Finite Group Databasebeta

← C5×#86274 #14481801

Subgroups — C5×#86274 #14481801

Order 1920.

1992 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#533C211yesmaximal, characteristic, complemented
2C5×#5729C214yesmaximal, complemented
3C5×#5689C214yesmaximal, complemented
4C5×#5634C214yesmaximal
5C5×#5623C214yesmaximal, complemented
6C5×#5578C218yesmaximal, complemented
7C5×D3×#117C212yesmaximal
8C5×#5517C212yesmaximal, complemented
9C2×C5×#872C211yesmaximal, characteristic
10C5×#5470C211yesmaximal, characteristic, complemented
11C5×#3458—31nomaximal, Hall
12C22×C4×C5×D3C2211yescharacteristic
13C3×C5×C4⋊Q8C2212yescomplemented
14C3×C5×#125C2212yescomplemented
15C3×C5×C22⋊Q8C2214yes—
16C3×C4×C5×Q8C2214yescomplemented
17C3×C5×#117C2212yes—
18C2×C3×C42×C5C2211yescharacteristic
19C5×#946C2218yes—
20C5×#944C2218yescomplemented
21C2×C5×(C2×D3)⋊C4C2212yes—
22C5×#926C2212yes—
23C5×#924C2214yes—
24C2×C5×Dic3⋊C4C2212yes—
25C2×C4×C5×Dic3C2211yescharacteristic
26C5×#897C2218yes—
27C5×#896C2218yes—
28C5×#896C2218yes—
29C5×#892C2214yes—
30C5×D3×C4⋊C4C2214yes—
31C5×#890C2214yescomplemented
32C5×#889C2218yescomplemented
33C5×#889C2218yescomplemented
34C5×#888C2214yescomplemented
35C5×#888C2218yescomplemented
36C5×D3×C22⋊C4C2214yes—
37C5×#878C2218yes—
38C5×#877C2214yes—
39C5×#872C2212yes—
40C5×#872C2212yes—
41C5×#872C2214yes—
42C5×#872C2214yes—
43C42×C5×D3C2214yes—
44C5×#870C2212yescomplemented
45C5×#869C2212yescomplemented
46C4×C5×Dic6C2214yescomplemented
47C4×C5×Dic6C2218yescomplemented
48#86274C511yesmaximal, characteristic, direct product factor, Hall
49C5×#544—34no—
50C5×#544—34no—
51C5×#544—34no—
52C5×#543—34no—
53C5×#533—38no—
54C5×#533—32no—
55C5×#533—31no—
56C5×#533D311yescharacteristic, complemented
57C2×C5×#117—32no—
58C2×C5×#117—31no—
59C23×C5×D3C2311yescharacteristic
60C2×C3×C5×Q8C2314yes—
61C22×C3×C4×C5C2312yes—
62C22×C3×C4×C5C2311yescharacteristic
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2311yescharacteristic
65C2×C4×C5×D3Q818yescomplemented
66C2×C4×C5×D3C2318yes—
67C2×C4×C5×D3C2312yes—
68C2×C4×C5×D3C2312yes—
69C2×C5×Dic6C2314yes—
70C2×C5×Dic6C2318yes—
71C3×C5×C4⋊C4C2318yes—
72C3×C5×C4⋊C4C2314yes—
73C3×C5×C4⋊C4C2314yes—
74C3×C5×C22⋊C4C2314yes—
75C3×C42×C5C2314yes—
76C3×C42×C5C2312yes—
77C3×C42×C5C2311yescharacteristic
78C3×C42×C5C2311yescharacteristic
79C5×#217C2314yes—
80C5×(C2×D3)⋊C4C2314yes—
81C5×(C2×D3)⋊C4C2314yes—
82C5×(C2×D3)⋊C4C2318yes—
83C5×C4⋊Dic3C2314yes—
84C5×C4⋊Dic3C2318yes—
85C5×C4⋊Dic3C2314yes—
86C5×Dic3⋊C4C2318yes—
87C5×Dic3⋊C4C2318yes—
88C5×Dic3⋊C4C2318yes—
89C5×Dic3⋊C4C2314yes—
90C5×Dic3⋊C4C2314yes—
91C5×Dic3⋊C4C2318yes—
92C4×C5×Dic3C2314yes—
93C4×C5×Dic3C2312yes—
94C4×C5×Dic3C2318yes—
95C4×C5×Dic3C2312yes—
