Subgroups — C₃×C₅×#3028 #14276062

Order 1920.

1432 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#585	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#536	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#458	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#410	C₂	1	3	yes	maximal, complemented
5	C₅×#3028	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₅×2₋¹⁺⁴	—	2	3	no	—
7	C₃×C₅×H(5,𝔽₂)	—	2	3	no	—
8	C₃×C₅×H(5,𝔽₂)	C₂²	1	4	yes	—
9	C₂×C₃×C₅×#41	—	2	6	no	—
10	C₂×C₃×C₅×#41	C₂²	1	3	yes	—
11	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₃×C₅×C₄⋊Q₈	—	2	3	no	—
13	C₃×C₅×#124	—	2	3	no	—
14	C₃×C₅×#123	C₂²	1	12	yes	complemented
15	C₃×C₅×C₂²⋊Q₈	—	2	6	no	—
16	C₃×C₅×#117	C₂²	1	3	yes	complemented
17	C₃×C₅×#99	C₂²	1	12	yes	complemented
18	C₃×#3028	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#585	C₂×C₃	1	1	yes	characteristic
20	C₅×#536	C₂×C₃	1	3	yes	complemented
21	C₅×#458	C₂×C₃	1	8	yes	complemented
22	C₅×#410	C₂×C₃	1	3	yes	complemented
23	C₃×C₅×#41	—	2	4	no	—
24	C₃×C₅×#41	—	2	12	no	—
25	C₃×C₅×#41	—	2	6	no	—
26	C₃×C₅×#41	—	2	12	no	—
27	C₃×C₅×#41	—	2	6	no	—
28	C₂×C₃×C₅×Q₈	—	2	6	no	—
29	C₂×C₃×C₅×Q₈	—	2	3	no	—
30	C₂×C₃×C₅×Q₈	—	2	3	no	—
31	C₂×C₃×C₅×Q₈	D₄	1	3	yes	—
32	C₂×C₃×C₅×D₄	—	2	6	no	—
33	C₂×C₃×C₅×D₄	—	2	12	no	—
34	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
35	C₂×C₃×C₅×D₄	C₂³	1	6	yes	—
36	C₂²×C₃×C₄×C₅	—	2	4	no	—
37	C₂²×C₃×C₄×C₅	D₄	1	6	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	—	2	3	no	—
41	C₃×C₅×C₄⋊C₄	—	2	12	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
43	C₃×C₅×C₂²⋊C₄	C₂³	1	6	yes	—
44	C₃×C₄²×C₅	—	2	3	no	—
45	C₃×#585	C₂×C₅	1	1	yes	characteristic
46	C₃×#536	C₂×C₅	1	3	yes	complemented
47	C₃×#458	C₂×C₅	1	8	yes	complemented
48	C₃×#410	C₂×C₅	1	3	yes	complemented
49	C₅×2₋¹⁺⁴	—	2	3	no	—
50	C₅×H(5,𝔽₂)	—	2	3	no	—
51	C₅×H(5,𝔽₂)	C₂²×C₃	1	4	yes	—
52	C₂×C₅×#41	—	2	6	no	—
53	C₂×C₅×#41	C₂²×C₃	1	3	yes	—
54	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
55	C₅×C₄⋊Q₈	—	2	3	no	—
56	C₅×#124	—	2	3	no	—
57	C₅×#123	C₂²×C₃	1	12	yes	complemented
58	C₅×C₂²⋊Q₈	—	2	6	no	—
59	C₅×#117	C₂²×C₃	1	3	yes	complemented
60	C₅×#99	C₂²×C₃	1	12	yes	complemented
61	#3028	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	—	2	4	no	—
63	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
64	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	2	4	no	—
66	C₃×C₅×Q₈	—	2	6	no	—
67	C₃×C₅×Q₈	—	4	1	no	—
68	C₃×C₅×Q₈	—	4	3	no	—
69	C₃×C₅×D₄	—	2	12	no	—
70	C₃×C₅×D₄	—	2	12	no	—
71	C₃×C₅×D₄	—	2	6	no	—
