Subgroups — C₃×C₅×#1981 #14275015

Order 1920.

648 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	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#103	C₂	1	4	yes	maximal
4	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₅×#1981	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
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	—	2	1	no	—
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×#103	—	2	8	no	—
17	C₃×C₅×#95	—	2	1	no	—
18	C₃×#1981	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#103	C₂×C₃	1	4	yes	—
22	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	—	2	4	no	—
25	C₂×C₃×C₅×Q₈	—	2	4	no	—
26	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	—	2	4	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₄⋊C₄	—	2	4	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	—
38	C₃×C₄²×C₅	—	4	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
42	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
43	C₂×C₃×#103	C₂×C₅	1	4	yes	—
44	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
45	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
46	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
47	C₅×C₄⋊Q₈	—	2	2	no	—
48	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₄⋊C₄	—	2	1	no	—
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×C₄²×C₅	—	2	1	no	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×C₄⋊C₈	—	2	2	no	—
55	C₅×#103	—	2	8	no	—
56	C₅×#95	—	2	1	no	—
57	#1981	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×Q₈	—	2	8	no	—
60	C₃×C₅×Q₈	—	4	2	no	—
61	C₃×C₅×Q₈	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	4	4	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
71	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	#41	1	2	yes	—
76	C₃×C₅×C₈	—	4	2	no	—
77	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
78	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
79	C₃×C₄⋊Q₈	—	2	2	no	—
80	C₃×C₄⋊Q₈	C₄×C₅	1	4	yes	complemented
81	C₂×C₃×C₄⋊C₄	—	2	1	no	—
82	C₂×C₃×C₄⋊C₄	—	2	1	no	—
83	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
84	C₂×C₃×C₄²	—	2	1	no	—
85	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄⋊C₈	—	2	2	no	—
87	C₃×#103	—	2	8	no	—
88	C₃×#95	—	2	1	no	—
89	C₂×C₅×Q₈	—	2	2	no	—
90	C₂×C₅×Q₈	—	2	4	no	—
91	C₂×C₅×Q₈	—	2	4	no	—
92	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₅×C₈	—	2	4	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	C₃×D₄	1	4	yes	—
104	C₄²×C₅	—	4	2	no	—
105	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
106	C₄²×C₅	C₃×D₄	1	2	yes	—
107	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
108	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
109	C₂×#103	C₂×C₃×C₅	1	4	yes	—
110	C₄×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	#121	1	2	yes	—
113	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
114	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
115	C₂²×C₃×C₅	#137	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	—	4	4	no	—
118	C₃×C₄×C₅	—	4	4	no	—
119	C₃×C₄×C₅	#103	1	4	yes	complemented
120	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
121	C₃×C₄×C₅	#124	1	1	yes	characteristic
122	C₃×C₄×C₅	#127	1	1	yes	characteristic
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	2	4	no	—
125	C₂×C₃×Q₈	—	2	4	no	—
126	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	—	2	2	no	—
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
132	C₂×C₃×C₈	—	2	4	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₄⋊C₄	—	2	4	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	—
138	C₃×C₄²	—	4	2	no	—
139	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
140	C₃×C₄²	C₅×D₄	1	2	yes	—
141	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₅×Q₈	—	2	8	no	—
143	C₅×Q₈	—	4	2	no	—
144	C₅×Q₈	—	4	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	4	4	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
154	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
155	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₃×#41	1	2	yes	—
159	C₅×C₈	—	4	2	no	—
160	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
161	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₃×C₄×C₅	1	4	yes	complemented
