Subgroups — C₂×C₅×#5073 #14438769

Order 1920.

1520 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	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×#213	C₂	1	4	yes	maximal, complemented
3	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5073	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#493	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₅×D₁₂	—	2	2	no	—
10	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
11	C₃×C₅×#127	—	2	2	no	—
12	C₃×C₅×#127	C₂²	1	4	yes	complemented
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₅×#213	—	2	16	no	—
15	C₂×C₅×#213	C₂²	1	16	yes	complemented
16	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
17	C₅×#874	—	2	2	no	—
18	C₅×#874	C₂²	1	4	yes	complemented
19	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₂×#5073	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂²×C₅×D₈	—	3	4	no	—
22	C₂×C₅×#127	—	3	1	no	—
23	C₂×C₅×#127	D₃	1	1	yes	characteristic, complemented
24	C₅×#493	—	3	8	no	—
25	C₂×C₄×C₅×C₈	—	3	1	no	—
26	C₂⁴×C₃×C₅	—	2	2	no	—
27	C₂³×C₅×D₃	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	2	8	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	4	8	no	—
32	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×D₁₂	—	2	8	no	—
36	C₂×C₅×D₁₂	—	2	2	no	—
37	C₂×C₅×D₁₂	—	2	8	no	—
38	C₂×C₅×D₁₂	—	4	8	no	—
39	C₂×C₅×D₁₂	C₂³	1	4	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄²×C₅	C₂³	1	2	yes	—
42	C₅×#213	—	2	32	no	—
43	C₂×C₅×C₃⋊C₈	D₄	1	8	yes	complemented
44	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
45	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂²×#213	C₂×C₅	1	4	yes	complemented
47	C₂×#874	C₂×C₅	1	1	yes	characteristic, complemented
48	#5073	C₂×C₅	1	8	yes	direct product factor
49	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×C₅×D₄	—	2	2	no	—
51	C₂²×C₅×D₄	—	3	2	no	—
52	C₂²×C₅×D₄	—	6	2	no	—
53	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
54	C₂×C₅×D₈	—	3	16	no	—
55	C₂×C₅×D₈	—	6	16	no	—
56	C₂²×C₅×C₈	—	3	2	no	—
57	C₅×#127	—	2	2	no	—
58	C₅×#127	—	3	4	no	—
59	C₅×#127	—	6	2	no	—
60	C₅×#127	C₂×D₃	1	4	yes	complemented
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×C₈	—	3	4	no	—
63	C₂×#493	—	3	1	no	Sylow
64	C₂³×C₃×C₅	—	2	4	no	—
65	C₂³×C₃×C₅	—	4	4	no	—
66	C₂³×C₃×C₅	—	4	8	no	—
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₅×D₃	—	2	4	no	—
69	C₂²×C₅×D₃	—	4	4	no	—
70	C₂²×C₅×D₃	—	4	8	no	—
71	C₃×C₅×D₄	—	2	8	no	—
72	C₃×C₅×D₄	—	4	4	no	—
73	C₃×C₅×D₄	—	4	16	no	—
74	C₂×C₃×C₄×C₅	D₈	1	8	yes	complemented
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₅×D₁₂	—	2	8	no	—
80	C₅×D₁₂	—	4	4	no	—
81	C₅×D₁₂	—	4	16	no	—
82	C₅×C₃⋊C₈	C₂×D₄	1	8	yes	complemented
83	C₂²×C₃×D₄	—	2	2	no	—
84	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
