Subgroups — C₃×C₅×#1987 #14275021

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#340	C₂	1	4	yes	maximal
5	C₅×#1987	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²×C₅	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×C₅×#95	—	2	1	no	—
17	C₃×#1987	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₅×#340	C₂×C₃	1	4	yes	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	4	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	4	no	—
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₃×C₅×C₄⋊C₄	Q₈	1	4	yes	complemented
36	C₃×C₄²×C₅	—	4	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
40	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
41	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
42	C₃×#340	C₂×C₅	1	4	yes	—
43	C₂²×C₅×Q₈	—	2	1	no	—
44	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
50	C₂×C₄²×C₅	—	2	1	no	—
51	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₅×C₄⋊C₈	—	2	2	no	—
53	C₅×#95	—	2	1	no	—
54	#1987	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	4	8	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
66	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₃×C₅×C₈	—	4	2	no	—
72	C₂²×C₃×Q₈	—	2	1	no	—
73	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
74	C₃×C₄⋊Q₈	—	2	2	no	—
75	C₃×C₄⋊Q₈	C₄×C₅	1	4	yes	complemented
76	C₂×C₃×C₄⋊C₄	—	2	1	no	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄²	—	2	1	no	—
80	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₄⋊C₈	—	2	2	no	—
82	C₃×#95	—	2	1	no	—
83	C₂×C₅×Q₈	—	2	4	no	—
84	C₂×C₅×Q₈	—	4	4	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
96	C₅×C₄⋊C₄	C₃×Q₈	1	4	yes	complemented
97	C₄²×C₅	—	4	2	no	—
98	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
99	C₄²×C₅	C₃×D₄	1	2	yes	—
100	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
101	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
102	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
103	#340	C₂×C₃×C₅	1	4	yes	—
104	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
106	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
108	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	4	4	no	—
111	C₃×C₄×C₅	—	4	4	no	—
112	C₃×C₄×C₅	#103	1	4	yes	—
113	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
114	C₃×C₄×C₅	#124	1	1	yes	characteristic
115	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₃×Q₈	—	2	4	no	—
117	C₂×C₃×Q₈	—	4	4	no	—
118	C₂²×C₃×C₄	—	2	2	no	—
119	C₂²×C₃×C₄	—	2	2	no	—
120	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₃×C₄⋊C₄	—	2	4	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
129	C₃×C₄⋊C₄	C₅×Q₈	1	4	yes	complemented
130	C₃×C₄²	—	4	2	no	—
131	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
132	C₃×C₄²	C₅×D₄	1	2	yes	—
133	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
134	C₅×Q₈	—	4	8	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
144	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
145	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₃×#41	1	2	yes	—
149	C₅×C₈	—	4	2	no	—
150	C₂²×Q₈	—	2	1	no	—
151	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	C₃×C₄×C₅	1	4	yes	complemented
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	#95	—	2	1	no	—
161	C₂×C₃×C₅	#391	1	1	yes	characteristic
162	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
163	C₂×C₃×C₅	#418	1	1	yes	characteristic
164	C₂×C₃×C₅	#476	1	2	yes	—
165	C₂×C₃×C₅	#477	1	2	yes	—
166	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
167	C₃×Q₈	—	4	8	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	4	2	no	—
173	C₂×C₃×C₄	—	4	4	no	—
174	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
175	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
176	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
177	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
181	C₂×C₃×C₄	C₅×#41	1	2	yes	—
182	C₃×C₈	—	4	2	no	—
183	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
185	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
186	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
187	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	4	4	no	—
190	C₄×C₅	—	4	4	no	—
191	C₄×C₅	C₃×#103	1	4	yes	—
192	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
193	C₄×C₅	C₃×#124	1	1	yes	characteristic
194	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
208	C₄⋊C₄	C₃×C₅×Q₈	1	4	yes	complemented
209	C₄²	—	4	2	no	—
210	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
211	C₄²	C₃×C₅×D₄	1	2	yes	—
212	C₃×C₅	#1987	1	1	yes	characteristic, direct product factor, Hall
213	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
215	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
216	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
217	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	—	4	4	no	—
221	C₃×C₄	C₅×#103	1	4	yes	—
222	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
223	C₃×C₄	C₅×#124	1	1	yes	characteristic
224	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
225	C₂×C₅	C₃×#391	1	1	yes	characteristic
226	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
227	C₂×C₅	C₃×#418	1	1	yes	characteristic
228	C₂×C₅	C₃×#476	1	2	yes	—
229	C₂×C₅	C₃×#477	1	2	yes	—
230	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
231	Q₈	—	4	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
239	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
240	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
241	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
242	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
244	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
245	C₂×C₄	C₃×C₅×#41	1	2	yes	—
246	C₈	—	4	2	no	—
247	C₂×C₃	C₅×#391	1	1	yes	characteristic
248	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
249	C₂×C₃	C₅×#418	1	1	yes	characteristic
250	C₂×C₃	C₅×#476	1	2	yes	—
251	C₂×C₃	C₅×#477	1	2	yes	—
252	C₅	C₃×#1987	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
255	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
256	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
257	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	4	4	no	—
260	C₄	—	4	4	no	—
261	C₄	C₃×C₅×#103	1	4	yes	—
262	C₄	C₃×C₄×C₅×Q₈	1	2	yes	—
263	C₄	C₃×C₅×#124	1	1	yes	characteristic
264	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
265	C₃	C₅×#1987	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	C₃×C₅×#391	1	1	yes	characteristic
267	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic
268	C₂	C₃×C₅×#418	1	1	yes	characteristic
269	C₂	C₃×C₅×#476	1	2	yes	—
270	C₂	C₃×C₅×#477	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
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
5. C₅×#1987	C₃	direct (N × H)
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
17. C₃×#1987	C₅	direct (N × H)
35. C₃×C₅×C₄⋊C₄	Q₈	semidirect
46. C₅×C₄⋊Q₈	C₃×C₄	semidirect
54. #1987	C₃×C₅	direct (N × H)
75. C₃×C₄⋊Q₈	C₄×C₅	semidirect
96. C₅×C₄⋊C₄	C₃×Q₈	semidirect
129. C₃×C₄⋊C₄	C₅×Q₈	semidirect
153. C₄⋊Q₈	C₃×C₄×C₅	semidirect
208. C₄⋊C₄	C₃×C₅×Q₈	semidirect
212. C₃×C₅	#1987	direct (N × H)
252. C₅	C₃×#1987	direct (N × H)
265. C₃	C₅×#1987	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×C₄⋊Q₈	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₅×#340	2
C₅×#1987	3
C₃×#1987	5

Subgroups — C3×C5×#1987 #14275021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1987 #14275021