Subgroups — C₅×#70808 #14284332

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#875	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4926	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4464	C₂	1	4	yes	maximal, complemented
5	C₅×#1533	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	C₄	1	1	yes	characteristic
7	C₂²×C₅×Dic₆	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
10	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
11	C₅×#875	—	2	1	no	—
12	C₅×#875	—	2	1	no	—
13	C₅×#875	C₄	1	2	yes	—
14	C₅×#875	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₅×#97	C₂²	1	2	yes	complemented
18	C₅×#830	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₅×#803	C₂²	1	2	yes	complemented
21	#70808	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#124	—	3	1	no	—
23	C₅×#432	—	3	1	no	—
24	C₅×#432	D₃	1	1	yes	characteristic, complemented
25	C₅×#329	—	3	4	no	—
26	C₂³×C₅×D₃	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₅×Dic₃	—	2	1	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	4	2	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₅×Dic₆	—	2	1	no	—
35	C₂×C₅×Dic₆	—	4	2	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
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×(C₂×D₃)⋊C₄	—	2	2	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×#875	C₂×C₅	1	1	yes	characteristic
48	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
49	#4926	C₂×C₅	1	1	yes	characteristic, complemented
50	#4464	C₂×C₅	1	4	yes	complemented
51	C₂²×C₅×Q₈	—	3	1	no	—
52	C₂²×C₅×D₄	—	3	1	no	—
53	C₅×#124	—	3	2	no	—
54	C₅×#124	—	3	2	no	—
55	C₅×#124	—	6	1	no	—
56	C₅×#124	—	6	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
58	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	C₅×C₄⋊C₈	—	2	1	no	—
61	C₅×C₄⋊C₈	—	6	1	no	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×C₂²⋊C₈	—	6	1	no	—
64	C₅×#97	—	3	2	no	—
65	C₅×#97	C₂×D₃	1	2	yes	complemented
66	#1533	—	3	1	no	Sylow
67	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅×D₃	—	2	2	no	—
69	C₂²×C₅×D₃	—	4	2	no	—
70	C₂²×C₅×D₃	—	4	2	no	—
71	C₂²×C₅×D₃	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₅×Dic₃	—	2	2	no	—
82	C₂×C₅×Dic₃	—	4	2	no	—
83	C₅×D₁₂	—	4	2	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×Dic₆	—	4	2	no	—
86	C₅×Dic₆	—	4	2	no	—
87	C₃×C₅×C₈	—	4	2	no	—
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₂²×D₁₂	C₄×C₅	1	1	yes	characteristic
90	C₂²×Dic₆	C₄×C₅	1	1	yes	characteristic
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
93	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
94	#875	—	2	1	no	—
95	#875	—	2	1	no	—
96	#875	C₄×C₅	1	2	yes	—
97	#875	C₂²×C₅	1	2	yes	—
98	C₃×C₄⋊C₈	—	2	1	no	—
99	C₃×C₂²⋊C₈	—	2	1	no	—
100	C₃×#97	C₂²×C₅	1	2	yes	complemented
101	#830	—	2	1	no	—
102	C₄⋊(C₃⋊C₈)	—	2	1	no	—
103	#803	C₂²×C₅	1	2	yes	complemented
104	C₂⁴×C₅	—	6	1	no	—
105	C₂×C₅×Q₈	—	3	2	no	—
106	C₂×C₅×Q₈	—	6	1	no	—
107	C₂×C₅×Q₈	—	12	2	no	—
108	C₂×C₅×D₄	—	3	2	no	—
109	C₂×C₅×D₄	—	6	1	no	—
110	C₂×C₅×D₄	—	12	2	no	—
111	C₂²×C₄×C₅	—	6	1	no	—
112	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
113	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
114	C₂²×C₄×C₅	#67	1	1	yes	characteristic
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	6	2	no	—
117	C₅×C₂²⋊C₄	—	6	2	no	—
118	C₅×C₂²⋊C₄	—	6	2	no	—
119	C₅×C₂²⋊C₄	—	6	2	no	—
120	C₅×C₂²⋊C₄	—	6	2	no	—
121	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
122	C₄²×C₅	D₁₂	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₂×#124	—	3	1	no	—
126	#432	—	3	1	no	—
127	#432	C₅×D₃	1	1	yes	characteristic, complemented
128	#329	—	3	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	#100	1	2	yes	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₅×D₃	—	4	2	no	—
134	C₂×C₅×D₃	—	4	2	no	—
135	C₂×C₅×D₃	—	4	2	no	—
136	C₂×C₅×D₃	—	8	4	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₅×Dic₃	—	4	2	no	—
144	C₂³×D₃	—	2	1	no	—
145	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
148	C₂²×Dic₃	—	2	1	no	—
149	C₂×D₁₂	—	2	1	no	—
150	C₂×D₁₂	—	4	2	no	—
151	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
152	C₂×Dic₆	—	2	1	no	—
153	C₂×Dic₆	—	4	2	no	—
154	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₃×C₈	—	2	2	no	—
156	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
157	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
159	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
160	(C₂×D₃)⋊C₄	—	2	2	no	—
161	(C₂×D₃)⋊C₄	—	2	2	no	—
162	(C₂×D₃)⋊C₄	—	2	2	no	—
163	(C₂×D₃)⋊C₄	—	2	2	no	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂³×C₅	—	6	2	no	—
