Subgroups — C₅×#68427 #14285168

Order 1920.

396 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	C₅×#5728	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic
3	C₅×#4833	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1554	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅×#946	—	2	1	no	—
11	C₅×Q₈×Dic₃	—	2	1	no	—
12	C₅×#897	C₄	1	2	yes	—
13	C₅×#892	—	2	1	no	—
14	C₅×#890	—	2	1	no	—
15	C₅×#887	C₄	1	2	yes	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
21	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₅×#815	—	2	1	no	—
23	C₅×C₈×Dic₃	—	2	1	no	—
24	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#68427	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#533	—	3	1	no	—
28	C₅×#432	—	3	1	no	—
29	C₅×#431	—	3	1	no	—
30	C₅×#333	—	3	1	no	—
31	C₅×#333	—	3	1	no	—
32	C₅×#332	—	3	1	no	—
33	C₅×#332	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	—	2	1	no	—
47	C₅×C₄⋊Dic₃	—	2	2	no	—
48	C₅×Dic₃⋊C₄	—	2	2	no	—
49	C₄×C₅×Dic₃	—	2	2	no	—
50	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
52	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
53	C₂×C₅×C₃⋊C₈	—	2	1	no	—
54	C₂×C₅×C₃⋊C₈	—	2	1	no	—
55	#5728	C₂×C₅	1	1	yes	characteristic
56	#4835	C₂×C₅	1	1	yes	characteristic
57	#4833	C₂×C₅	1	1	yes	characteristic
58	C₃×#332	C₂×C₅	1	1	yes	characteristic, complemented
59	#4547	C₂×C₅	1	1	yes	characteristic, complemented
60	#4481	C₂×C₅	1	1	yes	characteristic, complemented
61	#4481	C₂×C₅	1	1	yes	characteristic, complemented
62	C₅×C₄⋊Q₈	—	3	1	no	—
63	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
64	C₅×#125	—	6	1	no	—
65	C₅×C₂²⋊Q₈	—	3	2	no	—
66	C₅×C₂²⋊Q₈	—	6	1	no	—
67	C₄×C₅×Q₈	—	3	2	no	—
68	C₄×C₅×Q₈	—	6	1	no	—
69	C₅×#117	—	6	1	no	—
70	C₂×C₄²×C₅	—	3	1	no	—
71	C₅×C₄⋊C₈	—	3	1	no	—
72	C₅×C₄⋊C₈	—	3	1	no	—
73	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
75	C₅×C₂²⋊C₈	—	6	1	no	—
76	C₅×C₂²⋊C₈	—	6	1	no	—
77	C₅×#97	—	6	1	no	—
78	C₄×C₅×C₈	—	6	1	no	—
79	#1554	—	3	1	no	Sylow
80	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₃×C₅×Q₈	—	4	1	no	—
82	C₃×C₅×Q₈	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₅×Dic₃	—	2	2	no	—
88	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₅×D₃	—	2	1	no	—
92	C₄×C₅×D₃	—	2	1	no	—
93	C₄×C₅×D₃	—	2	1	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₄×C₅×D₃	—	4	1	no	—
96	C₅×Dic₆	—	4	2	no	—
97	C₃×C₅×C₈	—	4	1	no	—
98	C₃×C₅×C₈	—	4	1	no	—
99	C₅×C₃⋊C₈	—	4	1	no	—
100	C₅×C₃⋊C₈	—	4	1	no	—
101	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
102	#946	—	2	1	no	—
103	Q₈×Dic₃	—	2	1	no	—
104	#897	C₄×C₅	1	2	yes	—
105	#892	—	2	1	no	—
106	#890	—	2	1	no	—
107	#887	C₄×C₅	1	2	yes	—
108	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
109	#869	C₂²×C₅	1	1	yes	characteristic
110	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	(C₂×D₃)⋊C₈	—	2	1	no	—
113	(C₂×D₃)⋊C₈	—	2	1	no	—
114	#815	—	2	1	no	—
115	C₈×Dic₃	—	2	1	no	—
116	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
117	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₅×Q₈	—	2	1	no	—
119	C₂×C₅×Q₈	—	3	2	no	—
120	C₂²×C₄×C₅	—	3	1	no	—
121	C₂²×C₄×C₅	—	3	1	no	—
122	C₂²×C₄×C₅	—	3	1	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	6	1	no	—
126	C₂×C₅×C₈	—	6	1	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	—	6	2	no	—
129	C₅×C₄⋊C₄	—	6	2	no	—
130	C₅×C₄⋊C₄	—	6	1	no	—
131	C₅×C₄⋊C₄	—	6	1	no	—
132	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
