Finite Group Databasebeta

← #14472060

Subgroups — #14472060

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756430C211yesmaximal, characteristic
2#11756328C211yesmaximal, characteristic, complemented
3C3×C5×#464C211yesmaximal, characteristic
4#11751283C211yesmaximal, characteristic, complemented
5Q16×Dic15C211yesmaximal, characteristic, complemented
6#11751267C211yesmaximal, characteristic
7SD16×Dic15C211yesmaximal, characteristic
8#11751157C211yesmaximal, characteristic
9C4×#6718C211yesmaximal, characteristic, complemented
10#11750931C211yesmaximal, characteristic, complemented
11#11750930C211yesmaximal, characteristic, complemented
12#11750900C211yesmaximal, characteristic, complemented
13#11750893C211yesmaximal, characteristic, complemented
14#11750869C211yesmaximal, characteristic
15#11750860C211yesmaximal, characteristic
16#10612311—31nomaximal, Hall
17C2×C3×C5×Q16C2211yescharacteristic
18C2×C3×C5×SD16C2211yescharacteristic
19C3×C5×#124C2211yescharacteristic
20C3×C4×C5×Q8C2211yescharacteristic
21Q8×Dic15C2211yescharacteristic
22Q8×Dic15C2211yescharacteristic
23Q8×Dic15C2211yescharacteristic
24#91833C2211yescharacteristic
25C2×#6718C2211yescharacteristic
26#91825C2211yescharacteristic
27D4×Dic15C2211yescharacteristic
28#91779—21no—
29#91779C2211yescharacteristic
30#91777—21no—
31#91777C2211yescharacteristic
32#91768—21no—
33#91768—21no—
34#91767—21no—
35#91766—21no—
36#91759—21no—
37#91759C2211yescharacteristic
38C4×Dic30—21no—
39C4×Dic30C2211yescharacteristic
40C4×Dic30C2211yescharacteristic
41C3×C5×C4⋊C8C2211yescharacteristic
42C3×C5×#103C2211yescharacteristic
43C3×C5×#102C2211yescharacteristic
44#91119C2211yescharacteristic
45#91119C2211yescharacteristic
46#91119C2211yescharacteristic
47#91116C2211yescharacteristic
48#91116C2211yescharacteristic
49#91102C2211yescharacteristic
50#91101C2211yescharacteristic
51#91100C2211yescharacteristic
52#91100C2211yescharacteristic
53#91098C2211yescharacteristic
54#91098C2211yescharacteristic
55C8×Dic15C2211yescharacteristic
56C8×Dic15C2211yescharacteristic
57#91094C2211yescharacteristic
58#91091C2211yescharacteristic
59C4×C3⋊#168C2211yescharacteristic
60#80580—51nomaximal, Hall
61#66005—31no—
62#65903—31no—
63C5×#464D311yescharacteristic
64Q16×Dic5—31no—
65#65471—31no—
66#65455—31no—
67SD16×Dic5—31no—
68#65345—31no—
69C4×#689—31no—
70#65119—31no—
71#65118—31no—
72#65088—31no—
73#65081—31no—
74#65057—31no—
75#65048—31no—
76C2×C3×C5×Q8C2311yescharacteristic
77C2×C3×C5×Q8C2311yescharacteristic
78C2×C3×C5×D4C2311yescharacteristic
79C22×Dic15—21no—
80C2×Dic30—21no—
81C2×Dic30D411yescharacteristic
82C2×Dic30C2311yescharacteristic
83C3×C5×Q16—22no—
84C3×C5×SD16—22no—
85C2×C3×C5×C8C2311yescharacteristic
86C2×C3×C5×C8C2311yescharacteristic
87C3×C5×C4⋊C4—21no—
88C3×C5×C4⋊C4C2311yescharacteristic
89C3×C5×C22⋊C4—21no—
90C3×C5×C22⋊C4—21no—
