Finite Group Databasebeta

← #14480690

Subgroups — #14480690

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×#6780C212yesmaximal, complemented
2C2×#91766C211yesmaximal, characteristic, complemented
3C5×#5493C211yesmaximal, characteristic, complemented
4C3×#65827C211yesmaximal, characteristic, complemented
5#11755426C212yesmaximal, complemented
6#11754387C214yesmaximal, complemented
7#11754386C214yesmaximal, complemented
8#11753083C214yesmaximal, complemented
9#11752883C214yesmaximal, complemented
10#11752882C214yesmaximal, complemented
11#11752880C214yesmaximal, complemented
12#10618195—31nomaximal, Hall
13C23×Dic15C2211yescharacteristic
14C22×C5×#67C2212yes—
15C22×C3×C22⋊D5C2212yes—
16C22×#1204C2212yescomplemented
17C22×#1201C2212yescomplemented
18C22×D3×Dic5C2212yescomplemented
19C2×#6780—216no—
20C2×#6780C2214yes—
21C2×#6780C2214yes—
22C2×#6780C2214yescomplemented
23C2×#6780C2214yescomplemented
24C22×D5×Dic3C2212yescomplemented
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26C2×#6719C2211yescharacteristic
27C2×#6712C2212yescomplemented
28C2×C4×Dic15C2212yescomplemented
29#91766C2214yescomplemented
30#91766C2214yescomplemented
31C5×#930C2214yescomplemented
32C5×#886C2214yescomplemented
33C5×#882C2214yescomplemented
34C3×#4076C2214yescomplemented
35C3×#4032C2214yescomplemented
36C3×#4028C2214yescomplemented
37#91571C2218yescomplemented
38#91568C2214yescomplemented
39#91563C2214yescomplemented
40#91539C2214yescomplemented
41#91536C2214yescomplemented
42#91470C2214yescomplemented
43#91464C2218yescomplemented
44#91463C2214yescomplemented
45#91453C2218yescomplemented
46#91442C2218yescomplemented
47#91436C2218yescomplemented
48#91435C2214yescomplemented
49#91434C2214yescomplemented
50#91430C2218yescomplemented
51#91408C2218yescomplemented
52#86294—51nomaximal, Hall
53C22×#711—32no—
54C5×#534—31no—
55#66138—32no—
56#65990—34no—
57#65982—34no—
58#65931—34no—
59#65930—34no—
60#65926—34no—
61#65920—34no—
62#65827D311yescharacteristic, complemented
63C2×#4023—31no—
64C24×C3×C5C2311yescharacteristic
65C23×C5×D3C2312yes—
66C23×C3×D5C2312yes—
67C22×C3×C4×C5C2312yes—
68C22×Dic15D418yescomplemented
69C22×Dic15C2312yes—
70C22×Dic15C2312yes—
71C22×Dic15C2312yes—
72C22×Dic15C2311yescharacteristic
73C22×Dic15C2311yescharacteristic
74C2×C5×#67—28no—
75C2×C5×#67C2314yes—
76C2×C5×#67C2314yes—
77C22×C5×Dic3C2312yes—
78C2×C5×D12C2314yes—
79C2×C3×C22⋊D5—28no—
80C2×C3×C22⋊D5C2314yes—
81C2×C3×C22⋊D5C2314yes—
82C22×C3×Dic5C2312yes—
83C2×C3×D20C2314yes—
84C2×#1204—28no—
85C2×#1204C2314yes—
86C2×#1204C2314yes—
87C2×#1201—28no—
88C2×#1201C2314yes—
89C2×#1201C2314yes—
90C2×#1201C2318yes—
91C2×D3×Dic5—28no—
92C2×D3×Dic5C2314yes—
93C2×D3×Dic5C2314yes—
94#6780—216no—
95#6780—216no—
96#6780—216no—
97#6780—216no—
98C2×D5×Dic3—28no—
