Finite Group Databasebeta

← #14495357

Subgroups — #14495357

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
1D15×#125C211yesmaximal, characteristic, complemented
2C5×#5700C211yesmaximal, characteristic, complemented
3C3×#66034C211yesmaximal, characteristic, complemented
4#11753867C214yesmaximal, complemented
5#11753850C214yesmaximal, complemented
6#11753844C214yesmaximal, complemented
7#11753814C214yesmaximal, complemented
8#11753643C214yesmaximal, complemented
9#11753638C214yesmaximal, complemented
10#11752751C212yesmaximal, complemented
11#11752708C212yesmaximal, complemented
12D5×#562—31nomaximal, Hall
13C2×#6777C2212yes—
14C2×#6777C2214yes—
15C2×C4×D3×D5C2214yes—
16C2×D15⋊Q8C2212yes—
17C3×C5×#125C2211yescharacteristic, complemented
18C4⋊C4×D15C2212yes—
19C4⋊C4×D15C2214yes—
20#91779C2212yescomplemented
21#91778C2214yescomplemented
22C42×D15C2211yescharacteristic
23#91759C2211yescharacteristic, complemented
24C5×#896C2214yescomplemented
25C5×#895C2214yescomplemented
26C5×#894C2214yescomplemented
27C5×#875C2212yescomplemented
28C3×#4042C2214yescomplemented
29C3×#4041C2214yescomplemented
30C3×#4040C2214yescomplemented
31C3×#4021C2212yescomplemented
32#91473C2214yes—
33#91473C2214yes—
34#91473C2214yes—
35#91464C2214yescomplemented
36#91464C2214yescomplemented
37#91457C2214yescomplemented
38#91456C2214yescomplemented
39#91453C2214yescomplemented
40#91453C2214yescomplemented
41#91452C2214yescomplemented
42#91449C2214yescomplemented
43#91447C2214yescomplemented
44#91423C2214yescomplemented
45#91422C2214yescomplemented
46#91421C2214yescomplemented
47#91417C2214yescomplemented
48#91414C2214yescomplemented
49#91410C2214yescomplemented
50#91408C2214yescomplemented
51#91408C2214yescomplemented
52#91407C2214yescomplemented
53#91404C2214yes—
54#91370C2212yescomplemented
55#91368C2212yescomplemented
56#91352C2214yescomplemented
57#91349C2214yescomplemented
58D3×#562—51nomaximal, Hall
59C5×#562—31no—
60#66034D311yescharacteristic, complemented
61D5×#125—31no—
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×C4×D15C2314yes—
72C2×C4×D15C2311yescharacteristic
73C2×C4×D15C2312yes—
74C2×C5×D12C2312yes—
75C2×C5×D12C2314yes—
76C2×C4×C5×D3C2314yes—
77C2×C5×Dic6C2312yes—
78C2×C3×D20C2314yes—
79C2×C3×D20C2312yes—
80C2×C3×C4×D5C2314yes—
81C2×C3×Dic10C2312yes—
82C2×#1204C2314yes—
83C2×#1201C2314yes—
84C2×#1201C2314yes—
85C2×#1201C2314yes—
86C2×#1200C2314yes—
87C2×D3×Dic5C2314yes—
88C2×D5×Dic3C2314yes—
89#6777—44no—
90#6777—48no—
91C4×D3×D5—48no—
92D15⋊Q8—44no—
93C3×C5×C4⋊C4C2312yes—
94C3×C5×C4⋊C4C2314yes—
95C3×C42×C5C2311yescharacteristic
96C4⋊Dic15C2312yes—
97C4⋊Dic15C2314yes—
98#6712C2314yes—
99#6712C2314yes—
100#6712C2314yes—