85	C₂²×D₁₂	—	2	2	no	—
86	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
87	C₃×#127	—	2	2	no	—
88	C₃×#127	C₂²×C₅	1	4	yes	complemented
89	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	C₂×#213	—	2	16	no	—
91	C₂×#213	C₂²×C₅	1	16	yes	complemented
92	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
93	#874	—	2	2	no	—
94	#874	C₂²×C₅	1	4	yes	complemented
95	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
96	C₂⁴×C₅	—	2	2	no	—
97	C₂⁴×C₅	—	6	2	no	—
98	C₂×C₅×D₄	—	2	8	no	—
99	C₂×C₅×D₄	—	2	2	no	—
100	C₂×C₅×D₄	—	2	8	no	—
101	C₂×C₅×D₄	—	3	4	no	—
102	C₂×C₅×D₄	—	4	8	no	—
103	C₂×C₅×D₄	—	6	8	no	—
104	C₂×C₅×D₄	—	6	8	no	—
105	C₂×C₅×D₄	—	6	2	no	—
106	C₂×C₅×D₄	—	12	8	no	—
107	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
108	C₂²×C₄×C₅	#67	1	2	yes	—
109	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
110	C₅×D₈	—	6	32	no	—
111	C₂×C₅×C₈	—	3	8	no	—
112	C₂×C₅×C₈	—	3	4	no	—
113	C₄²×C₅	#67	1	2	yes	—
114	C₄²×C₅	C₂²×D₃	1	2	yes	—
115	C₂²×D₈	—	3	4	no	—
116	C₂×#127	—	3	1	no	—
117	C₂×#127	C₅×D₃	1	1	yes	characteristic, complemented
118	#493	—	3	8	no	—
119	C₂×C₄×C₈	—	3	1	no	—
120	C₂²×C₃×C₅	—	4	4	no	—
121	C₂²×C₃×C₅	—	4	8	no	—
122	C₂²×C₃×C₅	—	8	8	no	—
123	C₂²×C₃×C₅	#127	1	2	yes	—
124	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
125	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₅×D₃	—	4	4	no	—
127	C₂×C₅×D₃	—	4	8	no	—
128	C₂×C₅×D₃	—	8	8	no	—
129	C₃×C₄×C₅	#127	1	2	yes	—
130	C₃×C₄×C₅	C₂×D₈	1	8	yes	complemented
131	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
132	C₂⁴×C₃	—	2	2	no	—
133	C₂³×D₃	—	2	2	no	—
134	C₂×C₃×D₄	—	2	8	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	—	2	8	no	—
137	C₂×C₃×D₄	—	4	8	no	—
138	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
139	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
140	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₂×D₁₂	—	2	8	no	—
142	C₂×D₁₂	—	2	2	no	—
143	C₂×D₁₂	—	2	8	no	—
144	C₂×D₁₂	—	4	8	no	—
145	C₂×D₁₂	C₂³×C₅	1	4	yes	—
146	C₃×C₄²	C₅×D₄	1	2	yes	—
147	C₃×C₄²	C₂³×C₅	1	2	yes	—
148	#213	—	2	32	no	—
149	C₂×C₃⋊C₈	C₅×D₄	1	8	yes	complemented
150	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
151	C₂³×C₅	—	2	4	no	—
152	C₂³×C₅	—	4	8	no	—
153	C₂³×C₅	—	4	4	no	—
154	C₂³×C₅	—	6	4	no	—
155	C₂³×C₅	—	12	8	no	—
156	C₂³×C₅	—	12	4	no	—
157	C₂³×C₅	C₂×#67	1	1	yes	characteristic
158	C₅×D₄	—	2	8	no	—
159	C₅×D₄	—	4	16	no	—
160	C₅×D₄	—	4	4	no	—
161	C₅×D₄	—	6	8	no	—
162	C₅×D₄	—	12	16	no	—
163	C₅×D₄	—	12	4	no	—
164	C₂×C₄×C₅	#213	1	8	yes	complemented
165	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
166	C₂×C₄×C₅	C₂×#67	1	4	yes	—
167	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
169	C₅×C₈	—	3	8	no	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	—	3	2	no	—
