Subgroups — C₅×#71156 #14289484

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5579	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5092	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5084	C₂	1	1	yes	maximal, characteristic
5	C₅×#4535	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4483	C₂	1	1	yes	maximal, characteristic
7	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1632	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×#946	—	2	1	no	—
12	C₂×C₅×#216	—	2	2	no	—
13	C₅×#896	—	2	2	no	—
14	C₅×D₃×C₄⋊C₄	—	2	1	no	—
15	C₄×C₅×D₁₂	—	2	1	no	—
16	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₅×#870	—	2	1	no	—
18	C₅×#869	C₂²	1	1	yes	characteristic
19	C₃×C₅×#107	—	2	1	no	—
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₅×#103	—	2	1	no	—
22	C₅×#809	—	2	1	no	—
23	C₅×#808	—	2	1	no	—
24	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#71156	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#554	—	3	1	no	—
28	C₅×#494	—	3	1	no	—
29	C₅×#477	D₃	1	1	yes	characteristic, complemented
30	C₅×#476	—	3	1	no	—
31	C₅×#333	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×D₄⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
36	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	—	2	1	no	—
38	C₂×C₄×C₅×D₃	—	2	2	no	—
39	C₂×C₅×Dic₆	D₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₅×#216	—	4	4	no	—
48	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
49	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×C₄⋊Dic₃	—	2	1	no	—
52	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
53	C₅×Dic₃⋊C₄	—	2	2	no	—
54	C₅×Dic₃⋊C₄	—	2	2	no	—
55	C₂×C₅×C₃⋊C₈	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	#5579	C₂×C₅	1	1	yes	characteristic
58	C₃×#477	C₂×C₅	1	1	yes	characteristic, complemented
59	#5092	C₂×C₅	1	1	yes	characteristic, complemented
60	#5084	C₂×C₅	1	1	yes	characteristic
61	#4535	C₂×C₅	1	1	yes	characteristic, complemented
62	#4483	C₂×C₅	1	1	yes	characteristic
63	#4481	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×Q₁₆	—	6	2	no	—
65	C₅×C₄⋊Q₈	—	3	1	no	—
66	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#125	—	6	1	no	—
68	C₅×C₂²⋊Q₈	—	6	2	no	—
69	C₅×C₂²⋊Q₈	—	6	1	no	—
70	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
71	C₄×C₅×D₄	—	3	1	no	—
72	C₄×C₅×D₄	—	6	1	no	—
73	C₂×C₅×C₄⋊C₄	—	6	1	no	—
74	C₅×#107	—	2	1	no	—
75	C₅×#106	—	6	1	no	—
76	C₅×C₄⋊C₈	—	3	1	no	—
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₅×#103	—	2	1	no	—
79	C₅×#102	—	6	1	no	—
80	C₄×C₅×C₈	—	3	1	no	—
81	#1632	—	3	1	no	Sylow
82	C₂²×C₅×D₃	—	2	1	no	—
83	C₃×C₅×Q₈	—	2	1	no	—
84	C₃×C₅×Q₈	—	2	1	no	—
85	C₃×C₅×Q₈	—	4	2	no	—
86	C₂×C₃×C₄×C₅	—	2	1	no	—
87	C₂×C₃×C₄×C₅	—	2	1	no	—
