Subgroups — C₃×C₅×#2675 #14275709

Order 1920.

1608 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×#121	C₂	1	6	yes	maximal, complemented
2	C₃×C₅×#396	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#392	C₂	1	3	yes	maximal, complemented
4	C₃×C₅×#392	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#385	C₂	1	3	yes	maximal, complemented
6	C₅×#2675	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
8	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
11	C₃×C₅×#121	—	2	24	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₃×C₅×#95	C₂²	1	3	yes	—
17	C₃×C₅×#95	C₂²	1	1	yes	characteristic
18	C₃×#2675	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#121	C₂×C₃	1	6	yes	complemented
20	C₅×#396	C₂×C₃	1	2	yes	complemented
21	C₅×#392	C₂×C₃	1	3	yes	complemented
22	C₅×#392	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#385	C₂×C₃	1	3	yes	complemented
24	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
25	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
26	C₂×C₃×C₅×D₄	—	2	12	no	—
27	C₂×C₃×C₅×D₄	—	2	12	no	—
28	C₂×C₃×C₅×D₄	—	2	12	no	—
29	C₂×C₃×C₅×D₄	—	4	6	no	—
30	C₂×C₃×C₅×D₄	—	4	3	no	—
31	C₂×C₃×C₅×D₄	—	4	3	no	—
32	C₂²×C₃×C₄×C₅	—	2	12	no	—
33	C₂²×C₃×C₄×C₅	—	4	3	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	12	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
39	C₃×C₅×C₂²⋊C₄	—	4	3	no	—
40	C₃×C₅×C₂²⋊C₄	D₄	1	12	yes	complemented
41	C₂×C₃×#121	C₂×C₅	1	6	yes	complemented
42	C₃×#396	C₂×C₅	1	2	yes	complemented
43	C₃×#392	C₂×C₅	1	3	yes	complemented
44	C₃×#392	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×#385	C₂×C₅	1	3	yes	complemented
46	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
47	C₂²×C₅×D₄	C₂²×C₃	1	6	yes	—
48	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
49	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
50	C₅×#121	—	2	24	no	—
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
52	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
53	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
54	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
55	C₅×#95	C₂²×C₃	1	3	yes	—
56	C₅×#95	C₂²×C₃	1	1	yes	characteristic
57	#2675	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	—	2	6	no	—
59	C₂³×C₃×C₅	—	2	3	no	—
60	C₂³×C₃×C₅	—	2	6	no	—
61	C₂³×C₃×C₅	—	2	3	no	—
62	C₂³×C₃×C₅	—	2	6	no	—
63	C₂³×C₃×C₅	—	2	6	no	—
64	C₂³×C₃×C₅	—	2	2	no	—
65	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
66	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
67	C₃×C₅×D₄	—	4	24	no	—
68	C₃×C₅×D₄	—	4	12	no	—
69	C₃×C₅×D₄	—	4	4	no	—
70	C₂×C₃×C₄×C₅	—	2	6	no	—
71	C₂×C₃×C₄×C₅	—	2	6	no	—
72	C₂×C₃×C₄×C₅	—	2	6	no	—
73	C₂×C₃×C₄×C₅	—	2	3	no	—
74	C₂×C₃×C₄×C₅	—	2	3	no	—
75	C₂×C₃×C₄×C₅	—	4	12	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
77	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
78	C₂²×C₃×D₄	C₂²×C₅	1	6	yes	—
79	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
80	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
81	C₃×#121	—	2	24	no	—
82	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
83	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
84	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
85	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
86	C₃×#95	C₂²×C₅	1	3	yes	—
87	C₃×#95	C₂²×C₅	1	1	yes	characteristic
88	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
89	C₂⁴×C₅	C₂³×C₃	1	3	yes	—
90	C₂×C₅×D₄	—	2	12	no	—
91	C₂×C₅×D₄	—	2	12	no	—
92	C₂×C₅×D₄	—	2	12	no	—
93	C₂×C₅×D₄	—	4	6	no	—
94	C₂×C₅×D₄	—	4	3	no	—
