Subgroups — C₃×#10599211 #14284655

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66062	C₂	1	1	yes	maximal, characteristic
2	C₃×#65122	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64767	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599211	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×#4092	—	2	1	no	—
11	C₃×Q₈×Dic₅	—	2	1	no	—
12	C₂×C₃×C₈×D₅	—	2	1	no	—
13	C₃×#4043	C₄	1	2	yes	—
14	C₃×#4038	—	2	1	no	—
15	C₃×#4036	—	2	1	no	—
16	C₃×#4033	C₄	1	2	yes	—
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₃×#4015	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#3952	—	2	1	no	—
22	C₃×#3946	—	2	1	no	—
23	C₃×C₈×Dic₅	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
26	C₃×#1542	—	5	1	no	maximal, Hall
27	#66062	C₂×C₃	1	1	yes	characteristic
28	#65122	C₂×C₃	1	1	yes	characteristic
29	D₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
30	C₅×#332	C₂×C₃	1	1	yes	characteristic, complemented
31	#64767	C₂×C₃	1	1	yes	characteristic, complemented
32	#64701	C₂×C₃	1	1	yes	characteristic, complemented
33	#64701	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₃×#685	—	2	2	no	—
45	C₃×C₄⋊Dic₅	—	2	1	no	—
46	C₃×C₄⋊Dic₅	—	2	1	no	—
47	C₃×C₄⋊Dic₅	—	2	2	no	—
48	C₃×#683	—	2	2	no	—
49	C₃×C₄×Dic₅	—	2	2	no	—
50	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
53	C₂×C₃×#168	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	C₃×C₈×D₅	—	4	2	no	—
56	C₃×#533	—	5	1	no	—
57	C₃×#431	—	5	1	no	—
58	C₂×C₃×C₄⋊C₈	—	5	1	no	—
59	C₃×#333	—	5	1	no	—
60	C₃×#333	—	5	1	no	—
61	C₃×#332	—	5	1	no	—
62	C₃×#332	D₅	1	1	yes	characteristic, complemented
63	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
64	#4092	—	2	1	no	—
65	Q₈×Dic₅	—	2	1	no	—
66	C₂×C₈×D₅	—	2	1	no	—
67	#4043	C₃×C₄	1	2	yes	—
68	#4038	—	2	1	no	—
69	#4036	—	2	1	no	—
70	#4033	C₃×C₄	1	2	yes	—
71	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
72	#4015	C₂²×C₃	1	1	yes	characteristic
73	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
74	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
75	#3952	—	2	1	no	—
76	#3946	—	2	1	no	—
77	C₈×Dic₅	—	2	1	no	—
78	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
80	#1542	—	5	1	no	Sylow
81	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₅×Q₈	—	4	1	no	—
83	C₃×C₅×Q₈	—	4	1	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×Dic₅	—	2	2	no	—
89	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₄×D₅	—	2	1	no	—
93	C₃×C₄×D₅	—	2	1	no	—
94	C₃×C₄×D₅	—	4	1	no	—
95	C₃×C₄×D₅	SD₁₆	1	2	yes	—
96	C₃×C₄×D₅	Q₁₆	1	2	yes	—
97	C₃×Dic₁₀	—	4	2	no	—
98	C₃×C₅×C₈	—	4	1	no	—
99	C₃×C₅×C₈	—	4	1	no	—
100	C₃×#168	—	4	1	no	—
101	C₃×#168	—	4	1	no	—
102	C₂²×C₃×C₈	—	10	1	no	—
103	C₃×C₄⋊Q₈	—	5	1	no	—
104	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
105	C₃×#125	—	10	1	no	—
106	C₃×C₂²⋊Q₈	—	5	2	no	—
107	C₃×C₂²⋊Q₈	—	10	1	no	—
108	C₃×C₄×Q₈	—	5	2	no	—
109	C₃×C₄×Q₈	—	10	1	no	—
110	C₃×#117	—	10	1	no	—
111	C₂×C₃×C₄²	—	5	1	no	—
112	C₃×C₄⋊C₈	—	5	1	no	—
113	C₃×C₄⋊C₈	—	5	1	no	—
114	C₃×C₄⋊C₈	—	10	1	no	—
115	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
116	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
117	C₃×C₂²⋊C₈	—	10	1	no	—
118	C₃×C₄×C₈	—	10	1	no	—
119	C₂×C₅×Q₈	—	2	1	no	—
120	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
122	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
123	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₅×C₄⋊C₄	—	2	1	no	—
127	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
128	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
129	#685	—	2	2	no	—
130	C₄⋊Dic₅	—	2	1	no	—
131	C₄⋊Dic₅	—	2	1	no	—
132	C₄⋊Dic₅	—	2	2	no	—
