Subgroups — C₃×C₅×#3027 #14276061

Order 1920.

1544 conjugacy classes of subgroups.

The subgroup lattice has 300 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, complemented
2	C₃×C₅×#535	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×B(4,𝔽₂)	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#410	C₂	1	3	yes	maximal, complemented
5	C₅×#3027	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×D₄	C₂²	1	1	yes	characteristic, complemented
12	C₃×C₅×#127	—	2	3	no	—
13	C₃×C₅×#124	—	2	3	no	—
14	C₃×C₅×#121	—	2	6	no	—
15	C₃×C₅×#120	C₂²	1	12	yes	complemented
16	C₃×C₅×#117	C₂²	1	3	yes	complemented
17	C₃×C₅×#99	C₂²	1	12	yes	complemented
18	C₃×#3027	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#585	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#535	C₂×C₃	1	3	yes	complemented
21	C₅×B(4,𝔽₂)	C₂×C₃	1	8	yes	complemented
22	C₅×#410	C₂×C₃	1	3	yes	complemented
23	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
24	C₃×C₅×#41	—	2	4	no	—
25	C₃×C₅×#41	—	2	12	no	—
26	C₃×C₅×#41	—	2	6	no	—
27	C₃×C₅×#41	—	2	12	no	—
28	C₃×C₅×#41	—	2	6	no	—
29	C₂×C₃×C₅×Q₈	—	2	6	no	—
30	C₂×C₃×C₅×Q₈	D₄	1	3	yes	complemented
31	C₂×C₃×C₅×D₄	—	2	6	no	—
32	C₂×C₃×C₅×D₄	—	2	12	no	—
33	C₂×C₃×C₅×D₄	—	2	12	no	—
34	C₂×C₃×C₅×D₄	—	2	3	no	—
35	C₂×C₃×C₅×D₄	—	2	3	no	—
36	C₂×C₃×C₅×D₄	D₄	1	3	yes	complemented
37	C₂×C₃×C₅×D₄	C₂³	1	6	yes	—
38	C₂²×C₃×C₄×C₅	—	2	4	no	—
39	C₂²×C₃×C₄×C₅	D₄	1	6	yes	complemented
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	3	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, complemented
46	C₃×#535	C₂×C₅	1	3	yes	complemented
47	C₃×B(4,𝔽₂)	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×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
55	C₅×#127	—	2	3	no	—
56	C₅×#124	—	2	3	no	—
57	C₅×#121	—	2	6	no	—
58	C₅×#120	C₂²×C₃	1	12	yes	complemented
59	C₅×#117	C₂²×C₃	1	3	yes	complemented
60	C₅×#99	C₂²×C₃	1	12	yes	complemented
61	#3027	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	—	2	4	no	—
63	C₂³×C₃×C₅	—	2	6	no	—
64	C₂³×C₃×C₅	—	8	2	no	—
65	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
66	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	2	6	no	—
69	C₃×C₅×D₄	—	2	12	no	—
70	C₃×C₅×D₄	—	2	12	no	—
71	C₃×C₅×D₄	—	2	6	no	—
72	C₃×C₅×D₄	—	4	1	no	—
73	C₃×C₅×D₄	—	4	3	no	—
74	C₃×C₅×D₄	—	8	12	no	—
75	C₂×C₃×C₄×C₅	—	2	12	no	—
76	C₂×C₃×C₄×C₅	—	2	6	no	—
77	C₂×C₃×C₄×C₅	—	2	6	no	—
78	C₂×C₃×C₄×C₅	—	2	6	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
82	C₃×2₋¹⁺⁴	—	2	3	no	—
83	C₃×H(5,𝔽₂)	—	2	3	no	—
84	C₃×H(5,𝔽₂)	C₂²×C₅	1	4	yes	—
85	C₂×C₃×#41	—	2	6	no	—
86	C₂×C₃×#41	C₂²×C₅	1	3	yes	—
87	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
88	C₃×#127	—	2	3	no	—
89	C₃×#124	—	2	3	no	—
