Finite Group Databasebeta

← #14472121

Subgroups — #14472121

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
1C5×#5569C211yesmaximal, characteristic, complemented
2#11752464C211yesmaximal, characteristic
3#11751129C211yesmaximal, characteristic
4C3×C4×#689C211yesmaximal, characteristic, complemented
5#11749360C211yesmaximal, characteristic, complemented
6#11749326C211yesmaximal, characteristic
7#11749257C211yesmaximal, characteristic, complemented
8#11749232C211yesmaximal, characteristic
9#11747422C211yesmaximal, characteristic, complemented
10#11747379C211yesmaximal, characteristic
11#11747290C211yesmaximal, characteristic, complemented
12#11747246C211yesmaximal, characteristic
13#11746919C211yesmaximal, characteristic, complemented
14#11746913C211yesmaximal, characteristic
15Dic6×#168C211yesmaximal, characteristic, complemented
16#10615420—31nomaximal, Hall
17C3×C4×C5×Q8C2211yescharacteristic
18C2×#6718C2211yescharacteristic
19C2×#6717C2211yescharacteristic
20#91764C2211yescharacteristic
21C5×Q8×Dic3C2211yescharacteristic
22C5×Q8×Dic3C2211yescharacteristic
23C5×#890—21no—
24C5×#890C2211yescharacteristic
25C5×#888—21no—
26C5×#888C2211yescharacteristic
27C5×#870—21no—
28C5×#870C2211yescharacteristic
29C4×C5×Dic6—21no—
30C4×C5×Dic6C2211yescharacteristic
31C2×C3×#689C2211yescharacteristic
32C3×C4×Dic10C2211yescharacteristic
33#91428C2211yescharacteristic
34#91424—21no—
35#91402—21no—
36#91389C2211yescharacteristic
37#91387—21no—
38#91374C2211yescharacteristic
39#91340C2211yescharacteristic
40#91337—21no—
41#91094C2211yescharacteristic
42#91093C2211yescharacteristic
43C4⋊(C3⋊#168)C2211yescharacteristic
44C3×#3967C2211yescharacteristic
45C3×#3942C2211yescharacteristic
46C3×#3939C2211yescharacteristic
47C3×C4×#168C2211yescharacteristic
48#90989C2211yescharacteristic
49#90986C2211yescharacteristic
50Dic3⋊#168C2211yescharacteristic
51Dic3⋊#168C2211yescharacteristic
52#90975C2211yescharacteristic
53#90975C2211yescharacteristic
54#90970C2211yescharacteristic
55#90970C2211yescharacteristic
56#90969C2211yescharacteristic
57#90969C2211yescharacteristic
58Dic3×#168C2211yescharacteristic
59Dic3×#168C2211yescharacteristic
60#79700—51nomaximal, Hall
61C5×#557—31no—
62#65918—31no—
63#65382—31no—
64#65376—31no—
65#65357—31no—
66#65357—31no—
67#65352—31no—
68#65352—31no—
69#65320—31no—
70C4×#689—31no—
71C4×#689D311yescharacteristic, complemented
72#65317—31no—
73#65317—31no—
74C4×#688—31no—
75Q8×#168—31no—
76C2×C3×C5×Q8C2311yescharacteristic
77C2×D60C2311yescharacteristic
78C2×Dic30C2311yescharacteristic
79C2×C5×Dic6—21no—
80C2×C5×Dic6D411yescharacteristic
81C2×C3×Dic10C2311yescharacteristic
82C2×#1200—21no—
83C3×C5×C4⋊C4—21no—
84C3×C5×C4⋊C4C2311yescharacteristic
85C3×C42×C5—21no—
86C3×C42×C5C2311yescharacteristic
87#6718—22no—
88#6717—22no—
89#6714—21no—
90#6714—21no—
91C2×C3⋊#168C2311yescharacteristic
