Subgroups — C₃×C₅×#2243 #14275277

Order 1920.

284 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#495	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#357	C₂	1	2	yes	maximal
5	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2243	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	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₃×C₅×#124	—	2	1	no	—
12	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
13	C₃×C₅×#107	—	2	1	no	—
14	C₃×C₅×#107	C₂²	1	2	yes	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#2243	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
18	C₅×#495	C₂×C₃	1	1	yes	characteristic
19	C₅×#358	C₂×C₃	1	2	yes	complemented
20	C₅×#357	C₂×C₃	1	2	yes	—
21	C₅×#346	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅×Q₈	—	2	1	no	—
23	C₂×C₃×C₅×Q₈	—	4	1	no	—
24	C₂×C₃×C₅×D₄	—	2	1	no	—
25	C₃×C₅×Q₁₆	—	2	2	no	—
26	C₃×C₅×SD₁₆	—	4	2	no	—
27	C₃×C₅×D₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	4	1	no	—
32	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
34	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₃×C₅×C₁₆	—	2	2	no	—
36	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×#495	C₂×C₅	1	1	yes	characteristic
38	C₃×#358	C₂×C₅	1	2	yes	complemented
39	C₃×#357	C₂×C₅	1	2	yes	—
40	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₅×SD₁₆	—	2	1	no	—
43	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×#124	—	2	1	no	—
46	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
47	C₅×#107	—	2	1	no	—
48	C₅×#107	C₂²×C₃	1	2	yes	—
49	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	#2243	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	—	4	1	no	—
52	C₃×C₅×Q₈	—	4	1	no	—
53	C₃×C₅×Q₈	—	4	1	no	—
54	C₃×C₅×Q₈	—	8	2	no	—
55	C₃×C₅×D₄	—	4	1	no	—
56	C₃×C₅×D₄	—	4	1	no	—
57	C₂×C₃×C₄×C₅	—	4	1	no	—
58	C₂×C₃×C₄×C₅	—	4	2	no	—
59	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₃×C₅×C₈	SD₁₆	1	2	yes	—
63	C₃×C₅×C₈	#41	1	2	yes	—
64	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
65	C₂×C₃×SD₁₆	—	2	1	no	—
66	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
67	C₃×C₄⋊Q₈	—	2	1	no	—
68	C₃×#124	—	2	1	no	—
69	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
70	C₃×#107	—	2	1	no	—
71	C₃×#107	C₂²×C₅	1	2	yes	—
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₅×Q₈	—	2	1	no	—
74	C₂×C₅×Q₈	—	4	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₅×Q₁₆	—	2	2	no	—
77	C₅×SD₁₆	—	4	2	no	—
78	C₅×D₈	—	2	2	no	—
79	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	4	1	no	—
83	C₅×C₂²⋊C₄	—	4	1	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
86	C₅×C₁₆	—	2	2	no	—
87	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	#495	C₂×C₃×C₅	1	1	yes	characteristic
89	#358	C₂×C₃×C₅	1	2	yes	complemented
90	#357	C₂×C₃×C₅	1	2	yes	—
91	#346	C₂×C₃×C₅	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₅	—	8	1	no	—
93	C₂²×C₃×C₅	—	8	1	no	—
94	C₂²×C₃×C₅	—	8	1	no	—
95	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	8	1	no	—
99	C₃×C₄×C₅	—	8	2	no	—
100	C₃×C₄×C₅	#124	1	1	yes	characteristic
101	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	2	1	no	—
103	C₂×C₃×Q₈	—	4	1	no	—
104	C₂×C₃×D₄	—	2	1	no	—
105	C₃×Q₁₆	—	2	2	no	—
106	C₃×SD₁₆	—	4	2	no	—
107	C₃×D₈	—	2	2	no	—
108	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	4	1	no	—
112	C₃×C₂²⋊C₄	—	4	1	no	—
113	C₃×C₂²⋊C₄	—	4	1	no	—
114	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
115	C₃×C₁₆	—	2	2	no	—
116	C₂³×C₅	—	4	1	no	—
117	C₅×Q₈	—	4	1	no	—
118	C₅×Q₈	—	4	1	no	—
119	C₅×Q₈	—	8	2	no	—
120	C₅×D₄	—	4	1	no	—
121	C₅×D₄	—	4	1	no	—
122	C₂×C₄×C₅	—	4	1	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
