Subgroups — C₂⁷×C₄²×C₉ #72196253

Order 18432.

116917560 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁸×C₄×C₉	C₂	1	3	yes	maximal
2	C₂⁶×C₄²×C₉	C₂	1	508	yes	maximal, direct product factor
3	C₂⁷×C₃×C₄²	C₃	1	1	yes	maximal, characteristic
4	C₂⁹×C₉	C₂²	1	1	yes	characteristic
5	C₂⁷×C₄×C₉	C₄	1	768	yes	direct product factor
6	C₂⁷×C₄×C₉	C₂²	1	762	yes	—
7	C₂⁵×C₄²×C₉	C₂²	1	42672	yes	direct product factor
8	C₂⁸×C₃×C₄	C₂×C₃	1	3	yes	—
9	C₂⁶×C₃×C₄²	C₂×C₃	1	508	yes	—
10	C₂⁸×C₉	C₂×C₄	1	384	yes	—
11	C₂⁸×C₉	C₂³	1	127	yes	—
12	C₂⁶×C₄×C₉	C₂×C₄	1	97536	yes	direct product factor
13	C₂⁶×C₄×C₉	C₂³	1	32004	yes	—
14	C₂⁴×C₄²×C₉	C₂³	1	755904	yes	direct product factor
15	C₂⁷×C₄²	C₉	1	1	yes	characteristic, direct product factor, Sylow
16	C₂⁷×C₃×C₄	C₃×C₄	1	768	yes	—
17	C₂⁷×C₃×C₄	C₂²×C₃	1	762	yes	—
18	C₂⁹×C₃	C₂²×C₃	1	1	yes	characteristic
19	C₂⁵×C₃×C₄²	C₂²×C₃	1	42672	yes	—
20	C₂⁶×C₄²	C₂×C₉	1	508	yes	direct product factor
21	C₂⁸×C₄	C₂×C₉	1	3	yes	—
22	C₂⁷×C₉	C₄²	1	16384	yes	direct product factor
23	C₂⁷×C₉	C₂²×C₄	1	24384	yes	—
24	C₂⁷×C₉	C₂⁴	1	2667	yes	—
25	C₂⁵×C₄×C₉	C₂²×C₄	1	2048256	yes	direct product factor
26	C₂⁵×C₄×C₉	C₂⁴	1	283464	yes	—
27	C₂³×C₄²×C₉	C₂⁴	1	3023616	yes	direct product factor
28	C₂⁸×C₃	C₂×C₃×C₄	1	384	yes	—
29	C₂⁸×C₃	C₂³×C₃	1	127	yes	—
30	C₂⁶×C₃×C₄	C₂×C₃×C₄	1	97536	yes	—
31	C₂⁶×C₃×C₄	C₂³×C₃	1	32004	yes	—
32	C₂⁴×C₃×C₄²	C₂³×C₃	1	755904	yes	—
33	C₂⁶×C₉	C₂×C₄²	1	520192	yes	direct product factor
34	C₂⁶×C₉	C₂³×C₄	1	256032	yes	—
35	C₂⁶×C₉	C₂⁵	1	11811	yes	—
36	C₂⁴×C₄×C₉	C₂³×C₄	1	9070848	yes	direct product factor
37	C₂⁴×C₄×C₉	C₂⁵	1	566928	yes	—
38	C₂²×C₄²×C₉	C₂⁵	1	2731008	yes	direct product factor
39	C₂⁹	C₂²×C₉	1	1	yes	characteristic
40	C₂⁷×C₄	C₄×C₉	1	768	yes	direct product factor
41	C₂⁷×C₄	C₂²×C₉	1	762	yes	—
42	C₂⁵×C₄²	C₂²×C₉	1	42672	yes	direct product factor
43	C₂⁷×C₃	C₃×C₄²	1	16384	yes	—
44	C₂⁷×C₃	C₂²×C₃×C₄	1	24384	yes	—
45	C₂⁷×C₃	C₂⁴×C₃	1	2667	yes	—
46	C₂⁵×C₃×C₄	C₂²×C₃×C₄	1	2048256	yes	—
