Subgroups — C₂²×C₃×Dic₁₄ #10621621

Order 672.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₇	C₂	1	2	yes	maximal
3	C₂×C₃×Dic₁₄	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₁₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄×C₇	C₂²	1	6	yes	—
7	C₂×C₃×Dic₇	C₂²	1	12	yes	—
8	C₃×Dic₁₄	C₂²	1	16	yes	direct product factor
9	C₂²×C₄×C₇	C₂×C₃	1	1	yes	characteristic
10	C₂²×Dic₇	C₂×C₃	1	2	yes	—
11	C₂×Dic₁₄	C₂×C₃	1	12	yes	direct product factor
12	C₂²×C₃×Q₈	—	7	1	no	maximal, Hall
13	C₂²×C₃×C₇	Q₈	1	4	yes	complemented
14	C₂²×C₃×C₇	C₂³	1	3	yes	—
15	C₃×C₄×C₇	C₂³	1	4	yes	—
16	C₃×Dic₇	C₂³	1	8	yes	—
17	C₂³×C₇	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂²×C₃	1	6	yes	—
19	C₂×Dic₇	C₂²×C₃	1	12	yes	—
20	Dic₁₄	C₂²×C₃	1	16	yes	direct product factor
21	C₂×C₃×Q₈	—	7	12	no	—
22	C₂²×C₃×C₄	—	7	2	no	—
23	C₂²×C₃×C₄	D₇	1	1	yes	characteristic
24	C₂×C₃×C₇	C₂×Q₈	1	6	yes	complemented
25	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
26	C₂²×Q₈	—	7	1	no	Sylow
27	C₂²×C₇	C₃×Q₈	1	4	yes	complemented
28	C₂²×C₇	C₂³×C₃	1	3	yes	—
29	C₄×C₇	C₂³×C₃	1	4	yes	—
30	Dic₇	C₂³×C₃	1	8	yes	—
31	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
32	C₃×Q₈	—	7	16	no	—
33	C₂×C₃×C₄	—	7	12	no	—
34	C₂×C₃×C₄	C₂×D₇	1	6	yes	—
35	C₃×C₇	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
36	C₂×Q₈	—	7	12	no	—
37	C₂²×C₄	—	7	2	no	—
38	C₂²×C₄	C₃×D₇	1	1	yes	characteristic
39	C₂×C₇	C₂×C₃×Q₈	1	6	yes	complemented
40	C₂×C₇	C₂⁴×C₃	1	1	yes	characteristic
41	C₂²×C₃	Dic₁₄	1	4	yes	direct product factor
42	C₂²×C₃	C₂²×D₇	1	3	yes	—
43	C₃×C₄	—	7	8	no	—
44	C₃×C₄	C₂²×D₇	1	4	yes	—
45	C₂³	C₂×C₃×D₇	1	1	yes	characteristic
46	Q₈	—	7	16	no	—
47	C₂×C₄	—	7	12	no	—
48	C₂×C₄	C₂×C₃×D₇	1	6	yes	—
49	C₇	C₂²×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	C₂×Dic₁₄	1	6	yes	direct product factor
51	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
52	C₂²	C₃×Dic₁₄	1	4	yes	direct product factor
53	C₂²	C₂²×C₃×D₇	1	3	yes	—
54	C₄	—	7	8	no	—
55	C₄	C₂²×C₃×D₇	1	4	yes	—
56	C₃	C₂²×Dic₁₄	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	C₂×C₃×Dic₁₄	1	6	yes	direct product factor
58	C₂	C₂³×C₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₇
commutator, cosocle	C₂×C₇	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₇
Fitting	C₂²×C₃×C₄×C₇	C₂
socle, prime radical	C₂³×C₃×C₇	C₂²
hypercenter	C₂²×C₃×C₄	D₇
hypocenter	C₇	C₂²×C₃×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×Dic₁₄	C₂	direct (N × H)
4. C₂²×Dic₁₄	C₃	direct (N × H)
8. C₃×Dic₁₄	C₂²	direct (N × H)
11. C₂×Dic₁₄	C₂×C₃	direct (N × H)
13. C₂²×C₃×C₇	Q₈	semidirect
20. Dic₁₄	C₂²×C₃	direct (N × H)
24. C₂×C₃×C₇	C₂×Q₈	semidirect
27. C₂²×C₇	C₃×Q₈	semidirect
35. C₃×C₇	C₂²×Q₈	semidirect
39. C₂×C₇	C₂×C₃×Q₈	semidirect
41. C₂²×C₃	Dic₁₄	direct (N × H)
49. C₇	C₂²×C₃×Q₈	semidirect
50. C₂×C₃	C₂×Dic₁₄	direct (N × H)
52. C₂²	C₃×Dic₁₄	direct (N × H)
56. C₃	C₂²×Dic₁₄	direct (N × H)
57. C₂	C₂×C₃×Dic₁₄	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×C₃×C₄×C₇	2
C₂²×C₃×Dic₇	2
C₂×C₃×Dic₁₄	2
C₂²×Dic₁₄	3
C₂²×C₃×Q₈	7

Subgroups — C22×C3×Dic14 #10621621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Dic₁₄ #10621621