Subgroups — C₃×C₅×#3253 #14276287

Order 1920.

860 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#476	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#469	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#459	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3253	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	2	no	—
8	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₅×#127	C₂²	1	2	yes	—
12	C₃×C₅×#124	—	2	2	no	—
13	C₃×C₅×#121	—	2	2	no	—
14	C₃×C₅×#121	C₂²	1	4	yes	—
15	C₃×C₄×C₅×D₄	—	2	2	no	—
16	C₃×C₄×C₅×D₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×#106	C₂²	1	4	yes	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
20	C₃×C₅×#102	C₂²	1	8	yes	—
21	C₃×#3253	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	C₂×C₃	1	2	yes	complemented
23	C₅×#476	C₂×C₃	1	4	yes	complemented
24	C₅×#469	C₂×C₃	1	4	yes	complemented
25	C₅×#459	C₂×C₃	1	4	yes	complemented
26	C₅×#423	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×C₅×#41	—	4	8	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	—	2	4	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₅×D₈	—	4	4	no	—
37	C₃×C₅×M₁₆	—	4	2	no	—
38	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	2	yes	—
46	C₃×#532	C₂×C₅	1	2	yes	complemented
47	C₃×#476	C₂×C₅	1	4	yes	complemented
48	C₃×#469	C₂×C₅	1	4	yes	complemented
49	C₃×#459	C₂×C₅	1	4	yes	complemented
50	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×C₅×#41	—	2	2	no	—
52	C₂×C₅×D₈	C₂²×C₃	1	4	yes	—
53	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
54	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
55	C₅×#127	C₂²×C₃	1	2	yes	—
56	C₅×#124	—	2	2	no	—
57	C₅×#121	—	2	2	no	—
58	C₅×#121	C₂²×C₃	1	4	yes	—
59	C₄×C₅×D₄	—	2	2	no	—
60	C₄×C₅×D₄	C₂²×C₃	1	4	yes	—
61	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
62	C₅×#106	C₂²×C₃	1	4	yes	—
63	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
64	C₅×#102	C₂²×C₃	1	8	yes	—
65	#3253	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₃×C₅	—	2	4	no	—
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₃×C₅×D₄	—	2	4	no	—
70	C₃×C₅×D₄	—	2	4	no	—
71	C₃×C₅×D₄	—	4	4	no	—
72	C₃×C₅×D₄	—	4	8	no	—
73	C₃×C₅×D₄	—	4	4	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	4	8	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₃×C₅×C₈	—	2	4	no	—
84	C₂×C₃×#41	—	2	2	no	—
85	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
86	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
87	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
88	C₃×#127	C₂²×C₅	1	2	yes	—
89	C₃×#124	—	2	2	no	—
90	C₃×#121	—	2	2	no	—
91	C₃×#121	C₂²×C₅	1	4	yes	—
92	C₃×C₄×D₄	—	2	2	no	—
93	C₃×C₄×D₄	C₂²×C₅	1	4	yes	—
94	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₃×#106	C₂²×C₅	1	4	yes	—
96	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
97	C₃×#102	C₂²×C₅	1	8	yes	—
98	C₅×#41	—	4	8	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×D₄	—	2	4	no	—
101	C₂×C₅×D₄	—	2	2	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
104	C₂²×C₄×C₅	—	2	4	no	—
105	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₅×D₈	—	4	4	no	—
108	C₅×M₁₆	—	4	2	no	—
109	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
112	C₅×C₂²⋊C₄	—	2	4	no	—
113	C₅×C₂²⋊C₄	—	2	4	no	—
114	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
115	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂³×C₃	1	2	yes	—
117	#532	C₂×C₃×C₅	1	2	yes	complemented
118	#476	C₂×C₃×C₅	1	4	yes	complemented
119	#469	C₂×C₃×C₅	1	4	yes	complemented
120	#459	C₂×C₃×C₅	1	4	yes	complemented
121	#423	C₂×C₃×C₅	1	1	yes	characteristic, complemented
122	C₂²×C₃×C₅	—	2	1	no	—
123	C₂²×C₃×C₅	—	2	2	no	—
124	C₂²×C₃×C₅	—	4	4	no	—
125	C₂²×C₃×C₅	—	4	4	no	—
126	C₂²×C₃×C₅	—	4	4	no	—
127	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₃×C₄×C₅	—	2	1	no	—
129	C₃×C₄×C₅	—	2	2	no	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
132	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
133	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
134	C₃×#41	—	4	8	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×D₄	—	2	4	no	—
137	C₂×C₃×D₄	—	2	2	no	—
138	C₂×C₃×D₄	—	2	2	no	—
139	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
140	C₂²×C₃×C₄	—	2	4	no	—
141	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
143	C₃×D₈	—	4	4	no	—
144	C₃×M₁₆	—	4	2	no	—
145	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
148	C₃×C₂²⋊C₄	—	2	4	no	—
149	C₃×C₂²⋊C₄	—	2	4	no	—
150	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
151	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₂³×C₅	1	2	yes	—
153	C₂³×C₅	—	2	4	no	—
