Subgroups — C₃×C₅×#2087 #14275121

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#389	C₂	1	2	yes	maximal
4	C₃×C₅×#340	C₂	1	2	yes	maximal
5	C₃×C₅×#340	C₂	1	1	yes	maximal, characteristic
6	C₅×#2087	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	—	2	2	no	—
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	C₃×C₅×#95	—	2	2	no	—
16	C₃×#2087	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₅×#389	C₂×C₃	1	2	yes	—
20	C₅×#340	C₂×C₃	1	2	yes	—
21	C₅×#340	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	4	4	no	—
35	C₃×C₅×C₄⋊C₄	Q₈	1	4	yes	—
36	C₃×C₄²×C₅	—	2	1	no	—
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic
39	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
40	C₃×#389	C₂×C₅	1	2	yes	—
41	C₃×#340	C₂×C₅	1	2	yes	—
42	C₃×#340	C₂×C₅	1	1	yes	characteristic
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	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×#107	—	2	2	no	—
50	C₅×C₄⋊C₈	—	2	2	no	—
51	C₅×#95	—	2	2	no	—
52	#2087	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	—	2	1	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	4	2	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
69	C₃×C₅×C₈	—	2	2	no	—
70	C₃×C₅×C₈	—	4	1	no	—
71	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₃×C₄⋊C₄	—	2	2	no	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
75	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
76	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
77	C₃×#107	—	2	2	no	—
78	C₃×C₄⋊C₈	—	2	2	no	—
79	C₃×#95	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	2	no	—
82	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	—	4	4	no	—
93	C₅×C₄⋊C₄	C₃×Q₈	1	4	yes	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	2	1	no	—
96	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic
97	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
98	#389	C₂×C₃×C₅	1	2	yes	—
99	#340	C₂×C₃×C₅	1	2	yes	—
100	#340	C₂×C₃×C₅	1	1	yes	characteristic
101	C₂²×C₃×C₅	#121	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
103	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
104	C₂²×C₃×C₅	#135	1	1	yes	characteristic
105	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	4	1	no	—
108	C₃×C₄×C₅	—	4	4	no	—
109	C₃×C₄×C₅	—	8	2	no	—
110	C₃×C₄×C₅	#123	1	1	yes	characteristic
111	C₃×C₄×C₅	#125	1	2	yes	—
112	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	4	4	no	—
126	C₃×C₄⋊C₄	C₅×Q₈	1	4	yes	—
127	C₃×C₄²	—	2	1	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₃×#41	1	2	yes	—
144	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	—	4	1	no	—
147	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
152	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
153	#107	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	#95	—	2	2	no	—
156	C₂×C₃×C₅	#398	1	1	yes	characteristic
157	C₂×C₃×C₅	#467	1	1	yes	characteristic
158	C₂×C₃×C₅	#469	1	1	yes	characteristic
159	C₂×C₃×C₅	#477	1	2	yes	—
160	C₂×C₃×C₅	#478	1	2	yes	—
161	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	4	2	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	—	4	2	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
175	C₂×C₃×C₄	C₅×#41	1	2	yes	—
176	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
177	C₃×C₈	—	2	2	no	—
178	C₃×C₈	—	4	1	no	—
179	C₂²×C₅	C₃×#121	1	1	yes	characteristic
180	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
181	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
182	C₂²×C₅	C₃×#135	1	1	yes	characteristic
183	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	—	8	2	no	—
188	C₄×C₅	C₃×#123	1	1	yes	characteristic
189	C₄×C₅	C₃×#125	1	2	yes	—
190	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	C₃×C₅×Q₈	1	4	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₃×C₅	#2087	1	1	yes	characteristic, direct product factor, Hall
208	C₂²×C₃	C₅×#121	1	1	yes	characteristic
209	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
210	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
211	C₂²×C₃	C₅×#135	1	1	yes	characteristic
212	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	4	1	no	—
215	C₃×C₄	—	4	4	no	—
216	C₃×C₄	—	8	2	no	—
217	C₃×C₄	C₅×#123	1	1	yes	characteristic
218	C₃×C₄	C₅×#125	1	2	yes	—
219	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
220	C₂×C₅	C₃×#398	1	1	yes	characteristic
221	C₂×C₅	C₃×#467	1	1	yes	characteristic
222	C₂×C₅	C₃×#469	1	1	yes	characteristic
223	C₂×C₅	C₃×#477	1	2	yes	—
224	C₂×C₅	C₃×#478	1	2	yes	—
225	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
237	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
239	C₂×C₄	C₃×C₅×#41	1	2	yes	—
240	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	4	1	no	—
243	C₂×C₃	C₅×#398	1	1	yes	characteristic
244	C₂×C₃	C₅×#467	1	1	yes	characteristic
245	C₂×C₃	C₅×#469	1	1	yes	characteristic
246	C₂×C₃	C₅×#477	1	2	yes	—
247	C₂×C₃	C₅×#478	1	2	yes	—
248	C₅	C₃×#2087	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	C₃×C₅×#121	1	1	yes	characteristic
250	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
251	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
252	C₂²	C₃×C₅×#135	1	1	yes	characteristic
253	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	2	no	—
258	C₄	C₃×C₅×#123	1	1	yes	characteristic
259	C₄	C₃×C₅×#125	1	2	yes	—
260	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
261	C₃	C₅×#2087	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	C₃×C₅×#398	1	1	yes	characteristic
263	C₂	C₃×C₅×#467	1	1	yes	characteristic
264	C₂	C₃×C₅×#469	1	1	yes	characteristic
265	C₂	C₃×C₅×#477	1	2	yes	—
266	C₂	C₃×C₅×#478	1	2	yes	—

	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×#2087	C₃	direct (N × H)
16. C₃×#2087	C₅	direct (N × H)
52. #2087	C₃×C₅	direct (N × H)
207. C₃×C₅	#2087	direct (N × H)
248. C₅	C₃×#2087	direct (N × H)
261. C₃	C₅×#2087	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×#107	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₅×#389	2
C₃×C₅×#340	2
C₅×#2087	3
C₃×#2087	5

Subgroups — C3×C5×#2087 #14275121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2087 #14275121