91C3×C42×C5—21no—
92C3×C42×C5C2311yescharacteristic
93#6719—21no—
94#6719—21no—
95#6719—21no—
96#6718—22no—
97C4⋊Dic15—21no—
98C4⋊Dic15—21no—
99C4⋊Dic15—21no—
100C4⋊Dic15—21no—
101C4⋊Dic15D411yescharacteristic
102C4⋊Dic15C2311yescharacteristic
103C4⋊Dic15C2311yescharacteristic
104C4⋊Dic15C2311yescharacteristic
105#6712—21no—
106#6712—21no—
107#6712—21no—
108#6712—21no—
109#6712—21no—
110#6712—21no—
111#6712—21no—
112#6712D411yescharacteristic
113#6712D411yescharacteristic
114C4×Dic15—21no—
115C4×Dic15—21no—
116C4×Dic15—21no—
117C4×Dic15—21no—
118C4×Dic15C2311yescharacteristic
119C4×Dic15C2311yescharacteristic
120C2×C3⋊#168C2311yescharacteristic
121C2×C3⋊#168C2311yescharacteristic
122#5671—51no—
123#5569—51no—
124C3×#464D511yescharacteristic
125#5182—51no—
126Dic3×Q16—51no—
127#5166—51no—
128Dic3×SD16—51no—
129#5056—51no—
130C4×#216—51no—
131#4830—51no—
132#4829—51no—
133#4799—51no—
134#4792—51no—
135#4768—51no—
136#4759—51no—
137C2×C5×Q16C2×D311yescharacteristic
138C2×C5×SD16C2×D311yescharacteristic
139C5×#124C2×D311yescharacteristic
140C4×C5×Q8C2×D311yescharacteristic
141Q8×Dic5—31no—
142Q8×Dic5—31no—
143Q8×Dic5—31no—
144#4090—31no—
145C2×#689—31no—
146#4082—31no—
147D4×Dic5—31no—
148#4036—31no—
149#4036—61no—
150#4034—31no—
151#4034—61no—
152#4025—61no—
153#4025—61no—
154#4024—61no—
155#4023—61no—
156#4016—31no—
157#4016—61no—
158C4×Dic10—31no—
159C4×Dic10—31no—
160C4×Dic10—61no—
161C5×C4⋊C8C2×D311yescharacteristic
162C5×#103C2×D311yescharacteristic
163C5×#102C2×D311yescharacteristic
164#3967—31no—
165#3967—31no—
166#3967—31no—
167#3964—31no—
168#3964—31no—
169#3950—31no—
170#3949—31no—
171#3948—31no—
172#3948—31no—
173#3946—31no—
174#3946—31no—
175C8×Dic5—31no—
176C8×Dic5—31no—
177#3942—31no—
178#3939—31no—
179C4×#168—31no—
180#3364—151noSylow
181C23×C3×C5—21no—
182C3×C5×Q8—21no—
183C3×C5×Q8—21no—
184C3×C5×Q8—21no—
185C3×C5×Q8#4112yes—
186C3×C5×D4—21no—
187C3×C5×D4—21no—
188C2×C3×C4×C5—21no—
189C2×C3×C4×C5—21no—
190C2×C3×C4×C5—21no—
191C2×C3×C4×C5C2×D411yescharacteristic
192C2×C3×C4×C5C2×D411yescharacteristic
193C2×C3×C4×C5C2411yescharacteristic
194C2×Dic15—21no—
195C2×Dic15—21no—
196C2×Dic15—21no—
197C2×Dic15—21no—
198C2×Dic15—42no—
199C2×Dic15C2×D411yescharacteristic
200C2×Dic15C2×D411yescharacteristic
201C2×Dic15C2×D411yescharacteristic
202C2×Dic15C2×D411yescharacteristic
203Dic30—22no—
204Dic30—21no—
205Dic30—21no—
206Dic30—42no—
207C3×C5×C8—21no—
208C3×C5×C8—21no—
209C3⋊#168—21no—
210C3⋊#168—21no—
211C2×C3×Q16C2×D511yescharacteristic
212C2×C3×SD16C2×D511yescharacteristic
213C3×#124C2×D511yescharacteristic
214C3×C4×Q8C2×D511yescharacteristic
215Q8×Dic3—51no—
216Q8×Dic3—51no—
217Q8×Dic3—51no—
218#944—51no—
219C2×#216—51no—
220#936—51no—
221D4×Dic3—51no—
222#890—51no—
223#890—101no—
224#888—51no—
225#888—101no—
226#879—101no—
227#879—101no—
228#878—101no—
229#877—101no—
230#870—51no—
