Subgroups — C₂³×C₄⋊(C₃⋊(C₃⋊D₃)) #13181752

Order 1728.

19068 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊(C₃⋊D₃)	—	3	13	no	maximal
5	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
6	C₂³×C₃⋊(C₃⋊D₃)	—	2	16	no	—
7	C₂³×C₃⋊(C₃⋊D₃)	C₂²	1	28	yes	complemented
8	C₂²×C₃³×C₄	C₂²	1	14	yes	complemented
9	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	112	yes	direct product factor
10	C₂⁴×C₃⋊D₃	—	3	26	no	—
11	C₂³×C₃²×C₄	D₃	1	13	yes	complemented
12	C₂²×C₄⋊(C₃⋊D₃)	—	3	364	no	—
13	C₂³×C₃³	D₄	1	8	yes	complemented
14	C₂³×C₃³	C₂³	1	7	yes	—
15	C₂²×C₃⋊(C₃⋊D₃)	—	2	112	no	—
16	C₂²×C₃⋊(C₃⋊D₃)	C₂³	1	56	yes	complemented
17	C₂×C₃³×C₄	C₂³	1	28	yes	complemented
18	C₄⋊(C₃⋊(C₃⋊D₃))	C₂³	1	64	yes	direct product factor
19	C₂³×D₁₂	—	9	13	no	—
20	C₂⁴×C₃²	C₂×D₃	1	13	yes	—
21	C₂³×C₃⋊D₃	—	3	364	no	—
22	C₂³×C₃⋊D₃	—	6	208	no	—
23	C₂²×C₃²×C₄	C₂×D₃	1	182	yes	complemented
24	C₂×C₄⋊(C₃⋊D₃)	—	3	1456	no	—
25	C₂²×C₃³	C₂×D₄	1	28	yes	complemented
26	C₂²×C₃³	C₂⁴	1	7	yes	—
27	C₂×C₃⋊(C₃⋊D₃)	—	2	112	no	—
28	C₂×C₃⋊(C₃⋊D₃)	C₂⁴	1	16	yes	complemented
29	C₃³×C₄	C₂⁴	1	8	yes	complemented
30	C₂⁴×D₃	—	9	26	no	—
31	C₂³×C₃×C₄	C₃⋊D₃	1	13	yes	complemented
32	C₂²×D₁₂	—	9	364	no	—
33	C₂³×C₃²	D₁₂	1	104	yes	complemented
34	C₂³×C₃²	C₂²×D₃	1	91	yes	—
35	C₂²×C₃⋊D₃	—	3	728	no	—
36	C₂²×C₃⋊D₃	—	6	1456	no	—
37	C₂×C₃²×C₄	C₂²×D₃	1	364	yes	complemented
38	C₄⋊(C₃⋊D₃)	—	3	832	no	—
39	C₂³×D₄	—	27	1	no	Sylow
40	C₂×C₃³	C₂²×D₄	1	14	yes	complemented
41	C₂×C₃³	C₂⁵	1	1	yes	characteristic
42	C₃⋊(C₃⋊D₃)	—	2	16	no	—
43	C₂⁴×C₃	C₂×C₃⋊D₃	1	13	yes	—
44	C₂³×D₃	—	9	364	no	—
45	C₂³×D₃	—	18	208	no	—
46	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	182	yes	complemented
47	C₂×D₁₂	—	9	1456	no	—
48	C₂²×C₃²	C₂×D₁₂	1	364	yes	complemented
49	C₂²×C₃²	C₂³×D₃	1	91	yes	—
50	C₂×C₃⋊D₃	—	3	208	no	—
51	C₂×C₃⋊D₃	—	6	1456	no	—
52	C₃²×C₄	C₂³×D₃	1	104	yes	complemented
53	C₂⁵	—	27	2	no	—
54	C₂²×D₄	—	27	28	no	—
55	C₂³×C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
56	C₃³	C₂³×D₄	1	1	yes	characteristic, complemented, Sylow
57	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	104	yes	complemented
58	C₂³×C₃	C₂²×C₃⋊D₃	1	91	yes	—
59	C₂²×D₃	—	9	728	no	—
60	C₂²×D₃	—	18	1456	no	—
61	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	364	yes	complemented
62	D₁₂	—	9	832	no	—
63	C₂×C₃²	C₂²×D₁₂	1	182	yes	complemented
64	C₂×C₃²	C₂⁴×D₃	1	13	yes	—
65	C₃⋊D₃	—	6	208	no	—
66	C₂⁴	—	27	28	no	—
67	C₂⁴	—	54	16	no	—
68	C₂⁴	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
69	C₂×D₄	—	27	112	no	—
70	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	14	yes	complemented
71	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	364	yes	complemented
72	C₂²×C₃	C₂³×C₃⋊D₃	1	91	yes	—
73	C₂×D₃	—	9	208	no	—
74	C₂×D₃	—	18	1456	no	—
75	C₃×C₄	C₂³×C₃⋊D₃	1	104	yes	complemented
76	C₃²	C₂³×D₁₂	1	13	yes	complemented
