Subgroups — D₄×Dic₇ #6497

Order 224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Dic₇	C₂	1	2	yes	maximal, complemented
4	C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇	C₂²	1	2	yes	—
7	C₇×D₄	C₄	1	4	yes	complemented
8	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₇	—	2	4	no	—
10	C₂×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₇	C₂²	1	2	yes	—
12	C₂×Dic₇	C₂²	1	1	yes	characteristic
13	C₄×D₄	—	7	1	no	maximal, Sylow
14	C₂²×C₇	—	2	2	no	—
15	C₂²×C₇	—	2	2	no	—
16	C₂²×C₇	C₂×C₄	1	4	yes	complemented
17	C₂²×C₇	C₂³	1	1	yes	characteristic
18	C₄×C₇	C₂×C₄	1	2	yes	complemented
19	Dic₇	—	2	2	no	—
20	Dic₇	—	2	1	no	—
21	Dic₇	D₄	1	2	yes	direct product factor
22	C₂×D₄	D₇	1	1	yes	characteristic
23	C₂²×C₄	—	7	2	no	—
24	C₄⋊C₄	—	7	1	no	—
25	C₂²⋊C₄	—	7	2	no	—
26	C₄²	—	7	1	no	—
27	C₂×C₇	—	2	4	no	—
28	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₇	#41	1	1	yes	characteristic
31	C₂³	C₂×D₇	1	2	yes	—
32	D₄	Dic₇	1	4	yes	direct product factor
33	C₂×C₄	—	7	1	no	—
34	C₂×C₄	—	7	1	no	—
35	C₂×C₄	—	7	2	no	—
36	C₂×C₄	—	14	4	no	—
37	C₂×C₄	C₂×D₇	1	1	yes	characteristic
38	C₇	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
39	C₂²	—	2	2	no	—
40	C₂²	—	2	2	no	—
41	C₂²	C₂×Dic₇	1	4	yes	complemented
42	C₂²	C₂²×D₇	1	1	yes	characteristic
43	C₄	—	7	2	no	—
44	C₄	—	14	2	no	—
45	C₄	—	14	1	no	—
46	C₄	C₂×Dic₇	1	2	yes	complemented
47	C₂	—	2	4	no	—
48	C₂	D₄×D₇	1	1	yes	characteristic
49	C₂	#1159	1	1	yes	characteristic
50	C₂	C₂²×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄
Fitting, prime radical	C₂×C₇×D₄	C₂
socle, cosocle	C₂²×C₇	C₂³
hypercenter	C₂×D₄	D₇
hypocenter	C₇	C₄×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
2. C₂²×Dic₇	C₂	semidirect
3. C₂²⋊Dic₇	C₂	semidirect
4. C₄⋊Dic₇	C₂	semidirect
5. C₄×Dic₇	C₂	semidirect
7. C₇×D₄	C₄	semidirect
16. C₂²×C₇	C₂×C₄	semidirect
18. C₄×C₇	C₂×C₄	semidirect
21. Dic₇	D₄	direct (N × H)
32. D₄	Dic₇	direct (N × H)
38. C₇	C₄×D₄	semidirect
41. C₂²	C₂×Dic₇	semidirect
46. C₄	C₂×Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₄	2
C₂²×Dic₇	2
C₂²⋊Dic₇	2
C₄⋊Dic₇	2
C₄×Dic₇	2
C₄×D₄	7

Subgroups — D4×Dic7 #6497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×Dic₇ #6497