90	C₃×#121	—	2	6	no	—
91	C₃×#120	C₂²×C₅	1	12	yes	complemented
92	C₃×#117	C₂²×C₅	1	3	yes	complemented
93	C₃×#99	C₂²×C₅	1	12	yes	complemented
94	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
95	C₅×#41	—	2	4	no	—
96	C₅×#41	—	2	12	no	—
97	C₅×#41	—	2	6	no	—
98	C₅×#41	—	2	12	no	—
99	C₅×#41	—	2	6	no	—
100	C₂×C₅×Q₈	—	2	6	no	—
101	C₂×C₅×Q₈	C₃×D₄	1	3	yes	complemented
102	C₂×C₅×D₄	—	2	6	no	—
103	C₂×C₅×D₄	—	2	12	no	—
104	C₂×C₅×D₄	—	2	12	no	—
105	C₂×C₅×D₄	—	2	3	no	—
106	C₂×C₅×D₄	—	2	3	no	—
107	C₂×C₅×D₄	C₃×D₄	1	3	yes	complemented
108	C₂×C₅×D₄	C₂³×C₃	1	6	yes	—
109	C₂²×C₄×C₅	—	2	4	no	—
110	C₂²×C₄×C₅	C₃×D₄	1	6	yes	complemented
111	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₅×C₄⋊C₄	—	2	3	no	—
113	C₅×C₂²⋊C₄	—	2	12	no	—
114	C₅×C₂²⋊C₄	C₂³×C₃	1	6	yes	—
115	C₄²×C₅	—	2	3	no	—
116	#585	C₂×C₃×C₅	1	1	yes	characteristic, complemented
117	#535	C₂×C₃×C₅	1	3	yes	complemented
118	B(4,𝔽₂)	C₂×C₃×C₅	1	8	yes	complemented
119	#410	C₂×C₃×C₅	1	3	yes	complemented
120	C₂²×C₃×C₅	—	2	6	no	—
121	C₂²×C₃×C₅	—	4	1	no	—
122	C₂²×C₃×C₅	—	4	6	no	—
123	C₂²×C₃×C₅	—	4	12	no	—
124	C₂²×C₃×C₅	—	4	2	no	—
125	C₂²×C₃×C₅	—	8	4	no	—
126	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
127	C₃×C₄×C₅	—	2	6	no	—
128	C₃×C₄×C₅	—	4	6	no	—
129	C₃×C₄×C₅	#120	1	1	yes	characteristic
130	C₃×C₄×C₅	#120	1	3	yes	—
131	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
132	C₃×#41	—	2	4	no	—
133	C₃×#41	—	2	12	no	—
134	C₃×#41	—	2	6	no	—
135	C₃×#41	—	2	12	no	—
136	C₃×#41	—	2	6	no	—
137	C₂×C₃×Q₈	—	2	6	no	—
138	C₂×C₃×Q₈	C₅×D₄	1	3	yes	complemented
139	C₂×C₃×D₄	—	2	6	no	—
140	C₂×C₃×D₄	—	2	12	no	—
141	C₂×C₃×D₄	—	2	12	no	—
142	C₂×C₃×D₄	—	2	3	no	—
143	C₂×C₃×D₄	—	2	3	no	—
144	C₂×C₃×D₄	C₅×D₄	1	3	yes	complemented
145	C₂×C₃×D₄	C₂³×C₅	1	6	yes	—
146	C₂²×C₃×C₄	—	2	4	no	—
147	C₂²×C₃×C₄	C₅×D₄	1	6	yes	complemented
148	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
149	C₃×C₄⋊C₄	—	2	3	no	—
150	C₃×C₂²⋊C₄	—	2	12	no	—
151	C₃×C₂²⋊C₄	C₂³×C₅	1	6	yes	—
152	C₃×C₄²	—	2	3	no	—
153	C₂³×C₅	—	2	4	no	—
154	C₂³×C₅	—	2	6	no	—
155	C₂³×C₅	—	8	2	no	—
156	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
157	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
158	C₅×Q₈	—	2	4	no	—
159	C₅×Q₈	—	2	6	no	—
160	C₅×D₄	—	2	12	no	—
161	C₅×D₄	—	2	12	no	—
162	C₅×D₄	—	2	6	no	—
163	C₅×D₄	—	4	1	no	—
164	C₅×D₄	—	4	3	no	—
165	C₅×D₄	—	8	12	no	—
166	C₂×C₄×C₅	—	2	12	no	—
167	C₂×C₄×C₅	—	2	6	no	—
168	C₂×C₄×C₅	—	2	6	no	—
169	C₂×C₄×C₅	—	2	6	no	—
170	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
172	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
173	2₋¹⁺⁴	—	2	3	no	—
174	H(5,𝔽₂)	—	2	3	no	—
175	H(5,𝔽₂)	C₂²×C₃×C₅	1	4	yes	—
176	C₂×#41	—	2	6	no	—
177	C₂×#41	C₂²×C₃×C₅	1	3	yes	—
