Subgroups — C₃×#10615006 #14415233

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4094	C₂	1	2	yes	maximal, complemented
3	C₃×#65523	C₂	1	2	yes	maximal, complemented
4	C₃×#65301	C₂	1	4	yes	maximal, complemented
5	C₃×D₄×#168	C₂	1	4	yes	maximal, complemented
6	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615006	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₃×C₅×#127	C₄	1	1	yes	characteristic
12	C₃×C₅×#124	C₄	1	2	yes	—
13	C₃×C₅×#121	C₄	1	4	yes	—
14	C₃×C₄×C₅×D₄	C₂²	1	4	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×#4094	—	2	8	no	—
17	C₂×C₃×#681	C₂²	1	4	yes	—
18	C₂×C₃×#681	C₂²	1	2	yes	—
19	C₂²×C₃×#168	C₂²	1	4	yes	—
20	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
21	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
22	C₃×C₂²⋊#168	C₂²	1	8	yes	—
23	C₃×C₄⋊#168	C₂²	1	4	yes	—
24	C₃×C₄×#168	C₂²	1	2	yes	—
25	C₃×C₄×#168	C₂²	1	2	yes	—
26	C₃×#2931	—	5	1	no	maximal, Hall
27	C₅×#532	C₂×C₃	1	1	yes	characteristic
28	C₂×#4094	C₂×C₃	1	2	yes	complemented
29	#65523	C₂×C₃	1	2	yes	complemented
30	#65301	C₂×C₃	1	4	yes	complemented
31	D₄×#168	C₂×C₃	1	4	yes	complemented
32	#65213	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₅×#41	—	2	4	no	—
35	C₃×C₅×#41	—	2	4	no	—
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
41	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
45	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	2	yes	—
48	C₃×#681	—	2	4	no	—
49	C₃×#681	—	2	8	no	—
50	C₂×C₃×#168	—	2	2	no	—
51	C₂×C₃×#168	—	2	8	no	—
52	C₂×C₃×#168	D₄	1	4	yes	complemented
53	C₂×C₃×#168	C₂³	1	2	yes	—
54	C₂×C₃×#168	C₂³	1	4	yes	—
55	C₂×C₃×#168	C₂³	1	1	yes	characteristic
56	C₂×C₃×#168	C₂³	1	1	yes	characteristic
57	C₂×C₃×#131	—	5	2	no	—
58	C₃×#532	D₅	1	1	yes	characteristic
59	C₃×#436	—	5	4	no	—
60	C₃×C₈×D₄	—	5	4	no	—
61	C₃×#423	—	5	1	no	—
62	C₃×#408	—	5	2	no	—
63	C₂×C₃×C₄×C₈	—	5	1	no	—
64	C₂×C₅×#41	C₂²×C₃	1	2	yes	—
65	C₅×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
66	C₅×#127	C₃×C₄	1	1	yes	characteristic
67	C₅×#124	C₃×C₄	1	2	yes	—
68	C₅×#121	C₃×C₄	1	4	yes	—
69	C₄×C₅×D₄	C₂²×C₃	1	4	yes	—
70	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
71	#4094	—	2	8	no	—
72	C₂×#681	C₂²×C₃	1	4	yes	—
73	C₂×#681	C₂²×C₃	1	2	yes	—
74	C₂²×#168	C₂²×C₃	1	4	yes	—
75	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
76	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
77	C₂²⋊#168	C₂²×C₃	1	8	yes	—
78	C₄⋊#168	C₂²×C₃	1	4	yes	—
79	C₄×#168	C₂²×C₃	1	2	yes	—
80	C₄×#168	C₂²×C₃	1	2	yes	—
81	#2931	—	5	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
83	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₅×Q₈	—	2	4	no	—
85	C₃×C₅×D₄	—	2	8	no	—
86	C₃×C₅×D₄	—	2	4	no	—
87	C₃×C₅×D₄	—	2	8	no	—
88	C₂×C₃×C₄×C₅	—	2	4	no	—
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	—	2	4	no	—
91	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
93	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	#41	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₃×#168	—	2	4	no	—
99	C₃×#168	—	2	2	no	—
100	C₃×#168	C₂×D₄	1	2	yes	—
101	C₃×#168	C₂×D₄	1	2	yes	—
102	C₂×C₃×#41	C₂×D₅	1	2	yes	—
103	C₃×#131	—	10	8	no	—
104	C₂×C₃×M₁₆	—	5	2	no	—
105	C₂×C₃×M₁₆	—	5	4	no	—
106	C₂²×C₃×C₈	—	5	1	no	—
107	C₂²×C₃×C₈	—	5	4	no	—
108	C₂²×C₃×C₈	—	5	1	no	—
109	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
110	C₃×#127	Dic₅	1	1	yes	characteristic
111	C₃×#124	Dic₅	1	2	yes	—
112	C₃×#121	Dic₅	1	4	yes	—
113	C₃×C₄×D₄	C₂×D₅	1	4	yes	—
114	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
115	C₃×C₄⋊C₈	—	5	4	no	—
116	C₃×C₂²⋊C₈	—	5	8	no	—
117	C₃×C₄×C₈	—	5	2	no	—
118	C₃×C₄×C₈	—	5	2	no	—
119	C₅×#41	—	2	4	no	—
120	C₅×#41	—	2	4	no	—
121	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
122	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
123	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
124	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
125	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
126	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
