Finite Group Databasebeta

← C5×#84084 #14424108

Subgroups — C5×#84084 #14424108

Order 1920.

1508 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×#41C211yesmaximal, characteristic
2C2×C5×#951C212yesmaximal, complemented
3C2×C5×#924C211yesmaximal, characteristic, complemented
4C5×#5239C214yesmaximal, complemented
5C5×#5232C212yesmaximal, complemented
6C5×#5228C212yesmaximal, complemented
7C5×#5218C212yesmaximal, complemented
8C5×#5207C211yesmaximal, characteristic, complemented
9C5×#3007—31nomaximal, Hall
10C2×C3×C5×#41—28no—
11C2×C3×C5×#41—21no—
12C2×C3×C5×#41—21no—
13C2×C3×C5×#41C2212yes—
14C2×C3×C5×#41C2212yes—
15C22×C3×C5×Q8C2211yescharacteristic
16C22×C3×C5×D4—21no—
17C22×C3×C5×D4C2211yescharacteristic
18C23×C3×C4×C5—21no—
19C23×C3×C4×C5C2211yescharacteristic
20C22×C5×Dic6C2211yescharacteristic, complemented
21C2×C5×#217—21no—
22C5×#951—28no—
23C2×C5×#216C2214yescomplemented
24C2×C5×#214C2214yescomplemented
25C2×C5×C4⋊Dic3C2211yescharacteristic, complemented
26C5×#924—21no—
27C5×#924—21no—
28C5×#924C2212yes—
29C5×#924C2212yescomplemented
30C2×C5×Dic3⋊C4—21no—
31C2×C5×#208C2212yescomplemented
32C5×#835C2214yescomplemented
33C5×#832C2214yescomplemented
34C5×#830C2212yes—
35C5×#830C2212yescomplemented
36#84084C511yesmaximal, characteristic, direct product factor, Hall
37C22×C5×#41D311yescharacteristic
38C2×C5×#137—32no—
39C2×C5×C22⋊Q8—31no—
40C5×#452—34no—
41C5×#451—32no—
42C5×#450—32no—
43C5×#417—32no—
44C5×#407—31no—
45C24×C3×C5—21no—
46C24×C3×C5D411yescharacteristic
47C3×C5×#41—28no—
48C3×C5×#41—28no—
49C3×C5×#41—24no—
50C3×C5×#41—24no—
51C3×C5×#41—24no—
52C3×C5×#41D418yes—
53C2×C3×C5×Q8—24no—
54C2×C3×C5×Q8—21no—
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×D4—28no—
57C2×C3×C5×D4—22no—
58C2×C3×C5×D4—22no—
59C2×C3×C5×D4—24no—
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4C2312yes—
62C22×C3×C4×C5—28no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5—22no—
65C22×C3×C4×C5—22no—
66C22×C3×C4×C5—24no—
67C22×C3×C4×C5D411yescharacteristic
68C22×C3×C4×C5D411yescharacteristic
69C22×C3×C4×C5D411yescharacteristic
70C22×C3×C4×C5C2311yescharacteristic
71C22×C3×C4×C5C2311yescharacteristic
72C22×C3×C4×C5C2311yescharacteristic
73C22×C5×Dic3—21no—
74C22×C5×Dic3—21no—
75C2×C5×Dic6—24no—
76C2×C5×Dic6—21no—
77C2×C5×Dic6C2312yes—
78C5×#217—22no—
79C5×#217—22no—
80C5×#216—28no—
81C5×#214—28no—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3C2312yes—
84C5×Dic3⋊C4—22no—
85C5×Dic3⋊C4—22no—
86C5×#208—24no—
87C2×C5×C3⋊C8C2314yes—
88C22×C3×#41C2×C511yescharacteristic
89C2×#951C2×C512yescomplemented
90C2×#924C2×C511yescharacteristic, complemented
91#5239C2×C514yescomplemented
92#5232C2×C512yescomplemented
93#5228C2×C512yescomplemented
94#5218C2×C512yescomplemented
95#5207C2×C511yescharacteristic, complemented
96C2×C5×#41—28no—
97C2×C5×#41—21no—
