Subgroups — C₅×#70805 #14284129

Order 1920.

432 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×#5479	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4465	C₂	1	2	yes	maximal, complemented
5	C₅×#4464	C₂	1	2	yes	maximal, complemented
6	C₅×#1529	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#930	C₄	1	1	yes	characteristic
9	C₅×#929	—	2	1	no	—
10	C₄×C₅×#67	—	2	1	no	—
11	C₅×#924	C₄	1	1	yes	characteristic
12	C₂×C₅×#208	—	2	1	no	—
13	C₅×#876	—	2	1	no	—
14	C₅×#875	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₁₂	C₄	1	1	yes	characteristic
16	C₅×#872	—	2	1	no	—
17	C₅×#870	C₂²	1	1	yes	characteristic
18	C₄×C₅×Dic₆	C₄	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	—	2	1	no	—
20	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
21	C₃×C₅×#97	C₂²	1	2	yes	—
22	C₅×#803	C₂²	1	2	yes	—
23	C₄×C₅×C₃⋊C₈	—	2	1	no	—
24	#70805	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#529	—	3	1	no	—
26	C₅×#432	D₃	1	1	yes	characteristic, complemented
27	C₄×C₅×M₁₆	—	3	1	no	—
28	C₅×#330	—	3	2	no	—
29	C₅×#329	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×#67	—	2	1	no	—
34	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×#217	—	2	1	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₅×Dic₃⋊C₄	—	2	1	no	—
50	C₅×Dic₃⋊C₄	—	2	1	no	—
51	C₅×Dic₃⋊C₄	—	2	1	no	—
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	1	no	—
54	C₅×#208	—	2	4	no	—
55	C₂×C₅×C₃⋊C₈	—	2	2	no	—
56	#5479	C₂×C₅	1	1	yes	characteristic
57	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
58	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
59	#4465	C₂×C₅	1	2	yes	complemented
60	#4464	C₂×C₅	1	2	yes	complemented
61	C₂×C₅×M₁₆	—	6	1	no	—
62	C₅×#126	—	6	1	no	—
63	C₅×#125	—	3	1	no	—
64	C₅×#124	—	3	1	no	—
65	C₅×#123	—	6	1	no	—
66	C₅×C₂²⋊Q₈	—	3	1	no	—
67	C₅×#121	—	3	1	no	—
68	C₄×C₅×Q₈	—	3	1	no	—
69	C₄×C₅×D₄	—	3	1	no	—
70	C₄×C₅×D₄	—	6	1	no	—
71	C₅×#117	—	6	1	no	—
72	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	C₅×C₄⋊C₈	—	2	1	no	—
74	C₅×C₂²⋊C₈	—	2	1	no	—
75	C₅×#97	—	3	2	no	—
76	C₅×#97	C₂×D₃	1	2	yes	—
77	C₄×C₅×C₈	—	6	1	no	—
78	#1529	—	3	1	no	Sylow
79	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×#67	—	4	2	no	—
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	1	no	—
94	C₅×D₁₂	—	4	1	no	—
95	C₄×C₅×D₃	—	4	2	no	—
96	C₅×Dic₆	—	4	1	no	—
97	C₃×C₅×C₈	—	4	2	no	—
98	C₅×C₃⋊C₈	—	2	4	no	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	#930	C₄×C₅	1	1	yes	characteristic
101	#929	—	2	1	no	—
102	C₄×#67	—	2	1	no	—
103	#924	C₄×C₅	1	1	yes	characteristic
104	C₂×#208	—	2	1	no	—
105	#876	—	2	1	no	—
106	#875	C₂²×C₅	1	1	yes	characteristic
107	C₄×D₁₂	C₄×C₅	1	1	yes	characteristic
108	#872	—	2	1	no	—
109	#870	C₂²×C₅	1	1	yes	characteristic
110	C₄×Dic₆	C₄×C₅	1	1	yes	characteristic
111	C₃×C₄⋊C₈	—	2	1	no	—
112	C₃×C₂²⋊C₈	—	2	1	no	—
113	C₃×#97	C₂²×C₅	1	2	yes	—
114	#803	C₂²×C₅	1	2	yes	—
115	C₄×C₃⋊C₈	—	2	1	no	—
116	C₂×C₅×Q₈	—	3	1	no	—
117	C₂×C₅×D₄	—	3	1	no	—
118	C₂×C₅×D₄	—	6	1	no	—
119	C₂²×C₄×C₅	—	6	1	no	—
120	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
121	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
122	C₂²×C₄×C₅	#67	1	1	yes	characteristic
123	C₅×M₁₆	—	6	4	no	—
124	C₂×C₅×C₈	—	2	2	no	—
125	C₂×C₅×C₈	—	6	2	no	—
126	C₅×C₄⋊C₄	—	3	1	no	—
127	C₅×C₄⋊C₄	—	3	1	no	—
128	C₅×C₄⋊C₄	—	6	1	no	—
129	C₅×C₄⋊C₄	—	6	1	no	—
130	C₅×C₄⋊C₄	—	6	1	no	—
131	C₅×C₄⋊C₄	—	6	1	no	—
132	C₅×C₂²⋊C₄	—	6	1	no	—
133	C₅×C₂²⋊C₄	—	6	1	no	—
134	C₅×C₂²⋊C₄	—	6	1	no	—
