Subgroups — C₅×#68492 #14289490

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5730	C₂	1	1	yes	maximal, characteristic
2	C₅×#5092	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4850	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4842	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4483	C₂	1	1	yes	maximal, characteristic
8	C₅×#1632	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅×#946	—	2	2	no	—
11	C₂×C₅×#216	—	2	2	no	—
12	C₅×#897	—	2	1	no	—
13	C₅×#893	—	2	1	no	—
14	C₅×D₃×C₄⋊C₄	—	2	1	no	—
15	C₅×#890	—	2	1	no	—
16	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₅×#821	—	2	1	no	—
22	C₅×#818	—	2	1	no	—
23	C₅×#817	—	2	1	no	—
24	C₅×#816	—	2	1	no	—
25	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	#68492	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#554	—	3	1	no	—
28	C₅×#494	—	3	1	no	—
29	C₅×#477	—	3	1	no	—
30	C₅×#476	—	3	1	no	—
31	C₅×#333	D₃	1	1	yes	characteristic, complemented
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×D₄⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
35	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	—	2	1	no	—
37	C₂×C₄×C₅×D₃	—	2	2	no	—
38	C₂×C₅×Dic₆	—	2	1	no	—
39	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#216	—	4	4	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
47	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
48	C₅×C₄⋊Dic₃	—	2	2	no	—
49	C₅×C₄⋊Dic₃	—	2	1	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
52	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
53	C₅×Dic₃⋊C₄	—	2	2	no	—
54	C₅×Dic₃⋊C₄	—	2	1	no	—
55	C₄×C₅×Dic₃	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	#5730	C₂×C₅	1	1	yes	characteristic
58	#5092	C₂×C₅	1	1	yes	characteristic, complemented
59	#4850	C₂×C₅	1	1	yes	characteristic, complemented
60	#4842	C₂×C₅	1	1	yes	characteristic
61	C₃×#333	C₂×C₅	1	1	yes	characteristic, complemented
62	#4484	C₂×C₅	1	1	yes	characteristic, complemented
63	#4483	C₂×C₅	1	1	yes	characteristic
64	C₂×C₅×Q₁₆	—	6	2	no	—
65	C₅×C₄⋊Q₈	—	3	1	no	—
66	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#125	—	6	1	no	—
68	C₅×C₂²⋊Q₈	—	6	1	no	—
69	C₅×C₂²⋊Q₈	—	6	2	no	—
70	C₄×C₅×Q₈	—	3	1	no	—
71	C₄×C₅×D₄	—	3	1	no	—
72	C₄×C₅×D₄	—	6	1	no	—
73	C₂×C₅×C₄⋊C₄	—	6	1	no	—
74	C₅×#107	—	6	1	no	—
75	C₅×#106	—	6	1	no	—
76	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
78	C₅×#103	—	6	1	no	—
79	C₅×#102	—	6	1	no	—
80	C₄×C₅×C₈	—	3	1	no	—
81	#1632	—	3	1	no	Sylow
82	C₂²×C₅×D₃	—	2	1	no	—
83	C₃×C₅×Q₈	—	4	2	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₅×Dic₃	—	2	1	no	—
89	C₂×C₅×Dic₃	—	2	1	no	—
90	C₂×C₅×Dic₃	—	2	1	no	—
91	C₂×C₅×Dic₃	—	4	1	no	—
92	C₅×D₁₂	—	2	1	no	—
93	C₅×D₁₂	Q₁₆	1	2	yes	complemented
94	C₄×C₅×D₃	—	4	1	no	—
95	C₄×C₅×D₃	—	4	1	no	—
96	C₄×C₅×D₃	—	8	2	no	—
