Finite Group Databasebeta

← #14426240

Subgroups — #14426240

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
1C5×#5716C211yesmaximal, characteristic, complemented
2C2×#91493C212yesmaximal, complemented
3C4×#6765C211yesmaximal, characteristic, complemented
4#11751176C211yesmaximal, characteristic, complemented
5C3×#65361C211yesmaximal, characteristic, complemented
6#11749012C212yesmaximal, complemented
7#11748982C212yesmaximal, complemented
8#11748965C212yesmaximal, complemented
9#11747105C211yesmaximal, characteristic, complemented
10#11747059C211yesmaximal, characteristic, complemented
11D3×C4⋊#168C211yesmaximal, characteristic, complemented
12#10615027—31nomaximal, Hall
13C2×C5×#237C2212yes—
14C2×#6765C2211yescharacteristic
15C3×C5×#127C2211yescharacteristic
16C2×#6716C2212yes—
17C4×Dic30C2211yescharacteristic
18C5×#938—21no—
19C5×#938—21no—
20C5×#936C2212yes—
21C5×D4×Dic3—21no—
22C5×D4×Dic3—21no—
23C42×C5×D3C2211yescharacteristic
24C5×#869C2211yescharacteristic
25C2×C3×#686C2212yes—
26C3×C4×D20C2211yescharacteristic
27#91493—28no—
28#91441—21no—
29C4×#1201—21no—
30Dic3×D20C2211yescharacteristic
31#91393—21no—
32C4×D5×Dic3—21no—
33#91340C2211yescharacteristic
34#91338C2211yescharacteristic
35C2×#6663C2212yes—
36C2×#6655C2212yes—
37C2×D3×#168C2212yes—
38#91116C2212yes—
39C4⋊(C3⋊#168)C2211yescharacteristic
40C3×#3964C2212yes—
41C3×C4⋊#168C2211yescharacteristic
42Dic3⋊#168C2212yes—
43#90973C2212yes—
44#90972C2212yes—
45#80331—51nomaximal, Hall
46C5×#532—31no—
47C2×#4096—32no—
48C4×#709—31no—
49#65522—32no—
50#65379—31no—
51#65372—31no—
52#65364—31no—
53#65361D311yescharacteristic, complemented
54#65346—32no—
55#65343—32no—
56C2×C4⋊#168—31no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4C2312yes—
60C2×Dic30C2311yescharacteristic
61C2×C5×#67—22no—
62C22×C5×Dic3—22no—
63C5×#237—24no—
64C5×#237—24no—
65C2×C4×C5×D3D411yescharacteristic
66C2×C4×C5×D3D411yescharacteristic
67C2×C4×C5×D3C2311yescharacteristic
68C2×C5×Dic6C2312yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×#1201—21no—
72C2×D5×Dic3—21no—
73C2×D5×Dic3—21no—
74#6765—22no—
75#6765D414yescomplemented
76C3×C42×C5C2311yescharacteristic
77#6716—24no—
78C4⋊Dic15C2311yescharacteristic
79#6712—21no—
80C4×Dic15—21no—
81C2×C3⋊#168C2312yes—
82C5×#217—22no—
83C5×#217—22no—
84C5×C4⋊Dic3—21no—
85C5×C4⋊Dic3—21no—
86C4×C5×Dic3D411yescharacteristic
87C4×C5×Dic3D411yescharacteristic
88C4×C5×Dic3C2311yescharacteristic
89C3×#686—24no—
90C3×#685—21no—
91C3×C4⋊Dic5C2311yescharacteristic
92C2×C3×#168C2312yes—
93#6669—21no—
94(C2×D3)⋊Dic5—21no—
95#6665—21no—
96Dic3×Dic5—21no—
97Dic3×Dic5—21no—
98#6663—24no—