101C4×Dic15C2311yescharacteristic
102C4×Dic15C2312yes—
103C5×(C2×D3)⋊C4C2314yes—
104C5×(C2×D3)⋊C4C2314yes—
105C5×(C2×D3)⋊C4C2314yes—
106C5×Dic3⋊C4C2314yes—
107C3×#685C2314yes—
108C3×#685C2314yes—
109C3×#685C2314yes—
110C3×#683C2314yes—
111Dic3⋊Dic5C2314yes—
112#6668C2314yes—
113(C2×D3)⋊Dic5C2314yes—
114(C2×D3)⋊Dic5C2314yes—
115(C2×D3)⋊Dic5C2314yes—
116#6665C2314yes—
117#6665C2314yes—
118#6665C2314yes—
119C3×#562—51no—
120#5700D511yescharacteristic, complemented
121D3×#125—51no—
122#5684—52no—
123D3×#124—52no—
124#5667—54no—
125D3×#123—54no—
126#5647—54no—
127D3×C22⋊Q8—54no—
128#5621—54no—
129D3×#121—54no—
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#4042C2×D314yescomplemented
152#4041C2×D314yescomplemented
153#4040—34no—
154#4040C2×D314yescomplemented
155D5×C4⋊C4—34no—
156D5×C4⋊C4—32no—
157D5×C4⋊C4—34no—
158#4036—32no—
159#4035—34no—
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#4021C2×D312yescomplemented
173C42×D5—31no—
174#4016—31no—
175C2×#562—151noSylow
176C22×D15C2411yescharacteristic
177C22×C5×D3C2414yes—
178C22×C3×D5C2414yes—
179C2×D3×D5—28no—
180C2×D3×D5—28no—
181C2×D3×D5—28no—
182C2×C3×C4×C5C2414yes—
183C2×C3×C4×C5C2411yescharacteristic
184C2×C3×C4×C5C2412yes—
185C2×Dic15C2411yescharacteristic
186C2×Dic15C2412yes—
187C2×Dic15C2414yes—
188C4×D15—24no—
189C4×D15—28no—
190C4×D15—22no—
191C2×C5×Dic3C2414yes—
192C5×D12—42no—
193C5×D12—44no—
194C4×C5×D3—44no—
195C5×Dic6—42no—
196C2×C3×Dic5C2414yes—
197C3×D20—42no—
198C3×D20—44no—
199C3×C4×D5—44no—
200C3×Dic10—42no—
201#1204—44no—
202#1201—44no—
203#1201—44no—
204#1201—44no—
205#1200—28no—
206D3×Dic5—44no—
207D5×Dic3—44no—
208C2×D3×Q8—52no—
209C2×D3×D4—54no—
210C2×D3×D4—52no—
211C22×C4×D3—54no—
212C3×#125C2×D511yescharacteristic, complemented
213C3×#124—52no—
214C3×#123—54no—
215C3×C22⋊Q8—54no—
216C3×#121—54no—
217#947—52no—
218#946—54no—
219#939—54no—
220#939—54no—
221#938—54no—
222#936—52no—
223#935—54no—
224#930—54no—
225#929—54no—
226#924—54no—
227#897—54no—
228#896—54no—
229#896C2×D514yescomplemented
230#895C2×D514yescomplemented
231#894—54no—
232#894C2×D514yescomplemented
233D3×C4⋊C4—52no—
234D3×C4⋊C4—54no—
235D3×C4⋊C4—54no—
236#890—52no—
237#889—54no—
238#886—54no—
239#885—54no—
240#885—54no—
241#884—54no—
242#884—54no—
243#883—54no—
244#883—54no—
245D3×C22⋊C4—54no—
246D3×C22⋊C4—54no—
247D3×C22⋊C4—54no—
248#878—54no—
249#878—54no—
250#875C2×D512yescomplemented
251C42×D3—51no—
252#870—51no—
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×D20C22×D312yes—
266C2×D20C22×D314yes—
267C2×C4×D5—34no—
268C2×C4×D5—32no—
269C2×C4×D5—31no—
270C2×C4×D5—34no—
271C2×C4×D5—34no—
272C2×C4×D5—128no—
273C2×C4×D5C22×D314yes—
274C2×Dic10—34no—
