Subgroups — C₅×#70818 #14285222

Order 1920.

664 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	C₂×C₅×#874	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×#4485	C₂	1	4	yes	maximal, complemented
5	C₅×#1555	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	1	no	—
7	C₂²×C₅×D₁₂	—	2	1	no	—
8	C₂²×C₅×D₁₂	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₅×#874	—	2	1	no	—
11	C₅×#874	—	2	1	no	—
12	C₅×#874	C₄	1	2	yes	—
13	C₅×#874	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
15	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₅×#830	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
19	C₅×#803	—	2	1	no	—
20	#70818	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×#127	—	3	1	no	—
22	C₅×#432	—	3	1	no	—
23	C₅×#432	D₃	1	1	yes	characteristic, complemented
24	C₅×#331	—	3	4	no	—
25	C₂³×C₅×D₃	—	2	2	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×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	2	no	—
31	C₂×C₅×D₁₂	—	2	2	no	—
32	C₂×C₅×D₁₂	—	2	2	no	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₅×D₁₂	—	4	4	no	—
35	C₂×C₅×D₁₂	—	4	4	no	—
36	C₂×C₅×D₁₂	—	4	4	no	—
37	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₂×#874	C₂×C₅	1	1	yes	characteristic
45	C₃×#432	C₂×C₅	1	1	yes	characteristic, complemented
46	#4926	C₂×C₅	1	1	yes	characteristic, complemented
47	#4485	C₂×C₅	1	4	yes	complemented
48	C₂²×C₅×D₄	—	3	2	no	—
49	C₂²×C₅×D₄	—	6	1	no	—
50	C₂²×C₅×D₄	—	6	1	no	—
51	C₅×#127	—	3	2	no	—
52	C₅×#127	—	3	2	no	—
53	C₅×#127	—	6	1	no	—
54	C₅×#127	—	6	1	no	—
55	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
56	C₅×C₄⋊C₈	—	3	2	no	—
57	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	complemented
58	C₅×C₂²⋊C₈	—	2	1	no	—
59	C₅×C₂²⋊C₈	—	6	1	no	—
60	C₅×#97	—	2	1	no	—
61	C₅×#97	—	6	1	no	—
62	#1555	—	3	1	no	Sylow
63	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₅×D₃	—	2	4	no	—
65	C₂²×C₅×D₃	—	4	4	no	—
66	C₂²×C₅×D₃	—	4	4	no	—
67	C₂²×C₅×D₃	—	4	4	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	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×D₁₂	—	4	4	no	—
78	C₅×D₁₂	—	4	4	no	—
79	C₅×D₁₂	—	4	4	no	—
80	C₅×D₁₂	—	4	4	no	—
81	C₅×D₁₂	—	4	4	no	—
82	C₅×D₁₂	—	4	4	no	—
83	C₃×C₅×C₈	—	4	2	no	—
84	C₅×C₃⋊C₈	—	4	2	no	—
85	C₂²×D₁₂	—	2	1	no	—
86	C₂²×D₁₂	—	2	1	no	—
87	C₂²×D₁₂	C₄×C₅	1	2	yes	—
88	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
89	#874	—	2	1	no	—
90	#874	—	2	1	no	—
91	#874	C₄×C₅	1	2	yes	—
92	#874	C₂²×C₅	1	2	yes	—
93	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
94	C₃×C₂²⋊C₈	—	2	1	no	—
95	C₃×#97	—	2	1	no	—
96	#830	—	2	1	no	—
97	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	complemented
98	#803	—	2	1	no	—
99	C₂⁴×C₅	—	6	2	no	—
100	C₂×C₅×D₄	—	3	4	no	—
101	C₂×C₅×D₄	—	6	2	no	—
102	C₂×C₅×D₄	—	6	2	no	—
103	C₂×C₅×D₄	—	6	2	no	—
104	C₂×C₅×D₄	—	6	2	no	—
105	C₂×C₅×D₄	—	6	2	no	—
106	C₂×C₅×D₄	—	12	4	no	—
107	C₂×C₅×D₄	—	12	4	no	—
108	C₂×C₅×D₄	—	12	4	no	—
109	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	6	2	no	—
114	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
115	C₄²×C₅	D₁₂	1	1	yes	characteristic
116	C₄²×C₅	#67	1	1	yes	characteristic
117	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₂×#127	—	3	1	no	—
119	#432	—	3	1	no	—
120	#432	C₅×D₃	1	1	yes	characteristic, complemented
121	#331	—	3	4	no	—
122	C₂²×C₃×C₅	—	2	1	no	—
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	#100	1	2	yes	—
125	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₅×D₃	—	4	4	no	—
127	C₂×C₅×D₃	—	4	4	no	—
128	C₂×C₅×D₃	—	4	4	no	—
129	C₂×C₅×D₃	—	8	8	no	—
130	C₃×C₄×C₅	—	2	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
135	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
136	C₂³×D₃	—	2	2	no	—
137	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂×D₁₂	—	2	2	no	—
141	C₂×D₁₂	—	2	2	no	—
142	C₂×D₁₂	—	2	2	no	—
143	C₂×D₁₂	—	2	2	no	—
144	C₂×D₁₂	—	2	2	no	—
145	C₂×D₁₂	—	4	4	no	—
146	C₂×D₁₂	—	4	4	no	—
147	C₂×D₁₂	—	4	4	no	—
148	C₂×D₁₂	C₂×C₄×C₅	1	4	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×C₃⋊C₈	—	2	2	no	—
155	C₂³×C₅	—	6	4	no	—
156	C₂³×C₅	—	12	4	no	—
157	C₂³×C₅	—	12	4	no	—
158	C₂³×C₅	—	12	4	no	—
159	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₅×D₄	—	12	4	no	—
161	C₅×D₄	—	12	4	no	—
162	C₅×D₄	—	12	4	no	—
163	C₅×D₄	—	12	4	no	—
164	C₅×D₄	—	12	4	no	—
165	C₅×D₄	—	12	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	—	12	2	no	—
