Subgroups — C₂³×C₉×A₄ #11724067

Order 864.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×A₄	C₂	1	7	yes	maximal, direct product factor
2	C₂³×C₃×A₄	C₃	1	1	yes	maximal, characteristic
3	C₂⁵×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₂²⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₂³×C₃×C₉	—	4	1	no	maximal
6	C₂×C₉×A₄	C₂²	1	7	yes	direct product factor
7	C₂²×C₃×A₄	C₂×C₃	1	7	yes	—
8	C₂⁴×C₉	—	3	1	no	—
9	C₂⁴×C₉	—	3	7	no	—
10	C₂⁴×C₉	C₂×C₃	1	7	yes	complemented
11	C₂²×C₂²⋊C₉	C₂×C₃	1	14	yes	complemented
12	C₂²×C₃×C₉	—	4	7	no	—
13	C₉×A₄	C₂³	1	1	yes	characteristic, direct product factor
14	C₂⁵×C₃	C₃²	1	1	yes	characteristic
15	C₂³×A₄	C₉	1	3	yes	direct product factor
16	C₂³×C₃²	—	4	1	no	—
17	C₂×C₃×A₄	C₂²×C₃	1	7	yes	—
18	C₂³×C₉	—	3	21	no	—
19	C₂³×C₉	—	3	14	no	—
20	C₂³×C₉	—	3	7	no	—
21	C₂³×C₉	—	3	7	no	—
22	C₂³×C₉	—	4	2	no	—
23	C₂³×C₉	A₄	1	1	yes	characteristic, direct product factor
24	C₂³×C₉	C₂²×C₃	1	7	yes	complemented
25	C₂×C₂²⋊C₉	C₂²×C₃	1	14	yes	complemented
26	C₂×C₃×C₉	—	4	7	no	—
27	C₂⁴×C₃	—	3	7	no	—
28	C₂⁴×C₃	—	3	1	no	—
29	C₂⁴×C₃	C₂×C₃²	1	7	yes	—
30	C₂²×A₄	C₂×C₉	1	21	yes	direct product factor
31	C₂²×C₃²	—	4	7	no	—
32	C₃×A₄	C₂³×C₃	1	1	yes	characteristic
33	C₂²×C₉	—	3	7	no	—
34	C₂²×C₉	—	3	14	no	—
35	C₂²×C₉	—	3	21	no	—
36	C₂²×C₉	—	3	7	no	—
37	C₂²×C₉	—	4	14	no	—
38	C₂²×C₉	C₂×A₄	1	7	yes	direct product factor
39	C₂²×C₉	C₂³×C₃	1	1	yes	characteristic, complemented
40	C₂²⋊C₉	C₂³×C₃	1	2	yes	complemented
41	C₂⁵	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
42	C₃×C₉	—	4	1	no	Sylow
43	C₂³×C₃	—	3	21	no	—
44	C₂³×C₃	—	3	14	no	—
45	C₂³×C₃	—	3	7	no	—
46	C₂³×C₃	—	3	7	no	—
47	C₂³×C₃	—	4	3	no	—
48	C₂³×C₃	C₃×A₄	1	1	yes	characteristic
49	C₂³×C₃	C₂²×C₃²	1	7	yes	—
50	C₂×A₄	C₂²×C₉	1	21	yes	direct product factor
51	C₂×C₃²	—	4	7	no	—
52	C₂×C₉	—	3	7	no	—
53	C₂×C₉	—	3	1	no	—
54	C₂×C₉	—	4	14	no	—
55	C₂×C₉	C₂²×A₄	1	7	yes	direct product factor
56	C₂⁴	—	3	7	no	—
57	C₂⁴	—	3	1	no	—
58	C₂⁴	C₂×C₃×C₉	1	7	yes	complemented
59	C₂²×C₃	—	3	7	no	—
60	C₂²×C₃	—	3	14	no	—
61	C₂²×C₃	—	3	7	no	—
62	C₂²×C₃	—	3	21	no	—
