Subgroups — C₃×#10599398 #14298706

Order 1920.

520 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	C₂×C₃×#3946	C₂	1	2	yes	maximal
2	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
4	C₃×#64770	C₂	1	2	yes	maximal
5	C₃×#64744	C₂	1	1	yes	maximal, characteristic
6	#10599398	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	2	yes	—
16	C₃×#3946	—	2	4	no	—
17	C₃×#1831	—	5	1	no	maximal, Hall
18	C₂×#3946	C₂×C₃	1	2	yes	—
19	C₄²×Dic₅	C₂×C₃	1	1	yes	characteristic
20	C₅×#336	C₂×C₃	1	1	yes	characteristic
21	#64770	C₂×C₃	1	2	yes	—
22	#64744	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄×Dic₅	—	2	2	no	—
35	C₃×C₄×Dic₅	—	2	2	no	—
36	C₃×C₄×Dic₅	—	2	2	no	—
37	C₃×C₄×Dic₅	—	2	2	no	—
38	C₃×C₄×Dic₅	C₈	1	4	yes	complemented
39	C₃×C₄×Dic₅	D₄	1	2	yes	—
40	C₃×C₄×Dic₅	Q₈	1	2	yes	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×#168	—	2	2	no	—
44	C₂×C₃×C₄⋊C₈	—	5	2	no	—
45	C₃×C₄³	—	5	1	no	—
46	C₃×#336	—	5	2	no	—
47	C₃×#336	—	5	1	no	—
48	C₃×#336	D₅	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
53	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅	C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
57	C₂²×#168	C₂²×C₃	1	2	yes	—
58	#3946	—	2	4	no	—
59	#1831	—	5	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	2	yes	—
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	1	no	—
74	C₂×C₃×Dic₅	—	2	1	no	—
75	C₂×C₃×Dic₅	—	2	1	no	—
76	C₂×C₃×Dic₅	—	2	1	no	—
77	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
78	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
79	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
80	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
81	C₂×C₃×Dic₅	M₁₆	1	2	yes	—
82	C₂×C₃×Dic₅	M₁₆	1	2	yes	—
83	C₃×C₅×C₈	—	4	2	no	—
84	C₃×#168	—	4	2	no	—
85	C₂²×C₃×C₈	—	5	2	no	—
86	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
87	C₂×C₃×C₄²	—	5	1	no	—
88	C₂×C₃×C₄²	—	5	1	no	—
89	C₂×C₃×C₄²	—	5	1	no	—
90	C₂×C₃×C₄²	—	5	1	no	—
91	C₂×C₃×C₄²	—	5	1	no	—
92	C₂×C₃×C₄²	—	5	1	no	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	C₃×C₄⋊C₈	—	10	4	no	—
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄×Dic₅	—	2	2	no	—
107	C₄×Dic₅	—	2	2	no	—
108	C₄×Dic₅	—	2	2	no	—
109	C₄×Dic₅	—	2	2	no	—
110	C₄×Dic₅	C₃×C₈	1	4	yes	complemented
111	C₄×Dic₅	C₃×D₄	1	2	yes	—
112	C₄×Dic₅	C₃×Q₈	1	2	yes	—
113	C₂×#168	—	2	2	no	—
114	C₂×#168	—	2	2	no	—
115	C₂×#168	—	2	2	no	—
116	C₂×C₄⋊C₈	—	5	2	no	—
117	C₄³	—	5	1	no	—
118	#336	—	5	2	no	—
119	#336	—	5	1	no	—
120	#336	C₃×D₅	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#117	1	1	yes	characteristic
123	C₂²×C₃×C₅	#117	1	1	yes	characteristic
124	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
125	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
127	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₃×C₄×C₅	—	2	2	no	—
129	C₃×C₄×C₅	—	2	2	no	—
130	C₃×C₄×C₅	#124	1	2	yes	—
131	C₃×C₄×C₅	#125	1	2	yes	—
132	C₃×Dic₅	—	2	2	no	—
133	C₃×Dic₅	—	2	2	no	—
134	C₃×Dic₅	—	2	2	no	—
135	C₃×Dic₅	C₄⋊C₈	1	4	yes	—
136	C₂²×C₃×C₄	—	5	1	no	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	1	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
