Subgroups — C₂×C₃×C₅×#490 #14276167

Order 1920.

976 conjugacy classes of subgroups.

The subgroup lattice has 256 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×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#490	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#102	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#490	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	—	2	2	no	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₅×#127	—	2	2	no	—
11	C₃×C₅×#127	C₂²	1	4	yes	—
12	C₃×C₅×#125	—	2	2	no	—
13	C₃×C₅×#125	C₂²	1	4	yes	complemented
14	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₃×C₅×#102	C₂²	1	16	yes	complemented
18	C₃×C₅×#97	C₂²	1	4	yes	complemented
19	C₂×C₃×#490	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#490	C₂×C₃	1	8	yes	direct product factor
23	C₂×C₅×#102	C₂×C₃	1	4	yes	complemented
24	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂⁴×C₃×C₅	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	8	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	4	8	no	—
30	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	—	2	4	no	—
35	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
36	C₃×C₅×C₄⋊C₄	—	2	8	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
39	C₃×C₄²×C₅	D₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
42	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#490	C₂×C₅	1	8	yes	direct product factor
44	C₂×C₃×#102	C₂×C₅	1	4	yes	complemented
45	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂²×C₅×D₄	—	2	2	no	—
47	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
49	C₅×#127	—	2	2	no	—
50	C₅×#127	C₂²×C₃	1	4	yes	—
51	C₅×#125	—	2	2	no	—
52	C₅×#125	C₂²×C₃	1	4	yes	complemented
53	C₂×C₅×C₄⋊C₄	—	2	2	no	—
54	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₅×#102	C₂²×C₃	1	16	yes	complemented
57	C₅×#97	C₂²×C₃	1	4	yes	complemented
58	C₂×#490	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
59	C₂³×C₃×C₅	—	2	4	no	—
60	C₂³×C₃×C₅	—	4	4	no	—
61	C₂³×C₃×C₅	—	4	8	no	—
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₃×C₅×D₄	—	2	8	no	—
64	C₃×C₅×D₄	—	4	16	no	—
65	C₃×C₅×D₄	—	4	4	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	#41	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
73	C₃×C₅×C₈	—	2	4	no	—
74	C₂²×C₃×D₄	—	2	2	no	—
75	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
76	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
77	C₃×#127	—	2	2	no	—
78	C₃×#127	C₂²×C₅	1	4	yes	—
79	C₃×#125	—	2	2	no	—
80	C₃×#125	C₂²×C₅	1	4	yes	complemented
81	C₂×C₃×C₄⋊C₄	—	2	2	no	—
82	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
84	C₃×#102	C₂²×C₅	1	16	yes	complemented
85	C₃×#97	C₂²×C₅	1	4	yes	complemented
86	C₂⁴×C₅	—	2	2	no	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	—	2	8	no	—
89	C₂×C₅×D₄	—	2	8	no	—
90	C₂×C₅×D₄	—	4	8	no	—
91	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
97	C₅×C₄⋊C₄	—	2	8	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
100	C₄²×C₅	C₃×D₄	1	2	yes	—
101	C₄²×C₅	C₂³×C₃	1	2	yes	—
102	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#490	C₂×C₃×C₅	1	8	yes	direct product factor
105	C₂×#102	C₂×C₃×C₅	1	4	yes	complemented
106	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic, complemented
107	C₂²×C₃×C₅	—	4	4	no	—
108	C₂²×C₃×C₅	—	4	8	no	—
109	C₂²×C₃×C₅	—	8	8	no	—
110	C₂²×C₃×C₅	#124	1	2	yes	—
111	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
112	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
113	C₃×C₄×C₅	—	2	4	no	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	#124	1	2	yes	—
116	C₃×C₄×C₅	C₂×#41	1	2	yes	—
117	C₂⁴×C₃	—	2	2	no	—
118	C₂×C₃×D₄	—	2	2	no	—
119	C₂×C₃×D₄	—	2	8	no	—
120	C₂×C₃×D₄	—	2	8	no	—
121	C₂×C₃×D₄	—	4	8	no	—
122	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
123	C₂²×C₃×C₄	—	2	2	no	—
124	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
125	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₃×C₈	—	2	4	no	—
127	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
128	C₃×C₄⋊C₄	—	2	8	no	—
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
131	C₃×C₄²	C₅×D₄	1	2	yes	—
132	C₃×C₄²	C₂³×C₅	1	2	yes	—
133	C₂³×C₅	—	2	4	no	—
134	C₂³×C₅	—	4	4	no	—
135	C₂³×C₅	—	4	8	no	—
136	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₅×D₄	—	2	8	no	—
138	C₅×D₄	—	4	16	no	—
139	C₅×D₄	—	4	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
144	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₃×#41	1	4	yes	—
146	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
147	C₅×C₈	—	2	4	no	—
148	C₂²×D₄	—	2	2	no	—
149	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
