Subgroups — C₉×#5110 #13143102

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×(C₂×D₃)⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₃×#5110	C₃	1	1	yes	maximal, characteristic
6	C₉×#408	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×#67	C₄	1	2	yes	—
9	C₉×#235	—	2	2	no	—
10	C₉×#235	—	2	2	no	—
11	C₂×C₉×D₁₂	C₄	1	1	yes	characteristic
12	C₂×C₄×D₃×C₉	C₂²	1	2	yes	—
13	C₂×C₉×Dic₆	C₄	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	—	2	2	no	—
15	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
16	C₉×#208	—	2	2	no	—
17	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃×#235	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#208	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×#41	—	3	1	no	—
22	C₃×(C₂×D₃)⋊C₈	C₂×C₃	1	4	yes	—
23	C₂×C₉×M₁₆	—	3	1	no	—
24	C₂²×C₈×C₉	—	2	1	no	—
25	C₂²×C₈×C₉	D₃	1	1	yes	characteristic, complemented
26	C₉×C₂²⋊C₈	—	3	4	no	—
27	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₃×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×#67	—	2	4	no	—
34	C₂×C₉×Dic₃	C₂×C₄	1	2	yes	—
35	C₉×D₁₂	—	2	2	no	—
36	C₄×D₃×C₉	—	2	2	no	—
37	C₄×D₃×C₉	—	2	2	no	—
38	C₉×Dic₆	—	2	2	no	—
39	C₃×C₈×C₉	—	2	2	no	—
40	C₉×C₃⋊C₈	—	2	2	no	—
41	C₃×#408	—	3	1	no	—
42	#5110	C₉	1	1	yes	characteristic, direct product factor
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×#67	C₃×C₄	1	2	yes	—
45	C₃×#235	—	2	2	no	—
46	C₃×#235	—	2	2	no	—
47	C₂×C₃×D₁₂	C₃×C₄	1	1	yes	characteristic
48	C₂×C₃×C₄×D₃	C₂²×C₃	1	2	yes	—
49	C₂×C₃×Dic₆	C₃×C₄	1	1	yes	characteristic
50	C₂×C₃²×C₈	—	2	2	no	—
51	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
52	C₃×#208	—	2	2	no	—
53	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
54	C₉×#41	—	6	2	no	—
55	C₉×#41	—	6	2	no	—
56	C₂×Q₈×C₉	—	3	1	no	—
57	C₂×D₄×C₉	—	3	2	no	—
58	C₂×D₄×C₉	—	3	1	no	—
59	C₂²×C₄×C₉	—	2	1	no	—
60	C₂²×C₄×C₉	—	3	2	no	—
61	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
62	C₉×M₁₆	—	6	2	no	—
63	C₂×C₈×C₉	—	2	4	no	—
64	C₂×C₈×C₉	—	2	2	no	—
65	C₂×C₈×C₉	—	2	2	no	—
66	C₂×C₈×C₉	—	3	2	no	—
67	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
68	C₂²×C₃×C₉	—	2	2	no	—
69	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×D₃×C₉	—	2	2	no	—
72	C₂×D₃×C₉	—	2	4	no	—
73	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
74	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	C₉×Dic₃	—	2	2	no	—
77	C₂×C₃×#41	—	3	1	no	—
78	C₂×#235	C₂×C₉	1	1	yes	characteristic
79	C₂×C₃×M₁₆	—	3	1	no	—
80	C₂²×C₃×C₈	—	2	1	no	—
81	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic, complemented
82	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
83	C₂×#208	C₂×C₉	1	1	yes	characteristic, complemented
84	C₃×C₂²⋊C₈	—	3	4	no	—
85	(C₂×D₃)⋊C₈	C₂×C₉	1	4	yes	complemented
86	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₃×D₃	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
90	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
91	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₃×#67	—	2	4	no	—
93	C₂×C₃×Dic₃	C₂×C₃×C₄	1	2	yes	—
94	C₃×D₁₂	—	2	2	no	—
95	C₃×C₄×D₃	—	2	2	no	—
96	C₃×C₄×D₃	—	2	2	no	—
97	C₃×Dic₆	—	2	2	no	—
98	C₂³×C₉	—	2	1	no	—
99	C₂³×C₉	—	3	2	no	—
100	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
101	C₃²×C₈	—	2	2	no	—
102	C₃×C₃⋊C₈	—	2	2	no	—
103	Q₈×C₉	—	6	2	no	—
104	D₄×C₉	—	6	4	no	—
105	D₄×C₉	—	6	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	1	no	—
