Finite Group Databasebeta

← #14424140

Subgroups — #14424140

Order 1920.

1508 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6765C211yesmaximal, characteristic
2D3×#4070C211yesmaximal, characteristic, complemented
3C2×#91274C212yesmaximal, complemented
4#11750840C211yesmaximal, characteristic, complemented
5C5×#4726C211yesmaximal, characteristic, complemented
6C3×#65020C211yesmaximal, characteristic, complemented
7#11748564C211yesmaximal, characteristic, complemented
8#11748530C211yesmaximal, characteristic, complemented
9#11748162C212yesmaximal, complemented
10#11748147C212yesmaximal, complemented
11#11748138C212yesmaximal, complemented
12#10610371—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C22×C4×C5×D3C2211yescharacteristic
15C22×C3×D20C2211yescharacteristic
16C22×#1201—21no—
17C22×D5×Dic3—21no—
18C2×D3×Dic10C2211yescharacteristic, complemented
19C2×#6765—21no—
20C2×#6765—21no—
21C2×#6765—28no—
22C2×#6765C2212yes—
23C2×#6765C2212yes—
24#91813C2211yescharacteristic
25C2×Dic60C2212yescomplemented
26C2×C5×ΓL(1,𝔽25)C2212yescomplemented
27C3×#4070C2211yescharacteristic
28C2×C3×#677C2212yescomplemented
29#91575—21no—
30D3×C22⋊Dic5—21no—
31D3×C4⋊Dic5C2211yescharacteristic, complemented
32D3×#683—21no—
33#91369C2211yescharacteristic, complemented
34#91365C2211yescharacteristic, complemented
35#91352—21no—
36C2×#6662C2212yescomplemented
37C2×#6657C2212yescomplemented
38#91274—28no—
39C3×C5×C22⋊C8C2211yescharacteristic
40#91100C2212yescomplemented
41C5×#830C2211yescharacteristic
42C5×(C2×D3)⋊C8C2212yescomplemented
43C3×#3953C2212yescomplemented
44#90974C2212yescomplemented
45#90965C2212yescomplemented
46#80147—51nomaximal, Hall
47C22×#709—31no—
48C2×#4070—31no—
49C2×#4055—32no—
50C5×#407—31no—
51#65422—32no—
52#65028—31no—
53#65028—31no—
54#65022—32no—
55#65022—32no—
56#65020—31no—
57#65020D311yescharacteristic, complemented
58C23×C5×D3D411yescharacteristic
59C23×C3×D5—21no—
60C22×C3×C4×C5C2311yescharacteristic
61C22×Dic15—21no—
62C2×Dic30—21no—
63C2×Dic30—24no—
64C2×Dic30C2311yescharacteristic
65C2×Dic30C2312yes—
66C22×C5×Dic3D411yescharacteristic
67C2×C4×C5×D3—24no—
68C2×C4×C5×D3D411yescharacteristic
69C2×C4×C5×D3D411yescharacteristic
70C2×C4×C5×D3C2311yescharacteristic
71C2×C4×C5×D3C2311yescharacteristic
72C2×C3×D20—21no—
73C2×C3×D20—24no—
74C2×C3×D20C2312yes—
75C2×C3×Dic10C2311yescharacteristic
76C2×#1204—21no—
77C2×#1201—28no—
78C2×#1201—22no—
79C2×#1201—22no—
80C2×D3×Dic5—21no—
81C2×D3×Dic5—21no—
82C2×D5×Dic3—28no—
83C2×D5×Dic3—22no—
84C2×D5×Dic3—22no—
