Finite Group Databasebeta

← D5×#5700 #14495356

Subgroups — D5×#5700 #14495356

Order 1920.

2304 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756459C211yesmaximal, characteristic, complemented
2C5×#5700C211yesmaximal, characteristic, complemented
3C3×D5×#125C211yesmaximal, characteristic, complemented
4D5×#896C214yesmaximal, complemented
5D5×#895C214yesmaximal, complemented
6D5×#894C214yesmaximal, complemented
7#11753662C214yesmaximal, complemented
8#11753654C214yesmaximal, complemented
9#11753374C214yesmaximal, complemented
10D5×#875C212yesmaximal, complemented
11#11752656C212yesmaximal, complemented
12D5×#562—31nomaximal, Hall
13C2×D5×D12C2212yes—
14C2×D5×D12C2214yes—
15C2×C4×D3×D5C2214yes—
16C2×D5×Dic6C2212yes—
17C3×C5×#125C2211yescharacteristic, complemented
18#91785C2214yescomplemented
19#91784C2214yescomplemented
20#91783C2214yescomplemented
21#91764C2212yescomplemented
22C5×#896C2214yescomplemented
23C5×#895C2214yescomplemented
24C5×#894C2214yescomplemented
25C5×#875C2212yescomplemented
26C3×D5×C4⋊C4C2212yes—
27C3×D5×C4⋊C4C2214yes—
28C3×#4036C2212yescomplemented
29C3×#4035C2214yescomplemented
30C3×C42×D5C2211yescharacteristic
31C3×#4016C2211yescharacteristic, complemented
32D5×(C2×D3)⋊C4C2214yes—
33D5×(C2×D3)⋊C4C2214yes—
34D5×(C2×D3)⋊C4C2214yes—
35#91466C2214yescomplemented
36#91460C2214yescomplemented
37#91438C2214yescomplemented
38#91436C2214yescomplemented
39#91433C2214yescomplemented
40#91432C2214yescomplemented
41#91429C2214yescomplemented
42#91427C2214yescomplemented
43#91424C2214yescomplemented
44#91423C2214yescomplemented
45#91420C2214yescomplemented
46#91420C2214yescomplemented
47#91416C2214yescomplemented
48#91416C2214yescomplemented
49D5×Dic3⋊C4C2214yes—
50#91377C2214yescomplemented
51#91375C2212yescomplemented
52#91373C2212yescomplemented
53#91355C2214yescomplemented
54#91352C2214yescomplemented
55#91351C2214yescomplemented
56#91350C2214yescomplemented
57#91350C2214yescomplemented
58C2×#5700—51nomaximal, Hall
59C5×#562—31no—
60#66034—31no—
61D5×#125D311yescharacteristic, complemented
62#66018—32no—
63D5×#124—32no—
64#66001—34no—
65D5×#123—34no—
66#65981—34no—
67D5×C22⋊Q8—34no—
68#65955—34no—
69D5×#121—34no—
70C22×D3×D5C2314yes—
71C2×D60C2312yes—
72C2×D60C2314yes—
73C2×C4×D15C2314yes—
74C2×Dic30C2312yes—
75C2×C5×D12C2312yes—
76C2×C5×D12C2314yes—
77C2×C4×C5×D3C2314yes—
78C2×C5×Dic6C2312yes—
79C2×C3×C4×D5C2314yes—
80C2×C3×C4×D5C2311yescharacteristic
81C2×C3×C4×D5C2312yes—
82C2×#1204C2314yes—