99#6655—24no—
100D3×#168—24no—
101C2×#912—52no—
102#5716D511yescharacteristic, complemented
103C4×#237—51no—
104C3×#459—51no—
105#5153—52no—
106#5075—51no—
107#5019—51no—
108#4849—51no—
109D3×C4⋊C8—51no—
110#4772—52no—
111#4767—52no—
112C2×C5×#41—32no—
113C2×#709—31no—
114C5×C4⋊Q8—31no—
115C5×#127C2×D311yescharacteristic
116C5×#124—32no—
117C5×#121—61no—
118C5×#121—61no—
119C4×C5×D4—61no—
120C4×C5×D4—61no—
121C2×C42×C5—31no—
122#4096—68no—
123C2×#689—32no—
124C2×#688—32no—
125C2×#687—32no—
126C2×#686C2×D312yes—
127C4×C22⋊D5—61no—
128C4×C22⋊D5—61no—
129#4072—31no—
130C22×#168—32no—
131C4×D20—31no—
132C4×D20C2×D311yescharacteristic
133#4018—61no—
134C42×D5—61no—
135C4×Dic10—31no—
136C4×Dic10—31no—
137#3967—32no—
138#3967—32no—
139#3964—32no—
140#3964C2×D312yes—
141C4⋊#168—31no—
142C4⋊#168—32no—
143C4⋊#168C2×D311yescharacteristic
144#3035—151noSylow
145C23×C3×C5—22no—
146C22×C5×D3C2×D411yescharacteristic
147C22×C3×D5—21no—
148C3×C5×D4—22no—
149C3×C5×D4—22no—
150C3×C5×D4—42no—
151C3×C5×D4—42no—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D411yescharacteristic
154C2×C3×C4×C5C2411yescharacteristic
155C2×Dic15—21no—
156C2×Dic15—21no—
157Dic30—21no—
158Dic30C2×D412yes—
159C5×#67—44no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—44no—
162C2×C5×Dic3C2×D411yescharacteristic
163C2×C5×Dic3C2×D411yescharacteristic
164C2×C5×Dic3C2×D411yescharacteristic
165C4×C5×D3—22no—
166C4×C5×D3—41no—
167C4×C5×D3C2×D412yes—
168C4×C5×D3#4112yes—
169C5×Dic6—22no—
170C5×Dic6—22no—
171C2×C3×Dic5—21no—
172C3×D20—21no—
173C3×D20C2×D412yes—
174C3×C4×D5—42no—
175#1201—24no—
176D5×Dic3—24no—
177D5×Dic3—42no—
178C3⋊#168—22no—
179C3×#168—22no—
180C2×#237—51no—
181C2×#237C2×D512yes—
182C2×C3×D8—52no—
183C3×#127C2×D511yescharacteristic
184C3×C4×D4—51no—
185#938—21no—
186#938—21no—
187#936C2×D512yes—
188D4×Dic3—21no—
189D4×Dic3—21no—
190D4×Dic3—51no—
191C2×#214—52no—
192C2×#214—52no—
193C4×#67—101no—
194C2×C4×Dic3—101no—
195#912—108no—
196C2×Dic12—52no—
197C2×D3×C8—52no—
198#892—51no—
199#887—51no—
200#67⋊C4—101no—
201#877—101no—
202C42×D3C2×D511yescharacteristic
203#869C2×D511yescharacteristic
204C4×Dic6—51no—
205C3×C4⋊C8—51no—
206C3×#102—52no—
207#832—52no—
208#816—52no—
209Dic3⋊C8—52no—
210#810—52no—
211C4⋊(C3⋊C8)—51no—
212C5×#41—64no—
213C5×#41—64no—
214C2×C5×Q8—32no—
215C2×C5×D4—21no—
216C2×C5×D4—21no—
217C2×C5×D4—62no—
218C2×C5×D4C22×D312yes—
219C22×C4×C5—31no—
220C22×C4×C5—31no—
221C22×C4×C5—31no—
222C22×C4×C5—62no—
223C2×C22⋊D5—61no—
224#709—34no—
225#709—62no—
226C2×D20C22×D311yescharacteristic
227C2×C4×D5—21no—
228C2×C4×D5—61no—
229C2×C4×D5—61no—
230C2×Dic10—31no—
