Finite Group Databasebeta

← #14426228

Subgroups — #14426228

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C5×#41C211yesmaximal, characteristic, complemented
2C2×#91839C212yesmaximal, complemented
3#11756236C211yesmaximal, characteristic, complemented
4#11751132C211yesmaximal, characteristic, complemented
5#11751129C212yesmaximal, complemented
6#11751128C211yesmaximal, characteristic, complemented
7#11751120C211yesmaximal, characteristic, complemented
8#11751117C212yesmaximal, complemented
9#11751116C211yesmaximal, characteristic, complemented
10#11751103C212yesmaximal, complemented
11C2×C4⋊(C3⋊#168)C211yesmaximal, characteristic, complemented
12#10614256—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C2×#6817C2212yes—
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×Q8C2211yescharacteristic
17C3×C4×C5×D4—21no—
18C3×C4×C5×D4—21no—
19C3×C4×C5×D4C2211yescharacteristic
20C3×C4×C5×D4C2211yescharacteristic
21C3×C5×#117—21no—
22C3×C5×#117C2211yescharacteristic
23C2×C3×C42×C5—21no—
24C2×C3×C42×C5C2211yescharacteristic
25#91839—28no—
26C2×#6718C2212yes—
27C2×#6717C2212yes—
28C2×#6716C2212yes—
29C2×#6715C2212yes—
30#91819—21no—
31#91819—21no—
32C22×C3⋊#168C2212yes—
33#91764C2212yes—
34#91763C2211yescharacteristic
35C4×D60—21no—
36C4×D60—21no—
37#91758C2211yescharacteristic
38#91094C2212yes—
39#91094C2212yes—
40#91093C2212yes—
41#91093C2212yes—
42C4⋊(C3⋊#168)C2211yescharacteristic
43C4⋊(C3⋊#168)C2212yes—
44C4⋊(C3⋊#168)C2211yescharacteristic
45#82525—51nomaximal, Hall
46C4×C5×#41D311yescharacteristic, complemented
47C2×#4096—32no—
48#65811—31no—
49#65320—31no—
50#65317—32no—
51#65316—31no—
52#65308—31no—
53#65305—32no—
54#65304—31no—
55#65291—32no—
56C2×C4⋊#168—31no—
57C3×C5×#41—22no—
58C3×C5×#41D414yescomplemented
59C2×C3×C5×Q8C2311yescharacteristic
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4C2311yescharacteristic
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5—21no—
65C22×C3×C4×C5D411yescharacteristic
66C22×C3×C4×C5D411yescharacteristic
67C22×C3×C4×C5C2311yescharacteristic
68C2×#1220—22no—
69#6817—24no—
70#6817—24no—
71C2×D60—21no—
72C2×D60—21no—
73C2×D60C2312yes—
74C2×C4×D15—22no—
75C2×Dic30C2312yes—
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4—21no—
78C3×C5×C4⋊C4C2311yescharacteristic
79C3×C5×C4⋊C4C2311yescharacteristic
80C3×C5×C22⋊C4—21no—
81C3×C5×C22⋊C4—21no—
82C3×C5×C22⋊C4—21no—
83C3×C42×C5—21no—
84C3×C42×C5—21no—
85C3×C42×C5—21no—
86C3×C42×C5D411yescharacteristic
87C3×C42×C5D411yescharacteristic
88C3×C42×C5C2311yescharacteristic
89C3×C42×C5C2311yescharacteristic
90#6718—24no—
91#6717—24no—
92#6716—24no—
93#6715—24no—
94#6714—22no—
95#6714—22no—
96C4⋊Dic15—21no—
97C4⋊Dic15—21no—
98C2×C3⋊#168—24no—
99C2×C3⋊#168C2312yes—
100C2×C3⋊#168C2312yes—
101C3×C4×#41D511yescharacteristic, complemented
102C2×#950—52no—
103#5477—51no—
104#5031—51no—