83C2×#1203C2314yes—
84C2×#1203C2314yes—
85C2×#1203C2314yes—
86C2×#1200C2314yes—
87C2×D3×Dic5C2314yes—
88C2×D5×Dic3C2314yes—
89D5×D12—44no—
90D5×D12—48no—
91C4×D3×D5—48no—
92D5×Dic6—44no—
93C3×C5×C4⋊C4C2312yes—
94C3×C5×C4⋊C4C2314yes—
95C3×C42×C5C2311yescharacteristic
96#6714C2314yes—
97#6714C2314yes—
98#6714C2314yes—
99#6712C2314yes—
100C5×(C2×D3)⋊C4C2314yes—
101C5×(C2×D3)⋊C4C2314yes—
102C5×(C2×D3)⋊C4C2314yes—
103C5×Dic3⋊C4C2314yes—
104C3×C4⋊Dic5C2312yes—
105C3×C4⋊Dic5C2314yes—
106C3×#683C2314yes—
107C3×#683C2314yes—
108C3×#683C2314yes—
109C3×C4×Dic5C2311yescharacteristic
110C3×C4×Dic5C2312yes—
111Dic3⋊Dic5C2314yes—
112#6669C2314yes—
113#6667C2314yes—
114#6667C2314yes—
115#6667C2314yes—
116(C2×D3)⋊Dic5C2314yes—
117(C2×D3)⋊Dic5C2314yes—
118(C2×D3)⋊Dic5C2314yes—
119C2×C3×#125—51no—
120#5700—51no—
121#5700—54no—
122#5700—54no—
123#5700—54no—
124#5700—52no—
125#5700D511yescharacteristic, direct product factor
126C2×#896—54no—
127C2×#895—54no—
128C2×#894—54no—
129C2×#875—52no—
130C2×Q8×D5—32no—
131C2×D4×D5—34no—
132C2×D4×D5—32no—
133C22×C4×D5—34no—
134C5×#125C2×D311yescharacteristic, complemented
135C5×#124—32no—
136C5×#123—34no—
137C5×C22⋊Q8—34no—
138C5×#121—34no—
139#4093—32no—
140#4092—34no—
141#4085—34no—
142#4085—34no—
143#4084—34no—
144#4082—32no—
145#4081—34no—
146#4076—34no—
147#4075—34no—
148#4070—34no—
149#4043—34no—
150#4042—34no—
151#4042—34no—
152#4041—34no—
153#4040—34no—
154#4040—34no—
155D5×C4⋊C4—34no—
156D5×C4⋊C4C2×D314yes—
157D5×C4⋊C4C2×D312yes—
158#4036C2×D312yescomplemented
159#4035C2×D314yescomplemented
160#4032—34no—
161#4031—34no—
162#4031—34no—
163#4030—34no—
164#4030—34no—
165#4029—34no—
166#4029—34no—
167D5×C22⋊C4—34no—
168D5×C22⋊C4—34no—
169D5×C22⋊C4—34no—
170#4024—34no—
171#4024—34no—
172#4021—32no—
173C42×D5C2×D311yescharacteristic
174#4016C2×D311yescharacteristic, complemented
175C2×#562—151noSylow
176C22×D15C2414yes—
177C22×C5×D3C2414yes—
178C22×C3×D5C2411yescharacteristic
179C2×D3×D5—28no—
180C2×D3×D5—28no—
181C2×D3×D5—28no—
182C2×C3×C4×C5C2412yes—
183C2×C3×C4×C5C2411yescharacteristic
184C2×C3×C4×C5C2414yes—
185C2×Dic15C2414yes—
186D60—42no—
187D60—44no—
188C4×D15—44no—
189Dic30—42no—
190C2×C5×Dic3C2414yes—
191C5×D12—42no—
192C5×D12—44no—
193C4×C5×D3—44no—
194C5×Dic6—42no—
195C2×C3×Dic5C2411yescharacteristic
196C2×C3×Dic5C2412yes—
197C2×C3×Dic5C2414yes—
198C3×C4×D5—24no—
199C3×C4×D5—28no—
200C3×C4×D5—22no—
201#1204—44no—
202#1203—44no—
203#1203—44no—
204#1203—44no—
205#1200—44no—
206D3×Dic5—44no—