231C5×C4⋊C4—61no—
232C5×C4⋊C4—61no—
233C5×C22⋊C4—62no—
234C5×C22⋊C4—62no—
235C42×C5—31no—
236C42×C5—31no—
237C42×C5—31no—
238C42×C5C22×D311yescharacteristic
239C22⋊Dic5—61no—
240#689—64no—
241#688—64no—
242#687—64no—
243#686—24no—
244#685—21no—
245#685—61no—
246C4⋊Dic5—31no—
247C4⋊Dic5C22×D311yescharacteristic
248#683—61no—
249#683—61no—
250C4×Dic5—61no—
251C4×Dic5—61no—
252C4×Dic5—61no—
253C2×#168—32no—
254C2×#168—64no—
255C2×#168C22×D312yes—
256C2×#135—152no—
257#532—31no—
258C4×#41—151no—
259#464—152no—
260#463—152no—
261#462—151no—
262#461—151no—
263#460—151no—
264#459—51no—
265C2×C4⋊C8—151no—
266#416—152no—
267C22×C3×C5—42no—
268C22×C3×C5—42no—
269C22×C3×C5—42no—
270C22×C3×C5C22×D411yescharacteristic
271C2×C5×D3—21no—
272C2×C5×D3—21no—
273C2×C5×D3#12112yes—
274C2×C3×D5—22no—
275C2×C3×D5—41no—
276C2×C3×D5—41no—
277C3×C4×C5—21no—
278C3×C4×C5#13512yescomplemented
279C3×C4×C5C22×D411yescharacteristic
280C3×C4×C5C2×#4111yescharacteristic
281Dic15—22no—
282Dic15—41no—
283C3×Dic5—41no—
284C5×Dic3—21no—
285C5×Dic3—21no—
286C5×Dic3—42no—
287C5×Dic3#12112yes—
288C2×C3×D4—21no—
289C2×C3×D4—21no—
290C2×C3×D4—51no—
291C2×C3×D4C22×D512yes—
292C22×C3×C4—101no—
293C2×#67—22no—
294C2×#67—101no—
295C22×Dic3—22no—
296C22×Dic3—101no—
297C22×Dic3—101no—
298#237—24no—
299#237—24no—
300#237—54no—
301#237—102no—
302C2×C4×D3C22⋊D511yescharacteristic
303C2×C4×D3C22⋊D511yescharacteristic
304C2×C4×D3C22×D511yescharacteristic
305C2×Dic6—51no—
306C2×Dic6C22×D512yes—
307C3×D8—104no—
308C2×C3×C8—52no—
309C3×C4⋊C4—51no—
310C3×C22⋊C4—101no—
311C3×C42C22×D511yescharacteristic
312#217—22no—
313#217—22no—
314#217—101no—
315#214—104no—
316#214—104no—
317(C2×D3)⋊C4—101no—
318C4⋊Dic3—21no—
319C4⋊Dic3—21no—
320C4⋊Dic3—51no—
321Dic3⋊C4—101no—
322Dic3⋊C4—101no—
323C4×Dic3—101no—
324C4×Dic3—101no—
325C4×Dic3—101no—
326C4×Dic3C22⋊D511yescharacteristic
327C4×Dic3C22⋊D511yescharacteristic
328C4×Dic3C22×D511yescharacteristic
329C2×C3⋊C8—52no—
330Dic12—104no—
331D3×C8—104no—
332C23×C5—22no—
333C23×C5—31no—
334C22×D5—21no—
335C5×Q8—62no—
336C5×Q8—62no—
337C5×D4—22no—
338C5×D4—22no—
339C5×D4—42no—
340C5×D4—42no—
341C5×D4—124no—
342C2×C4×C5—31no—
343C2×C4×C5—32no—
344C2×C4×C5—32no—
345C2×C4×C5—31no—
346C2×C4×C5—31no—
347C2×C4×C5—62no—
348C2×C4×C5—62no—
349C2×C4×C5—121no—
350C2×C4×C5—124no—
351C2×C4×C5D3×D411yescharacteristic
352C2×C4×C5D3×D411yescharacteristic
353C2×C4×C5C23×D311yescharacteristic
354C22⋊D5—64no—
355C2×Dic5—21no—
356C2×Dic5—61no—
357C2×Dic5—61no—
358D20—21no—
359D20D3×D412yes—
360C4×D5—42no—
361C4×D5—64no—
362C4×D5—122no—
363Dic10—32no—
364Dic10—61no—
365#168—22no—
366#168—62no—
367C2×#41—32no—
