Finite Group Databasebeta

← C2×C5×#5728 #14480120

Subgroups — C2×C5×#5728 #14480120

Order 1920.

2336 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C5×#946C212yesmaximal
3C2×C5×Q8×Dic3C212yesmaximal, complemented
4C5×#5728C2116yesmaximal, direct product factor
5C2×C5×#897C212yesmaximal
6C2×C5×#892C212yesmaximal
7C2×C5×#890C212yesmaximal, complemented
8C2×C5×#887C212yesmaximal, complemented
9C2×C42×C5×D3C211yesmaximal, characteristic
10C2×C5×#869C211yesmaximal, characteristic, complemented
11C2×C5×#533—31nomaximal, Hall
12C22×C3×C5×Q8C2212yes—
13C22×C4×C5×D3C2211yescharacteristic
14C22×C4×C5×D3C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C22×C5×Dic6C2212yes—
17C3×C5×C4⋊Q8C2218yescomplemented
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C42×C5C2211yescharacteristic
21C5×#946C22116yes—
22C5×Q8×Dic3C22116yescomplemented
23C2×C5×(C2×D3)⋊C4C2214yes—
24C2×C5×C4⋊Dic3C2212yes—
25C2×C5×C4⋊Dic3C2212yes—
26C2×C5×C4⋊Dic3C2214yes—
27C2×C5×Dic3⋊C4C2214yes—
28C2×C4×C5×Dic3C2211yescharacteristic
29C2×C4×C5×Dic3C2211yescharacteristic
30C2×C4×C5×Dic3C2211yescharacteristic
31C2×C4×C5×Dic3C2214yes—
32C5×#897C22116yes—
33C5×#892C22116yes—
34C5×#890C22116yescomplemented
35C5×#887C22116yescomplemented
36C42×C5×D3C2218yes—
37C5×#869C2218yescomplemented
38C2×#5728C511yesmaximal, characteristic, direct product factor, Hall
39C5×#533—316no—
40C2×C5×C4⋊Q8—31no—
41C2×C5×C4⋊Q8D311yescharacteristic, complemented
42C2×C5×#125—32no—
43C2×C5×C22⋊Q8—32no—
44C2×C5×C22⋊Q8—32no—
45C2×C4×C5×Q8—32no—
46C2×C4×C5×Q8—32no—
47C2×C5×#117—32no—
48C22×C42×C5—31no—
49C23×C5×D3C2311yescharacteristic
50C2×C3×C5×Q8—28no—
51C2×C3×C5×Q8C2318yes—
52C22×C3×C4×C5C2314yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C5×Dic3C2311yescharacteristic
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2314yes—
59C22×C5×Dic3C2311yescharacteristic
60C2×C4×C5×D3—24no—
61C2×C4×C5×D3—24no—
62C2×C4×C5×D3Q818yescomplemented
63C2×C4×C5×D3C2314yes—
64C2×C4×C5×D3C2314yes—
65C2×C4×C5×D3C2314yes—
66C2×C5×Dic6—28no—
67C2×C5×Dic6C2318yes—
68C3×C5×C4⋊C4C2318yes—
69C3×C5×C4⋊C4C2318yes—
70C3×C42×C5C2314yes—
71C5×(C2×D3)⋊C4C23116yes—
72C5×C4⋊Dic3C2318yes—
73C5×C4⋊Dic3C2318yes—
74C5×C4⋊Dic3C23116yes—
75C5×Dic3⋊C4C23116yes—
76C4×C5×Dic3C2314yes—
77C4×C5×Dic3C2314yes—
78C4×C5×Dic3C2314yes—
79C4×C5×Dic3C23116yes—
80C2×C3×C4⋊Q8C2×C511yescharacteristic, complemented
81C2×#946C2×C512yes—
82C2×Q8×Dic3C2×C512yescomplemented
83#5728C2×C5116yesdirect product factor
84C2×#897C2×C512yes—
85C2×#892C2×C512yes—