154	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₅×Q₈	—	4	4	no	—
156	C₅×D₄	—	2	4	no	—
157	C₅×D₄	—	2	4	no	—
158	C₅×D₄	—	4	4	no	—
159	C₅×D₄	—	4	8	no	—
160	C₅×D₄	—	4	4	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	4	8	no	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
168	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
169	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
170	C₅×C₈	—	2	4	no	—
171	C₂×#41	—	2	2	no	—
172	C₂×D₈	C₂²×C₃×C₅	1	4	yes	—
173	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
174	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
175	#127	C₂²×C₃×C₅	1	2	yes	—
176	#124	—	2	2	no	—
177	#121	—	2	2	no	—
178	#121	C₂²×C₃×C₅	1	4	yes	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₂²×C₃×C₅	1	4	yes	—
181	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
182	#106	C₂²×C₃×C₅	1	4	yes	—
183	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
184	#102	C₂²×C₃×C₅	1	8	yes	—
185	C₂×C₃×C₅	—	4	1	no	—
186	C₂×C₃×C₅	—	8	4	no	—
187	C₂×C₃×C₅	#537	1	1	yes	characteristic
188	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
189	C₂³×C₃	—	2	4	no	—
190	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
191	C₃×Q₈	—	4	4	no	—
192	C₃×D₄	—	2	4	no	—
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	—	4	4	no	—
195	C₃×D₄	—	4	8	no	—
196	C₃×D₄	—	4	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	4	8	no	—
202	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
206	C₃×C₈	—	2	4	no	—
207	C₂²×C₅	—	2	2	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	4	4	no	—
210	C₂²×C₅	—	4	4	no	—
211	C₂²×C₅	—	4	4	no	—
212	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	4	4	no	—
216	C₄×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
217	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
218	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
219	#41	—	4	8	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
227	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
228	D₈	—	4	4	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
236	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
237	C₄²	C₂³×C₃×C₅	1	2	yes	—
238	C₃×C₅	#3253	1	1	yes	characteristic, direct product factor, Hall
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	4	4	no	—
242	C₂²×C₃	—	4	4	no	—
243	C₂²×C₃	—	4	4	no	—
244	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
249	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
250	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×C₅	—	4	1	no	—
252	C₂×C₅	—	8	4	no	—
253	C₂×C₅	C₃×#537	1	1	yes	characteristic
254	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
255	C₂³	—	2	4	no	—
256	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	8	no	—
262	D₄	—	4	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
270	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
271	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₂×C₃	—	4	1	no	—
274	C₂×C₃	—	8	4	no	—
275	C₂×C₃	C₅×#537	1	1	yes	characteristic
276	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
277	C₅	C₃×#3253	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	4	no	—
287	C₄	C₃×C₅×AGL(1,ℤ/8)	1	4	yes	—
288	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
289	C₄	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
290	C₃	C₅×#3253	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	4	1	no	—
292	C₂	—	8	4	no	—
293	C₂	C₃×C₅×#537	1	1	yes	characteristic
294	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
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
1. C₃×C₅×#532	C₂	semidirect
2. C₃×C₅×#476	C₂	semidirect
3. C₃×C₅×#469	C₂	semidirect
4. C₃×C₅×#459	C₂	semidirect
5. C₃×C₅×#423	C₂	semidirect
6. C₅×#3253	C₃	direct (N × H)
21. C₃×#3253	C₅	direct (N × H)
22. C₅×#532	C₂×C₃	semidirect
23. C₅×#476	C₂×C₃	semidirect
24. C₅×#469	C₂×C₃	semidirect
25. C₅×#459	C₂×C₃	semidirect
26. C₅×#423	C₂×C₃	semidirect
46. C₃×#532	C₂×C₅	semidirect
47. C₃×#476	C₂×C₅	semidirect
48. C₃×#469	C₂×C₅	semidirect
49. C₃×#459	C₂×C₅	semidirect
50. C₃×#423	C₂×C₅	semidirect
65. #3253	C₃×C₅	direct (N × H)
117. #532	C₂×C₃×C₅	semidirect
118. #476	C₂×C₃×C₅	semidirect
119. #469	C₂×C₃×C₅	semidirect
120. #459	C₂×C₃×C₅	semidirect
121. #423	C₂×C₃×C₅	semidirect
238. C₃×C₅	#3253	direct (N × H)
277. C₅	C₃×#3253	direct (N × H)
290. C₃	C₅×#3253	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×#532	2
C₃×C₅×#476	2
C₃×C₅×#469	2
C₃×C₅×#459	2
C₃×C₅×#423	2
C₅×#3253	3
C₃×#3253	5

Subgroups — C3×C5×#3253 #14276287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3253 #14276287