368C2×#41—151no—
369#135—308no—
370C2×Q16—152no—
371C2×SD16—152no—
372C2×SD16—152no—
373C2×D8—52no—
374C22×C8—152no—
375C4⋊Q8—31no—
376#127D3×D511yescharacteristic
377#124—32no—
378#121—61no—
379#121—61no—
380C4×Q8—151no—
381C4×Q8—151no—
382C4×D4—51no—
383C4×D4—61no—
384C4×D4—61no—
385C4×D4—151no—
386C4×D4—301no—
387C4×D4—301no—
388#117—151no—
389#117—301no—
390C2×C42—31no—
391C2×C42—301no—
392C4⋊C8—51no—
393C4⋊C8—152no—
394C4⋊C8—151no—
395#103—152no—
396#103—152no—
397#102—52no—
398#102—152no—
399C2×C3×C5—82no—
400C2×C3×C5C2×#12111yescharacteristic
401C2×C3×C5C2×#13511yescharacteristic
402C2×C3×C5#57511yescharacteristic
403C3×D5—42no—
404C5×D3—22no—
405C23×C3—22no—
406C23×C3—101no—
407C22×D3C2×C22⋊D511yescharacteristic
408C3×D4—22no—
409C3×D4—22no—
410C3×D4—42no—
411C3×D4—42no—
412C3×D4—52no—
413C3×D4—101no—
414C2×C3×C4—101no—
415C2×C3×C4—202no—
416C2×C3×C4C2×C22⋊D511yescharacteristic
417C2×C3×C4C2×C22⋊D511yescharacteristic
418C2×C3×C4C23×D511yescharacteristic
419#67—44no—
420#67—104no—
421C2×Dic3—22no—
422C2×Dic3—44no—
423C2×Dic3—101no—
424C2×Dic3—104no—
425C2×Dic3—101no—
426C2×Dic3—202no—
427C2×Dic3C2×C22⋊D511yescharacteristic
428C2×Dic3C2×C22⋊D511yescharacteristic
429C2×Dic3C2×C22⋊D511yescharacteristic
430C4×D3—22no—
431C4×D3—41no—
432C4×D3D4×D512yes—
433C4×D3#71112yes—
434Dic6—22no—
435Dic6—22no—
436Dic6—52no—
437Dic6—101no—
438C3×C8—102no—
439C3⋊C8—102no—
440C22×C5—32no—
441C22×C5—42no—
442C22×C5—42no—
443C22×C5—42no—
444C22×C5—61no—
445C22×C5—61no—
446C22×C5C2×D3×D411yescharacteristic
447C2×D5—22no—
448C2×D5—41no—
449C2×D5—41no—
450C4×C5—21no—
451C4×C5—32no—
452C4×C5—61no—
453C4×C5—61no—
454C4×C5—122no—
455C4×C5#91212yescomplemented
456C4×C5C2×D3×D411yescharacteristic
457C4×C5D3×#4111yescharacteristic
458Dic5—41no—
459Dic5—62no—
460Dic5—121no—
461#41—64no—
462#41—64no—
463#41—154no—
464#41—302no—
465C2×Q8—32no—
466C2×Q8—151no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—51no—
470C2×D4—62no—
471C2×D4—301no—
472C2×D4C2×D3×D512yes—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—62no—
477C22×C4—101no—
478C22×C4—301no—
479C22×C4—301no—
480Q16—304no—
481SD16—304no—
482SD16—304no—
483D8—104no—
484C2×C8—52no—
485C2×C8—152no—
486C2×C8—304no—
487C4⋊C4—51no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—151no—
491C4⋊C4—301no—
492C4⋊C4—301no—
493C22⋊C4—62no—
494C22⋊C4—62no—
495C22⋊C4—101no—
496C22⋊C4—301no—
497C22⋊C4—301no—
498C42—31no—
499C42—31no—
500C42—31no—
501C42—301no—
502C42—301no—
503C42—301no—
504C42C2×D3×D511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—102no—
510C22×C3—201no—
511C22×C3—201no—
512C22×C3C22×C22⋊D511yescharacteristic