207D5×Dic3—28no—
208C22×D12—52no—
209C22×D12—54no—
210C22×C4×D3—54no—
211C22×Dic6—52no—
212C3×#125—54no—
213C3×#125—52no—
214C3×#125—51no—
215C3×#125C2×D511yescharacteristic, complemented
216C2×C3×C4⋊C4—52no—
217C2×C3×C4⋊C4—54no—
218C2×C3×C42—51no—
219C2×(C2×D3)⋊C4—54no—
220C2×(C2×D3)⋊C4—54no—
221C2×(C2×D3)⋊C4—54no—
222C2×Dic3⋊C4—54no—
223#896—54no—
224#896—54no—
225#896—54no—
226#896—54no—
227#896—54no—
228#896—54no—
229#896—54no—
230#896C2×D514yescomplemented
231#895—54no—
232#895—54no—
233#895—54no—
234#895—54no—
235#895—54no—
236#895—54no—
237#895—54no—
238#895C2×D514yescomplemented
239#894—54no—
240#894—54no—
241#894—54no—
242#894—54no—
243#894—54no—
244#894—54no—
245#894—54no—
246#894C2×D514yescomplemented
247#875—52no—
248#875—52no—
249#875—52no—
250#875—54no—
251#875—54no—
252#875C2×D512yescomplemented
253C23×D5—34no—
254C2×C5×Q8—32no—
255C2×C5×D4—34no—
256C2×C5×D4—32no—
257C22×C4×C5—34no—
258C2×C22⋊D5—34no—
259C2×C22⋊D5—34no—
260C2×C22⋊D5—34no—
261C22×Dic5—34no—
262Q8×D5—124no—
263D4×D5—128no—
264D4×D5—124no—
265C2×D20—32no—
266C2×D20—34no—
267C2×C4×D5—34no—
268C2×C4×D5—34no—
269C2×C4×D5—34no—
270C2×C4×D5—128no—
271C2×C4×D5C22×D314yes—
272C2×C4×D5C22×D312yes—
273C2×C4×D5C22×D311yescharacteristic
274C2×Dic10—32no—
275C2×Dic10—34no—
276C5×C4⋊C4—34no—
277C5×C4⋊C4C22×D314yes—
278C5×C4⋊C4C22×D312yes—
279C5×C22⋊C4—34no—
280C5×C22⋊C4—34no—
281C5×C22⋊C4—34no—
282C42×C5C22×D311yescharacteristic
283C22⋊Dic5—34no—
284C22⋊Dic5—34no—
285C22⋊Dic5—34no—
286#685—34no—
287#685—34no—
288#685—34no—
289#685—34no—
290#685—34no—
291#685—34no—
292C4⋊Dic5—34no—
293C4⋊Dic5C22×D312yes—
294C4⋊Dic5C22×D314yes—
295#683—34no—
296#683—34no—
297#683C22×D314yes—
298#683C22×D314yes—
299#683C22×D314yes—
300C4×Dic5C22×D311yescharacteristic
301C4×Dic5C22×D312yes—
302#562—31no—
303#562—154no—
304#562—154no—
305#562—154no—
306#562—152no—
307#562—151no—
308C2×#125—51no—
309C2×#124—152no—
310C2×#123—154no—
311C2×C22⋊Q8—154no—
312C2×#121—154no—
313C22×C3×C5C2511yescharacteristic
314C2×D15—24no—
315C2×D15—24no—
316C2×D15—24no—
317C2×C5×D3—24no—
318C2×C5×D3—24no—
319C2×C5×D3—24no—
320C2×C3×D5H(5,𝔽2)14yes—
321C2×C3×D52−1+412yes—
322D3×D5—416no—
323C3×C4×C5—22no—
324C3×C4×C5—24no—
325C3×C4×C5—21no—
326Dic15—24no—
327C3×Dic5—24no—
328C3×Dic5—22no—
329C3×Dic5—21no—
330C5×Dic3—24no—
331C23×D3—54no—
332C22×C3×C4—54no—
333C22×C3×C4—51no—
334C22×C3×C4—52no—
335C22×Dic3—54no—
336C2×D12—54no—
337C2×D12—54no—
338C2×D12—54no—
339C2×D12—52no—
340C2×D12—54no—
