Subgroups — C₅×#69485 #14304990

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
5	C₅×#4524	C₂	1	2	yes	maximal
6	C₅×#1940	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#95	C₄	1	1	yes	characteristic
14	C₃×C₅×#95	C₄	1	1	yes	characteristic
15	C₅×C₄⋊(C₃⋊C₈)	—	2	2	no	—
16	C₅×C₄⋊(C₃⋊C₈)	—	2	4	no	—
17	C₅×#803	—	2	2	no	—
18	#69485	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×C₄⋊C₈	—	3	2	no	—
20	C₂×C₅×C₄⋊C₈	—	3	1	no	—
21	C₂×C₅×#97	—	3	1	no	—
22	C₄×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic
23	C₅×#336	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₄²×C₅	—	2	2	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₃×C₄²×C₅	Q₈	1	2	yes	—
34	C₂×C₅×C₃⋊C₈	—	2	1	no	—
35	C₂×C₅×C₃⋊C₈	—	2	1	no	—
36	C₂×C₅×C₃⋊C₈	—	2	1	no	—
37	C₂×C₅×C₃⋊C₈	—	2	1	no	—
38	C₂×C₅×C₃⋊C₈	—	2	2	no	—
39	C₂×C₅×C₃⋊C₈	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
42	C₂×C₅×C₃⋊C₈	Q₈	1	2	yes	—
43	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
44	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
45	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	2	yes	—
46	C₂×#803	C₂×C₅	1	1	yes	characteristic
47	#4524	C₂×C₅	1	2	yes	—
48	C₂²×C₅×C₈	—	3	1	no	—
49	C₂²×C₅×C₈	—	3	1	no	—
50	C₂²×C₅×C₈	—	3	2	no	—
51	C₂×C₅×C₄⋊C₄	Dic₃	1	2	yes	—
52	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
53	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
54	C₅×C₄⋊C₈	—	6	4	no	—
55	C₅×C₄⋊C₈	—	6	2	no	—
56	C₅×#97	—	6	2	no	—
57	C₅×#95	Dic₃	1	1	yes	characteristic
58	C₅×#95	Dic₃	1	1	yes	characteristic
59	#1940	—	3	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₅×C₃⋊C₈	—	2	2	no	—
78	C₅×C₃⋊C₈	—	2	2	no	—
79	C₅×C₃⋊C₈	—	4	2	no	—
80	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
81	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
82	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
83	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
84	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
86	C₃×#95	C₄×C₅	1	1	yes	characteristic
87	C₃×#95	C₄×C₅	1	1	yes	characteristic
88	C₄⋊(C₃⋊C₈)	—	2	2	no	—
89	C₄⋊(C₃⋊C₈)	—	2	4	no	—
90	#803	—	2	2	no	—
91	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
92	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
93	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	—	6	1	no	—
99	C₂×C₅×C₈	—	6	1	no	—
100	C₂×C₅×C₈	—	6	1	no	—
101	C₂×C₅×C₈	—	6	1	no	—
102	C₂×C₅×C₈	—	6	2	no	—
103	C₂×C₅×C₈	—	6	2	no	—
104	C₂×C₅×C₈	—	6	2	no	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	Dic₆	1	2	yes	—
109	C₄²×C₅	D₁₂	1	2	yes	—
110	C₂×C₄⋊C₈	—	3	2	no	—
111	C₂×C₄⋊C₈	—	3	1	no	—
112	C₂×#97	—	3	1	no	—
113	C₄×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic
114	#336	—	3	2	no	—
115	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
119	C₂²×C₃×C₅	#131	1	2	yes	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	—	2	4	no	—
123	C₃×C₄×C₅	—	4	2	no	—
124	C₃×C₄×C₅	#121	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
126	C₃×C₄×C₅	#125	1	1	yes	characteristic
127	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
129	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
132	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	—	2	2	no	—
136	C₃×C₄²	C₅×D₄	1	2	yes	—
137	C₃×C₄²	C₅×Q₈	1	2	yes	—
138	C₂×C₃⋊C₈	—	2	1	no	—
139	C₂×C₃⋊C₈	—	2	1	no	—
140	C₂×C₃⋊C₈	—	2	1	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂×C₃⋊C₈	—	2	2	no	—
143	C₂×C₃⋊C₈	—	2	2	no	—
144	C₂×C₃⋊C₈	—	2	2	no	—
145	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
146	C₂×C₃⋊C₈	C₅×Q₈	1	2	yes	—
147	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	#208	1	4	yes	—
156	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
157	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
158	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
160	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	#237	1	1	yes	characteristic
162	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅	#239	1	1	yes	characteristic
164	C₅×C₈	—	6	2	no	—
165	C₅×C₈	—	6	2	no	—
166	C₅×C₈	—	12	2	no	—
167	C₂²×C₈	—	3	1	no	—
168	C₂²×C₈	—	3	1	no	—
169	C₂²×C₈	—	3	2	no	—
170	C₂×C₄⋊C₄	C₅×Dic₃	1	2	yes	—
171	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
