Subgroups — C₅×#68406 #14289946

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5713	C₂	1	1	yes	maximal, characteristic
2	C₅×#5070	C₂	1	1	yes	maximal, characteristic
3	C₅×#4851	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4849	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	1	yes	maximal, characteristic
6	C₅×#4484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4484	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1640	—	3	1	no	maximal, Hall
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₅×#936	—	2	1	no	—
11	C₅×#936	—	2	1	no	—
12	C₅×#935	—	2	2	no	—
13	C₅×D₄×Dic₃	—	2	1	no	—
14	C₅×D₄×Dic₃	—	2	1	no	—
15	C₂×C₅×Dic₁₂	—	2	1	no	—
16	C₅×#869	C₂²	1	1	yes	characteristic
17	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₅×#832	—	2	2	no	—
22	C₅×#817	—	2	1	no	—
23	C₅×#816	—	2	1	no	—
24	C₅×#816	—	2	1	no	—
25	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#68406	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#559	—	3	1	no	—
28	C₅×#486	—	3	1	no	—
29	C₅×#475	—	3	1	no	—
30	C₅×#462	—	3	1	no	—
31	C₅×#331	D₃	1	1	yes	characteristic
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×Q₈⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
37	C₂²×C₅×Dic₃	—	2	2	no	—
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
40	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#217	—	2	2	no	—
45	C₅×#217	—	2	2	no	—
46	C₅×#217	—	2	2	no	—
47	C₅×C₄⋊Dic₃	—	2	1	no	—
48	C₅×C₄⋊Dic₃	—	2	1	no	—
49	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
50	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
51	C₅×Dic₃⋊C₄	—	2	1	no	—
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	1	no	—
54	C₄×C₅×Dic₃	—	2	1	no	—
55	C₂×C₅×C₃⋊C₈	—	2	1	no	—
56	C₅×Dic₁₂	—	4	2	no	—
57	#5713	C₂×C₅	1	1	yes	characteristic
58	#5070	C₂×C₅	1	1	yes	characteristic
59	#4851	C₂×C₅	1	1	yes	characteristic, complemented
60	#4849	C₂×C₅	1	1	yes	characteristic, complemented
61	C₃×#331	C₂×C₅	1	1	yes	characteristic
62	#4484	C₂×C₅	1	1	yes	characteristic, complemented
63	#4484	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×Q₁₆	—	6	1	no	—
65	C₅×C₄⋊Q₈	—	3	1	no	—
66	C₅×#127	C₂×D₃	1	1	yes	characteristic
67	C₅×#124	—	6	1	no	—
68	C₅×#124	—	6	1	no	—
69	C₅×#123	—	6	2	no	—
70	C₄×C₅×Q₈	—	3	1	no	—
71	C₄×C₅×Q₈	—	3	1	no	—
72	C₄×C₅×D₄	—	6	1	no	—
73	C₄×C₅×D₄	—	6	1	no	—
74	C₅×#106	—	6	1	no	—
75	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
76	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
77	C₅×#103	—	6	1	no	—
78	C₅×#103	—	6	1	no	—
79	C₅×#102	—	6	2	no	—
80	C₄×C₅×C₈	—	3	1	no	—
81	#1640	—	3	1	no	Sylow
82	C₂³×C₃×C₅	—	2	2	no	—
83	C₃×C₅×D₄	—	4	2	no	—
84	C₃×C₅×D₄	—	4	1	no	—
85	C₃×C₅×D₄	—	4	1	no	—
86	C₃×C₅×D₄	—	4	1	no	—
87	C₃×C₅×D₄	—	4	1	no	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	1	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₂×C₅×Dic₃	—	4	1	no	—
