Finite Group Databasebeta

← #14426207

Subgroups — #14426207

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
1C2×#91801C212yesmaximal, complemented
2#11756475C211yesmaximal, characteristic, complemented
3C4×#6819C211yesmaximal, characteristic, complemented
4C3×C5×#459C211yesmaximal, characteristic, complemented
5#11751254C212yesmaximal, complemented
6#11751176C211yesmaximal, characteristic, complemented
7#11751120C211yesmaximal, characteristic, complemented
8#11750950C211yesmaximal, characteristic, complemented
9D15×C4⋊C8C211yesmaximal, characteristic, complemented
10#11750873C212yesmaximal, complemented
11#11750868C212yesmaximal, complemented
12#10612062—31nomaximal, Hall
13C2×#6819C2212yes—
14C2×#6819C2211yescharacteristic
15C2×C3×C5×D8C2212yes—
16C3×C5×#127C2211yescharacteristic
17C3×C4×C5×D4C2211yescharacteristic
18#91827—21no—
19#91827—21no—
20#91825C2212yes—
21D4×Dic15—21no—
22D4×Dic15—21no—
23D4×Dic15C2211yescharacteristic
24C2×#6716C2212yes—
25C2×#6716C2212yes—
26C4×#1220—21no—
27C2×C4×Dic15—21no—
28#91801—28no—
29C2×Dic60C2212yes—
30C2×C8×D15C2212yes—
31#91781C2211yescharacteristic
32#91776C2211yescharacteristic
33#91770—21no—
34#91766—21no—
35C42×D15C2211yescharacteristic
36#91758C2211yescharacteristic
37C4×Dic30C2211yescharacteristic
38C3×C5×C4⋊C8C2211yescharacteristic
39C3×C5×#102C2212yes—
40#91116C2212yes—
41#91100C2212yes—
42#91098C2212yes—
43#91094C2212yes—
44C4⋊(C3⋊#168)C2211yescharacteristic
45#80331—51nomaximal, Hall
46C2×#4058—32no—
47#66050—31no—
48C4×#711—31no—
49C5×#459D311yescharacteristic, complemented
50#65442—32no—
51#65364—31no—
52#65308—31no—
53#65138—31no—
54D5×C4⋊C8—31no—
55#65061—32no—
56#65056—32no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4C2312yes—
60C2×C3×C5×D4C2311yescharacteristic
61C22×C3×C4×C5—21no—
62C2×#1220—21no—
63C2×#1220—22no—
64C22×Dic15—21no—
65C22×Dic15—21no—
66C22×Dic15—22no—
67#6819—24no—
68#6819—22no—
69#6819—24no—
70#6819D414yescomplemented
71C2×C4×D15D411yescharacteristic
72C2×C4×D15D411yescharacteristic
73C2×C4×D15C2311yescharacteristic
74C2×Dic30C2312yes—
75C2×Dic30C2311yescharacteristic
76C3×C5×D8—24no—
77C2×C3×C5×C8C2312yes—
78C3×C5×C4⋊C4C2311yescharacteristic
79C3×C5×C22⋊C4—21no—
80C3×C42×C5C2311yescharacteristic
81#6719—21no—
82#6719—22no—
83#6719—22no—
84#6716—24no—
85#6716—24no—
86#6714—21no—
87C4⋊Dic15—21no—
88C4⋊Dic15—21no—
89C4⋊Dic15C2311yescharacteristic
90#6712—21no—
91#6712—21no—
92C4×Dic15—21no—
93C4×Dic15—21no—
94C4×Dic15—21no—
95C4×Dic15D411yescharacteristic
96C4×Dic15D411yescharacteristic
97C4×Dic15C2311yescharacteristic
98C2×C3⋊#168C2312yes—
