Subgroups — C₃×C₅×#1957 #14274991

Order 1920.

872 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#102	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1957	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	—	2	2	no	—
7	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₅×#127	—	2	2	no	—
10	C₃×C₅×#127	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×#102	—	2	8	no	—
15	C₃×C₄×C₅×C₈	—	2	2	no	—
16	C₃×#1957	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#102	C₂×C₃	1	4	yes	complemented
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂⁴×C₃×C₅	—	2	2	no	—
22	C₂×C₃×C₅×D₄	—	2	8	no	—
23	C₂×C₃×C₅×D₄	—	2	4	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	2	4	no	—
26	C₂×C₃×C₅×D₄	—	4	4	no	—
27	C₂×C₃×C₅×D₄	—	4	4	no	—
28	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₄×C₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	D₄	1	4	yes	complemented
35	C₃×C₅×C₄⋊C₄	—	2	4	no	—
36	C₃×C₅×C₄⋊C₄	—	4	4	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
40	C₂×C₃×#102	C₂×C₅	1	4	yes	complemented
41	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂²×C₅×D₄	—	2	2	no	—
44	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
45	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
46	C₅×#127	—	2	2	no	—
47	C₅×#127	C₃×C₄	1	4	yes	complemented
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₅×#102	—	2	8	no	—
52	C₄×C₅×C₈	—	2	2	no	—
53	#1957	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	—	2	4	no	—
55	C₂³×C₃×C₅	—	4	2	no	—
56	C₂³×C₃×C₅	—	4	2	no	—
57	C₂³×C₃×C₅	—	4	4	no	—
58	C₂³×C₃×C₅	—	4	4	no	—
59	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
60	C₃×C₅×D₄	—	2	8	no	—
61	C₃×C₅×D₄	—	4	16	no	—
62	C₃×C₅×D₄	—	4	4	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	4	4	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
66	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
67	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	#41	1	2	yes	—
72	C₃×C₅×C₈	—	2	4	no	—
73	C₂²×C₃×D₄	—	2	2	no	—
74	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
75	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
76	C₃×#127	—	2	2	no	—
77	C₃×#127	C₄×C₅	1	4	yes	complemented
78	C₂×C₃×C₄⋊C₄	—	2	2	no	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
81	C₃×#102	—	2	8	no	—
82	C₃×C₄×C₈	—	2	2	no	—
83	C₂⁴×C₅	—	2	2	no	—
84	C₂×C₅×D₄	—	2	8	no	—
85	C₂×C₅×D₄	—	2	4	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	2	4	no	—
88	C₂×C₅×D₄	—	4	4	no	—
89	C₂×C₅×D₄	—	4	4	no	—
90	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	C₃×D₄	1	4	yes	complemented
97	C₅×C₄⋊C₄	—	2	4	no	—
98	C₅×C₄⋊C₄	—	4	4	no	—
99	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
100	C₄²×C₅	C₃×D₄	1	2	yes	—
101	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
102	C₂×#102	C₂×C₃×C₅	1	4	yes	complemented
103	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#384	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	C₂²×C₃×C₅	—	4	4	no	—
106	C₂²×C₃×C₅	—	4	8	no	—
107	C₂²×C₃×C₅	—	8	4	no	—
108	C₂²×C₃×C₅	—	8	2	no	—
109	C₂²×C₃×C₅	—	8	2	no	—
110	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₃×C₅	#124	1	1	yes	characteristic
112	C₂²×C₃×C₅	#127	1	1	yes	characteristic
113	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
114	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
115	C₃×C₄×C₅	—	8	2	no	—
