Subgroups — C₂×C₅×#5070 #14437638

Order 1920.

1056 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×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#832	C₂	1	4	yes	maximal, complemented
4	C₅×#5070	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#486	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	—	2	1	no	—
8	C₂²×C₃×C₅×D₄	—	2	1	no	—
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₆	—	2	1	no	—
11	C₃×C₅×#127	—	2	2	no	—
12	C₃×C₅×#127	C₂²	1	4	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊Dic₃	—	2	1	no	—
15	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₅×#869	—	2	2	no	—
18	C₅×#869	C₂²	1	4	yes	complemented
19	C₅×#832	C₂²	1	16	yes	complemented
20	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
21	C₂×#5070	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
23	C₂×C₅×#127	D₃	1	1	yes	characteristic
24	C₅×#486	—	3	8	no	—
25	C₂×C₅×#102	—	3	4	no	—
26	C₂×C₄×C₅×C₈	—	3	1	no	—
27	C₂⁴×C₃×C₅	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	4	no	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	8	no	—
32	C₂×C₃×C₅×D₄	—	4	4	no	—
33	C₂×C₃×C₅×D₄	—	4	4	no	—
34	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂²×C₅×Dic₃	—	2	2	no	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	4	4	no	—
41	C₃×C₄²×C₅	D₄	1	2	yes	—
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₅×C₄⋊Dic₃	—	2	2	no	—
44	C₅×C₄⋊Dic₃	—	2	4	no	—
45	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₂×C₅×C₃⋊C₈	—	2	4	no	—
47	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
48	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
49	C₂×#869	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#832	C₂×C₅	1	4	yes	complemented
51	#5070	C₂×C₅	1	8	yes	direct product factor
52	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂²×C₅×Q₈	—	6	1	no	—
54	C₂²×C₅×D₄	—	2	1	no	—
55	C₂²×C₅×D₄	—	2	1	no	—
56	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
57	C₂²×C₅×C₈	—	3	2	no	—
58	C₅×C₄⋊Q₈	—	3	4	no	—
59	C₅×C₄⋊Q₈	—	6	2	no	—
60	C₅×#127	—	2	2	no	—
61	C₅×#127	C₂×D₃	1	4	yes	—
62	C₂×C₅×C₄⋊C₄	—	3	2	no	—
63	C₂×C₅×C₄⋊C₄	—	6	1	no	—
64	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
65	C₅×#102	—	3	16	no	—
66	C₄×C₅×C₈	—	3	4	no	—
67	C₂×#486	—	3	1	no	Sylow
68	C₂³×C₃×C₅	—	2	4	no	—
69	C₂³×C₃×C₅	—	4	4	no	—
70	C₂³×C₃×C₅	—	4	4	no	—
71	C₂³×C₃×C₅	—	4	4	no	—
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×D₄	—	2	8	no	—
74	C₃×C₅×D₄	—	4	4	no	—
75	C₃×C₅×D₄	—	4	8	no	—
76	C₃×C₅×D₄	—	4	8	no	—
77	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
78	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	#41	1	4	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₂×C₅×Dic₃	—	2	4	no	—
85	C₂×C₅×Dic₃	—	4	4	no	—
86	C₅×Dic₆	—	4	8	no	—
87	C₅×C₃⋊C₈	—	2	4	no	—
88	C₂²×C₃×D₄	—	2	1	no	—
89	C₂²×C₃×D₄	—	2	1	no	—
90	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
91	C₂²×Dic₆	—	2	1	no	—
92	C₃×#127	—	2	2	no	—
93	C₃×#127	C₂²×C₅	1	4	yes	—
94	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄⋊Dic₃	—	2	1	no	—
96	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
97	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
98	#869	—	2	2	no	—
99	#869	C₂²×C₅	1	4	yes	complemented
100	#832	C₂²×C₅	1	16	yes	complemented
101	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
102	C₂⁴×C₅	—	2	2	no	—
103	C₂×C₅×Q₈	—	6	4	no	—
104	C₂×C₅×Q₈	—	12	4	no	—
105	C₂×C₅×D₄	—	2	8	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	4	no	—
108	C₂×C₅×D₄	—	2	4	no	—
109	C₂×C₅×D₄	—	4	4	no	—
110	C₂×C₅×D₄	—	4	4	no	—
111	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
112	C₂²×C₄×C₅	—	6	2	no	—
