Subgroups — C₃×C₅×#1810 #14274844

Order 1920.

584 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	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#407	C₂	1	2	yes	maximal
4	C₃×C₅×#341	C₂	1	2	yes	maximal, complemented
5	C₅×#1810	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×C₈	—	2	4	no	—
9	C₂²×C₃×C₅×C₈	C₄	1	4	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	4	no	—
13	C₃×C₅×C₂²⋊C₈	—	2	4	no	—
14	C₃×#1810	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂³×C₅×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×C₅×#108	C₂×C₃	1	2	yes	complemented
17	C₅×#407	C₂×C₃	1	2	yes	—
18	C₅×#341	C₂×C₃	1	2	yes	complemented
19	C₂⁴×C₃×C₅	Q₈	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₃×C₅×M₁₆	—	2	4	no	—
27	C₃×C₅×M₁₆	—	2	4	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	8	no	—
30	C₂×C₃×C₅×C₈	—	2	4	no	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
34	C₂³×C₃×C₈	C₂×C₅	1	1	yes	characteristic
35	C₂×C₃×#108	C₂×C₅	1	2	yes	complemented
36	C₃×#407	C₂×C₅	1	2	yes	—
37	C₃×#341	C₂×C₅	1	2	yes	complemented
38	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₅×M₁₆	C₂²×C₃	1	4	yes	—
40	C₂²×C₅×C₈	—	2	4	no	—
41	C₂²×C₅×C₈	C₃×C₄	1	4	yes	—
42	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₅×#108	—	2	4	no	—
45	C₅×C₂²⋊C₈	—	2	4	no	—
46	#1810	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₅	—	2	4	no	—
48	C₂³×C₃×C₅	—	2	2	no	—
49	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
50	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	2	yes	—
60	C₃×C₅×C₈	—	2	2	no	—
61	C₃×C₅×C₈	—	4	4	no	—
62	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	—
63	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃×M₁₆	C₂²×C₅	1	4	yes	—
65	C₂²×C₃×C₈	—	2	4	no	—
66	C₂²×C₃×C₈	C₄×C₅	1	4	yes	—
67	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
68	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
69	C₃×#108	—	2	4	no	—
70	C₃×C₂²⋊C₈	—	2	4	no	—
71	C₂⁴×C₅	C₃×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	8	no	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
86	C₂³×C₈	C₂×C₃×C₅	1	1	yes	characteristic
87	C₂×#108	C₂×C₃×C₅	1	2	yes	complemented
88	#407	C₂×C₃×C₅	1	2	yes	—
89	#341	C₂×C₃×C₅	1	2	yes	complemented
90	C₂²×C₃×C₅	—	2	8	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	2	no	—
93	C₂²×C₃×C₅	—	2	2	no	—
94	C₂²×C₃×C₅	#108	1	4	yes	complemented
95	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₃×C₄×C₅	#117	1	1	yes	characteristic
100	C₃×C₄×C₅	#121	1	2	yes	—
101	C₂⁴×C₃	C₅×Q₈	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	—	2	2	no	—
104	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
105	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
106	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
108	C₃×M₁₆	—	2	4	no	—
109	C₃×M₁₆	—	2	4	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	8	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
114	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
115	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
116	C₂³×C₅	—	2	4	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
119	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
127	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₃×#41	1	2	yes	—
129	C₅×C₈	—	2	2	no	—
130	C₅×C₈	—	4	4	no	—
131	C₅×C₈	C₃×C₂²⋊C₄	1	4	yes	—
132	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
133	C₂×M₁₆	C₂²×C₃×C₅	1	4	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₃×C₄×C₅	1	4	yes	—
136	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
138	#108	—	2	4	no	—
139	C₂²⋊C₈	—	2	4	no	—
140	C₂×C₃×C₅	—	2	4	no	—
141	C₂×C₃×C₅	—	2	2	no	—
142	C₂×C₃×C₅	#380	1	1	yes	characteristic
143	C₂×C₃×C₅	C₂×#108	1	2	yes	—
144	C₂³×C₃	—	2	2	no	—
145	C₂³×C₃	—	2	4	no	—
146	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
147	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
153	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×#41	1	2	yes	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	4	4	no	—
159	C₃×C₈	C₅×C₂²⋊C₄	1	4	yes	—
160	C₂²×C₅	—	2	8	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	C₃×#108	1	4	yes	complemented
165	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₄×C₅	C₃×#117	1	1	yes	characteristic
170	C₄×C₅	C₃×#121	1	2	yes	—
171	C₂⁴	C₃×C₅×Q₈	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
175	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
176	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
178	M₁₆	—	2	4	no	—
179	M₁₆	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
184	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
185	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
186	C₃×C₅	#1810	1	1	yes	characteristic, direct product factor, Hall
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	C₅×#108	1	4	yes	complemented
192	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
193	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₃×C₄	C₅×#117	1	1	yes	characteristic
197	C₃×C₄	C₅×#121	1	2	yes	—
198	C₂×C₅	—	2	4	no	—
199	C₂×C₅	—	2	2	no	—
200	C₂×C₅	C₃×#380	1	1	yes	characteristic
201	C₂×C₅	C₂×C₃×#108	1	2	yes	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	4	no	—
204	C₂³	C₃×C₅×C₄⋊C₄	1	2	yes	—
205	C₂³	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
211	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
212	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
214	C₂×C₄	C₃×C₅×#41	1	2	yes	—
215	C₈	—	2	2	no	—
216	C₈	—	4	4	no	—
217	C₈	C₃×C₅×C₂²⋊C₄	1	4	yes	—
218	C₂×C₃	—	2	2	no	—
219	C₂×C₃	—	2	4	no	—
220	C₂×C₃	C₅×#380	1	1	yes	characteristic
221	C₂×C₃	C₂×C₅×#108	1	2	yes	—
222	C₅	C₃×#1810	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	2	8	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₃×C₅×#108	1	4	yes	complemented
228	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
229	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
230	C₄	—	2	2	no	—
231	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₄	C₃×C₅×#117	1	1	yes	characteristic
233	C₄	C₃×C₅×#121	1	2	yes	—
234	C₃	C₅×#1810	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	2	2	no	—
236	C₂	—	2	4	no	—
237	C₂	C₃×C₅×#380	1	1	yes	characteristic
238	C₂	C₂×C₃×C₅×#108	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃×C₅	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#108	C₂	semidirect
4. C₃×C₅×#341	C₂	semidirect
5. C₅×#1810	C₃	direct (N × H)
14. C₃×#1810	C₅	direct (N × H)
16. C₂×C₅×#108	C₂×C₃	semidirect
18. C₅×#341	C₂×C₃	semidirect
35. C₂×C₃×#108	C₂×C₅	semidirect
37. C₃×#341	C₂×C₅	semidirect
46. #1810	C₃×C₅	direct (N × H)
87. C₂×#108	C₂×C₃×C₅	semidirect
89. #341	C₂×C₃×C₅	semidirect
94. C₂²×C₃×C₅	#108	semidirect
164. C₂²×C₅	C₃×#108	semidirect
186. C₃×C₅	#1810	direct (N × H)
191. C₂²×C₃	C₅×#108	semidirect
222. C₅	C₃×#1810	direct (N × H)
227. C₂²	C₃×C₅×#108	semidirect
234. C₃	C₅×#1810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₈	2
C₂×C₃×C₅×#108	2
C₃×C₅×#407	2
C₃×C₅×#341	2
C₅×#1810	3
C₃×#1810	5

Subgroups — C3×C5×#1810 #14274844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1810 #14274844