Subgroups — C₃²×C₄×C₉³ #72203402

Order 26244.

180234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₉³	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄×C₉²	C₃	1	13	yes	maximal
3	C₃×C₄×C₉³	C₃	1	108	yes	maximal, direct product factor
4	C₃²×C₉³	C₄	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×C₃³×C₉²	C₂×C₃	1	13	yes	—
6	C₂×C₃×C₉³	C₂×C₃	1	108	yes	—
7	C₃⁴×C₄×C₉	C₃²	1	13	yes	—
8	C₃²×C₄×C₉²	C₉	1	1053	yes	direct product factor
9	C₃²×C₄×C₉²	C₃²	1	468	yes	—
10	C₄×C₉³	C₃²	1	729	yes	direct product factor
11	C₃³×C₉²	C₃×C₄	1	13	yes	—
12	C₃×C₉³	C₃×C₄	1	108	yes	direct product factor
13	C₂×C₃⁴×C₉	C₂×C₃²	1	13	yes	—
14	C₂×C₃²×C₉²	C₂×C₉	1	1053	yes	—
15	C₂×C₃²×C₉²	C₂×C₃²	1	468	yes	—
16	C₂×C₉³	C₂×C₃²	1	729	yes	—
17	C₃⁵×C₄	C₃³	1	1	yes	characteristic
18	C₃³×C₄×C₉	C₃×C₉	1	1404	yes	—
19	C₃³×C₄×C₉	C₃³	1	156	yes	—
20	C₃×C₄×C₉²	C₃×C₉	1	12636	yes	direct product factor
21	C₃×C₄×C₉²	C₃³	1	1053	yes	—
22	C₃⁴×C₉	C₃²×C₄	1	13	yes	—
23	C₃²×C₉²	C₄×C₉	1	1053	yes	direct product factor
24	C₃²×C₉²	C₃²×C₄	1	468	yes	—
25	C₉³	C₃²×C₄	1	729	yes	direct product factor
26	C₂×C₃⁵	C₂×C₃³	1	1	yes	characteristic
27	C₂×C₃³×C₉	C₂×C₃×C₉	1	1404	yes	—
28	C₂×C₃³×C₉	C₂×C₃³	1	156	yes	—
29	C₂×C₃×C₉²	C₂×C₃×C₉	1	12636	yes	—
30	C₂×C₃×C₉²	C₂×C₃³	1	1053	yes	—
31	C₃⁴×C₄	C₃²×C₉	1	117	yes	—
32	C₃⁴×C₄	C₃⁴	1	4	yes	—
33	C₃²×C₄×C₉	C₉²	1	9477	yes	direct product factor
34	C₃²×C₄×C₉	C₃²×C₉	1	5616	yes	—
35	C₃²×C₄×C₉	C₃⁴	1	117	yes	—
36	C₄×C₉²	C₃²×C₉	1	9477	yes	direct product factor
37	C₃⁵	C₃³×C₄	1	1	yes	characteristic
38	C₃³×C₉	C₃×C₄×C₉	1	1404	yes	—
39	C₃³×C₉	C₃³×C₄	1	156	yes	—
40	C₃×C₉²	C₃×C₄×C₉	1	12636	yes	direct product factor
41	C₃×C₉²	C₃³×C₄	1	1053	yes	—
42	C₂×C₃⁴	C₂×C₃²×C₉	1	117	yes	—
43	C₂×C₃⁴	C₂×C₃⁴	1	4	yes	—
44	C₂×C₃²×C₉	C₂×C₉²	1	9477	yes	—
45	C₂×C₃²×C₉	C₂×C₃²×C₉	1	5616	yes	—
46	C₂×C₃²×C₉	C₂×C₃⁴	1	117	yes	—
47	C₂×C₉²	C₂×C₃²×C₉	1	9477	yes	—
48	C₃³×C₄	C₃×C₉²	1	1053	yes	—
49	C₃³×C₄	C₃³×C₉	1	156	yes	—
50	C₃³×C₄	C₃⁵	1	1	yes	characteristic
51	C₃×C₄×C₉	C₃×C₉²	1	12636	yes	direct product factor
52	C₃×C₄×C₉	C₃³×C₉	1	1404	yes	—
53	C₃⁴	C₃²×C₄×C₉	1	117	yes	—
54	C₃⁴	C₃⁴×C₄	1	4	yes	—
55	C₃²×C₉	C₄×C₉²	1	9477	yes	direct product factor
56	C₃²×C₉	C₃²×C₄×C₉	1	5616	yes	—
57	C₃²×C₉	C₃⁴×C₄	1	117	yes	—
