Subgroups — C₃×#10601279 #14297772

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3948	C₂	1	2	yes	maximal
4	C₃×#65225	C₂	1	1	yes	maximal, characteristic
5	C₃×#64771	C₂	1	2	yes	maximal
6	#10601279	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×Dic₁₀	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×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×#683	—	2	1	no	—
13	C₂×C₃×C₄×Dic₅	—	2	1	no	—
14	C₃×C₄×Dic₁₀	—	2	2	no	—
15	C₃×C₄×Dic₁₀	C₄	1	4	yes	complemented
16	C₃×C₅×C₄⋊C₈	—	2	2	no	—
17	C₃×#3963	—	2	1	no	—
18	C₃×#3963	—	2	1	no	—
19	C₃×#3948	—	2	4	no	—
20	C₃×#1805	—	5	1	no	maximal, Hall
21	C₂×C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×#3948	C₂×C₃	1	2	yes	—
24	#65225	C₂×C₃	1	1	yes	characteristic
25	#64771	C₂×C₃	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	—	2	1	no	—
30	C₂²×C₃×Dic₅	—	2	1	no	—
31	C₂²×C₃×Dic₅	—	2	2	no	—
32	C₂×C₃×Dic₁₀	—	2	1	no	—
33	C₂×C₃×Dic₁₀	—	2	1	no	—
34	C₂×C₃×Dic₁₀	—	2	1	no	—
35	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
36	C₂×C₃×Dic₁₀	D₄	1	2	yes	—
37	C₂×C₃×Dic₁₀	Q₈	1	2	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₂×C₃×C₅×C₈	—	2	2	no	—
41	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₃×C₄²×C₅	D₄	1	2	yes	—
43	C₃×C₄⋊Dic₅	—	2	1	no	—
44	C₃×C₄⋊Dic₅	—	2	2	no	—
45	C₃×C₄⋊Dic₅	—	2	2	no	—
46	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
47	C₃×#683	—	2	4	no	—
48	C₃×C₄×Dic₅	—	2	4	no	—
49	C₂×C₃×C₄×Q₈	—	5	1	no	—
50	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic
51	C₂×C₃×#103	—	5	2	no	—
52	C₃×#382	—	5	1	no	—
53	C₃×#340	—	5	2	no	—
54	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
56	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
58	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×#683	—	2	1	no	—
60	C₂×C₄×Dic₅	—	2	1	no	—
61	C₄×Dic₁₀	—	2	2	no	—
62	C₄×Dic₁₀	C₃×C₄	1	4	yes	complemented
63	C₅×C₄⋊C₈	—	2	2	no	—
64	#3963	—	2	1	no	—
65	#3963	—	2	1	no	—
66	#3948	—	2	4	no	—
67	#1805	—	5	1	no	Sylow
68	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₂×C₃×Dic₅	—	2	2	no	—
82	C₂×C₃×Dic₅	—	2	2	no	—
83	C₂×C₃×Dic₅	—	4	2	no	—
84	C₂×C₃×Dic₅	—	4	2	no	—
85	C₂×C₃×Dic₅	—	4	2	no	—
86	C₂×C₃×Dic₅	—	4	2	no	—
87	C₃×Dic₁₀	—	2	2	no	—
88	C₃×Dic₁₀	—	2	2	no	—
89	C₃×Dic₁₀	—	2	2	no	—
90	C₃×Dic₁₀	C₄⋊C₄	1	4	yes	—
91	C₃×C₅×C₈	—	4	2	no	—
92	C₂²×C₃×Q₈	—	5	1	no	—
93	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
94	C₃×C₄×Q₈	—	5	4	no	—
95	C₃×C₄×Q₈	—	10	2	no	—
96	C₂×C₃×C₄⋊C₄	—	5	1	no	—
97	C₂×C₃×C₄⋊C₄	—	5	2	no	—
98	C₂×C₃×C₄⋊C₄	—	10	1	no	—
99	C₂×C₃×C₄²	—	10	1	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	—	2	2	no	—
102	C₃×#103	—	10	4	no	—
103	C₃×#95	—	10	1	no	—
104	C₃×#95	—	10	1	no	—
105	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂²×Dic₅	—	2	1	no	—
109	C₂²×Dic₅	—	2	1	no	—
110	C₂²×Dic₅	—	2	2	no	—
111	C₂×Dic₁₀	—	2	1	no	—
112	C₂×Dic₁₀	—	2	1	no	—
113	C₂×Dic₁₀	—	2	1	no	—
114	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
115	C₂×Dic₁₀	C₃×D₄	1	2	yes	—
116	C₂×Dic₁₀	C₃×Q₈	1	2	yes	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
121	C₄²×C₅	C₃×D₄	1	2	yes	—
122	C₄⋊Dic₅	—	2	1	no	—
123	C₄⋊Dic₅	—	2	2	no	—
124	C₄⋊Dic₅	—	2	2	no	—
125	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
126	#683	—	2	4	no	—
127	C₄×Dic₅	—	2	4	no	—
128	C₂×C₄×Q₈	—	5	1	no	—
129	C₂×C₄⋊C₈	C₃×D₅	1	1	yes	characteristic
130	C₂×#103	—	5	2	no	—
131	#382	—	5	1	no	—
132	#340	—	5	2	no	—
133	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₃×C₅	#121	1	1	yes	characteristic
