Subgroups — C₉×#5101 #13143084

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#926	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
4	C₉×#815	C₂	1	2	yes	maximal, complemented
5	C₈×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₃×#5101	C₃	1	1	yes	maximal, characteristic
7	C₉×#405	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₉×#217	C₄	1	2	yes	—
13	C₉×C₄⋊Dic₃	C₄	1	2	yes	—
14	C₄×C₉×Dic₃	C₂²	1	2	yes	—
15	C₉×#208	C₄	1	4	yes	complemented
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×#926	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#208	C₂×C₃	1	1	yes	characteristic
20	C₃×#815	C₂×C₃	1	2	yes	—
21	C₃×C₈×Dic₃	C₂×C₃	1	2	yes	—
22	C₂×C₉×M₁₆	—	3	1	no	—
23	C₂²×C₈×C₉	—	2	1	no	—
24	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
25	C₉×#117	—	3	1	no	—
26	C₉×#97	—	3	2	no	—
27	C₄×C₈×C₉	—	3	2	no	—
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
34	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
35	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
36	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₃×#405	—	3	1	no	—
38	#5101	C₉	1	1	yes	characteristic, direct product factor
39	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
41	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₃×#217	C₃×C₄	1	2	yes	—
44	C₃×C₄⋊Dic₃	C₃×C₄	1	2	yes	—
45	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
46	C₃×#208	C₃×C₄	1	4	yes	—
47	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
50	C₉×M₁₆	—	3	4	no	—
51	C₂×C₈×C₉	—	2	4	no	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₂×C₈×C₉	—	2	1	no	—
54	C₂×C₈×C₉	—	3	2	no	—
55	C₂×C₈×C₉	Dic₃	1	4	yes	complemented
56	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
57	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
58	C₉×C₄⋊C₄	—	3	2	no	—
59	C₉×C₂²⋊C₄	—	3	2	no	—
60	C₄²×C₉	—	3	2	no	—
61	C₂²×C₃×C₉	—	2	2	no	—
62	C₂²×C₃×C₉	C₄²	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₄²	1	2	yes	—
65	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₉×Dic₃	—	2	4	no	—
68	C₂×C₃×M₁₆	—	3	1	no	—
69	C₂²×C₃×C₈	—	2	1	no	—
70	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
72	C₃×#117	—	3	1	no	—
73	#926	C₂×C₉	1	1	yes	characteristic
74	C₂×#208	C₂×C₉	1	1	yes	characteristic
75	C₃×#97	—	3	2	no	—
76	C₃×C₄×C₈	—	3	2	no	—
77	#815	C₂×C₉	1	2	yes	complemented
78	C₈×Dic₃	C₂×C₉	1	2	yes	complemented
79	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
87	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
88	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
89	C₃×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	3	4	no	—
95	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
96	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
97	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
98	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	2	no	—
101	C₈×C₉	—	3	4	no	—
102	C₈×C₉	C₂×Dic₃	1	2	yes	—
103	C₈×C₉	C₂×Dic₃	1	2	yes	—
104	#405	—	3	1	no	Sylow
105	C₂×C₃×C₉	—	2	2	no	—
106	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₃×C₉	#131	1	2	yes	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
111	C₃×M₁₆	—	3	4	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	3	2	no	—
116	C₂×C₃×C₈	C₄×C₉	1	4	yes	complemented
117	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
119	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	3	2	no	—
123	C₃×C₂²⋊C₄	—	3	2	no	—
124	C₃×C₄²	—	3	2	no	—
125	#217	C₄×C₉	1	2	yes	—
126	C₄⋊Dic₃	C₄×C₉	1	2	yes	—
127	C₄×Dic₃	C₂²×C₉	1	2	yes	—
128	#208	C₄×C₉	1	4	yes	complemented
129	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	C₃×C₄²	1	2	yes	—
132	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₃²×C₄	C₃×C₄²	1	2	yes	—
134	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₃×Dic₃	—	2	4	no	—
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₄×Dic₃	1	2	yes	—
142	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	—	6	4	no	—
147	C₄×C₉	C₄×Dic₃	1	2	yes	—
148	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
