Finite Group Databasebeta

← #14480631

Subgroups — #14480631

Order 1920.

3112 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6819C212yesmaximal, complemented
2C3×C5×#534C211yesmaximal, characteristic, complemented
3#11756563C212yesmaximal, complemented
4#11756415C214yesmaximal, complemented
5#11756407C214yesmaximal, complemented
6#11756356C214yesmaximal, complemented
7#11756355C214yesmaximal, complemented
8#11756351C214yesmaximal, complemented
9#11756345C214yesmaximal, complemented
10#11756252C211yesmaximal, characteristic, complemented
11C2×#91766C211yesmaximal, characteristic, complemented
12#10618195—31nomaximal, Hall
13C22×C3×C5×D4C2212yes—
14C22×#1220C2212yes—
15C22×#1220C2212yescomplemented
16C23×Dic15C2211yescharacteristic
17C23×Dic15C2212yescomplemented
18C2×#6819—216no—
19C2×#6819C2214yes—
20C2×#6819C2214yes—
21C2×#6819C2214yescomplemented
22C2×#6819C2214yescomplemented
23C22×C4×D15C2212yescomplemented
24C22×Dic30C2212yescomplemented
25C3×C5×#123C2214yescomplemented
26C3×C5×#121C2214yescomplemented
27C3×C5×#120C2214yescomplemented
28C2×C3×C5×C22⋊C4C2211yescharacteristic
29#91842C2214yescomplemented
30C2×#6719C2211yescharacteristic
31#91829C2214yescomplemented
32#91828C2218yescomplemented
33#91828C2218yescomplemented
34#91826C2214yescomplemented
35#91824C2214yescomplemented
36D4×Dic15C2214yescomplemented
37#91819C2214yescomplemented
38#91818C2214yescomplemented
39C2×#6712C2212yescomplemented
40C2×C4×Dic15C2212yescomplemented
41#91785C2218yescomplemented
42#91782C2214yescomplemented
43#91777C2214yescomplemented
44#91775C2214yescomplemented
45#91774C2218yescomplemented
46#91773C2218yescomplemented
47#91771C2214yescomplemented
48#91770C2218yescomplemented
49#91767C2218yescomplemented
50#91766C2214yescomplemented
51#91766C2214yescomplemented
52#86294—51nomaximal, Hall
53C22×#711—32no—
54C5×#534D311yescharacteristic, complemented
55#66138—32no—
56#65990—34no—
57#65982—34no—
58#65931—34no—
59#65930—34no—
60#65926—34no—
61#65920—34no—
62#65827—31no—
63C2×#4023—31no—
64C24×C3×C5C2312yes—
65C24×C3×C5C2311yescharacteristic
66C23×D15C2312yes—
67C2×C3×C5×D4—28no—
68C2×C3×C5×D4C2314yes—
69C2×C3×C5×D4C2314yes—
70C2×C3×C5×D4C2314yes—
71C22×C3×C4×C5C2312yes—
72C22×C3×C4×C5C2312yes—
73C2×#1220—28no—
74C2×#1220—28no—
75C2×#1220C2314yes—
76C2×#1220C2314yes—
77C2×#1220C2314yes—
78C2×#1220C2314yes—
79C2×#1220C2318yes—
80C22×Dic15—28no—
81C22×Dic15D418yescomplemented
82C22×Dic15C2314yes—
83C22×Dic15C2314yes—
84C22×Dic15C2312yes—
85C22×Dic15C2312yes—