97	C₅×Dic₆	—	2	1	no	—
98	C₅×Dic₆	—	2	1	no	—
99	C₅×Dic₆	—	4	2	no	—
100	C₃×C₅×C₈	—	4	1	no	—
101	C₃×C₅×C₈	—	4	1	no	—
102	C₅×C₃⋊C₈	—	2	2	no	—
103	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
104	#946	—	2	2	no	—
105	C₂×#216	—	2	2	no	—
106	#897	—	2	1	no	—
107	#893	—	2	1	no	—
108	D₃×C₄⋊C₄	—	2	1	no	—
109	#890	—	2	1	no	—
110	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
111	#869	C₂²×C₅	1	1	yes	characteristic
112	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
113	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
114	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
115	#821	—	2	1	no	—
116	#818	—	2	1	no	—
117	#817	—	2	1	no	—
118	#816	—	2	1	no	—
119	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₅×Q₈	—	3	1	no	—
121	C₂×C₅×Q₈	—	6	1	no	—
122	C₂×C₅×Q₈	D₁₂	1	2	yes	—
123	C₂×C₅×D₄	—	3	1	no	—
124	C₂²×C₄×C₅	—	6	2	no	—
125	C₂²×C₄×C₅	—	6	1	no	—
126	C₅×Q₁₆	—	12	4	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	—	6	1	no	—
130	C₅×C₄⋊C₄	—	2	1	no	—
131	C₅×C₄⋊C₄	—	2	1	no	—
132	C₅×C₄⋊C₄	—	3	1	no	—
133	C₅×C₄⋊C₄	—	3	1	no	—
134	C₅×C₄⋊C₄	—	6	2	no	—
135	C₅×C₄⋊C₄	—	6	2	no	—
136	C₅×C₄⋊C₄	—	6	1	no	—
137	C₅×C₄⋊C₄	—	6	1	no	—
138	C₅×C₄⋊C₄	—	6	1	no	—
139	C₅×C₂²⋊C₄	—	6	2	no	—
140	C₅×C₂²⋊C₄	—	6	1	no	—
141	C₄²×C₅	—	6	1	no	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	#554	—	3	1	no	—
144	#494	—	3	1	no	—
145	#477	—	3	1	no	—
146	#476	—	3	1	no	—
147	#333	C₅×D₃	1	1	yes	characteristic, complemented
148	Q₈⋊C₈	—	3	1	no	—
149	D₄⋊C₈	—	3	1	no	—
150	C₂²×C₃×C₅	#120	1	1	yes	characteristic
151	C₂×C₅×D₃	—	2	2	no	—
152	C₂×C₅×D₃	—	4	1	no	—
153	C₂×C₅×D₃	—	4	1	no	—
154	C₃×C₄×C₅	—	2	1	no	—
155	C₃×C₄×C₅	—	4	2	no	—
156	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
157	C₃×C₄×C₅	#135	1	1	yes	characteristic
158	C₃×C₄×C₅	#137	1	1	yes	characteristic
159	C₃×C₄×C₅	#137	1	1	yes	characteristic
160	C₅×Dic₃	—	4	1	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₅×Dic₃	—	4	1	no	—
163	C₅×Dic₃	—	8	1	no	—
164	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
165	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
166	C₂×C₄×D₃	—	2	1	no	—
167	C₂×C₄×D₃	—	2	2	no	—
168	C₂×Dic₆	—	2	1	no	—
169	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
170	C₂×C₃×C₈	—	2	1	no	—
171	C₂×C₃×C₈	—	2	1	no	—
172	C₃×C₄⋊C₄	—	2	1	no	—
173	C₃×C₄⋊C₄	—	2	1	no	—
174	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
175	#216	—	4	4	no	—
176	(C₂×D₃)⋊C₄	—	2	2	no	—
177	(C₂×D₃)⋊C₄	—	2	1	no	—
178	C₄⋊Dic₃	—	2	2	no	—
179	C₄⋊Dic₃	—	2	1	no	—
180	C₄⋊Dic₃	—	2	1	no	—
181	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
182	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
183	Dic₃⋊C₄	—	2	2	no	—
184	Dic₃⋊C₄	—	2	1	no	—
185	C₄×Dic₃	—	2	1	no	—
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂³×C₅	—	6	1	no	—