96C3×#533C2×C511yescharacteristic, complemented
97#5729C2×C514yescomplemented
98#5689C2×C514yescomplemented
99#5634C2×C514yes—
100#5623C2×C514yescomplemented
101#5578C2×C518yescomplemented
102D3×#117C2×C512yes—
103#5517C2×C512yescomplemented
104C2×#872C2×C511yescharacteristic
105#5470C2×C511yescharacteristic, complemented
106C23×C4×C5—31no—
107C5×C4⋊Q8—38no—
108C5×C4⋊Q8—38no—
109C5×C4⋊Q8—34no—
110C5×C4⋊Q8—32no—
111C5×C4⋊Q8C2×D312yescomplemented
112C5×#125—38no—
113C5×#125—38no—
114C5×#125—34no—
115C5×#125—32no—
116C5×#125C2×D312yescomplemented
117C5×C22⋊Q8—38no—
118C5×C22⋊Q8—34no—
119C5×C22⋊Q8—38no—
120C5×C22⋊Q8—38no—
121C5×C22⋊Q8—38no—
122C5×C22⋊Q8—38no—
123C5×C22⋊Q8C2×D314yes—
124C4×C5×Q8—38no—
125C4×C5×Q8—34no—
126C4×C5×Q8C2×D314yescomplemented
127C5×#117—32no—
128C5×#117—34no—
129C5×#117—34no—
130C5×#117—34no—
131C5×#117—34no—
132C5×#117—32no—
133C5×#117—32no—
134C5×#117C2×D312yes—
135C2×C5×C4⋊C4—32no—
136C2×C5×C4⋊C4—34no—
137C2×C5×C22⋊C4—32no—
138C2×C5×C22⋊C4—34no—
139C2×C42×C5—31no—
140C2×C42×C5—34no—
141C2×C42×C5C2×D311yescharacteristic
142#3458—31noSylow
143C23×C3×C5C2411yescharacteristic
144C22×C5×D3—42no—
145C22×C5×D3C2×Q814yes—
146C22×C5×D3C2412yes—
147C3×C5×Q8—44no—
148C2×C3×C4×C5—42no—
149C2×C3×C4×C5C2×Q814yes—
150C2×C3×C4×C5C2418yes—
151C2×C3×C4×C5C2412yes—
152C2×C3×C4×C5C2412yes—
153C2×C3×C4×C5C2411yescharacteristic
154C2×C3×C4×C5C2411yescharacteristic
155C2×C5×Dic3—42no—
156C2×C5×Dic3C2×Q814yes—
157C2×C5×Dic3C2414yes—
158C2×C5×Dic3C2418yes—
159C2×C5×Dic3C2412yes—
160C4×C5×D3—48no—
161C4×C5×D3C2×Q818yes—
162C4×C5×D3C2×Q818yes—
163C5×Dic6—44no—
164C5×Dic6—48no—
165C22×C4×D3C22×C511yescharacteristic
166C3×C4⋊Q8C22×C512yescomplemented
167C3×#125C22×C512yescomplemented
168C3×C22⋊Q8C22×C514yes—
169C3×C4×Q8C22×C514yescomplemented
170C3×#117C22×C512yes—
171C2×C3×C42C22×C511yescharacteristic
172#946C22×C518yes—
173#944C22×C518yescomplemented
174C2×(C2×D3)⋊C4C22×C512yes—
175#926C22×C512yes—
176#924C22×C514yes—
177C2×Dic3⋊C4C22×C512yes—
178C2×C4×Dic3C22×C511yescharacteristic
179#897C22×C518yes—
180#896C22×C518yes—
181#896C22×C518yes—
182#892C22×C514yes—
183D3×C4⋊C4C22×C514yes—
184#890C22×C514yescomplemented
185#889C22×C518yescomplemented
186#889C22×C518yescomplemented
187#888C22×C514yescomplemented
188#888C22×C518yescomplemented
189D3×C22⋊C4C22×C514yes—
190#878C22×C518yes—
191#877C22×C514yes—
192#872C22×C512yes—
193#872C22×C512yes—
194#872C22×C514yes—
195#872C22×C514yes—
196C42×D3C22×C514yes—
197#870C22×C512yescomplemented
198#869C22×C512yescomplemented
199C4×Dic6C22×C514yescomplemented
200C4×Dic6C22×C518yescomplemented
201C24×C5—31no—
202C2×C5×Q8—38no—
203C2×C5×Q8—34no—
204C2×C5×Q8C22×D314yes—
205C22×C4×C5—31no—
206C22×C4×C5—38no—