92C2×C3⋊#168C2311yescharacteristic
93C5×C4⋊Dic3—21no—
94C5×C4⋊Dic3—21no—
95C5×C4⋊Dic3—21no—
96C5×C4⋊Dic3D411yescharacteristic
97C5×C4⋊Dic3C2311yescharacteristic
98C5×C4⋊Dic3C2311yescharacteristic
99C5×Dic3⋊C4—21no—
100C5×Dic3⋊C4—21no—
101C5×Dic3⋊C4—21no—
102C5×Dic3⋊C4—21no—
103C5×Dic3⋊C4D411yescharacteristic
104C5×Dic3⋊C4D411yescharacteristic
105C4×C5×Dic3—21no—
106C4×C5×Dic3—21no—
107C4×C5×Dic3C2311yescharacteristic
108C4×C5×Dic3C2311yescharacteristic
109C3×#689—22no—
110C3×C4⋊Dic5C2311yescharacteristic
111C3×#683—21no—
112C3×C4×Dic5—21no—
113C2×C3×#168C2311yescharacteristic
114C2×C3×#168C2311yescharacteristic
115Dic3⋊Dic5—21no—
116Dic3⋊Dic5—21no—
117#6669—21no—
118#6667—21no—
119#6667—21no—
120#6667—21no—
121Dic3×Dic5—21no—
122#5569D511yescharacteristic, complemented
123#5568—51no—
124C3×C4×Q16—51no—
125#5185—51no—
126#5182—51no—
127#5028—51no—
128#5022—51no—
129#4890—51no—
130#4877—51no—
131#4800—51no—
132#4792—51no—
133#4790—51no—
134#4788—51no—
135#4697—51no—
136C8×Dic6—51no—
137C5×C4⋊Q8—31no—
138C5×C4⋊Q8—61no—
139C5×#125—31no—
140C5×#125—31no—
141C5×#125—61no—
142C5×#125—61no—
143C4×C5×Q8—31no—
144C4×C5×Q8—31no—
145C4×C5×Q8—31no—
146C4×C5×Q8—61no—
147C4×C5×Q8C2×D311yescharacteristic
148#4092—61no—
149C2×#689—31no—
150C2×#689C2×D311yescharacteristic
151C2×#688—31no—
152#4044—61no—
153#4044—61no—
154#4038—61no—
155#4021—31no—
156#4021—31no—
157C4×D20—31no—
158#4015—31no—
159C4×Dic10—31no—
160C4×Dic10C2×D311yescharacteristic
161#3967C2×D311yescharacteristic
162#3942—31no—
163#3942—31no—
164#3942—31no—
165#3942—31no—
166#3942—31no—
167#3942C2×D311yescharacteristic
168#3941—31no—
169#3941—31no—
170#3941—31no—
171#3940—31no—
172#3939—31no—
173#3939C2×D311yescharacteristic
174C4⋊#168—31no—
175C4⋊#168—31no—
176C4⋊#168—31no—
177C4×#168—31no—
178C4×#168—31no—
179C4×#168C2×D311yescharacteristic
180#3364—151noSylow
181C22×D15—21no—
182C3×C5×Q8—21no—
183C3×C5×Q8#4112yes—
184C2×C3×C4×C5—21no—
185C2×C3×C4×C5—21no—
186C2×C3×C4×C5C2×D411yescharacteristic
187C2×C3×C4×C5C2×D411yescharacteristic
188C2×C3×C4×C5C2411yescharacteristic
189C2×Dic15—21no—
190D60—21no—
191D60—21no—
192Dic30—21no—
193Dic30—21no—
194C2×C5×Dic3—21no—
195C2×C5×Dic3—21no—
196C2×C5×Dic3C2×D411yescharacteristic
197C2×C5×Dic3C2×D411yescharacteristic
198C2×C5×Dic3C2×D411yescharacteristic
199C2×C5×Dic3C2×D411yescharacteristic
200C5×Dic6—22no—
201C5×Dic6—42no—
202C2×C3×Dic5—21no—
203C2×C3×Dic5—21no—
204C3×Dic10—21no—
205C3×Dic10—21no—
206#1200—42no—
207C3⋊#168—21no—
208C3⋊#168—21no—
209C3×#168—21no—
210C3×#168—21no—
211C2×C3×Q16—51no—
212C3×C4×Q8—51no—
213C3×C4×Q8C2×D511yescharacteristic
214#947—51no—
215Q8×Dic3C2×D511yescharacteristic
216Q8×Dic3C2×D511yescharacteristic
217#944—51no—
218C2×#216—51no—
219C2×#215—51no—
220#898—101no—
221#896—101no—
222#893—51no—
223#890—21no—
224#890C2×D511yescharacteristic
