Subgroups — C₃×#10599456 #14297471

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4018	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3952	C₂	1	2	yes	maximal
3	C₂×C₃×#3946	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64744	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599456	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×#685	C₄	1	2	yes	—
11	C₂×C₃×#683	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	C₃×#4018	—	2	4	no	—
15	C₃×#3952	—	2	4	no	—
16	C₃×#3946	—	2	4	no	—
17	C₃×#1798	—	5	1	no	maximal, Hall
18	C₂×#4018	C₂×C₃	1	1	yes	characteristic
19	C₂×#3952	C₂×C₃	1	2	yes	—
20	C₂×#3946	C₂×C₃	1	2	yes	complemented
21	C₅×#336	C₂×C₃	1	1	yes	characteristic, complemented
22	#64744	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×D₅	—	2	2	no	—
29	C₂×C₃×C₄×D₅	—	2	2	no	—
30	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
31	C₂×C₃×C₄×D₅	Q₈	1	2	yes	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₄²×C₅	—	2	2	no	—
36	C₃×#685	—	2	4	no	—
37	C₃×#683	—	2	4	no	—
38	C₃×C₄×Dic₅	D₄	1	4	yes	—
39	C₂×C₃×#168	—	2	2	no	—
40	C₂×C₃×#168	—	2	2	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#117	—	5	1	no	—
43	C₂×C₃×C₄⋊C₈	—	5	2	no	—
44	C₂×C₃×C₂²⋊C₈	—	5	2	no	—
45	C₃×#336	—	5	1	no	—
46	C₃×#336	D₅	1	1	yes	characteristic, complemented
47	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
49	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×#685	C₃×C₄	1	2	yes	—
51	C₂×#683	C₃×C₄	1	2	yes	—
52	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
53	C₂²×#168	C₂²×C₃	1	2	yes	—
54	#4018	—	2	4	no	—
55	#3952	—	2	4	no	—
56	#3946	—	2	4	no	—
57	#1798	—	5	1	no	Sylow
58	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₃×D₅	—	2	1	no	—
60	C₂²×C₃×D₅	—	2	1	no	—
61	C₂²×C₃×D₅	—	2	2	no	—
62	C₂²×C₃×D₅	—	2	2	no	—
63	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₂×C₃×Dic₅	—	2	1	no	—
72	C₂×C₃×Dic₅	—	2	1	no	—
73	C₂×C₃×Dic₅	—	2	2	no	—
74	C₂×C₃×Dic₅	—	2	2	no	—
75	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
76	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
77	C₃×C₄×D₅	—	2	4	no	—
78	C₃×C₄×D₅	—	2	4	no	—
79	C₃×C₅×C₈	—	4	2	no	—
80	C₃×#168	—	4	2	no	—
81	C₂³×C₃×C₄	—	5	1	no	—
82	C₂²×C₃×C₈	—	5	2	no	—
83	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
84	C₃×#117	—	10	4	no	—
85	C₂×C₃×C₄⋊C₄	—	5	2	no	—
86	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
87	C₂×C₃×C₄²	—	5	1	no	—
88	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
89	C₃×C₄⋊C₈	—	10	4	no	—
90	C₃×C₂²⋊C₈	—	10	4	no	—
91	C₂³×D₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
95	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₄×D₅	—	2	2	no	—
97	C₂×C₄×D₅	—	2	2	no	—
98	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
99	C₂×C₄×D₅	C₃×Q₈	1	2	yes	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₄²×C₅	—	2	2	no	—
104	#685	—	2	4	no	—
105	#683	—	2	4	no	—
106	C₄×Dic₅	C₃×D₄	1	4	yes	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#117	—	5	1	no	—
111	C₂×C₄⋊C₈	—	5	2	no	—
112	C₂×C₂²⋊C₈	—	5	2	no	—
113	#336	—	5	1	no	—
114	#336	C₃×D₅	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#117	1	1	yes	characteristic
118	C₂²×C₃×C₅	#131	1	2	yes	—
119	C₂²×C₃×C₅	#131	1	2	yes	—