149	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
150	C₂×M₁₆	—	3	1	no	—
151	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
152	#117	—	3	1	no	—
153	#97	—	3	2	no	—
154	C₄×C₈	—	3	2	no	—
155	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
158	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	3	4	no	—
164	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
166	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
167	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
171	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
172	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
173	C₃×C₈	—	2	2	no	—
174	C₃×C₈	—	2	2	no	—
175	C₃×C₈	—	3	4	no	—
176	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
177	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
178	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
179	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
180	C₃⋊C₈	C₂×C₄×C₉	1	4	yes	—
181	C₂×C₃²	—	2	2	no	—
182	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
183	C₂×C₃²	C₃×#131	1	2	yes	—
184	C₂×C₉	—	2	4	no	—
185	C₂×C₉	—	2	2	no	—
186	C₂×C₉	—	2	2	no	—
187	C₂×C₉	—	2	1	no	—
188	C₂×C₉	#901	1	2	yes	—
189	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
191	M₁₆	—	3	4	no	—
192	C₂×C₈	—	3	2	no	—
193	C₂×C₈	C₉×Dic₃	1	4	yes	complemented
194	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
195	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
196	C₄⋊C₄	—	3	2	no	—
197	C₂²⋊C₄	—	3	2	no	—
198	C₄²	—	3	2	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	C₄²×C₉	1	2	yes	—
205	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
206	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
207	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	6	4	no	—
212	C₃×C₄	C₄²×C₉	1	2	yes	—
213	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
214	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
215	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
216	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
217	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
218	Dic₃	—	2	4	no	—
219	C₃²	C₃×#405	1	1	yes	characteristic
220	C₉	—	2	1	no	—
221	C₉	#5101	1	1	yes	characteristic, direct product factor
222	C₂³	C₄×D₃×C₉	1	1	yes	characteristic
223	C₂×C₄	—	3	4	no	—
224	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
225	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
226	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
227	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
228	C₈	—	3	4	no	—
229	C₈	C₂×C₉×Dic₃	1	2	yes	—
230	C₈	C₂×C₉×Dic₃	1	2	yes	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	—	2	2	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
237	C₂×C₃	C₉×#131	1	2	yes	—
238	C₂×C₃	C₃×#901	1	2	yes	—
239	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	C₄×C₉×Dic₃	1	2	yes	—
242	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
243	C₄	—	6	4	no	—
244	C₄	C₄×C₉×Dic₃	1	2	yes	—
245	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
246	C₄	C₂²×C₉×Dic₃	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	C₉×#405	1	1	yes	characteristic, complemented
249	C₃	C₃×#5101	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	C₉×#901	1	2	yes	—
252	C₂	C₂×C₄×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₉	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₉×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
4. C₉×#815	C₂	semidirect
5. C₈×C₉×Dic₃	C₂	semidirect
9. C₂×C₃×C₈×C₉	C₄	semidirect
15. C₉×#208	C₄	semidirect
38. #5101	C₉	direct (N × H)
55. C₂×C₈×C₉	Dic₃	semidirect
77. #815	C₂×C₉	semidirect
78. C₈×Dic₃	C₂×C₉	semidirect
116. C₂×C₃×C₈	C₄×C₉	semidirect
128. #208	C₄×C₉	semidirect
155. C₃×C₉	#405	semidirect
193. C₂×C₈	C₉×Dic₃	semidirect
221. C₉	#5101	direct (N × H)
248. C₃	C₉×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₉×#926	2
C₂×C₉×#208	2
C₉×#815	2
C₈×C₉×Dic₃	2
C₃×#5101	3
C₉×#405	3

Subgroups — C9×#5101 #13143084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5101 #13143084