Subgroups — C₂×C₃×C₅×#355 #14275164

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#101	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#355	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×#355	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₅×D₄	C₄	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	—	2	2	no	—
8	C₃×C₅×#124	—	2	2	no	—
9	C₃×C₅×#124	C₄	1	2	yes	—
10	C₃×C₅×#124	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₄	1	1	yes	characteristic
13	C₃×C₅×#101	—	2	2	no	—
14	C₃×C₅×#101	C₂²	1	4	yes	complemented
15	C₂×C₃×#355	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×#124	C₂×C₃	1	1	yes	characteristic
17	C₂×C₅×#101	C₂×C₃	1	2	yes	complemented
18	C₅×#355	C₂×C₃	1	4	yes	direct product factor
19	C₂⁴×C₃×C₅	—	2	1	no	—
20	C₂×C₃×C₅×Q₈	—	4	1	no	—
21	C₂×C₃×C₅×Q₈	—	4	1	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
24	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	—	2	1	no	—
26	C₂×C₃×C₅×D₄	—	4	1	no	—
27	C₂×C₃×C₅×D₄	—	4	1	no	—
28	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	—	2	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₃×C₅×M₁₆	—	2	4	no	—
33	C₃×C₅×M₁₆	—	4	2	no	—
34	C₂×C₃×C₅×C₈	—	4	2	no	—
35	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
36	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic
40	C₂×C₃×#101	C₂×C₅	1	2	yes	complemented
41	C₃×#355	C₂×C₅	1	4	yes	direct product factor
42	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₅×D₄	C₃×C₄	1	1	yes	characteristic
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₅×#124	—	2	2	no	—
46	C₅×#124	C₃×C₄	1	2	yes	—
47	C₅×#124	C₂²×C₃	1	2	yes	—
48	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₅	C₃×C₄	1	1	yes	characteristic
50	C₅×#101	—	2	2	no	—
51	C₅×#101	C₂²×C₃	1	4	yes	complemented
52	C₂×#355	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₃×C₅	—	2	2	no	—
54	C₂³×C₃×C₅	—	4	4	no	—
55	C₂³×C₃×C₅	—	4	1	no	—
56	C₂³×C₃×C₅	—	4	1	no	—
57	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₃×C₅×Q₈	—	4	1	no	—
59	C₃×C₅×Q₈	—	4	2	no	—
60	C₃×C₅×Q₈	—	4	1	no	—
61	C₃×C₅×D₄	—	4	2	no	—
62	C₃×C₅×D₄	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₃×C₅×C₈	—	4	4	no	—
74	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₃×D₄	C₄×C₅	1	1	yes	characteristic
76	C₂×C₃×M₁₆	—	2	2	no	—
77	C₃×#124	—	2	2	no	—
78	C₃×#124	C₄×C₅	1	2	yes	—
79	C₃×#124	C₂²×C₅	1	2	yes	—
80	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
81	C₂×C₃×C₄²	C₄×C₅	1	1	yes	characteristic
82	C₃×#101	—	2	2	no	—
83	C₃×#101	C₂²×C₅	1	4	yes	complemented
84	C₂⁴×C₅	—	2	1	no	—
85	C₂×C₅×Q₈	—	4	1	no	—
86	C₂×C₅×Q₈	—	4	1	no	—
87	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
89	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₅×D₄	—	2	1	no	—
91	C₂×C₅×D₄	—	4	1	no	—
92	C₂×C₅×D₄	—	4	1	no	—
93	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
94	C₂²×C₄×C₅	—	2	1	no	—
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
97	C₅×M₁₆	—	2	4	no	—
98	C₅×M₁₆	—	4	2	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₅×C₂²⋊C₄	—	2	4	no	—
101	C₅×C₂²⋊C₄	—	2	4	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
104	C₂×#124	C₂×C₃×C₅	1	1	yes	characteristic
105	C₂×#101	C₂×C₃×C₅	1	2	yes	complemented
106	#355	C₂×C₃×C₅	1	4	yes	direct product factor
107	C₂²×C₃×C₅	—	2	2	no	—
108	C₂²×C₃×C₅	—	4	4	no	—
109	C₂²×C₃×C₅	—	4	2	no	—
110	C₂²×C₃×C₅	—	8	2	no	—
111	C₂²×C₃×C₅	—	8	2	no	—
112	C₂²×C₃×C₅	#99	1	1	yes	characteristic
113	C₂²×C₃×C₅	#99	1	1	yes	characteristic
114	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	—	2	2	no	—
116	C₃×C₄×C₅	—	2	1	no	—
117	C₃×C₄×C₅	—	2	2	no	—
118	C₃×C₄×C₅	—	2	1	no	—
119	C₃×C₄×C₅	—	4	2	no	—
120	C₂⁴×C₃	—	2	1	no	—
121	C₂×C₃×Q₈	—	4	1	no	—
122	C₂×C₃×Q₈	—	4	1	no	—
123	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
125	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₃×D₄	—	2	1	no	—
127	C₂×C₃×D₄	—	4	1	no	—
128	C₂×C₃×D₄	—	4	1	no	—
129	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
133	C₃×M₁₆	—	2	4	no	—
134	C₃×M₁₆	—	4	2	no	—
135	C₂×C₃×C₈	—	4	2	no	—
136	C₃×C₂²⋊C₄	—	2	4	no	—
137	C₃×C₂²⋊C₄	—	2	4	no	—
138	C₃×C₄²	—	2	1	no	—
139	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
140	C₂³×C₅	—	2	2	no	—
141	C₂³×C₅	—	4	4	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	—	4	1	no	—