96	C₂×C₅×Dic₃	—	8	2	no	—
97	C₅×Dic₆	—	2	1	no	—
98	C₅×Dic₆	—	2	1	no	—
99	C₅×Dic₆	—	2	1	no	—
100	C₅×Dic₆	—	2	1	no	—
101	C₅×Dic₆	—	4	1	no	—
102	C₅×Dic₆	—	4	1	no	—
103	C₃×C₅×C₈	—	4	1	no	—
104	C₃×C₅×C₈	—	4	1	no	—
105	C₅×C₃⋊C₈	—	4	1	no	—
106	C₃×#127	C₂²×C₅	1	1	yes	characteristic
107	#936	—	2	1	no	—
108	#936	—	2	1	no	—
109	#935	—	2	2	no	—
110	D₄×Dic₃	—	2	1	no	—
111	D₄×Dic₃	—	2	1	no	—
112	C₂×Dic₁₂	—	2	1	no	—
113	#869	C₂²×C₅	1	1	yes	characteristic
114	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
115	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
116	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
117	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
118	#832	—	2	2	no	—
119	#817	—	2	1	no	—
120	#816	—	2	1	no	—
121	#816	—	2	1	no	—
122	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
123	C₂×C₅×Q₈	—	3	1	no	—
124	C₂×C₅×Q₈	—	3	1	no	—
125	C₂×C₅×Q₈	—	6	1	no	—
126	C₂×C₅×D₄	—	2	1	no	—
127	C₂×C₅×D₄	—	2	1	no	—
128	C₂×C₅×D₄	D₁₂	1	2	yes	—
129	C₂²×C₄×C₅	—	6	2	no	—
130	C₅×Q₁₆	—	12	2	no	—
131	C₂×C₅×C₈	—	2	1	no	—
132	C₂×C₅×C₈	—	2	1	no	—
133	C₂×C₅×C₈	—	6	1	no	—
134	C₅×C₄⋊C₄	—	3	1	no	—
135	C₅×C₄⋊C₄	—	3	1	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₅×C₄⋊C₄	—	6	1	no	—
138	C₅×C₄⋊C₄	—	6	1	no	—
139	C₅×C₄⋊C₄	—	6	1	no	—
140	C₅×C₂²⋊C₄	—	6	2	no	—
141	C₅×C₂²⋊C₄	—	6	2	no	—
142	C₅×C₂²⋊C₄	—	6	2	no	—
143	C₄²×C₅	—	6	1	no	—
144	C₄²×C₅	—	6	1	no	—
145	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
146	#559	—	3	1	no	—
147	#486	—	3	1	no	—
148	#475	—	3	1	no	—
149	#462	—	3	1	no	—
150	#331	C₅×D₃	1	1	yes	characteristic
151	Q₈⋊C₈	—	3	1	no	—
152	Q₈⋊C₈	—	3	1	no	—
153	C₂²×C₃×C₅	—	4	2	no	—
154	C₂²×C₃×C₅	—	4	2	no	—
155	C₂²×C₃×C₅	—	4	2	no	—
156	C₂²×C₃×C₅	#120	1	1	yes	characteristic
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	#135	1	1	yes	characteristic
159	C₃×C₄×C₅	#135	1	1	yes	characteristic
160	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₃×C₄×C₅	#137	1	1	yes	characteristic
162	C₅×Dic₃	—	4	1	no	—
163	C₅×Dic₃	—	4	1	no	—
164	C₅×Dic₃	—	4	1	no	—
165	C₅×Dic₃	—	4	1	no	—
166	C₅×Dic₃	—	8	1	no	—
167	C₂×C₃×D₄	—	2	1	no	—
168	C₂×C₃×D₄	—	2	1	no	—
169	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
170	C₂²×Dic₃	—	2	2	no	—
171	C₂×Dic₆	—	2	1	no	—
172	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
173	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
174	C₂×C₃×C₈	—	2	1	no	—
175	C₂×C₃×C₈	—	2	1	no	—
176	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
177	#217	—	2	2	no	—
178	#217	—	2	2	no	—
179	#217	—	2	2	no	—
180	C₄⋊Dic₃	—	2	1	no	—
181	C₄⋊Dic₃	—	2	1	no	—
182	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
183	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
184	Dic₃⋊C₄	—	2	1	no	—
185	Dic₃⋊C₄	—	2	1	no	—
186	C₄×Dic₃	—	2	1	no	—
