Subgroups — C₃×#10599661 #14297874

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#65276	C₂	1	2	yes	maximal
3	C₂×C₃×#3940	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#340	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64771	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599661	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	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×#685	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×D₅	—	2	4	no	—
14	C₂×C₃×C₈×D₅	C₄	1	4	yes	complemented
15	C₃×#3963	—	2	2	no	—
16	C₃×#3940	—	2	4	no	—
17	C₃×#1807	—	5	1	no	maximal, Hall
18	C₂²×C₈×D₅	C₂×C₃	1	1	yes	characteristic
19	#65276	C₂×C₃	1	2	yes	—
20	C₂×#3940	C₂×C₃	1	2	yes	complemented
21	C₅×#340	C₂×C₃	1	1	yes	characteristic, complemented
22	#64771	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂³×C₃×D₅	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	—	2	2	no	—
27	C₂²×C₃×Dic₅	D₄	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	C₂×C₄	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	C₂×C₄	1	2	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×#685	—	4	4	no	—
37	C₃×C₄⋊Dic₅	—	2	2	no	—
38	C₃×C₄⋊Dic₅	—	2	2	no	—
39	C₂×C₃×#168	C₂×C₄	1	2	yes	—
40	C₂×C₃×#168	D₄	1	4	yes	—
41	C₃×C₈×D₅	—	2	8	no	—
42	C₂³×C₃×C₈	—	5	1	no	—
43	C₂×C₃×#106	—	5	2	no	—
44	C₃×#380	—	5	2	no	—
45	C₃×#340	—	5	1	no	—
46	C₃×#340	D₅	1	1	yes	characteristic, complemented
47	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×#685	—	2	2	no	—
51	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
52	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₈×D₅	—	2	4	no	—
54	C₂×C₈×D₅	C₃×C₄	1	4	yes	complemented
55	#3963	—	2	2	no	—
56	#3940	—	2	4	no	—
57	#1807	—	5	1	no	Sylow
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₃×D₅	—	2	2	no	—
60	C₂²×C₃×D₅	—	2	1	no	—
61	C₂²×C₃×D₅	—	2	1	no	—
62	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
63	C₂²×C₃×D₅	SD₁₆	1	4	yes	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	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	2	no	—
72	C₂×C₃×Dic₅	—	4	2	no	—
73	C₂×C₃×Dic₅	—	4	2	no	—
74	C₂×C₃×Dic₅	—	4	2	no	—
75	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
76	C₃×C₄×D₅	—	2	2	no	—
77	C₃×C₄×D₅	—	2	2	no	—
78	C₃×C₄×D₅	—	2	4	no	—
79	C₃×C₅×C₈	—	2	2	no	—
80	C₃×#168	C₂²⋊C₄	1	4	yes	complemented
81	C₂³×C₃×C₄	—	5	1	no	—
82	C₂²×C₃×C₈	—	5	1	no	—
83	C₂²×C₃×C₈	—	5	4	no	—
84	C₂²×C₃×C₈	—	10	4	no	—
85	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
86	C₂×C₃×C₄⋊C₄	—	5	2	no	—
87	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
88	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
89	C₃×#106	—	10	4	no	—
90	C₃×#95	—	10	2	no	—
91	C₂³×D₅	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	2	2	no	—
95	C₂²×Dic₅	C₃×D₄	1	1	yes	characteristic
96	C₂×C₄×D₅	—	2	2	no	—
97	C₂×C₄×D₅	—	2	2	no	—
98	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₄×D₅	C₃×Q₈	1	2	yes	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	#685	—	4	4	no	—
105	C₄⋊Dic₅	—	2	2	no	—
106	C₄⋊Dic₅	—	2	2	no	—
107	C₂×#168	C₂×C₃×C₄	1	2	yes	—
108	C₂×#168	C₃×D₄	1	4	yes	—
109	C₈×D₅	—	2	8	no	—
110	C₂³×C₈	—	5	1	no	—
111	C₂×#106	—	5	2	no	—
112	#380	—	5	2	no	—
113	#340	—	5	1	no	—
114	#340	C₃×D₅	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	#121	1	2	yes	—
117	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
118	C₂²×C₃×C₅	#135	1	1	yes	characteristic
