Subgroups — C₅×#69909 #14297467

Order 1920.

624 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	C₂×C₅×#926	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#830	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₅×#4550	C₂	1	2	yes	maximal, complemented
6	C₅×#1798	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₅×#217	C₄	1	2	yes	—
10	C₅×#926	—	2	4	no	—
11	C₂×C₅×C₄⋊Dic₃	C₄	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
16	C₅×#830	—	2	2	no	—
17	C₅×Dic₃⋊C₈	—	2	4	no	—
18	#69909	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#117	—	3	1	no	—
20	C₂×C₅×C₄⋊C₈	—	3	2	no	—
21	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
22	C₂×C₅×C₂²⋊C₈	D₃	1	1	yes	characteristic
23	C₅×#336	—	3	2	no	—
24	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
32	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₅×#217	—	2	4	no	—
37	C₅×C₄⋊Dic₃	—	2	4	no	—
38	C₄×C₅×Dic₃	—	2	2	no	—
39	C₄×C₅×Dic₃	D₄	1	4	yes	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×C₅×C₃⋊C₈	—	2	2	no	—
42	C₂×C₅×C₃⋊C₈	—	2	2	no	—
43	C₂×#926	C₂×C₅	1	1	yes	characteristic
44	C₂×C₃×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
45	C₂×#830	C₂×C₅	1	1	yes	characteristic
46	C₂×Dic₃⋊C₈	C₂×C₅	1	2	yes	complemented
47	#4550	C₂×C₅	1	2	yes	complemented
48	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₅×C₈	—	3	2	no	—
50	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
51	C₅×#117	—	6	4	no	—
52	C₂×C₅×C₄⋊C₄	—	3	2	no	—
53	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
54	C₂×C₄²×C₅	—	3	1	no	—
55	C₂×C₄²×C₅	—	3	1	no	—
56	C₅×C₄⋊C₈	—	6	4	no	—
57	C₅×C₂²⋊C₈	—	2	2	no	—
58	C₅×C₂²⋊C₈	—	6	2	no	—
59	#1798	—	3	1	no	Sylow
60	C₂³×C₃×C₅	—	2	1	no	—
61	C₂³×C₃×C₅	—	2	1	no	—
62	C₂³×C₃×C₅	—	2	2	no	—
63	C₂³×C₃×C₅	—	2	2	no	—
64	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
65	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	2	1	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₂×C₅×Dic₃	—	2	2	no	—
76	C₂×C₅×Dic₃	—	2	2	no	—
77	C₂×C₅×Dic₃	—	2	2	no	—
78	C₂×C₅×Dic₃	—	2	2	no	—
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₂×C₅×Dic₃	C₂²⋊C₄	1	4	yes	—
81	C₃×C₅×C₈	—	4	2	no	—
82	C₅×C₃⋊C₈	—	4	2	no	—
83	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
84	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
85	C₂×#217	C₄×C₅	1	2	yes	—
86	#926	—	2	4	no	—
87	C₂×C₄⋊Dic₃	C₄×C₅	1	2	yes	—
88	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
91	C₃×C₂²⋊C₈	—	2	2	no	—
92	#830	—	2	2	no	—
93	Dic₃⋊C₈	—	2	4	no	—
94	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	3	2	no	—
98	C₂²×C₄×C₅	—	3	2	no	—
99	C₂²×C₄×C₅	Dic₆	1	2	yes	—
100	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
101	C₂²×C₄×C₅	#67	1	2	yes	—
102	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	6	2	no	—
107	C₂×C₅×C₈	—	6	2	no	—
108	C₂×C₅×C₈	—	6	2	no	—
109	C₅×C₄⋊C₄	—	6	4	no	—
110	C₅×C₂²⋊C₄	—	6	4	no	—
111	C₄²×C₅	—	3	4	no	—
112	C₄²×C₅	—	6	2	no	—
113	C₂×#117	—	3	1	no	—
114	C₂×C₄⋊C₈	—	3	2	no	—
115	C₂×C₂²⋊C₈	—	3	1	no	—
116	C₂×C₂²⋊C₈	C₅×D₃	1	1	yes	characteristic
117	#336	—	3	2	no	—
118	C₂²×C₃×C₅	—	2	2	no	—
119	C₂²×C₃×C₅	—	2	2	no	—
120	C₂²×C₃×C₅	—	2	2	no	—
121	C₂²×C₃×C₅	—	4	2	no	—
122	C₂²×C₃×C₅	—	4	2	no	—
123	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₃×C₅	#117	1	1	yes	characteristic
126	C₂²×C₃×C₅	#131	1	2	yes	—
127	C₂²×C₃×C₅	#131	1	2	yes	—
128	C₃×C₄×C₅	—	2	2	no	—
129	C₃×C₄×C₅	#121	1	2	yes	—
130	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₅×Dic₃	—	2	4	no	—
132	C₅×Dic₃	—	4	2	no	—
133	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
138	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
139	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
141	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
142	C₂×C₃×C₈	—	2	2	no	—
143	C₂×C₃×C₈	—	2	2	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	#217	—	2	4	no	—
146	C₄⋊Dic₃	—	2	4	no	—
147	C₄×Dic₃	—	2	2	no	—
148	C₄×Dic₃	C₅×D₄	1	4	yes	—
149	C₂×C₃⋊C₈	—	2	2	no	—
150	C₂×C₃⋊C₈	—	2	2	no	—
151	C₂×C₃⋊C₈	—	2	2	no	—
152	C₂³×C₅	—	2	1	no	—
153	C₂³×C₅	—	2	1	no	—
154	C₂³×C₅	—	2	2	no	—
155	C₂³×C₅	—	2	2	no	—
156	C₂³×C₅	Dic₃⋊C₄	1	2	yes	—
157	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	3	4	no	—
163	C₂×C₄×C₅	—	6	2	no	—
164	C₂×C₄×C₅	—	6	2	no	—
165	C₂×C₄×C₅	—	6	2	no	—
166	C₂×C₄×C₅	—	6	2	no	—
167	C₂×C₄×C₅	—	6	2	no	—
168	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
169	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
