Subgroups — C₂³×D₈₄ #13062370

Order 1344.

2724 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×D₈₄	C₂	1	28	yes	maximal, direct product factor
4	C₂³×D₂₈	—	3	1	no	maximal, Hall
5	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
6	C₂³×D₂₁	—	2	16	no	—
7	C₂³×D₂₁	C₂²	1	28	yes	complemented
8	C₂²×C₃×C₄×C₇	C₂²	1	14	yes	complemented
9	C₂×D₈₄	C₂²	1	112	yes	direct product factor
10	C₂⁴×D₇	—	3	2	no	—
11	C₂³×C₄×C₇	D₃	1	1	yes	characteristic, complemented
12	C₂²×D₂₈	—	3	28	no	—
13	C₂³×D₁₂	—	7	1	no	maximal, Hall
14	C₂³×C₃×C₇	D₄	1	8	yes	complemented
15	C₂³×C₃×C₇	C₂³	1	7	yes	—
16	C₂²×D₂₁	—	2	112	no	—
17	C₂²×D₂₁	C₂³	1	56	yes	complemented
18	C₂×C₃×C₄×C₇	C₂³	1	28	yes	complemented
19	D₈₄	C₂³	1	64	yes	direct product factor
20	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
21	C₂³×D₇	—	3	28	no	—
22	C₂³×D₇	—	6	16	no	—
23	C₂²×C₄×C₇	C₂×D₃	1	14	yes	complemented
24	C₂×D₂₈	—	3	112	no	—
25	C₂⁴×D₃	—	7	2	no	—
26	C₂³×C₃×C₄	D₇	1	1	yes	characteristic, complemented
27	C₂²×D₁₂	—	7	28	no	—
28	C₂²×C₃×C₇	C₂×D₄	1	28	yes	complemented
29	C₂²×C₃×C₇	C₂⁴	1	7	yes	—
30	C₂×D₂₁	—	2	112	no	—
31	C₂×D₂₁	C₂⁴	1	16	yes	complemented
32	C₃×C₄×C₇	C₂⁴	1	8	yes	complemented
33	C₂³×D₄	—	21	1	no	Sylow
34	C₂³×C₇	D₁₂	1	8	yes	complemented
35	C₂³×C₇	C₂²×D₃	1	7	yes	—
36	C₂²×D₇	—	3	56	no	—
37	C₂²×D₇	—	6	112	no	—
38	C₂×C₄×C₇	C₂²×D₃	1	28	yes	complemented
39	D₂₈	—	3	64	no	—
40	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
41	C₂³×D₃	—	7	28	no	—
42	C₂³×D₃	—	14	16	no	—
43	C₂²×C₃×C₄	C₂×D₇	1	14	yes	complemented
44	C₂×D₁₂	—	7	112	no	—
45	C₂×C₃×C₇	C₂²×D₄	1	14	yes	complemented
46	C₂×C₃×C₇	C₂⁵	1	1	yes	characteristic
47	D₂₁	—	2	16	no	—
48	C₂⁵	—	21	2	no	—
49	C₂²×D₄	—	21	28	no	—
50	C₂³×C₄	D₂₁	1	1	yes	characteristic, complemented
51	C₂²×C₇	C₂×D₁₂	1	28	yes	complemented
52	C₂²×C₇	C₂³×D₃	1	7	yes	—
53	C₂×D₇	—	3	16	no	—
54	C₂×D₇	—	6	112	no	—
55	C₄×C₇	C₂³×D₃	1	8	yes	complemented
56	C₂³×C₃	D₂₈	1	8	yes	complemented
57	C₂³×C₃	C₂²×D₇	1	7	yes	—
58	C₂²×D₃	—	7	56	no	—
59	C₂²×D₃	—	14	112	no	—
60	C₂×C₃×C₄	C₂²×D₇	1	28	yes	complemented
61	D₁₂	—	7	64	no	—
62	C₃×C₇	C₂³×D₄	1	1	yes	characteristic, complemented, Hall
63	C₂⁴	—	21	28	no	—
64	C₂⁴	—	42	16	no	—
65	C₂⁴	C₂×D₂₁	1	1	yes	characteristic
66	C₂×D₄	—	21	112	no	—
67	C₂²×C₄	C₂×D₂₁	1	14	yes	complemented
68	C₂×C₇	C₂²×D₁₂	1	14	yes	complemented
69	C₂×C₇	C₂⁴×D₃	1	1	yes	characteristic