77	C₂³	—	27	56	no	—
78	C₂³	—	54	112	no	—
79	C₂³	C₄⋊(C₃⋊(C₃⋊D₃))	1	8	yes	direct product factor
80	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	7	yes	—
81	D₄	—	27	64	no	—
82	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	28	yes	complemented
83	C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	1	182	yes	complemented
84	C₂×C₃	C₂⁴×C₃⋊D₃	1	13	yes	—
85	D₃	—	18	208	no	—
86	C₂²	—	27	16	no	—
87	C₂²	—	54	112	no	—
88	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	28	yes	direct product factor
89	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	7	yes	—
90	C₄	C₂³×C₃⋊(C₃⋊D₃)	1	8	yes	complemented
91	C₃	C₂³×C₄⋊(C₃⋊D₃)	1	13	yes	complemented
92	C₂	—	54	16	no	—
93	C₂	C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	1	14	yes	direct product factor
94	C₂	C₂⁴×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³	C₂⁵
Frattini	C₂	C₂⁴×C₃⋊(C₃⋊D₃)
Fitting	C₂³×C₃³×C₄	C₂
socle	C₂⁴×C₃³	C₂²
hypercenter	C₂³×C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂³×D₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃⋊(C₃⋊D₃)	C₂	semidirect
2. C₂³×C₃³×C₄	C₂	semidirect
3. C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	direct (N × H)
7. C₂³×C₃⋊(C₃⋊D₃)	C₂²	semidirect
8. C₂²×C₃³×C₄	C₂²	semidirect
9. C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	direct (N × H)
11. C₂³×C₃²×C₄	D₃	semidirect
13. C₂³×C₃³	D₄	semidirect
16. C₂²×C₃⋊(C₃⋊D₃)	C₂³	semidirect
17. C₂×C₃³×C₄	C₂³	semidirect
18. C₄⋊(C₃⋊(C₃⋊D₃))	C₂³	direct (N × H)
23. C₂²×C₃²×C₄	C₂×D₃	semidirect
25. C₂²×C₃³	C₂×D₄	semidirect
28. C₂×C₃⋊(C₃⋊D₃)	C₂⁴	semidirect
29. C₃³×C₄	C₂⁴	semidirect
31. C₂³×C₃×C₄	C₃⋊D₃	semidirect
33. C₂³×C₃²	D₁₂	semidirect
37. C₂×C₃²×C₄	C₂²×D₃	semidirect
40. C₂×C₃³	C₂²×D₄	semidirect
46. C₂²×C₃×C₄	C₂×C₃⋊D₃	semidirect
48. C₂²×C₃²	C₂×D₁₂	semidirect
52. C₃²×C₄	C₂³×D₃	semidirect
55. C₂³×C₄	C₃⋊(C₃⋊D₃)	semidirect
56. C₃³	C₂³×D₄	semidirect
57. C₂³×C₃	C₄⋊(C₃⋊D₃)	semidirect
61. C₂×C₃×C₄	C₂²×C₃⋊D₃	semidirect
63. C₂×C₃²	C₂²×D₁₂	semidirect
70. C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	semidirect
71. C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	semidirect
75. C₃×C₄	C₂³×C₃⋊D₃	semidirect
76. C₃²	C₂³×D₁₂	semidirect
79. C₂³	C₄⋊(C₃⋊(C₃⋊D₃))	direct (N × H)
82. C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	semidirect
83. C₂×C₃	C₂²×C₄⋊(C₃⋊D₃)	semidirect
88. C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	direct (N × H)
90. C₄	C₂³×C₃⋊(C₃⋊D₃)	semidirect
91. C₃	C₂³×C₄⋊(C₃⋊D₃)	semidirect
93. C₂	C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	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×C₃⋊(C₃⋊D₃)	2
C₂³×C₃³×C₄	2
C₂²×C₄⋊(C₃⋊(C₃⋊D₃))	2
C₂³×C₄⋊(C₃⋊D₃)	3

Subgroups — C23×C4⋊(C3⋊(C3⋊D3)) #13181752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊(C₃⋊(C₃⋊D₃)) #13181752