Subgroups — C₃×C₅×#1808 #14274842

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#380	C₂	1	2	yes	maximal
4	C₃×C₅×#340	C₂	1	2	yes	maximal, complemented
5	C₅×#1808	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	—	2	4	no	—
8	C₂²×C₃×C₅×C₈	C₄	1	4	yes	complemented
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
13	C₃×C₅×#107	—	2	4	no	—
14	C₃×C₅×#95	—	2	2	no	—
15	C₃×#1808	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₅×C₈	C₂×C₃	1	1	yes	characteristic
17	C₂×C₅×#107	C₂×C₃	1	2	yes	complemented
18	C₅×#380	C₂×C₃	1	2	yes	—
19	C₅×#340	C₂×C₃	1	2	yes	complemented
20	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	8	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
33	C₃×C₅×C₄⋊C₄	—	2	4	no	—
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
36	C₂³×C₃×C₈	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×#107	C₂×C₅	1	2	yes	complemented
38	C₃×#380	C₂×C₅	1	2	yes	—
39	C₃×#340	C₂×C₅	1	2	yes	complemented
40	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	2	4	no	—
42	C₂²×C₅×C₈	C₃×C₄	1	4	yes	complemented
43	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
47	C₅×#107	—	2	4	no	—
48	C₅×#95	—	2	2	no	—
49	#1808	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃×C₅	D₈	1	2	yes	—
54	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
55	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	4	4	no	—
61	C₂×C₃×C₄×C₅	—	4	4	no	—
62	C₂×C₃×C₄×C₅	—	4	4	no	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	complemented
70	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
71	C₂²×C₃×C₈	—	2	4	no	—
72	C₂²×C₃×C₈	C₄×C₅	1	4	yes	complemented
73	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
76	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
77	C₃×#107	—	2	4	no	—
78	C₃×#95	—	2	2	no	—
79	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	—	2	4	no	—
83	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₃×Q₈	1	2	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	8	no	—
89	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₂²⋊C₄	—	4	4	no	—
95	C₂³×C₈	C₂×C₃×C₅	1	1	yes	characteristic
96	C₂×#107	C₂×C₃×C₅	1	2	yes	complemented
97	#380	C₂×C₃×C₅	1	2	yes	—
98	#340	C₂×C₃×C₅	1	2	yes	complemented
99	C₂²×C₃×C₅	—	2	2	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	4	no	—
102	C₂²×C₃×C₅	—	2	4	no	—
103	C₂²×C₃×C₅	#107	1	4	yes	complemented
104	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	#121	1	2	yes	—
106	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
108	C₂²×C₃×C₅	#135	1	2	yes	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	8	4	no	—
111	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₃×C₄×C₅	#117	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
114	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	2	4	no	—
118	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
119	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
121	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	8	no	—
124	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
125	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
127	C₃×C₄⋊C₄	—	2	4	no	—
128	C₃×C₄⋊C₄	—	2	4	no	—
129	C₃×C₂²⋊C₄	—	4	4	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
133	C₂³×C₅	C₃×D₈	1	2	yes	—
134	C₂³×C₅	C₃×Q₁₆	1	2	yes	—
135	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	4	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅	C₃×#41	1	2	yes	—
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	C₃×C₂²⋊C₄	1	4	yes	complemented
150	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂²×C₈	—	2	4	no	—
152	C₂²×C₈	C₃×C₄×C₅	1	4	yes	complemented
153	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
156	C₂×C₂²⋊C₄	—	2	2	no	—
157	#107	—	2	4	no	—
158	#95	—	2	2	no	—
159	C₂×C₃×C₅	—	2	4	no	—
160	C₂×C₃×C₅	#380	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
162	C₂×C₃×C₅	#427	1	1	yes	characteristic
163	C₂×C₃×C₅	#466	1	2	yes	—
164	C₂×C₃×C₅	#467	1	2	yes	—
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	—	2	2	no	—
167	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
168	C₂³×C₃	C₅×D₈	1	2	yes	—
169	C₂³×C₃	C₅×Q₁₆	1	2	yes	—
170	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	—	4	4	no	—
178	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
179	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₅×#41	1	2	yes	—
183	C₃×C₈	—	2	2	no	—
184	C₃×C₈	C₅×C₂²⋊C₄	1	4	yes	complemented
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	2	4	no	—
189	C₂²×C₅	C₃×#107	1	4	yes	complemented
190	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
191	C₂²×C₅	C₃×#121	1	2	yes	—
192	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
193	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
194	C₂²×C₅	C₃×#135	1	2	yes	—
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	8	4	no	—
197	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₄×C₅	C₃×#117	1	1	yes	characteristic
199	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
200	C₂⁴	C₃×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
205	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₃×C₅×Q₈	1	2	yes	—
207	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
211	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
212	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₃×C₅	#1808	1	1	yes	characteristic, direct product factor, Hall
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	2	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	2	4	no	—
221	C₂²×C₃	C₅×#107	1	4	yes	complemented
222	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
223	C₂²×C₃	C₅×#121	1	2	yes	—
224	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
225	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
226	C₂²×C₃	C₅×#135	1	2	yes	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	8	4	no	—
229	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₃×C₄	C₅×#117	1	1	yes	characteristic
231	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
232	C₂×C₅	—	2	4	no	—
233	C₂×C₅	C₃×#380	1	1	yes	characteristic
234	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
235	C₂×C₅	C₃×#427	1	1	yes	characteristic
236	C₂×C₅	C₃×#466	1	2	yes	—
237	C₂×C₅	C₃×#467	1	2	yes	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₃×C₅×C₄⋊C₄	1	2	yes	—
241	C₂³	C₃×C₅×D₈	1	2	yes	—
242	C₂³	C₃×C₅×Q₁₆	1	2	yes	—
243	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
252	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
254	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
255	C₂×C₄	C₃×C₅×#41	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	C₃×C₅×C₂²⋊C₄	1	4	yes	complemented
258	C₂×C₃	—	2	4	no	—
259	C₂×C₃	C₅×#380	1	1	yes	characteristic
260	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
261	C₂×C₃	C₅×#427	1	1	yes	characteristic
262	C₂×C₃	C₅×#466	1	2	yes	—
263	C₂×C₃	C₅×#467	1	2	yes	—
264	C₅	C₃×#1808	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	—	2	4	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	C₃×C₅×#107	1	4	yes	complemented
270	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂²	C₃×C₅×#121	1	2	yes	—
272	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
273	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
274	C₂²	C₃×C₅×#135	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₄	C₃×C₅×#117	1	1	yes	characteristic
279	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
280	C₃	C₅×#1808	1	1	yes	characteristic, direct product factor, Sylow
281	C₂	—	2	4	no	—
282	C₂	C₃×C₅×#380	1	1	yes	characteristic
283	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
284	C₂	C₃×C₅×#427	1	1	yes	characteristic
285	C₂	C₃×C₅×#466	1	2	yes	—
286	C₂	C₃×C₅×#467	1	2	yes	—