275C2×Dic10C22×D312yes—
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#685C22×D314yes—
290#685C22×D314yes—
291#685C22×D314yes—
292C4⋊Dic5—32no—
293C4⋊Dic5—34no—
294C4⋊Dic5—34no—
295#683—34no—
296#683—34no—
297#683—34no—
298#683—34no—
299#683C22×D314yes—
300C4×Dic5—31no—
301C4×Dic5—32no—
302#562—31no—
303#562—51no—
304#562—154no—
305#562—154no—
306#562—154no—
307#562—152no—
308C2×#125—151no—
309C2×#124—152no—
310C2×#123—154no—
311C2×C22⋊Q8—154no—
312C2×#121—154no—
313C22×C3×C5C2511yescharacteristic
314C2×D15H(5,𝔽2)14yes—
315C2×D152−1+412yes—
316C2×C5×D3—24no—
317C2×C5×D3—24no—
318C2×C5×D3—24no—
319C2×C3×D5—24no—
320C2×C3×D5—24no—
321C2×C3×D5—24no—
322D3×D5—416no—
323C3×C4×C5—22no—
324C3×C4×C5—24no—
325C3×C4×C5—21no—
326Dic15—24no—
327Dic15—22no—
328Dic15—21no—
329C3×Dic5—24no—
330C5×Dic3—24no—
331C23×D3—54no—
332C2×C3×Q8—52no—
333C2×C3×D4—54no—
334C2×C3×D4—52no—
335C22×C3×C4—54no—
336C2×#67—54no—
337C2×#67—54no—
338C2×#67—54no—
339C22×Dic3—54no—
340D3×Q8—204no—
341D3×D4—208no—
342D3×D4—204no—
343C2×D12C22×D512yes—
344C2×D12C22×D514yes—
345C2×C4×D3—51no—
346C2×C4×D3—52no—
347C2×C4×D3—54no—
348C2×C4×D3—54no—
349C2×C4×D3—54no—
350C2×C4×D3—208no—
351C2×C4×D3C22×D514yes—
352C2×Dic6—54no—
353C2×Dic6C22×D512yes—
354C3×C4⋊C4—54no—
355C3×C4⋊C4C22×D514yes—
356C3×C4⋊C4C22×D512yes—
357C3×C22⋊C4—54no—
358C3×C22⋊C4—54no—
359C3×C22⋊C4—54no—
360C3×C42C22×D511yescharacteristic
361#217—54no—
362#217—54no—
363#217—54no—
364(C2×D3)⋊C4—54no—
365(C2×D3)⋊C4—54no—
366(C2×D3)⋊C4—54no—
367(C2×D3)⋊C4C22×D514yes—
368(C2×D3)⋊C4C22×D514yes—
369(C2×D3)⋊C4C22×D514yes—
370C4⋊Dic3—52no—
371C4⋊Dic3—54no—
372C4⋊Dic3—54no—
373Dic3⋊C4—54no—
374Dic3⋊C4—54no—
375Dic3⋊C4—54no—
376Dic3⋊C4—54no—
377Dic3⋊C4C22×D514yes—
378C4×Dic3—51no—
379C4×Dic3—52no—
380C23×C5—34no—
381C22×D5—31no—
382C22×D5—68no—
383C22×D5—68no—
384C22×D5—68no—
385C22×D5C23×D314yes—
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—31no—
398C2×Dic5—32no—
399C2×Dic5—34no—
400C2×Dic5—124no—
401C2×Dic5C23×D314yes—
402D20—42no—
403D20—44no—
404C4×D5—44no—
405C4×D5—68no—
406C4×D5—62no—
407C4×D5—68no—
408C4×D5—64no—
409C4×D5—124no—
410Dic10—42no—
411Dic10—124no—
412C22×Q8—152no—
413C22×D4—154no—
414C22×D4—152no—
415C23×C4—154no—
416#125—154no—
417#125—152no—
418#125—151no—
419#125D3×D511yescharacteristic, complemented
420#124—32no—
421#124—52no—
422#124—154no—
423#124—154no—
424#124—152no—
425#124—152no—
426#123—34no—
427#123—54no—
428#123—154no—
429#123—154no—
430#123—154no—
431#123—154no—
432#123—154no—
433#123—154no—
434C22⋊Q8—34no—
435C22⋊Q8—54no—
436C22⋊Q8—154no—
437C22⋊Q8—154no—
438C22⋊Q8—154no—
439C22⋊Q8—154no—
440C22⋊Q8—154no—
441C22⋊Q8—154no—
442#121—34no—
443#121—54no—
444#121—154no—
