Subgroups — C₅×#83288 #14439216

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#951	C₂	1	4	yes	maximal, complemented
3	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5092	C₂	1	2	yes	maximal, complemented
5	C₅×#5076	C₂	1	2	yes	maximal, complemented
6	C₅×#5060	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3152	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×#41	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₃×C₅×#124	C₂²	1	2	yes	—
14	C₃×C₅×#121	—	2	2	no	—
15	C₃×C₄×C₅×D₄	—	2	2	no	—
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₅×#951	—	2	16	no	—
18	C₂×C₅×#216	C₂²	1	8	yes	—
19	C₂×C₅×#214	C₂²	1	8	yes	—
20	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
21	C₂×C₅×#208	C₂²	1	2	yes	—
22	C₅×#869	—	2	2	no	—
23	C₅×#869	C₂²	1	2	yes	—
24	C₅×#869	C₂²	1	2	yes	—
25	C₅×#803	C₂²	1	2	yes	—
26	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
27	#83288	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₅×#137	—	3	4	no	—
29	C₅×#532	D₃	1	1	yes	characteristic
30	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
31	C₅×#497	—	3	4	no	—
32	C₅×#494	—	3	2	no	—
33	C₅×#492	—	3	2	no	—
34	C₄×C₅×M₁₆	—	3	1	no	—
35	C₃×C₅×#41	—	2	8	no	—
36	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
37	C₂×C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₅×D₄	—	2	2	no	—
39	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
40	C₂²×C₃×C₄×C₅	—	2	2	no	—
41	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂²×C₅×Dic₃	—	2	2	no	—
44	C₂×C₅×Dic₆	—	2	8	no	—
45	C₂×C₅×Dic₆	—	2	2	no	—
46	C₂×C₅×Dic₆	C₂³	1	4	yes	—
47	C₃×C₅×C₄⋊C₄	—	2	2	no	—
48	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×#216	—	2	16	no	—
53	C₅×#214	—	2	16	no	—
54	C₅×C₄⋊Dic₃	—	2	4	no	—
55	C₅×C₄⋊Dic₃	—	2	4	no	—
56	C₅×#208	D₄	1	8	yes	complemented
57	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
58	C₃×#532	C₂×C₅	1	1	yes	characteristic
59	C₂×#951	C₂×C₅	1	4	yes	complemented
60	C₂×#869	C₂×C₅	1	1	yes	characteristic, complemented
61	#5092	C₂×C₅	1	2	yes	complemented
62	#5076	C₂×C₅	1	2	yes	complemented
63	#5060	C₂×C₅	1	4	yes	complemented
64	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×C₅×#41	C₂×D₃	1	2	yes	—
66	C₂²×C₅×Q₈	—	3	2	no	—
67	C₅×#137	—	6	16	no	—
68	C₂×C₅×Q₁₆	—	3	8	no	—
69	C₂×C₅×SD₁₆	—	3	8	no	—
70	C₂×C₅×M₁₆	—	3	2	no	—
71	C₅×C₄⋊Q₈	—	3	2	no	—
72	C₅×C₄⋊Q₈	—	3	2	no	—
73	C₅×C₄⋊Q₈	—	6	2	no	—
74	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
75	C₅×#127	C₂×D₃	1	1	yes	characteristic
76	C₅×#124	C₂×D₃	1	2	yes	—
77	C₅×#121	—	2	2	no	—
78	C₄×C₅×D₄	—	2	2	no	—
79	C₂×C₅×C₄⋊C₄	—	6	2	no	—
80	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
81	C₅×#97	—	3	2	no	—
82	C₄×C₅×C₈	—	3	2	no	—
83	#3152	—	3	1	no	Sylow
84	C₂³×C₃×C₅	—	2	2	no	—
85	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
86	C₃×C₅×Q₈	—	2	4	no	—
87	C₃×C₅×D₄	—	2	4	no	—
88	C₃×C₅×D₄	—	4	4	no	—
89	C₃×C₅×D₄	—	4	4	no	—
90	C₂×C₃×C₄×C₅	—	2	4	no	—
91	C₂×C₃×C₄×C₅	—	2	2	no	—
92	C₂×C₃×C₄×C₅	—	4	4	no	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂×C₅×Dic₃	—	2	4	no	—
99	C₂×C₅×Dic₃	—	4	4	no	—
100	C₅×Dic₆	—	2	8	no	—
101	C₅×Dic₆	—	4	4	no	—
102	C₅×C₃⋊C₈	C₂×D₄	1	8	yes	—
103	C₂×C₃×#41	C₂²×C₅	1	2	yes	—
104	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
105	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
106	C₃×#127	C₂²×C₅	1	1	yes	characteristic
107	C₃×#124	C₂²×C₅	1	2	yes	—
108	C₃×#121	—	2	2	no	—
109	C₃×C₄×D₄	—	2	2	no	—
110	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
111	#951	—	2	16	no	—
112	C₂×#216	C₂²×C₅	1	8	yes	—
113	C₂×#214	C₂²×C₅	1	8	yes	—
114	C₂×C₄⋊Dic₃	—	2	2	no	—
115	C₂×#208	C₂²×C₅	1	2	yes	—
116	#869	—	2	2	no	—
117	#869	C₂²×C₅	1	2	yes	—
118	#869	C₂²×C₅	1	2	yes	—