88	C₂×C₃×C₄×C₅	—	4	1	no	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₅×D₁₂	—	2	1	no	—
95	C₅×D₁₂	Q₁₆	1	2	yes	complemented
96	C₄×C₅×D₃	—	4	1	no	—
97	C₄×C₅×D₃	—	4	1	no	—
98	C₄×C₅×D₃	—	8	2	no	—
99	C₅×Dic₆	—	4	2	no	—
100	C₃×C₅×C₈	—	4	1	no	—
101	C₅×C₃⋊C₈	—	2	2	no	—
102	C₅×C₃⋊C₈	—	4	1	no	—
103	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
105	#946	—	2	1	no	—
106	C₂×#216	—	2	2	no	—
107	#896	—	2	2	no	—
108	D₃×C₄⋊C₄	—	2	1	no	—
109	C₄×D₁₂	—	2	1	no	—
110	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
111	#870	—	2	1	no	—
112	#869	C₂²×C₅	1	1	yes	characteristic
113	C₃×#107	—	2	1	no	—
114	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
115	C₃×#103	—	2	1	no	—
116	#809	—	2	1	no	—
117	#808	—	2	1	no	—
118	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
119	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₅×Q₈	—	2	1	no	—
121	C₂×C₅×Q₈	—	3	2	no	—
122	C₂×C₅×Q₈	#67	1	1	yes	characteristic
123	C₂×C₅×D₄	—	3	1	no	—
124	C₂²×C₄×C₅	—	6	2	no	—
125	C₂²×C₄×C₅	—	6	1	no	—
126	C₅×Q₁₆	—	12	4	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	6	1	no	—
129	C₂×C₅×C₈	—	6	1	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	2	1	no	—
133	C₅×C₄⋊C₄	—	3	1	no	—
134	C₅×C₄⋊C₄	—	6	2	no	—
135	C₅×C₄⋊C₄	—	6	2	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₅×C₄⋊C₄	—	6	1	no	—
138	C₅×C₄⋊C₄	#67	1	1	yes	characteristic
139	C₅×C₂²⋊C₄	—	6	2	no	—
140	C₅×C₂²⋊C₄	—	6	1	no	—
141	C₄²×C₅	—	2	1	no	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	#554	—	3	1	no	—
144	#494	—	3	1	no	—
145	#477	C₅×D₃	1	1	yes	characteristic, complemented
146	#476	—	3	1	no	—
147	#333	—	3	1	no	—
148	Q₈⋊C₈	—	3	1	no	—
149	D₄⋊C₈	—	3	1	no	—
150	C₂²×C₃×C₅	#120	1	1	yes	characteristic
151	C₂×C₅×D₃	—	2	2	no	—
152	C₂×C₅×D₃	—	4	1	no	—
153	C₂×C₅×D₃	—	4	1	no	—
154	C₃×C₄×C₅	—	2	1	no	—
155	C₃×C₄×C₅	—	4	1	no	—
156	C₃×C₄×C₅	—	4	1	no	—
157	C₃×C₄×C₅	—	8	1	no	—
158	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
159	C₃×C₄×C₅	#135	1	1	yes	characteristic
160	C₃×C₄×C₅	#137	1	1	yes	characteristic
161	C₃×C₄×C₅	#137	1	1	yes	characteristic
162	C₅×Dic₃	—	4	1	no	—
163	C₅×Dic₃	—	4	2	no	—
164	C₂×C₃×Q₈	—	2	1	no	—
165	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
166	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
167	C₂×C₄×D₃	—	2	1	no	—
168	C₂×C₄×D₃	—	2	2	no	—
169	C₂×Dic₆	C₅×D₄	1	2	yes	—
170	C₂×C₃×C₈	—	2	1	no	—
171	C₃×C₄⋊C₄	—	2	1	no	—
172	C₃×C₄⋊C₄	—	2	1	no	—
173	C₃×C₄⋊C₄	—	2	1	no	—
174	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
175	C₃×C₄²	—	2	1	no	—
176	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
177	#216	—	4	4	no	—
178	(C₂×D₃)⋊C₄	—	2	2	no	—
179	(C₂×D₃)⋊C₄	—	2	1	no	—
180	C₄⋊Dic₃	—	2	1	no	—
181	C₄⋊Dic₃	—	2	1	no	—
182	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
183	Dic₃⋊C₄	—	2	2	no	—