231#870—101no—
232C4×Dic6—51no—
233C4×Dic6—51no—
234C4×Dic6—101no—
235C3×C4⋊C8C2×D511yescharacteristic
236C3×#103C2×D511yescharacteristic
237C3×#102C2×D511yescharacteristic
238#835—51no—
239#835—51no—
240#835—51no—
241#832—51no—
242#832—51no—
243#818—51no—
244#817—51no—
245#816—51no—
246#816—51no—
247Dic3⋊C8—51no—
248Dic3⋊C8—51no—
249C8×Dic3—51no—
250C8×Dic3—51no—
251#810—51no—
252#807—51no—
253C4×C3⋊C8—51no—
254C2×C5×Q8C22×D311yescharacteristic
255C2×C5×Q8C22×D311yescharacteristic
256C2×C5×D4C22×D311yescharacteristic
257C22×Dic5—61no—
258C2×Dic10—31no—
259C2×Dic10—31no—
260C2×Dic10—61no—
261C5×Q16—22no—
262C5×SD16—22no—
263C2×C5×C8C22×D311yescharacteristic
264C2×C5×C8C22×D311yescharacteristic
265C5×C4⋊C4—21no—
266C5×C4⋊C4C22×D311yescharacteristic
267C5×C22⋊C4—21no—
268C5×C22⋊C4—21no—
269C42×C5—21no—
270C42×C5C22×D311yescharacteristic
271C22⋊Dic5—61no—
272C22⋊Dic5—61no—
273C22⋊Dic5—61no—
274#689—62no—
275C4⋊Dic5—31no—
276C4⋊Dic5—31no—
277C4⋊Dic5—31no—
278C4⋊Dic5—31no—
279C4⋊Dic5—61no—
280C4⋊Dic5—61no—
281C4⋊Dic5—61no—
282C4⋊Dic5—61no—
283#683—31no—
284#683—31no—
285#683—61no—
286#683—61no—
287#683—61no—
288#683—61no—
289#683—61no—
290#683—61no—
291#683—61no—
292C4×Dic5—31no—
293C4×Dic5—31no—
294C4×Dic5—61no—
295C4×Dic5—61no—
296C4×Dic5—61no—
297C4×Dic5—61no—
298C2×#168—31no—
299C2×#168—31no—
300#557—151no—
301#556—151no—
302#488—151no—
303#488—151no—
304#487—151no—
305#479—151no—
306#479—151no—
307#474—151no—
308#464—151no—
309#464D1511yescharacteristic
310#462—151no—
311C8×Q8—151no—
312C4×Q16—151no—
313C4×Q16—151no—
314C4×SD16—151no—
315C22×C3×C5—41no—
316C22×C3×C5—41no—
317C22×C3×C5—41no—
318C22×C3×C5C22×D411yescharacteristic
319C3×C4×C5—22no—
320C3×C4×C5—21no—
321C3×C4×C5—21no—
322C3×C4×C5—41no—
323C3×C4×C5—41no—
324C3×C4×C5C2×#4111yescharacteristic
325C3×C4×C52−1+411yescharacteristic
326Dic15—21no—
327Dic15—21no—
328Dic15—21no—
329Dic15—41no—
330Dic15—41no—
331Dic15—41no—
332Dic15—41no—
333Dic15#13512yes—
334C2×C3×Q8C22×D511yescharacteristic
335C2×C3×Q8C22×D511yescharacteristic
336C2×C3×D4C22×D511yescharacteristic
337C22×Dic3—101no—
338C2×Dic6—51no—
339C2×Dic6—51no—
340C2×Dic6—101no—
341C3×Q16—22no—
342C3×SD16—22no—
343C2×C3×C8C22×D511yescharacteristic
344C2×C3×C8C22×D511yescharacteristic
345C3×C4⋊C4—21no—
346C3×C4⋊C4C22×D511yescharacteristic
347C3×C22⋊C4—21no—
348C3×C22⋊C4—21no—
349C3×C42—21no—
350C3×C42C22×D511yescharacteristic
351#217—101no—
352#217—101no—
353#217—101no—
354#216—102no—
355C4⋊Dic3—51no—
356C4⋊Dic3—51no—
357C4⋊Dic3—51no—
358C4⋊Dic3—51no—
359C4⋊Dic3—101no—
360C4⋊Dic3—101no—
361C4⋊Dic3—101no—
362C4⋊Dic3—101no—
363Dic3⋊C4—51no—
364Dic3⋊C4—51no—
365Dic3⋊C4—101no—
366Dic3⋊C4—101no—
367Dic3⋊C4—101no—
368Dic3⋊C4—101no—
369Dic3⋊C4—101no—
370Dic3⋊C4—101no—
371Dic3⋊C4—101no—
372C4×Dic3—51no—
373C4×Dic3—51no—
374C4×Dic3—101no—
375C4×Dic3—101no—
376C4×Dic3—101no—