86C22×Dic15C2312yes—
87C22×Dic15C2312yes—
88C22×Dic15C2311yescharacteristic
89C22×Dic15C2311yescharacteristic
90#6819—216no—
91#6819—216no—
92#6819—216no—
93#6819—216no—
94C2×D60C2314yes—
95C2×C4×D15—28no—
96C2×C4×D15C2314yes—
97C2×C4×D15C2314yes—
98C2×Dic30—28no—
99C2×Dic30C2314yes—
100C2×Dic30C2314yes—
101C3×C5×C4⋊C4C2314yes—
102C3×C5×C22⋊C4C2314yes—
103C3×C5×C22⋊C4C2318yes—
104#6719C2314yes—
105#6719C2318yes—
106#6714C2318yes—
107#6714C2318yes—
108C4⋊Dic15C2314yes—
109#6712C2318yes—
110#6712C2318yes—
111C4×Dic15C2314yes—
112C4×Dic15C2314yes—
113C22×#237—52no—
114C3×#534D511yescharacteristic, complemented
115#5804—52no—
116#5656—54no—
117#5648—54no—
118#5597—54no—
119#5596—54no—
120#5592—54no—
121#5586—54no—
122#5493—51no—
123C2×#877—51no—
124C22×C5×D4C2×D312yes—
125C22×C22⋊D5—32no—
126C22×C22⋊D5—32no—
127C23×Dic5—31no—
128C23×Dic5—32no—
129C2×#711—34no—
130C2×#711—34no—
131C2×#711—34no—
132C2×#711—34no—
133C2×#711—616no—
134C22×C4×D5—32no—
135C22×Dic10—32no—
136C5×#123C2×D314yescomplemented
137C5×#121C2×D314yescomplemented
138C5×#120C2×D314yescomplemented
139C2×C5×C22⋊C4C2×D311yescharacteristic
140#4099—34no—
141C2×C22⋊Dic5—31no—
142#4086—34no—
143#4085—38no—
144#4085—38no—
145#4083—34no—
146#4081—34no—
147D4×Dic5—34no—
148#4076—34no—
149#4075—34no—
150C2×#683—32no—
151C2×C4×Dic5—32no—
152#4042—38no—
153#4039—34no—
154#4034—34no—
155#4032—34no—
156#4031—38no—
157#4030—38no—
158#4028—34no—
159#4027—38no—
160#4024—38no—
161#4023—34no—
162#4023—34no—
163#3449—151noSylow
164C23×C3×C5—24no—
165C23×C3×C5—42no—
166C23×C3×C5—44no—
167C23×C3×C5C2×D414yes—
168C23×C3×C5C2414yes—
169C23×C3×C5C2412yes—
170C23×C3×C5C2411yescharacteristic
171C22×D15—24no—
172C22×D15—44no—
173C22×D15C2414yes—
174C3×C5×D4—28no—
175C3×C5×D4—28no—
176C3×C5×D4—44no—
177C2×C3×C4×C5—24no—
178C2×C3×C4×C5—42no—
179C2×C3×C4×C5C2414yes—
180C2×C3×C4×C5C2414yes—
181#1220—28no—
182#1220—28no—
183#1220—28no—
184#1220—28no—
185#1220—48no—
186C2×Dic15—24no—
187C2×Dic15—28no—
188C2×Dic15—28no—
189C2×Dic15—42no—
190C2×Dic15C2×D418yes—
191C2×Dic15C2×D418yes—
192C2×Dic15C2×D414yes—
193C2×Dic15C2×D414yes—
194C2×Dic15C2414yes—
195C2×Dic15C2414yes—
196C2×Dic15C2411yescharacteristic
197C2×Dic15C2411yescharacteristic
198C2×Dic15C2412yes—
199D60—44no—
200C4×D15—28no—
201C4×D15—28no—
202Dic30—28no—
203Dic30—28no—
204C22×C3×D4C2×D512yes—
205C22×#67—52no—
206C22×#67—52no—
207C23×Dic3—51no—
208C23×Dic3—52no—
209C2×#237—54no—
210C2×#237—54no—
211C2×#237—54no—
212C2×#237—54no—
213C2×#237—1016no—
214C22×C4×D3—52no—
