Subgroups — C₃×#10589933 #13153696

Order 1728.

1034 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×Q₈×D₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63500	C₂	1	3	yes	maximal, complemented
5	C₃×#63498	C₂	1	3	yes	maximal, complemented
6	C₃×C₄×#611	C₂	1	3	yes	maximal, complemented
7	C₃×#63486	C₂	1	3	yes	maximal, complemented
8	C₃×#5831	—	3	1	no	maximal
9	#10589933	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₃×C₉×#41	—	2	4	no	—
11	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₄×C₉	C₂²	1	3	yes	—
13	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
14	C₂×C₃×#611	C₂²	1	3	yes	—
15	C₃×Q₈×D₉	—	2	4	no	—
16	C₂×C₃×C₄×D₉	C₂²	1	3	yes	—
17	C₂×C₃×Dic₁₈	C₂²	1	3	yes	—
18	C₃×#3676	C₂²	1	6	yes	—
19	C₃×#3676	C₂²	1	3	yes	—
20	C₃×(C₂×D₉)⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊Dic₉	C₂²	1	3	yes	—
22	C₃×Dic₉⋊C₄	C₂²	1	3	yes	—
23	C₃×C₄×Dic₉	C₂²	1	3	yes	—
24	C₂×C₃²×#41	D₃	1	1	yes	characteristic
25	C₂×C₃×D₃×Q₈	—	3	1	no	—
26	C₃×Q₈×Dic₃	—	3	1	no	—
27	C₃×#938	—	3	3	no	—
28	C₃×#936	—	3	3	no	—
29	C₃×C₄×#67	—	3	3	no	—
30	C₃×#924	—	3	3	no	—
31	C₂×C₉×#41	—	2	1	no	—
32	C₂×C₉×#41	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×Q₈×D₉	C₂×C₃	1	1	yes	characteristic, complemented
34	Q₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
35	#63500	C₂×C₃	1	3	yes	complemented
36	#63498	C₂×C₃	1	3	yes	complemented
37	C₄×#611	C₂×C₃	1	3	yes	complemented
38	#63486	C₂×C₃	1	3	yes	complemented
39	C₂³×C₃×C₉	C₂³	1	3	yes	—
40	C₂²×C₃×D₉	C₂³	1	1	yes	characteristic
41	C₃×Q₈×C₉	D₄	1	4	yes	complemented
42	C₃×D₄×C₉	—	2	6	no	—
43	C₂×C₃×C₄×C₉	—	2	6	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
46	C₃×#611	—	2	6	no	—
47	C₂×C₃×Dic₉	C₂³	1	3	yes	—
48	C₂×C₃×Dic₉	C₂³	1	3	yes	—
49	C₂×C₃×Dic₉	C₂³	1	1	yes	characteristic
50	C₃×C₄×D₉	—	2	6	no	—
51	C₃×Dic₁₈	—	2	6	no	—
52	C₃×#548	—	9	1	no	—
53	#5831	—	3	1	no	—
54	C₃²×#41	—	2	4	no	—
55	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃²×D₄	C₂×D₃	1	3	yes	—
57	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
58	C₂×C₃×#67	—	3	3	no	—
59	C₃×D₃×Q₈	—	6	4	no	—
60	C₂×C₃×C₄×D₃	—	3	3	no	—
61	C₂×C₃×Dic₆	—	3	3	no	—
62	C₃×#217	—	3	6	no	—
63	C₃×#217	—	3	3	no	—
64	C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
65	C₃×C₄⋊Dic₃	—	3	3	no	—
66	C₃×Dic₃⋊C₄	—	3	3	no	—
67	C₃×C₄×Dic₃	—	3	3	no	—