99C2×D5×Dic3C2314yes—
100C2×D5×Dic3C2314yes—
101C3×C5×C22⋊C4C2314yes—
102#6719C2314yes—
103#6712C2318yes—
104C4×Dic15C2314yes—
105C4×Dic15C2314yes—
106C5×(C2×D3)⋊C4C2318yes—
107C5×C4⋊Dic3C2314yes—
108C3×#685C2318yes—
109C3×C4⋊Dic5C2314yes—
110#6669C2318yes—
111(C2×D3)⋊Dic5C2318yes—
112#6665C2318yes—
113C22×#237—52no—
114C3×#534—51no—
115#5804—52no—
116#5656—54no—
117#5648—54no—
118#5597—54no—
119#5596—54no—
120#5592—54no—
121#5586—54no—
122#5493D511yescharacteristic, complemented
123C2×#877—51no—
124C22×C5×D4—32no—
125C22×C22⋊D5—32no—
126C22×C22⋊D5C2×D312yes—
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×#123—34no—
137C5×#121—34no—
138C5×#120—34no—
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#4076C2×D314yescomplemented
149#4075—34no—
150C2×#683—32no—
151C2×C4×Dic5—32no—
152#4042—38no—
153#4039—34no—
154#4034—34no—
155#4032C2×D314yescomplemented
156#4031—38no—
157#4030—38no—
158#4028C2×D314yescomplemented
159#4027—38no—
160#4024—38no—
161#4023—34no—
162#4023—34no—
163#3449—151noSylow
164C23×C3×C5—42no—
165C23×C3×C5C2×D414yes—
166C23×C3×C5C2412yes—
167C23×C3×C5C2411yescharacteristic
168C22×C5×D3—24no—
169C22×C5×D3—44no—
170C22×C5×D3C2414yes—
171C22×C3×D5—24no—
172C22×C3×D5—44no—
173C22×C3×D5C2414yes—
174C2×C3×C4×C5—42no—
175C2×C3×C4×C5C2414yes—
176C2×Dic15—42no—
177C2×Dic15C2×D418yes—
178C2×Dic15C2×D414yes—
179C2×Dic15C2×D414yes—
180C2×Dic15C2×D418yes—
181C2×Dic15C2414yes—
182C2×Dic15C2411yescharacteristic
183C2×Dic15C2411yescharacteristic
184C2×Dic15C2412yes—
185C5×#67—28no—
186C5×#67—28no—
187C2×C5×Dic3—24no—
188C2×C5×Dic3C2414yes—
189C5×D12—44no—
190C3×C22⋊D5—28no—
191C3×C22⋊D5—28no—
192C2×C3×Dic5—24no—
193C2×C3×Dic5C2414yes—
194C3×D20—44no—
195#1204—28no—
196#1204—28no—
197#1201—28no—
198#1201—28no—
199#1201—48no—
200D3×Dic5—28no—
201D3×Dic5—28no—
202D5×Dic3—28no—
203D5×Dic3—28no—
204C22×C3×D4—52no—
205C22×#67—52no—
206C22×#67C2×D512yes—
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×#123—54no—
217C3×#121—54no—
218C3×#120—54no—
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#930C2×D514yescomplemented
229#929—54no—
230C2×Dic3⋊C4—52no—
231C2×C4×Dic3—52no—
232#896—58no—
233#893—54no—
234#888—54no—
235#886C2×D514yescomplemented
236#885—58no—
237#884—58no—
238#882C2×D514yescomplemented
239#67⋊C4—58no—
240#878—58no—
241#877—54no—
242#877—54no—
243C24×C5—32no—
244C24×C5C22×D311yescharacteristic
245C23×D5C22×D312yes—
246C2×C5×D4—34no—
247C2×C5×D4—34no—
248C2×C5×D4—34no—
249C2×C5×D4—68no—
250C22×C4×C5—32no—
251C22×C4×C5C22×D312yes—
252C2×C22⋊D5—28no—
253C2×C22⋊D5—34no—
254C2×C22⋊D5—34no—
255C2×C22⋊D5—38no—
256C2×C22⋊D5—68no—
257C2×C22⋊D5C22×D314yes—
258C2×C22⋊D5C22×D314yes—
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—68no—
268C22×Dic5C22×D312yes—
269#711—616no—
270#711—616no—
271#711—616no—
