Subgroups — C₂×C₉×#830 #13143510

Order 1728.

842 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₉×#830	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#830	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×C₂²⋊C₈	—	3	1	no	maximal
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₃⋊C₈	—	2	4	no	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	4	yes	complemented
14	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
15	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	2	yes	—
16	C₂³×C₄×C₉	—	2	1	no	—
17	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
18	C₃×#830	C₂×C₃	1	4	yes	—
19	C₂²×C₈×C₉	—	3	2	no	—
20	C₉×C₂²⋊C₈	—	3	4	no	—
21	C₂³×C₃×C₉	—	2	2	no	—
22	C₂³×C₃×C₉	—	2	2	no	—
23	C₂³×C₃×C₉	C₈	1	4	yes	complemented
24	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
25	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	—	2	2	no	—
27	C₂×C₃×C₄×C₉	—	2	4	no	—
28	C₂×C₃×C₄×C₉	—	2	4	no	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×C₃⋊C₈	—	2	4	no	—
34	C₂×C₃×C₂²⋊C₈	—	3	1	no	—
35	C₂×#830	C₉	1	1	yes	characteristic, direct product factor
36	C₂⁴×C₃²	C₃×C₄	1	1	yes	characteristic
37	C₂²×C₃²×C₄	—	2	4	no	—
38	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
39	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
40	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂⁴×C₉	—	2	1	no	—
43	C₂⁴×C₉	Dic₃	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	—	2	4	no	—
45	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
46	C₂²×C₄×C₉	—	2	8	no	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	—	2	2	no	—
49	C₂²×C₄×C₉	—	2	2	no	—
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	—	2	4	no	—
52	C₂²×C₄×C₉	Dic₃	1	1	yes	characteristic
53	C₂²×C₄×C₉	Dic₃	1	2	yes	—
54	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
55	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂×C₈×C₉	—	3	4	no	—
57	C₂×C₈×C₉	—	6	4	no	—
58	C₂²×C₃×C₉	—	2	8	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
63	C₂²×C₃×C₉	C₂×C₈	1	4	yes	complemented
64	C₂²×C₃×C₉	M₁₆	1	2	yes	—
65	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	—	2	2	no	—
67	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
68	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
69	C₂³×C₃×C₄	—	2	1	no	—
70	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
71	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
72	C₂²×C₃×C₈	—	3	2	no	—
73	C₂²×C₃⋊C₈	C₂×C₉	1	2	yes	complemented
74	C₃×C₂²⋊C₈	—	3	4	no	—
75	#830	C₂×C₉	1	4	yes	direct product factor
76	C₂³×C₃²	—	2	2	no	—
77	C₂³×C₃²	—	2	2	no	—
78	C₂³×C₃²	C₃×C₈	1	4	yes	—
79	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
80	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
87	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂³×C₉	—	2	4	no	—
89	C₂³×C₉	—	2	4	no	—
90	C₂³×C₉	—	2	2	no	—
91	C₂³×C₉	—	2	2	no	—
92	C₂³×C₉	—	2	4	no	—
93	C₂³×C₉	—	2	2	no	—
94	C₂³×C₉	—	2	1	no	—
95	C₂³×C₉	C₃⋊C₈	1	4	yes	complemented
96	C₂³×C₉	C₂×Dic₃	1	2	yes	—
97	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
98	C₃×C₃⋊C₈	—	2	4	no	—
99	C₂×C₄×C₉	—	2	4	no	—
100	C₂×C₄×C₉	—	2	8	no	—
101	C₂×C₄×C₉	—	2	8	no	—
102	C₂×C₄×C₉	—	2	2	no	—
103	C₂×C₄×C₉	—	2	4	no	—
104	C₂×C₄×C₉	—	2	4	no	—
105	C₂×C₄×C₉	—	2	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
110	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
111	C₂×C₄×C₉	#67	1	4	yes	—
112	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
113	C₈×C₉	—	6	4	no	—
114	C₂×C₂²⋊C₈	—	3	1	no	Sylow
115	C₂×C₃×C₉	—	2	4	no	—