68	C₉×#41	—	2	8	no	—
69	C₉×#41	—	2	4	no	—
70	C₂×Q₈×C₉	—	2	1	no	—
71	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
72	C₂×D₄×C₉	—	2	3	no	—
73	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
74	C₂²×C₄×C₉	—	2	3	no	—
75	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
76	C₂×#611	C₂²×C₃	1	3	yes	—
77	Q₈×D₉	—	2	4	no	—
78	C₂×C₄×D₉	C₂²×C₃	1	3	yes	—
79	C₂×Dic₁₈	C₂²×C₃	1	3	yes	—
80	#3676	C₂²×C₃	1	3	yes	—
81	#3676	C₂²×C₃	1	6	yes	—
82	(C₂×D₉)⋊C₄	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
84	Dic₉⋊C₄	C₂²×C₃	1	3	yes	—
85	C₄×Dic₉	C₂²×C₃	1	3	yes	—
86	C₂²×C₃×C₉	—	2	3	no	—
87	C₂²×C₃×C₉	—	2	3	no	—
88	C₂²×C₃×C₉	—	2	3	no	—
89	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₃×D₉	—	2	1	no	—
91	C₂×C₃×D₉	—	2	1	no	—
92	C₂×C₃×D₉	#41	1	2	yes	complemented
93	C₃×C₄×C₉	—	2	1	no	—
94	C₃×C₄×C₉	C₂×D₄	1	6	yes	—
95	C₃×Dic₉	—	2	3	no	—
96	C₃×Dic₉	—	2	3	no	—
97	C₃×Dic₉	—	2	1	no	—
98	C₂×C₃×#41	—	2	1	no	—
99	C₂×C₃×#41	D₉	1	1	yes	characteristic, complemented
100	C₂×C₃×#41	C₃×D₃	1	1	yes	characteristic
101	C₂²×C₃×Q₈	—	9	1	no	—
102	C₂×D₃×Q₈	—	3	1	no	—
103	C₃×#124	—	9	3	no	—
104	C₃×C₂²⋊Q₈	—	9	3	no	—
105	C₃×#121	—	9	3	no	—
106	C₃×C₄×Q₈	—	9	1	no	—
107	C₃×C₄×D₄	—	9	3	no	—
108	Q₈×Dic₃	—	3	1	no	—
109	#938	—	3	3	no	—
110	#936	—	3	3	no	—
111	C₄×#67	—	3	3	no	—
112	#924	—	3	3	no	—
113	C₂³×C₃²	C₂²×D₃	1	3	yes	—
114	C₂²×C₃×D₃	—	3	1	no	—
115	C₃²×Q₈	#67	1	4	yes	—
116	C₃²×D₄	—	2	6	no	—
117	C₂×C₃²×C₄	—	2	6	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
120	C₃×#67	—	6	6	no	—
121	C₂×C₃×Dic₃	—	3	1	no	—
122	C₂×C₃×Dic₃	—	3	3	no	—
123	C₂×C₃×Dic₃	—	3	3	no	—
124	C₃×C₄×D₃	—	6	6	no	—
125	C₃×Dic₆	—	6	6	no	—
126	C₂³×C₉	—	2	3	no	—
127	C₂³×C₉	C₂³×C₃	1	3	yes	—
128	C₂²×D₉	C₂³×C₃	1	1	yes	characteristic
129	Q₈×C₉	—	2	4	no	—
130	Q₈×C₉	C₃×D₄	1	4	yes	complemented
131	D₄×C₉	—	2	12	no	—
132	D₄×C₉	—	2	6	no	—
133	C₂×C₄×C₉	—	2	12	no	—
134	C₂×C₄×C₉	—	2	6	no	—
135	C₂×C₄×C₉	—	2	1	no	—
136	C₂×C₄×C₉	—	2	3	no	—
137	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
139	#611	—	2	6	no	—
140	C₂×Dic₉	C₂³×C₃	1	1	yes	characteristic
141	C₂×Dic₉	C₂³×C₃	1	3	yes	—
142	C₂×Dic₉	C₂³×C₃	1	3	yes	—
143	C₄×D₉	—	2	6	no	—
144	Dic₁₈	—	2	6	no	—
145	#548	—	9	1	no	Sylow
146	C₂×C₃×C₉	—	4	3	no	—
147	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
149	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