127	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
128	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
129	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
130	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
131	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
132	C₄²×C₅	C₂³×C₃	1	2	yes	—
133	#681	—	2	4	no	—
134	#681	—	2	8	no	—
135	C₂×#168	—	2	2	no	—
136	C₂×#168	—	2	8	no	—
137	C₂×#168	C₃×D₄	1	4	yes	complemented
138	C₂×#168	C₂³×C₃	1	2	yes	—
139	C₂×#168	C₂³×C₃	1	4	yes	—
140	C₂×#168	C₂³×C₃	1	1	yes	characteristic
141	C₂×#168	C₂³×C₃	1	1	yes	characteristic
142	C₂×#131	—	5	2	no	—
143	#532	C₃×D₅	1	1	yes	characteristic
144	#436	—	5	4	no	—
145	C₈×D₄	—	5	4	no	—
146	#423	—	5	1	no	—
147	#408	—	5	2	no	—
148	C₂×C₄×C₈	—	5	1	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	—	2	4	no	—
151	C₂²×C₃×C₅	—	2	8	no	—
152	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
153	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
154	C₃×C₄×C₅	—	2	4	no	—
155	C₃×C₄×C₅	—	2	2	no	—
156	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₃×C₄×C₅	#131	1	4	yes	complemented
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
160	C₃×#41	—	2	4	no	—
161	C₃×#41	—	2	4	no	—
162	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
163	C₂×C₃×D₄	C₂×Dic₅	1	4	yes	—
164	C₂×C₃×D₄	C₂×Dic₅	1	4	yes	—
165	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
166	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
167	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₃×M₁₆	—	10	8	no	—
170	C₃×M₁₆	—	10	4	no	—
171	C₂×C₃×C₈	—	5	4	no	—
172	C₂×C₃×C₈	—	5	1	no	—
173	C₂×C₃×C₈	—	5	1	no	—
174	C₂×C₃×C₈	—	5	2	no	—
175	C₂×C₃×C₈	—	5	4	no	—
176	C₂×C₃×C₈	—	10	8	no	—
177	C₂×C₃×C₈	—	10	2	no	—
178	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
179	C₃×C₂²⋊C₄	C₂×Dic₅	1	8	yes	—
180	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
181	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
182	C₃×C₄²	C₂²×D₅	1	2	yes	—
183	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
184	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₅×Q₈	—	2	4	no	—
186	C₅×D₄	—	2	8	no	—
187	C₅×D₄	—	2	4	no	—
188	C₅×D₄	—	2	8	no	—
189	C₂×C₄×C₅	—	2	4	no	—
190	C₂×C₄×C₅	—	2	4	no	—
191	C₂×C₄×C₅	—	2	4	no	—
192	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
193	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
194	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
196	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₄×C₅	C₃×#41	1	2	yes	—
198	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
199	#168	—	2	4	no	—
200	#168	—	2	2	no	—
201	#168	C₂×C₃×D₄	1	2	yes	—
202	#168	C₂×C₃×D₄	1	2	yes	—
203	C₂×#41	C₂×C₃×D₅	1	2	yes	—
204	#131	—	10	8	no	—
205	C₂×M₁₆	—	5	2	no	—
206	C₂×M₁₆	—	5	4	no	—
207	C₂²×C₈	—	5	1	no	—
208	C₂²×C₈	—	5	4	no	—
209	C₂²×C₈	—	5	1	no	—
210	C₄⋊Q₈	C₃×Dic₅	1	1	yes	characteristic
211	#127	C₃×Dic₅	1	1	yes	characteristic
212	#124	C₃×Dic₅	1	2	yes	—
213	#121	C₃×Dic₅	1	4	yes	—
214	C₄×D₄	C₂×C₃×D₅	1	4	yes	—
215	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
216	C₄⋊C₈	—	5	4	no	—
217	C₂²⋊C₈	—	5	8	no	—
218	C₄×C₈	—	5	2	no	—
219	C₄×C₈	—	5	2	no	—
220	C₂×C₃×C₅	—	2	2	no	—
221	C₂×C₃×C₅	—	4	4	no	—
222	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂×C₃×C₅	C₂×#131	1	2	yes	—
224	C₂³×C₃	C₂²×Dic₅	1	4	yes	—
225	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
226	C₃×Q₈	—	2	4	no	—
227	C₃×D₄	—	2	8	no	—
228	C₃×D₄	—	2	4	no	—
229	C₃×D₄	—	2	8	no	—
230	C₂×C₃×C₄	—	2	4	no	—
231	C₂×C₃×C₄	—	2	4	no	—
232	C₂×C₃×C₄	—	2	4	no	—
233	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
234	C₂×C₃×C₄	#711	1	2	yes	—
235	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
236	C₂×C₃×C₄	C₂²×Dic₅	1	4	yes	—
237	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
239	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
240	C₃×C₈	—	5	2	no	—
241	C₃×C₈	—	5	2	no	—
242	C₃×C₈	—	10	4	no	—
243	C₃×C₈	—	10	2	no	—
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	—	2	8	no	—