105#5028—52no—
106#5027—51no—
107#5019—51no—
108#5016—52no—
109#5015—51no—
110#5002—52no—
111C2×C4⋊(C3⋊C8)—51no—
112C2×C5×#41C2×D311yescharacteristic
113C2×#709—32no—
114C4×C5×Q8C2×D311yescharacteristic
115C4×C5×Q8C2×D311yescharacteristic
116C4×C5×D4—21no—
117C4×C5×D4—21no—
118C4×C5×D4C2×D311yescharacteristic
119C4×C5×D4C2×D311yescharacteristic
120C5×#117—21no—
121C5×#117C2×D311yescharacteristic
122C2×C42×C5—21no—
123C2×C42×C5C2×D311yescharacteristic
124#4096—68no—
125C2×#689—32no—
126C2×#688—32no—
127C2×#687—32no—
128C2×#686—32no—
129#4076—61no—
130#4076—61no—
131C22×#168—32no—
132#4021—32no—
133#4020—31no—
134C4×D20—61no—
135C4×D20—61no—
136#4015—31no—
137#3942—32no—
138#3942—32no—
139#3941—32no—
140#3941—32no—
141C4⋊#168—31no—
142C4⋊#168—32no—
143C4⋊#168—31no—
144#3035—151noSylow
145C23×C3×C5—21no—
146C23×C3×C5C2×D411yescharacteristic
147C22×D15—22no—
148C3×C5×Q8—21no—
149C3×C5×Q8C2×D412yes—
150C3×C5×D4—24no—
151C3×C5×D4—21no—
152C3×C5×D4C2×D412yes—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—24no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—21no—
157C2×C3×C4×C5—21no—
158C2×C3×C4×C5—41no—
159C2×C3×C4×C5—42no—
160C2×C3×C4×C5—42no—
161C2×C3×C4×C5C2×D411yescharacteristic
162C2×C3×C4×C5C2×D411yescharacteristic
163C2×C3×C4×C5C2×D411yescharacteristic
164C2×C3×C4×C5C2×D411yescharacteristic
165C2×C3×C4×C5C2×D412yes—
166C2×C3×C4×C5C2×D411yescharacteristic
167C2×C3×C4×C5#4112yes—
168C2×C3×C4×C5C2411yescharacteristic
169#1220—44no—
170C2×Dic15—22no—
171D60—22no—
172D60—22no—
173D60—42no—
174D60—42no—
175C4×D15—44no—
176Dic30—22no—
177Dic30—22no—
178C3⋊#168—22no—
179C3⋊#168—22no—
180C2×C3×#41C2×D511yescharacteristic
181C2×#235—52no—
182C3×C4×Q8C2×D511yescharacteristic
183C3×C4×Q8C2×D511yescharacteristic
184C3×C4×D4—21no—
185C3×C4×D4—21no—
186C3×C4×D4C2×D511yescharacteristic
187C3×C4×D4C2×D511yescharacteristic
188C3×#117—21no—
189C3×#117C2×D511yescharacteristic
190C2×C3×C42—21no—
191C2×C3×C42C2×D511yescharacteristic
192#950—108no—
193C2×#216—52no—
194C2×#215—52no—
195C2×#214—52no—
196C2×#213—52no—
197#930—101no—
198#930—101no—
199C22×C3⋊C8—52no—
200#875—52no—
201#874—51no—
202C4×D12—101no—
203C4×D12—101no—
204#869—51no—
205#810—52no—
206#810—52no—
207#809—52no—
208#809—52no—
209C4⋊(C3⋊C8)—51no—
210C4⋊(C3⋊C8)—52no—
211C4⋊(C3⋊C8)—51no—
212C5×#41—22no—
213C5×#41D1214yescomplemented
214C2×C5×Q8C22×D311yescharacteristic
215C2×C5×D4—21no—
216C2×C5×D4C22×D311yescharacteristic
217C22×C4×C5—21no—
218C22×C4×C5—21no—
219C22×C4×C5—21no—
220C22×C4×C5#6711yescharacteristic
221C22×C4×C5#6711yescharacteristic
222C22×C4×C5C22×D311yescharacteristic
223C2×C22⋊D5—62no—
224#709—64no—
225#709—64no—
226C2×D20—32no—
227C2×D20—61no—
228C2×D20—61no—
229C2×C4×D5—62no—
230C2×Dic10—32no—
231C5×C4⋊C4—21no—
232C5×C4⋊C4—21no—
233C5×C4⋊C4C22×D311yescharacteristic
234C5×C4⋊C4C22×D311yescharacteristic