135	C₅×C₂²⋊C₄	—	6	1	no	—
136	C₅×C₂²⋊C₄	—	6	1	no	—
137	C₄²×C₅	—	6	1	no	—
138	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
139	C₄²×C₅	D₁₂	1	1	yes	characteristic
140	C₄²×C₅	#67	1	1	yes	characteristic
141	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
142	#529	—	3	1	no	—
143	#432	C₅×D₃	1	1	yes	characteristic, complemented
144	C₄×M₁₆	—	3	1	no	—
145	#330	—	3	2	no	—
146	#329	—	3	2	no	—
147	C₂²×C₃×C₅	—	2	1	no	—
148	C₂²×C₃×C₅	—	2	1	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₅×D₃	—	4	1	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	—	2	1	no	—
156	C₃×C₄×C₅	—	2	1	no	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	#104	1	2	yes	—
160	C₅×Dic₃	—	4	1	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₅×Dic₃	—	4	1	no	—
163	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
166	C₂×#67	—	2	1	no	—
167	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
168	C₂×C₄×D₃	—	2	1	no	—
169	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂×C₃×C₈	—	2	2	no	—
171	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
172	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
173	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
174	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
175	#217	—	2	1	no	—
176	(C₂×D₃)⋊C₄	—	2	1	no	—
177	(C₂×D₃)⋊C₄	—	2	1	no	—
178	(C₂×D₃)⋊C₄	—	2	1	no	—
179	(C₂×D₃)⋊C₄	—	2	1	no	—
180	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
181	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
182	Dic₃⋊C₄	—	2	1	no	—
183	Dic₃⋊C₄	—	2	1	no	—
184	Dic₃⋊C₄	—	2	1	no	—
185	Dic₃⋊C₄	—	2	1	no	—
186	C₄×Dic₃	—	2	1	no	—
187	#208	—	2	4	no	—
188	C₂×C₃⋊C₈	—	2	2	no	—
189	C₂³×C₅	—	6	1	no	—
190	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₅×Q₈	—	12	1	no	—
192	C₅×D₄	—	12	1	no	—
193	C₅×D₄	—	12	2	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	4	1	no	—
196	C₂×C₄×C₅	—	4	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	6	1	no	—
199	C₂×C₄×C₅	—	6	1	no	—
200	C₂×C₄×C₅	—	12	2	no	—
201	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
207	C₅×C₈	—	4	2	no	—
208	C₅×C₈	—	6	4	no	—
209	C₂×M₁₆	—	6	1	no	—
210	#126	—	6	1	no	—
211	#125	—	3	1	no	—
212	#124	—	3	1	no	—
213	#123	—	6	1	no	—
214	C₂²⋊Q₈	—	3	1	no	—
215	#121	—	3	1	no	—
216	C₄×Q₈	—	3	1	no	—
217	C₄×D₄	—	3	1	no	—
218	C₄×D₄	—	6	1	no	—
219	#117	—	6	1	no	—
220	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
221	C₄⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	#97	—	3	2	no	—
224	#97	C₂×C₅×D₃	1	2	yes	—
225	C₄×C₈	—	6	1	no	—
226	C₂×C₃×C₅	—	4	1	no	—
227	C₂×C₃×C₅	#413	1	1	yes	characteristic
228	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
229	C₂×C₃×C₅	#421	1	1	yes	characteristic
230	C₅×D₃	—	8	1	no	—
231	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×D₃	—	2	1	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	4	1	no	—
235	C₂×C₃×C₄	—	4	1	no	—
236	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
242	#67	—	4	2	no	—
243	C₂×Dic₃	—	2	1	no	—
244	C₂×Dic₃	—	2	1	no	—
245	C₂×Dic₃	—	2	1	no	—
246	D₁₂	—	4	1	no	—
247	C₄×D₃	—	4	2	no	—
248	Dic₆	—	4	1	no	—
249	C₃×C₈	—	4	2	no	—
250	C₃⋊C₈	—	2	4	no	—
251	C₂²×C₅	—	2	1	no	—
252	C₂²×C₅	—	2	1	no	—
253	C₂²×C₅	—	2	1	no	—
254	C₂²×C₅	—	12	1	no	—
255	C₂²×C₅	—	12	1	no	—
256	C₂²×C₅	—	12	1	no	—
257	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	2	1	no	—
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	12	1	no	—
264	C₄×C₅	—	12	1	no	—
265	C₄×C₅	—	12	1	no	—
266	C₄×C₅	#825	1	2	yes	—