72	C₂×C₃×C₄×C₅	—	2	12	no	—
73	C₂×C₃×C₄×C₅	—	2	6	no	—
74	C₂×C₃×C₄×C₅	—	2	6	no	—
75	C₂×C₃×C₄×C₅	—	2	6	no	—
76	C₂×C₃×C₄×C₅	—	2	6	no	—
77	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
80	C₃×2₋¹⁺⁴	—	2	3	no	—
81	C₃×H(5,𝔽₂)	—	2	3	no	—
82	C₃×H(5,𝔽₂)	C₂²×C₅	1	4	yes	—
83	C₂×C₃×#41	—	2	6	no	—
84	C₂×C₃×#41	C₂²×C₅	1	3	yes	—
85	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
86	C₃×C₄⋊Q₈	—	2	3	no	—
87	C₃×#124	—	2	3	no	—
88	C₃×#123	C₂²×C₅	1	12	yes	complemented
89	C₃×C₂²⋊Q₈	—	2	6	no	—
90	C₃×#117	C₂²×C₅	1	3	yes	complemented
91	C₃×#99	C₂²×C₅	1	12	yes	complemented
92	C₅×#41	—	2	4	no	—
93	C₅×#41	—	2	12	no	—
94	C₅×#41	—	2	6	no	—
95	C₅×#41	—	2	12	no	—
96	C₅×#41	—	2	6	no	—
97	C₂×C₅×Q₈	—	2	6	no	—
98	C₂×C₅×Q₈	—	2	3	no	—
99	C₂×C₅×Q₈	—	2	3	no	—
100	C₂×C₅×Q₈	C₃×D₄	1	3	yes	—
101	C₂×C₅×D₄	—	2	6	no	—
102	C₂×C₅×D₄	—	2	12	no	—
103	C₂×C₅×D₄	C₃×D₄	1	3	yes	—
104	C₂×C₅×D₄	C₂³×C₃	1	6	yes	—
105	C₂²×C₄×C₅	—	2	4	no	—
106	C₂²×C₄×C₅	C₃×D₄	1	6	yes	—
107	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₅×C₄⋊C₄	—	2	3	no	—
110	C₅×C₄⋊C₄	—	2	12	no	—
111	C₅×C₂²⋊C₄	—	2	12	no	—
112	C₅×C₂²⋊C₄	C₂³×C₃	1	6	yes	—
113	C₄²×C₅	—	2	3	no	—
114	#585	C₂×C₃×C₅	1	1	yes	characteristic
115	#536	C₂×C₃×C₅	1	3	yes	complemented
116	#458	C₂×C₃×C₅	1	8	yes	complemented
117	#410	C₂×C₃×C₅	1	3	yes	complemented
118	C₂²×C₃×C₅	—	2	6	no	—
119	C₂²×C₃×C₅	—	4	1	no	—
120	C₂²×C₃×C₅	—	4	6	no	—
121	C₂²×C₃×C₅	—	8	4	no	—
122	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
123	C₃×C₄×C₅	—	2	6	no	—
124	C₃×C₄×C₅	—	4	6	no	—
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	#120	1	1	yes	characteristic
127	C₃×C₄×C₅	#120	1	3	yes	—
128	C₃×#41	—	2	4	no	—
129	C₃×#41	—	2	12	no	—
130	C₃×#41	—	2	6	no	—
131	C₃×#41	—	2	12	no	—
132	C₃×#41	—	2	6	no	—
133	C₂×C₃×Q₈	—	2	6	no	—
134	C₂×C₃×Q₈	—	2	3	no	—
135	C₂×C₃×Q₈	—	2	3	no	—
136	C₂×C₃×Q₈	C₅×D₄	1	3	yes	—
137	C₂×C₃×D₄	—	2	6	no	—
138	C₂×C₃×D₄	—	2	12	no	—
139	C₂×C₃×D₄	C₅×D₄	1	3	yes	—
140	C₂×C₃×D₄	C₂³×C₅	1	6	yes	—
141	C₂²×C₃×C₄	—	2	4	no	—
142	C₂²×C₃×C₄	C₅×D₄	1	6	yes	—
143	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	2	3	no	—
146	C₃×C₄⋊C₄	—	2	12	no	—
147	C₃×C₂²⋊C₄	—	2	12	no	—
148	C₃×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
149	C₃×C₄²	—	2	3	no	—
150	C₂³×C₅	—	2	4	no	—
151	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
152	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
153	C₅×Q₈	—	2	4	no	—
154	C₅×Q₈	—	2	6	no	—
155	C₅×Q₈	—	4	1	no	—
156	C₅×Q₈	—	4	3	no	—
157	C₅×D₄	—	2	12	no	—
158	C₅×D₄	—	2	12	no	—
