Subgroups — C₅×#70505 #14285413

Order 1920.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4485	C₂	1	2	yes	maximal, complemented
5	C₅×#4481	C₂	1	2	yes	maximal, complemented
6	C₅×#1560	—	3	1	no	maximal, Hall
7	C₂×C₅×#235	C₄	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#930	—	2	1	no	—
11	C₅×#930	—	2	1	no	—
12	C₂×C₅×#208	—	2	1	no	—
13	C₅×#875	C₄	1	2	yes	—
14	C₅×#874	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₁₂	—	2	1	no	—
16	C₄×C₅×D₁₂	—	2	1	no	—
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
20	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
22	#70505	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#532	—	3	1	no	—
24	C₅×#423	—	3	1	no	—
25	C₅×#423	D₃	1	1	yes	characteristic, complemented
26	C₅×#333	—	3	2	no	—
27	C₅×#331	—	3	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₅×#67	—	2	2	no	—
32	C₅×#235	—	4	4	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₅×D₃	—	2	2	no	—
37	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
38	C₃×C₅×M₁₆	—	4	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
46	C₅×C₄⋊Dic₃	—	2	1	no	—
47	C₅×C₄⋊Dic₃	—	2	1	no	—
48	C₅×#208	—	4	2	no	—
49	C₂×C₅×C₃⋊C₈	—	2	2	no	—
50	#5477	C₂×C₅	1	1	yes	characteristic
51	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
52	#4924	C₂×C₅	1	1	yes	characteristic, complemented
53	#4485	C₂×C₅	1	2	yes	complemented
54	#4481	C₂×C₅	1	2	yes	complemented
55	C₂×C₅×#41	—	3	2	no	—
56	C₂×C₅×M₁₆	—	2	1	no	—
57	C₂×C₅×M₁₆	—	6	1	no	—
58	C₅×C₄⋊Q₈	—	3	1	no	—
59	C₅×#127	—	3	1	no	—
60	C₅×#124	—	3	2	no	—
61	C₅×#121	—	6	1	no	—
62	C₅×#121	—	6	1	no	—
63	C₄×C₅×D₄	—	6	1	no	—
64	C₄×C₅×D₄	—	6	1	no	—
65	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
66	C₅×C₄⋊C₈	—	2	1	no	—
67	C₅×C₄⋊C₈	—	3	2	no	—
68	C₅×C₄⋊C₈	—	6	1	no	—
69	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
70	#1560	—	3	1	no	Sylow
71	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₅×D₃	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₅×#67	—	4	4	no	—
85	C₂×C₅×Dic₃	—	2	2	no	—
86	C₅×D₁₂	—	4	2	no	—
87	C₅×D₁₂	—	4	1	no	—
88	C₅×D₁₂	—	4	1	no	—
89	C₅×D₁₂	—	4	1	no	—
90	C₅×D₁₂	—	4	1	no	—
91	C₄×C₅×D₃	—	4	4	no	—
92	C₅×Dic₆	—	4	2	no	—
93	C₃×C₅×C₈	—	4	2	no	—
94	C₅×C₃⋊C₈	—	4	2	no	—
95	C₂×#235	C₄×C₅	1	2	yes	—
96	C₂×C₃×M₁₆	—	2	1	no	—
97	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	#930	—	2	1	no	—
99	#930	—	2	1	no	—
100	C₂×#208	—	2	1	no	—
101	#875	C₄×C₅	1	2	yes	—
102	#874	C₂²×C₅	1	1	yes	characteristic
103	C₄×D₁₂	—	2	1	no	—
104	C₄×D₁₂	—	2	1	no	—
105	#869	C₂²×C₅	1	1	yes	characteristic
106	C₃×C₄⋊C₈	—	2	1	no	—
107	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
108	C₄⋊(C₃⋊C₈)	—	2	1	no	—
109	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
110	C₅×#41	—	12	4	no	—
111	C₂×C₅×Q₈	—	3	2	no	—
112	C₂×C₅×D₄	—	3	2	no	—
113	C₂×C₅×D₄	—	6	2	no	—
114	C₂×C₅×D₄	—	6	1	no	—
115	C₂×C₅×D₄	—	6	1	no	—
116	C₂²×C₄×C₅	—	6	2	no	—
117	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
118	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
119	C₂²×C₄×C₅	#67	1	1	yes	characteristic
120	C₅×M₁₆	—	4	2	no	—
121	C₅×M₁₆	—	12	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	—	6	2	no	—
124	C₅×C₄⋊C₄	—	6	1	no	—
125	C₅×C₄⋊C₄	—	6	1	no	—
126	C₅×C₂²⋊C₄	—	6	2	no	—
127	C₅×C₂²⋊C₄	—	6	2	no	—
128	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