116	C₃×C₄×C₅	#102	1	8	yes	complemented
117	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₂⁴×C₃	—	2	2	no	—
120	C₂×C₃×D₄	—	2	8	no	—
121	C₂×C₃×D₄	—	2	4	no	—
122	C₂×C₃×D₄	—	2	2	no	—
123	C₂×C₃×D₄	—	2	4	no	—
124	C₂×C₃×D₄	—	4	4	no	—
125	C₂×C₃×D₄	—	4	4	no	—
126	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
129	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₅×D₄	1	4	yes	complemented
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	4	4	no	—
135	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
136	C₃×C₄²	C₅×D₄	1	2	yes	—
137	C₂³×C₅	—	2	4	no	—
138	C₂³×C₅	—	4	2	no	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	—	4	4	no	—
141	C₂³×C₅	—	4	4	no	—
142	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
143	C₅×D₄	—	2	8	no	—
144	C₅×D₄	—	4	16	no	—
145	C₅×D₄	—	4	4	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	C₃×D₈	1	4	yes	—
150	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
151	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×#41	1	2	yes	—
155	C₅×C₈	—	2	4	no	—
156	C₂²×D₄	—	2	2	no	—
157	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
158	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
159	#127	—	2	2	no	—
160	#127	C₃×C₄×C₅	1	4	yes	complemented
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
163	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
164	#102	—	2	8	no	—
165	C₄×C₈	—	2	2	no	—
166	C₂×C₃×C₅	—	8	4	no	—
167	C₂×C₃×C₅	#390	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×#102	1	2	yes	—
169	C₂×C₃×C₅	#486	1	2	yes	—
170	C₂×C₃×C₅	#492	1	1	yes	characteristic
171	C₂×C₃×C₅	#493	1	1	yes	characteristic
172	C₂³×C₃	—	2	4	no	—
173	C₂³×C₃	—	4	2	no	—
174	C₂³×C₃	—	4	2	no	—
175	C₂³×C₃	—	4	4	no	—
176	C₂³×C₃	—	4	4	no	—
177	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
178	C₃×D₄	—	2	8	no	—
179	C₃×D₄	—	4	4	no	—
180	C₃×D₄	—	4	16	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	4	4	no	—
183	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
184	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
185	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
186	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
188	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₅×#41	1	2	yes	—
190	C₃×C₈	—	2	4	no	—
191	C₂²×C₅	—	4	4	no	—
192	C₂²×C₅	—	4	8	no	—
193	C₂²×C₅	—	8	4	no	—
194	C₂²×C₅	—	8	2	no	—
195	C₂²×C₅	—	8	2	no	—
196	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₅	C₃×#124	1	1	yes	characteristic
198	C₂²×C₅	C₃×#127	1	1	yes	characteristic
199	C₂²×C₅	C₂×C₃×D₈	1	2	yes	—
200	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	C₃×#102	1	8	yes	complemented
203	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
204	C₄×C₅	C₃×#121	1	2	yes	—
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	8	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
215	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	C₃×C₅×D₄	1	4	yes	complemented
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
222	C₄²	C₃×C₅×D₄	1	2	yes	—
223	C₃×C₅	#1957	1	1	yes	characteristic, direct product factor, Hall
224	C₂²×C₃	—	4	4	no	—
225	C₂²×C₃	—	4	8	no	—
226	C₂²×C₃	—	8	4	no	—
227	C₂²×C₃	—	8	2	no	—
228	C₂²×C₃	—	8	2	no	—
229	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₃	C₅×#124	1	1	yes	characteristic
231	C₂²×C₃	C₅×#127	1	1	yes	characteristic