116	C₂×C₃×C₉	C₂²⋊C₈	1	4	yes	complemented
117	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
120	C₂⁴×C₃	—	2	1	no	—
121	C₂⁴×C₃	C₄×C₉	1	1	yes	characteristic
122	C₂⁴×C₃	C₃×Dic₃	1	1	yes	characteristic
123	C₂²×C₃×C₄	—	2	8	no	—
124	C₂²×C₃×C₄	—	2	4	no	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	—	2	2	no	—
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	C₄×C₉	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₄×C₉	1	2	yes	—
132	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
133	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	—	3	4	no	—
139	C₂×C₃×C₈	—	6	4	no	—
140	C₂×C₃⋊C₈	—	2	4	no	—
141	C₂×C₃⋊C₈	C₂²×C₉	1	4	yes	complemented
142	C₂²×C₃²	—	2	8	no	—
143	C₂²×C₃²	—	2	2	no	—
144	C₂²×C₃²	—	2	2	no	—
145	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
146	C₂²×C₃²	C₂×C₃×C₈	1	2	yes	—
147	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
148	C₂²×C₃²	C₃×M₁₆	1	2	yes	—
149	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₃²×C₄	—	2	2	no	—
151	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
152	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
153	C₂²×C₉	—	2	4	no	—
154	C₂²×C₉	—	2	4	no	—
155	C₂²×C₉	—	2	16	no	—
156	C₂²×C₉	—	2	8	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	—	2	2	no	—
159	C₂²×C₉	—	2	2	no	—
160	C₂²×C₉	—	2	4	no	—
161	C₂²×C₉	—	2	2	no	—
162	C₂²×C₉	—	2	2	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	C₂×C₃⋊C₈	1	2	yes	—
165	C₂²×C₉	C₂×C₃⋊C₈	1	4	yes	complemented
166	C₂²×C₉	#208	1	2	yes	—
167	C₂²×C₉	#217	1	2	yes	—
168	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
169	C₄×C₉	—	2	4	no	—
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	#217	1	2	yes	—
174	C₄×C₉	C₂×#67	1	2	yes	—
175	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
176	C₂²×C₈	—	3	2	no	—
177	C₂²⋊C₈	—	3	4	no	—
178	C₃×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₃	—	2	4	no	—
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	—	2	2	no	—
182	C₂³×C₃	—	2	2	no	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	—	2	4	no	—
186	C₂³×C₃	—	2	2	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	C₈×C₉	1	4	yes	complemented
189	C₂³×C₃	C₂×C₄×C₉	1	2	yes	—
190	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
191	C₂³×C₃	C₃×C₃⋊C₈	1	4	yes	—
192	C₂³×C₃	C₂×C₃×Dic₃	1	2	yes	—
193	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	8	no	—
196	C₂×C₃×C₄	—	2	8	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
208	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
209	C₂×C₃×C₄	D₄×C₉	1	4	yes	—
210	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
212	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
213	C₂×C₃×C₄	C₃×#67	1	4	yes	—
214	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
215	C₃×C₈	—	6	4	no	—
216	C₃⋊C₈	—	2	4	no	—
217	C₂×C₃²	—	2	4	no	—
218	C₂×C₃²	C₃×C₂²⋊C₈	1	4	yes	—
219	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₃²	C₂²×C₃×C₈	1	1	yes	characteristic
221	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
222	C₂×C₉	—	2	8	no	—
223	C₂×C₉	—	2	4	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	—	2	4	no	—
228	C₂×C₉	#830	1	4	yes	direct product factor
229	C₂×C₉	C₂²×C₃⋊C₈	1	1	yes	characteristic
230	C₂×C₉	C₂×#208	1	1	yes	characteristic
231	C₂×C₉	C₂×#217	1	1	yes	characteristic
232	C₂⁴	C₉×Dic₃	1	1	yes	characteristic
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₉×Dic₃	1	1	yes	characteristic
235	C₂²×C₄	C₉×Dic₃	1	2	yes	—
236	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
237	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
