Subgroups — C₅×#71601 #14280896

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	2	yes	maximal, complemented
3	C₅×#4525	C₂	1	4	yes	maximal, complemented
4	C₅×#1460	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×D₄	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
7	C₃×C₄×C₅×D₄	—	2	2	no	—
8	C₃×C₄×C₅×D₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₅×#830	C₂²	1	4	yes	—
13	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
14	C₄×C₅×C₃⋊C₈	—	2	2	no	—
15	#71601	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₄×C₅×D₄	D₃	1	1	yes	characteristic
17	C₅×#431	—	3	2	no	—
18	C₅×#323	—	3	4	no	—
19	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₅×D₄	—	2	2	no	—
21	C₂×C₃×C₅×D₄	—	2	2	no	—
22	C₂×C₃×C₅×D₄	C₈	1	4	yes	complemented
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	—	4	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₂×C₅×C₃⋊C₈	—	2	4	no	—
35	C₂×C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
36	#4925	C₂×C₅	1	2	yes	complemented
37	#4525	C₂×C₅	1	4	yes	complemented
38	C₂²×C₅×D₄	Dic₃	1	1	yes	characteristic
39	C₂³×C₄×C₅	Dic₃	1	2	yes	—
40	C₄×C₅×D₄	—	2	2	no	—
41	C₄×C₅×D₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	Dic₃	1	1	yes	characteristic
43	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
45	C₅×C₄⋊C₈	—	6	2	no	—
46	C₅×C₂²⋊C₈	—	3	4	no	—
47	C₄×C₅×C₈	—	6	2	no	—
48	#1460	—	3	1	no	Sylow
49	C₂³×C₃×C₅	—	2	2	no	—
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	—	4	2	no	—
52	C₂³×C₃×C₅	—	4	2	no	—
53	C₂³×C₃×C₅	C₂×C₈	1	4	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₅×D₄	—	2	8	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	4	1	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	4	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₅×C₃⋊C₈	—	4	4	no	—
69	C₂²×C₃×D₄	C₄×C₅	1	1	yes	characteristic
70	C₂³×C₃×C₄	C₄×C₅	1	2	yes	—
71	C₃×C₄×D₄	—	2	2	no	—
72	C₃×C₄×D₄	—	2	2	no	—
73	C₂×C₃×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
74	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	#830	C₂²×C₅	1	4	yes	—
77	C₄⋊(C₃⋊C₈)	—	2	2	no	—
78	C₄×C₃⋊C₈	—	2	2	no	—
79	C₂⁴×C₅	C₂×Dic₃	1	2	yes	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	C₃⋊C₈	1	4	yes	complemented
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	—	4	4	no	—
86	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
88	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
90	C₂×C₅×C₈	—	6	4	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₂²⋊C₄	—	2	4	no	—
93	C₄²×C₅	#67	1	2	yes	—
94	C₄²×C₅	#67	1	2	yes	—
95	C₂×C₄×D₄	C₅×D₃	1	1	yes	characteristic
96	#431	—	3	2	no	—
97	#323	—	3	4	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	4	no	—
101	C₂²×C₃×C₅	—	4	4	no	—
102	C₂²×C₃×C₅	—	4	4	no	—
103	C₂²×C₃×C₅	—	4	2	no	—
104	C₂²×C₃×C₅	—	4	2	no	—
105	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	4	2	no	—
111	C₃×C₄×C₅	C₂²⋊C₈	1	4	yes	—
112	C₃×C₄×C₅	#100	1	2	yes	—
113	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	C₅×C₈	1	4	yes	complemented
117	C₂²×C₃×C₄	—	2	2	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	—	4	4	no	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₂²⋊C₄	—	2	4	no	—
126	C₃×C₄²	C₅×D₄	1	2	yes	—
127	C₃×C₄²	C₅×D₄	1	2	yes	—
128	C₂×C₃⋊C₈	—	2	4	no	—
129	C₂³×C₅	—	2	2	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	4	2	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	C₂×C₃⋊C₈	1	4	yes	—
134	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
135	C₅×D₄	—	2	8	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
143	C₂×C₄×C₅	#208	1	2	yes	—
144	C₂×C₄×C₅	#217	1	1	yes	characteristic
145	C₂×C₄×C₅	#217	1	2	yes	—
146	C₂×C₄×C₅	#217	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×#67	1	2	yes	—
148	C₅×C₈	—	12	4	no	—
149	C₂²×D₄	C₅×Dic₃	1	1	yes	characteristic
150	C₂³×C₄	C₅×Dic₃	1	2	yes	—
151	C₄×D₄	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₅×Dic₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
155	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
156	C₄⋊C₈	—	6	2	no	—