85D3×Dic10—24no—
86#6765—28no—
87#6765—28no—
88#6765—24no—
89#6765—24no—
90#6765—24no—
91#6765D418yes—
92C2×C3×C5×C8C2312yes—
93#6719—21no—
94C4⋊Dic15C2311yescharacteristic
95#6712—21no—
96Dic60—24no—
97C2×C5×C3⋊C8C2312yes—
98C5×ΓL(1,𝔽25)—24no—
99C3×C22⋊Dic5—21no—
100C3×C4⋊Dic5C2311yescharacteristic
101C3×#683—21no—
102C3×#677—24no—
103Dic3⋊Dic5—21no—
104#6669—21no—
105#6668—21no—
106(C2×D3)⋊Dic5—21no—
107(C2×D3)⋊Dic5—21no—
108#6662—24no—
109#6657—24no—
110C22×#237—51no—
111C2×#915—52no—
112D3×C22⋊Q8—51no—
113C3×#450—51no—
114#5218—51no—
115#5173—52no—
116#5050—51no—
117#4783—52no—
118#4771—52no—
119#4739—51no—
120#4726D511yescharacteristic, complemented
121C23×C4×C5—31no—
122C22×C22⋊D5—61no—
123C2×#709—32no—
124C2×#709—32no—
125C2×#709—68no—
126C2×#709—61no—
127C2×#709—61no—
128C22×D20C2×D311yescharacteristic
129C22×C4×D5—61no—
130C22×Dic10—31no—
131C22×Dic10—31no—
132C2×C5×M16—32no—
133C2×C22⋊Dic5—61no—
134C2×C4⋊Dic5—31no—
135#4070—31no—
136#4070—31no—
137#4070—31no—
138#4070—61no—
139#4070—61no—
140#4070C2×D311yescharacteristic
141C2×#683—61no—
142#4055—68no—
143C2×Dic20—32no—
144C2×Dic20—32no—
145C2×#677—32no—
146C2×#677C2×D312yescomplemented
147C5×C22⋊C8—31no—
148C5×C22⋊C8—32no—
149C5×C22⋊C8C2×D311yescharacteristic
150#3953—32no—
151#3953C2×D312yescomplemented
152#3948—32no—
153#3948—32no—
154#3007—151noSylow
155C23×C3×C5C2×D411yescharacteristic
156C22×C5×D3—22no—
157C22×C5×D3—22no—
158C22×C5×D3—41no—
159C22×C5×D3C2×D411yescharacteristic
160C22×C5×D3C2×D411yescharacteristic
161C22×C3×D5—24no—
162C22×C3×D5—22no—
163C22×C3×D5—42no—
164C22×C3×D5—42no—
165C2×C3×C4×C5—22no—
166C2×C3×C4×C5C2×D411yescharacteristic
167C2×C3×C4×C5C2411yescharacteristic
168C2×Dic15—24no—
169C2×Dic15—22no—
170C2×Dic15—21no—
171C2×Dic15—21no—
172Dic30—24no—
173Dic30—22no—
174Dic30—22no—
175Dic30C2×D414yes—
176C2×C5×Dic3—22no—
177C2×C5×Dic3C2×D411yescharacteristic
178C2×C5×Dic3C2×D411yescharacteristic
179C4×C5×D3—22no—
180C4×C5×D3—22no—
181C4×C5×D3—22no—
182C4×C5×D3C2×D414yes—
183C2×C3×Dic5—21no—
184C2×C3×Dic5—21no—
185C3×D20—24no—
186C3×D20—22no—
187C3×D20C2×D414yes—
188C3×Dic10—22no—
189#1204—42no—
190#1201—28no—
191#1201—28no—
192D3×Dic5—42no—
193D3×Dic5—42no—
194D5×Dic3—28no—
195D5×Dic3—28no—
196C3×C5×C8—22no—
197C5×C3⋊C8—22no—
198C22×C3×D4—51no—
199C22×#67—101no—
200C23×Dic3—101no—
201C2×D3×Q8—51no—
202C2×#237—52no—
203C2×#237—52no—
204C2×#237—108no—
205C2×#237—101no—
206C2×#237—101no—
207C22×C4×D3C2×D511yescharacteristic
208C22×Dic6—51no—
209C2×C3×SD16—52no—
210C3×C22⋊Q8—51no—
211#946—51no—