513C2×D3—21no—
514C2×D3—21no—
515C2×D3#408412yes—
516C3×C4—21no—
517C3×C4—201no—
518C3×C4#409612yescomplemented
519C3×C4C2×D4×D511yescharacteristic
520C3×C4C2×#71111yescharacteristic
521Dic3—21no—
522Dic3—21no—
523Dic3—42no—
524Dic3—102no—
525Dic3—201no—
526Dic3#408412yes—
527C2×C5—62no—
528C2×C5—82no—
529C2×C5D3×#12111yescharacteristic
530C2×C5C2×#91211yescharacteristic
531C2×C5#577311yescharacteristic
532D5—42no—
533C23—22no—
534C23—31no—
535C23—101no—
536Q8—62no—
537Q8—62no—
538Q8—152no—
539Q8—301no—
540D4—22no—
541D4—22no—
542D4—42no—
543D4—42no—
544D4—52no—
545D4—101no—
546D4—124no—
547D4—304no—
548C2×C4—31no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—101no—
556C2×C4—121no—
557C2×C4—124no—
558C2×C4—202no—
559C2×C4—304no—
560C2×C4—301no—
561C2×C4—301no—
562C2×C4—602no—
563C2×C4D3×C22⋊D511yescharacteristic
564C2×C4D3×C22⋊D511yescharacteristic
565C2×C4C22×D3×D511yescharacteristic
566C8—102no—
567C8—302no—
568C2×C3—82no—
569C2×C3—202no—
570C2×C3C2×#408411yescharacteristic
571C2×C3C2×#409611yescharacteristic
572C2×C3#6615611yescharacteristic
573D3—22no—
574C5#8033111yescharacteristic, complemented, Sylow
575C22—32no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—61no—
580C22—61no—
581C22—102no—
582C22—201no—
583C22—201no—
584C22C2×D3×C22⋊D511yescharacteristic
585C4—21no—
586C4—32no—
587C4—61no—
588C4—61no—
589C4—122no—
590C4—201no—
591C4—302no—
592C4—601no—
593C4#9149312yescomplemented
594C4D3×D4×D511yescharacteristic
595C4D3×#71111yescharacteristic
596C3#1061502711yescharacteristic, complemented, Sylow
597C2—62no—
598C2—82no—
599C2—202no—
600C2#1175469611yescharacteristic
601C2C2×#9149311yescharacteristic
602C2D3×#408411yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#127C22
socleC22×C3×C5C22×D4
hypercenter#127D3×D5
hypocenterC3×C5#3035
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C5×#5716C2semidirect
2. C2×#91493C2semidirect
3. C4×#6765C2semidirect
4. #11751176C2semidirect
5. C3×#65361C2semidirect
6. #11749012C2semidirect
7. #11748982C2semidirect
8. #11748965C2semidirect
9. #11747105C2semidirect
10. #11747059C2semidirect
11. D3×C4⋊#168C2semidirect
53. #65361D3semidirect
75. #6765D4semidirect
102. #5716D5semidirect
278. C3×C4×C5#135semidirect
455. C4×C5#912semidirect
505. C3×C5#3035semidirect
518. C3×C4#4096semidirect
574. C5#80331semidirect
593. C4#91493semidirect
596. C3#10615027semidirect

Maximal subgroups

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

subgroupindex
C5×#57162
C2×#914932
C4×#67652
#117511762
C3×#653612
#117490122
#117489822
#117489652
#117471052
#117470592
D3×C4⋊#1682
#106150273
#803315