Subgroups — C₂³×C₄⁴×C₉ #72196249

Order 18432.

4185144 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄³×C₉	C₂	1	15	yes	maximal
2	C₂²×C₄⁴×C₉	C₂	1	112	yes	maximal, direct product factor
3	C₂³×C₃×C₄⁴	C₃	1	1	yes	maximal, characteristic
4	C₂⁵×C₄²×C₉	C₂²	1	35	yes	—
5	C₂³×C₄³×C₉	C₄	1	960	yes	direct product factor
6	C₂³×C₄³×C₉	C₂²	1	840	yes	—
7	C₂×C₄⁴×C₉	C₂²	1	1792	yes	direct product factor
8	C₂⁴×C₃×C₄³	C₂×C₃	1	15	yes	—
9	C₂²×C₃×C₄⁴	C₂×C₃	1	112	yes	—
10	C₂⁶×C₄×C₉	C₂³	1	15	yes	—
11	C₂⁴×C₄²×C₉	C₂×C₄	1	3360	yes	—
12	C₂⁴×C₄²×C₉	C₂³	1	980	yes	—
13	C₂²×C₄³×C₉	C₂×C₄	1	26880	yes	direct product factor
14	C₂²×C₄³×C₉	C₂³	1	6720	yes	—
15	C₄⁴×C₉	C₂³	1	4096	yes	direct product factor
16	C₂³×C₄⁴	C₉	1	1	yes	characteristic, direct product factor, Sylow
17	C₂⁵×C₃×C₄²	C₂²×C₃	1	35	yes	—
18	C₂³×C₃×C₄³	C₃×C₄	1	960	yes	—
19	C₂³×C₃×C₄³	C₂²×C₃	1	840	yes	—
20	C₂×C₃×C₄⁴	C₂²×C₃	1	1792	yes	—
21	C₂⁴×C₄³	C₂×C₉	1	15	yes	—
22	C₂²×C₄⁴	C₂×C₉	1	112	yes	direct product factor
23	C₂⁷×C₉	C₂⁴	1	1	yes	characteristic
24	C₂⁵×C₄×C₉	C₂²×C₄	1	1680	yes	—
25	C₂⁵×C₄×C₉	C₂⁴	1	210	yes	—
26	C₂³×C₄²×C₉	C₄²	1	35840	yes	direct product factor
27	C₂³×C₄²×C₉	C₂²×C₄	1	47040	yes	—
28	C₂³×C₄²×C₉	C₂⁴	1	3920	yes	—
29	C₂×C₄³×C₉	C₂²×C₄	1	107520	yes	direct product factor
30	C₂×C₄³×C₉	C₂⁴	1	7680	yes	—
31	C₂⁶×C₃×C₄	C₂³×C₃	1	15	yes	—
32	C₂⁴×C₃×C₄²	C₂×C₃×C₄	1	3360	yes	—
33	C₂⁴×C₃×C₄²	C₂³×C₃	1	980	yes	—
34	C₂²×C₃×C₄³	C₂×C₃×C₄	1	26880	yes	—
35	C₂²×C₃×C₄³	C₂³×C₃	1	6720	yes	—
36	C₃×C₄⁴	C₂³×C₃	1	4096	yes	—
37	C₂⁶×C₉	C₂³×C₄	1	120	yes	—
38	C₂⁶×C₉	C₂⁵	1	7	yes	—
39	C₂⁴×C₄×C₉	C₂×C₄²	1	26880	yes	—
40	C₂⁴×C₄×C₉	C₂³×C₄	1	11760	yes	—
41	C₂⁴×C₄×C₉	C₂⁵	1	420	yes	—
42	C₂²×C₄²×C₉	C₂×C₄²	1	250880	yes	direct product factor
43	C₂²×C₄²×C₉	C₂³×C₄	1	94080	yes	—
44	C₂²×C₄²×C₉	C₂⁵	1	2240	yes	—
45	C₄³×C₉	C₂³×C₄	1	61440	yes	direct product factor
46	C₂⁵×C₄²	C₂²×C₉	1	35	yes	—
47	C₂³×C₄³	C₄×C₉	1	960	yes	direct product factor
48	C₂³×C₄³	C₂²×C₉	1	840	yes	—
49	C₂×C₄⁴	C₂²×C₉	1	1792	yes	direct product factor
50	C₂⁷×C₃	C₂⁴×C₃	1	1	yes	characteristic
51	C₂⁵×C₃×C₄	C₂²×C₃×C₄	1	1680	yes	—
52	C₂⁵×C₃×C₄	C₂⁴×C₃	1	210	yes	—
53	C₂³×C₃×C₄²	C₃×C₄²	1	35840	yes	—