235C5×C22⋊C4—21no—
236C5×C22⋊C4—21no—
237C5×C22⋊C4—21no—
238C42×C5—21no—
239C42×C5—21no—
240C42×C5—21no—
241C42×C5#6711yescharacteristic
242C42×C5#6711yescharacteristic
243C42×C5C22×D311yescharacteristic
244C42×C5C22×D311yescharacteristic
245#689—64no—
246#688—64no—
247#687—64no—
248#686—64no—
249#685—62no—
250#685—62no—
251C4⋊Dic5—61no—
252C4⋊Dic5—61no—
253C2×#168—32no—
254C2×#168—32no—
255C2×#168—64no—
256C2×#135—152no—
257#532—151no—
258C4×#41D1511yescharacteristic, complemented
259#464—152no—
260#463—152no—
261#462—151no—
262#461—151no—
263#460—151no—
264#459—151no—
265C2×C4⋊C8—151no—
266#416—152no—
267C22×C3×C5—21no—
268C22×C3×C5—21no—
269C22×C3×C5—22no—
270C22×C3×C5—41no—
271C22×C3×C5—41no—
272C22×C3×C5#12112yes—
273C22×C3×C5C22×D411yescharacteristic
274C2×D15—42no—
275C2×D15—42no—
276C2×D15—42no—
277C3×C4×C5—22no—
278C3×C4×C5—21no—
279C3×C4×C5—21no—
280C3×C4×C5—21no—
281C3×C4×C5—41no—
282C3×C4×C5—41no—
283C3×C4×C5#12112yes—
284C3×C4×C5#13512yescomplemented
285C3×C4×C5C22×D411yescharacteristic
286C3×C4×C5C2×#4111yescharacteristic
287Dic15—42no—
288C3×#41—22no—
289C3×#41D2014yescomplemented
290C2×C3×Q8C22×D511yescharacteristic
291C2×C3×D4—21no—
292C2×C3×D4C22×D511yescharacteristic
293C22×C3×C4—21no—
294C22×C3×C4—21no—
295C22×C3×C4—21no—
296C22×C3×C4C22⋊D511yescharacteristic
297C22×C3×C4C22⋊D511yescharacteristic
298C22×C3×C4C22×D511yescharacteristic
299C2×#67—102no—
300#235—104no—
301#235—104no—
302C2×D12—52no—
303C2×D12—101no—
304C2×D12—101no—
305C2×C4×D3—102no—
306C2×Dic6—52no—
307C3×C4⋊C4—21no—
308C3×C4⋊C4—21no—
309C3×C4⋊C4C22×D511yescharacteristic
310C3×C4⋊C4C22×D511yescharacteristic
311C3×C22⋊C4—21no—
312C3×C22⋊C4—21no—
313C3×C22⋊C4—21no—
314C3×C42—21no—
315C3×C42—21no—
316C3×C42—21no—
317C3×C42C22⋊D511yescharacteristic
318C3×C42C22⋊D511yescharacteristic
319C3×C42C22×D511yescharacteristic
320C3×C42C22×D511yescharacteristic
321#216—104no—
322#215—104no—
323#214—104no—
324#213—104no—
325(C2×D3)⋊C4—102no—
326(C2×D3)⋊C4—102no—
327C4⋊Dic3—101no—
328C4⋊Dic3—101no—
329C2×C3⋊C8—52no—
330C2×C3⋊C8—52no—
331C2×C3⋊C8—104no—
332C23×C5—21no—
333C23×C5C2×#6711yescharacteristic
334C22×D5—62no—
335C5×Q8—21no—
336C5×Q8C2×D1212yes—
337C5×D4—24no—
338C5×D4—21no—
339C5×D4C2×D1212yes—
340C2×C4×C5—21no—
341C2×C4×C5—24no—
342C2×C4×C5—22no—
343C2×C4×C5—21no—
344C2×C4×C5—21no—
345C2×C4×C5—41no—
346C2×C4×C5—42no—
347C2×C4×C5—42no—
348C2×C4×C5C2×D1212yes—
349C2×C4×C5#23512yes—
350C2×C4×C5C2×#6711yescharacteristic
351C2×C4×C5C2×#6711yescharacteristic
352C2×C4×C5C2×#6711yescharacteristic
353C2×C4×C5C2×#6711yescharacteristic
354C2×C4×C5C2×#6711yescharacteristic
355C2×C4×C5C23×D311yescharacteristic
356C22⋊D5—124no—
357C2×Dic5—62no—
358D20—62no—
359D20—62no—
360D20—122no—
361D20—122no—
362C4×D5—124no—
363Dic10—62no—
364Dic10—62no—
365#168—62no—
366#168—62no—
367C2×#41—152no—