150	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
151	#127	—	2	2	no	—
152	#127	C₂²×C₃×C₅	1	4	yes	—
153	#125	—	2	2	no	—
154	#125	C₂²×C₃×C₅	1	4	yes	complemented
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
157	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
158	#102	C₂²×C₃×C₅	1	16	yes	complemented
159	#97	C₂²×C₃×C₅	1	4	yes	complemented
160	C₂×C₃×C₅	—	8	4	no	—
161	C₂×C₃×C₅	#490	1	4	yes	direct product factor
162	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
164	C₂³×C₃	—	2	4	no	—
165	C₂³×C₃	—	4	4	no	—
166	C₂³×C₃	—	4	8	no	—
167	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
168	C₃×D₄	—	2	8	no	—
169	C₃×D₄	—	4	16	no	—
170	C₃×D₄	—	4	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	4	4	no	—
174	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
175	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₅×#41	1	4	yes	—
177	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
178	C₃×C₈	—	2	4	no	—
179	C₂²×C₅	—	4	8	no	—
180	C₂²×C₅	—	4	4	no	—
181	C₂²×C₅	—	8	8	no	—
182	C₂²×C₅	C₃×#124	1	2	yes	—
183	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
184	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	C₃×#124	1	2	yes	—
188	C₄×C₅	C₂×C₃×#41	1	2	yes	—
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	8	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
197	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
203	C₄²	C₃×C₅×D₄	1	2	yes	—
204	C₄²	C₂³×C₃×C₅	1	2	yes	—
205	C₃×C₅	C₂×#490	1	1	yes	characteristic, direct product factor, Hall
206	C₂²×C₃	—	4	4	no	—
207	C₂²×C₃	—	4	8	no	—
208	C₂²×C₃	—	8	8	no	—
209	C₂²×C₃	C₅×#124	1	2	yes	—
210	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	4	yes	—
211	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
212	C₃×C₄	—	2	4	no	—
213	C₃×C₄	—	4	4	no	—
214	C₃×C₄	C₅×#124	1	2	yes	—
215	C₃×C₄	C₂×C₅×#41	1	2	yes	—
216	C₂×C₅	—	8	4	no	—
217	C₂×C₅	C₃×#490	1	4	yes	direct product factor
218	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
219	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	8	no	—
223	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
224	D₄	—	2	8	no	—
225	D₄	—	4	16	no	—
226	D₄	—	4	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
231	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
232	C₂×C₄	C₃×C₅×#41	1	4	yes	—
233	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
234	C₈	—	2	4	no	—
235	C₂×C₃	—	8	4	no	—
236	C₂×C₃	C₅×#490	1	4	yes	direct product factor
237	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
238	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
239	C₅	C₂×C₃×#490	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	4	4	no	—
241	C₂²	—	4	8	no	—
242	C₂²	—	8	8	no	—
243	C₂²	C₃×C₅×#124	1	2	yes	—
244	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	4	yes	—
245	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	C₃×C₅×#124	1	2	yes	—
249	C₄	C₂×C₃×C₅×#41	1	2	yes	—
250	C₃	C₂×C₅×#490	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	8	4	no	—
252	C₂	C₃×C₅×#490	1	4	yes	direct product factor
253	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
254	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#125	C₂	semidirect
3. C₃×C₅×#490	C₂	direct (N × H)
4. C₂×C₃×C₅×#102	C₂	semidirect
5. C₂×C₃×C₅×#97	C₂	semidirect
6. C₂×C₅×#490	C₃	direct (N × H)
13. C₃×C₅×#125	C₂²	semidirect
17. C₃×C₅×#102	C₂²	semidirect
18. C₃×C₅×#97	C₂²	semidirect
19. C₂×C₃×#490	C₅	direct (N × H)
21. C₂×C₅×#125	C₂×C₃	semidirect
22. C₅×#490	C₂×C₃	direct (N × H)
23. C₂×C₅×#102	C₂×C₃	semidirect
24. C₂×C₅×#97	C₂×C₃	semidirect
42. C₂×C₃×#125	C₂×C₅	semidirect
43. C₃×#490	C₂×C₅	direct (N × H)
44. C₂×C₃×#102	C₂×C₅	semidirect
45. C₂×C₃×#97	C₂×C₅	semidirect
52. C₅×#125	C₂²×C₃	semidirect
56. C₅×#102	C₂²×C₃	semidirect
57. C₅×#97	C₂²×C₃	semidirect
58. C₂×#490	C₃×C₅	direct (N × H)
80. C₃×#125	C₂²×C₅	semidirect
84. C₃×#102	C₂²×C₅	semidirect
85. C₃×#97	C₂²×C₅	semidirect
103. C₂×#125	C₂×C₃×C₅	semidirect
104. #490	C₂×C₃×C₅	direct (N × H)
105. C₂×#102	C₂×C₃×C₅	semidirect
106. C₂×#97	C₂×C₃×C₅	semidirect
154. #125	C₂²×C₃×C₅	semidirect
158. #102	C₂²×C₃×C₅	semidirect
159. #97	C₂²×C₃×C₅	semidirect
161. C₂×C₃×C₅	#490	direct (N × H)
205. C₃×C₅	C₂×#490	direct (N × H)
217. C₂×C₅	C₃×#490	direct (N × H)
236. C₂×C₃	C₅×#490	direct (N × H)
239. C₅	C₂×C₃×#490	direct (N × H)
250. C₃	C₂×C₅×#490	direct (N × H)
252. C₂	C₃×C₅×#490	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×C₃×C₅×#125	2
C₃×C₅×#490	2
C₂×C₃×C₅×#102	2
C₂×C₃×C₅×#97	2
C₂×C₅×#490	3
C₂×C₃×#490	5

Subgroups — C2×C3×C5×#490 #14276167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#490 #14276167