70	D₇	—	6	16	no	—
71	C₂²×C₃	C₂×D₂₈	1	28	yes	complemented
72	C₂²×C₃	C₂³×D₇	1	7	yes	—
73	C₂×D₃	—	7	16	no	—
74	C₂×D₃	—	14	112	no	—
75	C₃×C₄	C₂³×D₇	1	8	yes	complemented
76	C₂³	—	21	56	no	—
77	C₂³	—	42	112	no	—
78	C₂³	D₈₄	1	8	yes	direct product factor
79	C₂³	C₂²×D₂₁	1	7	yes	—
80	D₄	—	21	64	no	—
81	C₂×C₄	C₂²×D₂₁	1	28	yes	complemented
82	C₇	C₂³×D₁₂	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	C₂²×D₂₈	1	14	yes	complemented
84	C₂×C₃	C₂⁴×D₇	1	1	yes	characteristic
85	D₃	—	14	16	no	—
86	C₂²	—	21	16	no	—
87	C₂²	—	42	112	no	—
88	C₂²	C₂×D₈₄	1	28	yes	direct product factor
89	C₂²	C₂³×D₂₁	1	7	yes	—
90	C₄	C₂³×D₂₁	1	8	yes	complemented
91	C₃	C₂³×D₂₈	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	42	16	no	—
93	C₂	C₂²×D₈₄	1	14	yes	direct product factor
94	C₂	C₂⁴×D₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₂₁
commutator, cosocle	C₂×C₃×C₇	C₂⁵
Frattini	C₂	C₂⁴×D₂₁
Fitting	C₂³×C₃×C₄×C₇	C₂
socle	C₂⁴×C₃×C₇	C₂²
hypercenter	C₂³×C₄	D₂₁
hypocenter	C₃×C₇	C₂³×D₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₁	C₂	semidirect
2. C₂³×C₃×C₄×C₇	C₂	semidirect
3. C₂²×D₈₄	C₂	direct (N × H)
7. C₂³×D₂₁	C₂²	semidirect
8. C₂²×C₃×C₄×C₇	C₂²	semidirect
9. C₂×D₈₄	C₂²	direct (N × H)
11. C₂³×C₄×C₇	D₃	semidirect
14. C₂³×C₃×C₇	D₄	semidirect
17. C₂²×D₂₁	C₂³	semidirect
18. C₂×C₃×C₄×C₇	C₂³	semidirect
19. D₈₄	C₂³	direct (N × H)
23. C₂²×C₄×C₇	C₂×D₃	semidirect
26. C₂³×C₃×C₄	D₇	semidirect
28. C₂²×C₃×C₇	C₂×D₄	semidirect
31. C₂×D₂₁	C₂⁴	semidirect
32. C₃×C₄×C₇	C₂⁴	semidirect
34. C₂³×C₇	D₁₂	semidirect
38. C₂×C₄×C₇	C₂²×D₃	semidirect
43. C₂²×C₃×C₄	C₂×D₇	semidirect
45. C₂×C₃×C₇	C₂²×D₄	semidirect
50. C₂³×C₄	D₂₁	semidirect
51. C₂²×C₇	C₂×D₁₂	semidirect
55. C₄×C₇	C₂³×D₃	semidirect
56. C₂³×C₃	D₂₈	semidirect
60. C₂×C₃×C₄	C₂²×D₇	semidirect
62. C₃×C₇	C₂³×D₄	semidirect
67. C₂²×C₄	C₂×D₂₁	semidirect
68. C₂×C₇	C₂²×D₁₂	semidirect
71. C₂²×C₃	C₂×D₂₈	semidirect
75. C₃×C₄	C₂³×D₇	semidirect
78. C₂³	D₈₄	direct (N × H)
81. C₂×C₄	C₂²×D₂₁	semidirect
82. C₇	C₂³×D₁₂	semidirect
83. C₂×C₃	C₂²×D₂₈	semidirect
88. C₂²	C₂×D₈₄	direct (N × H)
90. C₄	C₂³×D₂₁	semidirect
91. C₃	C₂³×D₂₈	semidirect
93. C₂	C₂²×D₈₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁	2
C₂³×C₃×C₄×C₇	2
C₂²×D₈₄	2
C₂³×D₂₈	3
C₂³×D₁₂	7

Subgroups — C23×D84 #13062370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₈₄ #13062370