Subgroups — C₅×#70811 #14284453

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#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×#1536	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#930	—	2	1	no	—
9	C₅×#930	—	2	1	no	—
10	C₅×#875	C₂²	1	2	yes	—
11	C₅×#874	C₄	1	1	yes	characteristic
12	C₄×C₅×D₁₂	—	2	1	no	—
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₅×#869	C₄	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₅×#97	C₂²	1	2	yes	—
18	C₅×#830	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₅×#803	C₂²	1	2	yes	—
21	#70811	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	—	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	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×#67	—	2	2	no	—
30	C₅×#235	—	4	4	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	#5477	C₂×C₅	1	1	yes	characteristic
47	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
48	#4926	C₂×C₅	1	1	yes	characteristic, complemented
49	#4464	C₂×C₅	1	4	yes	complemented
50	C₂×C₅×#41	—	3	2	no	—
51	C₅×C₄⋊Q₈	—	3	1	no	—
52	C₅×#127	—	3	1	no	—
53	C₅×#124	—	3	2	no	—
54	C₅×#121	—	6	1	no	—
55	C₅×#121	—	6	1	no	—
56	C₄×C₅×D₄	—	6	1	no	—
57	C₄×C₅×D₄	—	6	1	no	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	—	2	1	no	—
60	C₅×C₄⋊C₈	—	6	1	no	—
61	C₅×C₂²⋊C₈	—	2	1	no	—
62	C₅×C₂²⋊C₈	—	6	1	no	—
63	C₅×#97	—	3	2	no	—
64	C₅×#97	C₂×D₃	1	2	yes	—
65	#1536	—	3	1	no	Sylow
66	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₅×D₃	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₅×#67	—	4	4	no	—
78	C₂×C₅×Dic₃	—	2	2	no	—
79	C₅×D₁₂	—	4	2	no	—
80	C₅×D₁₂	—	4	2	no	—
81	C₅×D₁₂	—	4	2	no	—
82	C₄×C₅×D₃	—	4	4	no	—
83	C₅×Dic₆	—	4	2	no	—
84	C₃×C₅×C₈	—	4	2	no	—
85	C₅×C₃⋊C₈	—	4	2	no	—
86	C₂×#235	C₄×C₅	1	2	yes	—
87	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
88	#930	—	2	1	no	—
89	#930	—	2	1	no	—
90	#875	C₂²×C₅	1	2	yes	—
91	#874	C₄×C₅	1	1	yes	characteristic
92	C₄×D₁₂	—	2	1	no	—
93	C₄×D₁₂	—	2	1	no	—
94	#869	C₄×C₅	1	1	yes	characteristic
95	C₃×C₄⋊C₈	—	2	1	no	—
96	C₃×C₂²⋊C₈	—	2	1	no	—
97	C₃×#97	C₂²×C₅	1	2	yes	—
98	#830	—	2	1	no	—
99	C₄⋊(C₃⋊C₈)	—	2	1	no	—
100	#803	C₂²×C₅	1	2	yes	—
101	C₅×#41	—	12	4	no	—
102	C₂×C₅×Q₈	—	3	2	no	—
103	C₂×C₅×D₄	—	3	2	no	—
104	C₂×C₅×D₄	—	6	2	no	—
105	C₂×C₅×D₄	—	6	1	no	—
106	C₂×C₅×D₄	—	6	1	no	—
107	C₂²×C₄×C₅	—	6	2	no	—
108	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
110	C₂²×C₄×C₅	#67	1	1	yes	characteristic
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	6	2	no	—
113	C₅×C₄⋊C₄	—	6	1	no	—
114	C₅×C₄⋊C₄	—	6	1	no	—
115	C₅×C₂²⋊C₄	—	6	2	no	—
116	C₅×C₂²⋊C₄	—	6	2	no	—
117	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
118	C₄²×C₅	D₁₂	1	1	yes	characteristic
119	C₄²×C₅	#67	1	1	yes	characteristic
120	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
121	#532	—	3	1	no	—
122	#432	—	3	1	no	—
123	#432	C₅×D₃	1	1	yes	characteristic, complemented
124	#329	—	3	4	no	—
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₅×D₃	—	4	2	no	—
130	C₂×C₅×D₃	—	4	2	no	—
131	C₂×C₅×D₃	—	4	2	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	#100	1	2	yes	—
138	C₅×Dic₃	—	4	2	no	—
139	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
142	C₂×#67	—	2	2	no	—
143	#235	—	4	4	no	—
144	C₂×D₁₂	—	2	1	no	—
145	C₂×D₁₂	—	2	1	no	—
146	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
147	C₂×C₄×D₃	—	2	2	no	—
148	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
149	C₂×C₃×C₈	—	2	2	no	—
150	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
151	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
154	(C₂×D₃)⋊C₄	—	2	2	no	—
155	(C₂×D₃)⋊C₄	—	2	2	no	—
156	C₄⋊Dic₃	—	2	1	no	—
157	C₄⋊Dic₃	—	2	1	no	—
158	C₂×C₃⋊C₈	—	2	2	no	—
159	C₂³×C₅	—	6	2	no	—
160	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
161	C₅×Q₈	—	12	2	no	—
162	C₅×D₄	—	12	2	no	—
163	C₅×D₄	—	12	2	no	—
164	C₅×D₄	—	12	2	no	—