143	C₂×C₃×C₈	—	2	2	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	—	10	2	no	—
147	C₂×C₃×C₈	—	10	2	no	—
148	C₂×C₃×C₈	—	10	2	no	—
149	C₃×C₄²	—	2	2	no	—
150	C₃×C₄²	—	5	4	no	—
151	C₃×C₄²	—	5	2	no	—
152	C₃×C₄²	—	5	2	no	—
153	C₃×C₄²	—	10	2	no	—
154	C₃×C₄²	—	10	2	no	—
155	C₃×C₄²	—	10	2	no	—
156	C₃×C₄²	—	10	2	no	—
157	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
163	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
166	C₂×C₄×C₅	C₃×#41	1	2	yes	—
167	C₂×C₄×C₅	C₃×#41	1	2	yes	—
168	C₂×Dic₅	—	2	1	no	—
169	C₂×Dic₅	—	2	1	no	—
170	C₂×Dic₅	—	2	1	no	—
171	C₂×Dic₅	—	2	1	no	—
172	C₂×Dic₅	—	2	1	no	—
173	C₂×Dic₅	—	2	1	no	—
174	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
175	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
176	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
177	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
178	C₂×Dic₅	C₃×M₁₆	1	2	yes	—
179	C₂×Dic₅	C₃×M₁₆	1	2	yes	—
180	C₅×C₈	—	4	2	no	—
181	#168	—	4	2	no	—
182	C₂²×C₈	—	5	2	no	—
183	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
184	C₂×C₄²	—	5	1	no	—
185	C₂×C₄²	—	5	1	no	—
186	C₂×C₄²	—	5	1	no	—
187	C₂×C₄²	—	5	1	no	—
188	C₂×C₄²	—	5	1	no	—
189	C₂×C₄²	—	5	1	no	—
190	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
191	C₄⋊C₈	—	10	4	no	—
192	C₂×C₃×C₅	#382	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
194	C₂×C₃×C₅	#423	1	1	yes	characteristic
195	C₂×C₃×C₅	#431	1	1	yes	characteristic
196	C₂×C₃×C₅	#431	1	1	yes	characteristic
197	C₂×C₃×C₅	#432	1	1	yes	characteristic
198	C₂×C₃×C₅	#432	1	1	yes	characteristic
199	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	5	2	no	—
205	C₂×C₃×C₄	—	5	2	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	5	2	no	—
208	C₂×C₃×C₄	—	5	2	no	—
209	C₂×C₃×C₄	—	5	2	no	—
210	C₂×C₃×C₄	—	10	1	no	—
211	C₂×C₃×C₄	—	10	1	no	—
212	C₂×C₃×C₄	—	10	1	no	—
213	C₂×C₃×C₄	—	10	1	no	—
214	C₂×C₃×C₄	—	10	1	no	—
215	C₂×C₃×C₄	—	10	1	no	—
216	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
217	C₂×C₃×C₄	#676	1	2	yes	—
218	C₂×C₃×C₄	#709	1	2	yes	—
219	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	#711	1	2	yes	—
221	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
222	C₃×C₈	—	4	2	no	—
223	C₃×C₈	—	20	2	no	—
224	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
225	C₂²×C₅	C₃×#117	1	1	yes	characteristic
226	C₂²×C₅	C₃×#117	1	1	yes	characteristic
227	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
228	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
229	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
230	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	C₃×#124	1	2	yes	—
234	C₄×C₅	C₃×#125	1	2	yes	—
235	Dic₅	—	2	2	no	—
236	Dic₅	—	2	2	no	—
237	Dic₅	—	2	2	no	—
238	Dic₅	C₃×C₄⋊C₈	1	4	yes	—
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
244	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	—	10	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	5	4	no	—
254	C₄²	—	5	2	no	—
255	C₄²	—	5	2	no	—
256	C₄²	—	10	2	no	—
257	C₄²	—	10	2	no	—
258	C₄²	—	10	2	no	—
259	C₄²	—	10	2	no	—
260	C₃×C₅	#1831	1	1	yes	characteristic, complemented, Hall
