Subgroups — C₃×C₅×#1978 #14275012

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#418	C₂	1	2	yes	maximal, complemented
4	C₂×C₃×C₅×#100	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1978	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₂×C₃×C₅×D₈	—	2	2	no	—
11	C₂×C₃×C₅×D₈	C₄	1	4	yes	complemented
12	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
13	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#117	C₂²	1	1	yes	characteristic
16	C₃×C₅×#108	—	2	2	no	—
17	C₃×C₅×#102	—	2	4	no	—
18	C₃×C₅×#100	—	2	4	no	—
19	C₃×C₅×#97	—	2	1	no	—
20	C₃×C₄×C₅×C₈	—	2	1	no	—
21	C₃×#1978	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₅×D₈	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#418	C₂×C₃	1	2	yes	complemented
25	C₂×C₅×#100	C₂×C₃	1	2	yes	complemented
26	C₅×#405	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂⁴×C₃×C₅	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₅×D₈	—	2	4	no	—
36	C₃×C₅×D₈	—	2	4	no	—
37	C₃×C₅×M₁₆	—	2	1	no	—
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₃×C₅×M₁₆	D₄	1	2	yes	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	2	no	—
43	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
47	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
48	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
49	C₃×C₄²×C₅	—	2	1	no	—
50	C₂²×C₃×D₈	C₂×C₅	1	1	yes	characteristic
51	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic, complemented
52	C₃×#418	C₂×C₅	1	2	yes	complemented
53	C₂×C₃×#100	C₂×C₅	1	2	yes	complemented
54	C₃×#405	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
56	C₂×C₅×D₈	—	2	1	no	—
57	C₂×C₅×D₈	—	2	1	no	—
58	C₂×C₅×D₈	—	2	2	no	—
59	C₂×C₅×D₈	C₃×C₄	1	4	yes	complemented
60	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
61	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₅×#117	C₂²×C₃	1	1	yes	characteristic
64	C₅×#108	—	2	2	no	—
65	C₅×#102	—	2	4	no	—
66	C₅×#100	—	2	4	no	—
67	C₅×#97	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	#1978	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
70	C₂³×C₃×C₅	—	2	4	no	—
71	C₂³×C₃×C₅	—	4	2	no	—
72	C₂³×C₃×C₅	—	4	2	no	—
73	C₂³×C₃×C₅	—	8	4	no	—
74	C₂³×C₃×C₅	#41	1	1	yes	characteristic
75	C₃×C₅×D₄	—	2	4	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	4	2	no	—
78	C₃×C₅×D₄	—	4	2	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₃×C₅×C₈	—	2	2	no	—
87	C₃×C₅×C₈	—	4	2	no	—
88	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
89	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×D₈	—	2	1	no	—
92	C₂×C₃×D₈	—	2	1	no	—
93	C₂×C₃×D₈	—	2	2	no	—
94	C₂×C₃×D₈	C₄×C₅	1	4	yes	complemented
95	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
96	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
97	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×#117	C₂²×C₅	1	1	yes	characteristic
99	C₃×#108	—	2	2	no	—
100	C₃×#102	—	2	4	no	—
101	C₃×#100	—	2	4	no	—
102	C₃×#97	—	2	1	no	—
103	C₃×C₄×C₈	—	2	1	no	—
104	C₂⁴×C₅	—	2	2	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
111	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
112	C₅×D₈	—	2	4	no	—
113	C₅×D₈	—	2	4	no	—
114	C₅×M₁₆	—	2	1	no	—
115	C₅×M₁₆	—	2	2	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₅×M₁₆	C₃×D₄	1	2	yes	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
121	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
123	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
124	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
125	C₅×C₂²⋊C₄	—	2	1	no	—
126	C₄²×C₅	—	2	1	no	—
127	C₂²×D₈	C₂×C₃×C₅	1	1	yes	characteristic
128	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic, complemented
129	#418	C₂×C₃×C₅	1	2	yes	complemented
130	C₂×#100	C₂×C₃×C₅	1	2	yes	complemented
131	#405	C₂×C₃×C₅	1	1	yes	characteristic, complemented
132	C₂²×C₃×C₅	—	2	2	no	—
133	C₂²×C₃×C₅	—	4	4	no	—
134	C₂²×C₃×C₅	—	8	4	no	—
135	C₂²×C₃×C₅	—	8	4	no	—
136	C₂²×C₃×C₅	—	8	4	no	—
137	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
138	C₂²×C₃×C₅	#121	1	1	yes	characteristic
139	C₂²×C₃×C₅	#124	1	1	yes	characteristic
140	C₃×C₄×C₅	—	4	2	no	—
141	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
143	C₃×C₄×C₅	#121	1	1	yes	characteristic
144	C₃×C₄×C₅	#127	1	1	yes	characteristic
145	C₂⁴×C₃	—	2	2	no	—
146	C₂×C₃×D₄	—	2	2	no	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	—	2	2	no	—
149	C₂×C₃×D₄	—	2	2	no	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
152	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₃×D₈	—	2	4	no	—
154	C₃×D₈	—	2	4	no	—
155	C₃×M₁₆	—	2	1	no	—
156	C₃×M₁₆	—	2	2	no	—
157	C₃×M₁₆	—	2	2	no	—
158	C₃×M₁₆	C₅×D₄	1	2	yes	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₂×C₃×C₈	—	2	2	no	—
161	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
162	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
164	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
165	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
166	C₃×C₂²⋊C₄	—	2	1	no	—
167	C₃×C₄²	—	2	1	no	—
168	C₂³×C₅	—	2	4	no	—
169	C₂³×C₅	—	4	2	no	—
170	C₂³×C₅	—	4	2	no	—
171	C₂³×C₅	—	8	4	no	—
172	C₂³×C₅	C₃×#41	1	1	yes	characteristic
173	C₅×D₄	—	2	4	no	—
