Subgroups — C₃×C₅×#2100 #14275134

Order 1920.

428 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×#389	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#340	C₂	1	2	yes	maximal
4	C₃×C₅×#340	C₂	1	2	yes	maximal
5	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic
6	C₅×#2100	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×#95	—	2	1	no	—
16	C₃×#2100	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#389	C₂×C₃	1	1	yes	characteristic
18	C₅×#384	C₂×C₃	1	1	yes	characteristic
19	C₅×#340	C₂×C₃	1	2	yes	—
20	C₅×#340	C₂×C₃	1	2	yes	—
21	C₅×#336	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	—	2	1	no	—
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	—	2	1	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	4	no	—
34	C₃×C₅×C₄⋊C₄	—	4	2	no	—
35	C₃×C₅×C₄⋊C₄	—	4	2	no	—
36	C₃×C₅×C₄⋊C₄	—	4	2	no	—
37	C₃×C₅×C₄⋊C₄	Q₈	1	2	yes	—
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₃×#389	C₂×C₅	1	1	yes	characteristic
40	C₃×#384	C₂×C₅	1	1	yes	characteristic
41	C₃×#340	C₂×C₅	1	2	yes	—
42	C₃×#340	C₂×C₅	1	2	yes	—
43	C₃×#336	C₂×C₅	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₅×C₄⋊C₄	—	2	1	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₅×#95	—	2	1	no	—
53	#2100	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	—	2	1	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
69	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	#41	1	2	yes	—
72	C₂×C₃×C₄×C₅	#41	1	2	yes	—
73	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
74	C₃×C₅×C₈	—	4	2	no	—
75	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
76	C₂×C₃×C₄⋊C₄	—	2	1	no	—
77	C₂×C₃×C₄⋊C₄	—	2	1	no	—
78	C₂×C₃×C₄⋊C₄	—	2	1	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
83	C₃×#95	—	2	1	no	—
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	1	no	—
87	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	—	4	2	no	—
99	C₅×C₄⋊C₄	C₃×Q₈	1	2	yes	—
100	C₄²×C₅	—	2	2	no	—
101	#389	C₂×C₃×C₅	1	1	yes	characteristic
102	#384	C₂×C₃×C₅	1	1	yes	characteristic
103	#340	C₂×C₃×C₅	1	2	yes	—
104	#340	C₂×C₃×C₅	1	2	yes	—
105	#336	C₂×C₃×C₅	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×C₃×C₅	#123	1	1	yes	characteristic
108	C₂²×C₃×C₅	#124	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
111	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	—	2	2	no	—
115	C₃×C₄×C₅	—	4	2	no	—
116	C₃×C₄×C₅	—	8	1	no	—
117	C₃×C₄×C₅	—	8	2	no	—
118	C₃×C₄×C₅	#125	1	2	yes	—
119	C₃×C₄×C₅	#126	1	2	yes	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	2	1	no	—
122	C₂²×C₃×C₄	—	2	2	no	—
123	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₃×C₄⋊C₄	—	2	1	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	2	4	no	—
132	C₃×C₄⋊C₄	—	4	2	no	—
133	C₃×C₄⋊C₄	—	4	2	no	—
134	C₃×C₄⋊C₄	—	4	2	no	—
135	C₃×C₄⋊C₄	C₅×Q₈	1	2	yes	—
136	C₃×C₄²	—	2	2	no	—
137	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	4	1	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
152	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×#41	1	2	yes	—
155	C₂×C₄×C₅	C₃×#41	1	2	yes	—
156	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
157	C₅×C₈	—	4	2	no	—
158	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
165	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
166	#95	—	2	1	no	—
167	C₂×C₃×C₅	#400	1	1	yes	characteristic
168	C₂×C₃×C₅	#476	1	1	yes	characteristic
169	C₂×C₃×C₅	#477	1	1	yes	characteristic
170	C₂×C₃×C₅	#481	1	1	yes	characteristic
171	C₂×C₃×C₅	#484	1	1	yes	characteristic
172	C₂×C₃×C₅	#487	1	1	yes	characteristic
