Subgroups — C₂×D₄×D₇ #6546

Order 224.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
4	D₄×D₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇	C₂²	1	2	yes	complemented
8	C₂²×D₇	—	2	8	no	—
9	C₂²×D₇	C₂²	1	1	yes	characteristic
10	C₂²×D₇	C₂²	1	8	yes	complemented
11	C₂²×D₇	C₂²	1	2	yes	complemented
12	C₇×D₄	C₂²	1	4	yes	complemented
13	C₂×C₄×C₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²⋊D₇	C₂²	1	8	yes	complemented
15	C₂×Dic₇	C₂²	1	1	yes	characteristic, complemented
16	D₂₈	C₂²	1	4	yes	complemented
17	C₄×D₇	C₂²	1	4	yes	complemented
18	C₂²×D₄	—	7	1	no	maximal, Sylow
19	C₂²×C₇	—	2	4	no	—
20	C₂²×C₇	C₂³	1	1	yes	characteristic
21	C₂²×C₇	C₂³	1	4	yes	complemented
22	C₂×D₇	—	2	4	no	—
23	C₂×D₇	—	2	16	no	—
24	C₂×D₇	D₄	1	4	yes	direct product factor
25	C₂×D₇	C₂³	1	2	yes	—
26	C₂×D₇	C₂³	1	4	yes	complemented
27	C₄×C₇	C₂³	1	2	yes	complemented
28	Dic₇	C₂³	1	2	yes	complemented
29	C₂⁴	—	7	2	no	—
30	C₂×D₄	—	7	2	no	—
31	C₂×D₄	—	7	8	no	—
32	C₂×D₄	—	7	1	no	—
33	C₂×D₄	D₇	1	1	yes	characteristic, direct product factor
34	C₂²×C₄	—	7	1	no	—
35	C₂×C₇	—	2	4	no	—
36	C₂×C₇	C₂×D₄	1	2	yes	complemented
37	C₂×C₇	C₂⁴	1	1	yes	characteristic
38	D₇	—	2	4	no	—
39	D₇	C₂×D₄	1	4	yes	direct product factor
40	C₂³	—	7	1	no	—
41	C₂³	—	7	2	no	—
42	C₂³	—	7	8	no	—
43	C₂³	—	14	8	no	—
44	C₂³	C₂×D₇	1	2	yes	complemented
45	D₄	—	7	8	no	—
46	D₄	—	7	4	no	—
47	D₄	C₂×D₇	1	4	yes	direct product factor
48	C₂×C₄	—	7	1	no	—
49	C₂×C₄	—	7	4	no	—
50	C₂×C₄	C₂×D₇	1	1	yes	characteristic, complemented
51	C₇	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	2	4	no	—
53	C₂²	—	7	4	no	—
54	C₂²	—	7	2	no	—
55	C₂²	—	7	4	no	—
56	C₂²	—	14	16	no	—
57	C₂²	—	14	4	no	—
58	C₂²	C₂²×D₇	1	1	yes	characteristic
59	C₂²	C₂²×D₇	1	4	yes	complemented
60	C₄	—	7	2	no	—
61	C₄	C₂²×D₇	1	2	yes	complemented
62	C₂	—	2	4	no	—
63	C₂	—	7	4	no	—
64	C₂	—	14	4	no	—
65	C₂	D₄×D₇	1	2	yes	direct product factor
66	C₂	C₂³×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇
commutator, cosocle	C₂×C₇	C₂⁴
Frattini	C₂	C₂³×D₇
Fitting	C₂×C₇×D₄	C₂
socle	C₂²×C₇	C₂³
hypercenter	C₂×D₄	D₇
hypocenter	C₇	C₂²×D₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×D₇	C₂	semidirect
2. C₂×C₇×D₄	C₂	semidirect
3. C₂×C₂²⋊D₇	C₂	semidirect
4. D₄×D₇	C₂	direct (N × H)
5. C₂×D₂₈	C₂	semidirect
6. C₂×C₄×D₇	C₂	semidirect
7. C₂³×C₇	C₂²	semidirect
10. C₂²×D₇	C₂²	semidirect
11. C₂²×D₇	C₂²	semidirect
12. C₇×D₄	C₂²	semidirect
13. C₂×C₄×C₇	C₂²	semidirect
14. C₂²⋊D₇	C₂²	semidirect
15. C₂×Dic₇	C₂²	semidirect
16. D₂₈	C₂²	semidirect
17. C₄×D₇	C₂²	semidirect
21. C₂²×C₇	C₂³	semidirect
24. C₂×D₇	D₄	direct (N × H)
26. C₂×D₇	C₂³	semidirect
27. C₄×C₇	C₂³	semidirect
28. Dic₇	C₂³	semidirect
33. C₂×D₄	D₇	direct (N × H)
36. C₂×C₇	C₂×D₄	semidirect
39. D₇	C₂×D₄	direct (N × H)
44. C₂³	C₂×D₇	semidirect
47. D₄	C₂×D₇	direct (N × H)
50. C₂×C₄	C₂×D₇	semidirect
51. C₇	C₂²×D₄	semidirect
59. C₂²	C₂²×D₇	semidirect
61. C₄	C₂²×D₇	semidirect
65. C₂	D₄×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
C₂³×D₇	2
C₂×C₇×D₄	2
C₂×C₂²⋊D₇	2
D₄×D₇	2
C₂×D₂₈	2
C₂×C₄×D₇	2
C₂²×D₄	7

Subgroups — C2×D4×D7 #6546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₇ #6546