212C2×#216—52no—
213C2×#214—52no—
214#924—51no—
215#915—108no—
216C2×Dic12—52no—
217C2×ΓL(1,𝔽25)C2×D512yescomplemented
218#897—51no—
219#896—101no—
220D3×C4⋊C4—51no—
221D3×C4⋊C4—101no—
222D3×C22⋊C4—101no—
223#878—101no—
224C3×#102—52no—
225C3×C22⋊C8C2×D511yescharacteristic
226#832—52no—
227#830C2×D511yescharacteristic
228(C2×D3)⋊C8C2×D512yescomplemented
229#816—52no—
230#810—52no—
231C24×C5—31no—
232C23×D5—21no—
233C22×C4×C5—31no—
234C22×C4×C5—31no—
235C22×C4×C5—31no—
236C22×C4×C5—31no—
237C22×C4×C5—31no—
238C22×C4×C5—64no—
239C22×C4×C5C22×D311yescharacteristic
240C2×C22⋊D5—62no—
241C2×C22⋊D5—62no—
242C2×C22⋊D5—68no—
243C22×Dic5—61no—
244C22×Dic5—61no—
245C22×Dic5—61no—
246#709—38no—
247#709—64no—
248#709—64no—
249#709—68no—
250#709—68no—
251#709—64no—
252C2×D20—21no—
253C2×D20—24no—
254C2×D20C22×D312yes—
255C2×C4×D5—62no—
256C2×C4×D5—62no—
257C2×C4×D5—68no—
258C2×Dic10—32no—
259C2×Dic10—31no—
260C2×Dic10—64no—
261C2×Dic10—64no—
262C2×Dic10—61no—
263C2×Dic10—61no—
264C2×Dic10C22×D311yescharacteristic
265C5×M16—64no—
266C2×C5×C8—32no—
267C2×C5×C8C22×D312yes—
268C22⋊Dic5—21no—
269C22⋊Dic5—61no—
270C22⋊Dic5—61no—
271C22⋊Dic5—61no—
272C4⋊Dic5—31no—
273C4⋊Dic5—61no—
274C4⋊Dic5C22×D311yescharacteristic
275#683—21no—
276#683—61no—
277#683—61no—
278#683—61no—
279Dic20—64no—
280Dic20—64no—
281#677—24no—
282#677—64no—
283C22×#41—151no—
284C2×#137—152no—
285C2×C22⋊Q8—151no—
286#452—152no—
287#452—152no—
288#451—151no—
289#451—151no—
290#450—51no—
291#450—151no—
292#417—152no—
293#407—31no—
294C22×C3×C5—21no—
295C22×C3×C5—21no—
296C22×C3×C5#13712yescomplemented
297C22×C3×C5C22×D411yescharacteristic
298C2×C5×D3—21no—
299C2×C5×D3—21no—
300C2×C5×D3—21no—
301C2×C5×D3—21no—
302C2×C5×D3—21no—
303C2×C5×D3—44no—
304C2×C5×D3#12012yes—
305C2×C3×D5—24no—
306C2×C3×D5—48no—
307C2×C3×D5—42no—
308C2×C3×D5—42no—
309C3×C4×C5—21no—
310C3×C4×C5C22×D412yes—
311Dic15—24no—
312Dic15—41no—
313Dic15—41no—
314C3×Dic5—41no—
315C3×Dic5—41no—
316C5×Dic3—21no—
317C5×Dic3#12012yes—
318C24×C3—101no—
319C23×D3D2011yescharacteristic
320C2×C3×Q8—51no—
321C2×C3×D4—52no—
322C2×C3×D4—104no—
323C2×C3×D4—101no—
324C22×C3×C4C22×D511yescharacteristic
325C2×#67—108no—
326C2×#67—102no—
327C2×#67—102no—
328C22×Dic3—108no—
329C22×Dic3—101no—
330C22×Dic3—102no—
331C22×Dic3—102no—
332C22×Dic3D2011yescharacteristic
333D3×Q8—104no—
334#237—58no—
335#237—104no—
336#237—104no—
337#237—108no—
338#237—108no—
339#237—104no—
340C2×C4×D3—24no—
341C2×C4×D3—101no—
342C2×C4×D3—101no—
343C2×C4×D3D2011yescharacteristic
344C2×C4×D3D2011yescharacteristic
