Subgroups — C₅×#71643 #14295630

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#837	C₂	1	2	yes	maximal
3	C₅×#4932	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4532	C₂	1	2	yes	maximal, complemented
6	C₅×#1759	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₄	1	2	yes	complemented
9	C₃×C₄×C₅×D₄	—	2	2	no	—
10	C₃×C₄×C₅×D₄	C₄	1	2	yes	complemented
11	C₃×C₅×#117	—	2	1	no	—
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₅×#837	C₄	1	8	yes	complemented
18	C₅×C₃⋊#95	—	2	2	no	—
19	C₅×#803	—	2	1	no	—
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	#71643	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₄×C₅×#41	D₃	1	1	yes	characteristic
23	C₂×C₅×#104	—	3	2	no	—
24	C₄×C₅×M₁₆	—	3	1	no	—
25	C₅×#376	—	3	1	no	—
26	C₅×#339	—	3	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	—	2	1	no	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₅×Dic₃	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₄²×C₅	—	2	1	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₄×C₅×Dic₃	—	2	2	no	—
47	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
48	C₅×#208	—	2	2	no	—
49	C₅×#208	C₂×C₄	1	4	yes	complemented
50	C₂×C₅×C₃⋊C₈	—	2	2	no	—
51	C₃×C₄×#41	C₂×C₅	1	1	yes	characteristic
52	C₂×#837	C₂×C₅	1	2	yes	—
53	#4932	C₂×C₅	1	1	yes	characteristic, complemented
54	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
55	#4532	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×M₁₆	—	3	2	no	—
58	C₄×C₅×Q₈	Dic₃	1	2	yes	complemented
59	C₄×C₅×D₄	—	2	2	no	—
60	C₄×C₅×D₄	Dic₃	1	2	yes	complemented
61	C₅×#117	—	2	1	no	—
62	C₅×#117	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄²×C₅	—	2	1	no	—
64	C₂×C₄²×C₅	—	3	2	no	—
65	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
66	C₅×#104	—	3	8	no	—
67	C₅×#97	—	6	1	no	—
68	C₄×C₅×C₈	—	6	1	no	—
69	C₅×#95	—	6	2	no	—
70	#1759	—	3	1	no	Sylow
71	C₂³×C₃×C₅	—	2	1	no	—
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	2	1	no	—
74	C₃×C₅×Q₈	C₄²	1	2	yes	—
75	C₃×C₅×D₄	—	2	4	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₄²	1	2	yes	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	—	4	4	no	—
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	2	yes	—
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	4	4	no	—
91	C₅×C₃⋊C₈	—	2	4	no	—
92	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×Q₈	C₄×C₅	1	2	yes	complemented
94	C₃×C₄×D₄	—	2	2	no	—
95	C₃×C₄×D₄	C₄×C₅	1	2	yes	complemented
96	C₃×#117	—	2	1	no	—
97	C₃×#117	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₃×C₄²	—	2	1	no	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
101	C₂×#208	C₂²×C₅	1	2	yes	—
102	#837	C₄×C₅	1	8	yes	complemented
103	C₃⋊#95	—	2	2	no	—
104	#803	—	2	1	no	—
105	C₄×C₃⋊C₈	—	2	1	no	—
106	C₅×#41	—	2	2	no	—
107	C₅×#41	C₄×D₃	1	4	yes	—
108	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₅×D₄	—	2	1	no	—
110	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	2	2	no	—
113	C₂²×C₄×C₅	—	6	2	no	—
114	C₂²×C₄×C₅	#67	1	2	yes	—
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₅×M₁₆	—	3	4	no	—
117	C₅×M₁₆	—	6	2	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
121	C₅×C₂²⋊C₄	—	2	1	no	—
122	C₅×C₂²⋊C₄	—	2	2	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	2	1	no	—
125	C₄²×C₅	—	3	4	no	—
126	C₄²×C₅	—	6	2	no	—
127	C₄²×C₅	C₂×Dic₃	1	2	yes	—
128	C₄²×C₅	#67	1	1	yes	characteristic
129	C₄²×C₅	#67	1	1	yes	characteristic
130	C₄×#41	C₅×D₃	1	1	yes	characteristic
131	C₂×#104	—	3	2	no	—
132	C₄×M₁₆	—	3	1	no	—
133	#376	—	3	1	no	—
134	#339	—	3	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	—	2	2	no	—
143	C₃×C₄×C₅	—	4	2	no	—
144	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
145	C₃×C₄×C₅	#117	1	1	yes	characteristic
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₅×Dic₃	—	4	4	no	—
148	C₃×#41	—	2	2	no	—
149	C₃×#41	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₂²×Dic₃	—	2	2	no	—
158	C₃×C₄⋊C₄	—	2	2	no	—
159	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₂²⋊C₄	—	2	2	no	—
162	C₃×C₄²	—	2	2	no	—
163	C₃×C₄²	—	2	1	no	—
164	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
165	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
166	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
167	C₄×Dic₃	—	2	2	no	—
168	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	complemented
169	#208	—	2	2	no	—
170	#208	C₂×C₄×C₅	1	4	yes	complemented
171	C₂×C₃⋊C₈	—	2	2	no	—
172	C₂³×C₅	—	2	1	no	—
173	C₂³×C₅	C₂×#67	1	1	yes	characteristic
174	C₅×Q₈	—	2	1	no	—
175	C₅×Q₈	C₄×Dic₃	1	2	yes	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	2	1	no	—
178	C₅×D₄	C₄×Dic₃	1	2	yes	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	—	4	4	no	—
184	C₂×C₄×C₅	—	6	4	no	—
185	C₂×C₄×C₅	—	12	4	no	—
186	C₂×C₄×C₅	#217	1	2	yes	—
187	C₂×C₄×C₅	#217	1	2	yes	—
188	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
189	C₂×C₄×C₅	#235	1	2	yes	—
190	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
192	C₅×C₈	—	6	4	no	—
193	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
194	C₂×M₁₆	—	3	2	no	—
195	C₄×Q₈	C₅×Dic₃	1	2	yes	complemented
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	C₅×Dic₃	1	2	yes	complemented
198	#117	—	2	1	no	—
199	#117	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	3	2	no	—
202	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
203	#104	—	3	8	no	—
204	#97	—	6	1	no	—
205	C₄×C₈	—	6	1	no	—
206	#95	—	6	2	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	—	4	2	no	—
209	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₅	#421	1	2	yes	—
211	C₂³×C₃	—	2	1	no	—
212	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
213	C₃×Q₈	—	2	1	no	—
214	C₃×Q₈	C₄²×C₅	1	2	yes	—
215	C₃×D₄	—	2	4	no	—
216	C₃×D₄	—	2	1	no	—
217	C₃×D₄	C₄²×C₅	1	2	yes	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
224	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
225	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
226	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₅×#41	1	2	yes	—
229	C₂×Dic₃	—	2	4	no	—
230	C₂×Dic₃	—	4	4	no	—
231	C₃⋊C₈	—	2	4	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	2	2	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	C₄×#67	1	2	yes	—
236	C₂²×C₅	C₂×#217	1	1	yes	characteristic
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	—	12	4	no	—
242	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
243	C₄×C₅	#926	1	1	yes	characteristic
244	C₄×C₅	C₄×#67	1	2	yes	—
245	#41	—	2	2	no	—
246	#41	C₄×C₅×D₃	1	4	yes	—
247	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	6	2	no	—
253	C₂²×C₄	C₅×#67	1	2	yes	—
254	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
255	M₁₆	—	3	4	no	—
256	M₁₆	—	6	2	no	—
257	C₂×C₈	—	6	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	3	4	no	—
265	C₄²	—	6	2	no	—
266	C₄²	C₂×C₅×Dic₃	1	2	yes	—
267	C₄²	C₅×#67	1	1	yes	characteristic
268	C₄²	C₅×#67	1	1	yes	characteristic
269	C₃×C₅	#1759	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
274	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
280	C₃×C₄	C₅×#117	1	1	yes	characteristic
281	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
282	Dic₃	—	4	4	no	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	—	4	2	no	—
285	C₂×C₅	C₄×#217	1	1	yes	characteristic
286	C₂×C₅	#5236	1	2	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	C₂×C₅×#67	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	C₄×C₅×Dic₃	1	2	yes	—
291	D₄	—	2	4	no	—
292	D₄	—	2	1	no	—
293	D₄	C₄×C₅×Dic₃	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	6	4	no	—
300	C₂×C₄	—	12	4	no	—
301	C₂×C₄	C₅×#217	1	2	yes	—
302	C₂×C₄	C₅×#217	1	2	yes	—
303	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
304	C₂×C₄	C₅×#235	1	2	yes	—
305	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
307	C₈	—	6	4	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₂×C₃	C₅×#421	1	2	yes	—
312	C₅	#71643	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₄×C₅×#67	1	2	yes	—
317	C₂²	C₂×C₅×#217	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	12	4	no	—
323	C₄	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
324	C₄	C₅×#926	1	1	yes	characteristic
325	C₄	C₄×C₅×#67	1	2	yes	—
326	C₃	C₅×#1759	1	1	yes	characteristic, complemented, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	4	2	no	—
329	C₂	C₄×C₅×#217	1	1	yes	characteristic
330	C₂	C₅×#5236	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₄×C₅×#41	D₃
hypocenter	C₃	C₅×#1759
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₄×C₅×#41

