Subgroups — C₅×#70486 #14297771

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×C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#816	C₂	1	2	yes	maximal
4	C₅×#4936	C₂	1	1	yes	maximal, characteristic
5	C₅×#4551	C₂	1	2	yes	maximal
6	C₅×#1805	—	3	1	no	maximal, Hall
7	C₂²×C₅×Dic₆	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×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₅×Dic₃⋊C₄	—	2	1	no	—
13	C₂×C₄×C₅×Dic₃	—	2	1	no	—
14	C₄×C₅×Dic₆	—	2	2	no	—
15	C₄×C₅×Dic₆	C₄	1	4	yes	complemented
16	C₃×C₅×C₄⋊C₈	—	2	2	no	—
17	C₅×C₃⋊#95	—	2	1	no	—
18	C₅×C₃⋊#95	—	2	1	no	—
19	C₅×#816	—	2	4	no	—
20	#70486	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₄×C₅×Q₈	—	3	1	no	—
22	C₂×C₅×C₄⋊C₈	D₃	1	1	yes	characteristic
23	C₂×C₅×#103	—	3	2	no	—
24	C₅×#382	—	3	1	no	—
25	C₅×#340	—	3	2	no	—
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×C₅×Dic₃	—	2	1	no	—
30	C₂²×C₅×Dic₃	—	2	1	no	—
31	C₂²×C₅×Dic₃	—	2	2	no	—
32	C₂×C₅×Dic₆	—	2	1	no	—
33	C₂×C₅×Dic₆	—	2	1	no	—
34	C₂×C₅×Dic₆	—	2	1	no	—
35	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
36	C₂×C₅×Dic₆	D₄	1	2	yes	—
37	C₂×C₅×Dic₆	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	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	2	no	—
46	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	C₅×Dic₃⋊C₄	—	2	4	no	—
48	C₄×C₅×Dic₃	—	2	4	no	—
49	C₂×C₄×Dic₆	C₂×C₅	1	1	yes	characteristic
50	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
51	C₂×#816	C₂×C₅	1	2	yes	—
52	#4936	C₂×C₅	1	1	yes	characteristic
53	#4551	C₂×C₅	1	2	yes	—
54	C₂²×C₅×Q₈	—	3	1	no	—
55	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
56	C₄×C₅×Q₈	—	3	4	no	—
57	C₄×C₅×Q₈	—	6	2	no	—
58	C₂×C₅×C₄⋊C₄	—	3	1	no	—
59	C₂×C₅×C₄⋊C₄	—	3	2	no	—
60	C₂×C₅×C₄⋊C₄	—	6	1	no	—
61	C₂×C₄²×C₅	—	6	1	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	C₅×C₄⋊C₈	—	2	2	no	—
64	C₅×#103	—	6	4	no	—
65	C₅×#95	—	6	1	no	—
66	C₅×#95	—	6	1	no	—
67	#1805	—	3	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×C₅×Dic₃	—	2	2	no	—
80	C₂×C₅×Dic₃	—	2	2	no	—
81	C₂×C₅×Dic₃	—	2	2	no	—
82	C₂×C₅×Dic₃	—	2	2	no	—
83	C₂×C₅×Dic₃	—	4	2	no	—
84	C₂×C₅×Dic₃	—	4	2	no	—
85	C₂×C₅×Dic₃	—	4	2	no	—
86	C₂×C₅×Dic₃	—	4	2	no	—
87	C₅×Dic₆	—	2	2	no	—
88	C₅×Dic₆	—	2	2	no	—
89	C₅×Dic₆	—	2	2	no	—
90	C₅×Dic₆	C₄⋊C₄	1	4	yes	—
91	C₃×C₅×C₈	—	4	2	no	—
92	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
94	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
96	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
97	C₂×Dic₃⋊C₄	—	2	1	no	—
98	C₂×C₄×Dic₃	—	2	1	no	—
99	C₄×Dic₆	—	2	2	no	—
100	C₄×Dic₆	C₄×C₅	1	4	yes	complemented
101	C₃×C₄⋊C₈	—	2	2	no	—
102	C₃⋊#95	—	2	1	no	—
103	C₃⋊#95	—	2	1	no	—
104	#816	—	2	4	no	—
105	C₂×C₅×Q₈	—	3	2	no	—
106	C₂×C₅×Q₈	—	3	2	no	—
107	C₂×C₅×Q₈	—	3	2	no	—
108	C₂×C₅×Q₈	—	6	1	no	—
109	C₂×C₅×Q₈	—	6	1	no	—
110	C₂×C₅×Q₈	—	6	1	no	—
111	C₂²×C₄×C₅	—	6	1	no	—
112	C₂²×C₄×C₅	—	6	1	no	—
113	C₂²×C₄×C₅	—	6	2	no	—
114	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
115	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₅×C₄⋊C₄	—	3	2	no	—
121	C₅×C₄⋊C₄	—	6	4	no	—
122	C₅×C₄⋊C₄	—	6	1	no	—
123	C₅×C₄⋊C₄	—	6	2	no	—
124	C₅×C₄⋊C₄	—	6	2	no	—
125	C₄²×C₅	—	6	4	no	—
126	C₄²×C₅	C₄×D₃	1	2	yes	—
127	C₄²×C₅	#67	1	2	yes	—
128	C₂×C₄×Q₈	—	3	1	no	—
