Subgroups — C₂×C₅×#4850 #14430075

Order 1920.

832 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#818	C₂	1	2	yes	maximal
5	C₂×C₅×#816	C₂	1	2	yes	maximal
6	C₅×#4850	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#477	—	3	1	no	maximal, Hall
8	C₂²×C₅×Dic₆	—	2	1	no	—
9	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊Dic₃	—	2	1	no	—
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₅×Dic₃⋊C₄	—	2	1	no	—
16	C₂×C₄×C₅×Dic₃	—	2	1	no	—
17	C₅×#869	—	2	2	no	—
18	C₅×#869	C₂²	1	4	yes	—
19	C₄×C₅×Dic₆	—	2	2	no	—
20	C₄×C₅×Dic₆	C₂²	1	4	yes	—
21	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
22	C₅×#818	C₂²	1	8	yes	—
23	C₅×#816	C₂²	1	8	yes	—
24	C₂×#4850	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
26	C₂×C₄×C₅×Q₈	—	3	1	no	—
27	C₅×#477	—	3	8	no	—
28	C₂×C₅×#107	—	3	2	no	—
29	C₂×C₅×C₄⋊C₈	D₃	1	1	yes	characteristic
30	C₂×C₅×#103	—	3	2	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×C₅×Dic₃	—	2	1	no	—
35	C₂²×C₅×Dic₃	—	2	1	no	—
36	C₂²×C₅×Dic₃	—	2	2	no	—
37	C₂×C₅×Dic₆	—	2	1	no	—
38	C₂×C₅×Dic₆	—	2	2	no	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	4	4	no	—
41	C₂×C₅×Dic₆	Q₈	1	4	yes	complemented
42	C₂×C₅×Dic₆	C₂³	1	2	yes	—
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	2	yes	—
47	C₅×C₄⋊Dic₃	—	2	1	no	—
48	C₅×C₄⋊Dic₃	—	2	2	no	—
49	C₅×C₄⋊Dic₃	—	2	4	no	—
50	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
51	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
52	C₅×Dic₃⋊C₄	—	2	4	no	—
53	C₄×C₅×Dic₃	—	2	4	no	—
54	C₂×#869	C₂×C₅	1	1	yes	characteristic
55	C₂×C₄×Dic₆	C₂×C₅	1	1	yes	characteristic
56	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
57	C₂×#818	C₂×C₅	1	2	yes	—
58	C₂×#816	C₂×C₅	1	2	yes	—
59	#4850	C₂×C₅	1	8	yes	direct product factor
60	C₂²×C₅×Q₈	—	3	1	no	—
61	C₂²×C₅×Q₈	—	6	1	no	—
62	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
63	C₅×C₄⋊Q₈	—	3	4	no	—
64	C₅×C₄⋊Q₈	—	6	2	no	—
65	C₄×C₅×Q₈	—	3	4	no	—
66	C₄×C₅×Q₈	—	6	2	no	—
67	C₂×C₅×C₄⋊C₄	—	3	2	no	—
68	C₂×C₅×C₄⋊C₄	—	3	1	no	—
69	C₂×C₅×C₄⋊C₄	—	6	1	no	—
70	C₂×C₅×C₄⋊C₄	—	6	1	no	—
71	C₂×C₄²×C₅	—	6	1	no	—
72	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	C₅×#107	—	3	8	no	—
74	C₅×C₄⋊C₈	C₂×D₃	1	4	yes	—
75	C₅×#103	—	3	8	no	—
76	C₂×#477	—	3	1	no	Sylow
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
84	C₂×C₃×C₄×C₅	#41	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×C₅×Dic₃	—	2	4	no	—
87	C₂×C₅×Dic₃	—	2	2	no	—
88	C₂×C₅×Dic₃	—	2	2	no	—
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	4	4	no	—
91	C₂×C₅×Dic₃	—	4	2	no	—
92	C₅×Dic₆	—	2	4	no	—
93	C₅×Dic₆	—	2	2	no	—
94	C₅×Dic₆	—	4	8	no	—
95	C₅×Dic₆	C₂×Q₈	1	4	yes	complemented
96	C₃×C₅×C₈	—	2	4	no	—
97	C₂²×Dic₆	—	2	1	no	—
98	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
99	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄⋊Dic₃	—	2	1	no	—
102	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
104	C₂×Dic₃⋊C₄	—	2	1	no	—
105	C₂×C₄×Dic₃	—	2	1	no	—
106	#869	—	2	2	no	—
107	#869	C₂²×C₅	1	4	yes	—
108	C₄×Dic₆	—	2	2	no	—
109	C₄×Dic₆	C₂²×C₅	1	4	yes	—
110	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
111	#818	C₂²×C₅	1	8	yes	—
112	#816	C₂²×C₅	1	8	yes	—
113	C₂×C₅×Q₈	—	3	4	no	—
114	C₂×C₅×Q₈	—	3	2	no	—
115	C₂×C₅×Q₈	—	6	1	no	—
116	C₂×C₅×Q₈	—	6	4	no	—
117	C₂×C₅×Q₈	—	6	2	no	—
118	C₂×C₅×Q₈	—	12	4	no	—
119	C₂²×C₄×C₅	—	6	1	no	—
120	C₂²×C₄×C₅	—	6	1	no	—
121	C₂²×C₄×C₅	—	6	2	no	—
122	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
124	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₅×C₈	—	2	4	no	—
126	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
