Subgroups — C₅×#70804 #14284040

Order 1920.

528 conjugacy classes of subgroups.

The subgroup lattice has 340 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	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4464	C₂	1	4	yes	maximal, complemented
5	C₅×#1527	—	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	C₂×C₅×#208	—	2	1	no	—
12	C₅×#875	—	2	1	no	—
13	C₅×#875	—	2	1	no	—
14	C₅×#875	C₄	1	2	yes	—
15	C₅×#875	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₈	—	2	1	no	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×#97	C₂²	1	2	yes	complemented
19	C₅×#803	C₂²	1	2	yes	complemented
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	#70804	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#124	—	3	1	no	—
23	C₅×#432	D₃	1	1	yes	characteristic, complemented
24	C₄×C₅×M₁₆	—	3	1	no	—
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	C₅×#208	—	2	4	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×#875	C₂×C₅	1	1	yes	characteristic
49	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
50	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
51	#4464	C₂×C₅	1	4	yes	complemented
52	C₂²×C₅×Q₈	—	3	1	no	—
53	C₂²×C₅×D₄	—	3	1	no	—
54	C₂×C₅×M₁₆	—	6	1	no	—
55	C₅×#124	—	3	2	no	—
56	C₅×#124	—	3	2	no	—
57	C₅×#124	—	6	1	no	—
58	C₅×#124	—	6	1	no	—
59	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
60	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	—	2	1	no	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₅×#97	—	3	2	no	—
65	C₅×#97	C₂×D₃	1	2	yes	complemented
66	C₄×C₅×C₈	—	6	1	no	—
67	#1527	—	3	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₅×D₃	—	2	2	no	—
70	C₂²×C₅×D₃	—	4	2	no	—
71	C₂²×C₅×D₃	—	4	2	no	—
72	C₂²×C₅×D₃	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×Dic₃	—	2	2	no	—
83	C₂×C₅×Dic₃	—	4	2	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×D₁₂	—	4	2	no	—
86	C₅×Dic₆	—	4	2	no	—
87	C₅×Dic₆	—	4	2	no	—
88	C₃×C₅×C₈	—	4	2	no	—
89	C₅×C₃⋊C₈	—	2	4	no	—
90	C₂²×D₁₂	C₄×C₅	1	1	yes	characteristic
91	C₂²×Dic₆	C₄×C₅	1	1	yes	characteristic
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
94	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
95	C₂×#208	—	2	1	no	—
96	#875	—	2	1	no	—
97	#875	—	2	1	no	—
98	#875	C₄×C₅	1	2	yes	—
99	#875	C₂²×C₅	1	2	yes	—
100	C₃×C₄⋊C₈	—	2	1	no	—
101	C₃×C₂²⋊C₈	—	2	1	no	—
102	C₃×#97	C₂²×C₅	1	2	yes	complemented
103	#803	C₂²×C₅	1	2	yes	complemented
104	C₄×C₃⋊C₈	—	2	1	no	—
105	C₂⁴×C₅	—	6	1	no	—
106	C₂×C₅×Q₈	—	3	2	no	—
107	C₂×C₅×Q₈	—	6	1	no	—
108	C₂×C₅×Q₈	—	12	2	no	—
109	C₂×C₅×D₄	—	3	2	no	—
110	C₂×C₅×D₄	—	6	1	no	—
111	C₂×C₅×D₄	—	12	2	no	—
112	C₂²×C₄×C₅	—	6	1	no	—
113	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
115	C₂²×C₄×C₅	#67	1	1	yes	characteristic
116	C₅×M₁₆	—	6	4	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₅×C₂²⋊C₄	—	6	2	no	—
120	C₅×C₂²⋊C₄	—	6	2	no	—
121	C₅×C₂²⋊C₄	—	6	2	no	—
122	C₅×C₂²⋊C₄	—	6	2	no	—
123	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
124	C₄²×C₅	D₁₂	1	1	yes	characteristic
125	C₄²×C₅	#67	1	1	yes	characteristic
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂×#124	—	3	1	no	—
128	#432	C₅×D₃	1	1	yes	characteristic, complemented
129	C₄×M₁₆	—	3	1	no	—
130	#329	—	3	4	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	#104	1	2	yes	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₅×D₃	—	4	2	no	—
136	C₂×C₅×D₃	—	4	2	no	—
137	C₂×C₅×D₃	—	4	2	no	—
138	C₂×C₅×D₃	—	8	4	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	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	#100	1	2	yes	—
145	C₅×Dic₃	—	4	2	no	—
146	C₂³×D₃	—	2	1	no	—
147	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂²×Dic₃	—	2	1	no	—
151	C₂×D₁₂	—	2	1	no	—
152	C₂×D₁₂	—	4	2	no	—
153	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
154	C₂×Dic₆	—	2	1	no	—
155	C₂×Dic₆	—	4	2	no	—
156	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
157	C₂×C₃×C₈	—	2	2	no	—
158	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
162	(C₂×D₃)⋊C₄	—	2	2	no	—
163	(C₂×D₃)⋊C₄	—	2	2	no	—
164	(C₂×D₃)⋊C₄	—	2	2	no	—
165	(C₂×D₃)⋊C₄	—	2	2	no	—
166	#208	—	2	4	no	—
167	C₂×C₃⋊C₈	—	2	2	no	—
168	C₂³×C₅	—	6	2	no	—
169	C₂³×C₅	—	12	2	no	—
170	C₂³×C₅	—	12	2	no	—
171	C₂³×C₅	—	12	2	no	—
172	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₅×Q₈	—	12	2	no	—
174	C₅×Q₈	—	12	2	no	—
175	C₅×D₄	—	12	2	no	—
176	C₅×D₄	—	12	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	—	12	2	no	—
