Subgroups — C₂⁴×C₇×A₄ #13062422

Order 1344.

2330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×A₄	C₂	1	15	yes	maximal, direct product factor
2	C₂⁶×C₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂⁴×C₃×C₇	—	4	1	no	maximal
4	C₂²×C₇×A₄	C₂²	1	35	yes	direct product factor
5	C₂⁵×C₇	—	3	15	no	—
6	C₂⁵×C₇	—	3	1	no	—
7	C₂⁵×C₇	C₂×C₃	1	15	yes	complemented
8	C₂⁴×A₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×C₇	—	4	15	no	—
10	C₂×C₇×A₄	C₂³	1	15	yes	direct product factor
11	C₂⁴×C₇	—	3	70	no	—
12	C₂⁴×C₇	—	3	15	no	—
13	C₂⁴×C₇	—	3	15	no	—
14	C₂⁴×C₇	—	3	105	no	—
15	C₂⁴×C₇	A₄	1	1	yes	characteristic, direct product factor
16	C₂⁴×C₇	C₂²×C₃	1	35	yes	complemented
17	C₂³×A₄	C₂×C₇	1	15	yes	direct product factor
18	C₂²×C₃×C₇	—	4	35	no	—
19	C₇×A₄	C₂⁴	1	1	yes	characteristic, direct product factor
20	C₂⁶	C₃×C₇	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₇	—	3	105	no	—
22	C₂³×C₇	—	3	210	no	—
23	C₂³×C₇	—	3	105	no	—
24	C₂³×C₇	—	3	35	no	—
25	C₂³×C₇	C₂×A₄	1	15	yes	direct product factor
26	C₂³×C₇	C₂³×C₃	1	15	yes	complemented
27	C₂⁴×C₃	—	4	1	no	—
28	C₂²×A₄	C₂²×C₇	1	35	yes	direct product factor
29	C₂×C₃×C₇	—	4	15	no	—
30	C₂⁵	—	3	15	no	—
31	C₂⁵	—	3	1	no	—
32	C₂⁵	C₂×C₃×C₇	1	15	yes	complemented
33	C₂²×C₇	—	3	15	no	—
34	C₂²×C₇	—	3	70	no	—
35	C₂²×C₇	—	3	105	no	—
36	C₂²×C₇	—	3	15	no	—
37	C₂²×C₇	C₂²×A₄	1	35	yes	direct product factor
38	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic, complemented
39	C₂³×C₃	—	4	15	no	—
40	C₂×A₄	C₂³×C₇	1	15	yes	direct product factor
41	C₃×C₇	—	4	1	no	Hall
42	C₂⁴	—	3	105	no	—
43	C₂⁴	—	3	70	no	—
44	C₂⁴	—	3	15	no	—
45	C₂⁴	—	3	15	no	—
46	C₂⁴	C₇×A₄	1	1	yes	characteristic, direct product factor
47	C₂⁴	C₂²×C₃×C₇	1	35	yes	complemented
48	C₂×C₇	—	3	15	no	—
49	C₂×C₇	—	3	1	no	—
50	C₂×C₇	C₂³×A₄	1	15	yes	direct product factor
51	C₂²×C₃	—	4	35	no	—
52	A₄	C₂⁴×C₇	1	1	yes	characteristic, direct product factor
53	C₂³	—	3	105	no	—
54	C₂³	—	3	210	no	—
55	C₂³	—	3	105	no	—
56	C₂³	—	3	35	no	—
57	C₂³	C₂×C₇×A₄	1	15	yes	direct product factor
58	C₂³	C₂³×C₃×C₇	1	15	yes	complemented
59	C₇	C₂⁴×A₄	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₃	—	4	15	no	—
61	C₂²	—	3	15	no	—
62	C₂²	—	3	70	no	—
63	C₂²	—	3	15	no	—
64	C₂²	—	3	105	no	—
65	C₂²	C₂²×C₇×A₄	1	35	yes	direct product factor
66	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic, complemented
67	C₃	—	4	1	no	Sylow
68	C₂	—	3	1	no	—
69	C₂	—	3	15	no	—
70	C₂	C₂³×C₇×A₄	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴×C₇	A₄
commutator, cosocle, hypocenter	C₂²	C₂⁴×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂⁶×C₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂⁴×C₇

Complements

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

N	H ≅ G/N	type
1. C₂³×C₇×A₄	C₂	direct (N × H)
2. C₂⁶×C₇	C₃	semidirect
4. C₂²×C₇×A₄	C₂²	direct (N × H)
7. C₂⁵×C₇	C₂×C₃	semidirect
8. C₂⁴×A₄	C₇	direct (N × H)
10. C₂×C₇×A₄	C₂³	direct (N × H)
15. C₂⁴×C₇	A₄	direct (N × H)
16. C₂⁴×C₇	C₂²×C₃	semidirect
17. C₂³×A₄	C₂×C₇	direct (N × H)
19. C₇×A₄	C₂⁴	direct (N × H)
20. C₂⁶	C₃×C₇	semidirect
25. C₂³×C₇	C₂×A₄	direct (N × H)
26. C₂³×C₇	C₂³×C₃	semidirect
28. C₂²×A₄	C₂²×C₇	direct (N × H)
32. C₂⁵	C₂×C₃×C₇	semidirect
37. C₂²×C₇	C₂²×A₄	direct (N × H)
38. C₂²×C₇	C₂⁴×C₃	semidirect
40. C₂×A₄	C₂³×C₇	direct (N × H)
46. C₂⁴	C₇×A₄	direct (N × H)
47. C₂⁴	C₂²×C₃×C₇	semidirect
50. C₂×C₇	C₂³×A₄	direct (N × H)
52. A₄	C₂⁴×C₇	direct (N × H)
57. C₂³	C₂×C₇×A₄	direct (N × H)
58. C₂³	C₂³×C₃×C₇	semidirect
59. C₇	C₂⁴×A₄	direct (N × H)
65. C₂²	C₂²×C₇×A₄	direct (N × H)
66. C₂²	C₂⁴×C₃×C₇	semidirect
70. C₂	C₂³×C₇×A₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₇×A₄	2
C₂⁶×C₇	3
C₂⁴×C₃×C₇	4
C₂⁴×A₄	7

Subgroups — C24×C7×A4 #13062422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₇×A₄ #13062422