445#121—154no—
446#121—154no—
447#121—154no—
448#121—154no—
449#121—154no—
450C2×C4⋊C4—154no—
451C2×C4⋊C4—154no—
452C2×C4⋊C4—152no—
453C2×C22⋊C4—154no—
454C2×C22⋊C4—154no—
455C2×C22⋊C4—154no—
456C2×C42—151no—
457C2×C3×C5C2×H(5,𝔽2)12yes—
458C2×C3×C5C2×2−1+411yescharacteristic
459D15#56214yescomplemented
460C3×D5—44no—
461C5×D3—44no—
462C23×C3—54no—
463C22×D3—51no—
464C22×D3—108no—
465C22×D3—108no—
466C22×D3—108no—
467C22×D3C23×D514yes—
468C3×Q8—202no—
469C3×D4—202no—
470C3×D4—204no—
471C2×C3×C4—54no—
472C2×C3×C4—204no—
473C2×C3×C4C23×D512yes—
474C2×C3×C4C23×D511yescharacteristic
475C2×C3×C4C23×D514yes—
476#67—204no—
477#67—204no—
478#67—204no—
479C2×Dic3—51no—
480C2×Dic3—52no—
481C2×Dic3—54no—
482C2×Dic3—204no—
483C2×Dic3C23×D514yes—
484D12—42no—
485D12—44no—
486C4×D3—44no—
487C4×D3—108no—
488C4×D3—102no—
489C4×D3—108no—
490C4×D3—104no—
491C4×D3—204no—
492Dic6—42no—
493Dic6—204no—
494C22×C5—64no—
495C22×C5—64no—
496C22×C5—64no—
497C22×C5C24×D311yescharacteristic
498C2×D5—24no—
499C2×D5—24no—
500C2×D5—24no—
501C2×D5—34no—
502C2×D5—32no—
503C2×D5—1216no—
504C4×C5—22no—
505C4×C5—24no—
506C4×C5—21no—
507C4×C5—64no—
508Dic5—24no—
509Dic5—62no—
510Dic5—64no—
511Dic5—61no—
512C24—154no—
513C2×Q8—32no—
514C2×Q8—52no—
515C2×Q8—154no—
516C2×Q8—604no—
517C2×D4—34no—
518C2×D4—32no—
519C2×D4—54no—
520C2×D4—52no—
521C2×D4—154no—
522C2×D4—154no—
523C2×D4—154no—
524C2×D4—608no—
525C2×D4—604no—
526C22×C4—34no—
527C22×C4—54no—
528C22×C4—154no—
529C22×C4—154no—
530C22×C4—154no—
531C22×C4—151no—
532C22×C4—152no—
533C22×C4—154no—
534C22×C4—608no—
535C4⋊C4—34no—
536C4⋊C4—54no—
537C4⋊C4—154no—
538C4⋊C4—154no—
539C4⋊C4—154no—
540C4⋊C4—154no—
541C4⋊C4—154no—
542C4⋊C4—154no—
543C4⋊C4—152no—
544C4⋊C4C2×D3×D514yes—
545C4⋊C4C2×D3×D512yes—
546C22⋊C4—34no—
547C22⋊C4—34no—
548C22⋊C4—34no—
549C22⋊C4—54no—
550C22⋊C4—54no—
551C22⋊C4—54no—
552C22⋊C4—154no—
553C22⋊C4—154no—
554C22⋊C4—154no—
555C22⋊C4—154no—
556C22⋊C4—154no—
557C22⋊C4—154no—
558C42—152no—
559C42—151no—
560C42C2×D3×D511yescharacteristic
561C3×C5C2×#56211yescharacteristic, complemented, Hall
562C22×C3—104no—
563C22×C3—104no—
564C22×C3—104no—
565C22×C3C24×D511yescharacteristic
566C2×D3—24no—
567C2×D3—24no—
568C2×D3—24no—
569C2×D3—54no—
570C2×D3—52no—
571C2×D3—2016no—
572C3×C4—22no—
573C3×C4—24no—
574C3×C4—21no—
575C3×C4—104no—
576Dic3—24no—
577Dic3—102no—
578Dic3—104no—
579Dic3—101no—
580C2×C5—124no—
581C2×C5D3×H(5,𝔽2)12yes—
582C2×C5D3×2−1+411yescharacteristic
583D5—34no—
584D5—44no—
585C23—34no—
586C23—54no—
587C23—151no—
588C23—308no—
589C23—308no—
590C23—308no—
591Q8—122no—
592Q8—202no—
593Q8—604no—
594D4—124no—
595D4—122no—
596D4—204no—
597D4—202no—
598D4—604no—
599D4—604no—
600D4—604no—