119	#803	C₂²×C₅	1	2	yes	—
120	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
121	C₅×#41	—	2	8	no	—
122	C₂×C₅×Q₈	—	3	4	no	—
123	C₂×C₅×Q₈	—	6	2	no	—
124	C₂×C₅×Q₈	—	6	8	no	—
125	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
126	C₂×C₅×D₄	—	2	2	no	—
127	C₂×C₅×D₄	—	2	2	no	—
128	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
129	C₂²×C₄×C₅	—	2	2	no	—
130	C₂²×C₄×C₅	—	6	2	no	—
131	C₂²×C₄×C₅	#67	1	2	yes	—
132	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₅×Q₁₆	—	6	16	no	—
134	C₅×SD₁₆	—	6	16	no	—
135	C₅×M₁₆	—	3	8	no	—
136	C₂×C₅×C₈	—	3	4	no	—
137	C₅×C₄⋊C₄	—	2	2	no	—
138	C₅×C₄⋊C₄	—	6	4	no	—
139	C₅×C₄⋊C₄	—	6	4	no	—
140	C₅×C₂²⋊C₄	—	2	4	no	—
141	C₄²×C₅	#67	1	2	yes	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×#137	—	3	4	no	—
145	#532	C₅×D₃	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	—	3	1	no	—
147	#497	—	3	4	no	—
148	#494	—	3	2	no	—
149	#492	—	3	2	no	—
150	C₄×M₁₆	—	3	1	no	—
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	—	4	2	no	—
153	C₂²×C₃×C₅	—	4	4	no	—
154	C₂²×C₃×C₅	#127	1	2	yes	—
155	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	—	4	2	no	—
158	C₃×C₄×C₅	#127	1	2	yes	—
159	C₃×C₄×C₅	#137	1	4	yes	complemented
160	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
161	C₅×Dic₃	—	4	4	no	—
162	C₃×#41	—	2	8	no	—
163	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
164	C₂×C₃×D₄	—	2	2	no	—
165	C₂×C₃×D₄	—	2	2	no	—
166	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
167	C₂²×C₃×C₄	—	2	2	no	—
168	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
169	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂²×Dic₃	—	2	2	no	—
171	C₂×Dic₆	—	2	8	no	—
172	C₂×Dic₆	—	2	2	no	—
173	C₂×Dic₆	C₂³×C₅	1	4	yes	—
174	C₃×C₄⋊C₄	—	2	2	no	—
175	C₃×C₂²⋊C₄	—	2	4	no	—
176	C₃×C₄²	C₅×D₄	1	2	yes	—
177	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
178	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
179	#216	—	2	16	no	—
180	#214	—	2	16	no	—
181	C₄⋊Dic₃	—	2	4	no	—
182	C₄⋊Dic₃	—	2	4	no	—
183	#208	C₅×D₄	1	8	yes	complemented
184	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
185	C₂³×C₅	—	2	2	no	—
186	C₂³×C₅	C₂×#67	1	1	yes	characteristic
187	C₅×Q₈	—	2	4	no	—
188	C₅×Q₈	—	6	8	no	—
189	C₅×Q₈	—	12	4	no	—
190	C₅×D₄	—	2	4	no	—
191	C₅×D₄	—	4	4	no	—
192	C₅×D₄	—	4	4	no	—
193	C₂×C₄×C₅	—	2	4	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	4	4	no	—
196	C₂×C₄×C₅	—	6	4	no	—
197	C₂×C₄×C₅	—	12	4	no	—
198	C₂×C₄×C₅	D₃×D₄	1	4	yes	—
199	C₂×C₄×C₅	C₂×#67	1	2	yes	—
200	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×#67	1	2	yes	—
202	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
203	C₅×C₈	—	3	8	no	—
204	C₂×#41	C₂×C₅×D₃	1	2	yes	—
205	C₂²×Q₈	—	3	2	no	—
206	#137	—	6	16	no	—
207	C₂×Q₁₆	—	3	8	no	—
208	C₂×SD₁₆	—	3	8	no	—
209	C₂×M₁₆	—	3	2	no	—
210	C₄⋊Q₈	—	3	2	no	—
211	C₄⋊Q₈	—	3	2	no	—
212	C₄⋊Q₈	—	6	2	no	—
213	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
214	#127	C₂×C₅×D₃	1	1	yes	characteristic
215	#124	C₂×C₅×D₃	1	2	yes	—
216	#121	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	6	2	no	—
219	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
220	#97	—	3	2	no	—
221	C₄×C₈	—	3	2	no	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	—	8	2	no	—
224	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
225	C₂×C₃×C₅	C₂×#137	1	2	yes	—
226	C₂³×C₃	—	2	2	no	—
227	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
228	C₃×Q₈	—	2	4	no	—
229	C₃×D₄	—	2	4	no	—
230	C₃×D₄	—	4	4	no	—
231	C₃×D₄	—	4	4	no	—
232	C₂×C₃×C₄	—	2	4	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	4	4	no	—
235	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
236	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