98C2×C5×#41—21no—
99C2×C5×#41C2×D312yes—
100C2×C5×#41C2×D312yes—
101C22×C5×Q8—31no—
102C22×C5×Q8C2×D311yescharacteristic
103C22×C5×D4—21no—
104C22×C5×D4C2×D311yescharacteristic
105C23×C4×C5—21no—
106C23×C4×C5C2×D311yescharacteristic
107C5×#137—68no—
108C2×C5×Q16—34no—
109C2×C5×SD16—34no—
110C2×C5×M16—32no—
111C5×C22⋊Q8—32no—
112C5×C22⋊Q8—32no—
113C5×C22⋊Q8—61no—
114C5×C22⋊Q8—61no—
115C2×C5×C4⋊C4—31no—
116C2×C5×C4⋊C4—61no—
117C2×C5×C22⋊C4—61no—
118C5×#103—34no—
119C5×#102—34no—
120C5×C22⋊C8—32no—
121C5×C22⋊C8—32no—
122#3007—31noSylow
123C23×C3×C5—24no—
124C23×C3×C5—22no—
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5—41no—
128C23×C3×C5—42no—
129C23×C3×C5—42no—
130C23×C3×C5C2×D411yescharacteristic
131C23×C3×C5C2×D411yescharacteristic
132C23×C3×C5C2×D411yescharacteristic
133C3×C5×Q8—24no—
134C3×C5×Q8—22no—
135C3×C5×Q8C2×D414yes—
136C3×C5×D4—28no—
137C3×C5×D4—24no—
138C3×C5×D4—28no—
139C3×C5×D4—22no—
140C3×C5×D4C2×D414yes—
141C2×C3×C4×C5—24no—
142C2×C3×C4×C5—28no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—28no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D414yes—
154C2×C3×C4×C5C2411yescharacteristic
155C2×C5×Dic3—22no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—42no—
158C2×C5×Dic3—42no—
159C5×Dic6—24no—
160C5×Dic6—42no—
161C5×C3⋊C8—24no—
162C2×C3×#41—28no—
163C2×C3×#41—21no—
164C2×C3×#41—21no—
165C2×C3×#41C22×C512yes—
166C2×C3×#41C22×C512yes—
167C22×C3×Q8C22×C511yescharacteristic
168C22×C3×D4—21no—
169C22×C3×D4C22×C511yescharacteristic
170C23×C3×C4—21no—
171C23×C3×C4C22×C511yescharacteristic
172C22×Dic6C22×C511yescharacteristic, complemented
173C2×#217—21no—
174#951—28no—
175C2×#216C22×C514yescomplemented
176C2×#214C22×C514yescomplemented
177C2×C4⋊Dic3C22×C511yescharacteristic, complemented
178#924—21no—
179#924—21no—
180#924C22×C512yes—
181#924C22×C512yescomplemented
182C2×Dic3⋊C4—21no—
183C2×#208C22×C512yescomplemented
184#835C22×C514yescomplemented
185#832C22×C514yescomplemented
186#830C22×C512yes—
187#830C22×C512yescomplemented
188C24×C5—21no—
189C24×C5#6711yescharacteristic
190C5×#41—28no—
191C5×#41—28no—
192C5×#41—24no—
193C5×#41—24no—
194C5×#41—24no—
195C5×#41#6718yes—
196C2×C5×Q8—24no—
197C2×C5×Q8—21no—
198C2×C5×Q8—32no—
199C2×C5×Q8—61no—
200C2×C5×Q8—64no—
201C2×C5×Q8C22×D312yes—
202C2×C5×D4—28no—
203C2×C5×D4—22no—
204C2×C5×D4—22no—
205C2×C5×D4—24no—
206C2×C5×D4—21no—
207C2×C5×D4C22×D312yes—
208C22×C4×C5—28no—
209C22×C4×C5—21no—
210C22×C4×C5—22no—
211C22×C4×C5—22no—
212C22×C4×C5—24no—
213C22×C4×C5—61no—
214C22×C4×C5—61no—
215C22×C4×C5#6711yescharacteristic
216C22×C4×C5#6711yescharacteristic
217C22×C4×C5#6711yescharacteristic
218C22×C4×C5C22×D311yescharacteristic
219C22×C4×C5C22×D311yescharacteristic
220C22×C4×C5C22×D311yescharacteristic
221C5×Q16—68no—
222C5×SD16—68no—
223C5×M16—64no—
224C2×C5×C8—34no—
225C5×C4⋊C4—32no—