54	C₂³×C₃×C₄²	C₂²×C₃×C₄	1	47040	yes	—
55	C₂³×C₃×C₄²	C₂⁴×C₃	1	3920	yes	—
56	C₂×C₃×C₄³	C₂²×C₃×C₄	1	107520	yes	—
57	C₂×C₃×C₄³	C₂⁴×C₃	1	7680	yes	—
58	C₂⁵×C₉	C₂²×C₄²	1	2240	yes	—
59	C₂⁵×C₉	C₂⁴×C₄	1	420	yes	—
60	C₂⁵×C₉	C₂⁶	1	7	yes	—
61	C₂³×C₄×C₉	C₄³	1	61440	yes	direct product factor
62	C₂³×C₄×C₉	C₂²×C₄²	1	94080	yes	—
63	C₂³×C₄×C₉	C₂⁴×C₄	1	11760	yes	—
64	C₂³×C₄×C₉	C₂⁶	1	120	yes	—
65	C₂×C₄²×C₉	C₂²×C₄²	1	250880	yes	direct product factor
66	C₂×C₄²×C₉	C₂⁴×C₄	1	26880	yes	—
67	C₂⁶×C₄	C₂³×C₉	1	15	yes	—
68	C₂⁴×C₄²	C₂×C₄×C₉	1	3360	yes	—
69	C₂⁴×C₄²	C₂³×C₉	1	980	yes	—
70	C₂²×C₄³	C₂×C₄×C₉	1	26880	yes	direct product factor
71	C₂²×C₄³	C₂³×C₉	1	6720	yes	—
72	C₄⁴	C₂³×C₉	1	4096	yes	direct product factor
73	C₂⁶×C₃	C₂³×C₃×C₄	1	120	yes	—
74	C₂⁶×C₃	C₂⁵×C₃	1	7	yes	—
75	C₂⁴×C₃×C₄	C₂×C₃×C₄²	1	26880	yes	—
76	C₂⁴×C₃×C₄	C₂³×C₃×C₄	1	11760	yes	—
77	C₂⁴×C₃×C₄	C₂⁵×C₃	1	420	yes	—
78	C₂²×C₃×C₄²	C₂×C₃×C₄²	1	250880	yes	—
79	C₂²×C₃×C₄²	C₂³×C₃×C₄	1	94080	yes	—
80	C₂²×C₃×C₄²	C₂⁵×C₃	1	2240	yes	—
81	C₃×C₄³	C₂³×C₃×C₄	1	61440	yes	—
82	C₂⁴×C₉	C₂×C₄³	1	7680	yes	—
83	C₂⁴×C₉	C₂³×C₄²	1	3920	yes	—
84	C₂⁴×C₉	C₂⁵×C₄	1	210	yes	—
85	C₂⁴×C₉	C₂⁷	1	1	yes	characteristic
86	C₂²×C₄×C₉	C₂×C₄³	1	107520	yes	direct product factor
87	C₂²×C₄×C₉	C₂³×C₄²	1	47040	yes	—
88	C₂²×C₄×C₉	C₂⁵×C₄	1	1680	yes	—
89	C₄²×C₉	C₂³×C₄²	1	35840	yes	direct product factor
90	C₂⁷	C₂⁴×C₉	1	1	yes	characteristic
91	C₂⁵×C₄	C₂²×C₄×C₉	1	1680	yes	—
92	C₂⁵×C₄	C₂⁴×C₉	1	210	yes	—
93	C₂³×C₄²	C₄²×C₉	1	35840	yes	direct product factor
94	C₂³×C₄²	C₂²×C₄×C₉	1	47040	yes	—
95	C₂³×C₄²	C₂⁴×C₉	1	3920	yes	—
96	C₂×C₄³	C₂²×C₄×C₉	1	107520	yes	direct product factor
97	C₂×C₄³	C₂⁴×C₉	1	7680	yes	—
98	C₂⁵×C₃	C₂²×C₃×C₄²	1	2240	yes	—
99	C₂⁵×C₃	C₂⁴×C₃×C₄	1	420	yes	—
100	C₂⁵×C₃	C₂⁶×C₃	1	7	yes	—
101	C₂³×C₃×C₄	C₃×C₄³	1	61440	yes	—
102	C₂³×C₃×C₄	C₂²×C₃×C₄²	1	94080	yes	—
103	C₂³×C₃×C₄	C₂⁴×C₃×C₄	1	11760	yes	—
104	C₂³×C₃×C₄	C₂⁶×C₃	1	120	yes	—
105	C₂×C₃×C₄²	C₂²×C₃×C₄²	1	250880	yes	—
106	C₂×C₃×C₄²	C₂⁴×C₃×C₄	1	26880	yes	—
107	C₂³×C₉	C₄⁴	1	4096	yes	direct product factor
108	C₂³×C₉	C₂²×C₄³	1	6720	yes	—
109	C₂³×C₉	C₂⁴×C₄²	1	980	yes	—
110	C₂³×C₉	C₂⁶×C₄	1	15	yes	—
111	C₂×C₄×C₉	C₂²×C₄³	1	26880	yes	direct product factor