86C2×#890C2×C512yescomplemented
87C2×#887C2×C512yescomplemented
88C2×C42×D3C2×C511yescharacteristic
89C2×#869C2×C511yescharacteristic, complemented
90C22×C5×Q8—32no—
91C22×C5×Q8C2×D312yes—
92C23×C4×C5—31no—
93C23×C4×C5—31no—
94C23×C4×C5—31no—
95C5×C4⋊Q8—38no—
96C5×C4⋊Q8C2×D318yescomplemented
97C5×#125—316no—
98C5×C22⋊Q8—316no—
99C5×C22⋊Q8—316no—
100C4×C5×Q8—316no—
101C4×C5×Q8—316no—
102C5×#117—316no—
103C2×C5×C4⋊C4—34no—
104C2×C5×C4⋊C4—32no—
105C2×C5×C4⋊C4—34no—
106C2×C5×C4⋊C4—32no—
107C2×C5×C4⋊C4C2×D312yes—
108C2×C5×C4⋊C4C2×D312yes—
109C2×C5×C22⋊C4—34no—
110C2×C42×C5—31no—
111C2×C42×C5—31no—
112C2×C42×C5—31no—
113C2×C42×C5—34no—
114C2×C42×C5—38no—
115C2×C42×C5C2×D311yescharacteristic
116C2×#533—31noSylow
117C23×C3×C5C2411yescharacteristic
118C22×C5×D3—22no—
119C22×C5×D3—22no—
120C22×C5×D3C2×Q814yes—
121C22×C5×D3C2412yes—
122C3×C5×Q8—216no—
123C2×C3×C4×C5—28no—
124C2×C3×C4×C5C2×Q814yes—
125C2×C3×C4×C5#4114yes—
126C2×C3×C4×C5#4114yes—
127C2×C3×C4×C5C2418yes—
128C2×C3×C4×C5C2412yes—
129C2×C3×C4×C5C2412yes—
130C2×C3×C4×C5C2412yes—
131C2×C5×Dic3—28no—
132C2×C5×Dic3—22no—
133C2×C5×Dic3—22no—
134C2×C5×Dic3C2×Q814yes—
135C2×C5×Dic3C2412yes—
136C2×C5×Dic3C2412yes—
137C2×C5×Dic3C2418yes—
138C2×C5×Dic3C2412yes—
139C4×C5×D3—28no—
140C4×C5×D3—28no—
141C4×C5×D3C2×Q8116yescomplemented
142C5×Dic6—216no—
143C22×C3×Q8C22×C512yes—
144C22×C4×D3C22×C511yescharacteristic
145C22×C4×D3C22×C511yescharacteristic
146C22×C4×D3C22×C511yescharacteristic
147C22×Dic6C22×C512yes—
148C3×C4⋊Q8C22×C518yescomplemented
149C2×C3×C4⋊C4C22×C512yes—
150C2×C3×C4⋊C4C22×C512yes—
151C2×C3×C42C22×C511yescharacteristic
152#946C22×C5116yes—
153Q8×Dic3C22×C5116yescomplemented
154C2×(C2×D3)⋊C4C22×C514yes—
155C2×C4⋊Dic3C22×C512yes—
156C2×C4⋊Dic3C22×C512yes—
157C2×C4⋊Dic3C22×C514yes—
158C2×Dic3⋊C4C22×C514yes—
159C2×C4×Dic3C22×C511yescharacteristic
160C2×C4×Dic3C22×C511yescharacteristic
161C2×C4×Dic3C22×C511yescharacteristic
162C2×C4×Dic3C22×C514yes—
163#897C22×C5116yes—
164#892C22×C5116yes—
165#890C22×C5116yescomplemented
166#887C22×C5116yescomplemented
167C42×D3C22×C518yes—
168#869C22×C518yescomplemented
169C24×C5—31no—
170C2×C5×Q8—28no—
171C2×C5×Q8—38no—
172C2×C5×Q8—68no—
173C2×C5×Q8C22×D318yes—
174C22×C4×C5—31no—
175C22×C4×C5—38no—
176C22×C4×C5—34no—
177C22×C4×C5—31no—
178C22×C4×C5—31no—
179C22×C4×C5—34no—
180C22×C4×C5—34no—
181C22×C4×C5—34no—
182C22×C4×C5—64no—
183C22×C4×C5—64no—
184C22×C4×C5C22×D314yes—
185C22×C4×C5C22×D311yescharacteristic
186C22×C4×C5C22×D311yescharacteristic
187C22×C4×C5C22×D311yescharacteristic