187	C₄×Dic₃	—	2	1	no	—
188	C₂×C₃⋊C₈	—	2	1	no	—
189	Dic₁₂	—	4	2	no	—
190	C₂³×C₅	—	2	2	no	—
191	C₅×Q₈	—	6	1	no	—
192	C₅×Q₈	—	6	1	no	—
193	C₅×Q₈	—	6	1	no	—
194	C₅×Q₈	—	6	1	no	—
195	C₅×Q₈	—	12	1	no	—
196	C₅×Q₈	—	12	1	no	—
197	C₅×D₄	—	4	2	no	—
198	C₅×D₄	—	4	1	no	—
199	C₅×D₄	—	4	1	no	—
200	C₅×D₄	—	4	1	no	—
201	C₅×D₄	—	4	1	no	—
202	C₂×C₄×C₅	—	6	1	no	—
203	C₂×C₄×C₅	—	6	1	no	—
204	C₂×C₄×C₅	—	6	1	no	—
205	C₂×C₄×C₅	—	6	1	no	—
206	C₂×C₄×C₅	—	12	1	no	—
207	C₂×C₄×C₅	—	24	2	no	—
208	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
209	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
210	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
211	C₅×C₈	—	4	1	no	—
212	C₅×C₈	—	4	1	no	—
213	C₅×C₈	—	12	1	no	—
214	C₂×Q₁₆	—	6	1	no	—
215	C₄⋊Q₈	—	3	1	no	—
216	#127	C₂×C₅×D₃	1	1	yes	characteristic
217	#124	—	6	1	no	—
218	#124	—	6	1	no	—
219	#123	—	6	2	no	—
220	C₄×Q₈	—	3	1	no	—
221	C₄×Q₈	—	3	1	no	—
222	C₄×D₄	—	6	1	no	—
223	C₄×D₄	—	6	1	no	—
224	#106	—	6	1	no	—
225	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
226	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
227	#103	—	6	1	no	—
228	#103	—	6	1	no	—
229	#102	—	6	2	no	—
230	C₄×C₈	—	3	1	no	—
231	C₂×C₃×C₅	—	8	2	no	—
232	C₂×C₃×C₅	#449	1	1	yes	characteristic
233	C₂×C₃×C₅	#452	1	1	yes	characteristic
234	C₂×C₃×C₅	#455	1	1	yes	characteristic
235	C₂³×C₃	—	2	2	no	—
236	C₃×D₄	—	4	2	no	—
237	C₃×D₄	—	4	1	no	—
238	C₃×D₄	—	4	1	no	—
239	C₃×D₄	—	4	1	no	—
240	C₃×D₄	—	4	1	no	—
241	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×Dic₃	—	2	1	no	—
245	C₂×Dic₃	—	2	1	no	—
246	C₂×Dic₃	—	2	1	no	—
247	C₂×Dic₃	—	2	1	no	—
248	C₂×Dic₃	—	4	1	no	—
249	C₂×Dic₃	—	8	2	no	—
250	Dic₆	—	2	1	no	—
251	Dic₆	—	2	1	no	—
252	Dic₆	—	2	1	no	—
253	Dic₆	—	2	1	no	—
254	Dic₆	—	4	1	no	—
255	Dic₆	—	4	1	no	—
256	C₃×C₈	—	4	1	no	—
257	C₃×C₈	—	4	1	no	—
258	C₃⋊C₈	—	4	1	no	—
259	C₂²×C₅	—	4	2	no	—
260	C₂²×C₅	—	4	2	no	—
261	C₂²×C₅	—	4	2	no	—
262	C₂²×C₅	#882	1	1	yes	characteristic
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	12	1	no	—
265	C₄×C₅	—	12	1	no	—
266	C₄×C₅	—	12	1	no	—
267	C₄×C₅	—	12	1	no	—
268	C₄×C₅	—	24	1	no	—
269	C₄×C₅	#911	1	1	yes	characteristic
270	C₄×C₅	#912	1	1	yes	characteristic
271	C₄×C₅	#915	1	1	yes	characteristic
272	C₄×C₅	#916	1	1	yes	characteristic
273	C₂×Q₈	—	3	1	no	—
274	C₂×Q₈	—	3	1	no	—
275	C₂×Q₈	—	6	1	no	—
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	C₅×D₁₂	1	2	yes	—
279	C₂²×C₄	—	6	2	no	—
280	Q₁₆	—	12	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	6	1	no	—
284	C₄⋊C₄	—	3	1	no	—
285	C₄⋊C₄	—	3	1	no	—
286	C₄⋊C₄	—	6	1	no	—
287	C₄⋊C₄	—	6	1	no	—
288	C₄⋊C₄	—	6	1	no	—
289	C₄⋊C₄	—	6	1	no	—
290	C₂²⋊C₄	—	6	2	no	—
291	C₂²⋊C₄	—	6	2	no	—
292	C₂²⋊C₄	—	6	2	no	—
293	C₄²	—	6	1	no	—
294	C₄²	—	6	1	no	—
295	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