47	C₂⁵×C₃×C₄	C₂⁴×C₃	1	283464	yes	—
48	C₂³×C₃×C₄²	C₂⁴×C₃	1	3023616	yes	—
49	C₂⁵×C₉	C₂²×C₄²	1	2731008	yes	direct product factor
50	C₂⁵×C₉	C₂⁴×C₄	1	566928	yes	—
51	C₂⁵×C₉	C₂⁶	1	11811	yes	—
52	C₂³×C₄×C₉	C₂⁴×C₄	1	9070848	yes	direct product factor
53	C₂³×C₄×C₉	C₂⁶	1	256032	yes	—
54	C₂×C₄²×C₉	C₂⁶	1	520192	yes	direct product factor
55	C₂⁸	C₂×C₄×C₉	1	384	yes	—
56	C₂⁸	C₂³×C₉	1	127	yes	—
57	C₂⁶×C₄	C₂×C₄×C₉	1	97536	yes	direct product factor
58	C₂⁶×C₄	C₂³×C₉	1	32004	yes	—
59	C₂⁴×C₄²	C₂³×C₉	1	755904	yes	direct product factor
60	C₂⁶×C₃	C₂×C₃×C₄²	1	520192	yes	—
61	C₂⁶×C₃	C₂³×C₃×C₄	1	256032	yes	—
62	C₂⁶×C₃	C₂⁵×C₃	1	11811	yes	—
63	C₂⁴×C₃×C₄	C₂³×C₃×C₄	1	9070848	yes	—
64	C₂⁴×C₃×C₄	C₂⁵×C₃	1	566928	yes	—
65	C₂²×C₃×C₄²	C₂⁵×C₃	1	2731008	yes	—
66	C₂⁴×C₉	C₂³×C₄²	1	3023616	yes	direct product factor
67	C₂⁴×C₉	C₂⁵×C₄	1	283464	yes	—
68	C₂⁴×C₉	C₂⁷	1	2667	yes	—
69	C₂²×C₄×C₉	C₂⁵×C₄	1	2048256	yes	direct product factor
70	C₂²×C₄×C₉	C₂⁷	1	24384	yes	—
71	C₄²×C₉	C₂⁷	1	16384	yes	direct product factor
72	C₂⁷	C₄²×C₉	1	16384	yes	direct product factor
73	C₂⁷	C₂²×C₄×C₉	1	24384	yes	—
74	C₂⁷	C₂⁴×C₉	1	2667	yes	—
75	C₂⁵×C₄	C₂²×C₄×C₉	1	2048256	yes	direct product factor
76	C₂⁵×C₄	C₂⁴×C₉	1	283464	yes	—
77	C₂³×C₄²	C₂⁴×C₉	1	3023616	yes	direct product factor
78	C₂⁵×C₃	C₂²×C₃×C₄²	1	2731008	yes	—
79	C₂⁵×C₃	C₂⁴×C₃×C₄	1	566928	yes	—
80	C₂⁵×C₃	C₂⁶×C₃	1	11811	yes	—
81	C₂³×C₃×C₄	C₂⁴×C₃×C₄	1	9070848	yes	—
82	C₂³×C₃×C₄	C₂⁶×C₃	1	256032	yes	—
83	C₂×C₃×C₄²	C₂⁶×C₃	1	520192	yes	—
84	C₂³×C₉	C₂⁴×C₄²	1	755904	yes	direct product factor
85	C₂³×C₉	C₂⁶×C₄	1	32004	yes	—
86	C₂³×C₉	C₂⁸	1	127	yes	—
87	C₂×C₄×C₉	C₂⁶×C₄	1	97536	yes	direct product factor
88	C₂×C₄×C₉	C₂⁸	1	384	yes	—
89	C₂⁶	C₂×C₄²×C₉	1	520192	yes	direct product factor
90	C₂⁶	C₂³×C₄×C₉	1	256032	yes	—
91	C₂⁶	C₂⁵×C₉	1	11811	yes	—
92	C₂⁴×C₄	C₂³×C₄×C₉	1	9070848	yes	direct product factor
93	C₂⁴×C₄	C₂⁵×C₉	1	566928	yes	—
94	C₂²×C₄²	C₂⁵×C₉	1	2731008	yes	direct product factor
