Subgroups — C₂×C₃×C₇×2₋¹⁺⁴ #13062393

Order 1344.

1588 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×2₋¹⁺⁴	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×C₇×#41	C₂	1	10	yes	maximal, complemented
3	C₂²×C₃×C₇×Q₈	C₂	1	5	yes	maximal, complemented
4	C₂×C₇×2₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₇×#41	C₂²	1	80	yes	complemented
6	C₂×C₃×C₇×Q₈	C₂²	1	10	yes	—
7	C₂×C₃×C₇×Q₈	C₂²	1	40	yes	complemented
8	C₂×C₃×C₇×D₄	C₂²	1	10	yes	—
9	C₂²×C₃×C₄×C₇	C₂²	1	15	yes	—
10	C₇×2₋¹⁺⁴	C₂×C₃	1	16	yes	direct product factor
11	C₂×C₇×#41	C₂×C₃	1	10	yes	complemented
12	C₂²×C₇×Q₈	C₂×C₃	1	5	yes	complemented
13	C₂×C₃×2₋¹⁺⁴	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₃×C₇	C₂³	1	5	yes	—
15	C₃×C₇×Q₈	C₂³	1	40	yes	—
16	C₃×C₇×D₄	C₂³	1	40	yes	—
17	C₂×C₃×C₄×C₇	C₂³	1	60	yes	—
18	C₂×C₃×C₄×C₇	C₂³	1	10	yes	—
19	C₇×#41	C₂²×C₃	1	80	yes	complemented
20	C₂×C₇×Q₈	C₂²×C₃	1	10	yes	—
21	C₂×C₇×Q₈	C₂²×C₃	1	40	yes	complemented
22	C₂×C₇×D₄	C₂²×C₃	1	10	yes	—
23	C₂²×C₄×C₇	C₂²×C₃	1	15	yes	—
24	C₃×2₋¹⁺⁴	C₂×C₇	1	16	yes	direct product factor
25	C₂×C₃×#41	C₂×C₇	1	10	yes	complemented
26	C₂²×C₃×Q₈	C₂×C₇	1	5	yes	complemented
27	C₂²×C₃×C₇	—	2	10	no	—
28	C₂²×C₃×C₇	C₂⁴	1	10	yes	—
29	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
30	C₃×C₄×C₇	C₂⁴	1	20	yes	—
31	C₂×2₋¹⁺⁴	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
32	C₂³×C₇	C₂³×C₃	1	5	yes	—
33	C₇×Q₈	C₂³×C₃	1	40	yes	—
34	C₇×D₄	C₂³×C₃	1	40	yes	—
35	C₂×C₄×C₇	C₂³×C₃	1	10	yes	—
36	C₂×C₄×C₇	C₂³×C₃	1	60	yes	—
37	C₃×#41	C₂²×C₇	1	80	yes	complemented
38	C₂×C₃×Q₈	C₂²×C₇	1	10	yes	—
39	C₂×C₃×Q₈	C₂²×C₇	1	40	yes	complemented
40	C₂×C₃×D₄	C₂²×C₇	1	10	yes	—
41	C₂²×C₃×C₄	C₂²×C₇	1	15	yes	—
42	C₂×C₃×C₇	—	2	10	no	—
43	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	direct product factor
44	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
45	2₋¹⁺⁴	C₂×C₃×C₇	1	16	yes	direct product factor
46	C₂×#41	C₂×C₃×C₇	1	10	yes	complemented
47	C₂²×Q₈	C₂×C₃×C₇	1	5	yes	complemented
48	C₂²×C₇	—	2	10	no	—
49	C₂²×C₇	C₂⁴×C₃	1	10	yes	—
50	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
51	C₄×C₇	C₂⁴×C₃	1	20	yes	—
52	C₂³×C₃	C₂³×C₇	1	5	yes	—
53	C₃×Q₈	C₂³×C₇	1	40	yes	—
54	C₃×D₄	C₂³×C₇	1	40	yes	—
55	C₂×C₃×C₄	C₂³×C₇	1	10	yes	—
56	C₂×C₃×C₄	C₂³×C₇	1	60	yes	—
57	C₃×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Hall
