Subgroups — C₅×#68220 #14286887

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#215	C₂	1	1	yes	maximal, characteristic
2	C₅×#5024	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4834	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4459	C₂	1	1	yes	maximal, characteristic
7	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1585	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#215	C₄	1	1	yes	characteristic
11	C₂×C₅×D₃×C₈	—	2	1	no	—
12	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#835	C₄	1	1	yes	characteristic
16	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
17	C₅×#809	C₄	1	1	yes	characteristic
18	C₅×#808	C₄	1	1	yes	characteristic
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₅×#803	—	2	1	no	—
22	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68220	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#446	—	3	1	no	—
26	C₄×C₅×SD₁₆	—	3	1	no	—
27	C₅×C₈×D₄	—	3	1	no	—
28	C₅×#335	—	3	1	no	—
29	C₅×Q₈⋊C₈	D₃	1	1	yes	characteristic, complemented
30	C₅×D₄⋊C₈	—	3	1	no	—
31	C₅×#322	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	—	2	1	no	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×#215	—	2	2	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
43	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	—	2	1	no	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
49	C₅×D₃×C₈	—	4	2	no	—
50	C₄×#215	C₂×C₅	1	1	yes	characteristic
51	#5024	C₂×C₅	1	1	yes	characteristic, complemented
52	#4834	C₂×C₅	1	1	yes	characteristic
53	C₃×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	#4479	C₂×C₅	1	1	yes	characteristic, complemented
55	#4459	C₂×C₅	1	1	yes	characteristic
56	#4455	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×SD₁₆	—	3	1	no	—
58	C₂²×C₅×C₈	—	6	1	no	—
59	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
60	C₄×C₅×D₄	—	3	1	no	—
61	C₅×#106	—	3	1	no	—
62	C₅×C₄⋊C₈	—	3	1	no	—
63	C₅×C₄⋊C₈	—	6	1	no	—
64	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₅×#103	—	3	1	no	—
66	C₅×#102	—	3	1	no	—
67	C₅×C₂²⋊C₈	—	6	1	no	—
68	C₅×#97	—	6	1	no	—
69	C₄×C₅×C₈	—	3	1	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#1585	—	3	1	no	Sylow
73	C₂²×C₅×D₃	—	2	1	no	—
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
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	#41	1	1	yes	characteristic
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₅×D₁₂	—	2	1	no	—
83	C₅×D₁₂	M₁₆	1	2	yes	—
84	C₄×C₅×D₃	—	4	1	no	—
85	C₄×C₅×D₃	—	4	1	no	—
86	C₃×C₅×C₈	—	2	2	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₅×C₃⋊C₈	—	2	1	no	—
90	C₅×C₃⋊C₈	—	2	1	no	—
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	—	4	1	no	—
93	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×#215	C₄×C₅	1	1	yes	characteristic
95	C₂×D₃×C₈	—	2	1	no	—
96	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	#835	C₄×C₅	1	1	yes	characteristic
100	(C₂×D₃)⋊C₈	—	2	1	no	—
101	#809	C₄×C₅	1	1	yes	characteristic
102	#808	C₄×C₅	1	1	yes	characteristic
103	C₄⋊(C₃⋊C₈)	—	2	1	no	—
104	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
105	#803	—	2	1	no	—
106	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
107	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
109	C₂×C₅×D₄	—	3	1	no	—
110	C₂²×C₄×C₅	—	6	1	no	—
111	C₅×SD₁₆	—	6	2	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	3	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	6	1	no	—
119	C₂×C₅×C₈	—	12	2	no	—
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	3	1	no	—
122	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
123	C₅×C₂²⋊C₄	—	6	1	no	—
124	C₄²×C₅	—	2	1	no	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	#446	—	3	1	no	—
127	C₄×SD₁₆	—	3	1	no	—
128	C₈×D₄	—	3	1	no	—
129	#335	—	3	1	no	—
130	Q₈⋊C₈	C₅×D₃	1	1	yes	characteristic, complemented
131	D₄⋊C₈	—	3	1	no	—
132	#322	—	3	1	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×C₅×D₃	—	2	2	no	—
135	C₂×C₅×D₃	—	4	1	no	—
136	C₂×C₅×D₃	—	4	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	#131	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₃×C₄×C₅	#137	1	1	yes	characteristic
144	C₅×Dic₃	—	4	1	no	—
145	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×C₄×D₃	—	2	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
154	#215	—	2	2	no	—
155	(C₂×D₃)⋊C₄	—	2	1	no	—
156	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₃⋊C₈	—	2	1	no	—
158	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
161	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
162	D₃×C₈	—	4	2	no	—
163	C₂³×C₅	—	6	1	no	—
164	C₅×Q₈	—	2	1	no	—
165	C₅×Q₈	—	2	1	no	—
166	C₅×D₄	—	3	2	no	—