184	Dic₃⋊C₄	—	2	2	no	—
185	C₂×C₃⋊C₈	—	2	1	no	—
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂³×C₅	—	6	1	no	—
188	C₅×Q₈	—	2	1	no	—
189	C₅×Q₈	—	2	1	no	—
190	C₅×Q₈	—	4	2	no	—
191	C₅×Q₈	—	12	2	no	—
192	C₅×D₄	—	3	2	no	—
193	C₅×D₄	—	6	1	no	—
194	C₂×C₄×C₅	—	2	1	no	—
195	C₂×C₄×C₅	—	2	1	no	—
196	C₂×C₄×C₅	—	4	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	6	2	no	—
199	C₂×C₄×C₅	—	12	1	no	—
200	C₂×C₄×C₅	—	12	1	no	—
201	C₂×C₄×C₅	—	24	2	no	—
202	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
205	C₅×C₈	—	4	1	no	—
206	C₅×C₈	—	6	2	no	—
207	C₅×C₈	—	12	1	no	—
208	C₂×Q₁₆	—	6	2	no	—
209	C₄⋊Q₈	—	3	1	no	—
210	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
211	#125	—	6	1	no	—
212	C₂²⋊Q₈	—	6	2	no	—
213	C₂²⋊Q₈	—	6	1	no	—
214	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
215	C₄×D₄	—	3	1	no	—
216	C₄×D₄	—	6	1	no	—
217	C₂×C₄⋊C₄	—	6	1	no	—
218	#107	—	2	1	no	—
219	#106	—	6	1	no	—
220	C₄⋊C₈	—	3	1	no	—
221	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
222	#103	—	2	1	no	—
223	#102	—	6	1	no	—
224	C₄×C₈	—	3	1	no	—
225	C₂×C₃×C₅	#451	1	1	yes	characteristic
226	C₂×C₃×C₅	#452	1	1	yes	characteristic
227	C₂×C₃×C₅	#453	1	1	yes	characteristic
228	C₅×D₃	—	4	2	no	—
229	C₂²×D₃	—	2	1	no	—
230	C₃×Q₈	—	2	1	no	—
231	C₃×Q₈	—	2	1	no	—
232	C₃×Q₈	—	4	2	no	—
233	C₂×C₃×C₄	—	2	1	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	4	1	no	—
236	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×Dic₃	—	2	1	no	—
240	C₂×Dic₃	—	2	2	no	—
241	D₁₂	—	2	1	no	—
242	D₁₂	C₅×Q₁₆	1	2	yes	complemented
243	C₄×D₃	—	4	1	no	—
244	C₄×D₃	—	4	1	no	—
245	C₄×D₃	—	8	2	no	—
246	Dic₆	—	4	2	no	—
247	C₃×C₈	—	4	1	no	—
248	C₃⋊C₈	—	2	2	no	—
249	C₃⋊C₈	—	4	1	no	—
250	C₂²×C₅	—	6	2	no	—
251	C₂²×C₅	—	12	1	no	—
252	C₂²×C₅	—	12	1	no	—
253	C₂²×C₅	#937	1	1	yes	characteristic
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	4	1	no	—
256	C₄×C₅	—	4	1	no	—
257	C₄×C₅	—	8	1	no	—
258	C₄×C₅	—	12	2	no	—
259	C₄×C₅	—	12	1	no	—
260	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
261	C₄×C₅	#918	1	1	yes	characteristic
262	C₄×C₅	#942	1	1	yes	characteristic
263	C₄×C₅	#950	1	1	yes	characteristic
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	3	2	no	—
266	C₂×Q₈	C₅×#67	1	1	yes	characteristic
267	C₂×D₄	—	3	1	no	—
268	C₂²×C₄	—	6	2	no	—
269	C₂²×C₄	—	6	1	no	—
270	Q₁₆	—	12	4	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	6	1	no	—
273	C₂×C₈	—	6	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	3	1	no	—
278	C₄⋊C₄	—	6	2	no	—
279	C₄⋊C₄	—	6	2	no	—
280	C₄⋊C₄	—	6	1	no	—
281	C₄⋊C₄	—	6	1	no	—
282	C₄⋊C₄	C₅×#67	1	1	yes	characteristic
283	C₂²⋊C₄	—	6	2	no	—
284	C₂²⋊C₄	—	6	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
287	C₃×C₅	#1632	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	C₅×#120	1	1	yes	characteristic
