Subgroups — C₃×C₅×#1714 #14274748

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#486	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1714	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
10	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×#127	C₂²	1	1	yes	characteristic
15	C₃×C₅×#107	—	2	2	no	—
16	C₃×C₅×#106	—	2	2	no	—
17	C₃×C₅×#103	—	2	2	no	—
18	C₃×C₅×#102	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×#1714	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×#492	C₂×C₃	1	1	yes	characteristic
26	C₅×#492	C₂×C₃	1	1	yes	characteristic
27	C₅×#487	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#486	C₂×C₃	1	1	yes	characteristic
29	C₅×#322	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₃×C₅×SD₁₆	—	4	4	no	—
37	C₃×C₅×SD₁₆	—	4	4	no	—
38	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
50	C₃×#492	C₂×C₅	1	1	yes	characteristic
51	C₃×#492	C₂×C₅	1	1	yes	characteristic
52	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×#486	C₂×C₅	1	1	yes	characteristic
54	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×C₅×SD₁₆	—	2	2	no	—
56	C₂×C₅×SD₁₆	—	2	2	no	—
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₅×#127	C₂²×C₃	1	1	yes	characteristic
61	C₅×#107	—	2	2	no	—
62	C₅×#106	—	2	2	no	—
63	C₅×#103	—	2	2	no	—
64	C₅×#102	—	2	2	no	—
65	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
67	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
68	#1714	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
69	C₂³×C₃×C₅	—	4	1	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₃×C₅×D₄	—	4	2	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	—	2	2	no	—
84	C₃×C₅×C₈	SD₁₆	1	4	yes	complemented
85	C₂×C₃×SD₁₆	—	2	2	no	—
86	C₂×C₃×SD₁₆	—	2	2	no	—
87	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
89	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
90	C₃×#127	C₂²×C₅	1	1	yes	characteristic
91	C₃×#107	—	2	2	no	—
92	C₃×#106	—	2	2	no	—
93	C₃×#103	—	2	2	no	—
94	C₃×#102	—	2	2	no	—
95	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	1	no	—
100	C₂×C₅×Q₈	—	2	1	no	—
101	C₂×C₅×D₄	—	2	1	no	—
102	C₂×C₅×D₄	—	2	1	no	—
103	C₂×C₅×D₄	—	2	1	no	—
104	C₅×SD₁₆	—	4	4	no	—
105	C₅×SD₁₆	—	4	4	no	—
106	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
107	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
108	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
109	C₅×C₄⋊C₄	—	2	1	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
116	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	#498	C₂×C₃×C₅	1	1	yes	characteristic, complemented
118	#492	C₂×C₃×C₅	1	1	yes	characteristic
119	#492	C₂×C₃×C₅	1	1	yes	characteristic
120	#487	C₂×C₃×C₅	1	1	yes	characteristic, complemented
121	#486	C₂×C₃×C₅	1	1	yes	characteristic
122	#322	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	C₂²×C₃×C₅	—	8	1	no	—
124	C₂²×C₃×C₅	—	8	1	no	—
125	C₂²×C₃×C₅	—	8	1	no	—
126	C₂²×C₃×C₅	#127	1	1	yes	characteristic
127	C₃×C₄×C₅	—	8	1	no	—
128	C₃×C₄×C₅	—	8	1	no	—
129	C₃×C₄×C₅	—	8	1	no	—
130	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
131	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
132	C₃×C₄×C₅	#137	1	2	yes	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×Q₈	—	2	1	no	—
135	C₂×C₃×Q₈	—	2	1	no	—
136	C₂×C₃×D₄	—	2	1	no	—
137	C₂×C₃×D₄	—	2	1	no	—
138	C₂×C₃×D₄	—	2	1	no	—
139	C₃×SD₁₆	—	4	4	no	—
140	C₃×SD₁₆	—	4	4	no	—
141	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
142	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
143	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
144	C₃×C₄⋊C₄	—	2	1	no	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
151	C₂³×C₅	—	4	1	no	—
152	C₅×Q₈	—	4	2	no	—
153	C₅×Q₈	—	4	2	no	—
154	C₅×Q₈	—	4	2	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	4	2	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	—	2	2	no	—
166	C₅×C₈	C₃×SD₁₆	1	4	yes	complemented