207C22×C4×C5—32no—
208C22×C4×C5—32no—
209C22×C4×C5—32no—
210C22×C4×C5—38no—
211C22×C4×C5C22×D312yes—
212C22×C4×C5C22×D311yescharacteristic
213C5×C4⋊C4—38no—
214C5×C4⋊C4—34no—
215C5×C4⋊C4—34no—
216C5×C4⋊C4—34no—
217C5×C4⋊C4—38no—
218C5×C4⋊C4—38no—
219C5×C4⋊C4—38no—
220C5×C4⋊C4—38no—
221C5×C4⋊C4—34no—
222C5×C4⋊C4C22×D314yes—
223C5×C4⋊C4C22×D314yes—
224C5×C4⋊C4C22×D318yes—
225C5×C22⋊C4—34no—
226C5×C22⋊C4—38no—
227C5×C22⋊C4—34no—
228C5×C22⋊C4—34no—
229C5×C22⋊C4C22×D314yes—
230C42×C5—38no—
231C42×C5—32no—
232C42×C5—32no—
233C42×C5—34no—
234C42×C5C22×D311yescharacteristic
235C42×C5C22×D311yescharacteristic
236C42×C5C22×D314yes—
237C42×C5C22×D312yes—
238#544—34no—
239#544—34no—
240#544—34no—
241#543—34no—
242#533—38no—
243#533—32no—
244#533—31no—
245#533C5×D311yescharacteristic, complemented
246C2×#117—32no—
247C2×#117—31no—
248C22×C3×C5—22no—
249C22×C3×C5C22×Q812yes—
250C22×C3×C5C2511yescharacteristic
251C2×C5×D3—24no—
252C2×C5×D3—44no—
253C2×C5×D3C22×Q814yes—
254C3×C4×C5—22no—
255C3×C4×C5—28no—
256C3×C4×C5—22no—
257C3×C4×C5C22×Q812yes—
258C3×C4×C5C22×Q812yes—
259C5×Dic3—28no—
260C5×Dic3—24no—
261C5×Dic3C22×Q814yes—
262C23×D3C23×C511yescharacteristic
263C2×C3×Q8C23×C514yes—
264C22×C3×C4C23×C512yes—
265C22×C3×C4C23×C511yescharacteristic
266C22×Dic3C23×C512yes—
267C22×Dic3C23×C511yescharacteristic
268C2×C4×D3C5×Q818yescomplemented
269C2×C4×D3C23×C518yes—
270C2×C4×D3C23×C512yes—
271C2×C4×D3C23×C512yes—
272C2×Dic6C23×C514yes—
273C2×Dic6C23×C518yes—
274C3×C4⋊C4C23×C518yes—
275C3×C4⋊C4C23×C514yes—
276C3×C4⋊C4C23×C514yes—
277C3×C22⋊C4C23×C514yes—
278C3×C42C23×C514yes—
279C3×C42C23×C512yes—
280C3×C42C23×C511yescharacteristic
281C3×C42C23×C511yescharacteristic
282#217C23×C514yes—
283(C2×D3)⋊C4C23×C514yes—
284(C2×D3)⋊C4C23×C514yes—
285(C2×D3)⋊C4C23×C518yes—
286C4⋊Dic3C23×C514yes—
287C4⋊Dic3C23×C518yes—
288C4⋊Dic3C23×C514yes—
289Dic3⋊C4C23×C518yes—
290Dic3⋊C4C23×C518yes—
291Dic3⋊C4C23×C518yes—
292Dic3⋊C4C23×C514yes—
293Dic3⋊C4C23×C514yes—
294Dic3⋊C4C23×C518yes—
295C4×Dic3C23×C514yes—
296C4×Dic3C23×C512yes—
297C4×Dic3C23×C518yes—
298C4×Dic3C23×C512yes—
299C23×C5—32no—
300C23×C5—34no—
301C23×C5—122no—
302C23×C5C23×D311yescharacteristic
303C5×Q8—44no—
304C5×Q8—124no—
305C5×Q8—128no—
306C2×C4×C5—34no—
307C2×C4×C5—32no—
308C2×C4×C5—38no—
309C2×C4×C5—38no—
310C2×C4×C5—38no—
311C2×C4×C5—34no—
312C2×C4×C5—42no—
313C2×C4×C5—128no—
314C2×C4×C5—122no—
315C2×C4×C5D3×Q814yes—
316C2×C4×C5C23×D318yes—
317C2×C4×C5C23×D312yes—
318C2×C4×C5C23×D312yes—
319C2×C4×C5C23×D311yescharacteristic
320C2×C4×C5C23×D311yescharacteristic
321C23×C4—31no—
322C4⋊Q8—38no—
323C4⋊Q8—38no—
324C4⋊Q8—34no—
325C4⋊Q8—32no—
326C4⋊Q8C2×C5×D312yescomplemented
327#125—38no—
328#125—38no—