112	C₂×C₄×C₉	C₂⁴×C₄²	1	3360	yes	—
113	C₂⁶	C₂³×C₄×C₉	1	120	yes	—
114	C₂⁶	C₂⁵×C₉	1	7	yes	—
115	C₂⁴×C₄	C₂×C₄²×C₉	1	26880	yes	—
116	C₂⁴×C₄	C₂³×C₄×C₉	1	11760	yes	—
117	C₂⁴×C₄	C₂⁵×C₉	1	420	yes	—
118	C₂²×C₄²	C₂×C₄²×C₉	1	250880	yes	direct product factor
119	C₂²×C₄²	C₂³×C₄×C₉	1	94080	yes	—
120	C₂²×C₄²	C₂⁵×C₉	1	2240	yes	—
121	C₄³	C₂³×C₄×C₉	1	61440	yes	direct product factor
122	C₂⁴×C₃	C₂×C₃×C₄³	1	7680	yes	—
123	C₂⁴×C₃	C₂³×C₃×C₄²	1	3920	yes	—
124	C₂⁴×C₃	C₂⁵×C₃×C₄	1	210	yes	—
125	C₂⁴×C₃	C₂⁷×C₃	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₃×C₄³	1	107520	yes	—
127	C₂²×C₃×C₄	C₂³×C₃×C₄²	1	47040	yes	—
128	C₂²×C₃×C₄	C₂⁵×C₃×C₄	1	1680	yes	—
129	C₃×C₄²	C₂³×C₃×C₄²	1	35840	yes	—
130	C₂²×C₉	C₂×C₄⁴	1	1792	yes	direct product factor
131	C₂²×C₉	C₂³×C₄³	1	840	yes	—
132	C₂²×C₉	C₂⁵×C₄²	1	35	yes	—
133	C₄×C₉	C₂³×C₄³	1	960	yes	direct product factor
134	C₂⁵	C₂²×C₄²×C₉	1	2240	yes	—
135	C₂⁵	C₂⁴×C₄×C₉	1	420	yes	—
136	C₂⁵	C₂⁶×C₉	1	7	yes	—
137	C₂³×C₄	C₄³×C₉	1	61440	yes	direct product factor
138	C₂³×C₄	C₂²×C₄²×C₉	1	94080	yes	—
139	C₂³×C₄	C₂⁴×C₄×C₉	1	11760	yes	—
140	C₂³×C₄	C₂⁶×C₉	1	120	yes	—
141	C₂×C₄²	C₂²×C₄²×C₉	1	250880	yes	direct product factor
142	C₂×C₄²	C₂⁴×C₄×C₉	1	26880	yes	—
143	C₂³×C₃	C₃×C₄⁴	1	4096	yes	—
144	C₂³×C₃	C₂²×C₃×C₄³	1	6720	yes	—
145	C₂³×C₃	C₂⁴×C₃×C₄²	1	980	yes	—
146	C₂³×C₃	C₂⁶×C₃×C₄	1	15	yes	—
147	C₂×C₃×C₄	C₂²×C₃×C₄³	1	26880	yes	—
148	C₂×C₃×C₄	C₂⁴×C₃×C₄²	1	3360	yes	—
149	C₂×C₉	C₂²×C₄⁴	1	112	yes	direct product factor
150	C₂×C₉	C₂⁴×C₄³	1	15	yes	—
151	C₂⁴	C₂×C₄³×C₉	1	7680	yes	—
152	C₂⁴	C₂³×C₄²×C₉	1	3920	yes	—
153	C₂⁴	C₂⁵×C₄×C₉	1	210	yes	—
154	C₂⁴	C₂⁷×C₉	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄³×C₉	1	107520	yes	direct product factor
156	C₂²×C₄	C₂³×C₄²×C₉	1	47040	yes	—
157	C₂²×C₄	C₂⁵×C₄×C₉	1	1680	yes	—
158	C₄²	C₂³×C₄²×C₉	1	35840	yes	direct product factor
159	C₂²×C₃	C₂×C₃×C₄⁴	1	1792	yes	—
160	C₂²×C₃	C₂³×C₃×C₄³	1	840	yes	—
161	C₂²×C₃	C₂⁵×C₃×C₄²	1	35	yes	—
162	C₃×C₄	C₂³×C₃×C₄³	1	960	yes	—
163	C₉	C₂³×C₄⁴	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³	C₄⁴×C₉	1	4096	yes	direct product factor
165	C₂³	C₂²×C₄³×C₉	1	6720	yes	—