110	C₂×C₄×C₉	—	3	2	no	—
111	C₂×C₄×C₉	—	6	2	no	—
112	C₂×C₄×C₉	—	6	2	no	—
113	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
114	C₂×C₄×C₉	D₁₂	1	2	yes	—
115	C₂×C₄×C₉	#67	1	2	yes	—
116	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
117	C₈×C₉	—	2	4	no	—
118	C₈×C₉	—	2	2	no	—
119	C₈×C₉	—	6	2	no	—
120	#408	—	3	1	no	Sylow
121	C₂×C₃×C₉	—	2	2	no	—
122	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₃×C₉	#131	1	2	yes	—
124	D₃×C₉	—	4	2	no	—
125	C₃×#41	—	6	2	no	—
126	C₃×#41	—	6	2	no	—
127	C₂×C₃×Q₈	—	3	1	no	—
128	C₂×C₃×D₄	—	3	2	no	—
129	C₂×C₃×D₄	—	3	1	no	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	—	3	2	no	—
132	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂×#67	C₄×C₉	1	2	yes	—
135	#235	—	2	2	no	—
136	#235	—	2	2	no	—
137	C₂×D₁₂	C₄×C₉	1	1	yes	characteristic
138	C₂×C₄×D₃	C₂²×C₉	1	2	yes	—
139	C₂×Dic₆	C₄×C₉	1	1	yes	characteristic
140	C₃×M₁₆	—	6	2	no	—
141	C₂×C₃×C₈	—	2	4	no	—
142	C₂×C₃×C₈	—	2	2	no	—
143	C₂×C₃×C₈	—	2	2	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	3	2	no	—
146	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
147	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
148	#208	—	2	2	no	—
149	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
150	C₂²×C₃²	—	2	2	no	—
151	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
152	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₂×C₃×D₃	—	2	2	no	—
154	C₂×C₃×D₃	—	2	4	no	—
155	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
156	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
157	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₃×Dic₃	—	2	2	no	—
159	C₂²×C₉	—	2	4	no	—
160	C₂²×C₉	—	2	2	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	6	4	no	—
164	C₂²×C₉	—	6	2	no	—
165	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
166	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
167	C₄×C₉	—	2	2	no	—
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	—	6	2	no	—
171	C₄×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
172	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
173	C₄×C₉	C₂×#67	1	1	yes	characteristic
174	C₂×#41	—	3	1	no	—
175	C₂×M₁₆	—	3	1	no	—
176	C₂²×C₈	D₃×C₉	1	1	yes	characteristic, complemented
177	C₂²⋊C₈	—	3	4	no	—
178	C₃×C₉	#408	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	1	no	—
180	C₂³×C₃	—	3	2	no	—
181	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
182	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
183	C₂²×D₃	C₂×C₄×C₉	1	2	yes	—
184	C₃×Q₈	—	6	2	no	—
185	C₃×D₄	—	6	4	no	—
186	C₃×D₄	—	6	2	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	3	2	no	—
192	C₂×C₃×C₄	—	6	2	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
195	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
196	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
197	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
198	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
200	C₂×C₃×C₄	C₃×#67	1	2	yes	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
202	#67	—	2	4	no	—
203	C₂×Dic₃	C₂×C₄×C₉	1	2	yes	—
204	D₁₂	—	2	2	no	—
205	C₄×D₃	—	2	2	no	—
206	C₄×D₃	—	2	2	no	—
207	Dic₆	—	2	2	no	—
208	C₃×C₈	—	2	4	no	—
209	C₃×C₈	—	2	2	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	—	6	2	no	—
212	C₃⋊C₈	—	2	2	no	—
213	C₂×C₃²	—	2	2	no	—
214	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₃²	C₃×#131	1	2	yes	—