120	C₂×C₃×D₅	—	2	2	no	—
121	C₂×C₃×D₅	—	2	4	no	—
122	C₂×C₃×D₅	—	4	2	no	—
123	C₂×C₃×D₅	—	4	1	no	—
124	C₂×C₃×D₅	—	4	1	no	—
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	#121	1	2	yes	—
127	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
128	C₃×Dic₅	—	2	2	no	—
129	C₃×Dic₅	—	2	4	no	—
130	C₂⁴×C₃	—	5	1	no	—
131	C₂²×C₃×C₄	—	5	2	no	—
132	C₂²×C₃×C₄	—	5	1	no	—
133	C₂²×C₃×C₄	—	5	2	no	—
134	C₂²×C₃×C₄	—	5	2	no	—
135	C₂²×C₃×C₄	—	10	2	no	—
136	C₂²×C₃×C₄	—	10	2	no	—
137	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
138	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	—	10	2	no	—
143	C₂×C₃×C₈	—	10	2	no	—
144	C₂×C₃×C₈	—	10	2	no	—
145	C₃×C₄⋊C₄	—	10	4	no	—
146	C₃×C₂²⋊C₄	—	10	4	no	—
147	C₃×C₄²	—	2	2	no	—
148	C₃×C₄²	—	5	4	no	—
149	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₂²×D₅	—	2	1	no	—
151	C₂²×D₅	—	2	1	no	—
152	C₂²×D₅	—	2	2	no	—
153	C₂²×D₅	—	2	2	no	—
154	C₂²×D₅	C₃×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
160	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅	C₃×#41	1	2	yes	—
162	C₂×Dic₅	—	2	1	no	—
163	C₂×Dic₅	—	2	1	no	—
164	C₂×Dic₅	—	2	2	no	—
165	C₂×Dic₅	—	2	2	no	—
166	C₂×Dic₅	C₃×C₂²⋊C₄	1	4	yes	—
167	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
168	C₄×D₅	—	2	4	no	—
169	C₄×D₅	—	2	4	no	—
170	C₅×C₈	—	4	2	no	—
171	#168	—	4	2	no	—
172	C₂³×C₄	—	5	1	no	—
173	C₂²×C₈	—	5	2	no	—
174	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
175	#117	—	10	4	no	—
176	C₂×C₄⋊C₄	—	5	2	no	—
177	C₂×C₂²⋊C₄	—	5	2	no	—
178	C₂×C₄²	—	5	1	no	—
179	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
180	C₄⋊C₈	—	10	4	no	—
181	C₂²⋊C₈	—	10	4	no	—
182	C₂×C₃×C₅	#380	1	1	yes	characteristic
183	C₂×C₃×C₅	#408	1	2	yes	—
184	C₂×C₃×C₅	#424	1	2	yes	—
185	C₂×C₃×C₅	#433	1	1	yes	characteristic
186	C₂×C₃×C₅	#433	1	1	yes	characteristic
187	C₃×D₅	—	4	2	no	—
188	C₂³×C₃	—	5	2	no	—
189	C₂³×C₃	—	10	1	no	—
190	C₂³×C₃	—	10	1	no	—
191	C₂³×C₃	—	10	2	no	—
192	C₂³×C₃	—	10	2	no	—
193	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	5	4	no	—
198	C₂×C₃×C₄	—	5	2	no	—
199	C₂×C₃×C₄	—	10	1	no	—
200	C₂×C₃×C₄	—	10	1	no	—
201	C₂×C₃×C₄	—	10	2	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	—	10	4	no	—
204	C₂×C₃×C₄	—	10	4	no	—
205	C₂×C₃×C₄	#709	1	2	yes	—
206	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
207	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
208	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
209	C₃×C₈	—	4	2	no	—
210	C₃×C₈	—	20	2	no	—
211	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₅	C₃×#117	1	1	yes	characteristic
214	C₂²×C₅	C₃×#131	1	2	yes	—
215	C₂²×C₅	C₃×#131	1	2	yes	—
216	C₂×D₅	—	2	2	no	—
217	C₂×D₅	—	2	4	no	—
218	C₂×D₅	—	4	2	no	—
219	C₂×D₅	—	4	1	no	—
220	C₂×D₅	—	4	1	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	C₃×#121	1	2	yes	—
223	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	4	no	—
226	C₂⁴	—	5	1	no	—
227	C₂²×C₄	—	5	2	no	—
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	5	2	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	—	10	2	no	—
232	C₂²×C₄	—	10	2	no	—
233	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
234	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	10	2	no	—