150	C₃×D₉	—	2	2	no	—
151	C₃×#41	—	2	8	no	—
152	C₃×#41	—	2	4	no	—
153	C₃×#41	—	2	4	no	—
154	C₂×C₃×Q₈	—	2	1	no	—
155	C₂×C₃×Q₈	—	9	3	no	—
156	C₂×C₃×Q₈	—	18	4	no	—
157	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
158	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
159	C₂×C₃×D₄	—	2	3	no	—
160	C₂×C₃×D₄	—	9	3	no	—
161	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
162	C₂×C₃×D₄	C₂×C₃×D₃	1	3	yes	—
163	C₂²×C₃×C₄	—	2	3	no	—
164	C₂²×C₃×C₄	—	9	3	no	—
165	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
166	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
167	C₂×#67	—	3	3	no	—
168	D₃×Q₈	—	6	4	no	—
169	C₂×C₄×D₃	—	3	3	no	—
170	C₂×Dic₆	—	3	3	no	—
171	C₃×C₄⋊C₄	—	9	3	no	—
172	C₃×C₄⋊C₄	—	9	3	no	—
173	C₃×C₂²⋊C₄	—	9	3	no	—
174	C₃×C₂²⋊C₄	—	9	6	no	—
175	C₃×C₂²⋊C₄	—	9	1	no	—
176	C₃×C₄²	—	9	3	no	—
177	#217	—	3	6	no	—
178	#217	—	3	3	no	—
179	(C₂×D₃)⋊C₄	—	3	1	no	—
180	C₄⋊Dic₃	—	3	3	no	—
181	Dic₃⋊C₄	—	3	3	no	—
182	C₄×Dic₃	—	3	3	no	—
183	C₂²×C₃²	—	2	3	no	—
184	C₂²×C₃²	—	2	3	no	—
185	C₂²×C₃²	—	2	3	no	—
186	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
187	C₂×C₃×D₃	—	3	2	no	—
188	C₂×C₃×D₃	—	6	1	no	—
189	C₂×C₃×D₃	—	6	1	no	—
190	C₃²×C₄	—	2	1	no	—
191	C₃²×C₄	C₂×#67	1	6	yes	—
192	C₃×Dic₃	—	6	3	no	—
193	C₃×Dic₃	—	6	1	no	—
194	C₃×Dic₃	—	6	3	no	—
195	C₂²×C₉	—	2	6	no	—
196	C₂²×C₉	—	2	3	no	—
197	C₂²×C₉	—	2	3	no	—
198	C₂²×C₉	—	2	3	no	—
199	C₂²×C₉	—	2	1	no	—
200	C₂²×C₉	—	4	3	no	—
201	C₂²×C₉	—	4	3	no	—
202	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
203	C₂×D₉	—	2	1	no	—
204	C₂×D₉	—	2	1	no	—
205	C₂×D₉	C₃×#41	1	2	yes	complemented
206	C₄×C₉	—	2	2	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	2	6	no	—
209	C₄×C₉	C₂×C₃×D₄	1	6	yes	—
210	Dic₉	—	2	3	no	—
211	Dic₉	—	2	3	no	—
212	Dic₉	—	2	1	no	—
213	C₂×#41	C₃×D₉	1	1	yes	characteristic, complemented
214	C₂²×Q₈	—	9	1	no	—
215	#124	—	9	3	no	—
216	C₂²⋊Q₈	—	9	3	no	—
217	#121	—	9	3	no	—
218	C₄×Q₈	—	9	1	no	—
219	C₄×D₄	—	9	3	no	—
220	C₃×C₉	#548	1	1	yes	characteristic, complemented, Sylow
221	C₂³×C₃	—	2	3	no	—
222	C₂³×C₃	—	9	1	no	—
223	C₂³×C₃	C₂²×D₉	1	3	yes	—
224	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
225	C₂²×D₃	—	3	1	no	—
226	C₃×Q₈	—	2	4	no	—
227	C₃×Q₈	—	18	6	no	—
228	C₃×Q₈	#611	1	4	yes	complemented
229	C₃×Q₈	C₃×#67	1	4	yes	—
230	C₃×D₄	—	2	12	no	—