289	C₂×D₃	—	2	2	no	—
290	C₂×D₃	—	4	1	no	—
291	C₂×D₃	—	4	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	4	1	no	—
294	C₃×C₄	—	4	1	no	—
295	C₃×C₄	—	8	1	no	—
296	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
297	C₃×C₄	C₅×#135	1	1	yes	characteristic
298	C₃×C₄	C₅×#137	1	1	yes	characteristic
299	C₃×C₄	C₅×#137	1	1	yes	characteristic
300	Dic₃	—	4	1	no	—
301	Dic₃	—	4	2	no	—
302	C₂×C₅	—	12	2	no	—
303	C₂×C₅	#5047	1	1	yes	characteristic
304	C₂×C₅	#5186	1	1	yes	characteristic
305	C₂×C₅	#5198	1	1	yes	characteristic
306	C₂³	—	6	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	12	2	no	—
311	D₄	—	3	2	no	—
312	D₄	—	6	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	6	2	no	—
318	C₂×C₄	—	12	1	no	—
319	C₂×C₄	—	12	1	no	—
320	C₂×C₄	—	24	2	no	—
321	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
322	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
323	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
324	C₈	—	4	1	no	—
325	C₈	—	6	2	no	—
326	C₈	—	12	1	no	—
327	C₂×C₃	C₅×#451	1	1	yes	characteristic
328	C₂×C₃	C₅×#452	1	1	yes	characteristic
329	C₂×C₃	C₅×#453	1	1	yes	characteristic
330	D₃	—	4	2	no	—
331	C₅	#71156	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²	—	6	2	no	—
333	C₂²	—	12	1	no	—
334	C₂²	—	12	1	no	—
335	C₂²	C₅×#937	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	8	1	no	—
340	C₄	—	12	2	no	—
341	C₄	—	12	1	no	—
342	C₄	C₅×D₃×Q₁₆	1	1	yes	characteristic
343	C₄	C₅×#918	1	1	yes	characteristic
344	C₄	C₅×#942	1	1	yes	characteristic
345	C₄	C₅×#950	1	1	yes	characteristic
346	C₃	C₅×#1632	1	1	yes	characteristic, complemented, Sylow
347	C₂	—	12	2	no	—
348	C₂	C₅×#5047	1	1	yes	characteristic
349	C₂	C₅×#5186	1	1	yes	characteristic
350	C₂	C₅×#5198	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#477	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₅×D₁₂	D₄
hypercenter	C₅×#477	D₃
hypocenter	C₃	C₅×#1632
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#477

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#477	C₂	semidirect
3. C₅×#5092	C₂	semidirect
5. C₅×#4535	C₂	semidirect
7. C₅×#4481	C₂	semidirect
26. #71156	C₅	direct (N × H)
29. C₅×#477	D₃	semidirect
58. C₃×#477	C₂×C₅	semidirect
59. #5092	C₂×C₅	semidirect
61. #4535	C₂×C₅	semidirect
63. #4481	C₂×C₅	semidirect
95. C₅×D₁₂	Q₁₆	semidirect
145. #477	C₅×D₃	semidirect
242. D₁₂	C₅×Q₁₆	semidirect
287. C₃×C₅	#1632	semidirect
331. C₅	#71156	direct (N × H)
346. C₃	C₅×#1632	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5579	2
C₃×C₅×#477	2
C₅×#5092	2
C₅×#5084	2
C₅×#4535	2
C₅×#4483	2
C₅×#4481	2
C₅×#1632	3
#71156	5

Subgroups — C5×#71156 #14289484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71156 #14289484