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₄⋊C₄	—	10	4	no	—
242	C₂²⋊C₄	—	10	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	5	4	no	—
245	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
246	C₂²×C₃	—	10	4	no	—
247	C₂²×C₃	—	10	2	no	—
248	C₂²×C₃	—	20	1	no	—
249	C₂²×C₃	—	20	1	no	—
250	C₂²×C₃	—	20	2	no	—
251	C₂²×C₃	#4018	1	1	yes	characteristic
252	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂²×C₃	#4047	1	2	yes	—
255	C₂²×C₃	#4053	1	2	yes	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	—	10	2	no	—
258	C₃×C₄	—	10	4	no	—
259	C₃×C₄	#4030	1	2	yes	—
260	C₃×C₄	#4042	1	2	yes	—
261	C₂×C₅	C₃×#380	1	1	yes	characteristic
262	C₂×C₅	C₃×#408	1	2	yes	—
263	C₂×C₅	C₃×#424	1	2	yes	—
264	C₂×C₅	C₃×#433	1	1	yes	characteristic
265	C₂×C₅	C₃×#433	1	1	yes	characteristic
266	D₅	—	4	2	no	—
267	C₂³	—	5	2	no	—
268	C₂³	—	10	1	no	—
269	C₂³	—	10	1	no	—
270	C₂³	—	10	2	no	—
271	C₂³	—	10	2	no	—
272	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	5	4	no	—
277	C₂×C₄	—	5	2	no	—
278	C₂×C₄	—	10	1	no	—
279	C₂×C₄	—	10	1	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	4	no	—
283	C₂×C₄	—	10	4	no	—
284	C₂×C₄	C₃×#709	1	2	yes	—
285	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
286	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
287	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
288	C₈	—	4	2	no	—
289	C₈	—	20	2	no	—
290	C₂×C₃	—	20	2	no	—
291	C₂×C₃	#64956	1	1	yes	characteristic
292	C₂×C₃	#64979	1	1	yes	characteristic
293	C₂×C₃	#64998	1	1	yes	characteristic
294	C₂×C₃	#65016	1	2	yes	—
295	C₂×C₃	#65128	1	2	yes	—
296	C₅	C₃×#1798	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	10	4	no	—
298	C₂²	—	10	2	no	—
299	C₂²	—	20	1	no	—
300	C₂²	—	20	1	no	—
301	C₂²	—	20	2	no	—
302	C₂²	C₃×#4018	1	1	yes	characteristic
303	C₂²	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
304	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
305	C₂²	C₃×#4047	1	2	yes	—
306	C₂²	C₃×#4053	1	2	yes	—
307	C₄	—	4	2	no	—
308	C₄	—	10	2	no	—
309	C₄	—	10	4	no	—
310	C₄	C₃×#4030	1	2	yes	—
311	C₄	C₃×#4042	1	2	yes	—
312	C₃	#10599456	1	1	yes	characteristic, direct product factor, Sylow
313	C₂	—	20	2	no	—
314	C₂	C₃×#64956	1	1	yes	characteristic
315	C₂	C₃×#64979	1	1	yes	characteristic
316	C₂	C₃×#64998	1	1	yes	characteristic
317	C₂	C₃×#65016	1	2	yes	—
318	C₂	C₃×#65128	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂³×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#336	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#336	D₅
hypocenter	C₅	C₃×#1798
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×#336

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#3946	C₂	semidirect
4. C₃×C₅×#336	C₂	semidirect
5. C₃×#64744	C₂	semidirect
6. #10599456	C₃	direct (N × H)
20. C₂×#3946	C₂×C₃	semidirect
21. C₅×#336	C₂×C₃	semidirect
22. #64744	C₂×C₃	semidirect
46. C₃×#336	D₅	semidirect
114. #336	C₃×D₅	semidirect
245. C₃×C₅	#1798	semidirect
296. C₅	C₃×#1798	semidirect
312. C₃	#10599456	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×#4018	2
C₂×C₃×#3952	2
C₂×C₃×#3946	2
C₃×C₅×#336	2
C₃×#64744	2
#10599456	3
C₃×#1798	5

Subgroups — C3×#10599456 #14297471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599456 #14297471