63	C₂²×C₃	—	4	21	no	—
64	C₂²×C₃	C₂×C₃×A₄	1	7	yes	—
65	C₂²×C₃	C₂³×C₃²	1	1	yes	characteristic
66	A₄	C₂³×C₉	1	3	yes	direct product factor
67	C₃²	—	4	1	no	—
68	C₉	—	4	2	no	—
69	C₉	C₂³×A₄	1	1	yes	characteristic, direct product factor
70	C₂³	—	3	21	no	—
71	C₂³	—	3	14	no	—
72	C₂³	—	3	7	no	—
73	C₂³	—	3	7	no	—
74	C₂³	C₉×A₄	1	1	yes	characteristic, direct product factor
75	C₂³	C₂²×C₃×C₉	1	7	yes	complemented
76	C₂×C₃	—	3	1	no	—
77	C₂×C₃	—	3	7	no	—
78	C₂×C₃	—	4	21	no	—
79	C₂×C₃	C₂²×C₃×A₄	1	7	yes	—
80	C₂²	—	3	7	no	—
81	C₂²	—	3	14	no	—
82	C₂²	—	3	7	no	—
83	C₂²	—	3	21	no	—
84	C₂²	C₂×C₉×A₄	1	7	yes	direct product factor
85	C₂²	C₂³×C₃×C₉	1	1	yes	characteristic, complemented
86	C₃	—	4	3	no	—
87	C₃	C₂³×C₃×A₄	1	1	yes	characteristic
88	C₂	—	3	1	no	—
89	C₂	—	3	7	no	—
90	C₂	C₂²×C₉×A₄	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₉	A₄
commutator, hypocenter	C₂²	C₂³×C₃×C₉
Frattini	C₃	C₂³×C₃×A₄
Fitting	C₂⁵×C₉	C₃
socle	C₂⁵×C₃	C₃²
prime radical	C₂³×C₃×A₄	C₃
cosocle	C₂²×C₃	C₂³×C₃²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂³×C₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×A₄	C₂	direct (N × H)
3. C₂⁵×C₉	C₃	semidirect
4. C₂³×C₂²⋊C₉	C₃	semidirect
6. C₂×C₉×A₄	C₂²	direct (N × H)
10. C₂⁴×C₉	C₂×C₃	semidirect
11. C₂²×C₂²⋊C₉	C₂×C₃	semidirect
13. C₉×A₄	C₂³	direct (N × H)
15. C₂³×A₄	C₉	direct (N × H)
23. C₂³×C₉	A₄	direct (N × H)
24. C₂³×C₉	C₂²×C₃	semidirect
25. C₂×C₂²⋊C₉	C₂²×C₃	semidirect
30. C₂²×A₄	C₂×C₉	direct (N × H)
38. C₂²×C₉	C₂×A₄	direct (N × H)
39. C₂²×C₉	C₂³×C₃	semidirect
40. C₂²⋊C₉	C₂³×C₃	semidirect
41. C₂⁵	C₃×C₉	semidirect
50. C₂×A₄	C₂²×C₉	direct (N × H)
55. C₂×C₉	C₂²×A₄	direct (N × H)
58. C₂⁴	C₂×C₃×C₉	semidirect
66. A₄	C₂³×C₉	direct (N × H)
69. C₉	C₂³×A₄	direct (N × H)
74. C₂³	C₉×A₄	direct (N × H)
75. C₂³	C₂²×C₃×C₉	semidirect
84. C₂²	C₂×C₉×A₄	direct (N × H)
85. C₂²	C₂³×C₃×C₉	semidirect
90. C₂	C₂²×C₉×A₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×A₄	2
C₂³×C₃×A₄	3
C₂⁵×C₉	3
C₂³×C₂²⋊C₉	3
C₂³×C₃×C₉	4

Subgroups — C23×C9×A4 #11724067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₉×A₄ #11724067