129	C₄²×C₅	D₁₂	1	1	yes	characteristic
130	C₄²×C₅	#67	1	1	yes	characteristic
131	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
132	#532	—	3	1	no	—
133	#423	—	3	1	no	—
134	#423	C₅×D₃	1	1	yes	characteristic, complemented
135	#333	—	3	2	no	—
136	#331	—	3	2	no	—
137	C₂²×C₃×C₅	—	2	1	no	—
138	C₂²×C₃×C₅	—	2	1	no	—
139	C₂²×C₃×C₅	—	2	1	no	—
140	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₅×D₃	—	4	2	no	—
142	C₂×C₅×D₃	—	4	2	no	—
143	C₂×C₅×D₃	—	4	2	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	2	1	no	—
147	C₃×C₄×C₅	#100	1	2	yes	—
148	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₃×C₄×C₅	#137	1	1	yes	characteristic
151	C₃×C₄×C₅	#137	1	1	yes	characteristic
152	C₅×Dic₃	—	4	2	no	—
153	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
155	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
156	C₂×#67	—	2	2	no	—
157	#235	—	4	4	no	—
158	C₂×D₁₂	—	2	1	no	—
159	C₂×D₁₂	—	2	1	no	—
160	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₄×D₃	—	2	2	no	—
162	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
163	C₃×M₁₆	—	4	2	no	—
164	C₂×C₃×C₈	—	2	2	no	—
165	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
166	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
167	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
168	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
169	(C₂×D₃)⋊C₄	—	2	2	no	—
170	(C₂×D₃)⋊C₄	—	2	2	no	—
171	C₄⋊Dic₃	—	2	1	no	—
172	C₄⋊Dic₃	—	2	1	no	—
173	#208	—	4	2	no	—
174	C₂×C₃⋊C₈	—	2	2	no	—
175	C₂³×C₅	—	6	2	no	—
176	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₅×Q₈	—	12	2	no	—
178	C₅×D₄	—	12	2	no	—
179	C₅×D₄	—	12	4	no	—
180	C₅×D₄	—	12	1	no	—
181	C₅×D₄	—	12	1	no	—
182	C₅×D₄	—	12	1	no	—
183	C₅×D₄	—	12	1	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	1	no	—
186	C₂×C₄×C₅	—	2	1	no	—
187	C₂×C₄×C₅	—	2	1	no	—
188	C₂×C₄×C₅	—	2	1	no	—
189	C₂×C₄×C₅	—	6	2	no	—
190	C₂×C₄×C₅	—	12	4	no	—
191	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
193	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
197	C₅×C₈	—	4	2	no	—
198	C₅×C₈	—	12	2	no	—
199	C₂×#41	—	3	2	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	—	6	1	no	—
202	C₄⋊Q₈	—	3	1	no	—
203	#127	—	3	1	no	—
204	#124	—	3	2	no	—
205	#121	—	6	1	no	—
206	#121	—	6	1	no	—
207	C₄×D₄	—	6	1	no	—
208	C₄×D₄	—	6	1	no	—
209	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	3	2	no	—
212	C₄⋊C₈	—	6	1	no	—
213	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
214	C₂×C₃×C₅	—	4	1	no	—
215	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
216	C₂×C₃×C₅	#417	1	1	yes	characteristic
217	C₂×C₃×C₅	#417	1	1	yes	characteristic
218	C₅×D₃	—	8	2	no	—
219	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×D₃	—	2	2	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	#67	—	4	4	no	—
233	C₂×Dic₃	—	2	2	no	—
234	D₁₂	—	4	2	no	—
235	D₁₂	—	4	1	no	—
236	D₁₂	—	4	1	no	—
237	D₁₂	—	4	1	no	—
238	D₁₂	—	4	1	no	—
239	C₄×D₃	—	4	4	no	—
240	Dic₆	—	4	2	no	—
241	C₃×C₈	—	4	2	no	—
242	C₃⋊C₈	—	4	2	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	12	2	no	—
247	C₂²×C₅	—	12	2	no	—
248	C₂²×C₅	—	12	2	no	—
249	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
250	C₄×C₅	—	2	1	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	12	2	no	—
254	C₄×C₅	#823	1	2	yes	—
255	C₄×C₅	#908	1	1	yes	characteristic
256	C₄×C₅	#909	1	1	yes	characteristic