119	C₂²×C₃×C₅	#135	1	1	yes	characteristic
120	C₂×C₃×D₅	—	2	2	no	—
121	C₂×C₃×D₅	—	2	2	no	—
122	C₂×C₃×D₅	—	2	4	no	—
123	C₂×C₃×D₅	—	2	4	no	—
124	C₂×C₃×D₅	#106	1	4	yes	complemented
125	C₃×C₄×C₅	—	8	2	no	—
126	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₃×C₄×C₅	#117	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
129	C₃×Dic₅	—	2	2	no	—
130	C₃×Dic₅	—	8	2	no	—
131	C₂⁴×C₃	—	5	1	no	—
132	C₂²×C₃×C₄	—	2	2	no	—
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₂²×C₃×C₄	—	5	2	no	—
135	C₂²×C₃×C₄	—	5	2	no	—
136	C₂²×C₃×C₄	—	10	2	no	—
137	C₂²×C₃×C₄	—	10	2	no	—
138	C₂²×C₃×C₄	—	10	2	no	—
139	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₂×C₃×C₈	—	5	4	no	—
143	C₂×C₃×C₈	—	10	8	no	—
144	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	—	10	2	no	—
148	C₃×C₄⋊C₄	—	10	2	no	—
149	C₃×C₂²⋊C₄	—	20	4	no	—
150	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂²×D₅	—	2	2	no	—
152	C₂²×D₅	—	2	1	no	—
153	C₂²×D₅	—	2	1	no	—
154	C₂²×D₅	C₃×C₄⋊C₄	1	2	yes	—
155	C₂²×D₅	C₃×SD₁₆	1	4	yes	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅	C₃×#41	1	2	yes	—
163	C₂×Dic₅	—	2	2	no	—
164	C₂×Dic₅	—	4	2	no	—
165	C₂×Dic₅	—	4	2	no	—
166	C₂×Dic₅	—	4	2	no	—
167	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
168	C₄×D₅	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	—	2	4	no	—
171	C₅×C₈	—	2	2	no	—
172	#168	C₃×C₂²⋊C₄	1	4	yes	complemented
173	C₂³×C₄	—	5	1	no	—
174	C₂²×C₈	—	5	1	no	—
175	C₂²×C₈	—	5	4	no	—
176	C₂²×C₈	—	10	4	no	—
177	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
178	C₂×C₄⋊C₄	—	5	2	no	—
179	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
180	C₂×C₂²⋊C₄	—	10	2	no	—
181	#106	—	10	4	no	—
182	#95	—	10	2	no	—
183	C₂×C₃×C₅	#380	1	1	yes	characteristic
184	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
185	C₂×C₃×C₅	#427	1	1	yes	characteristic
186	C₂×C₃×C₅	#465	1	2	yes	—
187	C₂×C₃×C₅	#465	1	2	yes	—
188	C₃×D₅	—	2	4	no	—
189	C₂³×C₃	—	5	2	no	—
190	C₂³×C₃	—	5	4	no	—
191	C₂³×C₃	—	10	2	no	—
192	C₂³×C₃	—	10	1	no	—
193	C₂³×C₃	—	10	1	no	—
194	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	4	2	no	—
198	C₂×C₃×C₄	—	5	2	no	—
199	C₂×C₃×C₄	—	10	2	no	—
200	C₂×C₃×C₄	—	10	2	no	—
201	C₂×C₃×C₄	—	10	2	no	—
202	C₂×C₃×C₄	—	10	4	no	—
203	C₂×C₃×C₄	—	20	2	no	—
204	C₂×C₃×C₄	—	20	2	no	—
205	C₂×C₃×C₄	—	20	2	no	—
206	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
207	C₂×C₃×C₄	#709	1	2	yes	—
208	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
209	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
210	C₃×C₈	—	2	2	no	—
211	C₃×C₈	—	5	4	no	—
212	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
213	C₂²×C₅	C₃×#121	1	2	yes	—
214	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
215	C₂²×C₅	C₃×#135	1	1	yes	characteristic
216	C₂²×C₅	C₃×#135	1	1	yes	characteristic
217	C₂×D₅	—	2	2	no	—
218	C₂×D₅	—	2	2	no	—
219	C₂×D₅	—	2	4	no	—
220	C₂×D₅	—	2	4	no	—
221	C₂×D₅	C₃×#106	1	4	yes	complemented
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₄×C₅	C₃×#117	1	1	yes	characteristic
225	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
226	Dic₅	—	2	2	no	—
227	Dic₅	—	8	2	no	—
228	C₂⁴	—	5	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	10	2	no	—
234	C₂²×C₄	—	10	2	no	—
235	C₂²×C₄	—	10	2	no	—
236	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	5	2	no	—
239	C₂×C₈	—	5	4	no	—
240	C₂×C₈	—	10	8	no	—