272#711—616no—
273C2×D20C22×D314yes—
274C2×C4×D5—34no—
275C2×C4×D5—34no—
276C2×C4×D5—68no—
277C2×Dic10—34no—
278C2×Dic10—34no—
279C2×Dic10—68no—
280C5×C4⋊C4—34no—
281C5×C22⋊C4—38no—
282C5×C22⋊C4C22×D314yes—
283C22⋊Dic5—34no—
284C22⋊Dic5—38no—
285#685—38no—
286#685C22×D318yes—
287C4⋊Dic5C22×D314yes—
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—31no—
297#534—51no—
298#534—152no—
299#532—154no—
300#525—154no—
301#525—154no—
302C2×#117—151no—
303C22×C3×C5—22no—
304C22×C3×C5—24no—
305C22×C3×C5—44no—
306C22×C3×C5C22×D414yes—
307C22×C3×C5C22×D412yes—
308C22×C3×C5C2511yescharacteristic
309C2×C5×D3—24no—
310C2×C5×D3—28no—
311C2×C5×D3—48no—
312C2×C3×D5—24no—
313C2×C3×D5—28no—
314C2×C3×D5—48no—
315C3×C4×C5—24no—
316Dic15—24no—
317Dic15C22×D412yes—
318Dic15C22×D414yes—
319Dic15C22×D412yes—
320C3×Dic5—24no—
321C5×Dic3—24no—
322C24×C3—52no—
323C24×C3C22×D511yescharacteristic
324C23×D3C22×D512yes—
325C2×C3×D4—54no—
326C2×C3×D4—54no—
327C2×C3×D4—54no—
328C2×C3×D4—108no—
329C22×C3×C4—52no—
330C22×C3×C4C22×D512yes—
331C2×#67—28no—
332C2×#67—54no—
333C2×#67—54no—
334C2×#67—58no—
335C2×#67—108no—
336C2×#67C22×D514yes—
337C2×#67C22×D514yes—
338C22×Dic3—58no—
339C22×Dic3—52no—
340C22×Dic3—52no—
341C22×Dic3—51no—
342C22×Dic3—51no—
343C22×Dic3—52no—
344C22×Dic3—54no—
345C22×Dic3—54no—
346C22×Dic3—108no—
347C22×Dic3C22×D512yes—
348#237—1016no—
349#237—1016no—
350#237—1016no—
351#237—1016no—
352C2×D12C22×D514yes—
353C2×C4×D3—54no—
354C2×C4×D3—54no—
355C2×C4×D3—108no—
356C2×Dic6—54no—
357C2×Dic6—54no—
358C2×Dic6—108no—
359C3×C4⋊C4—54no—
360C3×C22⋊C4—58no—
361C3×C22⋊C4C22×D514yes—
362#217—54no—
363#217—58no—
364(C2×D3)⋊C4—58no—
365(C2×D3)⋊C4C22×D518yes—
366C4⋊Dic3C22×D514yes—
367Dic3⋊C4—58no—
368Dic3⋊C4—58no—
369C4×Dic3—54no—
370C4×Dic3—54no—
371C23×C5—34no—
372C23×C5—42no—
373C23×C5—64no—
374C23×C5—124no—
375C23×C5D3×D414yes—
376C23×C5C23×D312yes—
377C23×C5C23×D311yescharacteristic
378C22×D5—24no—
379C22×D5—44no—
380C22×D5C23×D314yes—
381C5×D4—68no—
382C5×D4—68no—
383C5×D4—124no—
384C2×C4×C5—34no—
385C2×C4×C5—42no—
386C2×C4×C5—64no—
387C2×C4×C5C23×D314yes—
388C22⋊D5—28no—
389C22⋊D5—28no—
390C22⋊D5—68no—
391C22⋊D5—68no—
392C22⋊D5—128no—
393C2×Dic5—24no—
394C2×Dic5—38no—
395C2×Dic5—38no—
396C2×Dic5—34no—
397C2×Dic5—34no—
398C2×Dic5—31no—
399C2×Dic5—31no—
400C2×Dic5—32no—
401C2×Dic5—34no—
402C2×Dic5—68no—
403C2×Dic5—68no—
404C2×Dic5—122no—
405C2×Dic5C23×D314yes—
406D20—44no—
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—32no—
418C22×D4—52no—
419C22×D4—152no—
420C23×C4—152no—
421C23×C4—152no—
422C23×C4—151no—
423C4⋊Q8—154no—
424#127—154no—
425#124—158no—
426#123—34no—
427#123—54no—
428#123—154no—
429#123—154no—
430C22⋊Q8—158no—
431C22⋊Q8—158no—
432#121—34no—
433#121—54no—
434#121—158no—
435#121—158no—
436#121—158no—
437#120—34no—