329#125—34no—
330#125—32no—
331#125C2×C5×D312yescomplemented
332C22⋊Q8—38no—
333C22⋊Q8—34no—
334C22⋊Q8—38no—
335C22⋊Q8—38no—
336C22⋊Q8—38no—
337C22⋊Q8—38no—
338C22⋊Q8C2×C5×D314yes—
339C4×Q8—38no—
340C4×Q8—34no—
341C4×Q8C2×C5×D314yescomplemented
342#117—32no—
343#117—34no—
344#117—34no—
345#117—34no—
346#117—34no—
347#117—32no—
348#117—32no—
349#117C2×C5×D312yes—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—34no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—34no—
354C2×C42—31no—
355C2×C42—34no—
356C2×C42C2×C5×D311yescharacteristic
357C2×C3×C5—22no—
358C2×C3×C5C23×Q811yescharacteristic
359C2×C3×C5#58512yes—
360C5×D3—24no—
361C23×C3C24×C511yescharacteristic
362C22×D3—42no—
363C22×D3C2×C5×Q814yes—
364C22×D3C24×C512yes—
365C3×Q8—44no—
366C2×C3×C4—42no—
367C2×C3×C4C2×C5×Q814yes—
368C2×C3×C4C24×C518yes—
369C2×C3×C4C24×C512yes—
370C2×C3×C4C24×C512yes—
371C2×C3×C4C24×C511yescharacteristic
372C2×C3×C4C24×C511yescharacteristic
373C2×Dic3—42no—
374C2×Dic3C2×C5×Q814yes—
375C2×Dic3C24×C514yes—
376C2×Dic3C24×C518yes—
377C2×Dic3C24×C512yes—
378C4×D3—48no—
379C4×D3C2×C5×Q818yes—
380C4×D3C2×C5×Q818yes—
381Dic6—44no—
382Dic6—48no—
383C22×C5—22no—
384C22×C5—34no—
385C22×C5—64no—
386C22×C5—124no—
387C22×C5C2×D3×Q812yes—
388C22×C5C24×D311yescharacteristic
389C4×C5—22no—
390C4×C5—22no—
391C4×C5—28no—
392C4×C5—34no—
393C4×C5—68no—
394C4×C5—64no—
395C4×C5C2×D3×Q812yes—
396C4×C5C2×D3×Q812yes—
397C24—31no—
398C2×Q8—38no—
399C2×Q8—34no—
400C2×Q8C22×C5×D314yes—
401C22×C4—31no—
402C22×C4—38no—
403C22×C4—32no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—38no—
407C22×C4C22×C5×D312yes—
408C22×C4C22×C5×D311yescharacteristic
409C4⋊C4—38no—
410C4⋊C4—34no—
411C4⋊C4—34no—
412C4⋊C4—34no—
413C4⋊C4—38no—
414C4⋊C4—38no—
415C4⋊C4—38no—
416C4⋊C4—38no—
417C4⋊C4—34no—
418C4⋊C4C22×C5×D314yes—
419C4⋊C4C22×C5×D314yes—
420C4⋊C4C22×C5×D318yes—
421C22⋊C4—34no—
422C22⋊C4—38no—
423C22⋊C4—34no—
424C22⋊C4—34no—
425C22⋊C4C22×C5×D314yes—
426C42—32no—
427C42—34no—
428C42—38no—
429C42—32no—
430C42C22×C5×D314yes—
431C42C22×C5×D312yes—
432C42C22×C5×D311yescharacteristic
433C42C22×C5×D311yescharacteristic
434C3×C5#345811yescharacteristic, complemented, Hall
435C22×C3—22no—
436C22×C3C22×C5×Q812yes—
437C22×C3C25×C511yescharacteristic
438C2×D3—24no—
439C2×D3—44no—
440C2×D3C22×C5×Q814yes—
441C3×C4—22no—
442C3×C4—28no—
443C3×C4—22no—
444C3×C4C22×C5×Q812yes—
445C3×C4C22×C5×Q812yes—
446Dic3—28no—
447Dic3—24no—
448Dic3C22×C5×Q814yes—
449C2×C5—22no—
450C2×C5—64no—
451C2×C5C22×D3×Q811yescharacteristic
452C2×C5#596412yes—
453C23—32no—
454C23—34no—
455C23—122no—
456C23C23×C5×D311yescharacteristic
457Q8—44no—
458Q8—124no—
459Q8—128no—
460C2×C4—34no—
461C2×C4—32no—
462C2×C4—38no—