237	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
239	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
240	C₂×Dic₃	—	2	4	no	—
241	C₂×Dic₃	—	4	4	no	—
242	Dic₆	—	2	8	no	—
243	Dic₆	—	4	4	no	—
244	C₃⋊C₈	C₂×C₅×D₄	1	8	yes	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	—	4	2	no	—
247	C₂²×C₅	—	4	4	no	—
248	C₂²×C₅	#940	1	2	yes	—
249	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	12	4	no	—
253	C₄×C₅	#940	1	2	yes	—
254	C₄×C₅	#951	1	4	yes	complemented
255	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
256	#41	—	2	8	no	—
257	C₂×Q₈	—	3	4	no	—
258	C₂×Q₈	—	6	2	no	—
259	C₂×Q₈	—	6	8	no	—
260	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	6	2	no	—
266	C₂²×C₄	C₅×#67	1	2	yes	—
267	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
268	Q₁₆	—	6	16	no	—
269	SD₁₆	—	6	16	no	—
270	M₁₆	—	3	8	no	—
271	C₂×C₈	—	3	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	6	4	no	—
274	C₄⋊C₄	—	6	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₄²	C₅×#67	1	2	yes	—
277	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
278	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	4	4	no	—
283	C₂²×C₃	C₅×#127	1	2	yes	—
284	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	C₅×#127	1	2	yes	—
288	C₃×C₄	C₅×#137	1	4	yes	complemented
289	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
290	Dic₃	—	4	4	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	8	2	no	—
293	C₂×C₅	C₂×#940	1	1	yes	characteristic
294	C₂×C₅	C₂×#951	1	2	yes	—
295	C₂³	—	2	2	no	—
296	C₂³	C₂×C₅×#67	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	Q₈	—	6	8	no	—
299	Q₈	—	12	4	no	—
300	D₄	—	2	4	no	—
301	D₄	—	4	4	no	—
302	D₄	—	4	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	6	4	no	—
307	C₂×C₄	—	12	4	no	—
308	C₂×C₄	C₅×D₃×D₄	1	4	yes	—
309	C₂×C₄	C₂×C₅×#67	1	2	yes	—
310	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×#67	1	2	yes	—
312	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
313	C₈	—	3	8	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	8	2	no	—
316	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
317	C₂×C₃	C₂×C₅×#137	1	2	yes	—
318	C₅	#83288	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	4	no	—
322	C₂²	C₅×#940	1	2	yes	—
323	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	12	4	no	—
327	C₄	C₅×#940	1	2	yes	—
328	C₄	C₅×#951	1	4	yes	complemented
329	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
330	C₃	C₅×#3152	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	8	2	no	—
333	C₂	C₂×C₅×#940	1	1	yes	characteristic
334	C₂	C₂×C₅×#951	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₃×C₅×#532	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#532	D₃
hypocenter	C₃	C₅×#3152
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#532

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#951	C₂	semidirect
3. C₂×C₅×#869	C₂	semidirect
4. C₅×#5092	C₂	semidirect
5. C₅×#5076	C₂	semidirect
6. C₅×#5060	C₂	semidirect
7. C₄×C₅×#208	C₂	semidirect
27. #83288	C₅	direct (N × H)
56. C₅×#208	D₄	semidirect
59. C₂×#951	C₂×C₅	semidirect
60. C₂×#869	C₂×C₅	semidirect
61. #5092	C₂×C₅	semidirect
62. #5076	C₂×C₅	semidirect
63. #5060	C₂×C₅	semidirect
64. C₄×#208	C₂×C₅	semidirect
159. C₃×C₄×C₅	#137	semidirect
183. #208	C₅×D₄	semidirect
254. C₄×C₅	#951	semidirect
279. C₃×C₅	#3152	semidirect
288. C₃×C₄	C₅×#137	semidirect
318. C₅	#83288	direct (N × H)
328. C₄	C₅×#951	semidirect
330. C₃	C₅×#3152	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#532	2
C₂×C₅×#951	2
C₂×C₅×#869	2
C₅×#5092	2
C₅×#5076	2
C₅×#5060	2
C₄×C₅×#208	2
C₅×#3152	3
#83288	5

Subgroups — C5×#83288 #14439216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83288 #14439216