Subgroups — C₂×C₄²×Dic₇ #11739496

Order 896.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₇	C₂	1	6	yes	maximal
3	C₄²×Dic₇	C₂	1	8	yes	maximal, direct product factor
4	C₂³×C₄×C₇	C₂²	1	3	yes	—
5	C₂³×Dic₇	C₂²	1	4	yes	—
6	C₂×C₄²×C₇	C₄	1	8	yes	complemented
7	C₂×C₄²×C₇	C₂²	1	4	yes	—
8	C₂×C₄×Dic₇	C₄	1	48	yes	direct product factor
9	C₂×C₄×Dic₇	C₂²	1	24	yes	—
10	C₂×C₄³	—	7	1	no	maximal, Sylow
11	C₂⁴×C₇	C₂³	1	1	yes	characteristic
12	C₂²×C₄×C₇	C₂×C₄	1	24	yes	—
13	C₂²×C₄×C₇	C₂×C₄	1	12	yes	—
14	C₂²×C₄×C₇	C₂³	1	6	yes	—
15	C₂²×Dic₇	C₂×C₄	1	48	yes	—
16	C₂²×Dic₇	C₂³	1	8	yes	—
17	C₄²×C₇	C₂×C₄	1	16	yes	complemented
18	C₄×Dic₇	C₂×C₄	1	96	yes	direct product factor
19	C₂²×C₄²	—	7	6	no	—
20	C₂²×C₄²	D₇	1	1	yes	characteristic
21	C₄³	—	7	8	no	—
22	C₂³×C₇	C₂²×C₄	1	12	yes	—
23	C₂³×C₇	C₂²×C₄	1	2	yes	—
24	C₂³×C₇	C₂⁴	1	1	yes	characteristic
25	C₂×C₄×C₇	C₄²	1	48	yes	complemented
26	C₂×C₄×C₇	C₂²×C₄	1	24	yes	—
27	C₂×C₄×C₇	C₂²×C₄	1	12	yes	—
28	C₂×Dic₇	C₄²	1	64	yes	direct product factor
29	C₂×Dic₇	C₂²×C₄	1	48	yes	—
30	C₂³×C₄	—	7	4	no	—
31	C₂³×C₄	C₂×D₇	1	3	yes	—
32	C₂×C₄²	—	7	48	no	—
33	C₂×C₄²	—	7	24	no	—
34	C₂×C₄²	Dic₇	1	8	yes	direct product factor
35	C₂×C₄²	C₂×D₇	1	4	yes	—
36	C₂²×C₇	C₂×C₄²	1	16	yes	—
37	C₂²×C₇	C₂×C₄²	1	12	yes	—
38	C₂²×C₇	C₂³×C₄	1	6	yes	—
39	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₇	C₂×C₄²	1	24	yes	complemented
41	Dic₇	C₂×C₄²	1	32	yes	direct product factor
42	C₂⁴	C₂²×D₇	1	1	yes	characteristic
43	C₂²×C₄	—	7	48	no	—
44	C₂²×C₄	—	7	8	no	—
45	C₂²×C₄	C₄×D₇	1	24	yes	—
46	C₂²×C₄	C₂×Dic₇	1	12	yes	—
47	C₂²×C₄	C₂²×D₇	1	6	yes	—
48	C₄²	—	7	96	no	—
49	C₄²	C₂×Dic₇	1	16	yes	direct product factor
50	C₂×C₇	C₄³	1	8	yes	complemented
51	C₂×C₇	C₂²×C₄²	1	4	yes	—
52	C₂×C₇	C₂²×C₄²	1	3	yes	—
53	C₂³	C₂×C₄×D₇	1	12	yes	—
54	C₂³	C₂²×Dic₇	1	2	yes	—
55	C₂³	C₂³×D₇	1	1	yes	characteristic
56	C₂×C₄	—	7	64	no	—
57	C₂×C₄	—	7	48	no	—
58	C₂×C₄	C₄×Dic₇	1	48	yes	direct product factor
59	C₂×C₄	C₂×C₄×D₇	1	24	yes	—
60	C₂×C₄	C₂²×Dic₇	1	12	yes	—
61	C₇	C₂×C₄³	1	1	yes	characteristic, complemented, Sylow
62	C₂²	C₄²×D₇	1	16	yes	—
63	C₂²	C₂×C₄×Dic₇	1	12	yes	—
64	C₂²	C₂²×C₄×D₇	1	6	yes	—
65	C₂²	C₂³×Dic₇	1	1	yes	characteristic
66	C₄	—	7	32	no	—
67	C₄	C₂×C₄×Dic₇	1	24	yes	direct product factor
68	C₂	C₄²×Dic₇	1	8	yes	direct product factor
69	C₂	C₂×C₄²×D₇	1	4	yes	—
70	C₂	C₂²×C₄×Dic₇	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₇
commutator, hypocenter	C₇	C₂×C₄³
Frattini	C₂³	C₂³×D₇
Fitting	C₂²×C₄²×C₇	C₂
socle, prime radical	C₂⁴×C₇	C₂³
cosocle	C₂³×C₇	C₂⁴
LCS	G ▷ C₇
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄²×Dic₇	C₂	direct (N × H)
6. C₂×C₄²×C₇	C₄	semidirect
8. C₂×C₄×Dic₇	C₄	direct (N × H)
17. C₄²×C₇	C₂×C₄	semidirect
18. C₄×Dic₇	C₂×C₄	direct (N × H)
25. C₂×C₄×C₇	C₄²	semidirect
28. C₂×Dic₇	C₄²	direct (N × H)
34. C₂×C₄²	Dic₇	direct (N × H)
40. C₄×C₇	C₂×C₄²	semidirect
41. Dic₇	C₂×C₄²	direct (N × H)
49. C₄²	C₂×Dic₇	direct (N × H)
50. C₂×C₇	C₄³	semidirect
58. C₂×C₄	C₄×Dic₇	direct (N × H)
61. C₇	C₂×C₄³	semidirect
67. C₄	C₂×C₄×Dic₇	direct (N × H)
68. C₂	C₄²×Dic₇	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×C₄²×C₇	2
C₂²×C₄×Dic₇	2
C₄²×Dic₇	2
C₂×C₄³	7

Subgroups — C2×C42×Dic7 #11739496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×Dic₇ #11739496