188C5×C4⋊C4—38no—
189C5×C4⋊C4—316no—
190C5×C4⋊C4—316no—
191C5×C4⋊C4—38no—
192C5×C4⋊C4C22×D318yes—
193C5×C4⋊C4C22×D318yes—
194C5×C22⋊C4—316no—
195C42×C5—34no—
196C42×C5—34no—
197C42×C5—34no—
198C42×C5—316no—
199C42×C5C22×D314yes—
200#533—316no—
201C2×C4⋊Q8—31no—
202C2×C4⋊Q8C5×D311yescharacteristic, complemented
203C2×#125—32no—
204C2×C22⋊Q8—32no—
205C2×C22⋊Q8—32no—
206C2×C4×Q8—32no—
207C2×C4×Q8—32no—
208C2×#117—32no—
209C22×C42—31no—
210C22×C3×C5C22×Q812yes—
211C22×C3×C5C2×#4112yes—
212C22×C3×C5C2×#4112yes—
213C22×C3×C5C2511yescharacteristic
214C2×C5×D3—28no—
215C2×C5×D3—22no—
216C2×C5×D3—22no—
217C2×C5×D3C22×Q814yes—
218C3×C4×C5—28no—
219C3×C4×C5C22×Q814yes—
220C3×C4×C5C2×#4114yes—
221C3×C4×C5C2×#4114yes—
222C5×Dic3—28no—
223C5×Dic3—22no—
224C5×Dic3—22no—
225C5×Dic3C22×Q814yes—
226C23×D3C23×C511yescharacteristic
227C2×C3×Q8—28no—
228C2×C3×Q8C23×C518yes—
229C22×C3×C4C23×C514yes—
230C22×C3×C4C23×C511yescharacteristic
231C22×C3×C4C23×C511yescharacteristic
232C22×C3×C4C23×C511yescharacteristic
233C22×Dic3C23×C511yescharacteristic
234C22×Dic3C23×C511yescharacteristic
235C22×Dic3C23×C514yes—
236C22×Dic3C23×C511yescharacteristic
237C2×C4×D3—24no—
238C2×C4×D3—24no—
239C2×C4×D3C5×Q818yescomplemented
240C2×C4×D3C23×C514yes—
241C2×C4×D3C23×C514yes—
242C2×C4×D3C23×C514yes—
243C2×Dic6—28no—
244C2×Dic6C23×C518yes—
245C3×C4⋊C4C23×C518yes—
246C3×C4⋊C4C23×C518yes—
247C3×C42C23×C514yes—
248(C2×D3)⋊C4C23×C5116yes—
249C4⋊Dic3C23×C518yes—
250C4⋊Dic3C23×C518yes—
251C4⋊Dic3C23×C5116yes—
252Dic3⋊C4C23×C5116yes—
253C4×Dic3C23×C514yes—
254C4×Dic3C23×C514yes—
255C4×Dic3C23×C514yes—
256C4×Dic3C23×C5116yes—
257C23×C5—34no—
258C23×C5—32no—
259C23×C5—62no—
260C23×C5—62no—
261C23×C5C23×D311yescharacteristic
262C5×Q8—216no—
263C5×Q8—616no—
264C2×C4×C5—28no—
265C2×C4×C5—34no—
266C2×C4×C5—32no—
267C2×C4×C5—316no—
268C2×C4×C5—32no—
269C2×C4×C5—32no—
270C2×C4×C5—38no—
271C2×C4×C5—62no—
272C2×C4×C5—62no—
273C2×C4×C5—68no—
274C2×C4×C5—68no—
275C2×C4×C5—68no—
276C2×C4×C5#23714yes—
277C2×C4×C5D3×Q814yes—
278C2×C4×C5#23914yes—
279C2×C4×C5C23×D318yes—
280C2×C4×C5C23×D312yes—
281C2×C4×C5C23×D312yes—
282C2×C4×C5C23×D312yes—
283C22×Q8—32no—
284C22×Q8C2×C5×D312yes—
285C23×C4—31no—
286C23×C4—31no—
287C23×C4—31no—
288C4⋊Q8—38no—
289C4⋊Q8C2×C5×D318yescomplemented
290#125—316no—
291C22⋊Q8—316no—
292C22⋊Q8—316no—
293C4×Q8—316no—
294C4×Q8—316no—
295#117—316no—
296C2×C4⋊C4—32no—
297C2×C4⋊C4—34no—
298C2×C4⋊C4—34no—
299C2×C4⋊C4—32no—
300C2×C4⋊C4C2×C5×D312yes—
301C2×C4⋊C4C2×C5×D312yes—
302C2×C22⋊C4—34no—
303C2×C42—34no—