267	C₂×Q₈	—	3	1	no	—
268	C₂×D₄	—	3	1	no	—
269	C₂×D₄	—	6	1	no	—
270	C₂²×C₄	—	6	1	no	—
271	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
272	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
273	C₂²×C₄	C₅×#67	1	1	yes	characteristic
274	M₁₆	—	6	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	6	2	no	—
277	C₄⋊C₄	—	3	1	no	—
278	C₄⋊C₄	—	3	1	no	—
279	C₄⋊C₄	—	6	1	no	—
280	C₄⋊C₄	—	6	1	no	—
281	C₄⋊C₄	—	6	1	no	—
282	C₄⋊C₄	—	6	1	no	—
283	C₂²⋊C₄	—	6	1	no	—
284	C₂²⋊C₄	—	6	1	no	—
285	C₂²⋊C₄	—	6	1	no	—
286	C₂²⋊C₄	—	6	1	no	—
287	C₂²⋊C₄	—	6	1	no	—
288	C₄²	—	6	1	no	—
289	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
290	C₄²	C₅×D₁₂	1	1	yes	characteristic
291	C₄²	C₅×#67	1	1	yes	characteristic
292	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
293	C₃×C₅	#1529	1	1	yes	characteristic, complemented, Hall
294	C₂²×C₃	—	2	1	no	—
295	C₂²×C₃	—	2	1	no	—
296	C₂²×C₃	—	2	1	no	—
297	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
298	C₂×D₃	—	4	1	no	—
299	C₂×D₃	—	4	1	no	—
300	C₂×D₃	—	4	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	C₅×#104	1	2	yes	—
307	Dic₃	—	4	1	no	—
308	Dic₃	—	4	1	no	—
309	Dic₃	—	4	1	no	—
310	C₂×C₅	—	4	1	no	—
311	C₂×C₅	—	24	1	no	—
312	C₂×C₅	#5005	1	1	yes	characteristic
313	C₂×C₅	#5132	1	1	yes	characteristic
314	C₂×C₅	C₂×#825	1	1	yes	characteristic
315	C₂³	—	6	1	no	—
316	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
317	Q₈	—	12	1	no	—
318	D₄	—	12	1	no	—
319	D₄	—	12	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	6	1	no	—
324	C₂×C₄	—	6	1	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	—	12	2	no	—
327	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
329	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
330	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
331	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
332	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	6	4	no	—
335	C₂×C₃	—	4	1	no	—
336	C₂×C₃	C₅×#413	1	1	yes	characteristic
337	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
338	C₂×C₃	C₅×#421	1	1	yes	characteristic
339	D₃	—	8	1	no	—
340	C₅	#70805	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	12	1	no	—
345	C₂²	—	12	1	no	—
346	C₂²	—	12	1	no	—
347	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	12	1	no	—
354	C₄	—	12	1	no	—
355	C₄	—	12	1	no	—
356	C₄	C₅×#825	1	2	yes	—
357	C₃	C₅×#1529	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	4	1	no	—
359	C₂	—	24	1	no	—
360	C₂	C₅×#5005	1	1	yes	characteristic
361	C₂	C₅×#5132	1	1	yes	characteristic
362	C₂	C₂×C₅×#825	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#432	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#930	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#432	D₃
hypocenter	C₃	C₅×#1529
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#432

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₄×C₅×#208	C₂	semidirect
4. C₅×#4465	C₂	semidirect
5. C₅×#4464	C₂	semidirect
24. #70805	C₅	direct (N × H)
26. C₅×#432	D₃	semidirect
57. C₃×#432	C₂×C₅	semidirect
58. C₄×#208	C₂×C₅	semidirect
59. #4465	C₂×C₅	semidirect
60. #4464	C₂×C₅	semidirect
143. #432	C₅×D₃	semidirect
293. C₃×C₅	#1529	semidirect
340. C₅	#70805	direct (N × H)
357. C₃	C₅×#1529	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5479	2
C₃×C₅×#432	2
C₄×C₅×#208	2
C₅×#4465	2
C₅×#4464	2
C₅×#1529	3
#70805	5

Subgroups — C5×#70805 #14284129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70805 #14284129