129	C₂×C₄⋊C₈	C₅×D₃	1	1	yes	characteristic
130	C₂×#103	—	3	2	no	—
131	#382	—	3	1	no	—
132	#340	—	3	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	C₅×Dic₃	—	2	4	no	—
145	C₅×Dic₃	—	4	2	no	—
146	C₅×Dic₃	—	8	2	no	—
147	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
150	C₂²×Dic₃	—	2	1	no	—
151	C₂²×Dic₃	—	2	1	no	—
152	C₂²×Dic₃	—	2	2	no	—
153	C₂×Dic₆	—	2	1	no	—
154	C₂×Dic₆	—	2	1	no	—
155	C₂×Dic₆	—	2	1	no	—
156	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
157	C₂×Dic₆	C₅×D₄	1	2	yes	—
158	C₂×Dic₆	C₅×Q₈	1	2	yes	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₂×C₃×C₈	—	2	2	no	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
163	C₃×C₄²	C₅×D₄	1	2	yes	—
164	C₄⋊Dic₃	—	2	1	no	—
165	C₄⋊Dic₃	—	2	2	no	—
166	C₄⋊Dic₃	—	2	2	no	—
167	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
168	Dic₃⋊C₄	—	2	4	no	—
169	C₄×Dic₃	—	2	4	no	—
170	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
171	C₅×Q₈	—	3	4	no	—
172	C₅×Q₈	—	6	2	no	—
173	C₅×Q₈	—	6	2	no	—
174	C₅×Q₈	—	6	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	6	2	no	—
178	C₂×C₄×C₅	—	6	2	no	—
179	C₂×C₄×C₅	—	6	2	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	—	12	2	no	—
182	C₂×C₄×C₅	—	12	2	no	—
183	C₂×C₄×C₅	—	12	2	no	—
184	C₂×C₄×C₅	—	12	2	no	—
185	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
186	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
187	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	#237	1	1	yes	characteristic
190	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
191	C₂×C₄×C₅	#239	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
193	C₅×C₈	—	4	2	no	—
194	C₂²×Q₈	—	3	1	no	—
195	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
196	C₄×Q₈	—	3	4	no	—
197	C₄×Q₈	—	6	2	no	—
198	C₂×C₄⋊C₄	—	3	1	no	—
199	C₂×C₄⋊C₄	—	3	2	no	—
200	C₂×C₄⋊C₄	—	6	1	no	—
201	C₂×C₄²	—	6	1	no	—
202	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	#103	—	6	4	no	—
205	#95	—	6	1	no	—
206	#95	—	6	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×C₅×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
216	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
217	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
222	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
223	C₂×Dic₃	—	2	2	no	—
224	C₂×Dic₃	—	2	2	no	—
225	C₂×Dic₃	—	2	2	no	—
226	C₂×Dic₃	—	2	2	no	—
227	C₂×Dic₃	—	4	2	no	—
228	C₂×Dic₃	—	4	2	no	—
229	C₂×Dic₃	—	4	2	no	—
230	C₂×Dic₃	—	4	2	no	—
231	Dic₆	—	2	2	no	—
232	Dic₆	—	2	2	no	—
233	Dic₆	—	2	2	no	—
234	Dic₆	C₅×C₄⋊C₄	1	4	yes	—
235	C₃×C₈	—	4	2	no	—
236	C₂²×C₅	#895	1	1	yes	characteristic
237	C₂²×C₅	#897	1	1	yes	characteristic
238	C₂²×C₅	#904	1	2	yes	—
239	C₂²×C₅	#909	1	2	yes	—
240	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	2	2	no	—
243	C₄×C₅	—	6	4	no	—
244	C₄×C₅	—	12	2	no	—
245	C₄×C₅	—	24	2	no	—
246	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
247	C₄×C₅	#892	1	1	yes	characteristic
248	C₄×C₅	#938	1	1	yes	characteristic
249	C₄×C₅	#946	1	1	yes	characteristic
250	C₂×Q₈	—	3	2	no	—
251	C₂×Q₈	—	3	2	no	—
252	C₂×Q₈	—	3	2	no	—
253	C₂×Q₈	—	6	1	no	—
254	C₂×Q₈	—	6	1	no	—
255	C₂×Q₈	—	6	1	no	—
256	C₂²×C₄	—	6	1	no	—
257	C₂²×C₄	—	6	1	no	—
258	C₂²×C₄	—	6	2	no	—