159	C₅×D₄	—	2	6	no	—
160	C₂×C₄×C₅	—	2	12	no	—
161	C₂×C₄×C₅	—	2	6	no	—
162	C₂×C₄×C₅	—	2	6	no	—
163	C₂×C₄×C₅	—	2	6	no	—
164	C₂×C₄×C₅	—	2	6	no	—
165	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
167	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
168	2₋¹⁺⁴	—	2	3	no	—
169	H(5,𝔽₂)	—	2	3	no	—
170	H(5,𝔽₂)	C₂²×C₃×C₅	1	4	yes	—
171	C₂×#41	—	2	6	no	—
172	C₂×#41	C₂²×C₃×C₅	1	3	yes	—
173	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
174	C₄⋊Q₈	—	2	3	no	—
175	#124	—	2	3	no	—
176	#123	C₂²×C₃×C₅	1	12	yes	complemented
177	C₂²⋊Q₈	—	2	6	no	—
178	#117	C₂²×C₃×C₅	1	3	yes	complemented
179	#99	C₂²×C₃×C₅	1	12	yes	complemented
180	C₂×C₃×C₅	—	2	3	no	—
181	C₂×C₃×C₅	—	4	6	no	—
182	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
183	C₂³×C₃	—	2	4	no	—
184	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
185	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
186	C₃×Q₈	—	2	4	no	—
187	C₃×Q₈	—	2	6	no	—
188	C₃×Q₈	—	4	1	no	—
189	C₃×Q₈	—	4	3	no	—
190	C₃×D₄	—	2	12	no	—
191	C₃×D₄	—	2	12	no	—
192	C₃×D₄	—	2	6	no	—
193	C₂×C₃×C₄	—	2	12	no	—
194	C₂×C₃×C₄	—	2	6	no	—
195	C₂×C₃×C₄	—	2	6	no	—
196	C₂×C₃×C₄	—	2	6	no	—
197	C₂×C₃×C₄	—	2	6	no	—
198	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
200	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
201	C₂²×C₅	—	2	6	no	—
202	C₂²×C₅	—	4	1	no	—
203	C₂²×C₅	—	4	6	no	—
204	C₂²×C₅	—	8	4	no	—
205	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
206	C₄×C₅	—	2	6	no	—
207	C₄×C₅	—	4	6	no	—
208	C₄×C₅	—	4	2	no	—
209	C₄×C₅	C₃×#120	1	1	yes	characteristic
210	C₄×C₅	C₃×#120	1	3	yes	—
211	#41	—	2	4	no	—
212	#41	—	2	12	no	—
213	#41	—	2	6	no	—
214	#41	—	2	12	no	—
215	#41	—	2	6	no	—
216	C₂×Q₈	—	2	6	no	—
217	C₂×Q₈	—	2	3	no	—
218	C₂×Q₈	—	2	3	no	—
219	C₂×Q₈	C₃×C₅×D₄	1	3	yes	—
220	C₂×D₄	—	2	6	no	—
221	C₂×D₄	—	2	12	no	—
222	C₂×D₄	C₃×C₅×D₄	1	3	yes	—
223	C₂×D₄	C₂³×C₃×C₅	1	6	yes	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₃×C₅×D₄	1	6	yes	—
226	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
227	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
228	C₄⋊C₄	—	2	3	no	—
229	C₄⋊C₄	—	2	12	no	—
230	C₂²⋊C₄	—	2	12	no	—
231	C₂²⋊C₄	C₂³×C₃×C₅	1	6	yes	—
232	C₄²	—	2	3	no	—
233	C₃×C₅	#3028	1	1	yes	characteristic, direct product factor, Hall
234	C₂²×C₃	—	2	6	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	6	no	—
237	C₂²×C₃	—	8	4	no	—
238	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
239	C₃×C₄	—	2	6	no	—
240	C₃×C₄	—	4	6	no	—
241	C₃×C₄	—	4	2	no	—
242	C₃×C₄	C₅×#120	1	1	yes	characteristic