172	C₂²×D₄	—	6	2	no	—
173	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
174	C₂×D₈	—	3	16	no	—
175	C₂×D₈	—	6	16	no	—
176	C₂²×C₈	—	3	2	no	—
177	#127	—	2	2	no	—
178	#127	—	3	4	no	—
179	#127	—	6	2	no	—
180	#127	C₂×C₅×D₃	1	4	yes	complemented
181	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
182	C₄×C₈	—	3	4	no	—
183	C₂×C₃×C₅	—	8	4	no	—
184	C₂×C₃×C₅	#493	1	4	yes	complemented
185	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
187	C₅×D₃	—	8	4	no	—
188	C₂³×C₃	—	2	4	no	—
189	C₂³×C₃	—	4	4	no	—
190	C₂³×C₃	—	4	8	no	—
191	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂²×D₃	—	2	4	no	—
193	C₂²×D₃	—	4	4	no	—
194	C₂²×D₃	—	4	8	no	—
195	C₃×D₄	—	2	8	no	—
196	C₃×D₄	—	4	4	no	—
197	C₃×D₄	—	4	16	no	—
198	C₂×C₃×C₄	C₅×D₈	1	8	yes	complemented
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	C₂×C₅×D₄	1	4	yes	—
202	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
203	D₁₂	—	2	8	no	—
204	D₁₂	—	4	4	no	—
205	D₁₂	—	4	16	no	—
206	C₃⋊C₈	C₂×C₅×D₄	1	8	yes	complemented
207	C₂²×C₅	—	4	8	no	—
208	C₂²×C₅	—	4	4	no	—
209	C₂²×C₅	—	8	8	no	—
210	C₂²×C₅	—	12	8	no	—
211	C₂²×C₅	—	12	4	no	—
212	C₂²×C₅	—	24	8	no	—
213	C₂²×C₅	C₂×#213	1	4	yes	—
214	C₂²×C₅	#940	1	2	yes	—
215	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
216	C₄×C₅	C₂×#213	1	8	yes	complemented
217	C₄×C₅	#940	1	2	yes	—
218	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
219	C₂⁴	—	2	2	no	—
220	C₂⁴	—	6	2	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	8	no	—
224	C₂×D₄	—	3	4	no	—
225	C₂×D₄	—	4	8	no	—
226	C₂×D₄	—	6	8	no	—
227	C₂×D₄	—	6	8	no	—
228	C₂×D₄	—	6	2	no	—
229	C₂×D₄	—	12	8	no	—
230	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
231	C₂²×C₄	C₅×#67	1	2	yes	—
232	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
233	D₈	—	6	32	no	—
234	C₂×C₈	—	3	8	no	—
235	C₂×C₈	—	3	4	no	—
236	C₄²	C₅×#67	1	2	yes	—
237	C₄²	C₂²×C₅×D₃	1	2	yes	—
238	C₃×C₅	C₂×#493	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	4	4	no	—
240	C₂²×C₃	—	4	8	no	—
241	C₂²×C₃	—	8	8	no	—
242	C₂²×C₃	C₅×#127	1	2	yes	—
243	C₂²×C₃	C₂×C₅×D₈	1	4	yes	—
244	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
245	C₂×D₃	—	4	4	no	—
246	C₂×D₃	—	4	8	no	—
247	C₂×D₃	—	8	8	no	—
248	C₃×C₄	C₅×#127	1	2	yes	—
249	C₃×C₄	C₂×C₅×D₈	1	8	yes	complemented
250	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
251	C₂×C₅	—	8	4	no	—
252	C₂×C₅	—	24	4	no	—
253	C₂×C₅	#5073	1	4	yes	direct product factor
254	C₂×C₅	C₂²×#213	1	2	yes	—
255	C₂×C₅	C₂×#940	1	1	yes	characteristic
256	C₂³	—	2	4	no	—
257	C₂³	—	4	8	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	6	4	no	—
260	C₂³	—	12	8	no	—
261	C₂³	—	12	4	no	—
262	C₂³	C₂×C₅×#67	1	1	yes	characteristic
263	D₄	—	2	8	no	—
264	D₄	—	4	16	no	—
265	D₄	—	4	4	no	—
266	D₄	—	6	8	no	—
267	D₄	—	12	16	no	—