226C5×C4⋊C4—61no—
227C5×C4⋊C4—62no—
228C5×C4⋊C4—62no—
229C5×C22⋊C4—62no—
230C5×C22⋊C4—62no—
231C22×#41C5×D311yescharacteristic
232C2×#137—32no—
233C2×C22⋊Q8—31no—
234#452—34no—
235#451—32no—
236#450—32no—
237#417—32no—
238#407—31no—
239C22×C3×C5—21no—
240C22×C3×C5—21no—
241C22×C3×C5—21no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5—24no—
245C22×C3×C5—21no—
246C22×C3×C5—21no—
247C22×C3×C5—44no—
248C22×C3×C5—48no—
249C22×C3×C5—42no—
250C22×C3×C5—42no—
251C22×C3×C5#12012yes—
252C22×C3×C5#13712yescomplemented
253C22×C3×C5C22×D411yescharacteristic
254C3×C4×C5—21no—
255C3×C4×C5—24no—
256C3×C4×C5—21no—
257C3×C4×C5#12012yes—
258C3×C4×C5C22×D412yes—
259C5×Dic3—42no—
260C5×Dic3—42no—
261C24×C3—21no—
262C24×C3C5×D411yescharacteristic
263C3×#41—28no—
264C3×#41—28no—
265C3×#41—24no—
266C3×#41—24no—
267C3×#41—24no—
268C3×#41C5×D418yes—
269C2×C3×Q8—24no—
270C2×C3×Q8—21no—
271C2×C3×Q8C23×C512yes—
272C2×C3×D4—28no—
273C2×C3×D4—22no—
274C2×C3×D4—22no—
275C2×C3×D4—24no—
276C2×C3×D4—21no—
277C2×C3×D4C23×C512yes—
278C22×C3×C4—28no—
279C22×C3×C4—21no—
280C22×C3×C4—22no—
281C22×C3×C4—22no—
282C22×C3×C4—24no—
283C22×C3×C4C5×D411yescharacteristic
284C22×C3×C4C5×D411yescharacteristic
285C22×C3×C4C5×D411yescharacteristic
286C22×C3×C4C23×C511yescharacteristic
287C22×C3×C4C23×C511yescharacteristic
288C22×C3×C4C23×C511yescharacteristic
289C22×Dic3—21no—
290C22×Dic3—21no—
291C2×Dic6—24no—
292C2×Dic6—21no—
293C2×Dic6C23×C512yes—
294#217—22no—
295#217—22no—
296#216—28no—
297#214—28no—
298C4⋊Dic3—21no—
299C4⋊Dic3C23×C512yes—
300Dic3⋊C4—22no—
301Dic3⋊C4—22no—
302#208—24no—
303C2×C3⋊C8C23×C514yes—
304C23×C5—24no—
305C23×C5—22no—
306C23×C5—22no—
307C23×C5—22no—
308C23×C5—41no—
309C23×C5—42no—
310C23×C5—42no—
311C23×C5C2×#6711yescharacteristic
312C23×C5C2×#6711yescharacteristic
313C23×C5C2×#6711yescharacteristic
314C5×Q8—24no—
315C5×Q8—22no—
316C5×Q8—64no—
317C5×Q8—122no—
318C5×Q8C2×#6714yes—
319C5×D4—28no—
320C5×D4—24no—
321C5×D4—28no—
322C5×D4—22no—
323C5×D4C2×#6714yes—
324C2×C4×C5—24no—
325C2×C4×C5—28no—
326C2×C4×C5—22no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5—28no—
330C2×C4×C5—22no—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—122no—
336C2×C4×C5—122no—
337C2×C4×C5C2×#6711yescharacteristic
338C2×C4×C5C2×#6711yescharacteristic
339C2×C4×C5C2×#6711yescharacteristic
340C2×C4×C5C2×#6714yes—
341C2×C4×C5C23×D311yescharacteristic
342C5×C8—64no—
343C2×#41—28no—
344C2×#41—21no—
345C2×#41—21no—
346C2×#41C2×C5×D312yes—
347C2×#41C2×C5×D312yes—
348C22×Q8—31no—
349C22×Q8C2×C5×D311yescharacteristic
350C22×D4—21no—
351C22×D4C2×C5×D311yescharacteristic
352C23×C4—21no—
353C23×C4C2×C5×D311yescharacteristic
354#137—68no—
355C2×Q16—34no—
356C2×SD16—34no—
357C2×M16—32no—
358C22⋊Q8—32no—
359C22⋊Q8—32no—
360C22⋊Q8—61no—
361C22⋊Q8—61no—
362C2×C4⋊C4—31no—
363C2×C4⋊C4—61no—
364C2×C22⋊C4—61no—