58	C₉²	C₃²×C₄×C₉	1	9477	yes	direct product factor
59	C₂×C₃³	C₂×C₃×C₉²	1	1053	yes	—
60	C₂×C₃³	C₂×C₃³×C₉	1	156	yes	—
61	C₂×C₃³	C₂×C₃⁵	1	1	yes	characteristic
62	C₂×C₃×C₉	C₂×C₃×C₉²	1	12636	yes	—
63	C₂×C₃×C₉	C₂×C₃³×C₉	1	1404	yes	—
64	C₃²×C₄	C₉³	1	729	yes	direct product factor
65	C₃²×C₄	C₃²×C₉²	1	468	yes	—
66	C₃²×C₄	C₃⁴×C₉	1	13	yes	—
67	C₄×C₉	C₃²×C₉²	1	1053	yes	direct product factor
68	C₃³	C₃×C₄×C₉²	1	1053	yes	—
69	C₃³	C₃³×C₄×C₉	1	156	yes	—
70	C₃³	C₃⁵×C₄	1	1	yes	characteristic
71	C₃×C₉	C₃×C₄×C₉²	1	12636	yes	direct product factor
72	C₃×C₉	C₃³×C₄×C₉	1	1404	yes	—
73	C₂×C₃²	C₂×C₉³	1	729	yes	—
74	C₂×C₃²	C₂×C₃²×C₉²	1	468	yes	—
75	C₂×C₃²	C₂×C₃⁴×C₉	1	13	yes	—
76	C₂×C₉	C₂×C₃²×C₉²	1	1053	yes	—
77	C₃×C₄	C₃×C₉³	1	108	yes	direct product factor
78	C₃×C₄	C₃³×C₉²	1	13	yes	—
79	C₃²	C₄×C₉³	1	729	yes	direct product factor
80	C₃²	C₃²×C₄×C₉²	1	468	yes	—
81	C₃²	C₃⁴×C₄×C₉	1	13	yes	—
82	C₉	C₃²×C₄×C₉²	1	1053	yes	direct product factor
83	C₂×C₃	C₂×C₃×C₉³	1	108	yes	—
84	C₂×C₃	C₂×C₃³×C₉²	1	13	yes	—
85	C₄	C₃²×C₉³	1	1	yes	characteristic, direct product factor, Sylow
86	C₃	C₃×C₄×C₉³	1	108	yes	direct product factor
87	C₃	C₃³×C₄×C₉²	1	13	yes	—
88	C₂	C₂×C₃²×C₉³	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃³	C₂×C₃⁵
socle, prime radical	C₂×C₃⁵	C₂×C₃³
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₉³	C₃	direct (N × H)
4. C₃²×C₉³	C₄	direct (N × H)
8. C₃²×C₄×C₉²	C₉	direct (N × H)
10. C₄×C₉³	C₃²	direct (N × H)
12. C₃×C₉³	C₃×C₄	direct (N × H)
20. C₃×C₄×C₉²	C₃×C₉	direct (N × H)
23. C₃²×C₉²	C₄×C₉	direct (N × H)
25. C₉³	C₃²×C₄	direct (N × H)
33. C₃²×C₄×C₉	C₉²	direct (N × H)
36. C₄×C₉²	C₃²×C₉	direct (N × H)
40. C₃×C₉²	C₃×C₄×C₉	direct (N × H)
51. C₃×C₄×C₉	C₃×C₉²	direct (N × H)
55. C₃²×C₉	C₄×C₉²	direct (N × H)
58. C₉²	C₃²×C₄×C₉	direct (N × H)
64. C₃²×C₄	C₉³	direct (N × H)
67. C₄×C₉	C₃²×C₉²	direct (N × H)
71. C₃×C₉	C₃×C₄×C₉²	direct (N × H)
77. C₃×C₄	C₃×C₉³	direct (N × H)
79. C₃²	C₄×C₉³	direct (N × H)
82. C₉	C₃²×C₄×C₉²	direct (N × H)
85. C₄	C₃²×C₉³	direct (N × H)
86. C₃	C₃×C₄×C₉³	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₉³	2
C₃³×C₄×C₉²	3
C₃×C₄×C₉³	3

Subgroups — C32×C4×C93 #72203402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₉³ #72203402