Subgroups — C₃×C₅×#3249 #14276283

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	2	yes	maximal
2	C₃×C₅×#477	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#467	C₂	1	4	yes	maximal
4	C₃×C₅×#462	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₅×#3249	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₅×#125	—	2	2	no	—
12	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
14	C₃×C₄×C₅×Q₈	—	2	2	no	—
15	C₃×C₄×C₅×Q₈	C₂²	1	4	yes	—
16	C₃×C₅×#117	—	2	2	no	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×#107	C₂²	1	4	yes	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
20	C₃×C₅×#103	C₂²	1	8	yes	—
21	C₃×#3249	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#533	C₂×C₃	1	2	yes	—
23	C₅×#477	C₂×C₃	1	4	yes	complemented
24	C₅×#467	C₂×C₃	1	4	yes	—
25	C₅×#462	C₂×C₃	1	4	yes	complemented
26	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₃×C₅×Q₁₆	—	4	4	no	—
33	C₂×C₃×C₅×C₈	—	4	2	no	—
34	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	2	4	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₄²×C₅	—	2	4	no	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₃×#533	C₂×C₅	1	2	yes	—
46	C₃×#477	C₂×C₅	1	4	yes	complemented
47	C₃×#467	C₂×C₅	1	4	yes	—
48	C₃×#462	C₂×C₅	1	4	yes	complemented
49	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
51	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
52	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
53	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	C₅×#125	—	2	2	no	—
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
57	C₄×C₅×Q₈	—	2	2	no	—
58	C₄×C₅×Q₈	C₂²×C₃	1	4	yes	—
59	C₅×#117	—	2	2	no	—
60	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
61	C₅×#107	C₂²×C₃	1	4	yes	—
62	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
63	C₅×#103	C₂²×C₃	1	8	yes	—
64	#3249	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
66	C₃×C₅×Q₈	—	2	4	no	—
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	complemented
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₃×C₅×C₈	—	2	4	no	—
80	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
81	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
82	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
83	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
84	C₃×#125	—	2	2	no	—
85	C₃×C₂²⋊Q₈	—	2	2	no	—
86	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
87	C₃×C₄×Q₈	—	2	2	no	—
88	C₃×C₄×Q₈	C₂²×C₅	1	4	yes	—
89	C₃×#117	—	2	2	no	—
90	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
91	C₃×#107	C₂²×C₅	1	4	yes	—
92	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
93	C₃×#103	C₂²×C₅	1	8	yes	—
94	C₂×C₅×Q₈	—	2	2	no	—
95	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
96	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₅×Q₁₆	—	4	4	no	—
100	C₂×C₅×C₈	—	4	2	no	—
101	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
108	C₅×C₂²⋊C₄	—	2	4	no	—
109	C₄²×C₅	—	2	4	no	—
110	C₄²×C₅	C₃×D₄	1	2	yes	—
111	C₄²×C₅	C₂³×C₃	1	2	yes	—
112	#533	C₂×C₃×C₅	1	2	yes	—
113	#477	C₂×C₃×C₅	1	4	yes	complemented
114	#467	C₂×C₃×C₅	1	4	yes	—
115	#462	C₂×C₃×C₅	1	4	yes	complemented
116	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₅	—	2	1	no	—
118	C₂²×C₃×C₅	—	2	1	no	—
119	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
120	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₃×C₄×C₅	—	2	1	no	—
122	C₃×C₄×C₅	—	2	2	no	—
123	C₃×C₄×C₅	—	4	4	no	—
124	C₃×C₄×C₅	—	4	4	no	—
125	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
126	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
127	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×C₃×Q₈	—	2	2	no	—
129	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
130	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₃×Q₁₆	—	4	4	no	—
134	C₂×C₃×C₈	—	4	2	no	—
135	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	4	no	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
142	C₃×C₂²⋊C₄	—	2	4	no	—
143	C₃×C₄²	—	2	4	no	—
144	C₃×C₄²	C₅×D₄	1	2	yes	—
145	C₃×C₄²	C₂³×C₅	1	2	yes	—
146	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₅×Q₈	—	2	4	no	—
148	C₅×Q₈	—	2	4	no	—