365#103—34no—
366#102—34no—
367C22⋊C8—32no—
368C22⋊C8—32no—
369C2×C3×C5—22no—
370C2×C3×C5—22no—
371C2×C3×C5—41no—
372C2×C3×C5—44no—
373C2×C3×C5C2×#12011yescharacteristic
374C2×C3×C5C2×#13711yescharacteristic
375C2×C3×C5#57511yescharacteristic
376C23×C3—24no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—41no—
381C23×C3—42no—
382C23×C3—42no—
383C23×C3C2×C5×D411yescharacteristic
384C23×C3C2×C5×D411yescharacteristic
385C23×C3C2×C5×D411yescharacteristic
386C3×Q8—24no—
387C3×Q8—22no—
388C3×Q8C2×C5×D414yes—
389C3×D4—28no—
390C3×D4—24no—
391C3×D4—28no—
392C3×D4—22no—
393C3×D4C2×C5×D414yes—
394C2×C3×C4—24no—
395C2×C3×C4—28no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—28no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4C2×C5×D411yescharacteristic
404C2×C3×C4C2×C5×D411yescharacteristic
405C2×C3×C4C2×C5×D411yescharacteristic
406C2×C3×C4C2×C5×D414yes—
407C2×C3×C4C24×C511yescharacteristic
408C2×Dic3—22no—
409C2×Dic3—22no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412Dic6—24no—
413Dic6—42no—
414C3⋊C8—24no—
415C22×C5—21no—
416C22×C5—21no—
417C22×C5—21no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5—21no—
421C22×C5—21no—
422C22×C5—24no—
423C22×C5—44no—
424C22×C5—48no—
425C22×C5—42no—
426C22×C5—42no—
427C22×C5#95112yescomplemented
428C22×C5#95312yes—
429C22×C5C22×#6711yescharacteristic
430C4×C5—21no—
431C4×C5—24no—
432C4×C5—21no—
433C4×C5—122no—
434C4×C5—122no—
435C4×C5#95312yes—
436C4×C5C22×#6712yes—
437C24—21no—
438C24C5×#6711yescharacteristic
439#41—28no—
440#41—28no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41C5×#6718yes—
445C2×Q8—24no—
446C2×Q8—21no—
447C2×Q8—32no—
448C2×Q8—61no—
449C2×Q8—64no—
450C2×Q8C22×C5×D312yes—
451C2×D4—28no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—21no—
456C2×D4C22×C5×D312yes—
457C22×C4—28no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4C5×#6711yescharacteristic
465C22×C4C5×#6711yescharacteristic
466C22×C4C5×#6711yescharacteristic
467C22×C4C22×C5×D311yescharacteristic
468C22×C4C22×C5×D311yescharacteristic
469C22×C4C22×C5×D311yescharacteristic
470Q16—68no—
471SD16—68no—
472M16—64no—
473C2×C8—34no—
474C4⋊C4—32no—
475C4⋊C4—61no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C3×C5#300711yescharacteristic, complemented, Hall
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—21no—
485C22×C3—21no—
486C22×C3—24no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—44no—
490C22×C3—48no—
491C22×C3—42no—
492C22×C3—42no—
493C22×C3C5×#12012yes—
494C22×C3C5×#13712yescomplemented
495C22×C3C22×C5×D411yescharacteristic
496C3×C4—21no—
497C3×C4—24no—
498C3×C4—21no—
499C3×C4C5×#12012yes—
500C3×C4C22×C5×D412yes—
501Dic3—42no—
502Dic3—42no—
503C2×C5—22no—
504C2×C5—22no—
505C2×C5—41no—
506C2×C5—44no—
507C2×C5#582211yescharacteristic
508C2×C5C2×#95111yescharacteristic
509C2×C5C2×#95311yescharacteristic
