Subgroups — C₂×C₅×#5056 #14437722

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#870	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#835	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#832	C₂	1	2	yes	maximal
5	C₅×#5056	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#488	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	—	2	2	no	—
11	C₃×C₅×#124	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
16	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₅×#870	—	2	2	no	—
18	C₅×#870	C₂²	1	4	yes	complemented
19	C₅×#835	C₂²	1	8	yes	complemented
20	C₅×#832	C₂²	1	8	yes	—
21	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
22	C₂×#5056	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×#125	—	3	1	no	—
24	C₂×C₅×#124	D₃	1	1	yes	characteristic
25	C₅×#488	—	3	8	no	—
26	C₂×C₅×#103	—	3	2	no	—
27	C₂×C₅×#102	—	3	2	no	—
28	C₂×C₄×C₅×C₈	—	3	1	no	—
29	C₂⁴×C₃×C₅	—	2	1	no	—
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	4	no	—
32	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	4	no	—
35	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	—	2	1	no	—
37	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×Dic₃	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	8	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	C₂³	1	4	yes	—
45	C₅×Dic₃⋊C₄	—	2	8	no	—
46	C₂×C₅×C₃⋊C₈	—	2	4	no	—
47	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
48	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic
49	C₂×#870	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#835	C₂×C₅	1	2	yes	complemented
51	C₂×#832	C₂×C₅	1	2	yes	—
52	#5056	C₂×C₅	1	8	yes	direct product factor
53	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₅×C₈	—	3	2	no	—
57	C₅×#125	—	3	4	no	—
58	C₅×#125	—	6	2	no	—
59	C₅×#124	—	2	2	no	—
60	C₅×#124	C₂×D₃	1	4	yes	—
61	C₂×C₅×C₄⋊C₄	—	3	2	no	—
62	C₂×C₅×C₄⋊C₄	—	6	2	no	—
63	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
64	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
65	C₅×#103	—	3	8	no	—
66	C₅×#102	—	3	8	no	—
67	C₄×C₅×C₈	—	3	4	no	—
68	C₂×#488	—	3	1	no	Sylow
69	C₂³×C₃×C₅	—	2	2	no	—
70	C₂³×C₃×C₅	—	4	2	no	—
71	C₂³×C₃×C₅	—	4	4	no	—
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	4	2	no	—
75	C₃×C₅×D₄	—	2	4	no	—
76	C₃×C₅×D₄	—	4	2	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	4	2	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
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×C₃⋊C₈	—	2	4	no	—
87	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
89	C₃×#124	—	2	2	no	—
90	C₃×#124	C₂²×C₅	1	4	yes	—
91	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
94	C₂×Dic₃⋊C₄	—	2	2	no	—
95	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
96	#870	—	2	2	no	—
97	#870	C₂²×C₅	1	4	yes	complemented
98	#835	C₂²×C₅	1	8	yes	complemented
99	#832	C₂²×C₅	1	8	yes	—
100	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
101	C₂⁴×C₅	—	2	1	no	—
102	C₂×C₅×Q₈	—	2	1	no	—
103	C₂×C₅×Q₈	—	2	4	no	—
104	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
105	C₂×C₅×D₄	—	2	4	no	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
108	C₂²×C₄×C₅	—	2	1	no	—