133	C₅×C₂²⋊C₄	—	6	2	no	—
134	C₄²×C₅	—	3	1	no	—
135	C₄²×C₅	—	3	1	no	—
136	C₄²×C₅	—	3	1	no	—
137	C₄²×C₅	—	6	2	no	—
138	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
139	#533	—	3	1	no	—
140	#432	—	3	1	no	—
141	#431	—	3	1	no	—
142	#333	—	3	1	no	—
143	#333	—	3	1	no	—
144	#332	—	3	1	no	—
145	#332	C₅×D₃	1	1	yes	characteristic, complemented
146	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₅×D₃	—	2	1	no	—
148	C₂×C₅×D₃	—	2	1	no	—
149	C₂×C₅×D₃	—	2	1	no	—
150	C₃×C₄×C₅	—	2	1	no	—
151	C₃×C₄×C₅	—	4	2	no	—
152	C₃×C₄×C₅	#135	1	1	yes	characteristic
153	C₃×C₄×C₅	#135	1	1	yes	characteristic
154	C₃×C₄×C₅	#137	1	1	yes	characteristic
155	C₃×C₄×C₅	#137	1	1	yes	characteristic
156	C₅×Dic₃	—	2	1	no	—
157	C₅×Dic₃	—	2	1	no	—
158	C₅×Dic₃	—	2	1	no	—
159	C₅×Dic₃	—	4	2	no	—
160	C₂×C₃×Q₈	—	2	1	no	—
161	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
163	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
164	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
165	C₂×C₃×C₈	—	2	1	no	—
166	C₂×C₃×C₈	—	2	1	no	—
167	C₃×C₄⋊C₄	—	2	1	no	—
168	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
169	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
170	(C₂×D₃)⋊C₄	—	2	2	no	—
171	C₄⋊Dic₃	—	2	1	no	—
172	C₄⋊Dic₃	—	2	1	no	—
173	C₄⋊Dic₃	—	2	2	no	—
174	Dic₃⋊C₄	—	2	2	no	—
175	C₄×Dic₃	—	2	2	no	—
176	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
177	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
178	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
179	C₂×C₃⋊C₈	—	2	1	no	—
180	C₂×C₃⋊C₈	—	2	1	no	—
181	C₂³×C₅	—	3	1	no	—
182	C₅×Q₈	—	4	1	no	—
183	C₅×Q₈	—	4	1	no	—
184	C₅×Q₈	—	12	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	3	1	no	—
187	C₂×C₄×C₅	—	3	1	no	—
188	C₂×C₄×C₅	—	3	1	no	—
189	C₂×C₄×C₅	—	6	1	no	—
190	C₂×C₄×C₅	—	6	1	no	—
191	C₂×C₄×C₅	—	6	1	no	—
192	C₂×C₄×C₅	—	6	1	no	—
193	C₂×C₄×C₅	—	6	2	no	—
194	C₂×C₄×C₅	—	12	1	no	—
195	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
196	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
198	C₅×C₈	—	4	1	no	—
199	C₅×C₈	—	4	1	no	—
200	C₅×C₈	—	12	1	no	—
201	C₅×C₈	—	12	1	no	—
202	C₄⋊Q₈	—	3	1	no	—
203	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
204	#125	—	6	1	no	—
205	C₂²⋊Q₈	—	3	2	no	—
206	C₂²⋊Q₈	—	6	1	no	—
207	C₄×Q₈	—	3	2	no	—
208	C₄×Q₈	—	6	1	no	—
209	#117	—	6	1	no	—
210	C₂×C₄²	—	3	1	no	—
211	C₄⋊C₈	—	3	1	no	—
212	C₄⋊C₈	—	3	1	no	—
213	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
214	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
215	C₂²⋊C₈	—	6	1	no	—
216	C₂²⋊C₈	—	6	1	no	—
217	#97	—	6	1	no	—
218	C₄×C₈	—	6	1	no	—
219	C₂×C₃×C₅	#413	1	1	yes	characteristic
220	C₂×C₃×C₅	#416	1	1	yes	characteristic
221	C₂×C₃×C₅	#419	1	1	yes	characteristic
222	C₅×D₃	—	4	1	no	—
223	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₃×Q₈	—	4	1	no	—
225	C₃×Q₈	—	4	1	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂×Dic₃	—	2	2	no	—
231	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₄×D₃	—	2	1	no	—
235	C₄×D₃	—	2	1	no	—
236	C₄×D₃	—	2	1	no	—
237	C₄×D₃	—	2	1	no	—
238	C₄×D₃	—	4	1	no	—
239	Dic₆	—	4	2	no	—
240	C₃×C₈	—	4	1	no	—
241	C₃×C₈	—	4	1	no	—
242	C₃⋊C₈	—	4	1	no	—
243	C₃⋊C₈	—	4	1	no	—
244	C₂²×C₅	—	6	1	no	—
245	C₂²×C₅	—	6	1	no	—
246	C₂²×C₅	—	6	1	no	—
247	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	4	2	no	—
250	C₄×C₅	—	6	1	no	—
251	C₄×C₅	—	6	1	no	—
252	C₄×C₅	—	6	1	no	—