99Dic60—24no—
100C8×D15—24no—
101C2×#912—52no—
102#5716—51no—
103C4×#237—51no—
104C3×#459D511yescharacteristic, complemented
105#5153—52no—
106#5075—51no—
107#5019—51no—
108#4849—51no—
109D3×C4⋊C8—51no—
110#4772—52no—
111#4767—52no—
112C2×#711—31no—
113C2×#711—32no—
114C2×C5×D8C2×D312yes—
115C5×#127C2×D311yescharacteristic
116C4×C5×D4C2×D311yescharacteristic
117#4084—61no—
118#4084—61no—
119#4082—32no—
120D4×Dic5—31no—
121D4×Dic5—61no—
122D4×Dic5—61no—
123C2×#687—32no—
124C2×#687—32no—
125C4×C22⋊D5—61no—
126C2×C4×Dic5—61no—
127#4058—68no—
128C2×Dic20—32no—
129C2×C8×D5—32no—
130#4038—31no—
131#4033—31no—
132#4027—61no—
133#4023—61no—
134C42×D5—31no—
135#4015—31no—
136C4×Dic10—31no—
137C5×C4⋊C8C2×D311yescharacteristic
138C5×#102C2×D312yes—
139#3964—32no—
140#3948—32no—
141#3946—32no—
142#3942—32no—
143C4⋊#168—31no—
144#3035—151noSylow
145C23×C3×C5—21no—
146C23×C3×C5—22no—
147C22×D15C2×D411yescharacteristic
148C3×C5×D4—22no—
149C3×C5×D4—21no—
150C3×C5×D4—22no—
151C3×C5×D4—42no—
152C3×C5×D4—42no—
153C3×C5×D4C2×D412yes—
154C2×C3×C4×C5—21no—
155C2×C3×C4×C5—42no—
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5C2×D411yescharacteristic
158C2×C3×C4×C5C2411yescharacteristic
159#1220—24no—
160#1220—44no—
161C2×Dic15—21no—
162C2×Dic15—24no—
163C2×Dic15—21no—
164C2×Dic15—22no—
165C2×Dic15—42no—
166C2×Dic15—44no—
167C2×Dic15C2×D411yescharacteristic
168C2×Dic15C2×D411yescharacteristic
169C2×Dic15C2×D411yescharacteristic
170C4×D15—22no—
171C4×D15—41no—
172C4×D15C2×D412yes—
173C4×D15#4112yes—
174Dic30—21no—
175Dic30—22no—
176Dic30—22no—
177Dic30C2×D412yes—
178C3×C5×C8—22no—
179C3⋊#168—22no—
180C2×#237—51no—
181C2×#237—52no—
182C2×C3×D8C2×D512yes—
183C3×#127C2×D511yescharacteristic
184C3×C4×D4C2×D511yescharacteristic
185#938—101no—
186#938—101no—
187#936—52no—
188D4×Dic3—51no—
189D4×Dic3—101no—
190D4×Dic3—101no—
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×D3—51no—
203#869—51no—
204C4×Dic6—51no—
205C3×C4⋊C8C2×D511yescharacteristic
206C3×#102C2×D512yes—
207#832—52no—
208#816—52no—
209Dic3⋊C8—52no—
210#810—52no—
211C4⋊(C3⋊C8)—51no—
212C2×C5×D4—21no—
213C2×C5×D4—21no—
214C2×C5×D4C22×D312yes—
215C2×C5×D4C22×D311yescharacteristic
216C22×C4×C5—21no—
217C2×C22⋊D5—61no—
218C2×C22⋊D5—62no—
219C22×Dic5—61no—
220C22×Dic5—62no—
221C22×Dic5—61no—
222#711—34no—
223#711—62no—
224#711—64no—
225#711—64no—
226C2×C4×D5—31no—
227C2×C4×D5—31no—
228C2×C4×D5—31no—
229C2×Dic10—31no—
230C2×Dic10—32no—
231C5×D8—24no—
232C2×C5×C8C22×D312yes—
233C5×C4⋊C4C22×D311yescharacteristic
234C5×C22⋊C4—21no—
235C42×C5C22×D311yescharacteristic
236C22⋊Dic5—61no—
