	C₂×C₃	—	2	6	no	—
59	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
60	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂²	—	2	3	no	—
63	C₂²	—	2	3	no	—
64	C₂²	C₂²×C₃×C₄	1	6	yes	—
65	C₂²	C₂⁴×C₃	1	1	yes	characteristic
66	C₄	—	2	6	no	—
67	C₄	—	2	2	no	—
68	C₄	C₂²×C₃×C₄	1	2	yes	—
69	C₄	C₂²×C₃×C₄	1	6	yes	—
70	C₃	#520	1	1	yes	characteristic, direct product factor, Sylow
71	C₂	—	2	6	no	—
72	C₂	C₂³×C₃×C₄	1	1	yes	characteristic
73	C₂	C₃×H(5,𝔽₂)	1	1	yes	characteristic
74	C₂	C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂×C₃×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Q₈	C₂	semidirect
3. C₃×C₄×D₄	C₂	semidirect
4. C₃×#117	C₂	semidirect
5. C₂×C₃×C₄⋊C₄	C₂	semidirect
6. #520	C₃	direct (N × H)
7. C₃×#41	C₄	semidirect
18. C₄×Q₈	C₂×C₃	semidirect
19. C₄×D₄	C₂×C₃	semidirect
20. #117	C₂×C₃	semidirect
21. C₂×C₄⋊C₄	C₂×C₃	semidirect
23. C₃×Q₈	C₂×C₄	semidirect
24. C₃×D₄	C₂×C₄	semidirect
26. C₂×C₃×C₄	C₂×C₄	semidirect
31. #41	C₃×C₄	semidirect
50. Q₈	C₂×C₃×C₄	semidirect
51. D₄	C₂×C₃×C₄	semidirect
53. C₂×C₄	C₂×C₃×C₄	semidirect
70. C₃	#520	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×C₄×Q₈	2
C₃×C₄×D₄	2
C₃×#117	2
C₂×C₃×C₄⋊C₄	2
#520	3

Subgroups — C3×#520 #5850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#520 #5850