Subgroups — C₃×C₅×#2659 #14275693

Order 1920.

1048 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#398	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#395	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#380	C₂	1	4	yes	maximal
5	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic
6	C₅×#2659	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
10	C₃×C₅×C₂²⋊Q₈	—	2	8	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₃×C₅×#95	C₂²	1	4	yes	—
17	C₃×C₅×#95	C₂²	1	8	yes	—
18	C₃×#2659	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
20	C₅×#398	C₂×C₃	1	4	yes	complemented
21	C₅×#395	C₂×C₃	1	4	yes	complemented
22	C₅×#380	C₂×C₃	1	4	yes	—
23	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₅×Q₈	—	2	4	no	—
26	C₂×C₃×C₅×Q₈	—	4	4	no	—
27	C₂²×C₃×C₄×C₅	—	4	2	no	—
28	C₂²×C₃×C₄×C₅	—	4	2	no	—
29	C₂²×C₃×C₄×C₅	—	4	2	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	Q₈	1	8	yes	complemented
32	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
36	C₃×C₅×C₄⋊C₄	—	2	4	no	—
37	C₃×C₅×C₄⋊C₄	—	2	8	no	—
38	C₃×C₅×C₄⋊C₄	—	4	4	no	—
39	C₃×C₅×C₄⋊C₄	—	4	4	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
41	C₂×C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	—
42	C₃×#398	C₂×C₅	1	4	yes	complemented
43	C₃×#395	C₂×C₅	1	4	yes	complemented
44	C₃×#380	C₂×C₅	1	4	yes	—
45	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic
46	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
47	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
49	C₅×C₂²⋊Q₈	—	2	8	no	—
50	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
53	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
54	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
55	C₅×#95	C₂²×C₃	1	4	yes	—
56	C₅×#95	C₂²×C₃	1	8	yes	—
57	#2659	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	—	2	2	no	—
59	C₂³×C₃×C₅	—	2	1	no	—
60	C₂³×C₃×C₅	—	2	1	no	—
61	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
62	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
63	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
64	C₃×C₅×Q₈	—	4	8	no	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	—	2	4	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	—	2	8	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	4	no	—
74	C₂×C₃×C₄×C₅	—	4	8	no	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	8	yes	—
77	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
78	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
79	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
80	C₃×C₂²⋊Q₈	—	2	8	no	—
81	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
82	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
85	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
86	C₃×#95	C₂²×C₅	1	4	yes	—
87	C₃×#95	C₂²×C₅	1	8	yes	—
88	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₅×Q₈	—	2	4	no	—
90	C₂×C₅×Q₈	—	4	4	no	—
91	C₂²×C₄×C₅	—	4	2	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂²×C₄×C₅	—	4	2	no	—
94	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
95	C₂²×C₄×C₅	C₃×Q₈	1	8	yes	complemented
96	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
97	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
98	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
99	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	8	no	—
102	C₅×C₄⋊C₄	—	4	4	no	—
103	C₅×C₄⋊C₄	—	4	4	no	—
104	C₅×C₂²⋊C₄	—	2	8	no	—
105	C₂×C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	—
106	#398	C₂×C₃×C₅	1	4	yes	complemented
107	#395	C₂×C₃×C₅	1	4	yes	complemented
108	#380	C₂×C₃×C₅	1	4	yes	—
109	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	—	2	4	no	—
113	C₂²×C₃×C₅	—	2	4	no	—
114	C₂²×C₃×C₅	C₄⋊Q₈	1	4	yes	—
115	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₂²×Q₈	1	2	yes	—
117	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
120	C₃×C₄×C₅	—	2	4	no	—
121	C₃×C₄×C₅	—	2	8	no	—
122	C₃×C₄×C₅	—	4	4	no	—
123	C₃×C₄×C₅	—	4	4	no	—
124	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
125	C₂×C₃×Q₈	—	2	4	no	—
126	C₂×C₃×Q₈	—	4	4	no	—
127	C₂²×C₃×C₄	—	4	2	no	—
128	C₂²×C₃×C₄	—	4	2	no	—
129	C₂²×C₃×C₄	—	4	2	no	—
130	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
131	C₂²×C₃×C₄	C₅×Q₈	1	8	yes	complemented
132	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
133	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
134	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
135	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	2	8	no	—
138	C₃×C₄⋊C₄	—	4	4	no	—
139	C₃×C₄⋊C₄	—	4	4	no	—
140	C₃×C₂²⋊C₄	—	2	8	no	—
141	C₂³×C₅	—	2	2	no	—
142	C₂³×C₅	—	2	1	no	—
143	C₂³×C₅	—	2	1	no	—
144	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
145	C₂³×C₅	C₂×C₃×Q₈	1	4	yes	—
146	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