177	C₂²×D₄	—	3	2	no	—
178	C₂²×D₄	—	6	1	no	—
179	C₂²×D₄	—	6	1	no	—
180	#127	—	3	2	no	—
181	#127	—	3	2	no	—
182	#127	—	6	1	no	—
183	#127	—	6	1	no	—
184	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
185	C₄⋊C₈	—	3	2	no	—
186	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	complemented
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	6	1	no	—
189	#97	—	2	1	no	—
190	#97	—	6	1	no	—
191	C₂×C₃×C₅	—	2	2	no	—
192	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
193	C₂×C₃×C₅	#418	1	1	yes	characteristic
194	C₂×C₃×C₅	#418	1	1	yes	characteristic
195	C₅×D₃	—	8	4	no	—
196	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×D₃	—	2	4	no	—
198	C₂²×D₃	—	4	4	no	—
199	C₂²×D₃	—	4	4	no	—
200	C₂²×D₃	—	4	4	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	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	D₁₂	—	4	4	no	—
211	D₁₂	—	4	4	no	—
212	D₁₂	—	4	4	no	—
213	D₁₂	—	4	4	no	—
214	D₁₂	—	4	4	no	—
215	D₁₂	—	4	4	no	—
216	C₃×C₈	—	4	2	no	—
217	C₃⋊C₈	—	4	2	no	—
218	C₂²×C₅	—	2	1	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	12	4	no	—
221	C₂²×C₅	—	12	4	no	—
222	C₂²×C₅	—	12	4	no	—
223	C₂²×C₅	—	24	8	no	—
224	C₂²×C₅	#823	1	2	yes	—
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	#908	1	2	yes	—
231	C₄×C₅	#949	1	2	yes	—
232	C₂⁴	—	6	2	no	—
233	C₂×D₄	—	3	4	no	—
234	C₂×D₄	—	6	2	no	—
235	C₂×D₄	—	6	2	no	—
236	C₂×D₄	—	6	2	no	—
237	C₂×D₄	—	6	2	no	—
238	C₂×D₄	—	6	2	no	—
239	C₂×D₄	—	12	4	no	—
240	C₂×D₄	—	12	4	no	—
241	C₂×D₄	—	12	4	no	—
242	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
243	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
244	C₂²×C₄	C₅×#67	1	1	yes	characteristic
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	6	2	no	—
247	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
248	C₄²	C₅×D₁₂	1	1	yes	characteristic
249	C₄²	C₅×#67	1	1	yes	characteristic
250	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
251	C₃×C₅	#1555	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	C₅×#100	1	2	yes	—
255	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×D₃	—	4	4	no	—
257	C₂×D₃	—	4	4	no	—
258	C₂×D₃	—	4	4	no	—
259	C₂×D₃	—	8	8	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
265	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
266	C₂×C₅	—	2	2	no	—
267	C₂×C₅	—	24	4	no	—
268	C₂×C₅	#5001	1	1	yes	characteristic
269	C₂×C₅	C₂×#823	1	1	yes	characteristic
270	C₂×C₅	#5131	1	1	yes	characteristic
271	C₂³	—	6	4	no	—
272	C₂³	—	12	4	no	—
273	C₂³	—	12	4	no	—
274	C₂³	—	12	4	no	—
275	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
276	D₄	—	12	4	no	—
277	D₄	—	12	4	no	—
278	D₄	—	12	4	no	—
279	D₄	—	12	4	no	—
280	D₄	—	12	4	no	—
281	D₄	—	12	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	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	—	2	2	no	—
294	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
295	C₂×C₃	C₅×#418	1	1	yes	characteristic
296	C₂×C₃	C₅×#418	1	1	yes	characteristic
297	D₃	—	8	4	no	—
298	C₅	#70818	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	12	4	no	—
302	C₂²	—	12	4	no	—
303	C₂²	—	12	4	no	—
304	C₂²	—	24	8	no	—
305	C₂²	C₅×#823	1	2	yes	—
306	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	C₅×#908	1	2	yes	—
312	C₄	C₅×#949	1	2	yes	—
313	C₃	C₅×#1555	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	24	4	no	—
316	C₂	C₅×#5001	1	1	yes	characteristic
317	C₂	C₂×C₅×#823	1	1	yes	characteristic
318	C₂	C₅×#5131	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	C₂×C₅×#874	C₂
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#432	D₃
hypocenter	C₃	C₅×#1555
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×#4485	C₂	semidirect
14. C₃×C₅×C₄⋊C₈	C₂²	semidirect
18. C₅×C₄⋊(C₃⋊C₈)	C₂²	semidirect
20. #70818	C₅	direct (N × H)
23. C₅×#432	D₃	semidirect
45. C₃×#432	C₂×C₅	semidirect
46. #4926	C₂×C₅	semidirect
47. #4485	C₂×C₅	semidirect
57. C₅×C₄⋊C₈	C₂×D₃	semidirect
93. C₃×C₄⋊C₈	C₂²×C₅	semidirect
97. C₄⋊(C₃⋊C₈)	C₂²×C₅	semidirect
120. #432	C₅×D₃	semidirect
186. C₄⋊C₈	C₂×C₅×D₃	semidirect
251. C₃×C₅	#1555	semidirect
298. C₅	#70818	direct (N × H)
313. C₃	C₅×#1555	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#874	2
C₃×C₅×#432	2
C₅×#4926	2
C₅×#4485	2
C₅×#1555	3
#70818	5

Subgroups — C5×#70818 #14285222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70818 #14285222