Subgroups — C₂×C₅×#5068 #14437731

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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#875	C₂	1	1	yes	maximal, characteristic
3	C₅×#5068	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#810	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#809	C₂	1	2	yes	maximal
6	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#488	—	3	1	no	maximal, Hall
8	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
10	C₃×C₅×#125	—	2	2	no	—
11	C₃×C₅×#125	C₂²	1	4	yes	complemented
12	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
16	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₅×#875	—	2	2	no	—
18	C₅×#875	C₂²	1	4	yes	—
19	C₅×#810	C₂²	1	8	yes	complemented
20	C₅×#809	C₂²	1	8	yes	—
21	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
22	C₂×#5068	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×#125	D₃	1	1	yes	characteristic, complemented
24	C₂×C₅×#124	—	3	1	no	—
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	C₂³×C₅×D₃	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	—	2	2	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×Dic₃	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	4	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	C₂³	1	2	yes	—
37	C₂×C₅×Dic₆	—	2	4	no	—
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₂×C₅×Dic₆	C₂³	1	2	yes	—
40	C₃×C₅×C₄⋊C₄	—	2	8	no	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
46	C₂×C₅×C₃⋊C₈	—	2	4	no	—
47	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
48	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂×#875	C₂×C₅	1	1	yes	characteristic
50	#5068	C₂×C₅	1	8	yes	direct product factor
51	C₂×#810	C₂×C₅	1	2	yes	complemented
52	C₂×#809	C₂×C₅	1	2	yes	—
53	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂²×C₅×Q₈	—	3	1	no	—
55	C₂²×C₅×D₄	—	3	1	no	—
56	C₂²×C₅×C₈	—	3	2	no	—
57	C₅×#125	—	2	2	no	—
58	C₅×#125	C₂×D₃	1	4	yes	complemented
59	C₅×#124	—	3	4	no	—
60	C₅×#124	—	6	2	no	—
61	C₂×C₅×C₄⋊C₄	—	2	2	no	—
62	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
63	C₂×C₅×C₂²⋊C₄	—	6	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	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₅×D₃	—	2	2	no	—
71	C₂²×C₅×D₃	—	4	2	no	—
72	C₂²×C₅×D₃	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	4	4	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×Dic₃	—	2	2	no	—
81	C₂×C₅×Dic₃	—	4	2	no	—
82	C₅×D₁₂	—	2	4	no	—
83	C₅×D₁₂	—	4	2	no	—
84	C₅×Dic₆	—	2	4	no	—
85	C₅×Dic₆	—	4	2	no	—
86	C₅×C₃⋊C₈	—	2	4	no	—
87	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
88	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
89	C₃×#125	—	2	2	no	—
90	C₃×#125	C₂²×C₅	1	4	yes	complemented
91	C₂×C₃×C₄⋊C₄	—	2	2	no	—
92	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
94	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
95	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
96	#875	—	2	2	no	—
97	#875	C₂²×C₅	1	4	yes	—
98	#810	C₂²×C₅	1	8	yes	complemented
99	#809	C₂²×C₅	1	8	yes	—
