Subgroups — C₂²×C₇×D₈ #90590

Order 448.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×D₄	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×D₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₇	—	2	2	no	—
5	C₂×C₇×D₄	—	2	6	no	—
6	C₂×C₇×D₄	C₂²	1	12	yes	complemented
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×D₈	C₂²	1	16	yes	direct product factor
9	C₂×C₇×C₈	C₂²	1	6	yes	complemented
10	C₂²×D₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₇	—	2	12	no	—
12	C₂³×C₇	—	4	8	no	—
13	C₂³×C₇	D₄	1	1	yes	characteristic
14	C₇×D₄	—	2	12	no	—
15	C₇×D₄	C₂³	1	8	yes	complemented
16	C₂×C₄×C₇	D₄	1	3	yes	—
17	C₂×C₄×C₇	C₂³	1	3	yes	—
18	C₇×C₈	C₂³	1	4	yes	complemented
19	C₂²×D₄	C₂×C₇	1	2	yes	complemented
20	C₂×D₈	C₂×C₇	1	12	yes	direct product factor
21	C₂²×C₈	C₂×C₇	1	1	yes	characteristic, complemented
22	C₂²×C₇	—	2	8	no	—
23	C₂²×C₇	—	4	24	no	—
24	C₂²×C₇	D₈	1	4	yes	direct product factor
25	C₂²×C₇	C₂×D₄	1	3	yes	—
26	C₄×C₇	C₂×D₄	1	3	yes	—
27	C₄×C₇	C₂⁴	1	1	yes	characteristic
28	C₂⁴	—	2	2	no	—
29	C₂×D₄	—	2	6	no	—
30	C₂×D₄	C₂²×C₇	1	12	yes	complemented
31	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
32	D₈	C₂²×C₇	1	16	yes	direct product factor
33	C₂×C₈	C₂²×C₇	1	6	yes	complemented
34	C₂×C₇	—	4	8	no	—
35	C₂×C₇	C₂×D₈	1	6	yes	direct product factor
36	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
37	C₂³	—	2	12	no	—
38	C₂³	—	4	8	no	—
39	C₂³	C₇×D₄	1	1	yes	characteristic
40	D₄	—	2	12	no	—
41	D₄	C₂³×C₇	1	8	yes	complemented
42	C₂×C₄	C₇×D₄	1	3	yes	—
43	C₂×C₄	C₂³×C₇	1	3	yes	—
44	C₈	C₂³×C₇	1	4	yes	complemented
45	C₇	C₂²×D₈	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	—	2	8	no	—
47	C₂²	—	4	24	no	—
48	C₂²	C₇×D₈	1	4	yes	direct product factor
49	C₂²	C₂×C₇×D₄	1	3	yes	—
50	C₄	C₂×C₇×D₄	1	3	yes	—
51	C₄	C₂⁴×C₇	1	1	yes	characteristic
52	C₂	—	4	8	no	—
53	C₂	C₂×C₇×D₈	1	6	yes	direct product factor
54	C₂	C₂²×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×D₄	C₂	semidirect
2. C₂×C₇×D₈	C₂	direct (N × H)
3. C₂²×C₇×C₈	C₂	semidirect
6. C₂×C₇×D₄	C₂²	semidirect
8. C₇×D₈	C₂²	direct (N × H)
9. C₂×C₇×C₈	C₂²	semidirect
10. C₂²×D₈	C₇	direct (N × H)
15. C₇×D₄	C₂³	semidirect
18. C₇×C₈	C₂³	semidirect
19. C₂²×D₄	C₂×C₇	semidirect
20. C₂×D₈	C₂×C₇	direct (N × H)
21. C₂²×C₈	C₂×C₇	semidirect
24. C₂²×C₇	D₈	direct (N × H)
30. C₂×D₄	C₂²×C₇	semidirect
32. D₈	C₂²×C₇	direct (N × H)
33. C₂×C₈	C₂²×C₇	semidirect
35. C₂×C₇	C₂×D₈	direct (N × H)
41. D₄	C₂³×C₇	semidirect
44. C₈	C₂³×C₇	semidirect
45. C₇	C₂²×D₈	direct (N × H)
48. C₂²	C₇×D₈	direct (N × H)
53. C₂	C₂×C₇×D₈	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×C₇×D₄	2
C₂×C₇×D₈	2
C₂²×C₇×C₈	2
C₂²×D₈	7

Subgroups — C22×C7×D8 #90590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×D₈ #90590