173	C₂×C₃×C₅	#490	1	1	yes	characteristic
174	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	—	4	1	no	—
182	C₂×C₃×C₄	—	4	1	no	—
183	C₂×C₃×C₄	—	4	2	no	—
184	C₂×C₃×C₄	—	4	1	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
189	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
190	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
191	C₂×C₃×C₄	C₅×#41	1	2	yes	—
192	C₂×C₃×C₄	C₅×#41	1	2	yes	—
193	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
194	C₃×C₈	—	4	2	no	—
195	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
196	C₂²×C₅	C₃×#123	1	1	yes	characteristic
197	C₂²×C₅	C₃×#124	1	1	yes	characteristic
198	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
199	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
200	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	4	2	no	—
205	C₄×C₅	—	8	1	no	—
206	C₄×C₅	—	8	2	no	—
207	C₄×C₅	C₃×#125	1	2	yes	—
208	C₄×C₅	C₃×#126	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	C₃×C₅×Q₈	1	2	yes	—
225	C₄²	—	2	2	no	—
226	C₃×C₅	#2100	1	1	yes	characteristic, direct product factor, Hall
227	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
228	C₂²×C₃	C₅×#123	1	1	yes	characteristic
229	C₂²×C₃	C₅×#124	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
232	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	4	2	no	—
237	C₃×C₄	—	8	1	no	—
238	C₃×C₄	—	8	2	no	—
239	C₃×C₄	C₅×#125	1	2	yes	—
240	C₃×C₄	C₅×#126	1	2	yes	—
241	C₂×C₅	C₃×#400	1	1	yes	characteristic
242	C₂×C₅	C₃×#476	1	1	yes	characteristic
243	C₂×C₅	C₃×#477	1	1	yes	characteristic
244	C₂×C₅	C₃×#481	1	1	yes	characteristic
245	C₂×C₅	C₃×#484	1	1	yes	characteristic
246	C₂×C₅	C₃×#487	1	1	yes	characteristic
247	C₂×C₅	C₃×#490	1	1	yes	characteristic
248	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
263	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
264	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
265	C₂×C₄	C₃×C₅×#41	1	2	yes	—
266	C₂×C₄	C₃×C₅×#41	1	2	yes	—
267	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₂×C₃	C₅×#400	1	1	yes	characteristic
270	C₂×C₃	C₅×#476	1	1	yes	characteristic
271	C₂×C₃	C₅×#477	1	1	yes	characteristic
272	C₂×C₃	C₅×#481	1	1	yes	characteristic
273	C₂×C₃	C₅×#484	1	1	yes	characteristic
274	C₂×C₃	C₅×#487	1	1	yes	characteristic
275	C₂×C₃	C₅×#490	1	1	yes	characteristic
276	C₅	C₃×#2100	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
278	C₂²	C₃×C₅×#123	1	1	yes	characteristic
279	C₂²	C₃×C₅×#124	1	1	yes	characteristic
280	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
281	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
282	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	1	no	—
288	C₄	—	8	2	no	—
289	C₄	C₃×C₅×#125	1	2	yes	—
290	C₄	C₃×C₅×#126	1	2	yes	—
291	C₃	C₅×#2100	1	1	yes	characteristic, direct product factor, Sylow
292	C₂	C₃×C₅×#400	1	1	yes	characteristic
293	C₂	C₃×C₅×#476	1	1	yes	characteristic
294	C₂	C₃×C₅×#477	1	1	yes	characteristic
295	C₂	C₃×C₅×#481	1	1	yes	characteristic
296	C₂	C₃×C₅×#484	1	1	yes	characteristic
297	C₂	C₃×C₅×#487	1	1	yes	characteristic
298	C₂	C₃×C₅×#490	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
6. C₅×#2100	C₃	direct (N × H)
16. C₃×#2100	C₅	direct (N × H)
53. #2100	C₃×C₅	direct (N × H)
226. C₃×C₅	#2100	direct (N × H)
276. C₅	C₃×#2100	direct (N × H)
291. C₃	C₅×#2100	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×#389	2
C₃×C₅×#384	2
C₃×C₅×#340	2
C₃×C₅×#336	2
C₅×#2100	3
C₃×#2100	5

Subgroups — C3×C5×#2100 #14275134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2100 #14275134