296	C₃×C₅	#1640	1	1	yes	characteristic, complemented, Hall
297	C₂²×C₃	—	4	2	no	—
298	C₂²×C₃	—	4	2	no	—
299	C₂²×C₃	—	4	2	no	—
300	C₂²×C₃	C₅×#120	1	1	yes	characteristic
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	C₅×#135	1	1	yes	characteristic
303	C₃×C₄	C₅×#135	1	1	yes	characteristic
304	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₃×C₄	C₅×#137	1	1	yes	characteristic
306	Dic₃	—	4	1	no	—
307	Dic₃	—	4	1	no	—
308	Dic₃	—	4	1	no	—
309	Dic₃	—	4	1	no	—
310	Dic₃	—	8	1	no	—
311	C₂×C₅	—	8	2	no	—
312	C₂×C₅	#4746	1	1	yes	characteristic
313	C₂×C₅	#4781	1	1	yes	characteristic
314	C₂×C₅	#4783	1	1	yes	characteristic
315	C₂³	—	2	2	no	—
316	Q₈	—	6	1	no	—
317	Q₈	—	6	1	no	—
318	Q₈	—	6	1	no	—
319	Q₈	—	6	1	no	—
320	Q₈	—	12	1	no	—
321	Q₈	—	12	1	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	1	no	—
324	D₄	—	4	1	no	—
325	D₄	—	4	1	no	—
326	D₄	—	4	1	no	—
327	C₂×C₄	—	6	1	no	—
328	C₂×C₄	—	6	1	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	6	1	no	—
331	C₂×C₄	—	12	1	no	—
332	C₂×C₄	—	24	2	no	—
333	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
334	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
335	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	12	1	no	—
339	C₂×C₃	—	8	2	no	—
340	C₂×C₃	C₅×#449	1	1	yes	characteristic
341	C₂×C₃	C₅×#452	1	1	yes	characteristic
342	C₂×C₃	C₅×#455	1	1	yes	characteristic
343	C₅	#68406	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	C₅×#882	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	12	1	no	—
350	C₄	—	12	1	no	—
351	C₄	—	12	1	no	—
352	C₄	—	12	1	no	—
353	C₄	—	24	1	no	—
354	C₄	C₅×#911	1	1	yes	characteristic
355	C₄	C₅×#912	1	1	yes	characteristic
356	C₄	C₅×#915	1	1	yes	characteristic
357	C₄	C₅×#916	1	1	yes	characteristic
358	C₃	C₅×#1640	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	8	2	no	—
360	C₂	C₅×#4746	1	1	yes	characteristic
361	C₂	C₅×#4781	1	1	yes	characteristic
362	C₂	C₅×#4783	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#331	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₃×C₅×#127	C₂²
hypercenter	C₅×#331	D₃
hypocenter	C₃	C₅×#1640
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#331

Complements

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

N	H ≅ G/N	type
3. C₅×#4851	C₂	semidirect
4. C₅×#4849	C₂	semidirect
6. C₅×#4484	C₂	semidirect
7. C₅×#4484	C₂	semidirect
26. #68406	C₅	direct (N × H)
59. #4851	C₂×C₅	semidirect
60. #4849	C₂×C₅	semidirect
62. #4484	C₂×C₅	semidirect
63. #4484	C₂×C₅	semidirect
296. C₃×C₅	#1640	semidirect
343. C₅	#68406	direct (N × H)
358. C₃	C₅×#1640	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5713	2
C₅×#5070	2
C₅×#4851	2
C₅×#4849	2
C₃×C₅×#331	2
C₅×#4484	2
C₅×#1640	3
#68406	5

Subgroups — C5×#68406 #14289946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68406 #14289946