144	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₅×Q₈	—	4	1	no	—
146	C₅×Q₈	—	4	2	no	—
147	C₅×Q₈	—	4	1	no	—
148	C₅×D₄	—	4	2	no	—
149	C₅×D₄	—	4	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
160	C₅×C₈	—	4	4	no	—
161	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₂²×D₄	C₃×C₄×C₅	1	1	yes	characteristic
163	C₂×M₁₆	—	2	2	no	—
164	#124	—	2	2	no	—
165	#124	C₃×C₄×C₅	1	2	yes	—
166	#124	C₂²×C₃×C₅	1	2	yes	—
167	C₂×C₂²⋊C₄	—	2	2	no	—
168	C₂×C₄²	C₃×C₄×C₅	1	1	yes	characteristic
169	#101	—	2	2	no	—
170	#101	C₂²×C₃×C₅	1	4	yes	complemented
171	C₂×C₃×C₅	—	2	1	no	—
172	C₂×C₃×C₅	—	2	1	no	—
173	C₂×C₃×C₅	—	8	2	no	—
174	C₂×C₃×C₅	#355	1	2	yes	direct product factor
175	C₂×C₃×C₅	C₂×#99	1	1	yes	characteristic
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	4	4	no	—
178	C₂³×C₃	—	4	1	no	—
179	C₂³×C₃	—	4	1	no	—
180	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₃×Q₈	—	4	1	no	—
182	C₃×Q₈	—	4	2	no	—
183	C₃×Q₈	—	4	1	no	—
184	C₃×D₄	—	4	2	no	—
185	C₃×D₄	—	4	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
194	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
196	C₃×C₈	—	4	4	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	4	4	no	—
199	C₂²×C₅	—	4	2	no	—
200	C₂²×C₅	—	8	2	no	—
201	C₂²×C₅	—	8	2	no	—
202	C₂²×C₅	C₃×#99	1	1	yes	characteristic
203	C₂²×C₅	C₃×#99	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	4	2	no	—
210	C₂⁴	—	2	1	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	—	4	1	no	—
213	C₂×Q₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
214	C₂×Q₈	C₃×C₅×D₄	1	2	yes	—
215	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
230	C₃×C₅	C₂×#355	1	1	yes	characteristic, direct product factor, Hall
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	4	4	no	—
233	C₂²×C₃	—	4	2	no	—
234	C₂²×C₃	—	8	2	no	—
235	C₂²×C₃	—	8	2	no	—
236	C₂²×C₃	C₅×#99	1	1	yes	characteristic
237	C₂²×C₃	C₅×#99	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	4	2	no	—
244	C₂×C₅	—	2	1	no	—
245	C₂×C₅	—	2	1	no	—
246	C₂×C₅	—	8	2	no	—
247	C₂×C₅	C₃×#355	1	2	yes	direct product factor
248	C₂×C₅	C₂×C₃×#99	1	1	yes	characteristic
249	C₂³	—	2	2	no	—
250	C₂³	—	4	4	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
254	Q₈	—	4	1	no	—
255	Q₈	—	4	2	no	—
256	Q₈	—	4	1	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
267	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
268	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
269	C₈	—	4	4	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	8	2	no	—
273	C₂×C₃	C₅×#355	1	2	yes	direct product factor
274	C₂×C₃	C₂×C₅×#99	1	1	yes	characteristic
275	C₅	C₂×C₃×#355	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	2	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	C₃×C₅×#99	1	1	yes	characteristic
282	C₂²	C₃×C₅×#99	1	1	yes	characteristic
283	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	2	no	—
289	C₃	C₂×C₅×#355	1	1	yes	characteristic, direct product factor, Sylow
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	8	2	no	—
293	C₂	C₃×C₅×#355	1	2	yes	direct product factor
294	C₂	C₂×C₃×C₅×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#99
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×Q₈	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂³×C₃×C₅ ◁ C₂²×C₃×C₅×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#101	C₂	semidirect
3. C₃×C₅×#355	C₂	direct (N × H)
4. C₂×C₅×#355	C₃	direct (N × H)
14. C₃×C₅×#101	C₂²	semidirect
15. C₂×C₃×#355	C₅	direct (N × H)
17. C₂×C₅×#101	C₂×C₃	semidirect
18. C₅×#355	C₂×C₃	direct (N × H)
40. C₂×C₃×#101	C₂×C₅	semidirect
41. C₃×#355	C₂×C₅	direct (N × H)
51. C₅×#101	C₂²×C₃	semidirect
52. C₂×#355	C₃×C₅	direct (N × H)
83. C₃×#101	C₂²×C₅	semidirect
105. C₂×#101	C₂×C₃×C₅	semidirect
106. #355	C₂×C₃×C₅	direct (N × H)
170. #101	C₂²×C₃×C₅	semidirect
174. C₂×C₃×C₅	#355	direct (N × H)
230. C₃×C₅	C₂×#355	direct (N × H)
247. C₂×C₅	C₃×#355	direct (N × H)
273. C₂×C₃	C₅×#355	direct (N × H)
275. C₅	C₂×C₃×#355	direct (N × H)
289. C₃	C₂×C₅×#355	direct (N × H)
293. C₂	C₃×C₅×#355	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×#124	2
C₂×C₃×C₅×#101	2
C₃×C₅×#355	2
C₂×C₅×#355	3
C₂×C₃×#355	5

Subgroups — C2×C3×C5×#355 #14275164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#355 #14275164