341C2×D12—204no—
342C2×D12—208no—
343C2×D12C22×D512yes—
344C2×D12C22×D514yes—
345C2×C4×D3—54no—
346C2×C4×D3—54no—
347C2×C4×D3—54no—
348C2×C4×D3—208no—
349C2×C4×D3C22×D514yes—
350C2×Dic6—52no—
351C2×Dic6—54no—
352C2×Dic6—204no—
353C2×Dic6C22×D512yes—
354C3×C4⋊C4—54no—
355C3×C4⋊C4—54no—
356C3×C4⋊C4—54no—
357C3×C4⋊C4—52no—
358C3×C4⋊C4—54no—
359C3×C4⋊C4C22×D512yes—
360C3×C4⋊C4C22×D514yes—
361C3×C42—52no—
362C3×C42—51no—
363C3×C42C22×D511yescharacteristic
364(C2×D3)⋊C4—54no—
365(C2×D3)⋊C4—54no—
366(C2×D3)⋊C4—54no—
367(C2×D3)⋊C4—54no—
368(C2×D3)⋊C4—54no—
369(C2×D3)⋊C4—54no—
370(C2×D3)⋊C4—54no—
371(C2×D3)⋊C4—54no—
372(C2×D3)⋊C4—54no—
373(C2×D3)⋊C4C22×D514yes—
374(C2×D3)⋊C4C22×D514yes—
375(C2×D3)⋊C4C22×D514yes—
376Dic3⋊C4—54no—
377Dic3⋊C4—54no—
378Dic3⋊C4—54no—
379Dic3⋊C4C22×D514yes—
380C23×C5—34no—
381C22×D5—34no—
382C22×D5—68no—
383C22×D5—68no—
384C22×D5—68no—
385C22×D5C23×D311yescharacteristic
386C5×Q8—122no—
387C5×D4—122no—
388C5×D4—124no—
389C2×C4×C5—34no—
390C2×C4×C5—124no—
391C2×C4×C5C23×D312yes—
392C2×C4×C5C23×D311yescharacteristic
393C2×C4×C5C23×D314yes—
394C22⋊D5—124no—
395C22⋊D5—124no—
396C22⋊D5—124no—
397C2×Dic5—34no—
398C2×Dic5—124no—
399C2×Dic5C23×D311yescharacteristic
400C2×Dic5C23×D312yes—
401C2×Dic5C23×D314yes—
402D20—122no—
403D20—124no—
404C4×D5—22no—
405C4×D5—28no—
406C4×D5—24no—
407C4×D5—68no—
408C4×D5—124no—
409C4×D5—124no—
410Dic10—124no—
411Dic10—122no—
412C22×Q8—152no—
413C22×D4—154no—
414C22×D4—152no—
415C23×C4—154no—
416#125—54no—
417#125—52no—
418#125—51no—
419#125D3×D511yescharacteristic, complemented
420#124—32no—
421#124—154no—
422#124—154no—
423#124—152no—
424#124—152no—
425#124—152no—
426#123—34no—
427#123—154no—
428#123—154no—
429#123—154no—
430#123—154no—
431#123—154no—
432#123—154no—
433#123—154no—
434C22⋊Q8—34no—
435C22⋊Q8—154no—
436C22⋊Q8—154no—
437C22⋊Q8—154no—
438C22⋊Q8—154no—
439C22⋊Q8—154no—
440C22⋊Q8—154no—
441C22⋊Q8—154no—
442#121—34no—
443#121—154no—
444#121—154no—
445#121—154no—
446#121—154no—
447#121—154no—
448#121—154no—
449#121—154no—
450C2×C4⋊C4—54no—
451C2×C4⋊C4—52no—
452C2×C4⋊C4—154no—
453C2×C22⋊C4—154no—
454C2×C22⋊C4—154no—
455C2×C22⋊C4—154no—
456C2×C42—51no—
457C2×C3×C5C2×H(5,𝔽2)12yes—
458C2×C3×C5C2×2−1+411yescharacteristic
459D15—44no—
460C3×D5#56214yescomplemented
461C5×D3—44no—
462C23×C3—51no—
463C22×D3—54no—
464C22×D3—108no—
465C22×D3—108no—
466C22×D3—108no—
467C22×D3C23×D514yes—
468C2×C3×C4—54no—
469C2×C3×C4—51no—
470C2×C3×C4—52no—
471C2×C3×C4—104no—
472C2×C3×C4—108no—