237C22⋊Dic5—62no—
238C22⋊Dic5—62no—
239#687—64no—
240#687—64no—
241#685—61no—
242C4⋊Dic5—31no—
243C4⋊Dic5—61no—
244C4⋊Dic5—61no—
245#683—61no—
246#683—61no—
247C4×Dic5—31no—
248C4×Dic5—31no—
249C4×Dic5—31no—
250C4×Dic5—61no—
251C4×Dic5—61no—
252C4×Dic5—61no—
253C2×#168—32no—
254Dic20—64no—
255C8×D5—64no—
256C2×#135—152no—
257#532—151no—
258C4×#41—151no—
259#464—152no—
260#463—152no—
261#462—151no—
262#461—151no—
263#460—151no—
264#459D1511yescharacteristic, complemented
265C2×C4⋊C8—151no—
266#416—152no—
267C22×C3×C5—22no—
268C22×C3×C5—41no—
269C22×C3×C5—41no—
270C22×C3×C5—42no—
271C22×C3×C5—42no—
272C22×C3×C5—42no—
273C22×C3×C5C22×D411yescharacteristic
274C2×D15—21no—
275C2×D15—21no—
276C2×D15#12112yes—
277C3×C4×C5—21no—
278C3×C4×C5—41no—
279C3×C4×C5#13512yescomplemented
280C3×C4×C5C22×D411yescharacteristic
281C3×C4×C5C2×#4111yescharacteristic
282Dic15—22no—
283Dic15—21no—
284Dic15—21no—
285Dic15—42no—
286Dic15—41no—
287Dic15#12112yes—
288C2×C3×D4—21no—
289C2×C3×D4—21no—
290C2×C3×D4C22×D512yes—
291C2×C3×D4C22×D511yescharacteristic
292C22×C3×C4—21no—
293C2×#67—101no—
294C2×#67—102no—
295C22×Dic3—101no—
296C22×Dic3—101no—
297C22×Dic3—102no—
298#237—54no—
299#237—102no—
300#237—104no—
301#237—104no—
302C2×C4×D3—51no—
303C2×C4×D3—51no—
304C2×C4×D3—51no—
305C2×Dic6—51no—
306C2×Dic6—52no—
307C3×D8—24no—
308C2×C3×C8C22×D512yes—
309C3×C4⋊C4C22×D511yescharacteristic
310C3×C22⋊C4—21no—
311C3×C42C22×D511yescharacteristic
312#217—101no—
313#217—102no—
314#217—102no—
315#214—104no—
316#214—104no—
317(C2×D3)⋊C4—101no—
318C4⋊Dic3—51no—
319C4⋊Dic3—101no—
320C4⋊Dic3—101no—
321Dic3⋊C4—101no—
322Dic3⋊C4—101no—
323C4×Dic3—51no—
324C4×Dic3—51no—
325C4×Dic3—51no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3—101no—
329C2×C3⋊C8—52no—
330Dic12—104no—
331D3×C8—104no—
332C23×C5—21no—
333C23×C5—22no—
334C22×D5—31no—
335C5×D4—22no—
336C5×D4—21no—
337C5×D4—22no—
338C5×D4—42no—
339C5×D4—42no—
340C5×D4D3×D412yes—
341C2×C4×C5—21no—
342C2×C4×C5—42no—
343C2×C4×C5D3×D411yescharacteristic
344C2×C4×C5D3×D411yescharacteristic
345C2×C4×C5C23×D311yescharacteristic
346C22⋊D5—64no—
347C22⋊D5—124no—
348C2×Dic5—31no—
349C2×Dic5—31no—
350C2×Dic5—31no—
351C2×Dic5—64no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—62no—
355C2×Dic5—124no—
356C2×Dic5—122no—
357C4×D5—32no—
358C4×D5—32no—
359C4×D5—62no—
360C4×D5—121no—
361Dic10—32no—
362Dic10—61no—
363Dic10—62no—
364Dic10—62no—
365C5×C8—22no—
366#168—62no—
367C2×#41—151no—
368C2×#41—152no—
369#135—308no—
370C2×Q16—152no—
371C2×SD16—152no—
372C2×SD16—152no—
373C2×D8C2×D1512yes—
374C22×C8—152no—
375C4⋊Q8—151no—