178	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
179	#127	—	2	3	no	—
180	#124	—	2	3	no	—
181	#121	—	2	6	no	—
182	#120	C₂²×C₃×C₅	1	12	yes	complemented
183	#117	C₂²×C₃×C₅	1	3	yes	complemented
184	#99	C₂²×C₃×C₅	1	12	yes	complemented
185	C₂×C₃×C₅	—	2	3	no	—
186	C₂×C₃×C₅	—	4	6	no	—
187	C₂×C₃×C₅	—	8	2	no	—
188	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
189	C₂³×C₃	—	2	4	no	—
190	C₂³×C₃	—	2	6	no	—
191	C₂³×C₃	—	8	2	no	—
192	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
193	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
194	C₃×Q₈	—	2	4	no	—
195	C₃×Q₈	—	2	6	no	—
196	C₃×D₄	—	2	12	no	—
197	C₃×D₄	—	2	12	no	—
198	C₃×D₄	—	2	6	no	—
199	C₃×D₄	—	4	1	no	—
200	C₃×D₄	—	4	3	no	—
201	C₃×D₄	—	8	12	no	—
202	C₂×C₃×C₄	—	2	12	no	—
203	C₂×C₃×C₄	—	2	6	no	—
204	C₂×C₃×C₄	—	2	6	no	—
205	C₂×C₃×C₄	—	2	6	no	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
208	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
209	C₂²×C₅	—	2	6	no	—
210	C₂²×C₅	—	4	1	no	—
211	C₂²×C₅	—	4	6	no	—
212	C₂²×C₅	—	4	12	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	—	8	4	no	—
215	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
216	C₄×C₅	—	2	6	no	—
217	C₄×C₅	—	4	6	no	—
218	C₄×C₅	C₃×#120	1	1	yes	characteristic
219	C₄×C₅	C₃×#120	1	3	yes	—
220	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
221	#41	—	2	4	no	—
222	#41	—	2	12	no	—
223	#41	—	2	6	no	—
224	#41	—	2	12	no	—
225	#41	—	2	6	no	—
226	C₂×Q₈	—	2	6	no	—
227	C₂×Q₈	C₃×C₅×D₄	1	3	yes	complemented
228	C₂×D₄	—	2	6	no	—
229	C₂×D₄	—	2	12	no	—
230	C₂×D₄	—	2	12	no	—
231	C₂×D₄	—	2	3	no	—
232	C₂×D₄	—	2	3	no	—
233	C₂×D₄	C₃×C₅×D₄	1	3	yes	complemented
234	C₂×D₄	C₂³×C₃×C₅	1	6	yes	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₃×C₅×D₄	1	6	yes	complemented
237	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
238	C₄⋊C₄	—	2	3	no	—
239	C₂²⋊C₄	—	2	12	no	—
240	C₂²⋊C₄	C₂³×C₃×C₅	1	6	yes	—
241	C₄²	—	2	3	no	—
242	C₃×C₅	#3027	1	1	yes	characteristic, direct product factor, Hall
243	C₂²×C₃	—	2	6	no	—
244	C₂²×C₃	—	4	1	no	—
245	C₂²×C₃	—	4	6	no	—
246	C₂²×C₃	—	4	12	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	8	4	no	—
249	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
250	C₃×C₄	—	2	6	no	—
251	C₃×C₄	—	4	6	no	—
252	C₃×C₄	C₅×#120	1	1	yes	characteristic
253	C₃×C₄	C₅×#120	1	3	yes	—
254	C₂×C₅	—	2	3	no	—
255	C₂×C₅	—	4	6	no	—
256	C₂×C₅	—	8	2	no	—
257	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
258	C₂³	—	2	4	no	—
259	C₂³	—	2	6	no	—
260	C₂³	—	8	2	no	—
261	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
262	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	2	6	no	—
265	D₄	—	2	12	no	—
266	D₄	—	2	12	no	—
267	D₄	—	2	6	no	—
268	D₄	—	4	1	no	—
269	D₄	—	4	3	no	—