253	C₄×C₅	—	12	2	no	—
254	C₄×C₅	#912	1	1	yes	characteristic
255	C₄×C₅	#915	1	1	yes	characteristic
256	C₄×C₅	#916	1	1	yes	characteristic
257	C₄×C₅	#918	1	1	yes	characteristic
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	3	2	no	—
260	C₂²×C₄	—	3	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	3	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	6	1	no	—
266	C₂×C₈	—	6	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	6	2	no	—
269	C₄⋊C₄	—	6	2	no	—
270	C₄⋊C₄	—	6	1	no	—
271	C₄⋊C₄	—	6	1	no	—
272	C₄⋊C₄	C₄×C₅×D₃	1	2	yes	—
273	C₂²⋊C₄	—	6	2	no	—
274	C₄²	—	3	1	no	—
275	C₄²	—	3	1	no	—
276	C₄²	—	3	1	no	—
277	C₄²	—	6	2	no	—
278	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
279	C₃×C₅	#1554	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×D₃	—	2	1	no	—
282	C₂×D₃	—	2	1	no	—
283	C₂×D₃	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	C₅×#135	1	1	yes	characteristic
287	C₃×C₄	C₅×#135	1	1	yes	characteristic
288	C₃×C₄	C₅×#137	1	1	yes	characteristic
289	C₃×C₄	C₅×#137	1	1	yes	characteristic
290	Dic₃	—	2	1	no	—
291	Dic₃	—	2	1	no	—
292	Dic₃	—	2	1	no	—
293	Dic₃	—	4	2	no	—
294	C₂×C₅	—	12	1	no	—
295	C₂×C₅	#4751	1	1	yes	characteristic
296	C₂×C₅	#4772	1	1	yes	characteristic
297	C₂×C₅	#4802	1	1	yes	characteristic
298	C₂³	—	3	1	no	—
299	Q₈	—	4	1	no	—
300	Q₈	—	4	1	no	—
301	Q₈	—	12	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	3	1	no	—
304	C₂×C₄	—	3	1	no	—
305	C₂×C₄	—	3	1	no	—
306	C₂×C₄	—	6	1	no	—
307	C₂×C₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	6	1	no	—
310	C₂×C₄	—	6	2	no	—
311	C₂×C₄	—	12	1	no	—
312	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
313	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
314	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	12	1	no	—
318	C₈	—	12	1	no	—
319	C₂×C₃	C₅×#413	1	1	yes	characteristic
320	C₂×C₃	C₅×#416	1	1	yes	characteristic
321	C₂×C₃	C₅×#419	1	1	yes	characteristic
322	D₃	—	4	1	no	—
323	C₅	#68427	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	6	1	no	—
325	C₂²	—	6	1	no	—
326	C₂²	—	6	1	no	—
327	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	—	6	1	no	—
331	C₄	—	6	1	no	—
332	C₄	—	6	1	no	—
333	C₄	—	12	2	no	—
334	C₄	C₅×#912	1	1	yes	characteristic
335	C₄	C₅×#915	1	1	yes	characteristic
336	C₄	C₅×#916	1	1	yes	characteristic
337	C₄	C₅×#918	1	1	yes	characteristic
338	C₃	C₅×#1554	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	12	1	no	—
340	C₂	C₅×#4751	1	1	yes	characteristic
341	C₂	C₅×#4772	1	1	yes	characteristic
342	C₂	C₅×#4802	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#332	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#332	D₃
hypocenter	C₃	C₅×#1554
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#332

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#332	C₂	semidirect
5. C₅×#4547	C₂	semidirect
6. C₅×#4481	C₂	semidirect
7. C₅×#4481	C₂	semidirect
26. #68427	C₅	direct (N × H)
33. C₅×#332	D₃	semidirect
58. C₃×#332	C₂×C₅	semidirect
59. #4547	C₂×C₅	semidirect
60. #4481	C₂×C₅	semidirect
61. #4481	C₂×C₅	semidirect
145. #332	C₅×D₃	semidirect
279. C₃×C₅	#1554	semidirect
323. C₅	#68427	direct (N × H)
338. C₃	C₅×#1554	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5728	2
C₅×#4835	2
C₅×#4833	2
C₃×C₅×#332	2
C₅×#4547	2
C₅×#4481	2
C₅×#1554	3
#68427	5

Subgroups — C5×#68427 #14285168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68427 #14285168