215C22×Dic6—52no—
216C3×#123C2×D514yescomplemented
217C3×#121C2×D514yescomplemented
218C3×#120C2×D514yescomplemented
219C2×C3×C22⋊C4C2×D511yescharacteristic
220#953—54no—
221C2×#217—51no—
222#940—54no—
223#939—58no—
224#939—58no—
225#937—54no—
226#935—54no—
227D4×Dic3—54no—
228#930—54no—
229#929—54no—
230C2×Dic3⋊C4—52no—
231C2×C4×Dic3—52no—
232#896—58no—
233#893—54no—
234#888—54no—
235#886—54no—
236#885—58no—
237#884—58no—
238#882—54no—
239#67⋊C4—58no—
240#878—58no—
241#877—54no—
242#877—54no—
243C24×C5C22×D312yes—
244C24×C5C22×D311yescharacteristic
245C23×D5—32no—
246C2×C5×D4—28no—
247C2×C5×D4C22×D314yes—
248C2×C5×D4C22×D314yes—
249C2×C5×D4C22×D314yes—
250C22×C4×C5C22×D312yes—
251C22×C4×C5C22×D312yes—
252C2×C22⋊D5—34no—
253C2×C22⋊D5—34no—
254C2×C22⋊D5—34no—
255C2×C22⋊D5—34no—
256C2×C22⋊D5—38no—
257C2×C22⋊D5—68no—
258C2×C22⋊D5—68no—
259C22×Dic5—38no—
260C22×Dic5—32no—
261C22×Dic5—32no—
262C22×Dic5—31no—
263C22×Dic5—31no—
264C22×Dic5—32no—
265C22×Dic5—34no—
266C22×Dic5—34no—
267C22×Dic5—32no—
268C22×Dic5—68no—
269#711—616no—
270#711—616no—
271#711—616no—
272#711—616no—
273C2×D20—34no—
274C2×C4×D5—34no—
275C2×C4×D5—34no—
276C2×C4×D5—68no—
277C2×Dic10—34no—
278C2×Dic10—34no—
279C2×Dic10—68no—
280C5×C4⋊C4C22×D314yes—
281C5×C22⋊C4C22×D314yes—
282C5×C22⋊C4C22×D318yes—
283C22⋊Dic5—34no—
284C22⋊Dic5—38no—
285#685—38no—
286#685—38no—
287C4⋊Dic5—34no—
288#683—38no—
289#683—38no—
290C4×Dic5—34no—
291C4×Dic5—34no—
292C22×#41—152no—
293#537—154no—
294#536—154no—
295#535—154no—
296#534—152no—
297#534—151no—
298#534D1511yescharacteristic, complemented
299#532—154no—
300#525—154no—
301#525—154no—
302C2×#117—151no—
303C22×C3×C5—22no—
304C22×C3×C5—24no—
305C22×C3×C5—24no—
306C22×C3×C5—28no—
307C22×C3×C5—44no—
308C22×C3×C5—48no—
309C22×C3×C5C22×D414yes—
310C22×C3×C5C22×D412yes—
311C22×C3×C5C2511yescharacteristic
312C2×D15—24no—
313C2×D15—28no—
314C2×D15—48no—
315C3×C4×C5—24no—
316C3×C4×C5—24no—
317Dic15—24no—
318Dic15—24no—
319Dic15C22×D412yes—
320Dic15C22×D414yes—
321Dic15C22×D412yes—
322C24×C3C22×D512yes—
323C24×C3C22×D511yescharacteristic
324C23×D3—52no—
325C2×C3×D4—28no—
326C2×C3×D4C22×D514yes—
327C2×C3×D4C22×D514yes—
328C2×C3×D4C22×D514yes—
329C22×C3×C4C22×D512yes—
330C22×C3×C4C22×D512yes—
331C2×#67—54no—
332C2×#67—54no—
333C2×#67—54no—
334C2×#67—54no—
335C2×#67—58no—
336C2×#67—108no—
337C2×#67—108no—
338C22×Dic3—58no—
339C22×Dic3—52no—
340C22×Dic3—52no—
341C22×Dic3—51no—
342C22×Dic3—51no—
343C22×Dic3—54no—
344C22×Dic3—54no—
345C22×Dic3—52no—