127	C₅×C₈	C₃×SD₁₆	1	2	yes	—
128	C₅×C₈	C₃×#41	1	2	yes	—
129	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
130	C₂×SD₁₆	—	2	1	no	—
131	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	1	no	—
133	#124	—	2	1	no	—
134	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
135	#107	—	2	1	no	—
136	#107	C₂²×C₃×C₅	1	2	yes	—
137	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
138	C₂×C₃×C₅	—	16	1	no	—
139	C₂×C₃×C₅	#486	1	1	yes	characteristic
140	C₂×C₃×C₅	#509	1	1	yes	characteristic
141	C₂×C₃×C₅	#510	1	1	yes	characteristic
142	C₂³×C₃	—	4	1	no	—
143	C₃×Q₈	—	4	1	no	—
144	C₃×Q₈	—	4	1	no	—
145	C₃×Q₈	—	8	2	no	—
146	C₃×D₄	—	4	1	no	—
147	C₃×D₄	—	4	1	no	—
148	C₂×C₃×C₄	—	4	1	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
151	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
153	C₃×C₈	C₅×SD₁₆	1	2	yes	—
154	C₃×C₈	C₅×#41	1	2	yes	—
155	C₂²×C₅	—	8	1	no	—
156	C₂²×C₅	—	8	1	no	—
157	C₂²×C₅	—	8	1	no	—
158	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	2	1	no	—
161	C₄×C₅	—	8	2	no	—
162	C₄×C₅	—	8	1	no	—
163	C₄×C₅	C₃×#124	1	1	yes	characteristic
164	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂×D₄	—	2	1	no	—
168	Q₁₆	—	2	2	no	—
169	SD₁₆	—	4	2	no	—
170	D₈	—	2	2	no	—
171	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₂²⋊C₄	—	4	1	no	—
177	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
178	C₁₆	—	2	2	no	—
179	C₃×C₅	#2243	1	1	yes	characteristic, direct product factor, Hall
180	C₂²×C₃	—	8	1	no	—
181	C₂²×C₃	—	8	1	no	—
182	C₂²×C₃	—	8	1	no	—
183	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	8	1	no	—
187	C₃×C₄	—	8	2	no	—
188	C₃×C₄	C₅×#124	1	1	yes	characteristic
189	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
190	C₂×C₅	—	16	1	no	—
191	C₂×C₅	C₃×#486	1	1	yes	characteristic
192	C₂×C₅	C₃×#509	1	1	yes	characteristic
193	C₂×C₅	C₃×#510	1	1	yes	characteristic
194	C₂³	—	4	1	no	—
195	Q₈	—	4	1	no	—
196	Q₈	—	4	1	no	—
197	Q₈	—	8	2	no	—
198	D₄	—	4	1	no	—
199	D₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
203	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
205	C₈	C₃×C₅×SD₁₆	1	2	yes	—
206	C₈	C₃×C₅×#41	1	2	yes	—
207	C₂×C₃	—	16	1	no	—
208	C₂×C₃	C₅×#486	1	1	yes	characteristic
209	C₂×C₃	C₅×#509	1	1	yes	characteristic
210	C₂×C₃	C₅×#510	1	1	yes	characteristic
211	C₅	C₃×#2243	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	8	1	no	—
213	C₂²	—	8	1	no	—
214	C₂²	—	8	1	no	—
215	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	1	no	—
219	C₄	—	8	2	no	—
220	C₄	C₃×C₅×#124	1	1	yes	characteristic
221	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
222	C₃	C₅×#2243	1	1	yes	characteristic, direct product factor, Sylow
223	C₂	—	16	1	no	—
224	C₂	C₃×C₅×#486	1	1	yes	characteristic
225	C₂	C₃×C₅×#509	1	1	yes	characteristic
226	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₈	C₂²
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. C₃×C₅×#500	C₂	semidirect
3. C₃×C₅×#358	C₂	semidirect
5. C₃×C₅×#346	C₂	semidirect
6. C₅×#2243	C₃	direct (N × H)
16. C₃×#2243	C₅	direct (N × H)
17. C₅×#500	C₂×C₃	semidirect
19. C₅×#358	C₂×C₃	semidirect
21. C₅×#346	C₂×C₃	semidirect
36. C₃×#500	C₂×C₅	semidirect
38. C₃×#358	C₂×C₅	semidirect
40. C₃×#346	C₂×C₅	semidirect
50. #2243	C₃×C₅	direct (N × H)
87. #500	C₂×C₃×C₅	semidirect
89. #358	C₂×C₃×C₅	semidirect
91. #346	C₂×C₃×C₅	semidirect
179. C₃×C₅	#2243	direct (N × H)
211. C₅	C₃×#2243	direct (N × H)
222. C₃	C₅×#2243	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#500	2
C₃×C₅×#495	2
C₃×C₅×#358	2
C₃×C₅×#357	2
C₃×C₅×#346	2
C₅×#2243	3
C₃×#2243	5

Subgroups — C3×C5×#2243 #14275277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2243 #14275277