Complements

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

N	H ≅ G/N	type
3. C₅×#4932	C₂	semidirect
4. C₄×C₅×#208	C₂	semidirect
5. C₅×#4532	C₂	semidirect
8. C₃×C₄×C₅×Q₈	C₄	semidirect
10. C₃×C₄×C₅×D₄	C₄	semidirect
17. C₅×#837	C₄	semidirect
21. #71643	C₅	direct (N × H)
47. C₄×C₅×Dic₃	C₂×C₄	semidirect
49. C₅×#208	C₂×C₄	semidirect
53. #4932	C₂×C₅	semidirect
54. C₄×#208	C₂×C₅	semidirect
55. #4532	C₂×C₅	semidirect
58. C₄×C₅×Q₈	Dic₃	semidirect
60. C₄×C₅×D₄	Dic₃	semidirect
93. C₃×C₄×Q₈	C₄×C₅	semidirect
95. C₃×C₄×D₄	C₄×C₅	semidirect
102. #837	C₄×C₅	semidirect
168. C₄×Dic₃	C₂×C₄×C₅	semidirect
170. #208	C₂×C₄×C₅	semidirect
195. C₄×Q₈	C₅×Dic₃	semidirect
197. C₄×D₄	C₅×Dic₃	semidirect
269. C₃×C₅	#1759	semidirect
312. C₅	#71643	direct (N × H)
326. C₃	C₅×#1759	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×#41	2
C₂×C₅×#837	2
C₅×#4932	2
C₄×C₅×#208	2
C₅×#4532	2
C₅×#1759	3
#71643	5

Subgroups — C5×#71643 #14295630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71643 #14295630