368C2×#41C2×D1511yescharacteristic
369#135—308no—
370C2×Q16—152no—
371C2×SD16—152no—
372C2×SD16—152no—
373C2×D8—152no—
374C22×C8—152no—
375C4⋊Q8—151no—
376#127—151no—
377#124—152no—
378#121—301no—
379#121—301no—
380C4×Q8C2×D1511yescharacteristic
381C4×Q8C2×D1511yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4C2×D1511yescharacteristic
387C4×D4C2×D1511yescharacteristic
388#117—21no—
389#117C2×D1511yescharacteristic
390C2×C42—21no—
391C2×C42C2×D1511yescharacteristic
392C4⋊C8—151no—
393C4⋊C8—151no—
394C4⋊C8—152no—
395#103—152no—
396#103—152no—
397#102—152no—
398#102—152no—
399C2×C3×C5—22no—
400C2×C3×C5—42no—
401C2×C3×C5C2×#12111yescharacteristic
402C2×C3×C5C2×#13511yescharacteristic
403C2×C3×C5#57511yescharacteristic
404D15—82no—
405C23×C3—21no—
406C23×C3C2×C22⋊D511yescharacteristic
407C22×D3—102no—
408C3×Q8—21no—
409C3×Q8C2×D2012yes—
410C3×D4—24no—
411C3×D4—21no—
412C3×D4C2×D2012yes—
413C2×C3×C4—21no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—41no—
419C2×C3×C4—42no—
420C2×C3×C4—42no—
421C2×C3×C4C2×D2012yes—
422C2×C3×C4#70912yes—
423C2×C3×C4C2×C22⋊D511yescharacteristic
424C2×C3×C4C2×C22⋊D511yescharacteristic
425C2×C3×C4C2×C22⋊D511yescharacteristic
426C2×C3×C4C2×C22⋊D511yescharacteristic
427C2×C3×C4C2×C22⋊D511yescharacteristic
428C2×C3×C4C23×D511yescharacteristic
429#67—204no—
430C2×Dic3—102no—
431D12—102no—
432D12—102no—
433D12—202no—
434D12—202no—
435C4×D3—204no—
436Dic6—102no—
437Dic6—102no—
438C3⋊C8—102no—
439C3⋊C8—102no—
440C22×C5—21no—
441C22×C5—21no—
442C22×C5—22no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5#93012yes—
446C22×C5C22×#6711yescharacteristic
447C2×D5—122no—
448C2×D5—122no—
449C2×D5—122no—
450C4×C5—22no—
451C4×C5—21no—
452C4×C5—21no—
453C4×C5—21no—
454C4×C5—41no—
455C4×C5—41no—
456C4×C5#93012yes—
457C4×C5#95012yescomplemented
458C4×C5C22×D1211yescharacteristic
459C4×C5C2×#23511yescharacteristic
460Dic5—122no—
461#41—22no—
462#41—304no—
463#41—304no—
464#41D6014yescomplemented
465C2×Q8—152no—
466C2×Q8C22×D1511yescharacteristic
467C2×D4—21no—
468C2×D4—152no—
469C2×D4—302no—
470C2×D4—301no—
471C2×D4—301no—
472C2×D4C22×D1511yescharacteristic
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—302no—
477C22×C4#122011yescharacteristic
478C22×C4#122011yescharacteristic
479C22×C4C22×D1511yescharacteristic
480Q16—304no—
481SD16—304no—
482SD16—304no—
483D8—304no—
484C2×C8—152no—
485C2×C8—152no—
486C2×C8—304no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—301no—
490C4⋊C4—301no—
491C4⋊C4C22×D1511yescharacteristic
492C4⋊C4C22×D1511yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—302no—
497C22⋊C4—302no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42#122011yescharacteristic
502C42#122011yescharacteristic
503C42C22×D1511yescharacteristic
504C42C22×D1511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—22no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3#407612yes—