463C2×C4—38no—
464C2×C4—38no—
465C2×C4—34no—
466C2×C4—42no—
467C2×C4—128no—
468C2×C4—122no—
469C2×C4C5×D3×Q814yes—
470C2×C4C23×C5×D318yes—
471C2×C4C23×C5×D312yes—
472C2×C4C23×C5×D312yes—
473C2×C4C23×C5×D311yescharacteristic
474C2×C4C23×C5×D311yescharacteristic
475C2×C3—22no—
476C2×C3C23×C5×Q811yescharacteristic
477C2×C3C5×#58512yes—
478D3—24no—
479C5#8627411yescharacteristic, direct product factor, Sylow
480C22—22no—
481C22—34no—
482C22—64no—
483C22—124no—
484C22C2×C5×D3×Q812yes—
485C22C24×C5×D311yescharacteristic
486C4—22no—
487C4—22no—
488C4—28no—
489C4—34no—
490C4—68no—
491C4—64no—
492C4C2×C5×D3×Q812yes—
493C4C2×C5×D3×Q812yes—
494C3C5×#345811yescharacteristic, complemented, Sylow
495C2—22no—
496C2—64no—
497C2C22×C5×D3×Q811yescharacteristic
498C2C5×#596412yes—
subgroupG/H
centerC2×C4×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC3×C5×#533C2
socleC22×C3×C5C25
prime radicalC23×C5×D3C23
hypercenterC5×#533D3
hypocenterC3C5×#3458
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C5×#533

Complements

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

NH ≅ G/Ntype
1. C3×C5×#533C2semidirect
2. C5×#5729C2semidirect
3. C5×#5689C2semidirect
5. C5×#5623C2semidirect
6. C5×#5578C2semidirect
8. C5×#5517C2semidirect
10. C5×#5470C2semidirect
13. C3×C5×C4⋊Q8C22semidirect
14. C3×C5×#125C22semidirect
16. C3×C4×C5×Q8C22semidirect
20. C5×#944C22semidirect
31. C5×#890C22semidirect
32. C5×#889C22semidirect
33. C5×#889C22semidirect
34. C5×#888C22semidirect
35. C5×#888C22semidirect
44. C5×#870C22semidirect
45. C5×#869C22semidirect
46. C4×C5×Dic6C22semidirect
47. C4×C5×Dic6C22semidirect
48. #86274C5direct (N × H)
56. C5×#533D3semidirect
65. C2×C4×C5×D3Q8semidirect
96. C3×#533C2×C5semidirect
97. #5729C2×C5semidirect
98. #5689C2×C5semidirect
100. #5623C2×C5semidirect
101. #5578C2×C5semidirect
103. #5517C2×C5semidirect
105. #5470C2×C5semidirect
111. C5×C4⋊Q8C2×D3semidirect
116. C5×#125C2×D3semidirect
126. C4×C5×Q8C2×D3semidirect
166. C3×C4⋊Q8C22×C5semidirect
167. C3×#125C22×C5semidirect
169. C3×C4×Q8C22×C5semidirect
173. #944C22×C5semidirect
184. #890C22×C5semidirect
185. #889C22×C5semidirect
186. #889C22×C5semidirect
187. #888C22×C5semidirect
188. #888C22×C5semidirect
197. #870C22×C5semidirect
198. #869C22×C5semidirect
199. C4×Dic6C22×C5semidirect
200. C4×Dic6C22×C5semidirect
245. #533C5×D3semidirect
268. C2×C4×D3C5×Q8semidirect
326. C4⋊Q8C2×C5×D3semidirect
331. #125C2×C5×D3semidirect
341. C4×Q8C2×C5×D3semidirect
434. C3×C5#3458semidirect
479. C5#86274direct (N × H)
494. C3C5×#3458semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5332
C5×#57292
C5×#56892
C5×#56342
C5×#56232
C5×#55782
C5×D3×#1172
C5×#55172
C2×C5×#8722
C5×#54702
C5×#34583
#862745