147	C₅×Q₈	—	4	8	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	8	no	—
155	C₂×C₄×C₅	—	2	4	no	—
156	C₂×C₄×C₅	—	4	4	no	—
157	C₂×C₄×C₅	—	4	8	no	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
159	C₂×C₄×C₅	C₂×C₃×Q₈	1	8	yes	—
160	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
161	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
163	C₂²⋊Q₈	—	2	8	no	—
164	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
165	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
167	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
169	#95	C₂²×C₃×C₅	1	4	yes	—
170	#95	C₂²×C₃×C₅	1	8	yes	—
171	C₂×C₃×C₅	—	2	4	no	—
172	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
173	C₂×C₃×C₅	#534	1	1	yes	characteristic
174	C₂×C₃×C₅	#537	1	1	yes	characteristic
175	C₂×C₃×C₅	#543	1	2	yes	—
176	C₂×C₃×C₅	#544	1	2	yes	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	—	2	1	no	—
179	C₂³×C₃	—	2	1	no	—
180	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
181	C₂³×C₃	C₂×C₅×Q₈	1	4	yes	—
182	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
183	C₃×Q₈	—	4	8	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	2	4	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	—	4	8	no	—
194	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
195	C₂×C₃×C₄	C₂×C₅×Q₈	1	8	yes	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	C₃×C₄⋊Q₈	1	4	yes	—
201	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
202	C₂²×C₅	C₂²×C₃×Q₈	1	2	yes	—
203	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
204	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
205	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
206	C₄×C₅	—	2	4	no	—
207	C₄×C₅	—	2	8	no	—
208	C₄×C₅	—	4	4	no	—
209	C₄×C₅	—	4	4	no	—
210	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
217	C₂²×C₄	C₃×C₅×Q₈	1	8	yes	complemented
218	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
219	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
220	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
221	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	8	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₂²⋊C₄	—	2	8	no	—
227	C₃×C₅	#2659	1	1	yes	characteristic, direct product factor, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	C₅×C₄⋊Q₈	1	4	yes	—
233	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₃	C₂²×C₅×Q₈	1	2	yes	—
235	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
236	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
237	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	8	no	—
240	C₃×C₄	—	4	4	no	—
241	C₃×C₄	—	4	4	no	—
242	C₂×C₅	—	2	4	no	—
243	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₅	C₃×#534	1	1	yes	characteristic
245	C₂×C₅	C₃×#537	1	1	yes	characteristic
246	C₂×C₅	C₃×#543	1	2	yes	—
247	C₂×C₅	C₃×#544	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
252	C₂³	C₂×C₃×C₅×Q₈	1	4	yes	—
253	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
254	Q₈	—	4	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
266	C₂×C₄	C₂×C₃×C₅×Q₈	1	8	yes	—
267	C₂×C₃	—	2	4	no	—
268	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
269	C₂×C₃	C₅×#534	1	1	yes	characteristic
270	C₂×C₃	C₅×#537	1	1	yes	characteristic
271	C₂×C₃	C₅×#543	1	2	yes	—
272	C₂×C₃	C₅×#544	1	2	yes	—
273	C₅	C₃×#2659	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	C₃×C₅×C₄⋊Q₈	1	4	yes	—
279	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
280	C₂²	C₂²×C₃×C₅×Q₈	1	2	yes	—
281	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
282	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	8	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₃	C₅×#2659	1	1	yes	characteristic, direct product factor, Sylow
289	C₂	—	2	4	no	—
290	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
291	C₂	C₃×C₅×#534	1	1	yes	characteristic
292	C₂	C₃×C₅×#537	1	1	yes	characteristic
293	C₂	C₃×C₅×#543	1	2	yes	—
294	C₂	C₃×C₅×#544	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁴×C₃×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#398	C₂	semidirect
3. C₃×C₅×#395	C₂	semidirect
6. C₅×#2659	C₃	direct (N × H)
18. C₃×#2659	C₅	direct (N × H)
20. C₅×#398	C₂×C₃	semidirect
21. C₅×#395	C₂×C₃	semidirect
31. C₂²×C₃×C₄×C₅	Q₈	semidirect
42. C₃×#398	C₂×C₅	semidirect
43. C₃×#395	C₂×C₅	semidirect
57. #2659	C₃×C₅	direct (N × H)
95. C₂²×C₄×C₅	C₃×Q₈	semidirect
106. #398	C₂×C₃×C₅	semidirect
107. #395	C₂×C₃×C₅	semidirect
131. C₂²×C₃×C₄	C₅×Q₈	semidirect
217. C₂²×C₄	C₃×C₅×Q₈	semidirect
227. C₃×C₅	#2659	direct (N × H)
273. C₅	C₃×#2659	direct (N × H)
288. C₃	C₅×#2659	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×C₂²⋊Q₈	2
C₃×C₅×#398	2
C₃×C₅×#395	2
C₃×C₅×#380	2
C₂×C₃×C₅×#95	2
C₅×#2659	3
C₃×#2659	5

Subgroups — C3×C5×#2659 #14275693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2659 #14275693