232	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
233	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
234	C₃×C₄	—	8	2	no	—
235	C₃×C₄	C₅×#102	1	8	yes	complemented
236	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
237	C₃×C₄	C₅×#121	1	2	yes	—
238	C₂×C₅	—	8	4	no	—
239	C₂×C₅	C₃×#390	1	1	yes	characteristic
240	C₂×C₅	C₂×C₃×#102	1	2	yes	—
241	C₂×C₅	C₃×#486	1	2	yes	—
242	C₂×C₅	C₃×#492	1	1	yes	characteristic
243	C₂×C₅	C₃×#493	1	1	yes	characteristic
244	C₂³	—	2	4	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	4	4	no	—
249	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
250	D₄	—	2	8	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	16	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	4	yes	—
256	C₂×C₄	C₃×C₅×D₈	1	4	yes	—
257	C₂×C₄	C₃×C₅×SD₁₆	1	4	yes	—
258	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
259	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
260	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
261	C₂×C₄	C₃×C₅×#41	1	2	yes	—
262	C₈	—	2	4	no	—
263	C₂×C₃	—	8	4	no	—
264	C₂×C₃	C₅×#390	1	1	yes	characteristic
265	C₂×C₃	C₂×C₅×#102	1	2	yes	—
266	C₂×C₃	C₅×#486	1	2	yes	—
267	C₂×C₃	C₅×#492	1	1	yes	characteristic
268	C₂×C₃	C₅×#493	1	1	yes	characteristic
269	C₅	C₃×#1957	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	4	4	no	—
271	C₂²	—	4	8	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
276	C₂²	C₃×C₅×#124	1	1	yes	characteristic
277	C₂²	C₃×C₅×#127	1	1	yes	characteristic
278	C₂²	C₂×C₃×C₅×D₈	1	2	yes	—
279	C₂²	C₂×C₃×C₅×SD₁₆	1	2	yes	—
280	C₄	—	8	2	no	—
281	C₄	C₃×C₅×#102	1	8	yes	complemented
282	C₄	C₃×C₄×C₅×D₄	1	2	yes	—
283	C₄	C₃×C₅×#121	1	2	yes	—
284	C₃	C₅×#1957	1	1	yes	characteristic, direct product factor, Sylow
285	C₂	—	8	4	no	—
286	C₂	C₃×C₅×#390	1	1	yes	characteristic
287	C₂	C₂×C₃×C₅×#102	1	2	yes	—
288	C₂	C₃×C₅×#486	1	2	yes	—
289	C₂	C₃×C₅×#492	1	1	yes	characteristic
290	C₂	C₃×C₅×#493	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#127	C₂
LCS	G ▷ C₂×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
2. C₂×C₃×C₅×#102	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. C₃×C₅×#384	C₂	semidirect
5. C₅×#1957	C₃	direct (N × H)
10. C₃×C₅×#127	C₄	semidirect
16. C₃×#1957	C₅	direct (N × H)
18. C₂×C₅×#102	C₂×C₃	semidirect
19. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
20. C₅×#384	C₂×C₃	semidirect
34. C₂×C₃×C₅×C₈	D₄	semidirect
40. C₂×C₃×#102	C₂×C₅	semidirect
41. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
42. C₃×#384	C₂×C₅	semidirect
47. C₅×#127	C₃×C₄	semidirect
53. #1957	C₃×C₅	direct (N × H)
77. C₃×#127	C₄×C₅	semidirect
96. C₂×C₅×C₈	C₃×D₄	semidirect
102. C₂×#102	C₂×C₃×C₅	semidirect
103. C₂×C₄×C₈	C₂×C₃×C₅	semidirect
104. #384	C₂×C₃×C₅	semidirect
116. C₃×C₄×C₅	#102	semidirect
132. C₂×C₃×C₈	C₅×D₄	semidirect
160. #127	C₃×C₄×C₅	semidirect
202. C₄×C₅	C₃×#102	semidirect
218. C₂×C₈	C₃×C₅×D₄	semidirect
223. C₃×C₅	#1957	direct (N × H)
235. C₃×C₄	C₅×#102	semidirect
269. C₅	C₃×#1957	direct (N × H)
281. C₄	C₃×C₅×#102	semidirect
284. C₃	C₅×#1957	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×#127	2
C₂×C₃×C₅×#102	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#384	2
C₅×#1957	3
C₃×#1957	5

Subgroups — C3×C5×#1957 #14274991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1957 #14274991