345C2×C4×D3C22×D511yescharacteristic
346C2×C4×D3C22×D511yescharacteristic
347C2×Dic6—52no—
348C2×Dic6—51no—
349C2×Dic6—104no—
350C2×Dic6—101no—
351C2×Dic6—101no—
352C3×SD16—104no—
353C2×C3×C8C22×D512yes—
354C3×C4⋊C4—51no—
355C3×C4⋊C4—101no—
356C3×C22⋊C4—101no—
357#217—101no—
358#216—104no—
359#214—104no—
360(C2×D3)⋊C4—101no—
361(C2×D3)⋊C4—101no—
362C4⋊Dic3—51no—
363C4⋊Dic3—101no—
364Dic3⋊C4—101no—
365Dic3⋊C4—101no—
366Dic3⋊C4—101no—
367C2×C3⋊C8C22×D512yes—
368Dic12—104no—
369ΓL(1,𝔽25)—24no—
370C23×C5—31no—
371C23×C5—31no—
372C23×C5—62no—
373C23×C5—62no—
374C23×C5—121no—
375C23×C5D3×D411yescharacteristic
376C22×D5—24no—
377C22×D5—22no—
378C22×D5—42no—
379C22×D5—42no—
380C2×C4×C5—22no—
381C2×C4×C5—31no—
382C2×C4×C5—34no—
383C2×C4×C5—31no—
384C2×C4×C5—62no—
385C2×C4×C5—62no—
386C2×C4×C5—62no—
387C2×C4×C5—62no—
388C2×C4×C5D3×D411yescharacteristic
389C2×C4×C5C23×D311yescharacteristic
390C22⋊D5—68no—
391C22⋊D5—68no—
392C2×Dic5—21no—
393C2×Dic5—21no—
394C2×Dic5—64no—
395C2×Dic5—62no—
396C2×Dic5—61no—
397C2×Dic5—61no—
398C2×Dic5—122no—
399C2×Dic5—122no—
400D20—24no—
401D20—22no—
402D20D3×D414yes—
403C4×D5—68no—
404C4×D5—68no—
405Dic10—22no—
406Dic10—34no—
407Dic10—64no—
408Dic10—62no—
409Dic10—62no—
410Dic10—122no—
411C5×C8—22no—
412C5×C8—62no—
413C2×#41—152no—
414C2×#41—152no—
415C2×#41—301no—
416C2×#41—301no—
417C2×#41—308no—
418C22×Q8—151no—
419C22×Q8—151no—
420C22×D4—51no—
421C22×D4—301no—
422C23×C4—31no—
423C23×C4—301no—
424#137—308no—
425C2×Q16—152no—
426C2×Q16—152no—
427C2×SD16—52no—
428C2×SD16—152no—
429C2×M16—32no—
430C22⋊Q8—51no—
431C22⋊Q8—151no—
432C22⋊Q8—151no—
433C22⋊Q8—151no—
434C22⋊Q8—301no—
435C22⋊Q8—301no—
436C2×C4⋊C4—151no—
437C2×C4⋊C4—301no—
438C2×C22⋊C4—301no—
439#103—152no—
440#103—152no—
441#102—52no—
442#102—152no—
443C22⋊C8—31no—
444C22⋊C8—32no—
445C22⋊C8D3×D511yescharacteristic
446C2×C3×C5—22no—
447C2×C3×C5C2×#12011yescharacteristic
448C2×C3×C5C2×#13711yescharacteristic
449C2×C3×C5#57511yescharacteristic
450C3×D5—44no—
451C5×D3—22no—
452C5×D3—41no—
453C23×C3—104no—
454C23×C3—102no—
455C23×C3—202no—
456C23×C3—202no—
457C23×C3C2×D2011yescharacteristic
458C22×D3—22no—
459C22×D3—22no—
460C22×D3—41no—
461C22×D3C2×D2011yescharacteristic
462C22×D3C2×D2011yescharacteristic
463C3×Q8—102no—
464C3×D4—54no—
465C3×D4—104no—
466C3×D4—102no—
467C2×C3×C4—22no—
468C2×C3×C4—101no—
469C2×C3×C4—101no—
470C2×C3×C4C2×D2011yescharacteristic
471C2×C3×C4C23×D511yescharacteristic
472#67—108no—
473#67—108no—
474C2×Dic3—22no—
475C2×Dic3—108no—
476C2×Dic3—104no—
477C2×Dic3—102no—