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
170	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
172	#127	C₂²×C₃×C₅	1	1	yes	characteristic
173	#107	—	2	2	no	—
174	#106	—	2	2	no	—
175	#103	—	2	2	no	—
176	#102	—	2	2	no	—
177	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
178	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
179	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
180	C₂×C₃×C₅	—	16	1	no	—
181	C₂×C₃×C₅	#492	1	1	yes	characteristic
182	C₂×C₃×C₅	#496	1	1	yes	characteristic
183	C₂×C₃×C₅	#497	1	1	yes	characteristic
184	C₂³×C₃	—	4	1	no	—
185	C₃×Q₈	—	4	2	no	—
186	C₃×Q₈	—	4	2	no	—
187	C₃×Q₈	—	4	2	no	—
188	C₃×D₄	—	4	2	no	—
189	C₃×D₄	—	4	2	no	—
190	C₃×D₄	—	4	2	no	—
191	C₂×C₃×C₄	—	4	1	no	—
192	C₂×C₃×C₄	—	4	1	no	—
193	C₂×C₃×C₄	—	4	1	no	—
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	C₅×SD₁₆	1	4	yes	complemented
200	C₂²×C₅	—	8	1	no	—
201	C₂²×C₅	—	8	1	no	—
202	C₂²×C₅	—	8	1	no	—
203	C₂²×C₅	C₃×#127	1	1	yes	characteristic
204	C₄×C₅	—	8	1	no	—
205	C₄×C₅	—	8	1	no	—
206	C₄×C₅	—	8	1	no	—
207	C₄×C₅	C₂×C₃×SD₁₆	1	2	yes	—
208	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
209	C₄×C₅	C₃×#137	1	2	yes	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	SD₁₆	—	4	4	no	—
217	SD₁₆	—	4	4	no	—
218	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
219	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
220	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
228	C₃×C₅	#1714	1	1	yes	characteristic, direct product factor, Hall
229	C₂²×C₃	—	8	1	no	—
230	C₂²×C₃	—	8	1	no	—
231	C₂²×C₃	—	8	1	no	—
232	C₂²×C₃	C₅×#127	1	1	yes	characteristic
233	C₃×C₄	—	8	1	no	—
234	C₃×C₄	—	8	1	no	—
235	C₃×C₄	—	8	1	no	—
236	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
237	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
238	C₃×C₄	C₅×#137	1	2	yes	—
239	C₂×C₅	—	16	1	no	—
240	C₂×C₅	C₃×#492	1	1	yes	characteristic
241	C₂×C₅	C₃×#496	1	1	yes	characteristic
242	C₂×C₅	C₃×#497	1	1	yes	characteristic
243	C₂³	—	4	1	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	C₃×C₅×SD₁₆	1	4	yes	complemented
259	C₂×C₃	—	16	1	no	—
260	C₂×C₃	C₅×#492	1	1	yes	characteristic
261	C₂×C₃	C₅×#496	1	1	yes	characteristic
262	C₂×C₃	C₅×#497	1	1	yes	characteristic
263	C₅	C₃×#1714	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	8	1	no	—
265	C₂²	—	8	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	C₃×C₅×#127	1	1	yes	characteristic
268	C₄	—	8	1	no	—
269	C₄	—	8	1	no	—
270	C₄	—	8	1	no	—
271	C₄	C₂×C₃×C₅×SD₁₆	1	2	yes	—
272	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
273	C₄	C₃×C₅×#137	1	2	yes	—
274	C₃	C₅×#1714	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	16	1	no	—
276	C₂	C₃×C₅×#492	1	1	yes	characteristic
277	C₂	C₃×C₅×#496	1	1	yes	characteristic
278	C₂	C₃×C₅×#497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#500	C₂	semidirect
2. C₃×C₅×#498	C₂	semidirect
5. C₃×C₅×#487	C₂	semidirect
7. C₃×C₅×#322	C₂	semidirect
8. C₅×#1714	C₃	direct (N × H)
22. C₃×#1714	C₅	direct (N × H)
23. C₅×#500	C₂×C₃	semidirect
24. C₅×#498	C₂×C₃	semidirect
27. C₅×#487	C₂×C₃	semidirect
29. C₅×#322	C₂×C₃	semidirect
48. C₃×#500	C₂×C₅	semidirect
49. C₃×#498	C₂×C₅	semidirect
52. C₃×#487	C₂×C₅	semidirect
54. C₃×#322	C₂×C₅	semidirect
68. #1714	C₃×C₅	direct (N × H)
84. C₃×C₅×C₈	SD₁₆	semidirect
116. #500	C₂×C₃×C₅	semidirect
117. #498	C₂×C₃×C₅	semidirect
120. #487	C₂×C₃×C₅	semidirect
122. #322	C₂×C₃×C₅	semidirect
166. C₅×C₈	C₃×SD₁₆	semidirect
199. C₃×C₈	C₅×SD₁₆	semidirect
228. C₃×C₅	#1714	direct (N × H)
258. C₈	C₃×C₅×SD₁₆	semidirect
263. C₅	C₃×#1714	direct (N × H)
274. C₃	C₅×#1714	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×#498	2
C₃×C₅×#492	2
C₃×C₅×#487	2
C₃×C₅×#486	2
C₃×C₅×#322	2
C₅×#1714	3
C₃×#1714	5

Subgroups — C3×C5×#1714 #14274748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1714 #14274748