Subgroups — C₃×C₅×#2211 #14275245

Order 1920.

316 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#495	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#357	C₂	1	1	yes	maximal, characteristic
8	C₅×#2211	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
12	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#124	—	2	1	no	—
14	C₃×C₄×C₅×Q₈	—	2	1	no	—
15	C₃×C₅×Q₃₂	—	2	2	no	—
16	C₃×C₅×SD₃₂	—	2	2	no	—
17	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×#107	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	—	2	1	no	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×#2211	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₅×SD₃₂	C₂×C₃	1	1	yes	characteristic
25	C₅×#495	C₂×C₃	1	1	yes	characteristic
26	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#358	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#357	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	1	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₃×C₅×Q₁₆	—	2	1	no	—
34	C₃×C₅×Q₁₆	—	2	1	no	—
35	C₃×C₅×Q₁₆	—	2	1	no	—
36	C₃×C₅×Q₁₆	D₄	1	2	yes	—
37	C₃×C₅×SD₁₆	—	4	2	no	—
38	C₃×C₅×D₈	—	2	1	no	—
39	C₃×C₅×D₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	4	1	no	—
44	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
45	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
46	C₃×C₄²×C₅	—	4	1	no	—
47	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₃×C₅×C₁₆	—	2	1	no	—
49	C₃×C₅×C₁₆	—	2	1	no	—
50	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×C₃×SD₃₂	C₂×C₅	1	1	yes	characteristic
52	C₃×#495	C₂×C₅	1	1	yes	characteristic
53	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#358	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#357	C₂×C₅	1	1	yes	characteristic
57	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₅×SD₁₆	—	2	1	no	—
60	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
61	C₅×#124	—	2	1	no	—
62	C₄×C₅×Q₈	—	2	1	no	—
63	C₅×Q₃₂	—	2	2	no	—
64	C₅×SD₃₂	—	2	2	no	—
65	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₅×#107	C₂²×C₃	1	1	yes	characteristic
68	C₅×#103	—	2	1	no	—
69	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
70	#2211	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
71	C₂³×C₃×C₅	—	4	1	no	—
72	C₃×C₅×Q₈	—	2	2	no	—
73	C₃×C₅×Q₈	—	4	1	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×D₄	—	4	1	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₃×C₅×C₈	#41	1	1	yes	characteristic
87	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₃×SD₁₆	—	2	1	no	—
90	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
91	C₃×#124	—	2	1	no	—
92	C₃×C₄×Q₈	—	2	1	no	—
93	C₃×Q₃₂	—	2	2	no	—
94	C₃×SD₃₂	—	2	2	no	—
95	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
97	C₃×#107	C₂²×C₅	1	1	yes	characteristic
98	C₃×#103	—	2	1	no	—
99	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₅×Q₈	—	2	1	no	—
101	C₂×C₅×Q₈	—	2	1	no	—
102	C₂×C₅×D₄	—	2	1	no	—
103	C₅×Q₁₆	—	2	1	no	—
104	C₅×Q₁₆	—	2	1	no	—
105	C₅×Q₁₆	—	2	1	no	—
106	C₅×Q₁₆	C₃×D₄	1	2	yes	—
107	C₅×SD₁₆	—	4	2	no	—
108	C₅×D₈	—	2	1	no	—
109	C₅×D₈	—	2	1	no	—
110	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	4	1	no	—
114	C₅×C₂²⋊C₄	—	4	1	no	—
115	C₅×C₂²⋊C₄	—	4	1	no	—
116	C₄²×C₅	—	4	1	no	—
117	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
118	C₅×C₁₆	—	2	1	no	—
119	C₅×C₁₆	—	2	1	no	—
120	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
121	C₂×SD₃₂	C₂×C₃×C₅	1	1	yes	characteristic
122	#495	C₂×C₃×C₅	1	1	yes	characteristic
123	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
124	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
125	#358	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	#357	C₂×C₃×C₅	1	1	yes	characteristic
127	C₂²×C₃×C₅	—	8	1	no	—
128	C₂²×C₃×C₅	—	8	1	no	—
129	C₂²×C₃×C₅	—	8	1	no	—
130	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	4	2	no	—
134	C₃×C₄×C₅	—	8	1	no	—
135	C₃×C₄×C₅	—	8	1	no	—
136	C₃×C₄×C₅	#121	1	1	yes	characteristic
137	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₃×Q₈	—	2	1	no	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×D₄	—	2	1	no	—
141	C₃×Q₁₆	—	2	1	no	—
142	C₃×Q₁₆	—	2	1	no	—
143	C₃×Q₁₆	—	2	1	no	—
144	C₃×Q₁₆	C₅×D₄	1	2	yes	—
145	C₃×SD₁₆	—	4	2	no	—
146	C₃×D₈	—	2	1	no	—
147	C₃×D₈	—	2	1	no	—
148	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
149	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	4	1	no	—
152	C₃×C₂²⋊C₄	—	4	1	no	—
153	C₃×C₂²⋊C₄	—	4	1	no	—
154	C₃×C₄²	—	4	1	no	—
155	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