166	C₂³×C₅	—	12	2	no	—
167	C₂³×C₅	—	12	2	no	—
168	C₂³×C₅	—	12	2	no	—
169	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₅×Q₈	—	12	2	no	—
171	C₅×Q₈	—	12	2	no	—
172	C₅×D₄	—	12	2	no	—
173	C₅×D₄	—	12	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	—	6	2	no	—
178	C₂×C₄×C₅	—	12	2	no	—
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
185	C₅×C₈	—	4	2	no	—
186	C₅×C₈	—	12	2	no	—
187	C₂²×Q₈	—	3	1	no	—
188	C₂²×D₄	—	3	1	no	—
189	#124	—	3	2	no	—
190	#124	—	3	2	no	—
191	#124	—	6	1	no	—
192	#124	—	6	1	no	—
193	C₂×C₂²⋊C₄	—	6	1	no	—
194	C₂×C₂²⋊C₄	—	6	1	no	—
195	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	6	1	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	6	1	no	—
200	#97	—	3	2	no	—
201	#97	C₂×C₅×D₃	1	2	yes	complemented
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
204	C₂×C₃×C₅	#421	1	1	yes	characteristic
205	C₂×C₃×C₅	#421	1	1	yes	characteristic
206	C₅×D₃	—	8	2	no	—
207	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×D₃	—	2	2	no	—
209	C₂²×D₃	—	4	2	no	—
210	C₂²×D₃	—	4	2	no	—
211	C₂²×D₃	—	4	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×Dic₃	—	2	2	no	—
222	C₂×Dic₃	—	4	2	no	—
223	D₁₂	—	4	2	no	—
224	D₁₂	—	4	2	no	—
225	Dic₆	—	4	2	no	—
226	Dic₆	—	4	2	no	—
227	C₃×C₈	—	4	2	no	—
228	C₃⋊C₈	—	4	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	12	2	no	—
232	C₂²×C₅	—	12	2	no	—
233	C₂²×C₅	—	12	2	no	—
234	C₂²×C₅	—	24	4	no	—
235	C₂²×C₅	#823	1	2	yes	—
236	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	12	2	no	—
244	C₂⁴	—	6	1	no	—
245	C₂×Q₈	—	3	2	no	—
246	C₂×Q₈	—	6	1	no	—
247	C₂×Q₈	—	12	2	no	—
248	C₂×D₄	—	3	2	no	—
249	C₂×D₄	—	6	1	no	—
250	C₂×D₄	—	12	2	no	—
251	C₂²×C₄	—	6	1	no	—
252	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
253	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
254	C₂²×C₄	C₅×#67	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	6	2	no	—
257	C₂²⋊C₄	—	6	2	no	—
258	C₂²⋊C₄	—	6	2	no	—
259	C₂²⋊C₄	—	6	2	no	—
260	C₂²⋊C₄	—	6	2	no	—
261	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
262	C₄²	C₅×D₁₂	1	1	yes	characteristic
263	C₄²	C₅×#67	1	1	yes	characteristic
264	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
265	C₃×C₅	#1533	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	C₅×#100	1	2	yes	—
269	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×D₃	—	4	2	no	—
271	C₂×D₃	—	4	2	no	—
272	C₂×D₃	—	4	2	no	—
273	C₂×D₃	—	8	4	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	Dic₃	—	4	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	—	24	2	no	—
283	C₂×C₅	#5005	1	1	yes	characteristic
284	C₂×C₅	C₂×#823	1	1	yes	characteristic
285	C₂×C₅	#5139	1	1	yes	characteristic
286	C₂³	—	6	2	no	—
287	C₂³	—	12	2	no	—
288	C₂³	—	12	2	no	—
289	C₂³	—	12	2	no	—
290	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
291	Q₈	—	12	2	no	—
292	Q₈	—	12	2	no	—
293	D₄	—	12	2	no	—
294	D₄	—	12	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	12	2	no	—
300	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
305	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	12	2	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
310	C₂×C₃	C₅×#421	1	1	yes	characteristic
311	C₂×C₃	C₅×#421	1	1	yes	characteristic
312	D₃	—	8	2	no	—
313	C₅	#70808	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	12	2	no	—
317	C₂²	—	12	2	no	—
318	C₂²	—	12	2	no	—
319	C₂²	—	24	4	no	—
320	C₂²	C₅×#823	1	2	yes	—
321	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	12	2	no	—
329	C₃	C₅×#1533	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	2	2	no	—
331	C₂	—	24	2	no	—
332	C₂	C₅×#5005	1	1	yes	characteristic
333	C₂	C₂×C₅×#823	1	1	yes	characteristic
334	C₂	C₅×#5139	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	C₂²×C₅×D₁₂	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#432	D₃
hypocenter	C₃	C₅×#1533
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. C₅×#4926	C₂	semidirect
4. C₅×#4464	C₂	semidirect
17. C₃×C₅×#97	C₂²	semidirect
20. C₅×#803	C₂²	semidirect
21. #70808	C₅	direct (N × H)
24. C₅×#432	D₃	semidirect
48. C₃×#432	C₂×C₅	semidirect
49. #4926	C₂×C₅	semidirect
50. #4464	C₂×C₅	semidirect
65. C₅×#97	C₂×D₃	semidirect
100. C₃×#97	C₂²×C₅	semidirect
103. #803	C₂²×C₅	semidirect
127. #432	C₅×D₃	semidirect
201. #97	C₂×C₅×D₃	semidirect
265. C₃×C₅	#1533	semidirect
313. C₅	#70808	direct (N × H)
329. C₃	C₅×#1533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#875	2
C₃×C₅×#432	2
C₅×#4926	2
C₅×#4464	2
C₅×#1533	3
#70808	5

Subgroups — C5×#70808 #14284332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70808 #14284332