95	C₂⁴×C₃	C₂³×C₃×C₄²	1	3023616	yes	—
96	C₂⁴×C₃	C₂⁵×C₃×C₄	1	283464	yes	—
97	C₂⁴×C₃	C₂⁷×C₃	1	2667	yes	—
98	C₂²×C₃×C₄	C₂⁵×C₃×C₄	1	2048256	yes	—
99	C₂²×C₃×C₄	C₂⁷×C₃	1	24384	yes	—
100	C₃×C₄²	C₂⁷×C₃	1	16384	yes	—
101	C₂²×C₉	C₂⁵×C₄²	1	42672	yes	direct product factor
102	C₂²×C₉	C₂⁷×C₄	1	762	yes	—
103	C₂²×C₉	C₂⁹	1	1	yes	characteristic
104	C₄×C₉	C₂⁷×C₄	1	768	yes	direct product factor
105	C₂⁵	C₂²×C₄²×C₉	1	2731008	yes	direct product factor
106	C₂⁵	C₂⁴×C₄×C₉	1	566928	yes	—
107	C₂⁵	C₂⁶×C₉	1	11811	yes	—
108	C₂³×C₄	C₂⁴×C₄×C₉	1	9070848	yes	direct product factor
109	C₂³×C₄	C₂⁶×C₉	1	256032	yes	—
110	C₂×C₄²	C₂⁶×C₉	1	520192	yes	direct product factor
111	C₂³×C₃	C₂⁴×C₃×C₄²	1	755904	yes	—
112	C₂³×C₃	C₂⁶×C₃×C₄	1	32004	yes	—
113	C₂³×C₃	C₂⁸×C₃	1	127	yes	—
114	C₂×C₃×C₄	C₂⁶×C₃×C₄	1	97536	yes	—
115	C₂×C₃×C₄	C₂⁸×C₃	1	384	yes	—
116	C₂×C₉	C₂⁸×C₄	1	3	yes	—
117	C₂×C₉	C₂⁶×C₄²	1	508	yes	direct product factor
118	C₂⁴	C₂³×C₄²×C₉	1	3023616	yes	direct product factor
119	C₂⁴	C₂⁵×C₄×C₉	1	283464	yes	—
120	C₂⁴	C₂⁷×C₉	1	2667	yes	—
121	C₂²×C₄	C₂⁵×C₄×C₉	1	2048256	yes	direct product factor
122	C₂²×C₄	C₂⁷×C₉	1	24384	yes	—
123	C₄²	C₂⁷×C₉	1	16384	yes	direct product factor
124	C₂²×C₃	C₂⁵×C₃×C₄²	1	42672	yes	—
125	C₂²×C₃	C₂⁹×C₃	1	1	yes	characteristic
126	C₂²×C₃	C₂⁷×C₃×C₄	1	762	yes	—
127	C₃×C₄	C₂⁷×C₃×C₄	1	768	yes	—
128	C₉	C₂⁷×C₄²	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₂⁴×C₄²×C₉	1	755904	yes	direct product factor
130	C₂³	C₂⁶×C₄×C₉	1	32004	yes	—
131	C₂³	C₂⁸×C₉	1	127	yes	—
132	C₂×C₄	C₂⁶×C₄×C₉	1	97536	yes	direct product factor
133	C₂×C₄	C₂⁸×C₉	1	384	yes	—
134	C₂×C₃	C₂⁶×C₃×C₄²	1	508	yes	—
135	C₂×C₃	C₂⁸×C₃×C₄	1	3	yes	—
136	C₂²	C₂⁵×C₄²×C₉	1	42672	yes	direct product factor
137	C₂²	C₂⁷×C₄×C₉	1	762	yes	—
138	C₂²	C₂⁹×C₉	1	1	yes	characteristic
139	C₄	C₂⁷×C₄×C₉	1	768	yes	direct product factor
140	C₃	C₂⁷×C₃×C₄²	1	1	yes	characteristic
141	C₂	C₂⁶×C₄²×C₉	1	508	yes	direct product factor