167	C₅×D₄	—	6	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	6	1	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	—	12	1	no	—
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	#237	1	1	yes	characteristic
176	C₅×C₈	—	2	2	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	—	6	1	no	—
179	C₅×C₈	—	6	1	no	—
180	C₅×C₈	—	6	1	no	—
181	C₅×C₈	—	6	1	no	—
182	C₅×C₈	—	12	1	no	—
183	C₂×SD₁₆	—	3	1	no	—
184	C₂²×C₈	—	6	1	no	—
185	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
186	C₄×D₄	—	3	1	no	—
187	#106	—	3	1	no	—
188	C₄⋊C₈	—	3	1	no	—
189	C₄⋊C₈	—	6	1	no	—
190	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
191	#103	—	3	1	no	—
192	#102	—	3	1	no	—
193	C₂²⋊C₈	—	6	1	no	—
194	#97	—	6	1	no	—
195	C₄×C₈	—	3	1	no	—
196	C₄×C₈	—	3	1	no	—
197	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
198	C₂×C₃×C₅	#435	1	1	yes	characteristic
199	C₂×C₃×C₅	#440	1	1	yes	characteristic
200	C₂×C₃×C₅	#443	1	1	yes	characteristic
201	C₅×D₃	—	4	2	no	—
202	C₂²×D₃	—	2	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
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	C₅×#41	1	1	yes	characteristic
210	C₂×Dic₃	—	2	1	no	—
211	D₁₂	—	2	1	no	—
212	D₁₂	C₅×M₁₆	1	2	yes	—
213	C₄×D₃	—	4	1	no	—
214	C₄×D₃	—	4	1	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃⋊C₈	—	2	1	no	—
218	C₃⋊C₈	—	2	1	no	—
219	C₃⋊C₈	—	2	1	no	—
220	C₃⋊C₈	—	2	1	no	—
221	C₃⋊C₈	—	4	1	no	—
222	C₂²×C₅	—	6	2	no	—
223	C₂²×C₅	—	12	1	no	—
224	C₂²×C₅	—	12	1	no	—
225	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	1	no	—
228	C₄×C₅	—	4	1	no	—
229	C₄×C₅	—	12	1	no	—
230	C₄×C₅	#901	1	1	yes	characteristic
231	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
232	C₄×C₅	#914	1	1	yes	characteristic
233	C₄×C₅	#918	1	1	yes	characteristic
234	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
235	C₂×D₄	—	3	1	no	—
236	C₂²×C₄	—	6	1	no	—
237	SD₁₆	—	6	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	12	2	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	3	1	no	—
248	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
249	C₂²⋊C₄	—	6	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₃×C₅	#1585	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
254	C₂×D₃	—	2	2	no	—
255	C₂×D₃	—	4	1	no	—
256	C₂×D₃	—	4	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
261	C₃×C₄	C₅×#131	1	1	yes	characteristic
262	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₃×C₄	C₅×#137	1	1	yes	characteristic
264	Dic₃	—	4	1	no	—
265	C₂×C₅	—	12	2	no	—
266	C₂×C₅	#4728	1	1	yes	characteristic
267	C₂×C₅	#4817	1	1	yes	characteristic
268	C₂×C₅	#4824	1	1	yes	characteristic
269	C₂³	—	6	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	D₄	—	3	2	no	—
273	D₄	—	6	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	—	12	1	no	—
279	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
280	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
281	C₂×C₄	C₅×#237	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	6	1	no	—
285	C₈	—	6	1	no	—
286	C₈	—	6	1	no	—
287	C₈	—	6	1	no	—
288	C₈	—	12	1	no	—
289	C₂×C₃	C₅×#435	1	1	yes	characteristic
290	C₂×C₃	C₅×#440	1	1	yes	characteristic
291	C₂×C₃	C₅×#443	1	1	yes	characteristic
292	D₃	—	4	2	no	—
293	C₅	#68220	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	6	2	no	—
295	C₂²	—	12	1	no	—
296	C₂²	—	12	1	no	—
297	C₂²	C₅×#67⋊C₄	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	12	1	no	—
302	C₄	C₅×#901	1	1	yes	characteristic
303	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
304	C₄	C₅×#914	1	1	yes	characteristic
305	C₄	C₅×#918	1	1	yes	characteristic
306	C₃	C₅×#1585	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	12	2	no	—
308	C₂	C₅×#4728	1	1	yes	characteristic
309	C₂	C₅×#4817	1	1	yes	characteristic
310	C₂	C₅×#4824	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×Q₈⋊C₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×D₁₂	C₂×C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×Q₈⋊C₈	D₃
hypocenter	C₃	C₅×#1585
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₅×#5024	C₂	semidirect
4. C₃×C₅×Q₈⋊C₈	C₂	semidirect
5. C₅×#4479	C₂	semidirect
7. C₅×#4455	C₂	semidirect
24. #68220	C₅	direct (N × H)
29. C₅×Q₈⋊C₈	D₃	semidirect
51. #5024	C₂×C₅	semidirect
53. C₃×Q₈⋊C₈	C₂×C₅	semidirect
54. #4479	C₂×C₅	semidirect
56. #4455	C₂×C₅	semidirect
130. Q₈⋊C₈	C₅×D₃	semidirect
252. C₃×C₅	#1585	semidirect
293. C₅	#68220	direct (N × H)
306. C₃	C₅×#1585	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#215	2
C₅×#5024	2
C₅×#4834	2
C₃×C₅×Q₈⋊C₈	2
C₅×#4479	2
C₅×#4459	2
C₅×#4455	2
C₅×#1585	3
#68220	5

Subgroups — C5×#68220 #14286887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68220 #14286887