172	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
173	C₄⋊C₈	—	6	4	no	—
174	C₄⋊C₈	—	6	2	no	—
175	#97	—	6	2	no	—
176	#95	C₅×Dic₃	1	1	yes	characteristic
177	#95	C₅×Dic₃	1	1	yes	characteristic
178	C₂×C₃×C₅	#389	1	1	yes	characteristic
179	C₂×C₃×C₅	#423	1	1	yes	characteristic
180	C₂×C₃×C₅	#424	1	1	yes	characteristic
181	C₂×C₃×C₅	#435	1	2	yes	—
182	C₂×C₃×C₅	#446	1	2	yes	—
183	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
192	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
193	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
199	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
200	C₃⋊C₈	—	2	2	no	—
201	C₃⋊C₈	—	2	2	no	—
202	C₃⋊C₈	—	4	2	no	—
203	C₂²×C₅	C₂×#208	1	2	yes	—
204	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
205	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
206	C₂²×C₅	Q₈×Dic₃	1	1	yes	characteristic
207	C₂²×C₅	#948	1	2	yes	—
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	4	2	no	—
212	C₄×C₅	#888	1	1	yes	characteristic
213	C₄×C₅	#890	1	1	yes	characteristic
214	C₄×C₅	#895	1	1	yes	characteristic
215	C₄×C₅	#897	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
217	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
218	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
221	C₂×C₈	—	3	2	no	—
222	C₂×C₈	—	3	2	no	—
223	C₂×C₈	—	6	1	no	—
224	C₂×C₈	—	6	1	no	—
225	C₂×C₈	—	6	1	no	—
226	C₂×C₈	—	6	1	no	—
227	C₂×C₈	—	6	2	no	—
228	C₂×C₈	—	6	2	no	—
229	C₂×C₈	—	6	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	C₅×Dic₆	1	2	yes	—
234	C₄²	C₅×D₁₂	1	2	yes	—
235	C₃×C₅	#1940	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
237	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
238	C₂²×C₃	C₄×C₅×Q₈	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
240	C₂²×C₃	C₅×#131	1	2	yes	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	4	2	no	—
245	C₃×C₄	C₅×#121	1	1	yes	characteristic
246	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
247	C₃×C₄	C₅×#125	1	1	yes	characteristic
248	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
249	C₂×C₅	#4924	1	1	yes	characteristic
250	C₂×C₅	#4984	1	1	yes	characteristic
251	C₂×C₅	#4999	1	1	yes	characteristic
252	C₂×C₅	#5011	1	2	yes	—
253	C₂×C₅	#5024	1	2	yes	—
254	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	C₅×#208	1	4	yes	—
263	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
264	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
265	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
266	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
267	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
268	C₂×C₄	C₅×#237	1	1	yes	characteristic
269	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
270	C₂×C₄	C₅×#239	1	1	yes	characteristic
271	C₈	—	6	2	no	—
272	C₈	—	6	2	no	—
273	C₈	—	12	2	no	—
274	C₂×C₃	C₅×#389	1	1	yes	characteristic
275	C₂×C₃	C₅×#423	1	1	yes	characteristic
276	C₂×C₃	C₅×#424	1	1	yes	characteristic
277	C₂×C₃	C₅×#435	1	2	yes	—
278	C₂×C₃	C₅×#446	1	2	yes	—
279	C₅	#69485	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²	C₂×C₅×#208	1	2	yes	—
281	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
282	C₂²	C₅×D₄×Dic₃	1	1	yes	characteristic
283	C₂²	C₅×Q₈×Dic₃	1	1	yes	characteristic
284	C₂²	C₅×#948	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	C₅×#888	1	1	yes	characteristic
290	C₄	C₅×#890	1	1	yes	characteristic
291	C₄	C₅×#895	1	1	yes	characteristic
292	C₄	C₅×#897	1	1	yes	characteristic
293	C₃	C₅×#1940	1	1	yes	characteristic, complemented, Sylow
294	C₂	C₅×#4924	1	1	yes	characteristic
295	C₂	C₅×#4984	1	1	yes	characteristic
296	C₂	C₅×#4999	1	1	yes	characteristic
297	C₂	C₅×#5011	1	2	yes	—
298	C₂	C₅×#5024	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₄×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#1940
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
18. #69485	C₅	direct (N × H)
235. C₃×C₅	#1940	semidirect
279. C₅	#69485	direct (N × H)
293. C₃	C₅×#1940	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×C₄⋊C₄	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₂×C₅×#803	2
C₅×#4524	2
C₅×#1940	3
#69485	5

Subgroups — C5×#69485 #14304990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69485 #14304990