473C2×C3×C4—102no—
474C2×C3×C4C23×D514yes—
475C2×C3×C4C23×D511yescharacteristic
476C2×C3×C4C23×D512yes—
477C2×Dic3—54no—
478C2×Dic3—108no—
479C2×Dic3C23×D514yes—
480D12—42no—
481D12—44no—
482D12—202no—
483D12—204no—
484D12—204no—
485D12—204no—
486D12—204no—
487C4×D3—44no—
488C4×D3—204no—
489C4×D3—204no—
490C4×D3—204no—
491Dic6—42no—
492Dic6—204no—
493Dic6—202no—
494C22×C5—64no—
495C22×C5—64no—
496C22×C5—64no—
497C22×C5C24×D311yescharacteristic
498C2×D5—64no—
499C2×D5—64no—
500C2×D5—64no—
501C2×D5—1216no—
502C2×D5#100712yes—
503C2×D5#100914yes—
504C4×C5—21no—
505C4×C5—22no—
506C4×C5—24no—
507C4×C5—64no—
508Dic5—21no—
509Dic5—22no—
510Dic5—24no—
511Dic5—64no—
512C24—154no—
513C2×Q8—32no—
514C2×Q8—154no—
515C2×Q8—152no—
516C2×Q8—604no—
517C2×D4—34no—
518C2×D4—32no—
519C2×D4—154no—
520C2×D4—154no—
521C2×D4—154no—
522C2×D4—154no—
523C2×D4—152no—
524C2×D4—608no—
525C2×D4—604no—
526C22×C4—34no—
527C22×C4—51no—
528C22×C4—52no—
529C22×C4—54no—
530C22×C4—154no—
531C22×C4—154no—
532C22×C4—154no—
533C22×C4—154no—
534C22×C4—608no—
535C4⋊C4—34no—
536C4⋊C4—54no—
537C4⋊C4—54no—
538C4⋊C4—54no—
539C4⋊C4—54no—
540C4⋊C4—52no—
541C4⋊C4—154no—
542C4⋊C4—154no—
543C4⋊C4—154no—
544C4⋊C4C2×D3×D514yes—
545C4⋊C4C2×D3×D512yes—
546C22⋊C4—34no—
547C22⋊C4—34no—
548C22⋊C4—34no—
549C22⋊C4—154no—
550C22⋊C4—154no—
551C22⋊C4—154no—
552C22⋊C4—154no—
553C22⋊C4—154no—
554C22⋊C4—154no—
555C22⋊C4—154no—
556C22⋊C4—154no—
557C22⋊C4—154no—
558C42—52no—
559C42—51no—
560C42C2×D3×D511yescharacteristic
561C3×C5C2×#56211yescharacteristic, complemented, Hall
562C22×C3—54no—
563C22×C3—52no—
564C22×C3C24×D511yescharacteristic
565C2×D3—24no—
566C2×D3—24no—
567C2×D3—24no—
568C2×D3—104no—
569C2×D3—104no—
570C2×D3—104no—
571C2×D3—2016no—
572C3×C4—22no—
573C3×C4—24no—
574C3×C4—21no—
575C3×C4—102no—
576C3×C4—104no—
577C3×C4—101no—
578Dic3—24no—
579Dic3—104no—
580C2×C5—124no—
581C2×C5C2×#100711yescharacteristic
582C2×C5C2×#100912yes—
583D5—124no—
584D5#570014yesdirect product factor
585C23—34no—
586C23—51no—
587C23—154no—
588C23—308no—
589C23—308no—
590C23—308no—
591Q8—122no—
592Q8—602no—
593Q8—604no—
594D4—124no—
595D4—122no—
596D4—604no—
597D4—602no—
598D4—604no—
599D4—604no—
600D4—604no—
601C2×C4—34no—
602C2×C4—51no—
603C2×C4—52no—
604C2×C4—54no—
605C2×C4—108no—
606C2×C4—104no—
607C2×C4—102no—
608C2×C4—124no—
609C2×C4—154no—
610C2×C4—308no—
611C2×C4—604no—
612C2×C4—604no—
613C2×C4—604no—
614C2×C4C22×D3×D511yescharacteristic
615C2×C4C22×D3×D512yes—
616C2×C4C22×D3×D514yes—
617C2×C3—54no—