346C22×Dic3—52no—
347C22×Dic3—108no—
348#237—1016no—
349#237—1016no—
350#237—1016no—
351#237—1016no—
352C2×D12—54no—
353C2×C4×D3—54no—
354C2×C4×D3—54no—
355C2×C4×D3—108no—
356C2×Dic6—54no—
357C2×Dic6—54no—
358C2×Dic6—108no—
359C3×C4⋊C4C22×D514yes—
360C3×C22⋊C4C22×D514yes—
361C3×C22⋊C4C22×D518yes—
362#217—54no—
363#217—58no—
364(C2×D3)⋊C4—58no—
365(C2×D3)⋊C4—58no—
366C4⋊Dic3—54no—
367Dic3⋊C4—58no—
368Dic3⋊C4—58no—
369C4×Dic3—54no—
370C4×Dic3—54no—
371C23×C5—24no—
372C23×C5—42no—
373C23×C5—44no—
374C23×C5D3×D414yes—
375C23×C5C23×D314yes—
376C23×C5C23×D312yes—
377C23×C5C23×D311yescharacteristic
378C22×D5—34no—
379C22×D5—64no—
380C22×D5—124no—
381C5×D4—28no—
382C5×D4—28no—
383C5×D4—44no—
384C2×C4×C5—24no—
385C2×C4×C5—42no—
386C2×C4×C5C23×D314yes—
387C2×C4×C5C23×D314yes—
388C22⋊D5—68no—
389C22⋊D5—68no—
390C22⋊D5—68no—
391C22⋊D5—68no—
392C22⋊D5—128no—
393C2×Dic5—38no—
394C2×Dic5—38no—
395C2×Dic5—34no—
396C2×Dic5—34no—
397C2×Dic5—31no—
398C2×Dic5—31no—
399C2×Dic5—32no—
400C2×Dic5—34no—
401C2×Dic5—34no—
402C2×Dic5—68no—
403C2×Dic5—68no—
404C2×Dic5—64no—
405C2×Dic5—122no—
406D20—124no—
407C4×D5—68no—
408C4×D5—68no—
409Dic10—68no—
410Dic10—68no—
411C2×#41—154no—
412C2×#41—154no—
413C2×#41—154no—
414C2×#41—154no—
415C2×#41—3016no—
416C22×Q8—152no—
417C22×D4—152no—
418C22×D4—152no—
419C22×D4C2×D1512yes—
420C23×C4—151no—
421C23×C4—152no—
422C23×C4—152no—
423C4⋊Q8—154no—
424#127—154no—
425#124—158no—
426#123—154no—
427#123—154no—
428#123—154no—
429#123C2×D1514yescomplemented
430C22⋊Q8—158no—
431C22⋊Q8—158no—
432#121—158no—
433#121—158no—
434#121—154no—
435#121—158no—
436#121C2×D1514yescomplemented
437#120—154no—
438#120—154no—
439#120—154no—
440#120C2×D1514yescomplemented
441C4×D4—154no—
442C4×D4—154no—
443C4×D4—158no—
444#117—154no—
445#117—154no—
446C2×C4⋊C4—152no—
447C2×C22⋊C4—151no—
448C2×C22⋊C4C2×D1511yescharacteristic
449C2×C42—152no—
450C2×C3×C5—24no—
451C2×C3×C5—22no—
452C2×C3×C5—44no—
453C2×C3×C5C23×D411yescharacteristic
454C2×C3×C5#58512yes—
455D15—44no—
456C23×C3—24no—
457C23×C3—42no—
458C23×C3—44no—
459C23×C3D4×D514yes—
460C23×C3C23×D514yes—
461C23×C3C23×D512yes—
462C23×C3C23×D511yescharacteristic
463C22×D3—54no—
464C22×D3—104no—
465C22×D3—204no—
466C3×D4—28no—
467C3×D4—28no—
468C3×D4—44no—
469C2×C3×C4—24no—
470C2×C3×C4—42no—
471C2×C3×C4C23×D514yes—
472C2×C3×C4C23×D514yes—
473#67—108no—
474#67—108no—
475#67—108no—
476#67—108no—
477#67—208no—
478C2×Dic3—58no—
479C2×Dic3—58no—
480C2×Dic3—54no—
481C2×Dic3—54no—
482C2×Dic3—51no—
483C2×Dic3—51no—
484C2×Dic3—52no—