270	D₄	—	8	12	no	—
271	C₂×C₄	—	2	12	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	6	no	—
275	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
276	C₂×C₄	C₂×C₃×C₅×D₄	1	3	yes	—
277	C₂×C₄	C₂×C₃×C₅×D₄	1	3	yes	—
278	C₂×C₃	—	2	3	no	—
279	C₂×C₃	—	4	6	no	—
280	C₂×C₃	—	8	2	no	—
281	C₂×C₃	C₂×C₅×#120	1	1	yes	characteristic
282	C₅	C₃×#3027	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	6	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	6	no	—
286	C₂²	—	4	12	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	8	4	no	—
289	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
290	C₄	—	2	6	no	—
291	C₄	—	4	6	no	—
292	C₄	C₃×C₅×#120	1	1	yes	characteristic
293	C₄	C₃×C₅×#120	1	3	yes	—
294	C₃	C₅×#3027	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	3	no	—
296	C₂	—	4	6	no	—
297	C₂	—	8	2	no	—
298	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
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
1. C₃×C₅×#585	C₂	semidirect
2. C₃×C₅×#535	C₂	semidirect
3. C₃×C₅×B(4,𝔽₂)	C₂	semidirect
4. C₃×C₅×#410	C₂	semidirect
5. C₅×#3027	C₃	direct (N × H)
11. C₂²×C₃×C₅×D₄	C₂²	semidirect
15. C₃×C₅×#120	C₂²	semidirect
16. C₃×C₅×#117	C₂²	semidirect
17. C₃×C₅×#99	C₂²	semidirect
18. C₃×#3027	C₅	direct (N × H)
19. C₅×#585	C₂×C₃	semidirect
20. C₅×#535	C₂×C₃	semidirect
21. C₅×B(4,𝔽₂)	C₂×C₃	semidirect
22. C₅×#410	C₂×C₃	semidirect
30. C₂×C₃×C₅×Q₈	D₄	semidirect
36. C₂×C₃×C₅×D₄	D₄	semidirect
39. C₂²×C₃×C₄×C₅	D₄	semidirect
45. C₃×#585	C₂×C₅	semidirect
46. C₃×#535	C₂×C₅	semidirect
47. C₃×B(4,𝔽₂)	C₂×C₅	semidirect
48. C₃×#410	C₂×C₅	semidirect
54. C₂²×C₅×D₄	C₂²×C₃	semidirect
58. C₅×#120	C₂²×C₃	semidirect
59. C₅×#117	C₂²×C₃	semidirect
60. C₅×#99	C₂²×C₃	semidirect
61. #3027	C₃×C₅	direct (N × H)
87. C₂²×C₃×D₄	C₂²×C₅	semidirect
91. C₃×#120	C₂²×C₅	semidirect
92. C₃×#117	C₂²×C₅	semidirect
93. C₃×#99	C₂²×C₅	semidirect
101. C₂×C₅×Q₈	C₃×D₄	semidirect
107. C₂×C₅×D₄	C₃×D₄	semidirect
110. C₂²×C₄×C₅	C₃×D₄	semidirect
116. #585	C₂×C₃×C₅	semidirect
117. #535	C₂×C₃×C₅	semidirect
118. B(4,𝔽₂)	C₂×C₃×C₅	semidirect
119. #410	C₂×C₃×C₅	semidirect
138. C₂×C₃×Q₈	C₅×D₄	semidirect
144. C₂×C₃×D₄	C₅×D₄	semidirect
147. C₂²×C₃×C₄	C₅×D₄	semidirect
178. C₂²×D₄	C₂²×C₃×C₅	semidirect
182. #120	C₂²×C₃×C₅	semidirect
183. #117	C₂²×C₃×C₅	semidirect
184. #99	C₂²×C₃×C₅	semidirect
227. C₂×Q₈	C₃×C₅×D₄	semidirect
233. C₂×D₄	C₃×C₅×D₄	semidirect
236. C₂²×C₄	C₃×C₅×D₄	semidirect
242. C₃×C₅	#3027	direct (N × H)
282. C₅	C₃×#3027	direct (N × H)
294. C₃	C₅×#3027	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×#535	2
C₃×C₅×B(4,𝔽₂)	2
C₃×C₅×#410	2
C₅×#3027	3
C₃×#3027	5

Subgroups — C3×C5×#3027 #14276061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3027 #14276061