241	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	10	2	no	—
245	C₄⋊C₄	—	10	2	no	—
246	C₂²⋊C₄	—	20	4	no	—
247	C₃×C₅	#1807	1	1	yes	characteristic, complemented, Hall
248	C₂²×C₃	—	5	4	no	—
249	C₂²×C₃	—	10	4	no	—
250	C₂²×C₃	—	10	4	no	—
251	C₂²×C₃	—	10	2	no	—
252	C₂²×C₃	—	10	2	no	—
253	C₂²×C₃	#4030	1	2	yes	—
254	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
255	C₂²×C₃	#4058	1	1	yes	characteristic
256	C₂²×C₃	D₅×SD₁₆	1	2	yes	—
257	C₂²×C₃	#4065	1	1	yes	characteristic
258	C₃×C₄	—	8	2	no	—
259	C₃×C₄	—	10	2	no	—
260	C₃×C₄	—	40	2	no	—
261	C₃×C₄	#4018	1	1	yes	characteristic
262	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₃×C₄	#4042	1	2	yes	—
264	C₂×C₅	C₃×#380	1	1	yes	characteristic
265	C₂×C₅	C₂×C₃×#106	1	1	yes	characteristic
266	C₂×C₅	C₃×#427	1	1	yes	characteristic
267	C₂×C₅	C₃×#465	1	2	yes	—
268	C₂×C₅	C₃×#465	1	2	yes	—
269	D₅	—	2	4	no	—
270	C₂³	—	5	2	no	—
271	C₂³	—	5	4	no	—
272	C₂³	—	10	2	no	—
273	C₂³	—	10	1	no	—
274	C₂³	—	10	1	no	—
275	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	4	no	—
284	C₂×C₄	—	20	2	no	—
285	C₂×C₄	—	20	2	no	—
286	C₂×C₄	—	20	2	no	—
287	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
288	C₂×C₄	C₃×#709	1	2	yes	—
289	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
290	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	5	4	no	—
293	C₂×C₃	—	10	4	no	—
294	C₂×C₃	#64956	1	1	yes	characteristic
295	C₂×C₃	#65067	1	2	yes	—
296	C₂×C₃	#65096	1	2	yes	—
297	C₂×C₃	D₅×#106	1	1	yes	characteristic
298	C₂×C₃	#65169	1	1	yes	characteristic
299	C₅	C₃×#1807	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	5	4	no	—
301	C₂²	—	10	4	no	—
302	C₂²	—	10	4	no	—
303	C₂²	—	10	2	no	—
304	C₂²	—	10	2	no	—
305	C₂²	C₃×#4030	1	2	yes	—
306	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
307	C₂²	C₃×#4058	1	1	yes	characteristic
308	C₂²	C₃×D₅×SD₁₆	1	2	yes	—
309	C₂²	C₃×#4065	1	1	yes	characteristic
310	C₄	—	8	2	no	—
311	C₄	—	10	2	no	—
312	C₄	—	40	2	no	—
313	C₄	C₃×#4018	1	1	yes	characteristic
314	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
315	C₄	C₃×#4042	1	2	yes	—
316	C₃	#10599661	1	1	yes	characteristic, direct product factor, Sylow
317	C₂	—	10	4	no	—
318	C₂	C₃×#64956	1	1	yes	characteristic
319	C₂	C₃×#65067	1	2	yes	—
320	C₂	C₃×#65096	1	2	yes	—
321	C₂	C₃×D₅×#106	1	1	yes	characteristic
322	C₂	C₃×#65169	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#340	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×D₅	D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#340	D₅
hypocenter	C₅	C₃×#1807
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#340

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#3940	C₂	semidirect
4. C₃×C₅×#340	C₂	semidirect
5. C₃×#64771	C₂	semidirect
6. #10599661	C₃	direct (N × H)
14. C₂×C₃×C₈×D₅	C₄	semidirect
20. C₂×#3940	C₂×C₃	semidirect
21. C₅×#340	C₂×C₃	semidirect
22. #64771	C₂×C₃	semidirect
46. C₃×#340	D₅	semidirect
54. C₂×C₈×D₅	C₃×C₄	semidirect
80. C₃×#168	C₂²⋊C₄	semidirect
114. #340	C₃×D₅	semidirect
124. C₂×C₃×D₅	#106	semidirect
172. #168	C₃×C₂²⋊C₄	semidirect
221. C₂×D₅	C₃×#106	semidirect
247. C₃×C₅	#1807	semidirect
299. C₅	C₃×#1807	semidirect
316. C₃	#10599661	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×D₅	2
C₃×#65276	2
C₂×C₃×#3940	2
C₃×C₅×#340	2
C₃×#64771	2
#10599661	3
C₃×#1807	5

Subgroups — C3×#10599661 #14297874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599661 #14297874