618C2×C3D5×H(5,𝔽2)12yes—
619C2×C3D5×2−1+411yescharacteristic
620D3—44no—
621D3—204no—
622C5C2×#570011yescharacteristic, complemented, Sylow
623C22—52no—
624C22—54no—
625C22—64no—
626C22—64no—
627C22—64no—
628C22—304no—
629C22—304no—
630C22—304no—
631C22—6016no—
632C22C23×D3×D511yescharacteristic
633C4—21no—
634C4—24no—
635C4—22no—
636C4—64no—
637C4—101no—
638C4—104no—
639C4—102no—
640C4—304no—
641C3D5×#56211yescharacteristic, complemented, Sylow
642C2—54no—
643C2—124no—
644C2—604no—
645C2D5×#100711yescharacteristic
646C2D5×#100912yes—
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#125C22
hypercenter#125D3×D5
hypocenterC3×C5C2×#562
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #11756459C2semidirect
2. C5×#5700C2semidirect
3. C3×D5×#125C2semidirect
4. D5×#896C2semidirect
5. D5×#895C2semidirect
6. D5×#894C2semidirect
7. #11753662C2semidirect
8. #11753654C2semidirect
9. #11753374C2semidirect
10. D5×#875C2semidirect
11. #11752656C2semidirect
17. C3×C5×#125C22semidirect
18. #91785C22semidirect
19. #91784C22semidirect
20. #91783C22semidirect
21. #91764C22semidirect
22. C5×#896C22semidirect
23. C5×#895C22semidirect
24. C5×#894C22semidirect
25. C5×#875C22semidirect
28. C3×#4036C22semidirect
29. C3×#4035C22semidirect
31. C3×#4016C22semidirect
35. #91466C22semidirect
36. #91460C22semidirect
37. #91438C22semidirect
38. #91436C22semidirect
39. #91433C22semidirect
40. #91432C22semidirect
41. #91429C22semidirect
42. #91427C22semidirect
43. #91424C22semidirect
44. #91423C22semidirect
45. #91420C22semidirect
46. #91420C22semidirect
47. #91416C22semidirect
48. #91416C22semidirect
50. #91377C22semidirect
51. #91375C22semidirect
52. #91373C22semidirect
53. #91355C22semidirect
54. #91352C22semidirect
55. #91351C22semidirect
56. #91350C22semidirect
57. #91350C22semidirect
61. D5×#125D3semidirect
125. #5700D5direct (N × H)
134. C5×#125C2×D3semidirect
158. #4036C2×D3semidirect
159. #4035C2×D3semidirect
174. #4016C2×D3semidirect
215. C3×#125C2×D5semidirect
230. #896C2×D5semidirect
238. #895C2×D5semidirect
246. #894C2×D5semidirect
252. #875C2×D5semidirect
419. #125D3×D5semidirect
460. C3×D5#562semidirect
561. C3×C5C2×#562semidirect
584. D5#5700direct (N × H)
622. C5C2×#5700semidirect
641. C3D5×#562semidirect

Maximal subgroups

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

subgroupindex
#117564592
C5×#57002
C3×D5×#1252
D5×#8962
D5×#8952
D5×#8942
#117536622
#117536542
#117533742
D5×#8752
#117526562
D5×#5623
C2×#57005