261	C₂²×C₃	#4018	1	1	yes	characteristic
262	C₂²×C₃	#4023	1	1	yes	characteristic
263	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
264	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
265	C₂²×C₃	C₂×#676	1	1	yes	characteristic
266	C₂²×C₃	D₅×M₁₆	1	1	yes	characteristic
267	C₂²×C₃	D₅×M₁₆	1	1	yes	characteristic
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	5	4	no	—
271	C₃×C₄	—	10	2	no	—
272	C₃×C₄	—	10	2	no	—
273	C₃×C₄	—	10	2	no	—
274	C₃×C₄	#4031	1	2	yes	—
275	C₃×C₄	#4035	1	2	yes	—
276	C₂×C₅	C₃×#382	1	1	yes	characteristic
277	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
278	C₂×C₅	C₃×#423	1	1	yes	characteristic
279	C₂×C₅	C₃×#431	1	1	yes	characteristic
280	C₂×C₅	C₃×#431	1	1	yes	characteristic
281	C₂×C₅	C₃×#432	1	1	yes	characteristic
282	C₂×C₅	C₃×#432	1	1	yes	characteristic
283	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	2	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	5	2	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	10	1	no	—
296	C₂×C₄	—	10	1	no	—
297	C₂×C₄	—	10	1	no	—
298	C₂×C₄	—	10	1	no	—
299	C₂×C₄	—	10	1	no	—
300	C₂×C₄	C₃×C₈×D₅	1	2	yes	—
301	C₂×C₄	C₃×#676	1	2	yes	—
302	C₂×C₄	C₃×#709	1	2	yes	—
303	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
304	C₂×C₄	C₃×#711	1	2	yes	—
305	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	20	2	no	—
308	C₂×C₃	#64939	1	1	yes	characteristic
309	C₂×C₃	#64974	1	1	yes	characteristic
310	C₂×C₃	#64994	1	1	yes	characteristic
311	C₂×C₃	#65007	1	1	yes	characteristic
312	C₂×C₃	#65008	1	1	yes	characteristic
313	C₂×C₃	D₅×C₄⋊C₈	1	1	yes	characteristic
314	C₂×C₃	#65126	1	1	yes	characteristic
315	C₅	C₃×#1831	1	1	yes	characteristic, complemented, Sylow
316	C₂²	C₃×#4018	1	1	yes	characteristic
317	C₂²	C₃×#4023	1	1	yes	characteristic
318	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
319	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
320	C₂²	C₂×C₃×#676	1	1	yes	characteristic
321	C₂²	C₃×D₅×M₁₆	1	1	yes	characteristic
322	C₂²	C₃×D₅×M₁₆	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	5	4	no	—
326	C₄	—	10	2	no	—
327	C₄	—	10	2	no	—
328	C₄	—	10	2	no	—
329	C₄	C₃×#4031	1	2	yes	—
330	C₄	C₃×#4035	1	2	yes	—
331	C₃	#10599398	1	1	yes	characteristic, direct product factor, Sylow
332	C₂	C₃×#64939	1	1	yes	characteristic
333	C₂	C₃×#64974	1	1	yes	characteristic
334	C₂	C₃×#64994	1	1	yes	characteristic
335	C₂	C₃×#65007	1	1	yes	characteristic
336	C₂	C₃×#65008	1	1	yes	characteristic
337	C₂	C₃×D₅×C₄⋊C₈	1	1	yes	characteristic
338	C₂	C₃×#65126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#336	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#336	D₅
hypocenter	C₅	C₃×#1831
LCS	G ▷ C₂²×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
6. #10599398	C₃	direct (N × H)
38. C₃×C₄×Dic₅	C₈	semidirect
110. C₄×Dic₅	C₃×C₈	semidirect
260. C₃×C₅	#1831	semidirect
315. C₅	C₃×#1831	semidirect
331. C₃	#10599398	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×#3946	2
C₃×C₄²×Dic₅	2
C₃×C₅×#336	2
C₃×#64770	2
C₃×#64744	2
#10599398	3
C₃×#1831	5

Subgroups — C3×#10599398 #14298706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599398 #14298706