Subgroups — C₂³×C₃×2₋¹⁺⁴ #10681706

Order 768.

59806 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×2₋¹⁺⁴	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₃×#41	C₂	1	10	yes	maximal, complemented
3	C₂⁴×C₃×Q₈	C₂	1	5	yes	maximal, complemented
4	C₂³×2₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂×C₃×2₋¹⁺⁴	C₂²	1	1792	yes	direct product factor
6	C₂²×C₃×#41	C₂²	1	560	yes	complemented
7	C₂³×C₃×Q₈	C₂²	1	10	yes	—
8	C₂³×C₃×Q₈	C₂²	1	280	yes	complemented
9	C₂³×C₃×D₄	C₂²	1	10	yes	—
10	C₂⁴×C₃×C₄	C₂²	1	15	yes	—
11	C₂²×2₋¹⁺⁴	C₂×C₃	1	112	yes	direct product factor
12	C₂³×#41	C₂×C₃	1	10	yes	complemented
13	C₂⁴×Q₈	C₂×C₃	1	5	yes	complemented
14	C₂⁵×C₃	C₂³	1	5	yes	—
15	C₃×2₋¹⁺⁴	C₂³	1	4096	yes	direct product factor
16	C₂×C₃×#41	C₂³	1	4480	yes	complemented
17	C₂²×C₃×Q₈	C₂³	1	280	yes	—
18	C₂²×C₃×Q₈	C₂³	1	2240	yes	complemented
19	C₂²×C₃×D₄	C₂³	1	280	yes	—
20	C₂³×C₃×C₄	C₂³	1	420	yes	—
21	C₂³×C₃×C₄	C₂³	1	10	yes	—
22	C₂×2₋¹⁺⁴	C₂²×C₃	1	1792	yes	direct product factor
23	C₂²×#41	C₂²×C₃	1	560	yes	complemented
24	C₂³×Q₈	C₂²×C₃	1	10	yes	—
25	C₂³×Q₈	C₂²×C₃	1	280	yes	complemented
26	C₂³×D₄	C₂²×C₃	1	10	yes	—
27	C₂⁴×C₄	C₂²×C₃	1	15	yes	—
28	C₂⁴×C₃	—	2	40	no	—
29	C₂⁴×C₃	C₂⁴	1	70	yes	—
30	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
31	C₃×#41	C₂⁴	1	5120	yes	complemented
32	C₂×C₃×Q₈	C₂⁴	1	1120	yes	—
33	C₂×C₃×Q₈	C₂⁴	1	2560	yes	complemented
34	C₂×C₃×D₄	C₂⁴	1	1120	yes	—
35	C₂²×C₃×C₄	C₂⁴	1	140	yes	—
36	C₂²×C₃×C₄	C₂⁴	1	1680	yes	—
37	C₂⁵	C₂³×C₃	1	5	yes	—
38	2₋¹⁺⁴	C₂³×C₃	1	4096	yes	direct product factor
39	C₂×#41	C₂³×C₃	1	4480	yes	complemented
40	C₂²×Q₈	C₂³×C₃	1	280	yes	—
41	C₂²×Q₈	C₂³×C₃	1	2240	yes	complemented
42	C₂²×D₄	C₂³×C₃	1	280	yes	—
43	C₂³×C₄	C₂³×C₃	1	10	yes	—
44	C₂³×C₄	C₂³×C₃	1	420	yes	—
45	C₂³×C₃	—	2	280	no	—
46	C₂³×C₃	2₋¹⁺⁴	1	8	yes	direct product factor
47	C₂³×C₃	C₂⁵	1	140	yes	—
48	C₂³×C₃	C₂⁵	1	7	yes	—
49	C₃×Q₈	C₂⁵	1	640	yes	—
50	C₃×D₄	C₂⁵	1	640	yes	—
51	C₂×C₃×C₄	C₂⁵	1	280	yes	—
52	C₂×C₃×C₄	C₂⁵	1	960	yes	—
53	C₂⁴	—	2	40	no	—
54	C₂⁴	C₂⁴×C₃	1	70	yes	—
55	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
56	#41	C₂⁴×C₃	1	5120	yes	complemented
57	C₂×Q₈	C₂⁴×C₃	1	1120	yes	—
