Subgroups — C₃×C₅×#1812 #14274846

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#427	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#410	C₂	1	2	yes	maximal
4	C₃×C₅×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
5	C₅×#1812	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
7	C₃×C₅×#131	—	2	2	no	—
8	C₃×C₅×#131	—	2	2	no	—
9	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₄	1	2	yes	complemented
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#117	C₂²	1	4	yes	—
14	C₃×C₅×#107	—	2	2	no	—
15	C₃×C₅×#106	—	2	2	no	—
16	C₃×C₅×#99	—	2	4	no	—
17	C₃×#1812	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#131	C₂×C₃	1	1	yes	characteristic
19	C₅×#427	C₂×C₃	1	2	yes	complemented
20	C₅×#410	C₂×C₃	1	2	yes	—
21	C₅×H(3,ℤ/4)	C₂×C₃	1	2	yes	complemented
22	C₃×C₅×#41	—	2	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
26	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₃×C₅×M₁₆	—	2	4	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
40	C₃×C₄²×C₅	—	2	4	no	—
41	C₂×C₃×#131	C₂×C₅	1	1	yes	characteristic
42	C₃×#427	C₂×C₅	1	2	yes	complemented
43	C₃×#410	C₂×C₅	1	2	yes	—
44	C₃×H(3,ℤ/4)	C₂×C₅	1	2	yes	complemented
45	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
46	C₅×#131	—	2	2	no	—
47	C₅×#131	—	2	2	no	—
48	C₂×C₅×M₁₆	C₃×C₄	1	2	yes	complemented
49	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₃×C₄	1	2	yes	complemented
51	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₅×#117	C₂²×C₃	1	4	yes	—
53	C₅×#107	—	2	2	no	—
54	C₅×#106	—	2	2	no	—
55	C₅×#99	—	2	4	no	—
56	#1812	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×Q₈	—	2	2	no	—
60	C₃×C₅×D₄	—	2	4	no	—
61	C₃×C₅×D₄	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	4	4	no	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	2	yes	—
70	C₃×C₅×C₈	—	2	2	no	—
71	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
75	C₃×#131	—	2	2	no	—
76	C₃×#131	—	2	2	no	—
77	C₂×C₃×M₁₆	C₄×C₅	1	2	yes	complemented
78	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₃×C₈	C₄×C₅	1	2	yes	complemented
80	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₃×#117	C₂²×C₅	1	4	yes	—
82	C₃×#107	—	2	2	no	—
83	C₃×#106	—	2	2	no	—
84	C₃×#99	—	2	4	no	—
85	C₅×#41	—	2	2	no	—
86	C₅×#41	—	2	2	no	—
87	C₂×C₅×Q₈	C₃×Q₈	1	1	yes	characteristic
88	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
89	C₂×C₅×D₄	C₃×Q₈	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	4	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
100	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₂²⋊C₄	—	2	4	no	—
103	C₄²×C₅	—	2	4	no	—
104	C₂×#131	C₂×C₃×C₅	1	1	yes	characteristic
105	#427	C₂×C₃×C₅	1	2	yes	complemented
106	#410	C₂×C₃×C₅	1	2	yes	—
107	H(3,ℤ/4)	C₂×C₃×C₅	1	2	yes	complemented
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	2	no	—
111	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	#121	1	2	yes	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	—	8	4	no	—
115	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₃×C₄×C₅	#117	1	1	yes	characteristic
117	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
118	C₃×#41	—	2	2	no	—
119	C₃×#41	—	2	2	no	—
120	C₂×C₃×Q₈	C₅×Q₈	1	1	yes	characteristic
121	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
122	C₂×C₃×D₄	C₅×Q₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
124	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
125	C₃×M₁₆	—	2	2	no	—
126	C₃×M₁₆	—	2	4	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
133	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₂²⋊C₄	—	2	4	no	—
136	C₃×C₄²	—	2	4	no	—
137	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
138	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₅×Q₈	—	2	2	no	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₃×#41	1	2	yes	—
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
152	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
154	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	C₂×M₁₆	C₃×C₄×C₅	1	2	yes	complemented
158	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₂²×C₈	C₃×C₄×C₅	1	2	yes	complemented