216	C₃×D₃	—	4	2	no	—
217	C₂×C₉	—	2	4	no	—
218	C₂×C₉	—	2	2	no	—
219	C₂×C₉	—	2	2	no	—
220	C₂×C₉	—	2	1	no	—
221	C₂×C₉	—	12	2	no	—
222	C₂×C₉	#901	1	2	yes	—
223	C₂×C₉	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
224	#41	—	6	2	no	—
225	#41	—	6	2	no	—
226	C₂×Q₈	—	3	1	no	—
227	C₂×D₄	—	3	2	no	—
228	C₂×D₄	—	3	1	no	—
229	C₂²×C₄	—	3	2	no	—
230	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
231	M₁₆	—	6	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	3	2	no	—
234	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	6	4	no	—
241	C₂²×C₃	—	6	2	no	—
242	C₂²×C₃	C₉×C₂²⋊C₄	1	2	yes	—
243	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
244	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
245	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
246	C₂×D₃	—	2	2	no	—
247	C₂×D₃	—	2	4	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	6	2	no	—
252	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
253	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
254	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
255	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
256	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
257	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
258	Dic₃	—	2	2	no	—
259	C₃²	C₃×#408	1	1	yes	characteristic
260	C₉	—	2	1	no	—
261	C₉	#5110	1	1	yes	characteristic, direct product factor
262	C₂³	—	3	2	no	—
263	C₂³	C₄×D₃×C₉	1	1	yes	characteristic
264	Q₈	—	6	2	no	—
265	D₄	—	6	4	no	—
266	D₄	—	6	2	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	6	2	no	—
269	C₂×C₄	—	6	2	no	—
270	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
271	C₂×C₄	C₉×D₁₂	1	2	yes	—
272	C₂×C₄	C₉×#67	1	2	yes	—
273	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	6	2	no	—
276	C₂×C₃	—	2	4	no	—
277	C₂×C₃	—	2	2	no	—
278	C₂×C₃	—	2	2	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	12	2	no	—
282	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₃	C₉×#131	1	2	yes	—
284	C₂×C₃	C₃×#901	1	2	yes	—
285	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	D₃	—	4	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	6	4	no	—
289	C₂²	—	6	2	no	—
290	C₂²	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
291	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
292	C₄	—	6	2	no	—
293	C₄	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
294	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
295	C₄	C₂×C₉×#67	1	1	yes	characteristic
296	C₃	—	2	1	no	—
297	C₃	C₉×#408	1	1	yes	characteristic, complemented
298	C₃	C₃×#5110	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	12	2	no	—
301	C₂	C₉×#901	1	2	yes	—
302	C₂	C₂×C₉×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂×C₃×#235	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₉×#408
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈×C₉	C₂	semidirect
3. C₂×C₉×#208	C₂	semidirect
4. C₉×(C₂×D₃)⋊C₈	C₂	semidirect
25. C₂²×C₈×C₉	D₃	semidirect
42. #5110	C₉	direct (N × H)
81. C₂²×C₃×C₈	C₂×C₉	semidirect
83. C₂×#208	C₂×C₉	semidirect
85. (C₂×D₃)⋊C₈	C₂×C₉	semidirect
176. C₂²×C₈	D₃×C₉	semidirect
178. C₃×C₉	#408	semidirect
261. C₉	#5110	direct (N × H)
297. C₃	C₉×#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#235	2
C₂²×C₃×C₈×C₉	2
C₂×C₉×#208	2
C₉×(C₂×D₃)⋊C₈	2
C₃×#5110	3
C₉×#408	3

Subgroups — C9×#5110 #13143102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5110 #13143102