257	C₄×C₅	#932	1	1	yes	characteristic
258	C₄×C₅	#942	1	1	yes	characteristic
259	#41	—	12	4	no	—
260	C₂×Q₈	—	3	2	no	—
261	C₂×D₄	—	3	2	no	—
262	C₂×D₄	—	6	2	no	—
263	C₂×D₄	—	6	1	no	—
264	C₂×D₄	—	6	1	no	—
265	C₂²×C₄	—	6	2	no	—
266	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
267	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
268	C₂²×C₄	C₅×#67	1	1	yes	characteristic
269	M₁₆	—	4	2	no	—
270	M₁₆	—	12	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	6	2	no	—
273	C₄⋊C₄	—	6	1	no	—
274	C₄⋊C₄	—	6	1	no	—
275	C₂²⋊C₄	—	6	2	no	—
276	C₂²⋊C₄	—	6	2	no	—
277	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
278	C₄²	C₅×D₁₂	1	1	yes	characteristic
279	C₄²	C₅×#67	1	1	yes	characteristic
280	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
281	C₃×C₅	#1560	1	1	yes	characteristic, complemented, Hall
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×D₃	—	4	2	no	—
287	C₂×D₃	—	4	2	no	—
288	C₂×D₃	—	4	2	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	C₅×#100	1	2	yes	—
293	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
295	C₃×C₄	C₅×#137	1	1	yes	characteristic
296	C₃×C₄	C₅×#137	1	1	yes	characteristic
297	Dic₃	—	4	2	no	—
298	C₂×C₅	—	4	1	no	—
299	C₂×C₅	—	24	2	no	—
300	C₂×C₅	#4966	1	1	yes	characteristic
301	C₂×C₅	C₂×#823	1	1	yes	characteristic
302	C₂×C₅	#5133	1	1	yes	characteristic
303	C₂³	—	6	2	no	—
304	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
305	Q₈	—	12	2	no	—
306	D₄	—	12	2	no	—
307	D₄	—	12	4	no	—
308	D₄	—	12	1	no	—
309	D₄	—	12	1	no	—
310	D₄	—	12	1	no	—
311	D₄	—	12	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	6	2	no	—
318	C₂×C₄	—	12	4	no	—
319	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
324	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₈	—	12	2	no	—
327	C₂×C₃	—	4	1	no	—
328	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
329	C₂×C₃	C₅×#417	1	1	yes	characteristic
330	C₂×C₃	C₅×#417	1	1	yes	characteristic
331	D₃	—	8	2	no	—
332	C₅	#70505	1	1	yes	characteristic, direct product factor, Sylow
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	12	2	no	—
337	C₂²	—	12	2	no	—
338	C₂²	—	12	2	no	—
339	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	12	2	no	—
344	C₄	C₅×#823	1	2	yes	—
345	C₄	C₅×#908	1	1	yes	characteristic
346	C₄	C₅×#909	1	1	yes	characteristic
347	C₄	C₅×#932	1	1	yes	characteristic
348	C₄	C₅×#942	1	1	yes	characteristic
349	C₃	C₅×#1560	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	4	1	no	—
351	C₂	—	24	2	no	—
352	C₂	C₅×#4966	1	1	yes	characteristic
353	C₂	C₂×C₅×#823	1	1	yes	characteristic
354	C₂	C₅×#5133	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×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#1560
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₅×#4924	C₂	semidirect
4. C₅×#4485	C₂	semidirect
5. C₅×#4481	C₂	semidirect
22. #70505	C₅	direct (N × H)
25. C₅×#423	D₃	semidirect
51. C₃×#423	C₂×C₅	semidirect
52. #4924	C₂×C₅	semidirect
53. #4485	C₂×C₅	semidirect
54. #4481	C₂×C₅	semidirect
134. #423	C₅×D₃	semidirect
281. C₃×C₅	#1560	semidirect
332. C₅	#70505	direct (N × H)
349. C₃	C₅×#1560	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₃×C₅×#423	2
C₅×#4924	2
C₅×#4485	2
C₅×#4481	2
C₅×#1560	3
#70505	5

Subgroups — C5×#70505 #14285413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70505 #14285413