113	C₂²×C₄×C₅	#67	1	1	yes	characteristic
114	C₂²×C₄×C₅	#67	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₅×C₈	—	3	4	no	—
117	C₂×C₅×C₈	—	6	4	no	—
118	C₅×C₄⋊C₄	—	3	4	no	—
119	C₅×C₄⋊C₄	—	6	2	no	—
120	C₅×C₄⋊C₄	—	6	4	no	—
121	C₄²×C₅	#67	1	2	yes	—
122	C₄²×C₅	C₂²×D₃	1	2	yes	—
123	C₂×C₄⋊Q₈	—	3	1	no	—
124	C₂×#127	C₅×D₃	1	1	yes	characteristic
125	#486	—	3	8	no	—
126	C₂×#102	—	3	4	no	—
127	C₂×C₄×C₈	—	3	1	no	—
128	C₂²×C₃×C₅	—	4	4	no	—
129	C₂²×C₃×C₅	—	4	4	no	—
130	C₂²×C₃×C₅	—	4	4	no	—
131	C₂²×C₃×C₅	—	8	8	no	—
132	C₂²×C₃×C₅	#124	1	2	yes	—
133	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
134	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
135	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄×C₅	#124	1	2	yes	—
137	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
138	C₃×C₄×C₅	C₂×SD₁₆	1	4	yes	—
139	C₃×C₄×C₅	C₂×#41	1	2	yes	—
140	C₅×Dic₃	—	4	4	no	—
141	C₂⁴×C₃	—	2	2	no	—
142	C₂×C₃×D₄	—	2	2	no	—
143	C₂×C₃×D₄	—	2	4	no	—
144	C₂×C₃×D₄	—	2	4	no	—
145	C₂×C₃×D₄	—	2	8	no	—
146	C₂×C₃×D₄	—	4	4	no	—
147	C₂×C₃×D₄	—	4	4	no	—
148	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
149	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
151	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂²×Dic₃	—	2	2	no	—
153	C₂×Dic₆	—	2	4	no	—
154	C₂×Dic₆	—	4	4	no	—
155	C₃×C₄²	C₅×D₄	1	2	yes	—
156	C₃×C₄²	C₂³×C₅	1	2	yes	—
157	C₄⋊Dic₃	—	2	2	no	—
158	C₄⋊Dic₃	—	2	4	no	—
159	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
160	C₂×C₃⋊C₈	—	2	4	no	—
161	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
162	C₂³×C₅	—	2	4	no	—
163	C₂³×C₅	—	4	4	no	—
164	C₂³×C₅	—	4	4	no	—
165	C₂³×C₅	—	4	4	no	—
166	C₂³×C₅	C₂×#67	1	1	yes	characteristic
167	C₅×Q₈	—	12	8	no	—
168	C₅×D₄	—	2	8	no	—
169	C₅×D₄	—	4	8	no	—
170	C₅×D₄	—	4	8	no	—
171	C₅×D₄	—	4	4	no	—
172	C₂×C₄×C₅	—	6	4	no	—
173	C₂×C₄×C₅	—	12	4	no	—
174	C₂×C₄×C₅	#213	1	4	yes	—
175	C₂×C₄×C₅	#214	1	4	yes	—
176	C₂×C₄×C₅	#237	1	4	yes	—
177	C₂×C₄×C₅	C₂×#67	1	2	yes	—
178	C₂×C₄×C₅	C₂×#67	1	2	yes	—
179	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
181	C₅×C₈	—	6	4	no	—
182	C₂²×Q₈	—	6	1	no	—
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
186	C₂²×C₈	—	3	2	no	—
187	C₄⋊Q₈	—	3	4	no	—
188	C₄⋊Q₈	—	6	2	no	—
189	#127	—	2	2	no	—
190	#127	C₂×C₅×D₃	1	4	yes	—
191	C₂×C₄⋊C₄	—	3	2	no	—
192	C₂×C₄⋊C₄	—	6	1	no	—
193	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
194	#102	—	3	16	no	—
195	C₄×C₈	—	3	4	no	—
196	C₂×C₃×C₅	—	8	4	no	—
197	C₂×C₃×C₅	#486	1	4	yes	complemented
198	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
201	C₂³×C₃	—	2	4	no	—
202	C₂³×C₃	—	4	4	no	—
203	C₂³×C₃	—	4	4	no	—
204	C₂³×C₃	—	4	4	no	—
205	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
206	C₃×D₄	—	2	8	no	—
207	C₃×D₄	—	4	4	no	—
208	C₃×D₄	—	4	8	no	—
209	C₃×D₄	—	4	8	no	—
210	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
211	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
212	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
215	C₂×C₃×C₄	C₅×#41	1	4	yes	—
216	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
217	C₂×Dic₃	—	2	4	no	—
218	C₂×Dic₃	—	4	4	no	—
219	Dic₆	—	4	8	no	—
220	C₃⋊C₈	—	2	4	no	—
221	C₂²×C₅	—	4	4	no	—
222	C₂²×C₅	—	4	4	no	—
223	C₂²×C₅	—	4	4	no	—
224	C₂²×C₅	—	8	8	no	—
225	C₂²×C₅	C₂×#213	1	2	yes	—
226	C₂²×C₅	C₂×#214	1	2	yes	—
227	C₂²×C₅	#936	1	2	yes	—
228	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
229	C₄×C₅	—	12	4	no	—
230	C₄×C₅	C₂×#213	1	4	yes	—
231	C₄×C₅	C₂×#214	1	4	yes	—
232	C₄×C₅	#936	1	2	yes	—
233	C₄×C₅	C₂×#237	1	2	yes	—
234	C₂⁴	—	2	2	no	—
235	C₂×Q₈	—	6	4	no	—