	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	C₂⁴×C₃×C₅	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#107	C₂	semidirect
4. C₃×C₅×#340	C₂	semidirect
5. C₅×#1808	C₃	direct (N × H)
8. C₂²×C₃×C₅×C₈	C₄	semidirect
15. C₃×#1808	C₅	direct (N × H)
17. C₂×C₅×#107	C₂×C₃	semidirect
19. C₅×#340	C₂×C₃	semidirect
37. C₂×C₃×#107	C₂×C₅	semidirect
39. C₃×#340	C₂×C₅	semidirect
42. C₂²×C₅×C₈	C₃×C₄	semidirect
49. #1808	C₃×C₅	direct (N × H)
69. C₃×C₅×C₈	C₂²⋊C₄	semidirect
72. C₂²×C₃×C₈	C₄×C₅	semidirect
96. C₂×#107	C₂×C₃×C₅	semidirect
98. #340	C₂×C₃×C₅	semidirect
103. C₂²×C₃×C₅	#107	semidirect
149. C₅×C₈	C₃×C₂²⋊C₄	semidirect
152. C₂²×C₈	C₃×C₄×C₅	semidirect
184. C₃×C₈	C₅×C₂²⋊C₄	semidirect
189. C₂²×C₅	C₃×#107	semidirect
216. C₃×C₅	#1808	direct (N × H)
221. C₂²×C₃	C₅×#107	semidirect
257. C₈	C₃×C₅×C₂²⋊C₄	semidirect
264. C₅	C₃×#1808	direct (N × H)
269. C₂²	C₃×C₅×#107	semidirect
280. C₃	C₅×#1808	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×C₈	2
C₂×C₃×C₅×#107	2
C₃×C₅×#380	2
C₃×C₅×#340	2
C₅×#1808	3
C₃×#1808	5

Subgroups — C3×C5×#1808 #14274842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1808 #14274842