438#120—54no—
439#120—154no—
440#120—154no—
441C4×D4—158no—
442C4×D4—154no—
443C4×D4—154no—
444#117—154no—
445#117—154no—
446C2×C4⋊C4—152no—
447C2×C22⋊C4—151no—
448C2×C22⋊C4D3×D511yescharacteristic
449C2×C42—152no—
450C2×C3×C5—24no—
451C2×C3×C5—22no—
452C2×C3×C5C23×D411yescharacteristic
453C2×C3×C5#58512yes—
454C3×D5—44no—
455C5×D3—44no—
456C23×C3—42no—
457C23×C3—54no—
458C23×C3—104no—
459C23×C3—204no—
460C23×C3D4×D514yes—
461C23×C3C23×D512yes—
462C23×C3C23×D511yescharacteristic
463C22×D3—24no—
464C22×D3—44no—
465C22×D3C23×D514yes—
466C3×D4—108no—
467C3×D4—108no—
468C3×D4—204no—
469C2×C3×C4—42no—
470C2×C3×C4—54no—
471C2×C3×C4—104no—
472C2×C3×C4C23×D514yes—
473#67—28no—
474#67—28no—
475#67—108no—
476#67—108no—
477#67—208no—
478C2×Dic3—24no—
479C2×Dic3—58no—
480C2×Dic3—58no—
481C2×Dic3—54no—
482C2×Dic3—54no—
483C2×Dic3—51no—
484C2×Dic3—51no—
485C2×Dic3—52no—
486C2×Dic3—54no—
487C2×Dic3—108no—
488C2×Dic3—108no—
489C2×Dic3—202no—
490C2×Dic3C23×D514yes—
491D12—44no—
492C4×D3—108no—
493C4×D3—108no—
494Dic6—108no—
495Dic6—108no—
496C22×C5—22no—
497C22×C5—24no—
498C22×C5—44no—
499C22×C5—64no—
500C22×C5—68no—
501C22×C5—128no—
502C22×C5C2×D3×D414yes—
503C22×C5C2×D3×D412yes—
504C22×C5C24×D311yescharacteristic
505C2×D5—24no—
506C2×D5—28no—
507C2×D5—48no—
508C4×C5—24no—
509C4×C5—64no—
510Dic5—24no—
511Dic5—32no—
512Dic5—34no—
513Dic5—32no—
514Dic5—64no—
515C24—32no—
516C24—52no—
517C24C2×D3×D511yescharacteristic
518#41—3016no—
519#41—3016no—
520#41—3016no—
521#41—3016no—
522C2×Q8—154no—
523C2×Q8—154no—
524C2×Q8—308no—
525C2×D4—34no—
526C2×D4—34no—
527C2×D4—34no—
528C2×D4—54no—
529C2×D4—54no—
530C2×D4—54no—
531C2×D4—68no—
532C2×D4—108no—
533C2×D4—154no—
534C2×D4—154no—
535C2×D4—158no—
536C2×D4—308no—
537C22×C4—32no—
538C22×C4—52no—
539C22×C4—154no—
540C22×C4—154no—
541C22×C4—154no—
542C22×C4—154no—
543C22×C4—158no—
544C22×C4—151no—
545C22×C4—151no—
546C22×C4—152no—
547C22×C4—152no—
548C22×C4—152no—
549C22×C4—308no—
550C22×C4—308no—
551C22×C4C2×D3×D512yes—
552C4⋊C4—34no—
553C4⋊C4—54no—
554C4⋊C4—158no—
555C4⋊C4—158no—
556C22⋊C4—38no—
557C22⋊C4—58no—
558C22⋊C4—158no—
559C22⋊C4—158no—
560C22⋊C4—154no—
561C22⋊C4C2×D3×D514yes—
562C42—154no—
563C42—154no—
564C3×C5#344911yescharacteristic, complemented, Hall
565C22×C3—22no—
566C22×C3—24no—
567C22×C3—44no—
568C22×C3—104no—
569C22×C3—108no—
570C22×C3—208no—
571C22×C3C2×D4×D514yes—
572C22×C3C2×D4×D512yes—
573C22×C3C24×D511yescharacteristic
574C2×D3—24no—
575C2×D3—28no—
576C2×D3—48no—
577C3×C4—24no—
578C3×C4—104no—
579Dic3—24no—
580Dic3—52no—
581Dic3—54no—
582Dic3—52no—
583Dic3—104no—
584C2×C5—24no—
585C2×C5—22no—
586C2×C5—124no—
587C2×C5C22×D3×D411yescharacteristic
588C2×C5#596612yes—
589D5—44no—
590C23—34no—
591C23—42no—
592C23—54no—
593C23—64no—
594C23—104no—
595C23—124no—
596C23—204no—
597C23#677714yes—