109	C₂²×C₄×C₅	—	6	2	no	—
110	C₂²×C₄×C₅	#67	1	2	yes	—
111	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₅×C₈	—	3	4	no	—
113	C₂×C₅×C₈	—	6	4	no	—
114	C₅×C₄⋊C₄	—	3	4	no	—
115	C₅×C₄⋊C₄	—	6	2	no	—
116	C₅×C₄⋊C₄	—	6	8	no	—
117	C₅×C₂²⋊C₄	—	2	8	no	—
118	C₄²×C₅	#67	1	2	yes	—
119	C₄²×C₅	C₂²×D₃	1	2	yes	—
120	C₂×#125	—	3	1	no	—
121	C₂×#124	C₅×D₃	1	1	yes	characteristic
122	#488	—	3	8	no	—
123	C₂×#103	—	3	2	no	—
124	C₂×#102	—	3	2	no	—
125	C₂×C₄×C₈	—	3	1	no	—
126	C₂²×C₃×C₅	—	4	2	no	—
127	C₂²×C₃×C₅	—	4	4	no	—
128	C₂²×C₃×C₅	—	8	4	no	—
129	C₂²×C₃×C₅	#124	1	2	yes	—
130	C₂²×C₃×C₅	#135	1	4	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	—	4	2	no	—
134	C₃×C₄×C₅	#124	1	2	yes	—
135	C₃×C₄×C₅	C₂×#41	1	2	yes	—
136	C₅×Dic₃	—	4	4	no	—
137	C₂⁴×C₃	—	2	1	no	—
138	C₂×C₃×Q₈	—	2	1	no	—
139	C₂×C₃×Q₈	—	2	4	no	—
140	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
141	C₂×C₃×D₄	—	2	1	no	—
142	C₂×C₃×D₄	—	2	4	no	—
143	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
144	C₂²×C₃×C₄	—	2	1	no	—
145	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
146	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	2	2	no	—
148	C₃×C₂²⋊C₄	—	2	8	no	—
149	C₃×C₄²	C₅×D₄	1	2	yes	—
150	C₃×C₄²	C₂³×C₅	1	2	yes	—
151	C₄⋊Dic₃	—	2	2	no	—
152	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
153	Dic₃⋊C₄	—	2	8	no	—
154	C₂×C₃⋊C₈	—	2	4	no	—
155	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
156	C₂³×C₅	—	2	2	no	—
157	C₂³×C₅	—	4	2	no	—
158	C₂³×C₅	—	4	4	no	—
159	C₂³×C₅	C₂×#67	1	1	yes	characteristic
160	C₅×Q₈	—	2	4	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×D₄	—	2	4	no	—
163	C₅×D₄	—	4	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	6	4	no	—
168	C₂×C₄×C₅	—	12	4	no	—
169	C₂×C₄×C₅	#237	1	4	yes	—
170	C₂×C₄×C₅	C₂×#67	1	4	yes	—
171	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
173	C₅×C₈	—	6	4	no	—
174	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
175	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
176	C₂²×C₈	—	3	2	no	—
177	#125	—	3	4	no	—
178	#125	—	6	2	no	—
179	#124	—	2	2	no	—
180	#124	C₂×C₅×D₃	1	4	yes	—
181	C₂×C₄⋊C₄	—	3	2	no	—
182	C₂×C₄⋊C₄	—	6	2	no	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
185	#103	—	3	8	no	—
186	#102	—	3	8	no	—
187	C₄×C₈	—	3	4	no	—
188	C₂×C₃×C₅	—	8	2	no	—
189	C₂×C₃×C₅	#488	1	4	yes	complemented
190	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#135	1	2	yes	—
192	C₂³×C₃	—	2	2	no	—
193	C₂³×C₃	—	4	2	no	—
194	C₂³×C₃	—	4	4	no	—
195	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
196	C₃×Q₈	—	2	4	no	—
197	C₃×Q₈	—	4	2	no	—
198	C₃×D₄	—	2	4	no	—
199	C₃×D₄	—	4	2	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₅×#41	1	4	yes	—
206	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
207	C₂×Dic₃	—	2	4	no	—
208	C₂×Dic₃	—	4	4	no	—
209	C₃⋊C₈	—	2	4	no	—
210	C₂²×C₅	—	4	2	no	—
211	C₂²×C₅	—	4	4	no	—
212	C₂²×C₅	—	8	4	no	—
213	C₂²×C₅	#936	1	2	yes	—
214	C₂²×C₅	#950	1	4	yes	—
215	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	4	2	no	—
218	C₄×C₅	—	12	4	no	—
219	C₄×C₅	#936	1	2	yes	—
220	C₄×C₅	C₂×#237	1	2	yes	—