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23C2×C5×#6711yescharacteristic
518C23C2×C5×#6711yescharacteristic
519C23C2×C5×#6711yescharacteristic
520Q8—24no—
521Q8—22no—
522Q8—64no—
523Q8—122no—
524Q8C2×C5×#6714yes—
525D4—28no—
526D4—24no—
527D4—28no—
528D4—22no—
529D4C2×C5×#6714yes—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4C2×C5×#6711yescharacteristic
544C2×C4C2×C5×#6711yescharacteristic
545C2×C4C2×C5×#6711yescharacteristic
546C2×C4C2×C5×#6714yes—
547C2×C4C23×C5×D311yescharacteristic
548C8—64no—
549C2×C3—22no—
550C2×C3—22no—
551C2×C3—41no—
552C2×C3—44no—
553C2×C3C2×C5×#12011yescharacteristic
554C2×C3C2×C5×#13711yescharacteristic
555C2×C3C5×#57511yescharacteristic
556C5#8408411yescharacteristic, direct product factor, Sylow
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—44no—
566C22—48no—
567C22—42no—
568C22—42no—
569C22C5×#95112yescomplemented
570C22C5×#95312yes—
571C22C22×C5×#6711yescharacteristic
572C4—21no—
573C4—24no—
574C4—21no—
575C4—122no—
576C4—122no—
577C4C5×#95312yes—
578C4C22×C5×#6712yes—
579C3C5×#300711yescharacteristic, complemented, Sylow
580C2—22no—
581C2—22no—
582C2—41no—
583C2—44no—
584C2C5×#582211yescharacteristic
585C2C2×C5×#95111yescharacteristic
586C2C2×C5×#95311yescharacteristic
subgroupG/H
centerC22×C5C22×#67
commutator, cosocleC2×C3×C4C24×C5
FrattiniC2×C4C23×C5×D3
Fitting, prime radicalC22×C3×C5×#41C2
socleC22×C3×C5C22×D4
hypercenterC22×C5×#41D3
hypocenterC3C5×#3007
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C22×C5×#41

Complements

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

NH ≅ G/Ntype
2. C2×C5×#951C2semidirect
3. C2×C5×#924C2semidirect
4. C5×#5239C2semidirect
5. C5×#5232C2semidirect
6. C5×#5228C2semidirect
7. C5×#5218C2semidirect
8. C5×#5207C2semidirect
20. C22×C5×Dic6C22semidirect
23. C2×C5×#216C22semidirect
24. C2×C5×#214C22semidirect
25. C2×C5×C4⋊Dic3C22semidirect
29. C5×#924C22semidirect
31. C2×C5×#208C22semidirect
32. C5×#835C22semidirect
33. C5×#832C22semidirect
35. C5×#830C22semidirect
36. #84084C5direct (N × H)
89. C2×#951C2×C5semidirect
90. C2×#924C2×C5semidirect
91. #5239C2×C5semidirect
92. #5232C2×C5semidirect
93. #5228C2×C5semidirect
94. #5218C2×C5semidirect
95. #5207C2×C5semidirect
172. C22×Dic6C22×C5semidirect
175. C2×#216C22×C5semidirect
176. C2×#214C22×C5semidirect
177. C2×C4⋊Dic3C22×C5semidirect
181. #924C22×C5semidirect
183. C2×#208C22×C5semidirect
184. #835C22×C5semidirect
185. #832C22×C5semidirect
187. #830C22×C5semidirect
252. C22×C3×C5#137semidirect
427. C22×C5#951semidirect
480. C3×C5#3007semidirect
494. C22×C3C5×#137semidirect
556. C5#84084direct (N × H)
569. C22C5×#951semidirect
579. C3C5×#3007semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C5×#412
C2×C5×#9512
C2×C5×#9242
C5×#52392
C5×#52322
C5×#52282
C5×#52182
C5×#52072
C5×#30073
#840845