601C2×C4—34no—
602C2×C4—54no—
603C2×C4—124no—
604C2×C4—151no—
605C2×C4—152no—
606C2×C4—154no—
607C2×C4—204no—
608C2×C4—308no—
609C2×C4—304no—
610C2×C4—302no—
611C2×C4—308no—
612C2×C4—604no—
613C2×C4—604no—
614C2×C4C22×D3×D511yescharacteristic
615C2×C4C22×D3×D512yes—
616C2×C4C22×D3×D514yes—
617C2×C3—204no—
618C2×C3D5×H(5,𝔽2)12yes—
619C2×C3D5×2−1+411yescharacteristic
620D3—44no—
621D3—54no—
622C5D3×#56211yescharacteristic, complemented, Sylow
623C22—64no—
624C22—64no—
625C22—64no—
626C22—104no—
627C22—104no—
628C22—104no—
629C22—152no—
630C22—154no—
631C22—6016no—
632C22C23×D3×D511yescharacteristic
633C4—21no—
634C4—24no—
635C4—22no—
636C4—64no—
637C4—104no—
638C4—301no—
639C4—304no—
640C4—302no—
641C3D5×#56211yescharacteristic, complemented, Sylow
642C2—124no—
643C2—154no—
644C2—204no—
645C2#1175704711yescharacteristic
646C2#1175705412yes—
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. D15×#125C2semidirect
2. C5×#5700C2semidirect
3. C3×#66034C2semidirect
4. #11753867C2semidirect
5. #11753850C2semidirect
6. #11753844C2semidirect
7. #11753814C2semidirect
8. #11753643C2semidirect
9. #11753638C2semidirect
10. #11752751C2semidirect
11. #11752708C2semidirect
17. C3×C5×#125C22semidirect
20. #91779C22semidirect
21. #91778C22semidirect
23. #91759C22semidirect
24. C5×#896C22semidirect
25. C5×#895C22semidirect
26. C5×#894C22semidirect
27. C5×#875C22semidirect
28. C3×#4042C22semidirect
29. C3×#4041C22semidirect
30. C3×#4040C22semidirect
31. C3×#4021C22semidirect
35. #91464C22semidirect
36. #91464C22semidirect
37. #91457C22semidirect
38. #91456C22semidirect
39. #91453C22semidirect
40. #91453C22semidirect
41. #91452C22semidirect
42. #91449C22semidirect
43. #91447C22semidirect
44. #91423C22semidirect
45. #91422C22semidirect
46. #91421C22semidirect
47. #91417C22semidirect
48. #91414C22semidirect
49. #91410C22semidirect
50. #91408C22semidirect
51. #91408C22semidirect
52. #91407C22semidirect
54. #91370C22semidirect
55. #91368C22semidirect
56. #91352C22semidirect
57. #91349C22semidirect
60. #66034D3semidirect
120. #5700D5semidirect
134. C5×#125C2×D3semidirect
151. #4042C2×D3semidirect
152. #4041C2×D3semidirect
154. #4040C2×D3semidirect
172. #4021C2×D3semidirect
212. C3×#125C2×D5semidirect
229. #896C2×D5semidirect
230. #895C2×D5semidirect
232. #894C2×D5semidirect
250. #875C2×D5semidirect
419. #125D3×D5semidirect
459. D15#562semidirect
561. C3×C5C2×#562semidirect
622. C5D3×#562semidirect
641. C3D5×#562semidirect

Maximal subgroups

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

subgroupindex
D15×#1252
C5×#57002
C3×#660342
#117538672
#117538502
#117538442
#117538142
#117536432
#117536382
#117527512
#117527082
D5×#5623
D3×#5625