377C4×Dic3—101no—
378C2×C3⋊C8—51no—
379C2×C3⋊C8—51no—
380C23×C5—21no—
381C5×Q8—21no—
382C5×Q8—21no—
383C5×Q8—21no—
384C5×Q8#23712yes—
385C5×D4—21no—
386C5×D4—21no—
387C2×C4×C5—21no—
388C2×C4×C5—21no—
389C2×C4×C5—21no—
390C2×C4×C5D3×D411yescharacteristic
391C2×C4×C5D3×D411yescharacteristic
392C2×C4×C5C23×D311yescharacteristic
393C2×Dic5—31no—
394C2×Dic5—31no—
395C2×Dic5—31no—
396C2×Dic5—31no—
397C2×Dic5—61no—
398C2×Dic5—61no—
399C2×Dic5—61no—
400C2×Dic5—61no—
401C2×Dic5—122no—
402Dic10—62no—
403Dic10—61no—
404Dic10—61no—
405Dic10—122no—
406C5×C8—21no—
407C5×C8—21no—
408#168—61no—
409#168—61no—
410C2×Q16—151no—
411C2×Q16C2×D1511yescharacteristic
412C2×SD16C2×D1511yescharacteristic
413C4⋊Q8—151no—
414C4⋊Q8—151no—
415C4⋊Q8—301no—
416#126—301no—
417#126—301no—
418#125—151no—
419#125—151no—
420#125—301no—
421#125—301no—
422#124—151no—
423#124C2×D1511yescharacteristic
424C22⋊Q8—301no—
425C4×Q8—151no—
426C4×Q8—151no—
427C4×Q8—151no—
428C4×Q8—151no—
429C4×Q8—151no—
430C4×Q8—301no—
431C4×Q8C2×D1511yescharacteristic
432C4×D4—151no—
433#117—301no—
434#107—151no—
435#107—151no—
436#106—151no—
437C4⋊C8—151no—
438C4⋊C8—151no—
439C4⋊C8C2×D1511yescharacteristic
440#103—151no—
441#103—151no—
442#103—151no—
443#103—151no—
444#103—151no—
445#103—151no—
446#103C2×D1511yescharacteristic
447#102—151no—
448#102—151no—
449#102C2×D1511yescharacteristic
450C4×C8—151no—
451C4×C8—151no—
452C4×C8—151no—
453C2×C3×C5—81no—
454C2×C3×C5#54811yescharacteristic
455C2×C3×C5C2×#13511yescharacteristic
456C2×C3×C5#57811yescharacteristic
457C23×C3—21no—
458C3×Q8—21no—
459C3×Q8—21no—
460C3×Q8—21no—
461C3×Q8#71112yes—
462C3×D4—21no—
463C3×D4—21no—
464C2×C3×C4—21no—
465C2×C3×C4—21no—
466C2×C3×C4—21no—
467C2×C3×C4D4×D511yescharacteristic
468C2×C3×C4D4×D511yescharacteristic
469C2×C3×C4C23×D511yescharacteristic
470C2×Dic3—51no—
471C2×Dic3—51no—
472C2×Dic3—51no—
473C2×Dic3—51no—
474C2×Dic3—101no—
475C2×Dic3—101no—
476C2×Dic3—101no—
477C2×Dic3—101no—
478C2×Dic3—202no—
479Dic6—102no—
480Dic6—101no—
481Dic6—101no—
482Dic6—202no—
483C3×C8—21no—
484C3×C8—21no—
485C3⋊C8—101no—
486C3⋊C8—101no—
487C22×C5—41no—
488C22×C5—41no—
489C22×C5—41no—
490C22×C5C2×D3×D411yescharacteristic
491C4×C5—21no—
492C4×C5—21no—
493C4×C5—22no—
494C4×C5—41no—
495C4×C5—41no—
496C4×C5C2×#23711yescharacteristic
497C4×C5#101011yescharacteristic
498Dic5—32no—
499Dic5—61no—
500Dic5—61no—
501Dic5—61no—
502Dic5—121no—
503Dic5—121no—
504Dic5—121no—
505Dic5—121no—
506C2×Q8—151no—
507C2×Q8—151no—
508C2×Q8—301no—
509C2×Q8C22×D1511yescharacteristic
510C2×Q8C22×D1511yescharacteristic
511C2×D4C22×D1511yescharacteristic
512C22×C4—301no—
513Q16—22no—
514Q16—302no—
515SD16—22no—
516C2×C8—151no—
517C2×C8—151no—
518C2×C8C22×D1511yescharacteristic
519C2×C8C22×D1511yescharacteristic
520C4⋊C4—21no—
521C4⋊C4—151no—
522C4⋊C4—151no—
523C4⋊C4—151no—
524C4⋊C4—151no—
525C4⋊C4—151no—
526C4⋊C4—151no—