243	C₃×C₄	C₅×#120	1	3	yes	—
244	C₂×C₅	—	2	3	no	—
245	C₂×C₅	—	4	6	no	—
246	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
247	C₂³	—	2	4	no	—
248	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
249	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	6	no	—
252	Q₈	—	4	1	no	—
253	Q₈	—	4	3	no	—
254	D₄	—	2	12	no	—
255	D₄	—	2	12	no	—
256	D₄	—	2	6	no	—
257	C₂×C₄	—	2	12	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	6	no	—
262	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
263	C₂×C₄	C₂×C₃×C₅×D₄	1	3	yes	—
264	C₂×C₄	C₂×C₃×C₅×D₄	1	3	yes	—
265	C₂×C₃	—	2	3	no	—
266	C₂×C₃	—	4	6	no	—
267	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
268	C₅	C₃×#3028	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	—	2	6	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	6	no	—
272	C₂²	—	8	4	no	—
273	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
274	C₄	—	2	6	no	—
275	C₄	—	4	6	no	—
276	C₄	—	4	2	no	—
277	C₄	C₃×C₅×#120	1	1	yes	characteristic
278	C₄	C₃×C₅×#120	1	3	yes	—
279	C₃	C₅×#3028	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	2	3	no	—
281	C₂	—	4	6	no	—
282	C₂	C₂×C₃×C₅×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₃×C₅×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#536	C₂	semidirect
3. C₃×C₅×#458	C₂	semidirect
4. C₃×C₅×#410	C₂	semidirect
5. C₅×#3028	C₃	direct (N × H)
11. C₂²×C₃×C₅×Q₈	C₂²	semidirect
14. C₃×C₅×#123	C₂²	semidirect
16. C₃×C₅×#117	C₂²	semidirect
17. C₃×C₅×#99	C₂²	semidirect
18. C₃×#3028	C₅	direct (N × H)
20. C₅×#536	C₂×C₃	semidirect
21. C₅×#458	C₂×C₃	semidirect
22. C₅×#410	C₂×C₃	semidirect
46. C₃×#536	C₂×C₅	semidirect
47. C₃×#458	C₂×C₅	semidirect
48. C₃×#410	C₂×C₅	semidirect
54. C₂²×C₅×Q₈	C₂²×C₃	semidirect
57. C₅×#123	C₂²×C₃	semidirect
59. C₅×#117	C₂²×C₃	semidirect
60. C₅×#99	C₂²×C₃	semidirect
61. #3028	C₃×C₅	direct (N × H)
85. C₂²×C₃×Q₈	C₂²×C₅	semidirect
88. C₃×#123	C₂²×C₅	semidirect
90. C₃×#117	C₂²×C₅	semidirect
91. C₃×#99	C₂²×C₅	semidirect
115. #536	C₂×C₃×C₅	semidirect
116. #458	C₂×C₃×C₅	semidirect
117. #410	C₂×C₃×C₅	semidirect
173. C₂²×Q₈	C₂²×C₃×C₅	semidirect
176. #123	C₂²×C₃×C₅	semidirect
178. #117	C₂²×C₃×C₅	semidirect
179. #99	C₂²×C₃×C₅	semidirect
233. C₃×C₅	#3028	direct (N × H)
268. C₅	C₃×#3028	direct (N × H)
279. C₃	C₅×#3028	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×#585	2
C₃×C₅×#536	2
C₃×C₅×#458	2
C₃×C₅×#410	2
C₅×#3028	3
C₃×#3028	5

Subgroups — C3×C5×#3028 #14276062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3028 #14276062