Subgroups — C₃×C₅×#2250 #14275284

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2250	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#127	—	2	1	no	—
13	C₃×C₅×#124	—	2	1	no	—
14	C₃×C₅×SD₃₂	—	2	4	no	—
15	C₃×C₅×D₁₆	—	2	4	no	—
16	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#2250	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	complemented
20	C₂×C₅×D₁₆	C₂×C₃	1	2	yes	complemented
21	C₅×#495	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#346	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×Q₈	—	2	1	no	—
25	C₂×C₃×C₅×D₄	—	2	1	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	4	2	no	—
28	C₃×C₅×Q₁₆	—	2	2	no	—
29	C₃×C₅×SD₁₆	—	4	2	no	—
30	C₃×C₅×D₈	—	2	2	no	—
31	C₃×C₅×D₈	—	2	2	no	—
32	C₃×C₅×D₈	—	2	2	no	—
33	C₃×C₅×D₈	—	4	2	no	—
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₁₆	D₄	1	4	yes	complemented
39	C₂×C₃×SD₃₂	C₂×C₅	1	2	yes	complemented
40	C₂×C₃×D₁₆	C₂×C₅	1	2	yes	complemented
41	C₃×#495	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×SD₁₆	—	2	1	no	—
46	C₂×C₅×D₈	—	2	1	no	—
47	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
48	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×#127	—	2	1	no	—
50	C₅×#124	—	2	1	no	—
51	C₅×SD₃₂	—	2	4	no	—
52	C₅×D₁₆	—	2	4	no	—
53	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	#2250	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	4	1	no	—
57	C₂³×C₃×C₅	—	4	2	no	—
58	C₃×C₅×Q₈	—	4	1	no	—
59	C₃×C₅×Q₈	—	4	1	no	—
60	C₃×C₅×D₄	—	4	1	no	—
61	C₃×C₅×D₄	—	4	2	no	—
62	C₃×C₅×D₄	—	4	2	no	—
63	C₃×C₅×D₄	—	4	1	no	—
64	C₃×C₅×D₄	—	8	4	no	—
65	C₂×C₃×C₄×C₅	—	4	1	no	—
66	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×C₈	D₈	1	2	yes	—
69	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃×SD₁₆	—	2	1	no	—
72	C₂×C₃×D₈	—	2	1	no	—
73	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
75	C₃×#127	—	2	1	no	—
76	C₃×#124	—	2	1	no	—
77	C₃×SD₃₂	—	2	4	no	—
78	C₃×D₁₆	—	2	4	no	—
79	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
80	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₅×Q₈	—	2	1	no	—
82	C₂×C₅×D₄	—	2	1	no	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂×C₅×D₄	—	4	2	no	—
85	C₅×Q₁₆	—	2	2	no	—
86	C₅×SD₁₆	—	4	2	no	—
87	C₅×D₈	—	2	2	no	—
88	C₅×D₈	—	2	2	no	—
89	C₅×D₈	—	2	2	no	—
90	C₅×D₈	—	4	2	no	—
91	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
95	C₅×C₁₆	C₃×D₄	1	4	yes	complemented
96	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	complemented
97	C₂×D₁₆	C₂×C₃×C₅	1	2	yes	complemented
98	#495	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	#346	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	C₂²×C₃×C₅	—	8	2	no	—
102	C₂²×C₃×C₅	—	8	1	no	—
103	C₂²×C₃×C₅	—	8	4	no	—
104	C₂²×C₃×C₅	—	8	2	no	—
105	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	8	1	no	—
108	C₃×C₄×C₅	#127	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₂×C₃×Q₈	—	2	1	no	—
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	—	4	2	no	—
114	C₃×Q₁₆	—	2	2	no	—
115	C₃×SD₁₆	—	4	2	no	—
116	C₃×D₈	—	2	2	no	—
117	C₃×D₈	—	2	2	no	—
118	C₃×D₈	—	2	2	no	—
119	C₃×D₈	—	4	2	no	—
120	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	—	4	2	no	—
123	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
124	C₃×C₁₆	C₅×D₄	1	4	yes	complemented
125	C₂³×C₅	—	4	1	no	—
126	C₂³×C₅	—	4	2	no	—
127	C₅×Q₈	—	4	1	no	—
128	C₅×Q₈	—	4	1	no	—
129	C₅×D₄	—	4	1	no	—
130	C₅×D₄	—	4	2	no	—
131	C₅×D₄	—	4	2	no	—
132	C₅×D₄	—	4	1	no	—
133	C₅×D₄	—	8	4	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₅×C₈	C₃×D₈	1	2	yes	—
138	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
139	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
140	C₂×SD₁₆	—	2	1	no	—
141	C₂×D₈	—	2	1	no	—
142	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
143	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
144	#127	—	2	1	no	—
145	#124	—	2	1	no	—
146	SD₃₂	—	2	4	no	—
147	D₁₆	—	2	4	no	—