259	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
260	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₅×D₃	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	—	3	2	no	—
266	C₄⋊C₄	—	6	4	no	—
267	C₄⋊C₄	—	6	1	no	—
268	C₄⋊C₄	—	6	2	no	—
269	C₄⋊C₄	—	6	2	no	—
270	C₄²	—	6	4	no	—
271	C₄²	C₄×C₅×D₃	1	2	yes	—
272	C₄²	C₅×#67	1	2	yes	—
273	C₃×C₅	#1805	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂²×C₃	C₅×#121	1	1	yes	characteristic
276	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂²×C₃	C₅×#135	1	2	yes	—
278	C₂²×C₃	C₅×#137	1	2	yes	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
282	C₃×C₄	C₅×#117	1	1	yes	characteristic
283	C₃×C₄	C₅×#121	1	1	yes	characteristic
284	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
285	Dic₃	—	2	4	no	—
286	Dic₃	—	4	2	no	—
287	Dic₃	—	8	2	no	—
288	C₂×C₅	#4829	1	2	yes	—
289	C₂×C₅	#4845	1	2	yes	—
290	C₂×C₅	#4987	1	1	yes	characteristic
291	C₂×C₅	#5113	1	1	yes	characteristic
292	C₂×C₅	#5120	1	1	yes	characteristic
293	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
294	Q₈	—	3	4	no	—
295	Q₈	—	6	2	no	—
296	Q₈	—	6	2	no	—
297	Q₈	—	6	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	6	2	no	—
301	C₂×C₄	—	6	2	no	—
302	C₂×C₄	—	6	2	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	12	2	no	—
305	C₂×C₄	—	12	2	no	—
306	C₂×C₄	—	12	2	no	—
307	C₂×C₄	—	12	2	no	—
308	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
309	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
310	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
311	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
312	C₂×C₄	C₅×#237	1	1	yes	characteristic
313	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
314	C₂×C₄	C₅×#239	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
316	C₈	—	4	2	no	—
317	C₂×C₃	C₅×#380	1	1	yes	characteristic
318	C₂×C₃	C₅×#416	1	1	yes	characteristic
319	C₂×C₃	C₅×#419	1	1	yes	characteristic
320	C₂×C₃	C₅×#464	1	2	yes	—
321	C₂×C₃	C₅×#479	1	2	yes	—
322	C₅	#70486	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	C₅×#895	1	1	yes	characteristic
324	C₂²	C₅×#897	1	1	yes	characteristic
325	C₂²	C₅×#904	1	2	yes	—
326	C₂²	C₅×#909	1	2	yes	—
327	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	6	4	no	—
331	C₄	—	12	2	no	—
332	C₄	—	24	2	no	—
333	C₄	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
334	C₄	C₅×#892	1	1	yes	characteristic
335	C₄	C₅×#938	1	1	yes	characteristic
336	C₄	C₅×#946	1	1	yes	characteristic
337	C₃	C₅×#1805	1	1	yes	characteristic, complemented, Sylow
338	C₂	C₅×#4829	1	2	yes	—
339	C₂	C₅×#4845	1	2	yes	—
340	C₂	C₅×#4987	1	1	yes	characteristic
341	C₂	C₅×#5113	1	1	yes	characteristic
342	C₂	C₅×#5120	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×C₄⋊C₈	D₃
hypocenter	C₃	C₅×#1805
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
15. C₄×C₅×Dic₆	C₄	semidirect
20. #70486	C₅	direct (N × H)
100. C₄×Dic₆	C₄×C₅	semidirect
273. C₃×C₅	#1805	semidirect
322. C₅	#70486	direct (N × H)
337. C₃	C₅×#1805	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Dic₆	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₅×#816	2
C₅×#4936	2
C₅×#4551	2
C₅×#1805	3
#70486	5

Subgroups — C5×#70486 #14297771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70486 #14297771