170	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
171	C₂×C₄×C₅	#237	1	2	yes	—
172	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
173	C₅×C₈	—	4	2	no	—
174	C₅×C₈	—	12	2	no	—
175	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
176	C₂²×C₈	—	3	2	no	—
177	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
178	#117	—	6	4	no	—
179	C₂×C₄⋊C₄	—	3	2	no	—
180	C₂×C₂²⋊C₄	—	3	2	no	—
181	C₂×C₄²	—	3	1	no	—
182	C₂×C₄²	—	3	1	no	—
183	C₄⋊C₈	—	6	4	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	6	2	no	—
186	C₂×C₃×C₅	—	4	2	no	—
187	C₂×C₃×C₅	#380	1	1	yes	characteristic
188	C₂×C₃×C₅	#408	1	2	yes	—
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	#424	1	1	yes	characteristic
191	C₂×C₃×C₅	#433	1	2	yes	—
192	C₂³×C₃	—	2	1	no	—
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	2	2	no	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
197	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
206	C₂×C₃×C₄	C₅×#41	1	2	yes	—
207	C₂×Dic₃	—	2	2	no	—
208	C₂×Dic₃	—	2	2	no	—
209	C₂×Dic₃	—	2	2	no	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	C₅×C₂²⋊C₄	1	4	yes	—
213	C₃×C₈	—	4	2	no	—
214	C₃⋊C₈	—	4	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	4	2	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	#877	1	1	yes	characteristic
221	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	#901	1	2	yes	—
223	C₂²×C₅	#907	1	2	yes	—
224	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	6	4	no	—
227	C₄×C₅	—	12	2	no	—
228	C₄×C₅	#878	1	2	yes	—
229	C₄×C₅	#939	1	2	yes	—
230	C₂⁴	C₄×C₅×D₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	3	2	no	—
234	C₂²×C₄	—	3	2	no	—
235	C₂²×C₄	C₅×Dic₆	1	2	yes	—
236	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
237	C₂²×C₄	C₅×#67	1	2	yes	—
238	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	6	2	no	—
243	C₂×C₈	—	6	2	no	—
244	C₂×C₈	—	6	2	no	—
245	C₄⋊C₄	—	6	4	no	—
246	C₂²⋊C₄	—	6	4	no	—
247	C₄²	—	3	4	no	—
248	C₄²	—	6	2	no	—
249	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
257	C₂²×C₃	C₅×#117	1	1	yes	characteristic
258	C₂²×C₃	C₅×#131	1	2	yes	—
259	C₂²×C₃	C₅×#131	1	2	yes	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	C₅×#121	1	2	yes	—
262	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
263	Dic₃	—	2	4	no	—
264	Dic₃	—	4	2	no	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	#4720	1	2	yes	—
267	C₂×C₅	#4727	1	2	yes	—
268	C₂×C₅	#4942	1	1	yes	characteristic
269	C₂×C₅	#5102	1	1	yes	characteristic
270	C₂×C₅	#5119	1	1	yes	characteristic
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	C₅×Dic₃⋊C₄	1	2	yes	—
276	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	3	4	no	—
282	C₂×C₄	—	6	2	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	—	6	2	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	C₅×Dic₃⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
289	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
290	C₂×C₄	C₅×#237	1	2	yes	—
291	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₈	—	12	2	no	—
294	C₂×C₃	—	4	2	no	—
295	C₂×C₃	C₅×#380	1	1	yes	characteristic
296	C₂×C₃	C₅×#408	1	2	yes	—
297	C₂×C₃	C₅×#424	1	1	yes	characteristic
298	C₂×C₃	C₅×#424	1	1	yes	characteristic
299	C₂×C₃	C₅×#433	1	2	yes	—
300	C₅	#69909	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₅×#877	1	1	yes	characteristic
307	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
308	C₂²	C₅×#901	1	2	yes	—
309	C₂²	C₅×#907	1	2	yes	—
310	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	6	4	no	—
313	C₄	—	12	2	no	—
314	C₄	C₅×#878	1	2	yes	—
315	C₄	C₅×#939	1	2	yes	—
316	C₃	C₅×#1798	1	1	yes	characteristic, complemented, Sylow
317	C₂	—	4	2	no	—
318	C₂	C₅×#4720	1	2	yes	—
319	C₂	C₅×#4727	1	2	yes	—
320	C₂	C₅×#4942	1	1	yes	characteristic
321	C₂	C₅×#5102	1	1	yes	characteristic
322	C₂	C₅×#5119	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×C₂²⋊C₈	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×C₂²⋊C₈	D₃
hypocenter	C₃	C₅×#1798
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×Dic₃⋊C₈	C₂	semidirect
5. C₅×#4550	C₂	semidirect
18. #69909	C₅	direct (N × H)
46. C₂×Dic₃⋊C₈	C₂×C₅	semidirect
47. #4550	C₂×C₅	semidirect
249. C₃×C₅	#1798	semidirect
300. C₅	#69909	direct (N × H)
316. C₃	C₅×#1798	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#926	2
C₂×C₃×C₅×C₂²⋊C₈	2
C₂×C₅×#830	2
C₂×C₅×Dic₃⋊C₈	2
C₅×#4550	2
C₅×#1798	3
#69909	5

Subgroups — C5×#69909 #14297467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69909 #14297467