485C2×Dic3—54no—
486C2×Dic3—54no—
487C2×Dic3—108no—
488C2×Dic3—108no—
489C2×Dic3—104no—
490C2×Dic3—202no—
491D12—204no—
492C4×D3—108no—
493C4×D3—108no—
494Dic6—108no—
495Dic6—108no—
496C22×C5—22no—
497C22×C5—24no—
498C22×C5—24no—
499C22×C5—28no—
500C22×C5—44no—
501C22×C5—48no—
502C22×C5C2×D3×D414yes—
503C22×C5C2×D3×D412yes—
504C22×C5C24×D311yescharacteristic
505C2×D5—64no—
506C2×D5—68no—
507C2×D5—128no—
508C4×C5—24no—
509C4×C5—24no—
510Dic5—32no—
511Dic5—34no—
512Dic5—32no—
513Dic5—64no—
514Dic5—64no—
515C24—152no—
516C24C22×D1512yes—
517C24C22×D1511yescharacteristic
518#41—3016no—
519#41—3016no—
520#41—3016no—
521#41—3016no—
522C2×Q8—154no—
523C2×Q8—154no—
524C2×Q8—308no—
525C2×D4—28no—
526C2×D4—154no—
527C2×D4—154no—
528C2×D4—154no—
529C2×D4—154no—
530C2×D4—158no—
531C2×D4—154no—
532C2×D4—308no—
533C2×D4—308no—
534C2×D4C22×D1514yes—
535C2×D4C22×D1514yes—
536C2×D4C22×D1514yes—
537C22×C4—158no—
538C22×C4—151no—
539C22×C4—151no—
540C22×C4—152no—
541C22×C4—152no—
542C22×C4—154no—
543C22×C4—154no—
544C22×C4—152no—
545C22×C4—152no—
546C22×C4—154no—
547C22×C4—154no—
548C22×C4—308no—
549C22×C4—308no—
550C22×C4C22×D1512yes—
551C22×C4C22×D1512yes—
552C4⋊C4—158no—
553C4⋊C4—158no—
554C4⋊C4—154no—
555C4⋊C4C22×D1514yes—
556C22⋊C4—154no—
557C22⋊C4—158no—
558C22⋊C4—158no—
559C22⋊C4—158no—
560C22⋊C4C22×D1514yes—
561C22⋊C4C22×D1518yes—
562C42—154no—
563C42—154no—
564C3×C5#344911yescharacteristic, complemented, Hall
565C22×C3—22no—
566C22×C3—24no—
567C22×C3—24no—
568C22×C3—28no—
569C22×C3—44no—
570C22×C3—48no—
571C22×C3C2×D4×D514yes—
572C22×C3C2×D4×D512yes—
573C22×C3C24×D511yescharacteristic
574C2×D3—104no—
575C2×D3—108no—
576C2×D3—208no—
577C3×C4—24no—
578C3×C4—24no—
579Dic3—52no—
580Dic3—54no—
581Dic3—52no—
582Dic3—104no—
583Dic3—104no—
584C2×C5—24no—
585C2×C5—22no—
586C2×C5—44no—
587C2×C5C22×D3×D411yescharacteristic
588C2×C5#596612yes—
589D5—124no—
590C23—24no—
591C23—42no—
592C23—44no—
593C23—154no—
594C23—304no—
595C23—604no—
596C23D4×D1514yes—
597C23C23×D1514yes—
598C23C23×D1512yes—
599C23C23×D1511yescharacteristic
600Q8—308no—
601Q8—308no—
602D4—28no—
603D4—28no—
604D4—44no—
605D4—308no—
606D4—308no—
607D4—308no—
608D4—308no—
609D4—604no—
610D4—608no—
611C2×C4—24no—
612C2×C4—42no—
613C2×C4—158no—
614C2×C4—158no—
615C2×C4—154no—
616C2×C4—154no—
617C2×C4—151no—
618C2×C4—151no—
619C2×C4—152no—
620C2×C4—154no—
621C2×C4—154no—
622C2×C4—308no—
623C2×C4—308no—
624C2×C4—308no—
625C2×C4—308no—
626C2×C4—304no—
627C2×C4—602no—
628C2×C4C23×D1514yes—
629C2×C4C23×D1514yes—