225#888—21no—
226#888C2×D511yescharacteristic
227#887—51no—
228#876—101no—
229#875—51no—
230#872—101no—
231#870—21no—
232#870C2×D511yescharacteristic
233C4×Dic6—21no—
234C4×Dic6C2×D511yescharacteristic
235C3×#107—51no—
236C3×#103—51no—
237C3×#103—51no—
238C3×C4×C8—51no—
239#835—51no—
240#835—51no—
241#821—51no—
242#821—51no—
243#816—51no—
244#816—51no—
245Dic3⋊C8—51no—
246Dic3⋊C8—51no—
247C8×Dic3—51no—
248C8×Dic3—51no—
249#810—51no—
250#809—51no—
251#808—51no—
252#807—51no—
253C4⋊(C3⋊C8)—51no—
254C2×C5×Q8—31no—
255C2×C5×Q8—61no—
256C2×C5×Q8C22×D311yescharacteristic
257C2×D20—31no—
258C2×C4×D5—61no—
259C2×Dic10—31no—
260C2×Dic10C22×D311yescharacteristic
261C5×C4⋊C4—21no—
262C5×C4⋊C4—31no—
263C5×C4⋊C4—31no—
264C5×C4⋊C4—31no—
265C5×C4⋊C4—31no—
266C5×C4⋊C4—31no—
267C5×C4⋊C4—61no—
268C5×C4⋊C4—61no—
269C5×C4⋊C4—61no—
270C5×C4⋊C4—61no—
271C5×C4⋊C4—61no—
272C5×C4⋊C4—61no—
273C5×C4⋊C4—61no—
274C5×C4⋊C4C22×D311yescharacteristic
275C42×C5—21no—
276C42×C5—31no—
277C42×C5—31no—
278C42×C5—61no—
279C42×C5—61no—
280C42×C5C22×D311yescharacteristic
281#689—22no—
282#689—62no—
283#688—62no—
284#685—61no—
285#685—61no—
286#685—61no—
287#685—61no—
288#685—61no—
289C4⋊Dic5—61no—
290C4⋊Dic5—61no—
291C4⋊Dic5C22×D311yescharacteristic
292#683—21no—
293#683—61no—
294C4×Dic5—21no—
295C4×Dic5—61no—
296C2×#168—31no—
297C2×#168—31no—
298C2×#168C22×D311yescharacteristic
299C2×#168C22×D311yescharacteristic
300#557—31no—
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#464—151no—
310#462—151no—
311C8×Q8—151no—
312C4×Q16—51no—
313C4×Q16—151no—
314C4×SD16—151no—
315C22×C3×C5C22×D411yescharacteristic
316C2×D15—41no—
317C2×D15—41no—
318C2×D15—41no—
319C3×C4×C5—22no—
320C3×C4×C5—21no—
321C3×C4×C5—21no—
322C3×C4×C5—41no—
323C3×C4×C5C2×#4111yescharacteristic
324C3×C4×C52−1+411yescharacteristic
325Dic15—41no—
326C3×Dic5—41no—
327C3×Dic5—41no—
328C5×Dic3—21no—
329C5×Dic3—21no—
330C5×Dic3—21no—
331C5×Dic3—41no—
332C5×Dic3—41no—
333C5×Dic3#13512yes—
334C2×C3×Q8—51no—
335C2×C3×Q8C22×D511yescharacteristic
336C2×D12—51no—
337C2×C4×D3—101no—
338C2×Dic6—21no—
339C2×Dic6—51no—
340C2×Dic6C22⋊D511yescharacteristic
341C3×Q16—102no—
342C2×C3×C8—51no—
343C2×C3×C8—51no—
344C3×C4⋊C4—21no—
345C3×C4⋊C4—51no—
346C3×C4⋊C4—101no—
347C3×C4⋊C4C22×D511yescharacteristic
348C3×C42—21no—
349C3×C42—101no—
350C3×C42C22×D511yescharacteristic
351#216—102no—
352#215—102no—
353(C2×D3)⋊C4—101no—
354(C2×D3)⋊C4—101no—
355(C2×D3)⋊C4—101no—
356(C2×D3)⋊C4—101no—
357(C2×D3)⋊C4—101no—
358C4⋊Dic3—21no—
359C4⋊Dic3—21no—
360C4⋊Dic3—21no—
361C4⋊Dic3C22⋊D511yescharacteristic
362C4⋊Dic3C22×D511yescharacteristic
363C4⋊Dic3C22×D511yescharacteristic
364Dic3⋊C4—21no—
365Dic3⋊C4—21no—
366Dic3⋊C4—21no—
367Dic3⋊C4—21no—
368Dic3⋊C4—101no—
369Dic3⋊C4—101no—
370Dic3⋊C4—101no—
371Dic3⋊C4C22⋊D511yescharacteristic
372Dic3⋊C4C22⋊D511yescharacteristic
373C4×Dic3—21no—
374C4×Dic3—21no—
