Subgroups — C₃×#10602600 #14285025

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 340 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	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64766	C₂	1	2	yes	maximal, complemented
5	C₃×#64764	C₂	1	2	yes	maximal, complemented
6	#10602600	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₄	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	C₄	1	2	yes	—
11	C₃×C₅×#121	—	2	1	no	—
12	C₃×C₅×#121	—	2	1	no	—
13	C₃×C₄×C₅×D₄	—	2	1	no	—
14	C₃×C₄×C₅×D₄	—	2	1	no	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂²×C₃×#168	—	2	1	no	—
17	C₃×C₂²⋊#168	—	2	1	no	—
18	C₃×C₄⋊#168	—	2	1	no	—
19	C₃×C₄⋊#168	C₂²	1	2	yes	—
20	C₃×C₄⋊#168	C₂²	1	2	yes	—
21	C₃×#3935	—	2	1	no	—
22	C₃×#1551	—	5	1	no	maximal, Hall
23	C₅×#532	C₂×C₃	1	1	yes	characteristic
24	#65215	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₄⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
26	#64766	C₂×C₃	1	2	yes	complemented
27	#64764	C₂×C₃	1	2	yes	complemented
28	C₃×C₅×#41	—	4	4	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×#168	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×#168	—	4	2	no	—
49	C₃×#532	D₅	1	1	yes	characteristic
50	C₃×#432	—	5	1	no	—
51	C₂×C₃×C₄⋊C₈	—	5	1	no	—
52	C₃×#333	—	5	2	no	—
53	C₃×#331	—	5	2	no	—
54	C₂×C₅×#41	C₃×C₄	1	2	yes	—
55	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₅×#127	C₂²×C₃	1	1	yes	characteristic
57	C₅×#124	C₃×C₄	1	2	yes	—
58	C₅×#121	—	2	1	no	—
59	C₅×#121	—	2	1	no	—
60	C₄×C₅×D₄	—	2	1	no	—
61	C₄×C₅×D₄	—	2	1	no	—
62	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
63	C₂²×#168	—	2	1	no	—
64	C₂²⋊#168	—	2	1	no	—
65	C₄⋊#168	—	2	1	no	—
66	C₄⋊#168	C₂²×C₃	1	2	yes	—
67	C₄⋊#168	C₂²×C₃	1	2	yes	—
68	#3935	—	2	1	no	—
69	#1551	—	5	1	no	Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	4	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₃×C₅×D₄	—	4	1	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	4	4	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₃×#168	—	4	2	no	—
91	C₃×#168	—	4	2	no	—
92	C₂×C₃×#41	Dic₅	1	2	yes	—
93	C₂²×C₃×C₈	—	10	1	no	—
94	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
95	C₃×#127	C₂×D₅	1	1	yes	characteristic
96	C₃×#124	Dic₅	1	2	yes	—
97	C₃×#121	—	2	1	no	—
98	C₃×#121	—	2	1	no	—
99	C₃×C₄×D₄	—	2	1	no	—
100	C₃×C₄×D₄	—	2	1	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₃×C₄⋊C₈	—	5	2	no	—
103	C₃×C₄⋊C₈	—	5	2	no	—
104	C₃×C₄⋊C₈	—	10	1	no	—
105	C₃×C₂²⋊C₈	—	10	1	no	—
106	C₃×#97	—	10	1	no	—
107	C₅×#41	—	4	4	no	—
108	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	—	2	1	no	—
112	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
113	C₂²×C₄×C₅	—	2	2	no	—
114	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₂²⋊C₄	—	2	2	no	—
120	C₅×C₂²⋊C₄	—	2	2	no	—
121	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
122	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
123	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
124	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
125	C₂×#168	—	2	2	no	—
126	C₂×#168	—	2	2	no	—
127	C₂×#168	—	4	2	no	—
128	#532	C₃×D₅	1	1	yes	characteristic
129	#432	—	5	1	no	—
130	C₂×C₄⋊C₈	—	5	1	no	—
131	#333	—	5	2	no	—
132	#331	—	5	2	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	—	4	2	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	#100	1	2	yes	—
139	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	4	2	no	—
145	C₃×C₄×C₅	#135	1	2	yes	—
146	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₃×C₄×C₅	#137	1	1	yes	characteristic
148	C₃×#41	—	4	4	no	—
149	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	—	2	1	no	—
153	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	—	10	2	no	—
159	C₂×C₃×C₈	—	10	2	no	—
160	C₂×C₃×C₈	—	20	2	no	—
161	C₃×C₄⋊C₄	—	2	1	no	—
162	C₃×C₄⋊C₄	—	2	1	no	—
163	C₃×C₂²⋊C₄	—	2	2	no	—
164	C₃×C₂²⋊C₄	—	2	2	no	—
165	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
167	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
169	C₂³×C₅	—	2	2	no	—
170	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
171	C₅×Q₈	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	4	no	—
174	C₅×D₄	—	4	2	no	—
175	C₅×D₄	—	4	1	no	—
176	C₅×D₄	—	4	1	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	4	no	—