188	C₅×Q₈	—	4	2	no	—
189	C₅×Q₈	—	6	1	no	—
190	C₅×Q₈	—	6	1	no	—
191	C₅×Q₈	—	12	2	no	—
192	C₅×D₄	—	3	2	no	—
193	C₅×D₄	—	6	1	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	6	1	no	—
196	C₂×C₄×C₅	—	6	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	12	1	no	—
199	C₂×C₄×C₅	—	12	1	no	—
200	C₂×C₄×C₅	—	12	1	no	—
201	C₂×C₄×C₅	—	24	2	no	—
202	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
205	C₅×C₈	—	4	1	no	—
206	C₅×C₈	—	4	1	no	—
207	C₅×C₈	—	6	2	no	—
208	C₂×Q₁₆	—	6	2	no	—
209	C₄⋊Q₈	—	3	1	no	—
210	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
211	#125	—	6	1	no	—
212	C₂²⋊Q₈	—	6	1	no	—
213	C₂²⋊Q₈	—	6	2	no	—
214	C₄×Q₈	—	3	1	no	—
215	C₄×D₄	—	3	1	no	—
216	C₄×D₄	—	6	1	no	—
217	C₂×C₄⋊C₄	—	6	1	no	—
218	#107	—	6	1	no	—
219	#106	—	6	1	no	—
220	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
221	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
222	#103	—	6	1	no	—
223	#102	—	6	1	no	—
224	C₄×C₈	—	3	1	no	—
225	C₂×C₃×C₅	#451	1	1	yes	characteristic
226	C₂×C₃×C₅	#452	1	1	yes	characteristic
227	C₂×C₃×C₅	#453	1	1	yes	characteristic
228	C₅×D₃	—	4	2	no	—
229	C₂²×D₃	—	2	1	no	—
230	C₃×Q₈	—	4	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×Dic₃	—	2	1	no	—
236	C₂×Dic₃	—	2	1	no	—
237	C₂×Dic₃	—	2	1	no	—
238	C₂×Dic₃	—	4	1	no	—
239	D₁₂	—	2	1	no	—
240	D₁₂	C₅×Q₁₆	1	2	yes	complemented
241	C₄×D₃	—	4	1	no	—
242	C₄×D₃	—	4	1	no	—
243	C₄×D₃	—	8	2	no	—
244	Dic₆	—	2	1	no	—
245	Dic₆	—	2	1	no	—
246	Dic₆	—	4	2	no	—
247	C₃×C₈	—	4	1	no	—
248	C₃×C₈	—	4	1	no	—
249	C₃⋊C₈	—	2	2	no	—
250	C₂²×C₅	—	6	2	no	—
251	C₂²×C₅	—	12	1	no	—
252	C₂²×C₅	—	12	1	no	—
253	C₂²×C₅	#882	1	1	yes	characteristic
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	4	2	no	—
256	C₄×C₅	—	12	1	no	—
257	C₄×C₅	—	12	1	no	—
258	C₄×C₅	—	12	1	no	—
259	C₄×C₅	—	24	1	no	—
260	C₄×C₅	#915	1	1	yes	characteristic
261	C₄×C₅	#916	1	1	yes	characteristic
262	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
263	C₄×C₅	#918	1	1	yes	characteristic
264	C₂×Q₈	—	3	1	no	—
265	C₂×Q₈	—	6	1	no	—
266	C₂×Q₈	C₅×D₁₂	1	2	yes	—
267	C₂×D₄	—	3	1	no	—
268	C₂²×C₄	—	6	2	no	—
269	C₂²×C₄	—	6	1	no	—
270	Q₁₆	—	12	4	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	6	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	3	1	no	—
277	C₄⋊C₄	—	3	1	no	—
278	C₄⋊C₄	—	6	2	no	—
279	C₄⋊C₄	—	6	2	no	—
280	C₄⋊C₄	—	6	1	no	—
281	C₄⋊C₄	—	6	1	no	—
282	C₄⋊C₄	—	6	1	no	—
283	C₂²⋊C₄	—	6	2	no	—
284	C₂²⋊C₄	—	6	1	no	—
285	C₄²	—	6	1	no	—
286	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
287	C₃×C₅	#1632	1	1	yes	characteristic, complemented, Hall
288	C₂²×C₃	C₅×#120	1	1	yes	characteristic
289	C₂×D₃	—	2	2	no	—
290	C₂×D₃	—	4	1	no	—
291	C₂×D₃	—	4	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	4	2	no	—
294	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