478C2×Dic3—108no—
479C2×Dic3—101no—
480C2×Dic3—101no—
481C2×Dic3C2×D2011yescharacteristic
482C2×Dic3C2×D2011yescharacteristic
483C4×D3—22no—
484C4×D3—22no—
485C4×D3—22no—
486C4×D3—202no—
487C4×D3—202no—
488C4×D3D4×D514yes—
489Dic6—54no—
490Dic6—104no—
491Dic6—102no—
492Dic6—102no—
493Dic6—202no—
494C3×C8—22no—
495C3⋊C8—22no—
496C22×C5—21no—
497C22×C5—21no—
498C22×C5—32no—
499C22×C5—61no—
500C22×C5—61no—
501C22×C5—61no—
502C22×C5—61no—
503C22×C5—61no—
504C22×C5—124no—
505C22×C5#91512yescomplemented
506C22×C5C2×D3×D411yescharacteristic
507C2×D5—24no—
508C2×D5—48no—
509C2×D5—42no—
510C2×D5—42no—
511C4×C5—21no—
512C4×C5—32no—
513C4×C5—61no—
514C4×C5C2×D3×D412yes—
515Dic5—41no—
516Dic5—41no—
517Dic5—64no—
518Dic5—121no—
519Dic5—121no—
520C24—31no—
521C24—101no—
522#41—158no—
523#41—304no—
524#41—304no—
525#41—304no—
526#41—308no—
527#41—308no—
528C2×Q8—51no—
529C2×Q8—152no—
530C2×Q8—151no—
531C2×Q8—304no—
532C2×Q8—301no—
533C2×Q8—304no—
534C2×Q8—301no—
535C2×D4—52no—
536C2×D4—104no—
537C2×D4—101no—
538C2×D4—302no—
539C2×D4—302no—
540C2×D4—308no—
541C22×C4—31no—
542C22×C4—31no—
543C22×C4—31no—
544C22×C4—31no—
545C22×C4—31no—
546C22×C4—64no—
547C22×C4—302no—
548C22×C4—302no—
549C22×C4—308no—
550C22×C4—301no—
551C22×C4—301no—
552C22×C4—301no—
553C22×C4C2×D3×D511yescharacteristic
554Q16—304no—
555Q16—304no—
556SD16—104no—
557SD16—304no—
558M16—64no—
559C2×C8—32no—
560C2×C8C2×D3×D512yes—
561C4⋊C4—51no—
562C4⋊C4—101no—
563C4⋊C4—151no—
564C4⋊C4—301no—
565C4⋊C4—301no—
566C4⋊C4—301no—
567C4⋊C4—301no—
568C22⋊C4—101no—
569C22⋊C4—301no—
570C22⋊C4—301no—
571C22⋊C4—301no—
572C3×C5#300711yescharacteristic, complemented, Hall
573C22×C3—21no—
574C22×C3—21no—
575C22×C3—104no—
576C22×C3—208no—
577C22×C3—202no—
578C22×C3—202no—
579C22×C3#405512yescomplemented
580C22×C3C22×D2011yescharacteristic
581C2×D3—21no—
582C2×D3—21no—
583C2×D3—21no—
584C2×D3—21no—
585C2×D3—21no—
586C2×D3—44no—
587C2×D3#402812yes—
588C3×C4—21no—
589C3×C4—201no—
590C3×C4—201no—
591C3×C4C2×D4×D512yes—
592Dic3—21no—
593Dic3—104no—
594Dic3—201no—
595Dic3—201no—
596Dic3#402812yes—
597C2×C5—22no—
598C2×C5—62no—
599C2×C5—121no—
600C2×C5D3×#12011yescharacteristic
601C2×C5C2×#91511yescharacteristic
602C2×C5#577311yescharacteristic
603D5—44no—
604C23—31no—
605C23—31no—
606C23—62no—
607C23—62no—
608C23—104no—
609C23—102no—
610C23—121no—
611C23—202no—
612C23—202no—
613C23D3×D2011yescharacteristic
614Q8—102no—
615Q8—154no—
616Q8—304no—
617Q8—302no—
618Q8—302no—
619Q8—602no—
620D4—54no—
621D4—104no—
622D4—102no—
623D4—308no—
624D4—308no—