375C4×Dic3—101no—
376C4×Dic3C22×D511yescharacteristic
377C4×Dic3C22×D511yescharacteristic
378C2×C3⋊C8—51no—
379C2×C3⋊C8—51no—
380C22×D5—61no—
381C5×Q8—21no—
382C5×Q8—62no—
383C5×Q8—122no—
384C5×Q8#23512yes—
385C2×C4×C5—21no—
386C2×C4×C5—21no—
387C2×C4×C5—31no—
388C2×C4×C5—31no—
389C2×C4×C5—31no—
390C2×C4×C5—31no—
391C2×C4×C5—61no—
392C2×C4×C5—61no—
393C2×C4×C5D3×D411yescharacteristic
394C2×C4×C5D3×D411yescharacteristic
395C2×C4×C5C23×D311yescharacteristic
396C2×Dic5—21no—
397C2×Dic5—21no—
398C2×Dic5—61no—
399D20—61no—
400D20—61no—
401C4×D5—122no—
402Dic10—21no—
403Dic10—21no—
404Dic10—61no—
405Dic10—61no—
406#168—21no—
407#168—21no—
408#168—61no—
409#168—61no—
410C2×Q16—51no—
411C2×Q16—151no—
412C2×SD16—151no—
413C4⋊Q8—31no—
414C4⋊Q8—61no—
415C4⋊Q8—151no—
416#126—301no—
417#126—301no—
418#125—31no—
419#125—31no—
420#125—61no—
421#125—61no—
422#124—151no—
423#124—151no—
424C22⋊Q8—301no—
425C4×Q8—31no—
426C4×Q8—31no—
427C4×Q8—31no—
428C4×Q8—51no—
429C4×Q8—61no—
430C4×Q8—151no—
431C4×Q8D3×D511yescharacteristic
432C4×D4—151no—
433#117—301no—
434#107—51no—
435#107—151no—
436#106—151no—
437C4⋊C8—151no—
438C4⋊C8—151no—
439C4⋊C8—151no—
440#103—51no—
441#103—51no—
442#103—151no—
443#103—151no—
444#103—151no—
445#103—151no—
446#103—151no—
447#102—151no—
448#102—151no—
449#102—151no—
450C4×C8—51no—
451C4×C8—151no—
452C4×C8—151no—
453C2×C3×C5#54811yescharacteristic
454C2×C3×C5C2×#13511yescharacteristic
455C2×C3×C5#57811yescharacteristic
456D15—81no—
457C22×D3—101no—
458C3×Q8—21no—
459C3×Q8—101no—
460C3×Q8—101no—
461C3×Q8#71312yes—
462C2×C3×C4—21no—
463C2×C3×C4—21no—
464C2×C3×C4—101no—
465C2×C3×C4—101no—
466C2×C3×C4C2×C22⋊D511yescharacteristic
467C2×C3×C4C2×C22⋊D511yescharacteristic
468C2×C3×C4C23×D511yescharacteristic
469C2×Dic3—21no—
470C2×Dic3—21no—
471C2×Dic3—101no—
472C2×Dic3C2×C22⋊D511yescharacteristic
473C2×Dic3C2×C22⋊D511yescharacteristic
474C2×Dic3C2×C22⋊D511yescharacteristic
475C2×Dic3C2×C22⋊D511yescharacteristic
476D12—101no—
477D12—101no—
478C4×D3—202no—
479Dic6—22no—
480Dic6—42no—
481Dic6—101no—
482Dic6—101no—
483C3×C8—101no—
484C3×C8—101no—
485C3⋊C8—101no—
486C3⋊C8—101no—
487C22×C5C2×D3×D411yescharacteristic
488C2×D5—121no—
489C2×D5—121no—
490C2×D5—121no—
491C4×C5—21no—
492C4×C5—21no—
493C4×C5—22no—
494C4×C5—32no—
495C4×C5—41no—
496C4×C5—61no—
497C4×C5—61no—
498C4×C5—61no—
499C4×C5—121no—
500C4×C5—121no—
501C4×C5C2×#23511yescharacteristic
502C4×C5#101011yescharacteristic
503Dic5—41no—
504Dic5—41no—
505Dic5—121no—
506C2×Q8—31no—
507C2×Q8—51no—
508C2×Q8—61no—
509C2×Q8—151no—
510C2×Q8C2×D3×D511yescharacteristic
511C2×D4—151no—
512C22×C4—301no—
513Q16—102no—
514Q16—302no—
515SD16—302no—
516C2×C8—51no—
517C2×C8—51no—
518C2×C8—151no—
519C2×C8—151no—
520C4⋊C4—21no—
521C4⋊C4—31no—
522C4⋊C4—31no—
523C4⋊C4—31no—
524C4⋊C4—31no—
525C4⋊C4—31no—