95	C₂×C₅×D₄	—	4	3	no	—
96	C₂²×C₄×C₅	—	2	12	no	—
97	C₂²×C₄×C₅	—	4	3	no	—
98	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
99	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
100	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
101	C₅×C₄⋊C₄	—	2	12	no	—
102	C₅×C₂²⋊C₄	—	2	12	no	—
103	C₅×C₂²⋊C₄	—	4	3	no	—
104	C₅×C₂²⋊C₄	C₃×D₄	1	12	yes	complemented
105	C₂×#121	C₂×C₃×C₅	1	6	yes	complemented
106	#396	C₂×C₃×C₅	1	2	yes	complemented
107	#392	C₂×C₃×C₅	1	3	yes	complemented
108	#392	C₂×C₃×C₅	1	1	yes	characteristic, complemented
109	#385	C₂×C₃×C₅	1	3	yes	complemented
110	C₂²×C₃×C₅	—	2	6	no	—
111	C₂²×C₃×C₅	—	2	12	no	—
112	C₂²×C₃×C₅	—	4	6	no	—
113	C₂²×C₃×C₅	—	4	3	no	—
114	C₂²×C₃×C₅	—	4	3	no	—
115	C₂²×C₃×C₅	—	4	6	no	—
116	C₂²×C₃×C₅	—	4	6	no	—
117	C₂²×C₃×C₅	—	4	2	no	—
118	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
119	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
120	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
121	C₃×C₄×C₅	—	2	12	no	—
122	C₃×C₄×C₅	—	4	3	no	—
123	C₃×C₄×C₅	—	4	1	no	—
124	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
125	C₂⁴×C₃	C₂³×C₅	1	3	yes	—
126	C₂×C₃×D₄	—	2	12	no	—
127	C₂×C₃×D₄	—	2	12	no	—
128	C₂×C₃×D₄	—	2	12	no	—
129	C₂×C₃×D₄	—	4	6	no	—
130	C₂×C₃×D₄	—	4	3	no	—
131	C₂×C₃×D₄	—	4	3	no	—
132	C₂²×C₃×C₄	—	2	12	no	—
133	C₂²×C₃×C₄	—	4	3	no	—
134	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
136	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
137	C₃×C₄⋊C₄	—	2	12	no	—
138	C₃×C₂²⋊C₄	—	2	12	no	—
139	C₃×C₂²⋊C₄	—	4	3	no	—
140	C₃×C₂²⋊C₄	C₅×D₄	1	12	yes	complemented
141	C₂³×C₅	—	2	6	no	—
142	C₂³×C₅	—	2	3	no	—
143	C₂³×C₅	—	2	6	no	—
144	C₂³×C₅	—	2	3	no	—
145	C₂³×C₅	—	2	6	no	—
146	C₂³×C₅	—	2	6	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
149	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
150	C₅×D₄	—	4	24	no	—
151	C₅×D₄	—	4	12	no	—
152	C₅×D₄	—	4	4	no	—
153	C₂×C₄×C₅	—	2	6	no	—
154	C₂×C₄×C₅	—	2	6	no	—
155	C₂×C₄×C₅	—	2	6	no	—
156	C₂×C₄×C₅	—	2	3	no	—
157	C₂×C₄×C₅	—	2	3	no	—
158	C₂×C₄×C₅	—	4	12	no	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
160	C₂²×D₄	C₂²×C₃×C₅	1	3	yes	—
161	C₂²×D₄	C₂²×C₃×C₅	1	6	yes	—
162	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
163	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
164	#121	—	2	24	no	—
165	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
169	#95	C₂²×C₃×C₅	1	3	yes	—
170	#95	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₂×C₃×C₅	—	4	6	no	—
172	C₂×C₃×C₅	—	8	2	no	—
173	C₂×C₃×C₅	#537	1	3	yes	—
174	C₂×C₃×C₅	D₄²	1	3	yes	—
175	C₂×C₃×C₅	#560	1	1	yes	characteristic
176	C₂³×C₃	—	2	6	no	—
177	C₂³×C₃	—	2	3	no	—
178	C₂³×C₃	—	2	6	no	—
179	C₂³×C₃	—	2	3	no	—
180	C₂³×C₃	—	2	6	no	—
181	C₂³×C₃	—	2	6	no	—
182	C₂³×C₃	—	2	2	no	—
183	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
184	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
185	C₃×D₄	—	4	24	no	—
186	C₃×D₄	—	4	12	no	—
187	C₃×D₄	—	4	4	no	—
188	C₂×C₃×C₄	—	2	6	no	—
189	C₂×C₃×C₄	—	2	6	no	—
190	C₂×C₃×C₄	—	2	6	no	—
191	C₂×C₃×C₄	—	2	3	no	—
192	C₂×C₃×C₄	—	2	3	no	—
193	C₂×C₃×C₄	—	4	12	no	—
194	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
195	C₂²×C₅	—	2	6	no	—
196	C₂²×C₅	—	2	12	no	—
197	C₂²×C₅	—	4	3	no	—
198	C₂²×C₅	—	4	6	no	—
199	C₂²×C₅	—	4	3	no	—
200	C₂²×C₅	—	4	2	no	—
201	C₂²×C₅	—	4	6	no	—
202	C₂²×C₅	—	4	6	no	—
203	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
204	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
205	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
206	C₄×C₅	—	2	12	no	—