246	C₂²×C₅	—	2	4	no	—
247	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
248	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
249	C₄×C₅	—	2	2	no	—
250	C₄×C₅	—	2	4	no	—
251	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
252	C₄×C₅	C₃×#131	1	4	yes	complemented
253	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
254	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
255	#41	—	2	4	no	—
256	#41	—	2	4	no	—
257	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
258	C₂×D₄	C₂×C₃×Dic₅	1	4	yes	—
259	C₂×D₄	C₂×C₃×Dic₅	1	4	yes	—
260	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
261	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
262	C₂²×C₄	C₂²×C₃×D₅	1	4	yes	—
263	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
264	M₁₆	—	10	8	no	—
265	M₁₆	—	10	4	no	—
266	C₂×C₈	—	5	4	no	—
267	C₂×C₈	—	5	1	no	—
268	C₂×C₈	—	5	1	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	4	no	—
271	C₂×C₈	—	10	8	no	—
272	C₂×C₈	—	10	2	no	—
273	C₄⋊C₄	C₂×C₃×Dic₅	1	4	yes	—
274	C₂²⋊C₄	C₂×C₃×Dic₅	1	8	yes	—
275	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
276	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
277	C₄²	C₂²×C₃×D₅	1	2	yes	—
278	C₃×C₅	#2931	1	1	yes	characteristic, complemented, Hall
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	—	2	4	no	—
281	C₂²×C₃	—	2	8	no	—
282	C₂²×C₃	D₄×Dic₅	1	2	yes	—
283	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	D₄×Dic₅	1	2	yes	—
287	C₃×C₄	#4094	1	4	yes	complemented
288	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
289	C₃×C₄	C₂×#711	1	1	yes	characteristic
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	—	4	4	no	—
292	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
293	C₂×C₅	C₂×C₃×#131	1	2	yes	—
294	C₂³	C₂²×C₃×Dic₅	1	4	yes	—
295	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	D₄	—	2	8	no	—
298	D₄	—	2	4	no	—
299	D₄	—	2	8	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
304	C₂×C₄	C₃×#711	1	2	yes	—
305	C₂×C₄	C₂²×C₃×Dic₅	1	2	yes	—
306	C₂×C₄	C₂²×C₃×Dic₅	1	4	yes	—
307	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₃×Dic₅	1	2	yes	—
309	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
310	C₈	—	5	2	no	—
311	C₈	—	5	2	no	—
312	C₈	—	10	4	no	—
313	C₈	—	10	2	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	4	4	no	—
316	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
317	C₂×C₃	C₂×#4094	1	2	yes	—
318	C₅	C₃×#2931	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	2	8	no	—
322	C₂²	C₃×D₄×Dic₅	1	2	yes	—
323	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	C₃×D₄×Dic₅	1	2	yes	—
327	C₄	C₃×#4094	1	4	yes	complemented
328	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
329	C₄	C₂×C₃×#711	1	1	yes	characteristic
330	C₃	#10615006	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	4	4	no	—
333	C₂	C₂×C₃×D₄×Dic₅	1	1	yes	characteristic
334	C₂	C₂×C₃×#4094	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#532	D₅
hypocenter	C₅	C₃×#2931
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#532

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4094	C₂	semidirect
3. C₃×#65523	C₂	semidirect
4. C₃×#65301	C₂	semidirect
5. C₃×D₄×#168	C₂	semidirect
6. C₃×#65213	C₂	semidirect
7. C₂×C₃×C₄×#168	C₂	semidirect
8. #10615006	C₃	direct (N × H)
28. C₂×#4094	C₂×C₃	semidirect
29. #65523	C₂×C₃	semidirect
30. #65301	C₂×C₃	semidirect
31. D₄×#168	C₂×C₃	semidirect
32. #65213	C₂×C₃	semidirect
33. C₂×C₄×#168	C₂×C₃	semidirect
52. C₂×C₃×#168	D₄	semidirect
137. C₂×#168	C₃×D₄	semidirect
157. C₃×C₄×C₅	#131	semidirect
252. C₄×C₅	C₃×#131	semidirect
278. C₃×C₅	#2931	semidirect
287. C₃×C₄	#4094	semidirect
318. C₅	C₃×#2931	semidirect
327. C₄	C₃×#4094	semidirect
330. C₃	#10615006	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×C₅×#532	2
C₂×C₃×#4094	2
C₃×#65523	2
C₃×#65301	2
C₃×D₄×#168	2
C₃×#65213	2
C₂×C₃×C₄×#168	2
#10615006	3
C₃×#2931	5

Subgroups — C3×#10615006 #14415233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615006 #14415233