295	C₃×C₄	C₅×#135	1	1	yes	characteristic
296	C₃×C₄	C₅×#137	1	1	yes	characteristic
297	C₃×C₄	C₅×#137	1	1	yes	characteristic
298	Dic₃	—	4	1	no	—
299	Dic₃	—	4	1	no	—
300	Dic₃	—	4	1	no	—
301	Dic₃	—	8	1	no	—
302	C₂×C₅	—	12	2	no	—
303	C₂×C₅	#4747	1	1	yes	characteristic
304	C₂×C₅	#4808	1	1	yes	characteristic
305	C₂×C₅	#4809	1	1	yes	characteristic
306	C₂³	—	6	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	6	1	no	—
309	Q₈	—	6	1	no	—
310	Q₈	—	12	2	no	—
311	D₄	—	3	2	no	—
312	D₄	—	6	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	6	1	no	—
315	C₂×C₄	—	6	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	12	1	no	—
318	C₂×C₄	—	12	1	no	—
319	C₂×C₄	—	12	1	no	—
320	C₂×C₄	—	24	2	no	—
321	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
322	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
323	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
324	C₈	—	4	1	no	—
325	C₈	—	4	1	no	—
326	C₈	—	6	2	no	—
327	C₂×C₃	C₅×#451	1	1	yes	characteristic
328	C₂×C₃	C₅×#452	1	1	yes	characteristic
329	C₂×C₃	C₅×#453	1	1	yes	characteristic
330	D₃	—	4	2	no	—
331	C₅	#68492	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²	—	6	2	no	—
333	C₂²	—	12	1	no	—
334	C₂²	—	12	1	no	—
335	C₂²	C₅×#882	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	2	no	—
338	C₄	—	12	1	no	—
339	C₄	—	12	1	no	—
340	C₄	—	12	1	no	—
341	C₄	—	24	1	no	—
342	C₄	C₅×#915	1	1	yes	characteristic
343	C₄	C₅×#916	1	1	yes	characteristic
344	C₄	C₅×D₃×Q₁₆	1	1	yes	characteristic
345	C₄	C₅×#918	1	1	yes	characteristic
346	C₃	C₅×#1632	1	1	yes	characteristic, complemented, Sylow
347	C₂	—	12	2	no	—
348	C₂	C₅×#4747	1	1	yes	characteristic
349	C₂	C₅×#4808	1	1	yes	characteristic
350	C₂	C₅×#4809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#882
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#333	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₅×D₁₂	D₄
hypercenter	C₅×#333	D₃
hypocenter	C₃	C₅×#1632
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#333

Complements

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

N	H ≅ G/N	type
2. C₅×#5092	C₂	semidirect
3. C₅×#4850	C₂	semidirect
5. C₃×C₅×#333	C₂	semidirect
6. C₅×#4484	C₂	semidirect
26. #68492	C₅	direct (N × H)
31. C₅×#333	D₃	semidirect
58. #5092	C₂×C₅	semidirect
59. #4850	C₂×C₅	semidirect
61. C₃×#333	C₂×C₅	semidirect
62. #4484	C₂×C₅	semidirect
93. C₅×D₁₂	Q₁₆	semidirect
147. #333	C₅×D₃	semidirect
240. D₁₂	C₅×Q₁₆	semidirect
287. C₃×C₅	#1632	semidirect
331. C₅	#68492	direct (N × H)
346. C₃	C₅×#1632	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5730	2
C₅×#5092	2
C₅×#4850	2
C₅×#4842	2
C₃×C₅×#333	2
C₅×#4484	2
C₅×#4483	2
C₅×#1632	3
#68492	5

Subgroups — C5×#68492 #14289490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68492 #14289490