238	C₂×C₈	—	3	4	no	—
239	C₂×C₈	—	6	4	no	—
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	2	16	no	—
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	8	no	—
248	C₂²×C₃	—	2	8	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	C₉×C₂²⋊C₄	1	2	yes	—
255	C₂²×C₃	C₂×C₈×C₉	1	2	yes	—
256	C₂²×C₃	C₂×C₈×C₉	1	4	yes	complemented
257	C₂²×C₃	C₉×M₁₆	1	2	yes	—
258	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
259	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	2	yes	—
260	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	4	yes	—
261	C₂²×C₃	C₃×#208	1	2	yes	—
262	C₂²×C₃	C₃×#217	1	2	yes	—
263	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	C₉×C₂²⋊C₄	1	2	yes	—
270	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
271	C₃×C₄	C₃×#217	1	2	yes	—
272	C₃×C₄	C₂×C₃×#67	1	2	yes	—
273	C₃²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
274	C₉	—	2	1	no	—
275	C₉	C₂×#830	1	1	yes	characteristic, direct product factor
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	C₉×C₃⋊C₈	1	4	yes	complemented
279	C₂³	C₂×C₉×Dic₃	1	2	yes	—
280	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
285	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
286	C₂×C₄	C₉×#67	1	4	yes	—
287	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
288	C₈	—	6	4	no	—
289	C₂×C₃	—	2	8	no	—
290	C₂×C₃	—	2	4	no	—
291	C₂×C₃	—	2	4	no	—
292	C₂×C₃	—	2	4	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	C₉×C₂²⋊C₈	1	4	yes	complemented
297	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
298	C₂×C₃	C₂²×C₈×C₉	1	1	yes	characteristic
299	C₂×C₃	C₂×C₉×M₁₆	1	1	yes	characteristic
300	C₂×C₃	C₃×#830	1	4	yes	—
301	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
303	C₂×C₃	C₂×C₃×#217	1	1	yes	characteristic
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	C₂×C₉×C₃⋊C₈	1	2	yes	—
308	C₂²	C₂×C₉×C₃⋊C₈	1	4	yes	complemented
309	C₂²	C₉×#208	1	2	yes	—
310	C₂²	C₉×#217	1	2	yes	—
311	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	C₉×#217	1	2	yes	—
314	C₄	C₂×C₉×#67	1	2	yes	—
315	C₃	—	2	1	no	—
316	C₃	C₂×C₉×C₂²⋊C₈	1	1	yes	characteristic, complemented
317	C₃	C₂×C₃×#830	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	C₉×#830	1	4	yes	direct product factor
320	C₂	C₂²×C₉×C₃⋊C₈	1	1	yes	characteristic
321	C₂	C₂×C₉×#208	1	1	yes	characteristic
322	C₂	C₂×C₉×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂²×C₈×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂³×C₃²	C₂×C₃×C₄
prime radical	C₂⁴×C₃²	C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂³×C₄×C₉	D₃
hypocenter	C₃	C₂×C₉×C₂²⋊C₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×C₃⋊C₈	C₂	semidirect
3. C₉×#830	C₂	direct (N × H)
13. C₂×C₉×C₃⋊C₈	C₂²	semidirect
23. C₂³×C₃×C₉	C₈	semidirect
35. C₂×#830	C₉	direct (N × H)
63. C₂²×C₃×C₉	C₂×C₈	semidirect
73. C₂²×C₃⋊C₈	C₂×C₉	semidirect
75. #830	C₂×C₉	direct (N × H)
95. C₂³×C₉	C₃⋊C₈	semidirect
116. C₂×C₃×C₉	C₂²⋊C₈	semidirect
141. C₂×C₃⋊C₈	C₂²×C₉	semidirect
165. C₂²×C₉	C₂×C₃⋊C₈	semidirect
178. C₃×C₉	C₂×C₂²⋊C₈	semidirect
188. C₂³×C₃	C₈×C₉	semidirect
228. C₂×C₉	#830	direct (N × H)
256. C₂²×C₃	C₂×C₈×C₉	semidirect
275. C₉	C₂×#830	direct (N × H)
278. C₂³	C₉×C₃⋊C₈	semidirect
296. C₂×C₃	C₉×C₂²⋊C₈	semidirect
308. C₂²	C₂×C₉×C₃⋊C₈	semidirect
316. C₃	C₂×C₉×C₂²⋊C₈	semidirect
319. C₂	C₉×#830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂²×C₉×C₃⋊C₈	2
C₉×#830	2
C₂×C₃×#830	3
C₂×C₉×C₂²⋊C₈	3

Subgroups — C2×C9×#830 #13143510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#830 #13143510