Subgroups — C₂²×Dic₃×D₇ #10621583

Order 672.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃×D₇	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₄×D₇	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₃×D₇	C₄	1	8	yes	complemented
8	C₂²×C₃×D₇	C₂²	1	6	yes	—
9	C₂×Dic₂₁	C₂²	1	6	yes	complemented
10	C₂×C₇×Dic₃	C₂²	1	6	yes	complemented
11	Dic₃×D₇	C₂²	1	16	yes	direct product factor
12	C₂³×D₇	D₃	1	1	yes	characteristic
13	C₂²×C₄×C₇	—	3	1	no	—
14	C₂²×Dic₇	—	3	1	no	—
15	C₂×C₄×D₇	—	3	12	no	—
16	C₂³×Dic₃	—	7	1	no	maximal, Hall
17	C₂²×C₃×C₇	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃×C₇	C₂³	1	3	yes	—
19	C₂×C₃×D₇	C₂×C₄	1	24	yes	complemented
20	C₂×C₃×D₇	C₂³	1	4	yes	—
21	Dic₂₁	C₂³	1	4	yes	complemented
22	C₇×Dic₃	C₂³	1	4	yes	complemented
23	C₂³×C₇	C₂×D₃	1	1	yes	characteristic
24	C₂²×D₇	Dic₃	1	8	yes	direct product factor
25	C₂²×D₇	C₂×D₃	1	6	yes	—
26	C₂×C₄×C₇	—	3	6	no	—
27	C₂×Dic₇	—	3	6	no	—
28	C₄×D₇	—	3	16	no	—
29	C₂⁴×C₃	—	7	1	no	—
30	C₂²×Dic₃	—	7	1	no	—
31	C₂²×Dic₃	—	7	12	no	—
32	C₂²×Dic₃	D₇	1	1	yes	characteristic, direct product factor
33	C₂×C₃×C₇	C₂²×C₄	1	6	yes	complemented
34	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
35	C₃×D₇	C₂²×C₄	1	8	yes	complemented
36	C₂³×C₄	—	21	1	no	Sylow
37	C₂²×C₇	C₂×Dic₃	1	4	yes	complemented
38	C₂²×C₇	C₂²×D₃	1	3	yes	—
39	C₂×D₇	C₂×Dic₃	1	24	yes	direct product factor
40	C₂×D₇	C₂²×D₃	1	4	yes	—
41	C₄×C₇	—	3	4	no	—
42	Dic₇	—	3	4	no	—
43	C₂³×C₃	—	7	8	no	—
44	C₂³×C₃	—	7	6	no	—
45	C₂³×C₃	C₂×D₇	1	1	yes	characteristic
46	C₂×Dic₃	—	7	6	no	—
47	C₂×Dic₃	—	7	16	no	—
48	C₂×Dic₃	C₂×D₇	1	6	yes	direct product factor
49	C₃×C₇	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
50	C₂⁴	—	7	1	no	—
51	C₂²×C₄	—	3	1	no	—
52	C₂²×C₄	—	21	12	no	—
53	C₂²×C₄	—	21	1	no	—
54	C₂×C₇	C₂²×Dic₃	1	6	yes	complemented
55	C₂×C₇	C₂³×D₃	1	1	yes	characteristic
56	D₇	C₂²×Dic₃	1	8	yes	direct product factor
57	C₂²×C₃	—	7	24	no	—
58	C₂²×C₃	—	7	4	no	—
59	C₂²×C₃	C₄×D₇	1	4	yes	complemented
60	C₂²×C₃	C₂²×D₇	1	3	yes	—
61	Dic₃	—	7	4	no	—
62	Dic₃	C₂²×D₇	1	4	yes	direct product factor
63	C₂³	—	7	8	no	—
64	C₂³	—	7	6	no	—
65	C₂³	D₃×D₇	1	1	yes	characteristic
66	C₂×C₄	—	3	6	no	—
67	C₂×C₄	—	21	16	no	—
68	C₂×C₄	—	21	6	no	—
69	C₇	C₂³×Dic₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	7	8	no	—
71	C₂×C₃	C₂×C₄×D₇	1	6	yes	complemented
72	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
73	C₂²	—	7	24	no	—
74	C₂²	—	7	4	no	—
75	C₂²	Dic₃×D₇	1	4	yes	direct product factor
76	C₂²	C₂×D₃×D₇	1	3	yes	—
77	C₄	—	3	4	no	—
78	C₄	—	21	4	no	—
79	C₃	C₂²×C₄×D₇	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	7	8	no	—
81	C₂	C₂×Dic₃×D₇	1	6	yes	direct product factor
82	C₂	C₂²×D₃×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂³×C₄
Frattini	C₂	C₂²×D₃×D₇
Fitting, socle	C₂³×C₃×C₇	C₂²
prime radical	C₂³×C₃×D₇	C₂
cosocle	C₂×C₃×C₇	C₂⁴
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₂₁	C₂	semidirect
3. C₂²×C₇×Dic₃	C₂	semidirect
4. C₂×Dic₃×D₇	C₂	direct (N × H)
7. C₂²×C₃×D₇	C₄	semidirect
9. C₂×Dic₂₁	C₂²	semidirect
10. C₂×C₇×Dic₃	C₂²	semidirect
11. Dic₃×D₇	C₂²	direct (N × H)
17. C₂²×C₃×C₇	C₂×C₄	semidirect
19. C₂×C₃×D₇	C₂×C₄	semidirect
21. Dic₂₁	C₂³	semidirect
22. C₇×Dic₃	C₂³	semidirect
24. C₂²×D₇	Dic₃	direct (N × H)
32. C₂²×Dic₃	D₇	direct (N × H)
33. C₂×C₃×C₇	C₂²×C₄	semidirect
35. C₃×D₇	C₂²×C₄	semidirect
37. C₂²×C₇	C₂×Dic₃	semidirect
39. C₂×D₇	C₂×Dic₃	direct (N × H)
48. C₂×Dic₃	C₂×D₇	direct (N × H)
49. C₃×C₇	C₂³×C₄	semidirect
54. C₂×C₇	C₂²×Dic₃	semidirect
56. D₇	C₂²×Dic₃	direct (N × H)
59. C₂²×C₃	C₄×D₇	semidirect
62. Dic₃	C₂²×D₇	direct (N × H)
69. C₇	C₂³×Dic₃	semidirect
71. C₂×C₃	C₂×C₄×D₇	semidirect
75. C₂²	Dic₃×D₇	direct (N × H)
79. C₃	C₂²×C₄×D₇	semidirect
81. C₂	C₂×Dic₃×D₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×D₇	2
C₂²×Dic₂₁	2
C₂²×C₇×Dic₃	2
C₂×Dic₃×D₇	2
C₂²×C₄×D₇	3
C₂³×Dic₃	7

Subgroups — C22×Dic3×D7 #10621583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₃×D₇ #10621583