133	#683	—	2	2	no	—
134	C₄×Dic₅	—	2	2	no	—
135	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
136	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
137	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
138	C₂×#168	—	2	1	no	—
139	C₂×#168	—	2	1	no	—
140	C₈×D₅	—	4	2	no	—
141	#533	—	5	1	no	—
142	#431	—	5	1	no	—
143	C₂×C₄⋊C₈	—	5	1	no	—
144	#333	—	5	1	no	—
145	#333	—	5	1	no	—
146	#332	—	5	1	no	—
147	#332	C₃×D₅	1	1	yes	characteristic, complemented
148	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₃×D₅	—	2	1	no	—
150	C₂×C₃×D₅	—	2	1	no	—
151	C₂×C₃×D₅	#103	1	2	yes	—
152	C₃×C₄×C₅	—	2	1	no	—
153	C₃×C₄×C₅	—	4	2	no	—
154	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
155	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
156	C₃×C₄×C₅	#135	1	1	yes	characteristic
157	C₃×C₄×C₅	#135	1	1	yes	characteristic
158	C₃×Dic₅	—	2	1	no	—
159	C₃×Dic₅	—	2	1	no	—
160	C₃×Dic₅	—	4	2	no	—
161	C₃×Dic₅	#103	1	2	yes	—
162	C₂×C₃×Q₈	—	2	1	no	—
163	C₂×C₃×Q₈	—	5	2	no	—
164	C₂²×C₃×C₄	—	5	1	no	—
165	C₂²×C₃×C₄	—	5	1	no	—
166	C₂²×C₃×C₄	—	5	1	no	—
167	C₂×C₃×C₈	—	2	1	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₂×C₃×C₈	—	10	1	no	—
170	C₂×C₃×C₈	—	10	1	no	—
171	C₂×C₃×C₈	—	20	2	no	—
172	C₃×C₄⋊C₄	—	2	1	no	—
173	C₃×C₄⋊C₄	—	10	2	no	—
174	C₃×C₄⋊C₄	—	10	2	no	—
175	C₃×C₄⋊C₄	—	10	1	no	—
176	C₃×C₄⋊C₄	—	10	1	no	—
177	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
178	C₃×C₂²⋊C₄	—	10	2	no	—
179	C₃×C₄²	—	5	1	no	—
180	C₃×C₄²	—	5	1	no	—
181	C₃×C₄²	—	5	1	no	—
182	C₃×C₄²	—	10	2	no	—
183	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
184	C₂²×D₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
185	C₅×Q₈	—	4	1	no	—
186	C₅×Q₈	—	4	1	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×Dic₅	—	2	2	no	—
192	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₄×D₅	—	2	1	no	—
196	C₄×D₅	—	2	1	no	—
197	C₄×D₅	—	4	1	no	—
198	C₄×D₅	C₃×SD₁₆	1	2	yes	—
199	C₄×D₅	C₃×Q₁₆	1	2	yes	—
200	Dic₁₀	—	4	2	no	—
201	C₅×C₈	—	4	1	no	—
202	C₅×C₈	—	4	1	no	—
203	#168	—	4	1	no	—
204	#168	—	4	1	no	—
205	C₂²×C₈	—	10	1	no	—
206	C₄⋊Q₈	—	5	1	no	—
207	C₄⋊Q₈	C₂×C₃×D₅	1	1	yes	characteristic
208	#125	—	10	1	no	—
209	C₂²⋊Q₈	—	5	2	no	—
210	C₂²⋊Q₈	—	10	1	no	—
211	C₄×Q₈	—	5	2	no	—
212	C₄×Q₈	—	10	1	no	—
213	#117	—	10	1	no	—
214	C₂×C₄²	—	5	1	no	—
215	C₄⋊C₈	—	5	1	no	—
216	C₄⋊C₈	—	5	1	no	—
217	C₄⋊C₈	—	10	1	no	—
218	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
219	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
220	C₂²⋊C₈	—	10	1	no	—
221	C₄×C₈	—	10	1	no	—
222	C₂×C₃×C₅	#413	1	1	yes	characteristic
223	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
224	C₂×C₃×C₅	#416	1	1	yes	characteristic
225	C₃×D₅	—	2	2	no	—
226	C₂³×C₃	—	5	1	no	—
227	C₃×Q₈	—	4	1	no	—
228	C₃×Q₈	—	4	1	no	—
229	C₃×Q₈	—	20	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	5	1	no	—
232	C₂×C₃×C₄	—	5	2	no	—
233	C₂×C₃×C₄	—	5	2	no	—
234	C₂×C₃×C₄	—	5	1	no	—
235	C₂×C₃×C₄	—	5	1	no	—
236	C₂×C₃×C₄	—	10	1	no	—
237	C₂×C₃×C₄	—	10	1	no	—
238	C₂×C₃×C₄	—	10	2	no	—
239	C₂×C₃×C₄	—	20	1	no	—
240	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
241	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
242	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
243	C₃×C₈	—	4	1	no	—
244	C₃×C₈	—	4	1	no	—
245	C₃×C₈	—	20	1	no	—
246	C₃×C₈	—	20	1	no	—
247	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×D₅	—	2	1	no	—
249	C₂×D₅	—	2	1	no	—
250	C₂×D₅	C₃×#103	1	2	yes	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
254	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
255	C₄×C₅	C₃×#135	1	1	yes	characteristic
256	C₄×C₅	C₃×#135	1	1	yes	characteristic