182	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
188	C₅×C₈	—	4	2	no	—
189	C₅×C₈	—	6	4	no	—
190	C₂²×Q₈	—	3	1	no	—
191	C₂²×D₄	—	3	1	no	—
192	C₂×M₁₆	—	6	1	no	—
193	#124	—	3	2	no	—
194	#124	—	3	2	no	—
195	#124	—	6	1	no	—
196	#124	—	6	1	no	—
197	C₂×C₂²⋊C₄	—	6	1	no	—
198	C₂×C₂²⋊C₄	—	6	1	no	—
199	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
200	C₄⋊C₈	—	2	1	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	#97	—	3	2	no	—
203	#97	C₂×C₅×D₃	1	2	yes	complemented
204	C₄×C₈	—	6	1	no	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
208	C₂×C₃×C₅	#421	1	1	yes	characteristic
209	C₅×D₃	—	8	2	no	—
210	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₂²×D₃	—	2	2	no	—
212	C₂²×D₃	—	4	2	no	—
213	C₂²×D₃	—	4	2	no	—
214	C₂²×D₃	—	4	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×Dic₃	—	2	2	no	—
225	C₂×Dic₃	—	4	2	no	—
226	D₁₂	—	4	2	no	—
227	D₁₂	—	4	2	no	—
228	Dic₆	—	4	2	no	—
229	Dic₆	—	4	2	no	—
230	C₃×C₈	—	4	2	no	—
231	C₃⋊C₈	—	2	4	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	12	2	no	—
235	C₂²×C₅	—	12	2	no	—
236	C₂²×C₅	—	12	2	no	—
237	C₂²×C₅	—	24	4	no	—
238	C₂²×C₅	#825	1	2	yes	—
239	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	12	2	no	—
246	C₄×C₅	#823	1	2	yes	—
247	C₂⁴	—	6	1	no	—
248	C₂×Q₈	—	3	2	no	—
249	C₂×Q₈	—	6	1	no	—
250	C₂×Q₈	—	12	2	no	—
251	C₂×D₄	—	3	2	no	—
252	C₂×D₄	—	6	1	no	—
253	C₂×D₄	—	12	2	no	—
254	C₂²×C₄	—	6	1	no	—
255	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
256	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
257	C₂²×C₄	C₅×#67	1	1	yes	characteristic
258	M₁₆	—	6	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	6	2	no	—
261	C₂²⋊C₄	—	6	2	no	—
262	C₂²⋊C₄	—	6	2	no	—
263	C₂²⋊C₄	—	6	2	no	—
264	C₂²⋊C₄	—	6	2	no	—
265	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
266	C₄²	C₅×D₁₂	1	1	yes	characteristic
267	C₄²	C₅×#67	1	1	yes	characteristic
268	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
269	C₃×C₅	#1527	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	C₅×#104	1	2	yes	—
273	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×D₃	—	4	2	no	—
275	C₂×D₃	—	4	2	no	—
276	C₂×D₃	—	4	2	no	—
277	C₂×D₃	—	8	4	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	C₅×#100	1	2	yes	—
284	Dic₃	—	4	2	no	—
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	—	24	2	no	—
287	C₂×C₅	#5005	1	1	yes	characteristic
288	C₂×C₅	C₂×#823	1	1	yes	characteristic
289	C₂×C₅	C₂×#825	1	1	yes	characteristic
290	C₂³	—	6	2	no	—
291	C₂³	—	12	2	no	—
292	C₂³	—	12	2	no	—
293	C₂³	—	12	2	no	—
294	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
295	Q₈	—	12	2	no	—
296	Q₈	—	12	2	no	—
297	D₄	—	12	2	no	—
298	D₄	—	12	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	6	2	no	—
303	C₂×C₄	—	12	2	no	—
304	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
309	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	6	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
314	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
315	C₂×C₃	C₅×#421	1	1	yes	characteristic
316	D₃	—	8	2	no	—
317	C₅	#70804	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	12	2	no	—
321	C₂²	—	12	2	no	—
322	C₂²	—	12	2	no	—
323	C₂²	—	24	4	no	—
324	C₂²	C₅×#825	1	2	yes	—
325	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	12	2	no	—
332	C₄	C₅×#823	1	2	yes	—
333	C₃	C₅×#1527	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	24	2	no	—
336	C₂	C₅×#5005	1	1	yes	characteristic
337	C₂	C₂×C₅×#823	1	1	yes	characteristic
338	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	C₂²×C₅×D₁₂	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#432	D₃
hypocenter	C₃	C₅×#1527
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×#4464	C₂	semidirect
18. C₃×C₅×#97	C₂²	semidirect
19. C₅×#803	C₂²	semidirect
21. #70804	C₅	direct (N × H)
23. C₅×#432	D₃	semidirect
49. C₃×#432	C₂×C₅	semidirect
50. C₄×#208	C₂×C₅	semidirect
51. #4464	C₂×C₅	semidirect
65. C₅×#97	C₂×D₃	semidirect
102. C₃×#97	C₂²×C₅	semidirect
103. #803	C₂²×C₅	semidirect
128. #432	C₅×D₃	semidirect
203. #97	C₂×C₅×D₃	semidirect
269. C₃×C₅	#1527	semidirect
317. C₅	#70804	direct (N × H)
333. C₃	C₅×#1527	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
C₄×C₅×#208	2
C₅×#4464	2
C₅×#1527	3
#70804	5

Subgroups — C5×#70804 #14284040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70804 #14284040