174	C₅×D₄	—	2	4	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	4	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
184	C₅×C₈	—	2	2	no	—
185	C₅×C₈	—	4	2	no	—
186	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
187	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
188	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	C₃×C₄×C₅	1	4	yes	complemented
193	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
194	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
195	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
196	#117	C₂²×C₃×C₅	1	1	yes	characteristic
197	#108	—	2	2	no	—
198	#102	—	2	4	no	—
199	#100	—	2	4	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₂×C₃×C₅	—	2	1	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	8	4	no	—
205	C₂×C₃×C₅	#390	1	1	yes	characteristic
206	C₂×C₃×C₅	#472	1	2	yes	—
207	C₂³×C₃	—	2	4	no	—
208	C₂³×C₃	—	4	2	no	—
209	C₂³×C₃	—	4	2	no	—
210	C₂³×C₃	—	8	4	no	—
211	C₂³×C₃	C₅×#41	1	1	yes	characteristic
212	C₃×D₄	—	2	4	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
223	C₃×C₈	—	2	2	no	—
224	C₃×C₈	—	4	2	no	—
225	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
226	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	4	4	no	—
229	C₂²×C₅	—	8	4	no	—
230	C₂²×C₅	—	8	4	no	—
231	C₂²×C₅	—	8	4	no	—
232	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
233	C₂²×C₅	C₃×#121	1	1	yes	characteristic
234	C₂²×C₅	C₃×#124	1	1	yes	characteristic
235	C₄×C₅	—	4	2	no	—
236	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
238	C₄×C₅	C₃×#121	1	1	yes	characteristic
239	C₄×C₅	C₃×#127	1	1	yes	characteristic
240	C₂⁴	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	—
247	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
248	D₈	—	2	4	no	—
249	D₈	—	2	4	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	C₃×C₅×D₄	1	2	yes	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
257	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
258	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
259	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
260	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₃×C₅	#1978	1	1	yes	characteristic, direct product factor, Hall
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	—	4	4	no	—
266	C₂²×C₃	—	8	4	no	—
267	C₂²×C₃	—	8	4	no	—
268	C₂²×C₃	—	8	4	no	—
269	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
270	C₂²×C₃	C₅×#121	1	1	yes	characteristic
271	C₂²×C₃	C₅×#124	1	1	yes	characteristic
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
274	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
275	C₃×C₄	C₅×#121	1	1	yes	characteristic
276	C₃×C₄	C₅×#127	1	1	yes	characteristic
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	8	4	no	—
280	C₂×C₅	C₃×#390	1	1	yes	characteristic
281	C₂×C₅	C₃×#472	1	2	yes	—
282	C₂³	—	2	4	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	8	4	no	—
286	C₂³	C₃×C₅×#41	1	1	yes	characteristic
287	D₄	—	2	4	no	—
288	D₄	—	2	4	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
293	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
295	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
297	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
301	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	8	4	no	—
305	C₂×C₃	C₅×#390	1	1	yes	characteristic
306	C₂×C₃	C₅×#472	1	2	yes	—
307	C₅	C₃×#1978	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	2	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	8	4	no	—
311	C₂²	—	8	4	no	—
312	C₂²	—	8	4	no	—
313	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
314	C₂²	C₃×C₅×#121	1	1	yes	characteristic
315	C₂²	C₃×C₅×#124	1	1	yes	characteristic
316	C₄	—	4	2	no	—
317	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
318	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
319	C₄	C₃×C₅×#121	1	1	yes	characteristic
320	C₄	C₃×C₅×#127	1	1	yes	characteristic
321	C₃	C₅×#1978	1	1	yes	characteristic, direct product factor, Sylow
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	8	4	no	—
325	C₂	C₃×C₅×#390	1	1	yes	characteristic
326	C₂	C₃×C₅×#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₈	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#108	C₂	semidirect
3. C₃×C₅×#418	C₂	semidirect
4. C₂×C₃×C₅×#100	C₂	semidirect
5. C₃×C₅×#405	C₂	semidirect
6. C₅×#1978	C₃	direct (N × H)
11. C₂×C₃×C₅×D₈	C₄	semidirect
21. C₃×#1978	C₅	direct (N × H)
23. C₂×C₅×#108	C₂×C₃	semidirect
24. C₅×#418	C₂×C₃	semidirect
25. C₂×C₅×#100	C₂×C₃	semidirect
26. C₅×#405	C₂×C₃	semidirect
51. C₂×C₃×#108	C₂×C₅	semidirect
52. C₃×#418	C₂×C₅	semidirect
53. C₂×C₃×#100	C₂×C₅	semidirect
54. C₃×#405	C₂×C₅	semidirect
59. C₂×C₅×D₈	C₃×C₄	semidirect
69. #1978	C₃×C₅	direct (N × H)
94. C₂×C₃×D₈	C₄×C₅	semidirect
128. C₂×#108	C₂×C₃×C₅	semidirect
129. #418	C₂×C₃×C₅	semidirect
130. C₂×#100	C₂×C₃×C₅	semidirect
131. #405	C₂×C₃×C₅	semidirect
192. C₂×D₈	C₃×C₄×C₅	semidirect
263. C₃×C₅	#1978	direct (N × H)
307. C₅	C₃×#1978	direct (N × H)
321. C₃	C₅×#1978	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×D₈	2
C₂×C₃×C₅×#108	2
C₃×C₅×#418	2
C₂×C₃×C₅×#100	2
C₃×C₅×#405	2
C₅×#1978	3
C₃×#1978	5

Subgroups — C3×C5×#1978 #14275012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1978 #14275012