257	Dic₅	—	2	1	no	—
258	Dic₅	—	2	1	no	—
259	Dic₅	—	4	2	no	—
260	Dic₅	C₃×#103	1	2	yes	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	5	2	no	—
263	C₂²×C₄	—	5	1	no	—
264	C₂²×C₄	—	5	1	no	—
265	C₂²×C₄	—	5	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	10	1	no	—
269	C₂×C₈	—	10	1	no	—
270	C₂×C₈	—	20	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	10	2	no	—
273	C₄⋊C₄	—	10	2	no	—
274	C₄⋊C₄	—	10	1	no	—
275	C₄⋊C₄	—	10	1	no	—
276	C₄⋊C₄	C₃×C₄×D₅	1	2	yes	—
277	C₂²⋊C₄	—	10	2	no	—
278	C₄²	—	5	1	no	—
279	C₄²	—	5	1	no	—
280	C₄²	—	5	1	no	—
281	C₄²	—	10	2	no	—
282	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
283	C₃×C₅	#1542	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	—	5	2	no	—
285	C₂²×C₃	—	10	1	no	—
286	C₂²×C₃	—	10	1	no	—
287	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	5	2	no	—
291	C₃×C₄	—	10	1	no	—
292	C₃×C₄	—	10	1	no	—
293	C₃×C₄	—	20	2	no	—
294	C₃×C₄	#4058	1	1	yes	characteristic
295	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
296	C₃×C₄	#4062	1	1	yes	characteristic
297	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
298	C₂×C₅	C₃×#413	1	1	yes	characteristic
299	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
300	C₂×C₅	C₃×#416	1	1	yes	characteristic
301	D₅	—	2	2	no	—
302	C₂³	—	5	1	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	20	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	5	1	no	—
308	C₂×C₄	—	5	2	no	—
309	C₂×C₄	—	5	2	no	—
310	C₂×C₄	—	5	1	no	—
311	C₂×C₄	—	5	1	no	—
312	C₂×C₄	—	10	1	no	—
313	C₂×C₄	—	10	1	no	—
314	C₂×C₄	—	10	2	no	—
315	C₂×C₄	—	20	1	no	—
316	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
317	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
318	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
319	C₈	—	4	1	no	—
320	C₈	—	4	1	no	—
321	C₈	—	20	1	no	—
322	C₈	—	20	1	no	—
323	C₂×C₃	—	10	2	no	—
324	C₂×C₃	#65040	1	1	yes	characteristic
325	C₂×C₃	#65061	1	1	yes	characteristic
326	C₂×C₃	D₅×#103	1	1	yes	characteristic
327	C₅	C₃×#1542	1	1	yes	characteristic, complemented, Sylow
328	C₂²	—	5	2	no	—
329	C₂²	—	10	1	no	—
330	C₂²	—	10	1	no	—
331	C₂²	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	5	2	no	—
335	C₄	—	10	1	no	—
336	C₄	—	10	1	no	—
337	C₄	—	20	2	no	—
338	C₄	C₃×#4058	1	1	yes	characteristic
339	C₄	C₃×D₅×SD₁₆	1	1	yes	characteristic
340	C₄	C₃×#4062	1	1	yes	characteristic
341	C₄	C₃×D₅×Q₁₆	1	1	yes	characteristic
342	C₃	#10599211	1	1	yes	characteristic, direct product factor, Sylow
343	C₂	—	10	2	no	—
344	C₂	C₃×#65040	1	1	yes	characteristic
345	C₂	C₃×#65061	1	1	yes	characteristic
346	C₂	C₃×D₅×#103	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×C₂²⋊C₄
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#332	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂²⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#332	D₅
hypocenter	C₅	C₃×#1542
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#332

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#332	C₂	semidirect
5. C₃×#64767	C₂	semidirect
6. C₃×#64701	C₂	semidirect
7. C₃×#64701	C₂	semidirect
8. #10599211	C₃	direct (N × H)
30. C₅×#332	C₂×C₃	semidirect
31. #64767	C₂×C₃	semidirect
32. #64701	C₂×C₃	semidirect
33. #64701	C₂×C₃	semidirect
62. C₃×#332	D₅	semidirect
147. #332	C₃×D₅	semidirect
283. C₃×C₅	#1542	semidirect
327. C₅	C₃×#1542	semidirect
342. C₃	#10599211	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#66062	2
C₃×#65122	2
C₃×D₅×C₄⋊C₈	2
C₃×C₅×#332	2
C₃×#64767	2
C₃×#64701	2
#10599211	3
C₃×#1542	5

Subgroups — C3×#10599211 #14284655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599211 #14284655