221	C₂⁴	—	2	1	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	6	2	no	—
230	C₂²×C₄	C₅×#67	1	2	yes	—
231	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
232	C₂×C₈	—	3	4	no	—
233	C₂×C₈	—	6	4	no	—
234	C₄⋊C₄	—	3	4	no	—
235	C₄⋊C₄	—	6	2	no	—
236	C₄⋊C₄	—	6	8	no	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₄²	C₅×#67	1	2	yes	—
239	C₄²	C₂²×C₅×D₃	1	2	yes	—
240	C₃×C₅	C₂×#488	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	4	2	no	—
242	C₂²×C₃	—	4	4	no	—
243	C₂²×C₃	—	8	4	no	—
244	C₂²×C₃	C₅×#124	1	2	yes	—
245	C₂²×C₃	C₅×#135	1	4	yes	—
246	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	4	2	no	—
249	C₃×C₄	C₅×#124	1	2	yes	—
250	C₃×C₄	C₂×C₅×#41	1	2	yes	—
251	Dic₃	—	4	4	no	—
252	C₂×C₅	—	8	2	no	—
253	C₂×C₅	#5056	1	4	yes	direct product factor
254	C₂×C₅	C₂×#936	1	1	yes	characteristic
255	C₂×C₅	C₂×#950	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	4	no	—
259	C₂³	C₂×C₅×#67	1	1	yes	characteristic
260	Q₈	—	2	4	no	—
261	Q₈	—	4	2	no	—
262	D₄	—	2	4	no	—
263	D₄	—	4	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	12	4	no	—
269	C₂×C₄	C₅×#237	1	4	yes	—
270	C₂×C₄	C₂×C₅×#67	1	4	yes	—
271	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
272	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
273	C₈	—	6	4	no	—
274	C₂×C₃	—	8	2	no	—
275	C₂×C₃	C₅×#488	1	4	yes	complemented
276	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
277	C₂×C₃	C₂×C₅×#135	1	2	yes	—
278	C₅	C₂×#5056	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	4	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₅×#936	1	2	yes	—
283	C₂²	C₅×#950	1	4	yes	—
284	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	4	2	no	—
287	C₄	—	12	4	no	—
288	C₄	C₅×#936	1	2	yes	—
289	C₄	C₂×C₅×#237	1	2	yes	—
290	C₃	C₂×C₅×#488	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	8	2	no	—
292	C₂	C₅×#5056	1	4	yes	direct product factor
293	C₂	C₂×C₅×#936	1	1	yes	characteristic
294	C₂	C₂×C₅×#950	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×C₅×D₄	C₂²
hypercenter	C₂×C₅×#124	D₃
hypocenter	C₃	C₂×C₅×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#124

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#870	C₂	semidirect
3. C₂×C₅×#835	C₂	semidirect
5. C₅×#5056	C₂	direct (N × H)
6. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
18. C₅×#870	C₂²	semidirect
19. C₅×#835	C₂²	semidirect
21. C₄×C₅×C₃⋊C₈	C₂²	semidirect
22. C₂×#5056	C₅	direct (N × H)
49. C₂×#870	C₂×C₅	semidirect
50. C₂×#835	C₂×C₅	semidirect
52. #5056	C₂×C₅	direct (N × H)
53. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
97. #870	C₂²×C₅	semidirect
98. #835	C₂²×C₅	semidirect
100. C₄×C₃⋊C₈	C₂²×C₅	semidirect
189. C₂×C₃×C₅	#488	semidirect
240. C₃×C₅	C₂×#488	semidirect
253. C₂×C₅	#5056	direct (N × H)
275. C₂×C₃	C₅×#488	semidirect
278. C₅	C₂×#5056	direct (N × H)
290. C₃	C₂×C₅×#488	semidirect
292. C₂	C₅×#5056	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×#124	2
C₂×C₅×#870	2
C₂×C₅×#835	2
C₂×C₅×#832	2
C₅×#5056	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₅×#488	3
C₂×#5056	5

Subgroups — C2×C5×#5056 #14437722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5056 #14437722