160	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
161	#117	C₂²×C₃×C₅	1	4	yes	—
162	#107	—	2	2	no	—
163	#106	—	2	2	no	—
164	#99	—	2	4	no	—
165	C₂×C₃×C₅	—	2	2	no	—
166	C₂×C₃×C₅	—	2	1	no	—
167	C₂×C₃×C₅	—	4	2	no	—
168	C₂×C₃×C₅	#380	1	1	yes	characteristic
169	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
170	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
171	C₃×Q₈	—	2	2	no	—
172	C₃×D₄	—	2	4	no	—
173	C₃×D₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
181	C₂×C₃×C₄	C₅×#41	1	2	yes	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
184	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	—	4	2	no	—
189	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	C₃×#121	1	2	yes	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	8	4	no	—
193	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₄×C₅	C₃×#117	1	1	yes	characteristic
195	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	C₃×C₅×Q₈	1	1	yes	characteristic
199	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
200	C₂×D₄	C₃×C₅×Q₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
202	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
208	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
209	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
210	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
211	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
212	C₄⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₄²	—	2	4	no	—
215	C₃×C₅	#1812	1	1	yes	characteristic, direct product factor, Hall
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
220	C₂²×C₃	C₅×#121	1	2	yes	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	8	4	no	—
223	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₃×C₄	C₅×#117	1	1	yes	characteristic
225	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
226	C₂×C₅	—	2	2	no	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	—	4	2	no	—
229	C₂×C₅	C₃×#380	1	1	yes	characteristic
230	C₂³	C₃×C₅×C₄⋊C₄	1	2	yes	—
231	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
239	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
240	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
241	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
242	C₂×C₄	C₃×C₅×#41	1	2	yes	—
243	C₈	—	2	2	no	—
244	C₈	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₈	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	C₅×#380	1	1	yes	characteristic
251	C₅	C₃×#1812	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	2	no	—
255	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
256	C₂²	C₃×C₅×#121	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	8	4	no	—
259	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₄	C₃×C₅×#117	1	1	yes	characteristic
261	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
262	C₃	C₅×#1812	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	2	2	no	—
264	C₂	—	2	1	no	—
265	C₂	—	4	2	no	—
266	C₂	C₃×C₅×#380	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#380
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂×C₃×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#427	C₂	semidirect
4. C₃×C₅×H(3,ℤ/4)	C₂	semidirect
5. C₅×#1812	C₃	direct (N × H)
9. C₂×C₃×C₅×M₁₆	C₄	semidirect
11. C₂²×C₃×C₅×C₈	C₄	semidirect
17. C₃×#1812	C₅	direct (N × H)
19. C₅×#427	C₂×C₃	semidirect
21. C₅×H(3,ℤ/4)	C₂×C₃	semidirect
42. C₃×#427	C₂×C₅	semidirect
44. C₃×H(3,ℤ/4)	C₂×C₅	semidirect
48. C₂×C₅×M₁₆	C₃×C₄	semidirect
50. C₂²×C₅×C₈	C₃×C₄	semidirect
56. #1812	C₃×C₅	direct (N × H)
77. C₂×C₃×M₁₆	C₄×C₅	semidirect
79. C₂²×C₃×C₈	C₄×C₅	semidirect
105. #427	C₂×C₃×C₅	semidirect
107. H(3,ℤ/4)	C₂×C₃×C₅	semidirect
157. C₂×M₁₆	C₃×C₄×C₅	semidirect
159. C₂²×C₈	C₃×C₄×C₅	semidirect
215. C₃×C₅	#1812	direct (N × H)
251. C₅	C₃×#1812	direct (N × H)
262. C₃	C₅×#1812	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×#131	2
C₃×C₅×#427	2
C₃×C₅×#410	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#1812	3
C₃×#1812	5

Subgroups — C3×C5×#1812 #14274846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1812 #14274846