58	C₂×Q₈	C₂⁴×C₃	1	2560	yes	complemented
59	C₂×D₄	C₂⁴×C₃	1	1120	yes	—
60	C₂²×C₄	C₂⁴×C₃	1	140	yes	—
61	C₂²×C₄	C₂⁴×C₃	1	1680	yes	—
62	C₂²×C₃	—	2	280	no	—
63	C₂²×C₃	C₂×2₋¹⁺⁴	1	28	yes	direct product factor
64	C₂²×C₃	C₂⁶	1	40	yes	—
65	C₂²×C₃	C₂⁶	1	7	yes	—
66	C₃×C₄	C₂⁶	1	80	yes	—
67	C₂³	—	2	280	no	—
68	C₂³	C₃×2₋¹⁺⁴	1	8	yes	direct product factor
69	C₂³	C₂⁵×C₃	1	140	yes	—
70	C₂³	C₂⁵×C₃	1	7	yes	—
71	Q₈	C₂⁵×C₃	1	640	yes	—
72	D₄	C₂⁵×C₃	1	640	yes	—
73	C₂×C₄	C₂⁵×C₃	1	280	yes	—
74	C₂×C₄	C₂⁵×C₃	1	960	yes	—
75	C₂×C₃	—	2	40	no	—
76	C₂×C₃	C₂²×2₋¹⁺⁴	1	14	yes	direct product factor
77	C₂×C₃	C₂⁷	1	1	yes	characteristic
78	C₂²	—	2	280	no	—
79	C₂²	C₂×C₃×2₋¹⁺⁴	1	28	yes	direct product factor
80	C₂²	C₂⁶×C₃	1	40	yes	—
81	C₂²	C₂⁶×C₃	1	7	yes	—
82	C₄	C₂⁶×C₃	1	80	yes	—
83	C₃	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	2	40	no	—
85	C₂	C₂²×C₃×2₋¹⁺⁴	1	14	yes	direct product factor
86	C₂	C₂⁷×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁷×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×2₋¹⁺⁴	C₂	direct (N × H)
2. C₂³×C₃×#41	C₂	semidirect
3. C₂⁴×C₃×Q₈	C₂	semidirect
4. C₂³×2₋¹⁺⁴	C₃	direct (N × H)
5. C₂×C₃×2₋¹⁺⁴	C₂²	direct (N × H)
6. C₂²×C₃×#41	C₂²	semidirect
8. C₂³×C₃×Q₈	C₂²	semidirect
11. C₂²×2₋¹⁺⁴	C₂×C₃	direct (N × H)
12. C₂³×#41	C₂×C₃	semidirect
13. C₂⁴×Q₈	C₂×C₃	semidirect
15. C₃×2₋¹⁺⁴	C₂³	direct (N × H)
16. C₂×C₃×#41	C₂³	semidirect
18. C₂²×C₃×Q₈	C₂³	semidirect
22. C₂×2₋¹⁺⁴	C₂²×C₃	direct (N × H)
23. C₂²×#41	C₂²×C₃	semidirect
25. C₂³×Q₈	C₂²×C₃	semidirect
31. C₃×#41	C₂⁴	semidirect
33. C₂×C₃×Q₈	C₂⁴	semidirect
38. 2₋¹⁺⁴	C₂³×C₃	direct (N × H)
39. C₂×#41	C₂³×C₃	semidirect
41. C₂²×Q₈	C₂³×C₃	semidirect
46. C₂³×C₃	2₋¹⁺⁴	direct (N × H)
56. #41	C₂⁴×C₃	semidirect
58. C₂×Q₈	C₂⁴×C₃	semidirect
63. C₂²×C₃	C₂×2₋¹⁺⁴	direct (N × H)
68. C₂³	C₃×2₋¹⁺⁴	direct (N × H)
76. C₂×C₃	C₂²×2₋¹⁺⁴	direct (N × H)
79. C₂²	C₂×C₃×2₋¹⁺⁴	direct (N × H)
83. C₃	C₂³×2₋¹⁺⁴	direct (N × H)
85. C₂	C₂²×C₃×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×2₋¹⁺⁴	2
C₂³×C₃×#41	2
C₂⁴×C₃×Q₈	2
C₂³×2₋¹⁺⁴	3

Subgroups — C23×C3×2−1+4 #10681706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×2₋¹⁺⁴ #10681706