100	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
101	C₂⁴×C₅	—	6	1	no	—
102	C₂×C₅×Q₈	—	3	2	no	—
103	C₂×C₅×Q₈	—	6	1	no	—
104	C₂×C₅×Q₈	—	6	4	no	—
105	C₂×C₅×D₄	—	3	2	no	—
106	C₂×C₅×D₄	—	6	4	no	—
107	C₂×C₅×D₄	—	6	1	no	—
108	C₂²×C₄×C₅	—	2	2	no	—
109	C₂²×C₄×C₅	—	6	1	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	—	2	8	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
117	C₅×C₂²⋊C₄	—	6	8	no	—
118	C₄²×C₅	#67	1	2	yes	—
119	C₄²×C₅	C₂²×D₃	1	2	yes	—
120	C₂×#125	C₅×D₃	1	1	yes	characteristic, complemented
121	C₂×#124	—	3	1	no	—
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	#124	1	2	yes	—
127	C₂²×C₃×C₅	#135	1	4	yes	—
128	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₅×D₃	—	4	2	no	—
130	C₂×C₅×D₃	—	4	4	no	—
131	C₂×C₅×D₃	—	8	4	no	—
132	C₃×C₄×C₅	—	2	4	no	—
133	C₃×C₄×C₅	—	4	4	no	—
134	C₃×C₄×C₅	#124	1	2	yes	—
135	C₃×C₄×C₅	C₂×#41	1	2	yes	—
136	C₅×Dic₃	—	4	2	no	—
137	C₂³×D₃	—	2	1	no	—
138	C₂²×C₃×C₄	—	2	2	no	—
139	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
140	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₂²×Dic₃	—	2	1	no	—
142	C₂×D₁₂	—	2	4	no	—
143	C₂×D₁₂	—	2	1	no	—
144	C₂×D₁₂	C₂³×C₅	1	2	yes	—
145	C₂×Dic₆	—	2	4	no	—
146	C₂×Dic₆	—	2	1	no	—
147	C₂×Dic₆	C₂³×C₅	1	2	yes	—
148	C₃×C₄⋊C₄	—	2	8	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
151	C₃×C₄²	C₅×D₄	1	2	yes	—
152	C₃×C₄²	C₂³×C₅	1	2	yes	—
153	(C₂×D₃)⋊C₄	—	2	8	no	—
154	C₂×C₃⋊C₈	—	2	4	no	—
155	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
156	C₂³×C₅	—	6	2	no	—
157	C₂³×C₅	—	12	2	no	—
158	C₂³×C₅	—	12	4	no	—
159	C₂³×C₅	C₂×#67	1	1	yes	characteristic
160	C₅×Q₈	—	6	4	no	—
161	C₅×Q₈	—	12	2	no	—
162	C₅×D₄	—	6	4	no	—
163	C₅×D₄	—	12	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	4	4	no	—
167	C₂×C₄×C₅	—	6	2	no	—
168	C₂×C₄×C₅	—	12	2	no	—
169	C₂×C₄×C₅	#239	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	—	3	1	no	—
175	C₂²×D₄	—	3	1	no	—
176	C₂²×C₈	—	3	2	no	—
177	#125	—	2	2	no	—
178	#125	C₂×C₅×D₃	1	4	yes	complemented
179	#124	—	3	4	no	—
180	#124	—	6	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
183	C₂×C₂²⋊C₄	—	6	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	#488	1	4	yes	complemented
189	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×#135	1	2	yes	—
191	C₅×D₃	—	8	2	no	—
192	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
193	C₂²×D₃	—	2	2	no	—
194	C₂²×D₃	—	4	2	no	—
195	C₂²×D₃	—	4	4	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	4	4	no	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₅×#41	1	4	yes	—
202	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
203	C₂×Dic₃	—	2	2	no	—
204	C₂×Dic₃	—	4	2	no	—
205	D₁₂	—	2	4	no	—
206	D₁₂	—	4	2	no	—
207	Dic₆	—	2	4	no	—
208	Dic₆	—	4	2	no	—
209	C₃⋊C₈	—	2	4	no	—
210	C₂²×C₅	—	12	2	no	—
211	C₂²×C₅	—	12	4	no	—
212	C₂²×C₅	—	24	4	no	—
213	C₂²×C₅	#947	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	4	no	—
218	C₄×C₅	—	12	2	no	—
219	C₄×C₅	#947	1	2	yes	—
220	C₄×C₅	C₂×#239	1	2	yes	—
221	C₂⁴	—	6	1	no	—
222	C₂×Q₈	—	3	2	no	—
223	C₂×Q₈	—	6	1	no	—