165	C₅×D₄	—	12	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	6	2	no	—
170	C₂×C₄×C₅	—	12	4	no	—
171	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
177	C₅×C₈	—	4	2	no	—
178	C₅×C₈	—	12	2	no	—
179	C₂×#41	—	3	2	no	—
180	C₄⋊Q₈	—	3	1	no	—
181	#127	—	3	1	no	—
182	#124	—	3	2	no	—
183	#121	—	6	1	no	—
184	#121	—	6	1	no	—
185	C₄×D₄	—	6	1	no	—
186	C₄×D₄	—	6	1	no	—
187	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	6	1	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	6	1	no	—
192	#97	—	3	2	no	—
193	#97	C₂×C₅×D₃	1	2	yes	—
194	C₂×C₃×C₅	—	4	1	no	—
195	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
196	C₂×C₃×C₅	#421	1	1	yes	characteristic
197	C₂×C₃×C₅	#421	1	1	yes	characteristic
198	C₅×D₃	—	8	2	no	—
199	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₂²×D₃	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	#67	—	4	4	no	—
211	C₂×Dic₃	—	2	2	no	—
212	D₁₂	—	4	2	no	—
213	D₁₂	—	4	2	no	—
214	D₁₂	—	4	2	no	—
215	C₄×D₃	—	4	4	no	—
216	Dic₆	—	4	2	no	—
217	C₃×C₈	—	4	2	no	—
218	C₃⋊C₈	—	4	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	12	2	no	—
223	C₂²×C₅	—	12	2	no	—
224	C₂²×C₅	—	12	2	no	—
225	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	12	2	no	—
232	C₄×C₅	#823	1	2	yes	—
233	#41	—	12	4	no	—
234	C₂×Q₈	—	3	2	no	—
235	C₂×D₄	—	3	2	no	—
236	C₂×D₄	—	6	2	no	—
237	C₂×D₄	—	6	1	no	—
238	C₂×D₄	—	6	1	no	—
239	C₂²×C₄	—	6	2	no	—
240	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
241	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
242	C₂²×C₄	C₅×#67	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	6	2	no	—
245	C₄⋊C₄	—	6	1	no	—
246	C₄⋊C₄	—	6	1	no	—
247	C₂²⋊C₄	—	6	2	no	—
248	C₂²⋊C₄	—	6	2	no	—
249	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
250	C₄²	C₅×D₁₂	1	1	yes	characteristic
251	C₄²	C₅×#67	1	1	yes	characteristic
252	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
253	C₃×C₅	#1536	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×D₃	—	4	2	no	—
259	C₂×D₃	—	4	2	no	—
260	C₂×D₃	—	4	2	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	C₅×#100	1	2	yes	—
267	Dic₃	—	4	2	no	—
268	C₂×C₅	—	4	1	no	—
269	C₂×C₅	—	24	2	no	—
270	C₂×C₅	#5005	1	1	yes	characteristic
271	C₂×C₅	C₂×#823	1	1	yes	characteristic
272	C₂×C₅	#5139	1	1	yes	characteristic
273	C₂³	—	6	2	no	—
274	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
275	Q₈	—	12	2	no	—
276	D₄	—	12	2	no	—
277	D₄	—	12	2	no	—
278	D₄	—	12	2	no	—
279	D₄	—	12	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	12	4	no	—
285	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
290	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	12	2	no	—
293	C₂×C₃	—	4	1	no	—
294	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
295	C₂×C₃	C₅×#421	1	1	yes	characteristic
296	C₂×C₃	C₅×#421	1	1	yes	characteristic
297	D₃	—	8	2	no	—
298	C₅	#70811	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	12	2	no	—
303	C₂²	—	12	2	no	—
304	C₂²	—	12	2	no	—
305	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	12	2	no	—
312	C₄	C₅×#823	1	2	yes	—
313	C₃	C₅×#1536	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	4	1	no	—
315	C₂	—	24	2	no	—
316	C₂	C₅×#5005	1	1	yes	characteristic
317	C₂	C₂×C₅×#823	1	1	yes	characteristic
318	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	C₅×#5477	C₂
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#432	D₃
hypocenter	C₃	C₅×#1536
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
21. #70811	C₅	direct (N × H)
24. C₅×#432	D₃	semidirect
47. C₃×#432	C₂×C₅	semidirect
48. #4926	C₂×C₅	semidirect
49. #4464	C₂×C₅	semidirect
123. #432	C₅×D₃	semidirect
253. C₃×C₅	#1536	semidirect
298. C₅	#70811	direct (N × H)
313. C₃	C₅×#1536	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×#432	2
C₅×#4926	2
C₅×#4464	2
C₅×#1536	3
#70811	5

Subgroups — C5×#70811 #14284453

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70811 #14284453