630C2×C3—24no—
631C2×C3—22no—
632C2×C3—44no—
633C2×C3C22×D4×D511yescharacteristic
634C2×C3#6628512yes—
635D3—204no—
636C5#8629411yescharacteristic, complemented, Sylow
637C22—22no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—44no—
642C22—48no—
643C22—304no—
644C22—308no—
645C22—608no—
646C22C2×D4×D1514yes—
647C22C2×D4×D1512yes—
648C22C24×D1511yescharacteristic
649C4—24no—
650C4—24no—
651C4—152no—
652C4—154no—
653C4—152no—
654C4—304no—
655C4—304no—
656C3#1061819511yescharacteristic, complemented, Sylow
657C2—24no—
658C2—22no—
659C2—44no—
660C2—604no—
661C2C22×D4×D1511yescharacteristic
662C2#1175714812yes—
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#534C2
hypercenter#534D15
hypocenterC3×C5#3449
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C22×#6819C2semidirect
2. C3×C5×#534C2semidirect
3. #11756563C2semidirect
4. #11756415C2semidirect
5. #11756407C2semidirect
6. #11756356C2semidirect
7. #11756355C2semidirect
8. #11756351C2semidirect
9. #11756345C2semidirect
10. #11756252C2semidirect
11. C2×#91766C2semidirect
15. C22×#1220C22semidirect
17. C23×Dic15C22semidirect
21. C2×#6819C22semidirect
22. C2×#6819C22semidirect
23. C22×C4×D15C22semidirect
24. C22×Dic30C22semidirect
25. C3×C5×#123C22semidirect
26. C3×C5×#121C22semidirect
27. C3×C5×#120C22semidirect
29. #91842C22semidirect
31. #91829C22semidirect
32. #91828C22semidirect
33. #91828C22semidirect
34. #91826C22semidirect
35. #91824C22semidirect
36. D4×Dic15C22semidirect
37. #91819C22semidirect
38. #91818C22semidirect
39. C2×#6712C22semidirect
40. C2×C4×Dic15C22semidirect
41. #91785C22semidirect
42. #91782C22semidirect
43. #91777C22semidirect
44. #91775C22semidirect
45. #91774C22semidirect
46. #91773C22semidirect
47. #91771C22semidirect
48. #91770C22semidirect
49. #91767C22semidirect
50. #91766C22semidirect
51. #91766C22semidirect
54. C5×#534D3semidirect
81. C22×Dic15D4semidirect
114. C3×#534D5semidirect
136. C5×#123C2×D3semidirect
137. C5×#121C2×D3semidirect
138. C5×#120C2×D3semidirect
216. C3×#123C2×D5semidirect
217. C3×#121C2×D5semidirect
218. C3×#120C2×D5semidirect
298. #534D15semidirect
429. #123C2×D15semidirect
436. #121C2×D15semidirect
440. #120C2×D15semidirect
564. C3×C5#3449semidirect
636. C5#86294semidirect
656. C3#10618195semidirect

Maximal subgroups

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

subgroupindex
C22×#68192
C3×C5×#5342
#117565632
#117564152
#117564072
#117563562
#117563552
#117563512
#117563452
#117562522
C2×#917662
#106181953
#862945