598C23C22×D3×D512yes—
599C23C22×D3×D511yescharacteristic
600Q8—308no—
601Q8—308no—
602D4—68no—
603D4—68no—
604D4—108no—
605D4—108no—
606D4—124no—
607D4—204no—
608D4—308no—
609D4—308no—
610D4—608no—
611C2×C4—34no—
612C2×C4—42no—
613C2×C4—54no—
614C2×C4—64no—
615C2×C4—104no—
616C2×C4—158no—
617C2×C4—158no—
618C2×C4—154no—
619C2×C4—154no—
620C2×C4—151no—
621C2×C4—151no—
622C2×C4—152no—
623C2×C4—154no—
624C2×C4—308no—
625C2×C4—308no—
626C2×C4—308no—
627C2×C4—308no—
628C2×C4—602no—
629C2×C4C22×D3×D514yes—
630C2×C3—24no—
631C2×C3—22no—
632C2×C3—204no—
633C2×C3C22×D4×D511yescharacteristic
634C2×C3#6628512yes—
635D3—44no—
636C5#8629411yescharacteristic, complemented, Sylow
637C22—22no—
638C22—24no—
639C22—44no—
640C22—64no—
641C22—68no—
642C22—104no—
643C22—108no—
644C22—128no—
645C22—208no—
646C22C2×#677714yes—
647C22C2×#677712yes—
648C22C23×D3×D511yescharacteristic
649C4—24no—
650C4—64no—
651C4—104no—
652C4—152no—
653C4—154no—
654C4—152no—
655C4—304no—
656C3#1061819511yescharacteristic, complemented, Sylow
657C2—24no—
658C2—22no—
659C2—124no—
660C2—204no—
661C2C22×#677711yescharacteristic
662C2#1175703612yes—
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC2×C3×C5×C22⋊C4C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#3449
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6780C2semidirect
2. C2×#91766C2semidirect
3. C5×#5493C2semidirect
4. C3×#65827C2semidirect
5. #11755426C2semidirect
6. #11754387C2semidirect
7. #11754386C2semidirect
8. #11753083C2semidirect
9. #11752883C2semidirect
10. #11752882C2semidirect
11. #11752880C2semidirect
16. C22×#1204C22semidirect
17. C22×#1201C22semidirect
18. C22×D3×Dic5C22semidirect
22. C2×#6780C22semidirect
23. C2×#6780C22semidirect
24. C22×D5×Dic3C22semidirect
27. C2×#6712C22semidirect
28. C2×C4×Dic15C22semidirect
29. #91766C22semidirect
30. #91766C22semidirect
31. C5×#930C22semidirect
32. C5×#886C22semidirect
33. C5×#882C22semidirect
34. C3×#4076C22semidirect
35. C3×#4032C22semidirect
36. C3×#4028C22semidirect
37. #91571C22semidirect
38. #91568C22semidirect
39. #91563C22semidirect
40. #91539C22semidirect
41. #91536C22semidirect
42. #91470C22semidirect
43. #91464C22semidirect
44. #91463C22semidirect
45. #91453C22semidirect
46. #91442C22semidirect
47. #91436C22semidirect
48. #91435C22semidirect
49. #91434C22semidirect
50. #91430C22semidirect
51. #91408C22semidirect
62. #65827D3semidirect
68. C22×Dic15D4semidirect
122. #5493D5semidirect
148. #4076C2×D3semidirect
155. #4032C2×D3semidirect
158. #4028C2×D3semidirect
228. #930C2×D5semidirect
235. #886C2×D5semidirect
238. #882C2×D5semidirect
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×#67802
C2×#917662
C5×#54932
C3×#658272
#117554262
#117543872
#117543862
#117530832
#117528832
#117528822
#117528802
#106181953
#862945