Subgroups — (C₂²⋊D₇)⋊C₄ #6444

Order 224.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₇	C₂²	1	1	yes	characteristic
9	C₂²×D₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₂²⋊D₇	C₄	1	4	yes	complemented
13	C₂×Dic₇	—	2	2	no	—
14	C₂×Dic₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₇	C₂²	1	1	yes	characteristic
17	C₄×D₇	—	2	2	no	—
18	C₄×D₄	—	7	1	no	maximal, Sylow
19	C₂²×C₇	—	2	1	no	—
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	C₂×C₄	1	2	yes	complemented
22	C₂²×C₇	C₂³	1	1	yes	characteristic
23	C₂×D₇	—	2	1	no	—
24	C₂×D₇	—	2	1	no	—
25	C₂×D₇	C₂×C₄	1	2	yes	complemented
26	C₄×C₇	—	2	1	no	—
27	C₄×C₇	—	2	1	no	—
28	Dic₇	—	2	1	no	—
29	Dic₇	C₂×C₄	1	2	yes	complemented
30	Dic₇	D₄	1	2	yes	complemented
31	C₂×D₄	—	7	1	no	—
32	C₂²×C₄	—	7	1	no	—
33	C₂²×C₄	—	7	1	no	—
34	C₄⋊C₄	—	7	1	no	—
35	C₂²⋊C₄	—	7	1	no	—
36	C₂²⋊C₄	D₇	1	1	yes	characteristic, complemented
37	C₄²	—	7	1	no	—
38	C₂×C₇	—	2	2	no	—
39	C₂×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₇	#41	1	1	yes	characteristic
42	D₇	—	2	2	no	—
43	C₂³	—	7	1	no	—
44	C₂³	C₂×D₇	1	1	yes	characteristic
45	D₄	—	7	4	no	—
46	C₂×C₄	—	7	1	no	—
47	C₂×C₄	—	7	1	no	—
48	C₂×C₄	—	7	1	no	—
49	C₂×C₄	—	14	2	no	—
50	C₂×C₄	—	14	2	no	—
51	C₂×C₄	C₂×D₇	1	1	yes	characteristic
52	C₂×C₄	C₂×D₇	1	1	yes	characteristic
53	C₇	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	7	2	no	—
57	C₂²	—	14	1	no	—
58	C₂²	—	14	1	no	—
59	C₂²	C₄×D₇	1	2	yes	complemented
60	C₂²	C₂²×D₇	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	2	1	no	—
63	C₄	—	7	2	no	—
64	C₄	—	7	2	no	—
65	C₄	—	14	1	no	—
66	C₂	—	2	2	no	—
67	C₂	—	14	2	no	—
68	C₂	C₂×C₄×D₇	1	1	yes	characteristic
69	C₂	D₄×D₇	1	1	yes	characteristic
70	C₂	#1159	1	1	yes	characteristic

	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	C₂×C₂²⋊D₇	C₂
hypercenter	C₂²⋊C₄	D₇
hypocenter	C₇	C₄×D₄
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. C₂²×Dic₇	C₂	semidirect
3. C₂×C₄×D₇	C₂	semidirect
4. C₇×C₂²⋊C₄	C₂	semidirect
5. (C₂×D₇)⋊C₄	C₂	semidirect
6. Dic₇⋊C₄	C₂	semidirect
7. C₄×Dic₇	C₂	semidirect
12. C₂²⋊D₇	C₄	semidirect
21. C₂²×C₇	C₂×C₄	semidirect
25. C₂×D₇	C₂×C₄	semidirect
29. Dic₇	C₂×C₄	semidirect
30. Dic₇	D₄	semidirect
36. C₂²⋊C₄	D₇	semidirect
53. C₇	C₄×D₄	semidirect
59. 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
C₂×C₂²⋊D₇	2
C₂²×Dic₇	2
C₂×C₄×D₇	2
C₇×C₂²⋊C₄	2
(C₂×D₇)⋊C₄	2
Dic₇⋊C₄	2
C₄×Dic₇	2
C₄×D₄	7

Subgroups — (C22⋊D7)⋊C4 #6444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₂²⋊D₇)⋊C₄ #6444