304C2×C42—31no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—38no—
308C2×C42C2×C5×D311yescharacteristic
309C2×C3×C5#53314yescomplemented
310C2×C3×C5C23×Q811yescharacteristic
311C2×C3×C5C22×#4111yescharacteristic
312C2×C3×C5C22×#4111yescharacteristic
313C5×D3—24no—
314C23×C3C24×C511yescharacteristic
315C22×D3—22no—
316C22×D3—22no—
317C22×D3C2×C5×Q814yes—
318C22×D3C24×C512yes—
319C3×Q8—216no—
320C2×C3×C4—28no—
321C2×C3×C4C2×C5×Q814yes—
322C2×C3×C4C5×#4114yes—
323C2×C3×C4C5×#4114yes—
324C2×C3×C4C24×C518yes—
325C2×C3×C4C24×C512yes—
326C2×C3×C4C24×C512yes—
327C2×C3×C4C24×C512yes—
328C2×Dic3—28no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3C2×C5×Q814yes—
332C2×Dic3C24×C512yes—
333C2×Dic3C24×C512yes—
334C2×Dic3C24×C518yes—
335C2×Dic3C24×C512yes—
336C4×D3—28no—
337C4×D3—28no—
338C4×D3C2×C5×Q8116yescomplemented
339Dic6—216no—
340C22×C5—34no—
341C22×C5—62no—
342C22×C5—62no—
343C22×C5—68no—
344C22×C5C2×#23712yes—
345C22×C5C2×D3×Q812yes—
346C22×C5C2×#23912yes—
347C22×C5C24×D311yescharacteristic
348C4×C5—28no—
349C4×C5—34no—
350C4×C5—62no—
351C4×C5—62no—
352C4×C5—68no—
353C4×C5C2×#23714yes—
354C4×C5C2×D3×Q814yes—
355C4×C5C2×#23914yes—
356C24—31no—
357C2×Q8—28no—
358C2×Q8—38no—
359C2×Q8—68no—
360C2×Q8C22×C5×D318yes—
361C22×C4—31no—
362C22×C4—38no—
363C22×C4—34no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—34no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—64no—
370C22×C4—64no—
371C22×C4C22×C5×D314yes—
372C22×C4C22×C5×D311yescharacteristic
373C22×C4C22×C5×D311yescharacteristic
374C22×C4C22×C5×D311yescharacteristic
375C4⋊C4—38no—
376C4⋊C4—316no—
377C4⋊C4—316no—
378C4⋊C4—38no—
379C4⋊C4C22×C5×D318yes—
380C4⋊C4C22×C5×D318yes—
381C22⋊C4—316no—
382C42—34no—
383C42—34no—
384C42—34no—
385C42—316no—
386C42C22×C5×D314yes—
387C3×C5C2×#53311yescharacteristic, complemented, Hall
388C22×C3C22×C5×Q812yes—
389C22×C3C2×C5×#4112yes—
390C22×C3C2×C5×#4112yes—
391C22×C3C25×C511yescharacteristic
392C2×D3—28no—
393C2×D3—22no—
394C2×D3—22no—
395C2×D3C22×C5×Q814yes—
396C3×C4—28no—
397C3×C4C22×C5×Q814yes—
398C3×C4C2×C5×#4114yes—
399C3×C4C2×C5×#4114yes—
400Dic3—28no—
401Dic3—22no—
402Dic3—22no—
403Dic3C22×C5×Q814yes—
404C2×C5—64no—
405C2×C5#572814yesdirect product factor
406C2×C5C22×#23711yescharacteristic
407C2×C5C22×D3×Q811yescharacteristic
408C2×C5C22×#23911yescharacteristic
409C23—34no—
410C23—32no—
411C23—62no—
412C23—62no—
413C23C23×C5×D311yescharacteristic
414Q8—216no—
415Q8—616no—
416C2×C4—28no—
417C2×C4—34no—
418C2×C4—32no—