149	C₅×Q₈	—	4	4	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	complemented
156	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
159	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
160	C₅×C₈	—	2	4	no	—
161	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	—
162	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
163	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
164	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
165	#125	—	2	2	no	—
166	C₂²⋊Q₈	—	2	2	no	—
167	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	C₂²×C₃×C₅	1	4	yes	—
170	#117	—	2	2	no	—
171	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
172	#107	C₂²×C₃×C₅	1	4	yes	—
173	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
174	#103	C₂²×C₃×C₅	1	8	yes	—
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	#537	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
178	C₂×C₃×C₅	#575	1	1	yes	characteristic
179	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
180	C₃×Q₈	—	2	4	no	—
181	C₃×Q₈	—	2	4	no	—
182	C₃×Q₈	—	4	4	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	complemented
189	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
192	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
193	C₃×C₈	—	2	4	no	—
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	C₂×C₃×Q₁₆	1	2	yes	—
197	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	—	4	4	no	—
202	C₄×C₅	C₂×C₃×Q₁₆	1	4	yes	—
203	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
204	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
207	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
210	Q₁₆	—	4	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	C₃×C₅×D₄	1	2	yes	—
222	C₄²	C₂³×C₃×C₅	1	2	yes	—
223	C₃×C₅	#3249	1	1	yes	characteristic, direct product factor, Hall
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
227	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	4	4	no	—
231	C₃×C₄	—	4	4	no	—
232	C₃×C₄	C₂×C₅×Q₁₆	1	4	yes	—
233	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
234	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂×C₅	—	2	2	no	—
236	C₂×C₅	C₃×#537	1	1	yes	characteristic
237	C₂×C₅	C₂²×C₃×Q₁₆	1	1	yes	characteristic
238	C₂×C₅	C₃×#575	1	1	yes	characteristic
239	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	2	4	no	—
242	Q₈	—	4	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₃×C₅×Q₁₆	1	4	yes	complemented
249	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
251	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
252	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	C₅×#537	1	1	yes	characteristic
256	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
257	C₂×C₃	C₅×#575	1	1	yes	characteristic
258	C₅	C₃×#3249	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	C₂×C₃×C₅×Q₁₆	1	2	yes	—
262	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	4	no	—
266	C₄	—	4	4	no	—
267	C₄	C₂×C₃×C₅×Q₁₆	1	4	yes	—
268	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
269	C₄	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
270	C₃	C₅×#3249	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	2	2	no	—
272	C₂	C₃×C₅×#537	1	1	yes	characteristic
273	C₂	C₂²×C₃×C₅×Q₁₆	1	1	yes	characteristic
274	C₂	C₃×C₅×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#477	C₂	semidirect
4. C₃×C₅×#462	C₂	semidirect
6. C₅×#3249	C₃	direct (N × H)
21. C₃×#3249	C₅	direct (N × H)
23. C₅×#477	C₂×C₃	semidirect
25. C₅×#462	C₂×C₃	semidirect
46. C₃×#477	C₂×C₅	semidirect
48. C₃×#462	C₂×C₅	semidirect
64. #3249	C₃×C₅	direct (N × H)
74. C₂×C₃×C₄×C₅	Q₁₆	semidirect
113. #477	C₂×C₃×C₅	semidirect
115. #462	C₂×C₃×C₅	semidirect
155. C₂×C₄×C₅	C₃×Q₁₆	semidirect
188. C₂×C₃×C₄	C₅×Q₁₆	semidirect
223. C₃×C₅	#3249	direct (N × H)
248. C₂×C₄	C₃×C₅×Q₁₆	semidirect
258. C₅	C₃×#3249	direct (N × H)
270. C₃	C₅×#3249	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×#533	2
C₃×C₅×#477	2
C₃×C₅×#467	2
C₃×C₅×#462	2
C₂×C₃×C₅×C₄⋊C₈	2
C₅×#3249	3
C₃×#3249	5

Subgroups — C3×C5×#3249 #14276283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3249 #14276283