127	C₅×C₄⋊C₄	—	3	2	no	—
128	C₅×C₄⋊C₄	—	3	4	no	—
129	C₅×C₄⋊C₄	—	6	1	no	—
130	C₅×C₄⋊C₄	—	6	2	no	—
131	C₅×C₄⋊C₄	—	6	4	no	—
132	C₅×C₄⋊C₄	—	6	4	no	—
133	C₄²×C₅	—	6	4	no	—
134	C₄²×C₅	D₁₂	1	2	yes	—
135	C₄²×C₅	C₂²×D₃	1	2	yes	—
136	C₂×C₄⋊Q₈	—	3	1	no	—
137	C₂×C₄×Q₈	—	3	1	no	—
138	#477	—	3	8	no	—
139	C₂×#107	—	3	2	no	—
140	C₂×C₄⋊C₈	C₅×D₃	1	1	yes	characteristic
141	C₂×#103	—	3	2	no	—
142	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
144	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
145	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
148	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
149	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
150	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
151	C₅×Dic₃	—	2	4	no	—
152	C₅×Dic₃	—	4	4	no	—
153	C₅×Dic₃	—	4	2	no	—
154	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
155	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₂²×Dic₃	—	2	1	no	—
158	C₂²×Dic₃	—	2	1	no	—
159	C₂²×Dic₃	—	2	2	no	—
160	C₂×Dic₆	—	2	1	no	—
161	C₂×Dic₆	—	2	2	no	—
162	C₂×Dic₆	—	2	4	no	—
163	C₂×Dic₆	—	4	4	no	—
164	C₂×Dic₆	C₅×Q₈	1	4	yes	complemented
165	C₂×Dic₆	C₂³×C₅	1	2	yes	—
166	C₂×C₃×C₈	—	2	4	no	—
167	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
168	C₃×C₄²	C₅×D₄	1	2	yes	—
169	C₃×C₄²	C₂³×C₅	1	2	yes	—
170	C₄⋊Dic₃	—	2	1	no	—
171	C₄⋊Dic₃	—	2	2	no	—
172	C₄⋊Dic₃	—	2	4	no	—
173	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
174	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
175	Dic₃⋊C₄	—	2	4	no	—
176	C₄×Dic₃	—	2	4	no	—
177	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
178	C₅×Q₈	—	3	4	no	—
179	C₅×Q₈	—	6	4	no	—
180	C₅×Q₈	—	6	2	no	—
181	C₅×Q₈	—	12	8	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	6	4	no	—
184	C₂×C₄×C₅	—	6	2	no	—
185	C₂×C₄×C₅	—	6	2	no	—
186	C₂×C₄×C₅	—	6	4	no	—
187	C₂×C₄×C₅	—	12	2	no	—
188	C₂×C₄×C₅	—	12	4	no	—
189	C₂×C₄×C₅	Dic₁₂	1	4	yes	—
190	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
191	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
193	C₂×C₄×C₅	#237	1	2	yes	—
194	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
195	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₅×C₈	—	2	4	no	—
197	C₂²×Q₈	—	3	1	no	—
198	C₂²×Q₈	—	6	1	no	—
199	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
200	C₄⋊Q₈	—	3	4	no	—
201	C₄⋊Q₈	—	6	2	no	—
202	C₄×Q₈	—	3	4	no	—
203	C₄×Q₈	—	6	2	no	—
204	C₂×C₄⋊C₄	—	3	1	no	—
205	C₂×C₄⋊C₄	—	3	2	no	—
206	C₂×C₄⋊C₄	—	6	1	no	—
207	C₂×C₄⋊C₄	—	6	1	no	—
208	C₂×C₄²	—	6	1	no	—
209	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
210	#107	—	3	8	no	—
211	C₄⋊C₈	C₂×C₅×D₃	1	4	yes	—
212	#103	—	3	8	no	—
213	C₂×C₃×C₅	#477	1	4	yes	complemented
214	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	—
220	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
224	C₂×C₃×C₄	C₅×#41	1	2	yes	—
225	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
226	C₂×Dic₃	—	2	4	no	—
227	C₂×Dic₃	—	2	2	no	—
228	C₂×Dic₃	—	2	2	no	—
229	C₂×Dic₃	—	2	4	no	—
230	C₂×Dic₃	—	4	4	no	—
231	C₂×Dic₃	—	4	2	no	—
232	Dic₆	—	2	4	no	—
233	Dic₆	—	2	2	no	—
234	Dic₆	—	4	8	no	—
235	Dic₆	C₂×C₅×Q₈	1	4	yes	complemented
236	C₃×C₈	—	2	4	no	—
237	C₂²×C₅	#897	1	2	yes	—
238	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
239	C₂²×C₅	#908	1	2	yes	—
240	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	6	4	no	—
243	C₄×C₅	—	12	2	no	—
244	C₄×C₅	—	12	4	no	—
245	C₄×C₅	#897	1	2	yes	—
246	C₄×C₅	C₂×Dic₁₂	1	4	yes	—
247	C₄×C₅	C₂×#237	1	1	yes	characteristic
248	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