142	C₂	C₂⁸×C₄×C₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₃	C₂⁹×C₃
socle, prime radical	C₂⁹×C₃	C₂²×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁶×C₄²×C₉	C₂	direct (N × H)
5. C₂⁷×C₄×C₉	C₄	direct (N × H)
7. C₂⁵×C₄²×C₉	C₂²	direct (N × H)
12. C₂⁶×C₄×C₉	C₂×C₄	direct (N × H)
14. C₂⁴×C₄²×C₉	C₂³	direct (N × H)
15. C₂⁷×C₄²	C₉	direct (N × H)
20. C₂⁶×C₄²	C₂×C₉	direct (N × H)
22. C₂⁷×C₉	C₄²	direct (N × H)
25. C₂⁵×C₄×C₉	C₂²×C₄	direct (N × H)
27. C₂³×C₄²×C₉	C₂⁴	direct (N × H)
33. C₂⁶×C₉	C₂×C₄²	direct (N × H)
36. C₂⁴×C₄×C₉	C₂³×C₄	direct (N × H)
38. C₂²×C₄²×C₉	C₂⁵	direct (N × H)
40. C₂⁷×C₄	C₄×C₉	direct (N × H)
42. C₂⁵×C₄²	C₂²×C₉	direct (N × H)
49. C₂⁵×C₉	C₂²×C₄²	direct (N × H)
52. C₂³×C₄×C₉	C₂⁴×C₄	direct (N × H)
54. C₂×C₄²×C₉	C₂⁶	direct (N × H)
57. C₂⁶×C₄	C₂×C₄×C₉	direct (N × H)
59. C₂⁴×C₄²	C₂³×C₉	direct (N × H)
66. C₂⁴×C₉	C₂³×C₄²	direct (N × H)
69. C₂²×C₄×C₉	C₂⁵×C₄	direct (N × H)
71. C₄²×C₉	C₂⁷	direct (N × H)
72. C₂⁷	C₄²×C₉	direct (N × H)
75. C₂⁵×C₄	C₂²×C₄×C₉	direct (N × H)
77. C₂³×C₄²	C₂⁴×C₉	direct (N × H)
84. C₂³×C₉	C₂⁴×C₄²	direct (N × H)
87. C₂×C₄×C₉	C₂⁶×C₄	direct (N × H)
89. C₂⁶	C₂×C₄²×C₉	direct (N × H)
92. C₂⁴×C₄	C₂³×C₄×C₉	direct (N × H)
94. C₂²×C₄²	C₂⁵×C₉	direct (N × H)
101. C₂²×C₉	C₂⁵×C₄²	direct (N × H)
104. C₄×C₉	C₂⁷×C₄	direct (N × H)
105. C₂⁵	C₂²×C₄²×C₉	direct (N × H)
108. C₂³×C₄	C₂⁴×C₄×C₉	direct (N × H)
110. C₂×C₄²	C₂⁶×C₉	direct (N × H)
117. C₂×C₉	C₂⁶×C₄²	direct (N × H)
118. C₂⁴	C₂³×C₄²×C₉	direct (N × H)
121. C₂²×C₄	C₂⁵×C₄×C₉	direct (N × H)
123. C₄²	C₂⁷×C₉	direct (N × H)
128. C₉	C₂⁷×C₄²	direct (N × H)
129. C₂³	C₂⁴×C₄²×C₉	direct (N × H)
132. C₂×C₄	C₂⁶×C₄×C₉	direct (N × H)
136. C₂²	C₂⁵×C₄²×C₉	direct (N × H)
139. C₄	C₂⁷×C₄×C₉	direct (N × H)
141. C₂	C₂⁶×C₄²×C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁸×C₄×C₉	2
C₂⁶×C₄²×C₉	2
C₂⁷×C₃×C₄²	3

Subgroups — C27×C42×C9 #72196253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁷×C₄²×C₉ #72196253