58	#41	C₂²×C₃×C₇	1	80	yes	complemented
59	C₂×Q₈	C₂²×C₃×C₇	1	10	yes	—
60	C₂×Q₈	C₂²×C₃×C₇	1	40	yes	complemented
61	C₂×D₄	C₂²×C₃×C₇	1	10	yes	—
62	C₂²×C₄	C₂²×C₃×C₇	1	15	yes	—
63	C₂×C₇	—	2	10	no	—
64	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	direct product factor
65	C₂×C₇	C₂⁵×C₃	1	1	yes	characteristic
66	C₂²×C₃	—	2	10	no	—
67	C₂²×C₃	C₂⁴×C₇	1	10	yes	—
68	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
69	C₃×C₄	C₂⁴×C₇	1	20	yes	—
70	C₂³	C₂³×C₃×C₇	1	5	yes	—
71	Q₈	C₂³×C₃×C₇	1	40	yes	—
72	D₄	C₂³×C₃×C₇	1	40	yes	—
73	C₂×C₄	C₂³×C₃×C₇	1	10	yes	—
74	C₂×C₄	C₂³×C₃×C₇	1	60	yes	—
75	C₇	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×C₃	—	2	10	no	—
77	C₂×C₃	C₇×2₋¹⁺⁴	1	2	yes	direct product factor
78	C₂×C₃	C₂⁵×C₇	1	1	yes	characteristic
79	C₂²	—	2	10	no	—
80	C₂²	C₂⁴×C₃×C₇	1	10	yes	—
81	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
82	C₄	C₂⁴×C₃×C₇	1	20	yes	—
83	C₃	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	2	10	no	—
85	C₂	C₃×C₇×2₋¹⁺⁴	1	2	yes	direct product factor
86	C₂	C₂⁵×C₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₃×C₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×2₋¹⁺⁴	C₂	direct (N × H)
2. C₂×C₃×C₇×#41	C₂	semidirect
3. C₂²×C₃×C₇×Q₈	C₂	semidirect
4. C₂×C₇×2₋¹⁺⁴	C₃	direct (N × H)
5. C₃×C₇×#41	C₂²	semidirect
7. C₂×C₃×C₇×Q₈	C₂²	semidirect
10. C₇×2₋¹⁺⁴	C₂×C₃	direct (N × H)
11. C₂×C₇×#41	C₂×C₃	semidirect
12. C₂²×C₇×Q₈	C₂×C₃	semidirect
13. C₂×C₃×2₋¹⁺⁴	C₇	direct (N × H)
19. C₇×#41	C₂²×C₃	semidirect
21. C₂×C₇×Q₈	C₂²×C₃	semidirect
24. C₃×2₋¹⁺⁴	C₂×C₇	direct (N × H)
25. C₂×C₃×#41	C₂×C₇	semidirect
26. C₂²×C₃×Q₈	C₂×C₇	semidirect
31. C₂×2₋¹⁺⁴	C₃×C₇	direct (N × H)
37. C₃×#41	C₂²×C₇	semidirect
39. C₂×C₃×Q₈	C₂²×C₇	semidirect
43. C₂×C₃×C₇	2₋¹⁺⁴	direct (N × H)
45. 2₋¹⁺⁴	C₂×C₃×C₇	direct (N × H)
46. C₂×#41	C₂×C₃×C₇	semidirect
47. C₂²×Q₈	C₂×C₃×C₇	semidirect
57. C₃×C₇	C₂×2₋¹⁺⁴	direct (N × H)
58. #41	C₂²×C₃×C₇	semidirect
60. C₂×Q₈	C₂²×C₃×C₇	semidirect
64. C₂×C₇	C₃×2₋¹⁺⁴	direct (N × H)
75. C₇	C₂×C₃×2₋¹⁺⁴	direct (N × H)
77. C₂×C₃	C₇×2₋¹⁺⁴	direct (N × H)
83. C₃	C₂×C₇×2₋¹⁺⁴	direct (N × H)
85. C₂	C₃×C₇×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×2₋¹⁺⁴	2
C₂×C₃×C₇×#41	2
C₂²×C₃×C₇×Q₈	2
C₂×C₇×2₋¹⁺⁴	3
C₂×C₃×2₋¹⁺⁴	7

Subgroups — C2×C3×C7×2−1+4 #13062393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×2₋¹⁺⁴ #13062393