148	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
149	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₂×C₃×C₅	—	16	2	no	—
151	C₂×C₃×C₅	—	16	1	no	—
152	C₂×C₃×C₅	#493	1	1	yes	characteristic
153	C₂×C₃×C₅	#509	1	2	yes	—
154	C₂³×C₃	—	4	1	no	—
155	C₂³×C₃	—	4	2	no	—
156	C₃×Q₈	—	4	1	no	—
157	C₃×Q₈	—	4	1	no	—
158	C₃×D₄	—	4	1	no	—
159	C₃×D₄	—	4	2	no	—
160	C₃×D₄	—	4	2	no	—
161	C₃×D₄	—	4	1	no	—
162	C₃×D₄	—	8	4	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
165	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₃×C₈	C₅×D₈	1	2	yes	—
167	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
168	C₂²×C₅	—	8	4	no	—
169	C₂²×C₅	—	8	2	no	—
170	C₂²×C₅	—	8	2	no	—
171	C₂²×C₅	—	8	1	no	—
172	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	8	1	no	—
175	C₄×C₅	C₃×#127	1	1	yes	characteristic
176	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
177	C₂×Q₈	—	2	1	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	Q₁₆	—	2	2	no	—
182	SD₁₆	—	4	2	no	—
183	D₈	—	2	2	no	—
184	D₈	—	2	2	no	—
185	D₈	—	2	2	no	—
186	D₈	—	4	2	no	—
187	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
189	C₂²⋊C₄	—	4	2	no	—
190	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
191	C₁₆	C₃×C₅×D₄	1	4	yes	complemented
192	C₃×C₅	#2250	1	1	yes	characteristic, direct product factor, Hall
193	C₂²×C₃	—	8	2	no	—
194	C₂²×C₃	—	8	1	no	—
195	C₂²×C₃	—	8	4	no	—
196	C₂²×C₃	—	8	2	no	—
197	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	8	1	no	—
200	C₃×C₄	C₅×#127	1	1	yes	characteristic
201	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
202	C₂×C₅	—	16	2	no	—
203	C₂×C₅	—	16	1	no	—
204	C₂×C₅	C₃×#493	1	1	yes	characteristic
205	C₂×C₅	C₃×#509	1	2	yes	—
206	C₂³	—	4	1	no	—
207	C₂³	—	4	2	no	—
208	Q₈	—	4	1	no	—
209	Q₈	—	4	1	no	—
210	D₄	—	4	1	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	1	no	—
214	D₄	—	8	4	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
217	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
218	C₈	C₃×C₅×D₈	1	2	yes	—
219	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
220	C₂×C₃	—	16	2	no	—
221	C₂×C₃	—	16	1	no	—
222	C₂×C₃	C₅×#493	1	1	yes	characteristic
223	C₂×C₃	C₅×#509	1	2	yes	—
224	C₅	C₃×#2250	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	—	8	2	no	—
226	C₂²	—	8	1	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	2	no	—
229	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	8	1	no	—
232	C₄	C₃×C₅×#127	1	1	yes	characteristic
233	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
234	C₃	C₅×#2250	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	16	2	no	—
236	C₂	—	16	1	no	—
237	C₂	C₃×C₅×#493	1	1	yes	characteristic
238	C₂	C₃×C₅×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×SD₃₂	C₂	semidirect
2. C₂×C₃×C₅×D₁₆	C₂	semidirect
3. C₃×C₅×#495	C₂	semidirect
4. C₃×C₅×#493	C₂	semidirect
5. C₃×C₅×#346	C₂	semidirect
6. C₅×#2250	C₃	direct (N × H)
18. C₃×#2250	C₅	direct (N × H)
19. C₂×C₅×SD₃₂	C₂×C₃	semidirect
20. C₂×C₅×D₁₆	C₂×C₃	semidirect
21. C₅×#495	C₂×C₃	semidirect
22. C₅×#493	C₂×C₃	semidirect
23. C₅×#346	C₂×C₃	semidirect
38. C₃×C₅×C₁₆	D₄	semidirect
39. C₂×C₃×SD₃₂	C₂×C₅	semidirect
40. C₂×C₃×D₁₆	C₂×C₅	semidirect
41. C₃×#495	C₂×C₅	semidirect
42. C₃×#493	C₂×C₅	semidirect
43. C₃×#346	C₂×C₅	semidirect
55. #2250	C₃×C₅	direct (N × H)
95. C₅×C₁₆	C₃×D₄	semidirect
96. C₂×SD₃₂	C₂×C₃×C₅	semidirect
97. C₂×D₁₆	C₂×C₃×C₅	semidirect
98. #495	C₂×C₃×C₅	semidirect
99. #493	C₂×C₃×C₅	semidirect
100. #346	C₂×C₃×C₅	semidirect
124. C₃×C₁₆	C₅×D₄	semidirect
191. C₁₆	C₃×C₅×D₄	semidirect
192. C₃×C₅	#2250	direct (N × H)
224. C₅	C₃×#2250	direct (N × H)
234. C₃	C₅×#2250	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×SD₃₂	2
C₂×C₃×C₅×D₁₆	2
C₃×C₅×#495	2
C₃×C₅×#493	2
C₃×C₅×#346	2
C₅×#2250	3
C₃×#2250	5

Subgroups — C3×C5×#2250 #14275284

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2250 #14275284