183	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
189	#168	—	4	2	no	—
190	#168	—	4	2	no	—
191	C₂×#41	C₃×Dic₅	1	2	yes	—
192	C₂²×C₈	—	10	1	no	—
193	C₄⋊Q₈	C₂×C₃×D₅	1	1	yes	characteristic
194	#127	C₂×C₃×D₅	1	1	yes	characteristic
195	#124	C₃×Dic₅	1	2	yes	—
196	#121	—	2	1	no	—
197	#121	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
201	C₄⋊C₈	—	5	2	no	—
202	C₄⋊C₈	—	5	2	no	—
203	C₄⋊C₈	—	10	1	no	—
204	C₂²⋊C₈	—	10	1	no	—
205	#97	—	10	1	no	—
206	C₂×C₃×C₅	—	2	2	no	—
207	C₂×C₃×C₅	—	8	2	no	—
208	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
209	C₂×C₃×C₅	#416	1	1	yes	characteristic
210	C₂×C₃×C₅	#417	1	1	yes	characteristic
211	C₂³×C₃	—	2	2	no	—
212	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₃×Q₈	—	4	2	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	4	no	—
216	C₃×D₄	—	4	2	no	—
217	C₃×D₄	—	4	1	no	—
218	C₃×D₄	—	4	1	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	4	4	no	—
225	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₃×C₈	—	20	2	no	—
232	C₃×C₈	—	20	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	4	2	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	—	4	2	no	—
238	C₂²×C₅	C₃×#100	1	2	yes	—
239	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	4	2	no	—
245	C₄×C₅	C₃×#135	1	2	yes	—
246	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₄×C₅	C₃×#137	1	1	yes	characteristic
248	#41	—	4	4	no	—
249	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	C₂×C₃×Dic₅	1	2	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
256	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
257	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	10	2	no	—
260	C₂×C₈	—	20	2	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₄²	C₂×C₃×Dic₅	1	1	yes	characteristic
266	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
267	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
268	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
269	C₃×C₅	#1551	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	4	2	no	—
274	C₂²×C₃	—	4	2	no	—
275	C₂²×C₃	#3965	1	2	yes	—
276	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	#4095	1	1	yes	characteristic
283	C₃×C₄	#4096	1	2	yes	—
284	C₃×C₄	#4097	1	1	yes	characteristic
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	8	2	no	—
287	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
288	C₂×C₅	C₃×#416	1	1	yes	characteristic
289	C₂×C₅	C₃×#417	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
305	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
306	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
309	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
310	C₈	—	20	2	no	—
311	C₈	—	20	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	8	2	no	—
314	C₂×C₃	C₂×#3965	1	1	yes	characteristic
315	C₂×C₃	#65521	1	1	yes	characteristic
316	C₂×C₃	#65522	1	1	yes	characteristic
317	C₅	C₃×#1551	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₃×#3965	1	2	yes	—
324	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	C₃×#4095	1	1	yes	characteristic
331	C₄	C₃×#4096	1	2	yes	—
332	C₄	C₃×#4097	1	1	yes	characteristic
333	C₃	#10602600	1	1	yes	characteristic, direct product factor, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	8	2	no	—
336	C₂	C₂×C₃×#3965	1	1	yes	characteristic
337	C₂	C₃×#65521	1	1	yes	characteristic
338	C₂	C₃×#65522	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊Dic₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#532	D₅
hypocenter	C₅	C₃×#1551
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#532

Complements

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

N	H ≅ G/N	type
2. C₃×#65215	C₂	semidirect
3. C₂×C₃×C₄⋊#168	C₂	semidirect
4. C₃×#64766	C₂	semidirect
5. C₃×#64764	C₂	semidirect
6. #10602600	C₃	direct (N × H)
24. #65215	C₂×C₃	semidirect
25. C₂×C₄⋊#168	C₂×C₃	semidirect
26. #64766	C₂×C₃	semidirect
27. #64764	C₂×C₃	semidirect
269. C₃×C₅	#1551	semidirect
317. C₅	C₃×#1551	semidirect
333. C₃	#10602600	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
C₃×#65215	2
C₂×C₃×C₄⋊#168	2
C₃×#64766	2
C₃×#64764	2
#10602600	3
C₃×#1551	5

Subgroups — C3×#10602600 #14285025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602600 #14285025