231	C₃×D₄	—	2	6	no	—
232	C₃×D₄	—	2	6	no	—
233	C₃×D₄	—	18	6	no	—
234	C₂×C₃×C₄	—	2	12	no	—
235	C₂×C₃×C₄	—	2	6	no	—
236	C₂×C₃×C₄	—	2	6	no	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	2	3	no	—
239	C₂×C₃×C₄	—	9	1	no	—
240	C₂×C₃×C₄	—	9	3	no	—
241	C₂×C₃×C₄	—	9	3	no	—
242	C₂×C₃×C₄	—	18	6	no	—
243	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
245	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
247	#67	—	6	6	no	—
248	C₂×Dic₃	—	3	1	no	—
249	C₂×Dic₃	—	3	3	no	—
250	C₂×Dic₃	—	3	3	no	—
251	C₄×D₃	—	6	6	no	—
252	Dic₆	—	6	6	no	—
253	C₂×C₃²	—	4	3	no	—
254	C₂×C₃²	D₃×#41	1	1	yes	characteristic
255	C₂×C₃²	#1010	1	1	yes	characteristic
256	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
257	C₃×D₃	—	6	2	no	—
258	C₂×C₉	—	2	1	no	—
259	C₂×C₉	—	2	1	no	—
260	C₂×C₉	—	2	1	no	—
261	C₂×C₉	—	4	6	no	—
262	C₂×C₉	—	4	3	no	—
263	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
264	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
265	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
266	D₉	—	2	2	no	—
267	#41	—	2	4	no	—
268	C₂×Q₈	—	9	3	no	—
269	C₂×Q₈	—	18	4	no	—
270	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
271	C₂×D₄	—	9	3	no	—
272	C₂×D₄	C₂×C₃×D₉	1	3	yes	—
273	C₂²×C₄	—	9	3	no	—
274	C₂²×C₄	C₂×C₃×D₉	1	3	yes	—
275	C₄⋊C₄	—	9	3	no	—
276	C₄⋊C₄	—	9	3	no	—
277	C₂²⋊C₄	—	9	3	no	—
278	C₂²⋊C₄	—	9	6	no	—
279	C₂²⋊C₄	—	9	1	no	—
280	C₄²	—	9	3	no	—
281	C₂²×C₃	—	2	6	no	—
282	C₂²×C₃	—	2	3	no	—
283	C₂²×C₃	—	2	3	no	—
284	C₂²×C₃	—	2	3	no	—
285	C₂²×C₃	—	2	3	no	—
286	C₂²×C₃	—	2	3	no	—
287	C₂²×C₃	—	2	3	no	—
288	C₂²×C₃	—	2	1	no	—
289	C₂²×C₃	—	4	3	no	—
290	C₂²×C₃	—	4	3	no	—
291	C₂²×C₃	—	9	2	no	—
292	C₂²×C₃	—	18	1	no	—
293	C₂²×C₃	—	18	1	no	—
294	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
295	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
296	C₂×D₃	—	3	2	no	—
297	C₂×D₃	—	6	1	no	—
298	C₂×D₃	—	6	1	no	—
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	—	2	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	6	no	—
303	C₃×C₄	—	18	3	no	—
304	C₃×C₄	—	18	1	no	—
305	C₃×C₄	—	18	3	no	—
306	C₃×C₄	C₂×#611	1	6	yes	—
307	C₃×C₄	C₂×C₃×#67	1	6	yes	—
308	Dic₃	—	6	3	no	—
309	Dic₃	—	6	1	no	—
310	Dic₃	—	6	3	no	—
311	C₃²	#5831	1	1	yes	characteristic
312	C₉	—	2	1	no	—
313	C₉	C₃×#548	1	1	yes	characteristic, complemented