625C2×C4—22no—
626C2×C4—31no—
627C2×C4—34no—
628C2×C4—31no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—101no—
634C2×C4—101no—
635C2×C4—308no—
636C2×C4—308no—
637C2×C4—304no—
638C2×C4—302no—
639C2×C4—301no—
640C2×C4—301no—
641C2×C4—602no—
642C2×C4—602no—
643C2×C4D3×D2011yescharacteristic
644C2×C4C22×D3×D511yescharacteristic
645C8—22no—
646C8—62no—
647C2×C3—22no—
648C2×C3—204no—
649C2×C3C2×#402811yescharacteristic
650C2×C3C2×#405511yescharacteristic
651C2×C3#6608211yescharacteristic
652D3—22no—
653D3—41no—
654C5#8014711yescharacteristic, complemented, Sylow
655C22—21no—
656C22—21no—
657C22—32no—
658C22—61no—
659C22—61no—
660C22—61no—
661C22—61no—
662C22—61no—
663C22—104no—
664C22—124no—
665C22—208no—
666C22—202no—
667C22—202no—
668C22#9127412yescomplemented
669C22C2×D3×D2011yescharacteristic
670C4—21no—
671C4—32no—
672C4—61no—
673C4—201no—
674C4—201no—
675C4—304no—
676C4—601no—
677C4—601no—
678C4D3×D4×D512yes—
679C3#1061037111yescharacteristic, complemented, Sylow
680C2—22no—
681C2—62no—
682C2—121no—
683C2—204no—
684C2D3×#402811yescharacteristic
685C2C2×#9127411yescharacteristic
686C2#1175402911yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×C22⋊C8C22
socleC22×C3×C5C22×D4
prime radicalC22×#6765C2
hypercenterC22⋊C8D3×D5
hypocenterC3×C5#3007
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. D3×#4070C2semidirect
3. C2×#91274C2semidirect
4. #11750840C2semidirect
5. C5×#4726C2semidirect
6. C3×#65020C2semidirect
7. #11748564C2semidirect
8. #11748530C2semidirect
9. #11748162C2semidirect
10. #11748147C2semidirect
11. #11748138C2semidirect
18. C2×D3×Dic10C22semidirect
25. C2×Dic60C22semidirect
26. C2×C5×ΓL(1,𝔽25)C22semidirect
28. C2×C3×#677C22semidirect
31. D3×C4⋊Dic5C22semidirect
33. #91369C22semidirect
34. #91365C22semidirect
36. C2×#6662C22semidirect
37. C2×#6657C22semidirect
40. #91100C22semidirect
42. C5×(C2×D3)⋊C8C22semidirect
43. C3×#3953C22semidirect
44. #90974C22semidirect
45. #90965C22semidirect
57. #65020D3semidirect
120. #4726D5semidirect
146. C2×#677C2×D3semidirect
151. #3953C2×D3semidirect
217. C2×ΓL(1,𝔽25)C2×D5semidirect
228. (C2×D3)⋊C8C2×D5semidirect
296. C22×C3×C5#137semidirect
505. C22×C5#915semidirect
572. C3×C5#3007semidirect
579. C22×C3#4055semidirect
654. C5#80147semidirect
668. C22#91274semidirect
679. C3#10610371semidirect

Maximal subgroups

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

subgroupindex
C22×#67652
D3×#40702
C2×#912742
#117508402
C5×#47262
C3×#650202
#117485642
#117485302
#117481622
#117481472
#117481382
#106103713
#801475