224	C₂×Q₈	—	6	4	no	—
225	C₂×D₄	—	3	2	no	—
226	C₂×D₄	—	6	4	no	—
227	C₂×D₄	—	6	1	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	6	1	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	—	2	8	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
237	C₂²⋊C₄	—	6	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	C₅×#124	1	2	yes	—
242	C₂²×C₃	C₅×#135	1	4	yes	—
243	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₂×D₃	—	4	2	no	—
245	C₂×D₃	—	4	4	no	—
246	C₂×D₃	—	8	4	no	—
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	4	4	no	—
249	C₃×C₄	C₅×#124	1	2	yes	—
250	C₃×C₄	C₂×C₅×#41	1	2	yes	—
251	Dic₃	—	4	2	no	—
252	C₂×C₅	—	24	2	no	—
253	C₂×C₅	#5068	1	4	yes	direct product factor
254	C₂×C₅	C₂×#947	1	1	yes	characteristic
255	C₂×C₅	C₂×#950	1	2	yes	—
256	C₂³	—	6	2	no	—
257	C₂³	—	12	2	no	—
258	C₂³	—	12	4	no	—
259	C₂³	C₂×C₅×#67	1	1	yes	characteristic
260	Q₈	—	6	4	no	—
261	Q₈	—	12	2	no	—
262	D₄	—	6	4	no	—
263	D₄	—	12	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	6	2	no	—
268	C₂×C₄	—	12	2	no	—
269	C₂×C₄	C₅×#239	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	C₅×#488	1	4	yes	complemented
275	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
276	C₂×C₃	C₂×C₅×#135	1	2	yes	—
277	D₃	—	8	2	no	—
278	C₅	C₂×#5068	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	12	2	no	—
280	C₂²	—	12	4	no	—
281	C₂²	—	24	4	no	—
282	C₂²	C₅×#947	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	4	no	—
287	C₄	—	12	2	no	—
288	C₄	C₅×#947	1	2	yes	—
289	C₄	C₂×C₅×#239	1	2	yes	—
290	C₃	C₂×C₅×#488	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	24	2	no	—
292	C₂	C₅×#5068	1	4	yes	direct product factor
293	C₂	C₂×C₅×#947	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×#125	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₅×D₁₂	C₂²
hypercenter	C₂×C₅×#125	D₃
hypocenter	C₃	C₂×C₅×#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#125	C₂	semidirect
3. C₅×#5068	C₂	direct (N × H)
4. C₂×C₅×#810	C₂	semidirect
6. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
11. C₃×C₅×#125	C₂²	semidirect
19. C₅×#810	C₂²	semidirect
21. C₄×C₅×C₃⋊C₈	C₂²	semidirect
22. C₂×#5068	C₅	direct (N × H)
23. C₂×C₅×#125	D₃	semidirect
48. C₂×C₃×#125	C₂×C₅	semidirect
50. #5068	C₂×C₅	direct (N × H)
51. C₂×#810	C₂×C₅	semidirect
53. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
58. C₅×#125	C₂×D₃	semidirect
90. C₃×#125	C₂²×C₅	semidirect
98. #810	C₂²×C₅	semidirect
100. C₄×C₃⋊C₈	C₂²×C₅	semidirect
120. C₂×#125	C₅×D₃	semidirect
178. #125	C₂×C₅×D₃	semidirect
188. C₂×C₃×C₅	#488	semidirect
240. C₃×C₅	C₂×#488	semidirect
253. C₂×C₅	#5068	direct (N × H)
274. C₂×C₃	C₅×#488	semidirect
278. C₅	C₂×#5068	direct (N × H)
290. C₃	C₂×C₅×#488	semidirect
292. C₂	C₅×#5068	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×#125	2
C₂×C₅×#875	2
C₅×#5068	2
C₂×C₅×#810	2
C₂×C₅×#809	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₅×#488	3
C₂×#5068	5

Subgroups — C2×C5×#5068 #14437731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5068 #14437731