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
169	C₄⋊C₈	—	2	2	no	—
170	#103	—	2	8	no	—
171	#95	—	2	1	no	—
172	C₂×C₃×C₅	#390	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
174	C₂×C₃×C₅	#419	1	1	yes	characteristic
175	C₂×C₃×C₅	#461	1	2	yes	—
176	C₂×C₃×C₅	#462	1	2	yes	—
177	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
178	C₃×Q₈	—	2	8	no	—
179	C₃×Q₈	—	4	2	no	—
180	C₃×Q₈	—	4	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	4	4	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
188	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
190	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
191	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
193	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₅×#41	1	2	yes	—
195	C₃×C₈	—	4	2	no	—
196	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₅	C₃×#121	1	2	yes	—
198	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
199	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
200	C₂²×C₅	C₃×#137	1	2	yes	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	4	4	no	—
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	C₃×#103	1	4	yes	complemented
205	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
206	C₄×C₅	C₃×#124	1	1	yes	characteristic
207	C₄×C₅	C₃×#127	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
216	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	C₃×C₅×D₄	1	4	yes	—
223	C₄²	—	4	2	no	—
224	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
225	C₄²	C₃×C₅×D₄	1	2	yes	—
226	C₃×C₅	#1981	1	1	yes	characteristic, direct product factor, Hall
227	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²×C₃	C₅×#121	1	2	yes	—
229	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
231	C₂²×C₃	C₅×#137	1	2	yes	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	4	4	no	—
234	C₃×C₄	—	4	4	no	—
235	C₃×C₄	C₅×#103	1	4	yes	complemented
236	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
237	C₃×C₄	C₅×#124	1	1	yes	characteristic
238	C₃×C₄	C₅×#127	1	1	yes	characteristic
239	C₂×C₅	C₃×#390	1	1	yes	characteristic
240	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
241	C₂×C₅	C₃×#419	1	1	yes	characteristic
242	C₂×C₅	C₃×#461	1	2	yes	—
243	C₂×C₅	C₃×#462	1	2	yes	—
244	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
245	Q₈	—	2	8	no	—
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
255	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
257	C₂×C₄	C₃×C₅×Q₁₆	1	2	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	—	4	2	no	—
263	C₂×C₃	C₅×#390	1	1	yes	characteristic
264	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
265	C₂×C₃	C₅×#419	1	1	yes	characteristic
266	C₂×C₃	C₅×#461	1	2	yes	—
267	C₂×C₃	C₅×#462	1	2	yes	—
268	C₅	C₃×#1981	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₂²	C₃×C₅×#121	1	2	yes	—
271	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
272	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
273	C₂²	C₃×C₅×#137	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	4	no	—
277	C₄	C₃×C₅×#103	1	4	yes	complemented
278	C₄	C₃×C₄×C₅×D₄	1	2	yes	—
279	C₄	C₃×C₅×#124	1	1	yes	characteristic
280	C₄	C₃×C₅×#127	1	1	yes	characteristic
281	C₃	C₅×#1981	1	1	yes	characteristic, direct product factor, Sylow
282	C₂	C₃×C₅×#390	1	1	yes	characteristic
283	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic
284	C₂	C₃×C₅×#419	1	1	yes	characteristic
285	C₂	C₃×C₅×#461	1	2	yes	—
286	C₂	C₃×C₅×#462	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×#1981	C₃	direct (N × H)
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
18. C₃×#1981	C₅	direct (N × H)
48. C₅×C₄⋊Q₈	C₃×C₄	semidirect
57. #1981	C₃×C₅	direct (N × H)
80. C₃×C₄⋊Q₈	C₄×C₅	semidirect
119. C₃×C₄×C₅	#103	semidirect
163. C₄⋊Q₈	C₃×C₄×C₅	semidirect
204. C₄×C₅	C₃×#103	semidirect
226. C₃×C₅	#1981	direct (N × H)
235. C₃×C₄	C₅×#103	semidirect
268. C₅	C₃×#1981	direct (N × H)
277. C₄	C₃×C₅×#103	semidirect
281. C₃	C₅×#1981	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
C₂×C₃×C₅×#103	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#1981	3
C₃×#1981	5

Subgroups — C3×C5×#1981 #14275015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1981 #14275015