376#127C2×D1511yescharacteristic
377#124—152no—
378#121—301no—
379#121—301no—
380C4×Q8—151no—
381C4×Q8—151no—
382C4×D4—151no—
383C4×D4—301no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4C2×D1511yescharacteristic
388#117—151no—
389#117—301no—
390C2×C42—151no—
391C2×C42—301no—
392C4⋊C8—152no—
393C4⋊C8—151no—
394C4⋊C8C2×D1511yescharacteristic
395#103—152no—
396#103—152no—
397#102—152no—
398#102C2×D1512yes—
399C2×C3×C5—42no—
400C2×C3×C5—82no—
401C2×C3×C5C2×#12111yescharacteristic
402C2×C3×C5C2×#13511yescharacteristic
403C2×C3×C5#57511yescharacteristic
404D15—22no—
405C23×C3—21no—
406C23×C3—22no—
407C22×D3—51no—
408C3×D4—22no—
409C3×D4—21no—
410C3×D4—22no—
411C3×D4—42no—
412C3×D4—42no—
413C3×D4D4×D512yes—
414C2×C3×C4—21no—
415C2×C3×C4—42no—
416C2×C3×C4D4×D511yescharacteristic
417C2×C3×C4D4×D511yescharacteristic
418C2×C3×C4C23×D511yescharacteristic
419#67—104no—
420#67—204no—
421C2×Dic3—51no—
422C2×Dic3—51no—
423C2×Dic3—51no—
424C2×Dic3—104no—
425C2×Dic3—101no—
426C2×Dic3—101no—
427C2×Dic3—102no—
428C2×Dic3—204no—
429C2×Dic3—202no—
430C4×D3—52no—
431C4×D3—52no—
432C4×D3—102no—
433C4×D3—201no—
434Dic6—52no—
435Dic6—101no—
436Dic6—102no—
437Dic6—102no—
438C3×C8—22no—
439C3⋊C8—102no—
440C22×C5—22no—
441C22×C5—41no—
442C22×C5—41no—
443C22×C5—42no—
444C22×C5—42no—
445C22×C5—42no—
446C22×C5C2×D3×D411yescharacteristic
447C2×D5—32no—
448C2×D5—61no—
449C2×D5—61no—
450C4×C5—21no—
451C4×C5—41no—
452C4×C5#91212yescomplemented
453C4×C5C2×D3×D411yescharacteristic
454C4×C5D3×#4111yescharacteristic
455Dic5—32no—
456Dic5—61no—
457Dic5—61no—
458Dic5—62no—
459Dic5—121no—
460Dic5—122no—
461#41—154no—
462#41—302no—
463#41—304no—
464#41—304no—
465C2×Q8—151no—
466C2×Q8—152no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—301no—
470C2×D4—302no—
471C2×D4C22×D1512yes—
472C2×D4C22×D1511yescharacteristic
473C22×C4—21no—
474C22×C4—151no—
475C22×C4—151no—
476C22×C4—151no—
477C22×C4—301no—
478C22×C4—301no—
479C22×C4—302no—
480Q16—304no—
481SD16—304no—
482SD16—304no—
483D8—24no—
484C2×C8—152no—
485C2×C8—304no—
486C2×C8C22×D1512yes—
487C4⋊C4—151no—
488C4⋊C4—301no—
489C4⋊C4—301no—
490C4⋊C4—301no—
491C4⋊C4—301no—
492C4⋊C4C22×D1511yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—301no—
495C22⋊C4—301no—
496C22⋊C4—302no—
497C22⋊C4—302no—
498C42—151no—
499C42—151no—
500C42—151no—
501C42—301no—
502C42—301no—
503C42—301no—
504C42C22×D1511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—22no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3C2×D4×D511yescharacteristic
513C2×D3—52no—
514C2×D3—101no—
515C2×D3—101no—
516C3×C4—21no—
517C3×C4—41no—