157	C₂²⋊C₈	—	3	4	no	—
158	C₄×C₈	—	6	2	no	—
159	C₂×C₃×C₅	—	2	1	no	—
160	C₂×C₃×C₅	—	2	1	no	—
161	C₂×C₃×C₅	—	4	4	no	—
162	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂×C₃×C₅	#410	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	—	2	2	no	—
167	C₂³×C₃	—	4	2	no	—
168	C₂³×C₃	—	4	2	no	—
169	C₂³×C₃	C₂×C₅×C₈	1	4	yes	—
170	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
171	C₃×D₄	—	2	8	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	4	1	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
180	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
182	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
184	C₃⋊C₈	—	4	4	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	—	2	1	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	4	4	no	—
189	C₂²×C₅	—	4	4	no	—
190	C₂²×C₅	—	4	2	no	—
191	C₂²×C₅	—	4	2	no	—
192	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
193	C₂²×C₅	C₂×#208	1	1	yes	characteristic
194	C₂²×C₅	C₂×#217	1	1	yes	characteristic
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	#830	1	4	yes	—
199	C₄×C₅	#833	1	2	yes	—
200	C₂⁴	C₂×C₅×Dic₃	1	2	yes	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₅×C₃⋊C₈	1	4	yes	complemented
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
209	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
211	C₂×C₈	—	6	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₄²	C₅×#67	1	2	yes	—
215	C₄²	C₅×#67	1	2	yes	—
216	C₃×C₅	#1460	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	4	4	no	—
221	C₂²×C₃	—	4	4	no	—
222	C₂²×C₃	—	4	2	no	—
223	C₂²×C₃	—	4	2	no	—
224	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
226	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	2	no	—
230	C₃×C₄	C₅×C₂²⋊C₈	1	4	yes	—
231	C₃×C₄	C₅×#100	1	2	yes	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	—	4	4	no	—
235	C₂×C₅	C₂×#830	1	1	yes	characteristic
236	C₂×C₅	C₂×#833	1	1	yes	characteristic
237	C₂×C₅	#5240	1	1	yes	characteristic
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₂×C₅×C₃⋊C₈	1	4	yes	—
243	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
244	D₄	—	2	8	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₂×C₅×C₃⋊C₈	1	2	yes	—
252	C₂×C₄	C₅×#208	1	2	yes	—
253	C₂×C₄	C₅×#217	1	1	yes	characteristic
254	C₂×C₄	C₅×#217	1	2	yes	—
255	C₂×C₄	C₅×#217	1	1	yes	characteristic
256	C₂×C₄	C₂×C₅×#67	1	2	yes	—
257	C₈	—	12	4	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	4	no	—
261	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
262	C₂×C₃	C₅×#410	1	1	yes	characteristic
263	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
264	C₅	#71601	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	C₂²×C₅×C₃⋊C₈	1	1	yes	characteristic
273	C₂²	C₂×C₅×#208	1	1	yes	characteristic
274	C₂²	C₂×C₅×#217	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	C₅×#830	1	4	yes	—
279	C₄	C₅×#833	1	2	yes	—
280	C₃	C₅×#1460	1	1	yes	characteristic, complemented, Sylow
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	4	4	no	—
284	C₂	C₂×C₅×#830	1	1	yes	characteristic
285	C₂	C₂×C₅×#833	1	1	yes	characteristic
286	C₂	C₅×#5240	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#217
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×C₅×D₄	C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×D₄	D₃
hypocenter	C₃	C₅×#1460
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×D₄

Complements

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

N	H ≅ G/N	type
2. C₅×#4925	C₂	semidirect
3. C₅×#4525	C₂	semidirect
15. #71601	C₅	direct (N × H)
22. C₂×C₃×C₅×D₄	C₈	semidirect
36. #4925	C₂×C₅	semidirect
37. #4525	C₂×C₅	semidirect
82. C₂×C₅×D₄	C₃⋊C₈	semidirect
116. C₂×C₃×D₄	C₅×C₈	semidirect
203. C₂×D₄	C₅×C₃⋊C₈	semidirect
216. C₃×C₅	#1460	semidirect
264. C₅	#71601	direct (N × H)
280. C₃	C₅×#1460	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₄	2
C₅×#4925	2
C₅×#4525	2
C₅×#1460	3
#71601	5

Subgroups — C5×#71601 #14280896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71601 #14280896