Subgroups — D₇×C₂²⋊C₄ #6443

Order 224.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₇	C₂	semidirect
3. C₇×C₂²⋊C₄	C₂	semidirect
4. C₂²⋊Dic₇	C₂	semidirect
5. (C₂×D₇)⋊C₄	C₂	semidirect
8. C₂²×D₇	C₄	semidirect
11. C₂×C₄×C₇	C₂²	semidirect
12. C₂×Dic₇	C₂²	semidirect
16. C₂²×C₇	C₂×C₄	semidirect
21. C₂×D₇	C₂×C₄	semidirect
29. C₂²⋊C₄	D₇	direct (N × H)
34. D₇	C₂²⋊C₄	direct (N × H)
42. C₂×C₄	C₂×D₇	semidirect
43. C₇	C₂×C₂²⋊C₄	semidirect
50. C₂²	C₄×D₇	semidirect

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×C₄×D₇	2
C₇×C₂²⋊C₄	2
C₂²⋊Dic₇	2
(C₂×D₇)⋊C₄	2
C₂×C₂²⋊C₄	7

Subgroups — D7×C22⋊C4 #6443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×C₂²⋊C₄ #6443