Subgroups — C₃×C₇×#543 #13061963

Order 1344.

484 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
2	C₂×C₃×C₇×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₇×#543	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₃×C₇×Q₈	C₂²	1	4	yes	complemented
6	C₂²×C₃×C₄×C₇	C₂²	1	6	yes	—
7	C₃×C₇×C₄⋊C₄	C₂²	1	12	yes	complemented
8	C₃×C₇×C₂²⋊C₄	C₂²	1	12	yes	—
9	C₇×C₂²⋊Q₈	C₂×C₃	1	12	yes	complemented
10	C₂×C₇×C₂²⋊C₄	C₂×C₃	1	3	yes	—
11	C₃×#543	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂³×C₃×C₇	—	4	2	no	—
13	C₂³×C₃×C₇	Q₈	1	4	yes	complemented
14	C₂³×C₃×C₇	C₂³	1	3	yes	—
15	C₃×C₇×Q₈	—	4	4	no	—
16	C₂×C₃×C₄×C₇	—	4	6	no	—
17	C₂×C₃×C₄×C₇	C₂³	1	12	yes	—
18	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
19	C₂×C₇×Q₈	C₂²×C₃	1	4	yes	complemented
20	C₂²×C₄×C₇	C₂²×C₃	1	6	yes	—
21	C₇×C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
22	C₇×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
23	C₃×C₂²⋊Q₈	C₂×C₇	1	12	yes	complemented
24	C₂×C₃×C₂²⋊C₄	C₂×C₇	1	3	yes	—
25	C₂²×C₃×C₇	—	2	6	no	—
26	C₂²×C₃×C₇	—	4	4	no	—
27	C₂²×C₃×C₇	C₂×Q₈	1	6	yes	—
28	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
29	C₃×C₄×C₇	—	2	12	no	—
30	#543	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
31	C₂³×C₇	—	4	2	no	—
32	C₂³×C₇	C₃×Q₈	1	4	yes	complemented
33	C₂³×C₇	C₂³×C₃	1	3	yes	—
34	C₇×Q₈	—	4	4	no	—
35	C₂×C₄×C₇	—	4	6	no	—
36	C₂×C₄×C₇	C₂³×C₃	1	12	yes	—
37	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
38	C₂×C₃×Q₈	C₂²×C₇	1	4	yes	complemented
39	C₂²×C₃×C₄	C₂²×C₇	1	6	yes	—
40	C₃×C₄⋊C₄	C₂²×C₇	1	12	yes	complemented
41	C₃×C₂²⋊C₄	C₂²×C₇	1	12	yes	—
42	C₂×C₃×C₇	—	2	6	no	—
43	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
45	C₂²⋊Q₈	C₂×C₃×C₇	1	12	yes	complemented
46	C₂×C₂²⋊C₄	C₂×C₃×C₇	1	3	yes	—
47	C₂²×C₇	—	2	6	no	—
48	C₂²×C₇	—	4	4	no	—
49	C₂²×C₇	C₂×C₃×Q₈	1	6	yes	—
50	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
51	C₄×C₇	—	2	12	no	—
52	C₂³×C₃	—	4	2	no	—
53	C₂³×C₃	C₇×Q₈	1	4	yes	complemented
54	C₂³×C₃	C₂³×C₇	1	3	yes	—
55	C₃×Q₈	—	4	4	no	—
56	C₂×C₃×C₄	—	4	6	no	—
57	C₂×C₃×C₄	C₂³×C₇	1	12	yes	—
58	C₃×C₇	#543	1	1	yes	characteristic, direct product factor, Hall
59	C₂⁴	C₂²×C₃×C₇	1	1	yes	characteristic
60	C₂×Q₈	C₂²×C₃×C₇	1	4	yes	complemented
61	C₂²×C₄	C₂²×C₃×C₇	1	6	yes	—
62	C₄⋊C₄	C₂²×C₃×C₇	1	12	yes	complemented
63	C₂²⋊C₄	C₂²×C₃×C₇	1	12	yes	—
64	C₂×C₇	—	2	6	no	—
65	C₂×C₇	C₂²×C₃×Q₈	1	1	yes	characteristic
66	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
67	C₂²×C₃	—	2	6	no	—
68	C₂²×C₃	—	4	4	no	—
69	C₂²×C₃	C₂×C₇×Q₈	1	6	yes	—
70	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
71	C₃×C₄	—	2	12	no	—
72	C₂³	—	4	2	no	—
73	C₂³	C₃×C₇×Q₈	1	4	yes	complemented
74	C₂³	C₂³×C₃×C₇	1	3	yes	—
75	Q₈	—	4	4	no	—
76	C₂×C₄	—	4	6	no	—
77	C₂×C₄	C₂³×C₃×C₇	1	12	yes	—
78	C₇	C₃×#543	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃	—	2	6	no	—
80	C₂×C₃	C₂²×C₇×Q₈	1	1	yes	characteristic
81	C₂×C₃	C₇×H(5,𝔽₂)	1	2	yes	—
82	C₂²	—	2	6	no	—
83	C₂²	—	4	4	no	—
84	C₂²	C₂×C₃×C₇×Q₈	1	6	yes	—
85	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
86	C₄	—	2	12	no	—
87	C₃	C₇×#543	1	1	yes	characteristic, direct product factor, Sylow
88	C₂	—	2	6	no	—
89	C₂	C₂²×C₃×C₇×Q₈	1	1	yes	characteristic
90	C₂	C₃×C₇×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
prime radical	C₂⁴×C₃×C₇	C₂²
LCS	G ▷ C₂² ▷ 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×C₂²⋊Q₈	C₂	semidirect
3. C₇×#543	C₃	direct (N × H)
5. C₂×C₃×C₇×Q₈	C₂²	semidirect
7. C₃×C₇×C₄⋊C₄	C₂²	semidirect
9. C₇×C₂²⋊Q₈	C₂×C₃	semidirect
11. C₃×#543	C₇	direct (N × H)
13. C₂³×C₃×C₇	Q₈	semidirect
19. C₂×C₇×Q₈	C₂²×C₃	semidirect
21. C₇×C₄⋊C₄	C₂²×C₃	semidirect
23. C₃×C₂²⋊Q₈	C₂×C₇	semidirect
30. #543	C₃×C₇	direct (N × H)
32. C₂³×C₇	C₃×Q₈	semidirect
38. C₂×C₃×Q₈	C₂²×C₇	semidirect
40. C₃×C₄⋊C₄	C₂²×C₇	semidirect
45. C₂²⋊Q₈	C₂×C₃×C₇	semidirect
53. C₂³×C₃	C₇×Q₈	semidirect
58. C₃×C₇	#543	direct (N × H)
60. C₂×Q₈	C₂²×C₃×C₇	semidirect
62. C₄⋊C₄	C₂²×C₃×C₇	semidirect
73. C₂³	C₃×C₇×Q₈	semidirect
78. C₇	C₃×#543	direct (N × H)
87. C₃	C₇×#543	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×C₂²⋊Q₈	2
C₂×C₃×C₇×C₂²⋊C₄	2
C₇×#543	3
C₃×#543	7

Subgroups — C3×C7×#543 #13061963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#543 #13061963