Subgroups — C₂³×C₄×D₇ #90604

Order 448.

1362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₇	C₂	1	28	yes	maximal, direct product factor
5	C₂⁴×C₇	C₂²	1	1	yes	characteristic
6	C₂³×D₇	C₄	1	16	yes	direct product factor
7	C₂³×D₇	C₂²	1	14	yes	—
8	C₂²×C₄×C₇	C₂²	1	14	yes	complemented
9	C₂²×Dic₇	C₂²	1	14	yes	complemented
10	C₂×C₄×D₇	C₂²	1	112	yes	direct product factor
11	C₂⁴×C₄	—	7	1	no	maximal, Sylow
12	C₂³×C₇	C₂×C₄	1	8	yes	complemented
13	C₂³×C₇	C₂³	1	7	yes	—
14	C₂²×D₇	C₂×C₄	1	112	yes	direct product factor
15	C₂²×D₇	C₂³	1	28	yes	—
16	C₂×C₄×C₇	C₂³	1	28	yes	complemented
17	C₂×Dic₇	C₂³	1	28	yes	complemented
18	C₄×D₇	C₂³	1	64	yes	direct product factor
19	C₂⁵	—	7	1	no	—
20	C₂³×C₄	—	7	1	no	—
21	C₂³×C₄	—	7	28	no	—
22	C₂³×C₄	D₇	1	1	yes	characteristic, direct product factor
23	C₂²×C₇	C₂²×C₄	1	28	yes	complemented
24	C₂²×C₇	C₂⁴	1	7	yes	—
25	C₂×D₇	C₂²×C₄	1	112	yes	direct product factor
26	C₂×D₇	C₂⁴	1	8	yes	—
27	C₄×C₇	C₂⁴	1	8	yes	complemented
28	Dic₇	C₂⁴	1	8	yes	complemented
29	C₂⁴	—	7	16	no	—
30	C₂⁴	—	7	14	no	—
31	C₂⁴	C₂×D₇	1	1	yes	characteristic
32	C₂²×C₄	—	7	14	no	—
33	C₂²×C₄	—	7	112	no	—
34	C₂²×C₄	C₂×D₇	1	14	yes	direct product factor
35	C₂×C₇	C₂³×C₄	1	14	yes	complemented
36	C₂×C₇	C₂⁵	1	1	yes	characteristic
37	D₇	C₂³×C₄	1	16	yes	direct product factor
38	C₂³	—	7	112	no	—
39	C₂³	—	7	28	no	—
40	C₂³	C₄×D₇	1	8	yes	direct product factor
41	C₂³	C₂²×D₇	1	7	yes	—
42	C₂×C₄	—	7	28	no	—
43	C₂×C₄	—	7	64	no	—
44	C₂×C₄	C₂²×D₇	1	28	yes	direct product factor
45	C₇	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	7	112	no	—
47	C₂²	—	7	8	no	—
48	C₂²	C₂×C₄×D₇	1	28	yes	direct product factor
49	C₂²	C₂³×D₇	1	7	yes	—
50	C₄	—	7	8	no	—
51	C₄	C₂³×D₇	1	8	yes	direct product factor
52	C₂	—	7	16	no	—
53	C₂	C₂²×C₄×D₇	1	14	yes	direct product factor
54	C₂	C₂⁴×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₇
commutator, hypocenter	C₇	C₂⁴×C₄
Frattini	C₂	C₂⁴×D₇
Fitting	C₂³×C₄×C₇	C₂
socle	C₂⁴×C₇	C₂²
prime radical	C₂⁴×D₇	C₂
cosocle	C₂×C₇	C₂⁵
LCS	G ▷ C₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×C₇	C₂	semidirect
3. C₂³×Dic₇	C₂	semidirect
4. C₂²×C₄×D₇	C₂	direct (N × H)
6. C₂³×D₇	C₄	direct (N × H)
8. C₂²×C₄×C₇	C₂²	semidirect
9. C₂²×Dic₇	C₂²	semidirect
10. C₂×C₄×D₇	C₂²	direct (N × H)
12. C₂³×C₇	C₂×C₄	semidirect
14. C₂²×D₇	C₂×C₄	direct (N × H)
16. C₂×C₄×C₇	C₂³	semidirect
17. C₂×Dic₇	C₂³	semidirect
18. C₄×D₇	C₂³	direct (N × H)
22. C₂³×C₄	D₇	direct (N × H)
23. C₂²×C₇	C₂²×C₄	semidirect
25. C₂×D₇	C₂²×C₄	direct (N × H)
27. C₄×C₇	C₂⁴	semidirect
28. Dic₇	C₂⁴	semidirect
34. C₂²×C₄	C₂×D₇	direct (N × H)
35. C₂×C₇	C₂³×C₄	semidirect
37. D₇	C₂³×C₄	direct (N × H)
40. C₂³	C₄×D₇	direct (N × H)
44. C₂×C₄	C₂²×D₇	direct (N × H)
45. C₇	C₂⁴×C₄	semidirect
48. C₂²	C₂×C₄×D₇	direct (N × H)
51. C₄	C₂³×D₇	direct (N × H)
53. C₂	C₂²×C₄×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₇	2
C₂³×C₄×C₇	2
C₂³×Dic₇	2
C₂²×C₄×D₇	2
C₂⁴×C₄	7

Subgroups — C23×C4×D7 #90604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₇ #90604