249	C₂×Q₈	—	3	4	no	—
250	C₂×Q₈	—	3	2	no	—
251	C₂×Q₈	—	6	1	no	—
252	C₂×Q₈	—	6	4	no	—
253	C₂×Q₈	—	6	2	no	—
254	C₂×Q₈	—	12	4	no	—
255	C₂²×C₄	—	6	1	no	—
256	C₂²×C₄	—	6	1	no	—
257	C₂²×C₄	—	6	2	no	—
258	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
259	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
260	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
263	C₄⋊C₄	—	3	4	no	—
264	C₄⋊C₄	—	3	2	no	—
265	C₄⋊C₄	—	6	1	no	—
266	C₄⋊C₄	—	6	2	no	—
267	C₄⋊C₄	—	6	4	no	—
268	C₄⋊C₄	—	6	4	no	—
269	C₄²	—	6	4	no	—
270	C₄²	C₅×D₁₂	1	2	yes	—
271	C₄²	C₂²×C₅×D₃	1	2	yes	—
272	C₃×C₅	C₂×#477	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
274	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
275	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
276	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
279	C₃×C₄	C₂×C₅×Q₁₆	1	4	yes	—
280	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
281	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
282	Dic₃	—	2	4	no	—
283	Dic₃	—	4	4	no	—
284	Dic₃	—	4	2	no	—
285	C₂×C₅	#4850	1	4	yes	direct product factor
286	C₂×C₅	C₂×#897	1	1	yes	characteristic
287	C₂×C₅	C₂²×Dic₁₂	1	1	yes	characteristic
288	C₂×C₅	C₂×#908	1	1	yes	characteristic
289	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
290	Q₈	—	3	4	no	—
291	Q₈	—	6	4	no	—
292	Q₈	—	6	2	no	—
293	Q₈	—	12	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	6	4	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	6	2	no	—
298	C₂×C₄	—	6	4	no	—
299	C₂×C₄	—	12	4	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	C₅×Dic₁₂	1	4	yes	—
302	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
303	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
305	C₂×C₄	C₅×#237	1	2	yes	—
306	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
307	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₂×C₃	C₅×#477	1	4	yes	complemented
310	C₂×C₃	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
311	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
312	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₅	C₂×#4850	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	C₅×#897	1	2	yes	—
315	C₂²	C₂×C₅×Dic₁₂	1	2	yes	—
316	C₂²	C₅×#908	1	2	yes	—
317	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	6	4	no	—
320	C₄	—	12	4	no	—
321	C₄	—	12	2	no	—
322	C₄	C₅×#897	1	2	yes	—
323	C₄	C₂×C₅×Dic₁₂	1	4	yes	—
324	C₄	C₂×C₅×#237	1	1	yes	characteristic
325	C₄	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
326	C₃	C₂×C₅×#477	1	1	yes	characteristic, complemented, Sylow
327	C₂	C₅×#4850	1	4	yes	direct product factor
328	C₂	C₂×C₅×#897	1	1	yes	characteristic
329	C₂	C₂²×C₅×Dic₁₂	1	1	yes	characteristic
330	C₂	C₂×C₅×#908	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₅×C₄⋊C₈	D₃
hypocenter	C₃	C₂×C₅×#477
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
6. C₅×#4850	C₂	direct (N × H)
24. C₂×#4850	C₅	direct (N × H)
41. C₂×C₅×Dic₆	Q₈	semidirect
59. #4850	C₂×C₅	direct (N × H)
95. C₅×Dic₆	C₂×Q₈	semidirect
164. C₂×Dic₆	C₅×Q₈	semidirect
213. C₂×C₃×C₅	#477	semidirect
235. Dic₆	C₂×C₅×Q₈	semidirect
272. C₃×C₅	C₂×#477	semidirect
285. C₂×C₅	#4850	direct (N × H)
309. C₂×C₃	C₅×#477	semidirect
313. C₅	C₂×#4850	direct (N × H)
326. C₃	C₂×C₅×#477	semidirect
327. C₂	C₅×#4850	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#869	2
C₂×C₄×C₅×Dic₆	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₅×#818	2
C₂×C₅×#816	2
C₅×#4850	2
C₂×C₅×#477	3
C₂×#4850	5

Subgroups — C2×C5×#4850 #14430075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4850 #14430075