Subgroups — C₂³×C₇×Dic₆ #13062348

Order 1344.

1700 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	C₂³×C₇×Dic₃	C₂	1	2	yes	maximal
3	C₂²×C₇×Dic₆	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₇×Q₈	—	3	1	no	maximal, Hall
5	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₄×C₇	C₂²	1	14	yes	—
7	C₂²×C₇×Dic₃	C₂²	1	28	yes	—
8	C₂×C₇×Dic₆	C₂²	1	112	yes	direct product factor
9	C₂²×C₇×Q₈	—	3	28	no	—
10	C₂³×C₄×C₇	—	3	2	no	—
11	C₂³×C₄×C₇	D₃	1	1	yes	characteristic
12	C₂³×Dic₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂³×C₃×C₇	Q₈	1	8	yes	complemented
14	C₂³×C₃×C₇	C₂³	1	7	yes	—
15	C₂×C₃×C₄×C₇	C₂³	1	28	yes	—
16	C₂×C₇×Dic₃	C₂³	1	56	yes	—
17	C₇×Dic₆	C₂³	1	64	yes	direct product factor
18	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
19	C₂×C₇×Q₈	—	3	112	no	—
20	C₂²×C₄×C₇	—	3	28	no	—
21	C₂²×C₄×C₇	C₂×D₃	1	14	yes	—
22	C₂³×C₃×C₄	C₂×C₇	1	1	yes	characteristic
23	C₂³×Dic₃	C₂×C₇	1	2	yes	—
24	C₂²×Dic₆	C₂×C₇	1	28	yes	direct product factor
25	C₂²×C₃×C₇	C₂×Q₈	1	28	yes	complemented
26	C₂²×C₃×C₇	C₂⁴	1	7	yes	—
27	C₃×C₄×C₇	C₂⁴	1	8	yes	—
28	C₇×Dic₃	C₂⁴	1	16	yes	—
29	C₂³×Q₈	—	3	1	no	Sylow
30	C₂³×C₇	Dic₆	1	8	yes	direct product factor
31	C₂³×C₇	C₂²×D₃	1	7	yes	—
32	C₇×Q₈	—	3	64	no	—
33	C₂×C₄×C₇	—	3	56	no	—
34	C₂×C₄×C₇	C₂²×D₃	1	28	yes	—
35	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂²×C₇	1	14	yes	—
37	C₂²×Dic₃	C₂²×C₇	1	28	yes	—
38	C₂×Dic₆	C₂²×C₇	1	112	yes	direct product factor
39	C₂×C₃×C₇	C₂²×Q₈	1	14	yes	complemented
40	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
41	C₂²×Q₈	—	3	28	no	—
42	C₂³×C₄	—	3	2	no	—
43	C₂³×C₄	D₃×C₇	1	1	yes	characteristic
44	C₂²×C₇	C₂×Dic₆	1	28	yes	direct product factor
45	C₂²×C₇	C₂³×D₃	1	7	yes	—
46	C₄×C₇	—	3	16	no	—
47	C₄×C₇	C₂³×D₃	1	8	yes	—
48	C₂³×C₃	C₇×Q₈	1	8	yes	complemented
49	C₂³×C₃	C₂³×C₇	1	7	yes	—
50	C₂×C₃×C₄	C₂³×C₇	1	28	yes	—
51	C₂×Dic₃	C₂³×C₇	1	56	yes	—
52	Dic₆	C₂³×C₇	1	64	yes	direct product factor
53	C₃×C₇	C₂³×Q₈	1	1	yes	characteristic, complemented, Hall
54	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic
55	C₂×Q₈	—	3	112	no	—
56	C₂²×C₄	—	3	28	no	—
57	C₂²×C₄	C₂×D₃×C₇	1	14	yes	—
58	C₂×C₇	C₂²×Dic₆	1	14	yes	direct product factor
59	C₂×C₇	C₂⁴×D₃	1	1	yes	characteristic
60	C₂²×C₃	C₂×C₇×Q₈	1	28	yes	complemented
61	C₂²×C₃	C₂⁴×C₇	1	7	yes	—
62	C₃×C₄	C₂⁴×C₇	1	8	yes	—
63	Dic₃	C₂⁴×C₇	1	16	yes	—
64	C₂³	C₇×Dic₆	1	8	yes	direct product factor
65	C₂³	C₂²×D₃×C₇	1	7	yes	—
66	Q₈	—	3	64	no	—
67	C₂×C₄	—	3	56	no	—
68	C₂×C₄	C₂²×D₃×C₇	1	28	yes	—
69	C₇	C₂³×Dic₆	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃	C₂²×C₇×Q₈	1	14	yes	complemented
71	C₂×C₃	C₂⁵×C₇	1	1	yes	characteristic
72	C₂²	C₂×C₇×Dic₆	1	28	yes	direct product factor
73	C₂²	C₂³×D₃×C₇	1	7	yes	—
74	C₄	—	3	16	no	—
75	C₄	C₂³×D₃×C₇	1	8	yes	—
76	C₃	C₂³×C₇×Q₈	1	1	yes	characteristic, complemented, Sylow
77	C₂	C₂²×C₇×Dic₆	1	14	yes	direct product factor
78	C₂	C₂⁴×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₇	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₇
Frattini	C₂	C₂⁴×D₃×C₇
Fitting	C₂³×C₃×C₄×C₇	C₂
socle, prime radical	C₂⁴×C₃×C₇	C₂²
hypercenter	C₂³×C₄×C₇	D₃
hypocenter	C₃	C₂³×C₇×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂³×C₄×C₇

Complements

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

N	H ≅ G/N	type
3. C₂²×C₇×Dic₆	C₂	direct (N × H)
8. C₂×C₇×Dic₆	C₂²	direct (N × H)
12. C₂³×Dic₆	C₇	direct (N × H)
13. C₂³×C₃×C₇	Q₈	semidirect
17. C₇×Dic₆	C₂³	direct (N × H)
24. C₂²×Dic₆	C₂×C₇	direct (N × H)
25. C₂²×C₃×C₇	C₂×Q₈	semidirect
30. C₂³×C₇	Dic₆	direct (N × H)
38. C₂×Dic₆	C₂²×C₇	direct (N × H)
39. C₂×C₃×C₇	C₂²×Q₈	semidirect
44. C₂²×C₇	C₂×Dic₆	direct (N × H)
48. C₂³×C₃	C₇×Q₈	semidirect
52. Dic₆	C₂³×C₇	direct (N × H)
53. C₃×C₇	C₂³×Q₈	semidirect
58. C₂×C₇	C₂²×Dic₆	direct (N × H)
60. C₂²×C₃	C₂×C₇×Q₈	semidirect
64. C₂³	C₇×Dic₆	direct (N × H)
69. C₇	C₂³×Dic₆	direct (N × H)
70. C₂×C₃	C₂²×C₇×Q₈	semidirect
72. C₂²	C₂×C₇×Dic₆	direct (N × H)
76. C₃	C₂³×C₇×Q₈	semidirect
77. C₂	C₂²×C₇×Dic₆	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
C₂³×C₇×Dic₃	2
C₂²×C₇×Dic₆	2
C₂³×C₇×Q₈	3
C₂³×Dic₆	7

Subgroups — C23×C7×Dic6 #13062348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×Dic₆ #13062348