166	C₂³	C₂⁴×C₄²×C₉	1	980	yes	—
167	C₂³	C₂⁶×C₄×C₉	1	15	yes	—
168	C₂×C₄	C₂²×C₄³×C₉	1	26880	yes	direct product factor
169	C₂×C₄	C₂⁴×C₄²×C₉	1	3360	yes	—
170	C₂×C₃	C₂²×C₃×C₄⁴	1	112	yes	—
171	C₂×C₃	C₂⁴×C₃×C₄³	1	15	yes	—
172	C₂²	C₂×C₄⁴×C₉	1	1792	yes	direct product factor
173	C₂²	C₂³×C₄³×C₉	1	840	yes	—
174	C₂²	C₂⁵×C₄²×C₉	1	35	yes	—
175	C₄	C₂³×C₄³×C₉	1	960	yes	direct product factor
176	C₃	C₂³×C₃×C₄⁴	1	1	yes	characteristic
177	C₂	C₂²×C₄⁴×C₉	1	112	yes	direct product factor
178	C₂	C₂⁴×C₄³×C₉	1	15	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂⁴×C₃	C₂⁷×C₃
socle, prime radical	C₂⁷×C₃	C₂⁴×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⁴×C₉	C₂	direct (N × H)
5. C₂³×C₄³×C₉	C₄	direct (N × H)
7. C₂×C₄⁴×C₉	C₂²	direct (N × H)
13. C₂²×C₄³×C₉	C₂×C₄	direct (N × H)
15. C₄⁴×C₉	C₂³	direct (N × H)
16. C₂³×C₄⁴	C₉	direct (N × H)
22. C₂²×C₄⁴	C₂×C₉	direct (N × H)
26. C₂³×C₄²×C₉	C₄²	direct (N × H)
29. C₂×C₄³×C₉	C₂²×C₄	direct (N × H)
42. C₂²×C₄²×C₉	C₂×C₄²	direct (N × H)
45. C₄³×C₉	C₂³×C₄	direct (N × H)
47. C₂³×C₄³	C₄×C₉	direct (N × H)
49. C₂×C₄⁴	C₂²×C₉	direct (N × H)
61. C₂³×C₄×C₉	C₄³	direct (N × H)
65. C₂×C₄²×C₉	C₂²×C₄²	direct (N × H)
70. C₂²×C₄³	C₂×C₄×C₉	direct (N × H)
72. C₄⁴	C₂³×C₉	direct (N × H)
86. C₂²×C₄×C₉	C₂×C₄³	direct (N × H)
89. C₄²×C₉	C₂³×C₄²	direct (N × H)
93. C₂³×C₄²	C₄²×C₉	direct (N × H)
96. C₂×C₄³	C₂²×C₄×C₉	direct (N × H)
107. C₂³×C₉	C₄⁴	direct (N × H)
111. C₂×C₄×C₉	C₂²×C₄³	direct (N × H)
118. C₂²×C₄²	C₂×C₄²×C₉	direct (N × H)
121. C₄³	C₂³×C₄×C₉	direct (N × H)
130. C₂²×C₉	C₂×C₄⁴	direct (N × H)
133. C₄×C₉	C₂³×C₄³	direct (N × H)
137. C₂³×C₄	C₄³×C₉	direct (N × H)
141. C₂×C₄²	C₂²×C₄²×C₉	direct (N × H)
149. C₂×C₉	C₂²×C₄⁴	direct (N × H)
155. C₂²×C₄	C₂×C₄³×C₉	direct (N × H)
158. C₄²	C₂³×C₄²×C₉	direct (N × H)
163. C₉	C₂³×C₄⁴	direct (N × H)
164. C₂³	C₄⁴×C₉	direct (N × H)
168. C₂×C₄	C₂²×C₄³×C₉	direct (N × H)
172. C₂²	C₂×C₄⁴×C₉	direct (N × H)
175. C₄	C₂³×C₄³×C₉	direct (N × H)
177. C₂	C₂²×C₄⁴×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
C₂⁴×C₄³×C₉	2
C₂²×C₄⁴×C₉	2
C₂³×C₃×C₄⁴	3

Subgroups — C23×C44×C9 #72196249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⁴×C₉ #72196249