268	D₄	—	12	4	no	—
269	C₂×C₄	C₅×#213	1	8	yes	complemented
270	C₂×C₄	C₅×D₃×D₄	1	4	yes	—
271	C₂×C₄	C₂×C₅×#67	1	4	yes	—
272	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
274	C₈	—	3	8	no	—
275	C₂×C₃	—	8	4	no	—
276	C₂×C₃	C₅×#493	1	4	yes	complemented
277	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
278	C₂×C₃	C₂²×C₅×D₈	1	2	yes	—
279	D₃	—	8	4	no	—
280	C₅	C₂×#5073	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	8	8	no	—
284	C₂²	—	12	8	no	—
285	C₂²	—	12	4	no	—
286	C₂²	—	24	8	no	—
287	C₂²	C₂×C₅×#213	1	4	yes	—
288	C₂²	C₅×#940	1	2	yes	—
289	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
290	C₄	C₂×C₅×#213	1	8	yes	complemented
291	C₄	C₅×#940	1	2	yes	—
292	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
293	C₃	C₂×C₅×#493	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	8	4	no	—
295	C₂	—	24	4	no	—
296	C₂	C₅×#5073	1	4	yes	direct product factor
297	C₂	C₂²×C₅×#213	1	2	yes	—
298	C₂	C₂×C₅×#940	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	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₅×#127	D₃
hypocenter	C₃	C₂×C₅×#493
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
1. C₂×C₃×C₅×#127	C₂	semidirect
2. C₂²×C₅×#213	C₂	semidirect
3. C₂×C₅×#874	C₂	semidirect
4. C₅×#5073	C₂	direct (N × H)
5. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
12. C₃×C₅×#127	C₂²	semidirect
15. C₂×C₅×#213	C₂²	semidirect
18. C₅×#874	C₂²	semidirect
19. C₄×C₅×C₃⋊C₈	C₂²	semidirect
20. C₂×#5073	C₅	direct (N × H)
23. C₂×C₅×#127	D₃	semidirect
43. C₂×C₅×C₃⋊C₈	D₄	semidirect
45. C₂×C₃×#127	C₂×C₅	semidirect
46. C₂²×#213	C₂×C₅	semidirect
47. C₂×#874	C₂×C₅	semidirect
48. #5073	C₂×C₅	direct (N × H)
49. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
60. C₅×#127	C₂×D₃	semidirect
74. C₂×C₃×C₄×C₅	D₈	semidirect
82. C₅×C₃⋊C₈	C₂×D₄	semidirect
88. C₃×#127	C₂²×C₅	semidirect
91. C₂×#213	C₂²×C₅	semidirect
94. #874	C₂²×C₅	semidirect
95. C₄×C₃⋊C₈	C₂²×C₅	semidirect
117. C₂×#127	C₅×D₃	semidirect
130. C₃×C₄×C₅	C₂×D₈	semidirect
149. C₂×C₃⋊C₈	C₅×D₄	semidirect
164. C₂×C₄×C₅	#213	semidirect
180. #127	C₂×C₅×D₃	semidirect
184. C₂×C₃×C₅	#493	semidirect
198. C₂×C₃×C₄	C₅×D₈	semidirect
206. C₃⋊C₈	C₂×C₅×D₄	semidirect
216. C₄×C₅	C₂×#213	semidirect
238. C₃×C₅	C₂×#493	semidirect
249. C₃×C₄	C₂×C₅×D₈	semidirect
253. C₂×C₅	#5073	direct (N × H)
269. C₂×C₄	C₅×#213	semidirect
276. C₂×C₃	C₅×#493	semidirect
280. C₅	C₂×#5073	direct (N × H)
290. C₄	C₂×C₅×#213	semidirect
293. C₃	C₂×C₅×#493	semidirect
296. C₂	C₅×#5073	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
C₂²×C₅×#213	2
C₂×C₅×#874	2
C₅×#5073	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₅×#493	3
C₂×#5073	5

Subgroups — C2×C5×#5073 #14438769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5073 #14438769