236	C₂×Q₈	—	12	4	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
244	C₂²×C₄	—	6	2	no	—
245	C₂²×C₄	C₅×#67	1	1	yes	characteristic
246	C₂²×C₄	C₅×#67	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
248	C₂×C₈	—	3	4	no	—
249	C₂×C₈	—	6	4	no	—
250	C₄⋊C₄	—	3	4	no	—
251	C₄⋊C₄	—	6	2	no	—
252	C₄⋊C₄	—	6	4	no	—
253	C₄²	C₅×#67	1	2	yes	—
254	C₄²	C₂²×C₅×D₃	1	2	yes	—
255	C₃×C₅	C₂×#486	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	4	4	no	—
257	C₂²×C₃	—	4	4	no	—
258	C₂²×C₃	—	4	4	no	—
259	C₂²×C₃	—	8	8	no	—
260	C₂²×C₃	C₅×#124	1	2	yes	—
261	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
262	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
263	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
264	C₃×C₄	C₅×#124	1	2	yes	—
265	C₃×C₄	C₂×C₅×D₈	1	4	yes	—
266	C₃×C₄	C₂×C₅×SD₁₆	1	4	yes	—
267	C₃×C₄	C₂×C₅×#41	1	2	yes	—
268	Dic₃	—	4	4	no	—
269	C₂×C₅	—	8	4	no	—
270	C₂×C₅	#5070	1	4	yes	direct product factor
271	C₂×C₅	C₂²×#213	1	1	yes	characteristic
272	C₂×C₅	C₂²×#214	1	1	yes	characteristic
273	C₂×C₅	C₂×#936	1	1	yes	characteristic
274	C₂³	—	2	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	4	no	—
278	C₂³	C₂×C₅×#67	1	1	yes	characteristic
279	Q₈	—	12	8	no	—
280	D₄	—	2	8	no	—
281	D₄	—	4	8	no	—
282	D₄	—	4	8	no	—
283	D₄	—	4	4	no	—
284	C₂×C₄	—	6	4	no	—
285	C₂×C₄	—	12	4	no	—
286	C₂×C₄	C₅×#213	1	4	yes	—
287	C₂×C₄	C₅×#214	1	4	yes	—
288	C₂×C₄	C₅×#237	1	4	yes	—
289	C₂×C₄	C₂×C₅×#67	1	2	yes	—
290	C₂×C₄	C₂×C₅×#67	1	2	yes	—
291	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
292	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
293	C₈	—	6	4	no	—
294	C₂×C₃	—	8	4	no	—
295	C₂×C₃	C₅×#486	1	4	yes	complemented
296	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
297	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
298	C₂×C₃	C₂²×C₅×SD₁₆	1	1	yes	characteristic
299	C₅	C₂×#5070	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	4	4	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	8	8	no	—
304	C₂²	C₂×C₅×#213	1	2	yes	—
305	C₂²	C₂×C₅×#214	1	2	yes	—
306	C₂²	C₅×#936	1	2	yes	—
307	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
308	C₄	—	12	4	no	—
309	C₄	C₂×C₅×#213	1	4	yes	—
310	C₄	C₂×C₅×#214	1	4	yes	—
311	C₄	C₅×#936	1	2	yes	—
312	C₄	C₂×C₅×#237	1	2	yes	—
313	C₃	C₂×C₅×#486	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	8	4	no	—
315	C₂	C₅×#5070	1	4	yes	direct product factor
316	C₂	C₂²×C₅×#213	1	1	yes	characteristic
317	C₂	C₂²×C₅×#214	1	1	yes	characteristic
318	C₂	C₂×C₅×#936	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₅×#127	D₃
hypocenter	C₃	C₂×C₅×#486
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#869	C₂	semidirect
3. C₂×C₅×#832	C₂	semidirect
4. C₅×#5070	C₂	direct (N × H)
5. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
18. C₅×#869	C₂²	semidirect
19. C₅×#832	C₂²	semidirect
20. C₄×C₅×C₃⋊C₈	C₂²	semidirect
21. C₂×#5070	C₅	direct (N × H)
49. C₂×#869	C₂×C₅	semidirect
50. C₂×#832	C₂×C₅	semidirect
51. #5070	C₂×C₅	direct (N × H)
52. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
99. #869	C₂²×C₅	semidirect
100. #832	C₂²×C₅	semidirect
101. C₄×C₃⋊C₈	C₂²×C₅	semidirect
197. C₂×C₃×C₅	#486	semidirect
255. C₃×C₅	C₂×#486	semidirect
270. C₂×C₅	#5070	direct (N × H)
295. C₂×C₃	C₅×#486	semidirect
299. C₅	C₂×#5070	direct (N × H)
313. C₃	C₂×C₅×#486	semidirect
315. C₂	C₅×#5070	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×C₃×C₅×#127	2
C₂×C₅×#869	2
C₂×C₅×#832	2
C₅×#5070	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₅×#486	3
C₂×#5070	5

Subgroups — C2×C5×#5070 #14437638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5070 #14437638