156	C₃×C₁₆	—	2	1	no	—
157	C₃×C₁₆	—	2	1	no	—
158	C₂³×C₅	—	4	1	no	—
159	C₅×Q₈	—	2	2	no	—
160	C₅×Q₈	—	4	1	no	—
161	C₅×Q₈	—	4	1	no	—
162	C₅×Q₈	—	4	1	no	—
163	C₅×D₄	—	4	1	no	—
164	C₅×D₄	—	4	1	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
173	C₅×C₈	C₃×#41	1	1	yes	characteristic
174	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₂×SD₁₆	—	2	1	no	—
177	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
178	#124	—	2	1	no	—
179	C₄×Q₈	—	2	1	no	—
180	Q₃₂	—	2	2	no	—
181	SD₃₂	—	2	2	no	—
182	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
183	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
184	#107	C₂²×C₃×C₅	1	1	yes	characteristic
185	#103	—	2	1	no	—
186	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
187	C₂×C₃×C₅	—	16	1	no	—
188	C₂×C₃×C₅	#459	1	1	yes	characteristic
189	C₂×C₃×C₅	#508	1	1	yes	characteristic
190	C₂×C₃×C₅	#510	1	1	yes	characteristic
191	C₂³×C₃	—	4	1	no	—
192	C₃×Q₈	—	2	2	no	—
193	C₃×Q₈	—	4	1	no	—
194	C₃×Q₈	—	4	1	no	—
195	C₃×Q₈	—	4	1	no	—
196	C₃×D₄	—	4	1	no	—
197	C₃×D₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	—	4	1	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
202	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
204	C₃×C₈	—	2	1	no	—
205	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
206	C₃×C₈	C₅×#41	1	1	yes	characteristic
207	C₂²×C₅	—	8	1	no	—
208	C₂²×C₅	—	8	1	no	—
209	C₂²×C₅	—	8	1	no	—
210	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
211	C₄×C₅	—	2	1	no	—
212	C₄×C₅	—	2	1	no	—
213	C₄×C₅	—	4	2	no	—
214	C₄×C₅	—	8	1	no	—
215	C₄×C₅	—	8	1	no	—
216	C₄×C₅	C₃×#121	1	1	yes	characteristic
217	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	Q₁₆	—	2	1	no	—
222	Q₁₆	—	2	1	no	—
223	Q₁₆	—	2	1	no	—
224	Q₁₆	C₃×C₅×D₄	1	2	yes	—
225	SD₁₆	—	4	2	no	—
226	D₈	—	2	1	no	—
227	D₈	—	2	1	no	—
228	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₂²⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
236	C₁₆	—	2	1	no	—
237	C₁₆	—	2	1	no	—
238	C₃×C₅	#2211	1	1	yes	characteristic, direct product factor, Hall
239	C₂²×C₃	—	8	1	no	—
240	C₂²×C₃	—	8	1	no	—
241	C₂²×C₃	—	8	1	no	—
242	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	8	1	no	—
247	C₃×C₄	—	8	1	no	—
248	C₃×C₄	C₅×#121	1	1	yes	characteristic
249	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂×C₅	—	16	1	no	—
251	C₂×C₅	C₃×#459	1	1	yes	characteristic
252	C₂×C₅	C₃×#508	1	1	yes	characteristic
253	C₂×C₅	C₃×#510	1	1	yes	characteristic
254	C₂³	—	4	1	no	—
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	1	no	—
259	D₄	—	4	1	no	—
260	D₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
265	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
269	C₈	C₃×C₅×#41	1	1	yes	characteristic
270	C₂×C₃	—	16	1	no	—
271	C₂×C₃	C₅×#459	1	1	yes	characteristic
272	C₂×C₃	C₅×#508	1	1	yes	characteristic
273	C₂×C₃	C₅×#510	1	1	yes	characteristic
274	C₅	C₃×#2211	1	1	yes	characteristic, direct product factor, Sylow
275	C₂²	—	8	1	no	—
276	C₂²	—	8	1	no	—
277	C₂²	—	8	1	no	—
278	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	2	no	—
282	C₄	—	8	1	no	—
283	C₄	—	8	1	no	—
284	C₄	C₃×C₅×#121	1	1	yes	characteristic
285	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₃	C₅×#2211	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	16	1	no	—
288	C₂	C₃×C₅×#459	1	1	yes	characteristic
289	C₂	C₃×C₅×#508	1	1	yes	characteristic
290	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. C₂×C₃×C₅×Q₃₂	C₂	semidirect
4. C₃×C₄×C₅×Q₁₆	C₂	semidirect
5. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₃×C₅×#358	C₂	semidirect
8. C₅×#2211	C₃	direct (N × H)
22. C₃×#2211	C₅	direct (N × H)
23. C₂×C₅×Q₃₂	C₂×C₃	semidirect
26. C₄×C₅×Q₁₆	C₂×C₃	semidirect
27. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
28. C₅×#358	C₂×C₃	semidirect
50. C₂×C₃×Q₃₂	C₂×C₅	semidirect
53. C₃×C₄×Q₁₆	C₂×C₅	semidirect
54. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
55. C₃×#358	C₂×C₅	semidirect
70. #2211	C₃×C₅	direct (N × H)
120. C₂×Q₃₂	C₂×C₃×C₅	semidirect
123. C₄×Q₁₆	C₂×C₃×C₅	semidirect
124. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
125. #358	C₂×C₃×C₅	semidirect
238. C₃×C₅	#2211	direct (N × H)
274. C₅	C₃×#2211	direct (N × H)
286. C₃	C₅×#2211	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×Q₃₂	2
C₂×C₃×C₅×SD₃₂	2
C₃×C₅×#495	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#358	2
C₃×C₅×#357	2
C₅×#2211	3
C₃×#2211	5

Subgroups — C3×C5×#2211 #14275245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2211 #14275245