526C4⋊C4—51no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C4⋊C4—61no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—101no—
535C4⋊C4—301no—
536C4⋊C4—301no—
537C4⋊C4—301no—
538C4⋊C4C2×D3×D511yescharacteristic
539C22⋊C4—301no—
540C22⋊C4—301no—
541C22⋊C4—301no—
542C22⋊C4—301no—
543C22⋊C4—301no—
544C42—21no—
545C42—31no—
546C42—31no—
547C42—61no—
548C42—61no—
549C42—101no—
550C42—301no—
551C42C2×D3×D511yescharacteristic
552C3×C5#336411yescharacteristic, complemented, Hall
553C22×C3C22×C22⋊D511yescharacteristic
554C2×D3—201no—
555C2×D3—201no—
556C2×D3—201no—
557C3×C4—22no—
558C3×C4—21no—
559C3×C4—21no—
560C3×C4—41no—
561C3×C4—201no—
562C3×C4—201no—
563C3×C4C2×#71311yescharacteristic
564C3×C4#414711yescharacteristic
565Dic3—21no—
566Dic3—21no—
567Dic3—21no—
568Dic3—41no—
569Dic3—41no—
570Dic3—201no—
571Dic3#409612yes—
572C2×C5#555211yescharacteristic
573C2×C5#576611yescharacteristic
574C2×C5D3×#13511yescharacteristic
575D5—241no—
576C23—301no—
577Q8—21no—
578Q8—62no—
579Q8—101no—
580Q8—101no—
581Q8—122no—
582Q8—301no—
583Q8—301no—
584Q8#677312yes—
585D4—301no—
586D4—301no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—31no—
590C2×C4—31no—
591C2×C4—31no—
592C2×C4—31no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—101no—
596C2×C4—101no—
597C2×C4—301no—
598C2×C4—602no—
599C2×C4D3×C22⋊D511yescharacteristic
600C2×C4D3×C22⋊D511yescharacteristic
601C2×C4C22×D3×D511yescharacteristic
602C8—101no—
603C8—101no—
604C8—301no—
605C8—301no—
606C2×C3#6615111yescharacteristic
607C2×C3C2×#409611yescharacteristic
608C2×C3#6617211yescharacteristic
609D3—401no—
610C5#7970011yescharacteristic, complemented, Sylow
611C22—601no—
612C22—601no—
613C22—601no—
614C22C2×D3×C22⋊D511yescharacteristic
615C4—22no—
616C4—21no—
617C4—21no—
618C4—32no—
619C4—41no—
620C4—61no—
621C4—61no—
622C4—61no—
623C4—121no—
624C4—121no—
625C4—201no—
626C4—201no—
627C4—601no—
628C4#9200011yescharacteristic
629C4C2×#677311yescharacteristic
630C3#1061542011yescharacteristic, complemented, Sylow
631C2—1201no—
632C2#1175454411yescharacteristic
633C2#1175510511yescharacteristic
634C2D3×#409611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC2×D60C23
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3364
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C5×#5569C2semidirect
4. C3×C4×#689C2semidirect
5. #11749360C2semidirect
7. #11749257C2semidirect
9. #11747422C2semidirect
11. #11747290C2semidirect
13. #11746919C2semidirect
15. Dic6×#168C2semidirect
71. C4×#689D3semidirect
122. #5569D5semidirect
552. C3×C5#3364semidirect
610. C5#79700semidirect
630. C3#10615420semidirect

Maximal subgroups

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

subgroupindex
C5×#55692
#117524642
#117511292
C3×C4×#6892
#117493602
#117493262
#117492572
#117492322
#117474222
#117473792
#117472902
#117472462
#117469192
#117469132
Dic6×#1682
#106154203
#797005