527C4⋊C4—301no—
528C4⋊C4—301no—
529C4⋊C4—301no—
530C4⋊C4—301no—
531C4⋊C4—301no—
532C4⋊C4—301no—
533C4⋊C4—301no—
534C4⋊C4—301no—
535C4⋊C4—301no—
536C4⋊C4—301no—
537C4⋊C4—301no—
538C4⋊C4C22×D1511yescharacteristic
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—301no—
542C22⋊C4—301no—
543C22⋊C4—301no—
544C42—21no—
545C42—151no—
546C42—151no—
547C42—301no—
548C42—301no—
549C42—301no—
550C42—301no—
551C42C22×D1511yescharacteristic
552C3×C5#336411yescharacteristic, complemented, Hall
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—41no—
556C22×C3C2×D4×D511yescharacteristic
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—22no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4C2×#71111yescharacteristic
563C3×C4#415011yescharacteristic
564Dic3—52no—
565Dic3—101no—
566Dic3—101no—
567Dic3—101no—
568Dic3—201no—
569Dic3—201no—
570Dic3—201no—
571Dic3—201no—
572C2×C5—81no—
573C2×C5#559811yescharacteristic
574C2×C5D3×#13511yescharacteristic
575C2×C5#577911yescharacteristic
576C23—21no—
577Q8—21no—
578Q8—21no—
579Q8—21no—
580Q8—302no—
581Q8—301no—
582Q8—301no—
583Q8—602no—
584Q8#681912yes—
585D4—21no—
586D4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—151no—
591C2×C4—151no—
592C2×C4—151no—
593C2×C4—151no—
594C2×C4—301no—
595C2×C4—301no—
596C2×C4—301no—
597C2×C4—301no—
598C2×C4—602no—
599C2×C4D4×D1511yescharacteristic
600C2×C4D4×D1511yescharacteristic
601C2×C4C23×D1511yescharacteristic
602C8—21no—
603C8—21no—
604C8—301no—
605C8—301no—
606C2×C3—81no—
607C2×C3#6593211yescharacteristic
608C2×C3D5×#13511yescharacteristic
609C2×C3#6611311yescharacteristic
610C5#8058011yescharacteristic, complemented, Sylow
611C22—41no—
612C22—41no—
613C22—41no—
614C22C2×D4×D1511yescharacteristic
615C4—22no—
616C4—21no—
617C4—21no—
618C4—41no—
619C4—41no—
620C4—152no—
621C4—301no—
622C4—301no—
623C4—301no—
624C4—601no—
625C4—601no—
626C4—601no—
627C4—601no—
628C4C2×#681911yescharacteristic
629C4#9210111yescharacteristic
630C3#1061231111yescharacteristic, complemented, Sylow
631C2—81no—
632C2#1175635711yescharacteristic
633C2D15×#13511yescharacteristic
634C2#1175653811yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#464C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×D4C23
hypercenter#464D15
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #11756328C2semidirect
4. #11751283C2semidirect
5. Q16×Dic15C2semidirect
9. C4×#6718C2semidirect
10. #11750931C2semidirect
11. #11750930C2semidirect
12. #11750900C2semidirect
13. #11750893C2semidirect
552. C3×C5#3364semidirect
610. C5#80580semidirect
630. C3#10612311semidirect

Maximal subgroups

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

subgroupindex
#117564302
#117563282
C3×C5×#4642
#117512832
Q16×Dic152
#117512672
SD16×Dic152
#117511572
C4×#67182
#117509312
#117509302
#117509002
#117508932
#117508692
#117508602
#106123113
#805805