518C3×C4#405812yescomplemented
519C3×C4C2×D4×D511yescharacteristic
520C3×C4D5×#4111yescharacteristic
521Dic3—52no—
522Dic3—101no—
523Dic3—101no—
524Dic3—102no—
525Dic3—201no—
526Dic3—202no—
527C2×C5—42no—
528C2×C5—82no—
529C2×C5D3×#12111yescharacteristic
530C2×C5C2×#91211yescharacteristic
531C2×C5#577311yescharacteristic
532D5—62no—
533C23—21no—
534C23—22no—
535C23—151no—
536Q8—152no—
537Q8—301no—
538Q8—302no—
539Q8—302no—
540D4—22no—
541D4—21no—
542D4—22no—
543D4—42no—
544D4—42no—
545D4—304no—
546D4—604no—
547D4D4×D1512yes—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—151no—
551C2×C4—152no—
552C2×C4—152no—
553C2×C4—151no—
554C2×C4—151no—
555C2×C4—304no—
556C2×C4—301no—
557C2×C4—302no—
558C2×C4—301no—
559C2×C4—302no—
560C2×C4—601no—
561C2×C4—604no—
562C2×C4—602no—
563C2×C4D4×D1511yescharacteristic
564C2×C4D4×D1511yescharacteristic
565C2×C4C23×D1511yescharacteristic
566C8—22no—
567C8—302no—
568C2×C3—42no—
569C2×C3—82no—
570C2×C3D5×#12111yescharacteristic
571C2×C3C2×#405811yescharacteristic
572C2×C3#6610711yescharacteristic
573D3—102no—
574C5#8033111yescharacteristic, complemented, Sylow
575C22—22no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—152no—
582C22—301no—
583C22—301no—
584C22C2×D4×D1511yescharacteristic
585C4—21no—
586C4—41no—
587C4—152no—
588C4—302no—
589C4—301no—
590C4—301no—
591C4—602no—
592C4—601no—
593C4#9180112yescomplemented
594C4C2×D4×D1511yescharacteristic
595C4#41×D1511yescharacteristic
596C3#1061206211yescharacteristic, complemented, Sylow
597C2—42no—
598C2—82no—
599C2—302no—
600C2D15×#12111yescharacteristic
601C2C2×#9180111yescharacteristic
602C2#1175653211yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#459C2
socleC22×C3×C5C22×D4
hypercenter#459D15
hypocenterC3×C5#3035
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#91801C2semidirect
2. #11756475C2semidirect
3. C4×#6819C2semidirect
4. C3×C5×#459C2semidirect
5. #11751254C2semidirect
6. #11751176C2semidirect
7. #11751120C2semidirect
8. #11750950C2semidirect
9. D15×C4⋊C8C2semidirect
10. #11750873C2semidirect
11. #11750868C2semidirect
49. C5×#459D3semidirect
70. #6819D4semidirect
104. C3×#459D5semidirect
264. #459D15semidirect
279. C3×C4×C5#135semidirect
452. C4×C5#912semidirect
505. C3×C5#3035semidirect
518. C3×C4#4058semidirect
574. C5#80331semidirect
593. C4#91801semidirect
596. C3#10612062semidirect

Maximal subgroups

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

subgroupindex
C2×#918012
#117564752
C4×#68192
C3×C5×#4592
#117512542
#117511762
#117511202
#117509502
D15×C4⋊C82
#117508732
#117508682
#106120623
#803315