512C22×C3C22×C22⋊D511yescharacteristic
513C2×D3—202no—
514C2×D3—202no—
515C2×D3—202no—
516C3×C4—22no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4#407612yes—
523C3×C4#409612yescomplemented
524C3×C4C22×D2011yescharacteristic
525C3×C4C2×#70911yescharacteristic
526Dic3—202no—
527C2×C5—22no—
528C2×C5—42no—
529C2×C5C2×#93011yescharacteristic
530C2×C5C2×#95011yescharacteristic
531C2×C5#582211yescharacteristic
532D5—242no—
533C23—21no—
534C23—302no—
535C23C2×#122011yescharacteristic
536Q8—21no—
537Q8—302no—
538Q8—302no—
539Q8C2×D6012yes—
540D4—24no—
541D4—21no—
542D4—302no—
543D4—302no—
544D4—602no—
545D4—604no—
546D4—602no—
547D4C2×D6012yes—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—302no—
557C2×C4—604no—
558C2×C4C2×D6012yes—
559C2×C4#681712yes—
560C2×C4C2×#122011yescharacteristic
561C2×C4C2×#122011yescharacteristic
562C2×C4C2×#122011yescharacteristic
563C2×C4C2×#122011yescharacteristic
564C2×C4C2×#122011yescharacteristic
565C2×C4C23×D1511yescharacteristic
566C8—302no—
567C8—302no—
568C2×C3—22no—
569C2×C3—42no—
570C2×C3C2×#407611yescharacteristic
571C2×C3C2×#409611yescharacteristic
572C2×C3#6615611yescharacteristic
573D3—402no—
574C5#8252511yescharacteristic, complemented, Sylow
575C22—21no—
576C22—21no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—602no—
581C22—602no—
582C22—602no—
583C22#9181912yes—
584C22C22×#122011yescharacteristic
585C4—22no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—602no—
592C4#9181912yes—
593C4#9183912yescomplemented
594C4C22×D6011yescharacteristic
595C4C2×#681711yescharacteristic
596C3#1061425611yescharacteristic, complemented, Sylow
597C2—22no—
598C2—42no—
599C2—1202no—
600C2C2×#9181911yescharacteristic
601C2C2×#9183911yescharacteristic
602C2#1175658111yescharacteristic
subgroupG/H
centerC2×C4C2×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C4×C5×#41C2
socleC22×C3×C5C22×D4
hypercenterC4×#41D15
hypocenterC3×C5#3035
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. C3×C4×C5×#41C2semidirect
2. C2×#91839C2semidirect
3. #11756236C2semidirect
4. #11751132C2semidirect
5. #11751129C2semidirect
6. #11751128C2semidirect
7. #11751120C2semidirect
8. #11751117C2semidirect
9. #11751116C2semidirect
10. #11751103C2semidirect
11. C2×C4⋊(C3⋊#168)C2semidirect
46. C4×C5×#41D3semidirect
58. C3×C5×#41D4semidirect
101. C3×C4×#41D5semidirect
213. C5×#41D12semidirect
258. C4×#41D15semidirect
284. C3×C4×C5#135semidirect
289. C3×#41D20semidirect
457. C4×C5#950semidirect
464. #41D60semidirect
505. C3×C5#3035semidirect
523. C3×C4#4096semidirect
574. C5#82525semidirect
593. C4#91839semidirect
596. C3#10614256semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C5×#412
C2×#918392
#117562362
#117511322
#117511292
#117511282
#117511202
#117511172
#117511162
#117511032
C2×C4⋊(C3⋊#168)2
#106142563
#825255