135	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₃×C₅	#135	1	2	yes	—
137	C₂²×C₃×C₅	#137	1	2	yes	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	2	2	no	—
140	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₃×C₄×C₅	#117	1	1	yes	characteristic
142	C₃×C₄×C₅	#121	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
144	C₃×Dic₅	—	2	4	no	—
145	C₃×Dic₅	—	4	2	no	—
146	C₃×Dic₅	—	8	2	no	—
147	C₂×C₃×Q₈	—	5	2	no	—
148	C₂×C₃×Q₈	—	5	2	no	—
149	C₂×C₃×Q₈	—	5	2	no	—
150	C₂×C₃×Q₈	—	10	1	no	—
151	C₂×C₃×Q₈	—	10	1	no	—
152	C₂×C₃×Q₈	—	10	1	no	—
153	C₂²×C₃×C₄	—	10	1	no	—
154	C₂²×C₃×C₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
157	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
158	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₂×C₃×C₈	—	2	2	no	—
160	C₂×C₃×C₈	—	2	2	no	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₃×C₄⋊C₄	—	5	2	no	—
163	C₃×C₄⋊C₄	—	10	4	no	—
164	C₃×C₄⋊C₄	—	10	1	no	—
165	C₃×C₄⋊C₄	—	10	2	no	—
166	C₃×C₄⋊C₄	—	10	2	no	—
167	C₃×C₄²	—	10	4	no	—
168	C₃×C₄²	C₄×D₅	1	2	yes	—
169	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
170	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
174	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
175	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
180	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
181	C₂×Dic₅	—	2	2	no	—
182	C₂×Dic₅	—	2	2	no	—
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	2	2	no	—
185	C₂×Dic₅	—	4	2	no	—
186	C₂×Dic₅	—	4	2	no	—
187	C₂×Dic₅	—	4	2	no	—
188	C₂×Dic₅	—	4	2	no	—
189	Dic₁₀	—	2	2	no	—
190	Dic₁₀	—	2	2	no	—
191	Dic₁₀	—	2	2	no	—
192	Dic₁₀	C₃×C₄⋊C₄	1	4	yes	—
193	C₅×C₈	—	4	2	no	—
194	C₂²×Q₈	—	5	1	no	—
195	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
196	C₄×Q₈	—	5	4	no	—
197	C₄×Q₈	—	10	2	no	—
198	C₂×C₄⋊C₄	—	5	1	no	—
199	C₂×C₄⋊C₄	—	5	2	no	—
200	C₂×C₄⋊C₄	—	10	1	no	—
201	C₂×C₄²	—	10	1	no	—
202	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	#103	—	10	4	no	—
205	#95	—	10	1	no	—
206	#95	—	10	1	no	—
207	C₂×C₃×C₅	#380	1	1	yes	characteristic
208	C₂×C₃×C₅	#416	1	1	yes	characteristic
209	C₂×C₃×C₅	#419	1	1	yes	characteristic
210	C₂×C₃×C₅	#464	1	2	yes	—
211	C₂×C₃×C₅	#479	1	2	yes	—
212	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
213	C₃×Q₈	—	5	4	no	—
214	C₃×Q₈	—	10	2	no	—
215	C₃×Q₈	—	10	2	no	—
216	C₃×Q₈	—	10	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	10	2	no	—
220	C₂×C₃×C₄	—	10	2	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	20	2	no	—
224	C₂×C₃×C₄	—	20	2	no	—
225	C₂×C₃×C₄	—	20	2	no	—
226	C₂×C₃×C₄	—	20	2	no	—
227	C₂×C₃×C₄	#685	1	2	yes	—
228	C₂×C₃×C₄	#685	1	2	yes	—
229	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
230	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	#711	1	1	yes	characteristic
232	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
233	C₂×C₃×C₄	#713	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	C₃×C₈	—	4	2	no	—
236	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₅	C₃×#121	1	1	yes	characteristic
238	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂²×C₅	C₃×#135	1	2	yes	—
240	C₂²×C₅	C₃×#137	1	2	yes	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	2	2	no	—
243	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
244	C₄×C₅	C₃×#117	1	1	yes	characteristic
245	C₄×C₅	C₃×#121	1	1	yes	characteristic
246	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
247	Dic₅	—	2	4	no	—
248	Dic₅	—	4	2	no	—
