Subgroups — C₂³×C₇×C₃⋊C₈ #13061496

Order 1344.

676 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₃⋊C₈	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₇×C₈	—	3	1	no	maximal, Hall
4	C₂⁴×C₃×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₄×C₇	C₄	1	7	yes	—
6	C₂²×C₃×C₄×C₇	C₂²	1	7	yes	—
7	C₂×C₇×C₃⋊C₈	C₂²	1	28	yes	direct product factor
8	C₂³×C₄×C₇	D₃	1	1	yes	characteristic
9	C₂²×C₇×C₈	—	3	14	no	—
10	C₂³×C₃⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₃×C₇	C₈	1	8	yes	complemented
12	C₂³×C₃×C₇	C₂×C₄	1	7	yes	—
13	C₂×C₃×C₄×C₇	C₂×C₄	1	21	yes	—
14	C₂×C₃×C₄×C₇	C₂³	1	7	yes	—
15	C₇×C₃⋊C₈	C₂³	1	8	yes	direct product factor
16	C₂⁴×C₇	Dic₃	1	1	yes	characteristic
17	C₂²×C₄×C₇	Dic₃	1	7	yes	—
18	C₂²×C₄×C₇	C₂×D₃	1	7	yes	—
19	C₂×C₇×C₈	—	3	28	no	—
20	C₂³×C₃×C₄	C₂×C₇	1	1	yes	characteristic
21	C₂²×C₃⋊C₈	C₂×C₇	1	14	yes	direct product factor
22	C₂²×C₃×C₇	C₂×C₈	1	28	yes	complemented
23	C₂²×C₃×C₇	C₂²×C₄	1	7	yes	—
24	C₃×C₄×C₇	C₂²×C₄	1	7	yes	—
25	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
26	C₂³×C₈	—	3	1	no	Sylow
27	C₂³×C₇	C₃⋊C₈	1	8	yes	direct product factor
28	C₂³×C₇	C₂×Dic₃	1	7	yes	—
29	C₂×C₄×C₇	C₂×Dic₃	1	21	yes	—
30	C₂×C₄×C₇	C₂²×D₃	1	7	yes	—
31	C₇×C₈	—	3	8	no	—
32	C₂⁴×C₃	C₄×C₇	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₄×C₇	1	7	yes	—
34	C₂²×C₃×C₄	C₂²×C₇	1	7	yes	—
35	C₂×C₃⋊C₈	C₂²×C₇	1	28	yes	direct product factor
36	C₂×C₃×C₇	C₂²×C₈	1	14	yes	complemented
37	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
38	C₂³×C₄	D₃×C₇	1	1	yes	characteristic
39	C₂²×C₈	—	3	14	no	—
40	C₂²×C₇	C₂×C₃⋊C₈	1	28	yes	direct product factor
41	C₂²×C₇	C₂²×Dic₃	1	7	yes	—
42	C₄×C₇	C₂²×Dic₃	1	7	yes	—
43	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
44	C₂³×C₃	C₇×C₈	1	8	yes	complemented
45	C₂³×C₃	C₂×C₄×C₇	1	7	yes	—
46	C₂×C₃×C₄	C₂×C₄×C₇	1	21	yes	—
47	C₂×C₃×C₄	C₂³×C₇	1	7	yes	—
48	C₃⋊C₈	C₂³×C₇	1	8	yes	direct product factor
49	C₃×C₇	C₂³×C₈	1	1	yes	characteristic, complemented, Hall
50	C₂⁴	C₇×Dic₃	1	1	yes	characteristic
51	C₂²×C₄	C₇×Dic₃	1	7	yes	—
52	C₂²×C₄	C₂×D₃×C₇	1	7	yes	—
53	C₂×C₈	—	3	28	no	—
54	C₂×C₇	C₂²×C₃⋊C₈	1	14	yes	direct product factor
55	C₂×C₇	C₂³×Dic₃	1	1	yes	characteristic
56	C₂²×C₃	C₂×C₇×C₈	1	28	yes	complemented
57	C₂²×C₃	C₂²×C₄×C₇	1	7	yes	—
58	C₃×C₄	C₂²×C₄×C₇	1	7	yes	—
59	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
60	C₂³	C₇×C₃⋊C₈	1	8	yes	direct product factor
61	C₂³	C₂×C₇×Dic₃	1	7	yes	—
62	C₂×C₄	C₂×C₇×Dic₃	1	21	yes	—
63	C₂×C₄	C₂²×D₃×C₇	1	7	yes	—
64	C₈	—	3	8	no	—
65	C₇	C₂³×C₃⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×C₃	C₂²×C₇×C₈	1	14	yes	complemented
67	C₂×C₃	C₂³×C₄×C₇	1	1	yes	characteristic
68	C₂²	C₂×C₇×C₃⋊C₈	1	28	yes	direct product factor
69	C₂²	C₂²×C₇×Dic₃	1	7	yes	—
70	C₄	C₂²×C₇×Dic₃	1	7	yes	—
71	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
72	C₃	C₂³×C₇×C₈	1	1	yes	characteristic, complemented, Sylow
73	C₂	C₂²×C₇×C₃⋊C₈	1	14	yes	direct product factor
74	C₂	C₂³×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄×C₇	D₃
commutator, hypocenter	C₃	C₂³×C₇×C₈
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂³×C₃×C₄×C₇	C₂
socle, prime radical	C₂⁴×C₃×C₇	C₄
cosocle	C₃×C₄	C₂⁴×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×C₃⋊C₈	C₂	direct (N × H)
7. C₂×C₇×C₃⋊C₈	C₂²	direct (N × H)
10. C₂³×C₃⋊C₈	C₇	direct (N × H)
11. C₂³×C₃×C₇	C₈	semidirect
15. C₇×C₃⋊C₈	C₂³	direct (N × H)
21. C₂²×C₃⋊C₈	C₂×C₇	direct (N × H)
22. C₂²×C₃×C₇	C₂×C₈	semidirect
27. C₂³×C₇	C₃⋊C₈	direct (N × H)
35. C₂×C₃⋊C₈	C₂²×C₇	direct (N × H)
36. C₂×C₃×C₇	C₂²×C₈	semidirect
40. C₂²×C₇	C₂×C₃⋊C₈	direct (N × H)
44. C₂³×C₃	C₇×C₈	semidirect
48. C₃⋊C₈	C₂³×C₇	direct (N × H)
49. C₃×C₇	C₂³×C₈	semidirect
54. C₂×C₇	C₂²×C₃⋊C₈	direct (N × H)
56. C₂²×C₃	C₂×C₇×C₈	semidirect
60. C₂³	C₇×C₃⋊C₈	direct (N × H)
65. C₇	C₂³×C₃⋊C₈	direct (N × H)
66. C₂×C₃	C₂²×C₇×C₈	semidirect
68. C₂²	C₂×C₇×C₃⋊C₈	direct (N × H)
72. C₃	C₂³×C₇×C₈	semidirect
73. C₂	C₂²×C₇×C₃⋊C₈	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×C₄×C₇	2
C₂²×C₇×C₃⋊C₈	2
C₂³×C₇×C₈	3
C₂³×C₃⋊C₈	7

Subgroups — C23×C7×C3⋊C8 #13061496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×C₃⋊C₈ #13061496