419C2×C4—316no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—38no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—68no—
426C2×C4—68no—
427C2×C4—68no—
428C2×C4C5×#23714yes—
429C2×C4C5×D3×Q814yes—
430C2×C4C5×#23914yes—
431C2×C4C23×C5×D318yes—
432C2×C4C23×C5×D312yes—
433C2×C4C23×C5×D312yes—
434C2×C4C23×C5×D312yes—
435C2×C3C5×#53314yescomplemented
436C2×C3C23×C5×Q811yescharacteristic
437C2×C3C22×C5×#4111yescharacteristic
438C2×C3C22×C5×#4111yescharacteristic
439D3—24no—
440C5C2×#572811yescharacteristic, direct product factor, Sylow
441C22—34no—
442C22—62no—
443C22—62no—
444C22—68no—
445C22C2×C5×#23712yes—
446C22C2×C5×D3×Q812yes—
447C22C2×C5×#23912yes—
448C22C24×C5×D311yescharacteristic
449C4—28no—
450C4—34no—
451C4—62no—
452C4—62no—
453C4—68no—
454C4C2×C5×#23714yes—
455C4C2×C5×D3×Q814yes—
456C4C2×C5×#23914yes—
457C3C2×C5×#53311yescharacteristic, complemented, Sylow
458C2—64no—
459C2C5×#572814yesdirect product factor
460C2C22×C5×#23711yescharacteristic
461C2C22×C5×D3×Q811yescharacteristic
462C2C22×C5×#23911yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C5×C4⋊Q8C2
socleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C5×C4⋊Q8D3
hypocenterC3C2×C5×#533
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C2×C5×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×C4⋊Q8C2semidirect
3. C2×C5×Q8×Dic3C2semidirect
4. C5×#5728C2direct (N × H)
7. C2×C5×#890C2semidirect
8. C2×C5×#887C2semidirect
10. C2×C5×#869C2semidirect
17. C3×C5×C4⋊Q8C22semidirect
22. C5×Q8×Dic3C22semidirect
34. C5×#890C22semidirect
35. C5×#887C22semidirect
37. C5×#869C22semidirect
38. C2×#5728C5direct (N × H)
41. C2×C5×C4⋊Q8D3semidirect
62. C2×C4×C5×D3Q8semidirect
80. C2×C3×C4⋊Q8C2×C5semidirect
82. C2×Q8×Dic3C2×C5semidirect
83. #5728C2×C5direct (N × H)
86. C2×#890C2×C5semidirect
87. C2×#887C2×C5semidirect
89. C2×#869C2×C5semidirect
96. C5×C4⋊Q8C2×D3semidirect
141. C4×C5×D3C2×Q8semidirect
148. C3×C4⋊Q8C22×C5semidirect
153. Q8×Dic3C22×C5semidirect
165. #890C22×C5semidirect
166. #887C22×C5semidirect
168. #869C22×C5semidirect
202. C2×C4⋊Q8C5×D3semidirect
239. C2×C4×D3C5×Q8semidirect
289. C4⋊Q8C2×C5×D3semidirect
309. C2×C3×C5#533semidirect
338. C4×D3C2×C5×Q8semidirect
387. C3×C5C2×#533semidirect
405. C2×C5#5728direct (N × H)
435. C2×C3C5×#533semidirect
440. C5C2×#5728direct (N × H)
457. C3C2×C5×#533semidirect
459. C2C5×#5728direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×C4⋊Q82
C2×C5×#9462
C2×C5×Q8×Dic32
C5×#57282
C2×C5×#8972
C2×C5×#8922
C2×C5×#8902
C2×C5×#8872
C2×C42×C5×D32
C2×C5×#8692
C2×C5×#5333
C2×#57285