249	Dic₅	—	8	2	no	—
250	C₂×Q₈	—	5	2	no	—
251	C₂×Q₈	—	5	2	no	—
252	C₂×Q₈	—	5	2	no	—
253	C₂×Q₈	—	10	1	no	—
254	C₂×Q₈	—	10	1	no	—
255	C₂×Q₈	—	10	1	no	—
256	C₂²×C₄	—	10	1	no	—
257	C₂²×C₄	—	10	1	no	—
258	C₂²×C₄	—	10	2	no	—
259	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
260	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₄⋊C₄	—	10	4	no	—
267	C₄⋊C₄	—	10	1	no	—
268	C₄⋊C₄	—	10	2	no	—
269	C₄⋊C₄	—	10	2	no	—
270	C₄²	—	10	4	no	—
271	C₄²	C₃×C₄×D₅	1	2	yes	—
272	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
273	C₃×C₅	#1805	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	#4041	1	1	yes	characteristic
275	C₂²×C₃	#4043	1	1	yes	characteristic
276	C₂²×C₃	#4050	1	2	yes	—
277	C₂²×C₃	#4055	1	2	yes	—
278	C₂²×C₃	C₂×#685	1	1	yes	characteristic
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	10	4	no	—
282	C₃×C₄	—	20	2	no	—
283	C₃×C₄	—	40	2	no	—
284	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
285	C₃×C₄	#4038	1	1	yes	characteristic
286	C₃×C₄	#4084	1	1	yes	characteristic
287	C₃×C₄	#4092	1	1	yes	characteristic
288	C₂×C₅	C₃×#380	1	1	yes	characteristic
289	C₂×C₅	C₃×#416	1	1	yes	characteristic
290	C₂×C₅	C₃×#419	1	1	yes	characteristic
291	C₂×C₅	C₃×#464	1	2	yes	—
292	C₂×C₅	C₃×#479	1	2	yes	—
293	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
294	Q₈	—	5	4	no	—
295	Q₈	—	10	2	no	—
296	Q₈	—	10	2	no	—
297	Q₈	—	10	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	10	2	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	2	no	—
305	C₂×C₄	—	20	2	no	—
306	C₂×C₄	—	20	2	no	—
307	C₂×C₄	—	20	2	no	—
308	C₂×C₄	C₃×#685	1	2	yes	—
309	C₂×C₄	C₃×#685	1	2	yes	—
310	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
311	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
312	C₂×C₄	C₃×#711	1	1	yes	characteristic
313	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
314	C₂×C₄	C₃×#713	1	1	yes	characteristic
315	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₂×C₃	#65118	1	2	yes	—
318	C₂×C₃	#65134	1	2	yes	—
319	C₂×C₃	#65276	1	1	yes	characteristic
320	C₂×C₃	#65402	1	1	yes	characteristic
321	C₂×C₃	#65409	1	1	yes	characteristic
322	C₅	C₃×#1805	1	1	yes	characteristic, complemented, Sylow
323	C₂²	C₃×#4041	1	1	yes	characteristic
324	C₂²	C₃×#4043	1	1	yes	characteristic
325	C₂²	C₃×#4050	1	2	yes	—
326	C₂²	C₃×#4055	1	2	yes	—
327	C₂²	C₂×C₃×#685	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	10	4	no	—
331	C₄	—	20	2	no	—
332	C₄	—	40	2	no	—
333	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
334	C₄	C₃×#4038	1	1	yes	characteristic
335	C₄	C₃×#4084	1	1	yes	characteristic
336	C₄	C₃×#4092	1	1	yes	characteristic
337	C₃	#10601279	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	C₃×#65118	1	2	yes	—
339	C₂	C₃×#65134	1	2	yes	—
340	C₂	C₃×#65276	1	1	yes	characteristic
341	C₂	C₃×#65402	1	1	yes	characteristic
342	C₂	C₃×#65409	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊C₈	D₅
hypocenter	C₅	C₃×#1805
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
6. #10601279	C₃	direct (N × H)
15. C₃×C₄×Dic₁₀	C₄	semidirect
62. C₄×Dic₁₀	C₃×C₄	semidirect
273. C₃×C₅	#1805	semidirect
322. C₅	C₃×#1805	semidirect
337. C₃	#10601279	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×C₄×Dic₁₀	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₃×#3948	2
C₃×#65225	2
C₃×#64771	2
#10601279	3
C₃×#1805	5

Subgroups — C3×#10601279 #14297772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601279 #14297772