207	C₄×C₅	—	4	1	no	—
208	C₄×C₅	—	4	3	no	—
209	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
210	C₂⁴	C₂³×C₃×C₅	1	3	yes	—
211	C₂×D₄	—	2	12	no	—
212	C₂×D₄	—	2	12	no	—
213	C₂×D₄	—	2	12	no	—
214	C₂×D₄	—	4	6	no	—
215	C₂×D₄	—	4	3	no	—
216	C₂×D₄	—	4	3	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	4	3	no	—
219	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
221	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
222	C₄⋊C₄	—	2	12	no	—
223	C₂²⋊C₄	—	2	12	no	—
224	C₂²⋊C₄	—	4	3	no	—
225	C₂²⋊C₄	C₃×C₅×D₄	1	12	yes	complemented
226	C₃×C₅	#2675	1	1	yes	characteristic, direct product factor, Hall
227	C₂²×C₃	—	2	6	no	—
228	C₂²×C₃	—	2	12	no	—
229	C₂²×C₃	—	4	6	no	—
230	C₂²×C₃	—	4	3	no	—
231	C₂²×C₃	—	4	3	no	—
232	C₂²×C₃	—	4	6	no	—
233	C₂²×C₃	—	4	6	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
236	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
237	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
238	C₃×C₄	—	2	12	no	—
239	C₃×C₄	—	4	3	no	—
240	C₃×C₄	—	4	1	no	—
241	C₂×C₅	—	4	6	no	—
242	C₂×C₅	—	8	2	no	—
243	C₂×C₅	C₃×#537	1	3	yes	—
244	C₂×C₅	C₃×D₄²	1	3	yes	—
245	C₂×C₅	C₃×#560	1	1	yes	characteristic
246	C₂³	—	2	6	no	—
247	C₂³	—	2	3	no	—
248	C₂³	—	2	6	no	—
249	C₂³	—	2	3	no	—
250	C₂³	—	2	6	no	—
251	C₂³	—	2	6	no	—
252	C₂³	—	2	2	no	—
253	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
254	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
255	D₄	—	4	24	no	—
256	D₄	—	4	12	no	—
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	3	no	—
262	C₂×C₄	—	2	3	no	—
263	C₂×C₄	—	4	12	no	—
264	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
265	C₂×C₃	—	4	6	no	—
266	C₂×C₃	—	8	2	no	—
267	C₂×C₃	C₅×#537	1	3	yes	—
268	C₂×C₃	C₅×D₄²	1	3	yes	—
269	C₂×C₃	C₅×#560	1	1	yes	characteristic
270	C₅	C₃×#2675	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	6	no	—
272	C₂²	—	2	12	no	—
273	C₂²	—	4	6	no	—
274	C₂²	—	4	3	no	—
275	C₂²	—	4	3	no	—
276	C₂²	—	4	6	no	—
277	C₂²	—	4	6	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
280	C₂²	C₃×C₅×H(5,𝔽₂)	1	3	yes	—
281	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
282	C₄	—	2	12	no	—
283	C₄	—	4	3	no	—
284	C₄	—	4	1	no	—
285	C₃	C₅×#2675	1	1	yes	characteristic, direct product factor, Sylow
286	C₂	—	4	6	no	—
287	C₂	—	8	2	no	—
288	C₂	C₃×C₅×#537	1	3	yes	—
289	C₂	C₃×C₅×D₄²	1	3	yes	—
290	C₂	C₃×C₅×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#121	C₂	semidirect
2. C₃×C₅×#396	C₂	semidirect
3. C₃×C₅×#392	C₂	semidirect
4. C₃×C₅×#392	C₂	semidirect
5. C₃×C₅×#385	C₂	semidirect
6. C₅×#2675	C₃	direct (N × H)
18. C₃×#2675	C₅	direct (N × H)
19. C₂×C₅×#121	C₂×C₃	semidirect
20. C₅×#396	C₂×C₃	semidirect
21. C₅×#392	C₂×C₃	semidirect
22. C₅×#392	C₂×C₃	semidirect
23. C₅×#385	C₂×C₃	semidirect
40. C₃×C₅×C₂²⋊C₄	D₄	semidirect
41. C₂×C₃×#121	C₂×C₅	semidirect
42. C₃×#396	C₂×C₅	semidirect
43. C₃×#392	C₂×C₅	semidirect
44. C₃×#392	C₂×C₅	semidirect
45. C₃×#385	C₂×C₅	semidirect
57. #2675	C₃×C₅	direct (N × H)
104. C₅×C₂²⋊C₄	C₃×D₄	semidirect
105. C₂×#121	C₂×C₃×C₅	semidirect
106. #396	C₂×C₃×C₅	semidirect
107. #392	C₂×C₃×C₅	semidirect
108. #392	C₂×C₃×C₅	semidirect
109. #385	C₂×C₃×C₅	semidirect
140. C₃×C₂²⋊C₄	C₅×D₄	semidirect
225. C₂²⋊C₄	C₃×C₅×D₄	semidirect
226. C₃×C₅	#2675	direct (N × H)
270. C₅	C₃×#2675	direct (N × H)
285. C₃	C₅×#2675	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×#121	2
C₃×C₅×#396	2
C₃×C₅×#392	2
C₃×C₅×#385	2
C₅×#2675	3
C₃×#2675	5

Subgroups — C3×C5×#2675 #14275709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2675 #14275709