314	C₂³	—	9	1	no	—
315	C₂³	C₂²×C₃×D₉	1	3	yes	—
316	Q₈	—	18	6	no	—
317	Q₈	C₃×#611	1	4	yes	complemented
318	D₄	—	2	6	no	—
319	D₄	—	18	6	no	—
320	C₂×C₄	—	2	6	no	—
321	C₂×C₄	—	9	1	no	—
322	C₂×C₄	—	9	3	no	—
323	C₂×C₄	—	9	3	no	—
324	C₂×C₄	—	18	6	no	—
325	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	4	6	no	—
331	C₂×C₃	—	4	3	no	—
332	C₂×C₃	—	4	3	no	—
333	C₂×C₃	—	18	2	no	—
334	C₂×C₃	#41×D₉	1	1	yes	characteristic
335	C₂×C₃	#63716	1	1	yes	characteristic
336	C₂×C₃	C₂²×#611	1	1	yes	characteristic
337	C₂×C₃	C₃×D₃×#41	1	1	yes	characteristic
338	C₂×C₃	C₃×#1010	1	1	yes	characteristic
339	C₂×C₃	C₂²×C₃×#67	1	1	yes	characteristic
340	D₃	—	6	2	no	—
341	C₂²	—	2	3	no	—
342	C₂²	—	2	3	no	—
343	C₂²	—	2	3	no	—
344	C₂²	—	9	2	no	—
345	C₂²	—	18	1	no	—
346	C₂²	—	18	1	no	—
347	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	18	3	no	—
350	C₄	—	18	1	no	—
351	C₄	—	18	3	no	—
352	C₄	C₂×C₃×#611	1	6	yes	—
353	C₃	—	2	1	no	—
354	C₃	#10589933	1	1	yes	characteristic, direct product factor
355	C₃	C₃×#5831	1	1	yes	characteristic
356	C₂	—	4	3	no	—
357	C₂	—	18	2	no	—
358	C₂	C₃×#41×D₉	1	1	yes	characteristic
359	C₂	C₃×#63716	1	1	yes	characteristic
360	C₂	C₂²×C₃×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₂×C₃×#41	D₉
hypocenter	C₉	C₃×#548
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×#41	C₂	semidirect
2. C₂×C₃×Q₈×D₉	C₂	semidirect
3. C₃×Q₈×Dic₉	C₂	semidirect
4. C₃×#63500	C₂	semidirect
5. C₃×#63498	C₂	semidirect
6. C₃×C₄×#611	C₂	semidirect
7. C₃×#63486	C₂	semidirect
9. #10589933	C₃	direct (N × H)
32. C₂×C₉×#41	C₂×C₃	semidirect
33. C₂×Q₈×D₉	C₂×C₃	semidirect
34. Q₈×Dic₉	C₂×C₃	semidirect
35. #63500	C₂×C₃	semidirect
36. #63498	C₂×C₃	semidirect
37. C₄×#611	C₂×C₃	semidirect
38. #63486	C₂×C₃	semidirect
41. C₃×Q₈×C₉	D₄	semidirect
92. C₂×C₃×D₉	#41	semidirect
99. C₂×C₃×#41	D₉	semidirect
130. Q₈×C₉	C₃×D₄	semidirect
205. C₂×D₉	C₃×#41	semidirect
213. C₂×#41	C₃×D₉	semidirect
220. C₃×C₉	#548	semidirect
228. C₃×Q₈	#611	semidirect
313. C₉	C₃×#548	semidirect
317. Q₈	C₃×#611	semidirect
354. C₃	#10589933	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×#41	2
C₂×C₃×Q₈×D₉	2
C₃×Q₈×Dic₉	2
C₃×#63500	2
C₃×#63498	2
C₃×C₄×#611	2
C₃×#63486	2
C₃×#5831	3
#10589933	3

Subgroups — C3×#10589933 #13153696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589933 #13153696