Subgroups — C₃×C₅×#1546 #14274580

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 292 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	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	4	yes	maximal, complemented
5	C₅×#1546	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
19	C₃×C₄×C₅×C₈	—	2	1	no	—
20	C₃×#1546	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#431	C₂×C₃	1	1	yes	characteristic
23	C₅×#423	C₂×C₃	1	1	yes	characteristic
24	C₅×#332	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	—	4	2	no	—
28	C₂×C₃×C₅×Q₈	—	4	2	no	—
29	C₂²×C₃×C₄×C₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₃×C₅×M₁₆	—	4	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
45	C₃×#431	C₂×C₅	1	1	yes	characteristic
46	C₃×#423	C₂×C₅	1	1	yes	characteristic
47	C₃×#332	C₂×C₅	1	4	yes	complemented
48	C₂²×C₅×Q₈	—	2	1	no	—
49	C₂×C₅×M₁₆	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	—	2	1	no	—
52	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	—
53	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
56	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	1	no	—
58	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
59	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
60	C₅×C₂²⋊C₈	—	2	1	no	—
61	C₄×C₅×C₈	—	2	1	no	—
62	#1546	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₃×C₅×Q₈	—	4	4	no	—
65	C₃×C₅×Q₈	—	4	2	no	—
66	C₃×C₅×Q₈	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	4	4	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
75	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₃×C₅×C₈	—	4	2	no	—
80	C₃×C₅×C₈	—	4	2	no	—
81	C₂²×C₃×Q₈	—	2	1	no	—
82	C₂×C₃×M₁₆	—	2	1	no	—
83	C₃×C₄⋊Q₈	—	2	1	no	—
84	C₃×C₄⋊Q₈	—	2	1	no	—
85	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
86	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
87	C₂×C₃×C₄⋊C₄	—	2	1	no	—
88	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
89	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
90	C₃×C₄⋊C₈	—	2	1	no	—
91	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
92	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
93	C₃×C₂²⋊C₈	—	2	1	no	—
94	C₃×C₄×C₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	4	2	no	—
98	C₂×C₅×Q₈	—	4	2	no	—
99	C₂²×C₄×C₅	—	2	2	no	—
100	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₅×M₁₆	—	4	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
110	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
113	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
115	#431	C₂×C₃×C₅	1	1	yes	characteristic
116	#423	C₂×C₃×C₅	1	1	yes	characteristic
117	#332	C₂×C₃×C₅	1	4	yes	complemented
118	C₂²×C₃×C₅	—	2	1	no	—
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	#102	1	2	yes	—
121	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₃×C₄×C₅	—	2	1	no	—
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	4	4	no	—
125	C₃×C₄×C₅	#101	1	2	yes	—
126	C₃×C₄×C₅	#102	1	2	yes	—
127	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
129	C₃×C₄×C₅	#137	1	2	yes	—
130	C₂×C₃×Q₈	—	2	2	no	—
131	C₂×C₃×Q₈	—	2	2	no	—
132	C₂×C₃×Q₈	—	4	2	no	—
133	C₂×C₃×Q₈	—	4	2	no	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
138	C₃×M₁₆	—	4	2	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
145	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
146	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
147	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
148	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
149	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
150	C₅×Q₈	—	4	4	no	—
151	C₅×Q₈	—	4	2	no	—
152	C₅×Q₈	—	4	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	4	4	no	—
157	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
161	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
162	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₅×C₈	—	4	2	no	—
166	C₅×C₈	—	4	2	no	—
167	C₂²×Q₈	—	2	1	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₄⋊Q₈	—	2	1	no	—
170	C₄⋊Q₈	—	2	1	no	—
171	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	—
172	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
175	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
178	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
179	C₂²⋊C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₂×C₃×C₅	—	2	2	no	—
182	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
184	C₂×C₃×C₅	#419	1	1	yes	characteristic
185	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₃×Q₈	—	4	2	no	—
187	C₃×Q₈	—	4	2	no	—
188	C₃×Q₈	—	4	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
197	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
198	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₃×C₈	—	4	2	no	—
202	C₃×C₈	—	4	2	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	—	2	1	no	—
205	C₂²×C₅	C₃×#102	1	2	yes	—
206	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	4	4	no	—
210	C₄×C₅	C₃×#101	1	2	yes	—
211	C₄×C₅	C₃×#102	1	2	yes	—
212	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
213	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
214	C₄×C₅	C₃×#137	1	2	yes	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
223	M₁₆	—	4	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
230	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
231	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
232	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
233	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
234	C₃×C₅	#1546	1	1	yes	characteristic, direct product factor, Hall
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	C₅×#102	1	2	yes	—
238	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	4	4	no	—
242	C₃×C₄	C₅×#101	1	2	yes	—
243	C₃×C₄	C₅×#102	1	2	yes	—
244	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
245	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
246	C₃×C₄	C₅×#137	1	2	yes	—
247	C₂×C₅	—	2	2	no	—
248	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
249	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
250	C₂×C₅	C₃×#419	1	1	yes	characteristic
251	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
263	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
264	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
265	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
266	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	C₈	—	4	2	no	—
268	C₈	—	4	2	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	C₂×C₅×#101	1	1	yes	characteristic
271	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
272	C₂×C₃	C₅×#419	1	1	yes	characteristic
273	C₅	C₃×#1546	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	C₃×C₅×#102	1	2	yes	—
277	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	C₃×C₅×#101	1	2	yes	—
282	C₄	C₃×C₅×#102	1	2	yes	—
283	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
284	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
285	C₄	C₃×C₅×#137	1	2	yes	—
286	C₃	C₅×#1546	1	1	yes	characteristic, direct product factor, Sylow
287	C₂	—	2	2	no	—
288	C₂	C₂×C₃×C₅×#101	1	1	yes	characteristic
289	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
290	C₂	C₃×C₅×#419	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#332	C₂	semidirect
5. C₅×#1546	C₃	direct (N × H)
20. C₃×#1546	C₅	direct (N × H)
24. C₅×#332	C₂×C₃	semidirect
47. C₃×#332	C₂×C₅	semidirect
62. #1546	C₃×C₅	direct (N × H)
117. #332	C₂×C₃×C₅	semidirect
234. C₃×C₅	#1546	direct (N × H)
273. C₅	C₃×#1546	direct (N × H)
286. C₃	C₅×#1546	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
C₃×C₅×#431	2
C₃×C₅×#423	2
C₃×C₅×#332	2
C₅×#1546	3
C₃×#1546	5

Subgroups — C3×C5×#1546 #14274580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1546 #14274580