Finite Group Databasebeta

← #14430221

Subgroups — #14430221

Order 1920.

776 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756281C211yesmaximal, characteristic
2#11756276C211yesmaximal, characteristic, complemented
3C3×C5×#427C211yesmaximal, characteristic, complemented
4#11751214C211yesmaximal, characteristic
5C2×#91098C211yesmaximal, characteristic, complemented
6#11751103C211yesmaximal, characteristic
7#11751099C211yesmaximal, characteristic, complemented
8#11750992C211yesmaximal, characteristic, complemented
9#11750991C211yesmaximal, characteristic, complemented
10#11750978C211yesmaximal, characteristic, complemented
11#11750976C211yesmaximal, characteristic, complemented
12#11750972C211yesmaximal, characteristic, complemented
13#11750971C211yesmaximal, characteristic, complemented
14#11750958C211yesmaximal, characteristic, complemented
15#11750955C211yesmaximal, characteristic, complemented
16#10610927—31nomaximal, Hall
17C2×#6817C2211yescharacteristic
18C22×C3×C5×C8C2211yescharacteristic
19C3×C5×#117C2211yescharacteristic
20C3×C5×#117C2211yescharacteristic
21#91815C2211yescharacteristic
22C2×C4×Dic15C2211yescharacteristic
23C22×C3⋊#168C2211yescharacteristic
24#91782C2211yescharacteristic
25#91782C2211yescharacteristic
26#91779C2211yescharacteristic
27#91779C2211yescharacteristic
28#91777C2211yescharacteristic
29#91777C2211yescharacteristic
30#91776C2211yescharacteristic
31#91776C2211yescharacteristic
32#91770—21no—
33#91770—21no—
34#91767—21no—
35#91767—21no—
36#91761—21no—
37C42×D15—21no—
38C4×Dic30—21no—
39C4×Dic30—21no—
40C3×C5×#107C2211yescharacteristic
41C3×C5×#107C2211yescharacteristic
42C3×C5×#106C2211yescharacteristic
43C3×C5×#106C2211yescharacteristic
44#91105C2211yescharacteristic
45#91105C2211yescharacteristic
46#91100C2211yescharacteristic
47#91100C2211yescharacteristic
48#91098C2211yescharacteristic
49#91098C2211yescharacteristic
50#91098C2211yescharacteristic
51#91098C2211yescharacteristic
52#91094C2211yescharacteristic
53#91094C2211yescharacteristic
54#91093C2211yescharacteristic
55#91093C2211yescharacteristic
56#91092C2211yescharacteristic
57#91092C2211yescharacteristic
58#91091C2211yescharacteristic
59#91091C2211yescharacteristic
60#79196—51nomaximal, Hall
61#65856—31no—
62#65851—31no—
63C5×#427D311yescharacteristic, complemented
64#65402—31no—
65C2×#3946—31no—
66#65291—31no—
67#65287—31no—
68#65180—31no—
69#65179—31no—
70#65166—31no—
71#65164—31no—
72#65160—31no—
73#65159—31no—
74#65146—31no—
75#65143—31no—
76C22×C3×C4×C5C2311yescharacteristic
77C2×#1220—21no—
78C22×Dic15D411yescharacteristic
79C22×Dic15D411yescharacteristic
80#6817—22no—
81#6817Q814yescomplemented
82C2×D60C2311yescharacteristic
83C2×C4×D15—21no—
84C2×C4×D15—21no—
85C2×C4×D15—21no—
86C2×Dic30—21no—
87C2×Dic30—21no—
88C2×Dic30C2311yescharacteristic
89C2×C3×C5×C8—22no—
90C2×C3×C5×C8C2311yescharacteristic
91C2×C3×C5×C8C2311yescharacteristic
92C3×C5×C4⋊C4C2311yescharacteristic
93C3×C5×C4⋊C4C2311yescharacteristic
94C3×C5×C4⋊C4C2311yescharacteristic
95C3×C5×C4⋊C4C2311yescharacteristic
96C3×C5×C22⋊C4—21no—
97C3×C5×C22⋊C4—21no—
98C3×C42×C5—21no—
99C3×C42×C5—21no—
100#6719—21no—
101#6714—21no—
102#6714—21no—
103C4⋊Dic15—21no—
104C4⋊Dic15—21no—
105C4⋊Dic15C2311yescharacteristic
106C4⋊Dic15C2311yescharacteristic
107#6712—21no—
108#6712—21no—
109#6712—21no—
110#6712—21no—
111#6712—21no—
112#6712—21no—
113C4×Dic15—21no—
114C4×Dic15—21no—
115C4×Dic15—21no—
116C4×Dic15D411yescharacteristic
117C4×Dic15D411yescharacteristic
118C4×Dic15C2311yescharacteristic
119C4×Dic15C2311yescharacteristic
120C2×C3⋊#168—22no—
121C2×C3⋊#168C2311yescharacteristic
122C2×C3⋊#168C2311yescharacteristic
123#5522—51no—
124#5517—51no—
125C3×#427D511yescharacteristic, complemented
126#5113—51no—
127C2×Dic3⋊C8—51no—
128#5002—51no—
129#4998—51no—
130#4891—51no—
131#4890—51no—
132#4877—51no—
133#4875—51no—
134#4871—51no—
135#4870—51no—
136#4857—51no—
137#4854—51no—
138C2×#709—31no—
139C22×C5×C8C2×D311yescharacteristic
140C5×#117C2×D311yescharacteristic
141C5×#117C2×D311yescharacteristic
142#4072—31no—
143C2×C4×Dic5—31no—
144C22×#168—31no—
145#4039—31no—
146#4039—31no—
147#4036—31no—
148#4036—31no—
149#4034—31no—
150#4034—31no—
151#4033—31no—
152#4033—31no—
153#4027—61no—
154#4027—61no—
155#4024—61no—
156#4024—61no—
157#4018—61no—
158C42×D5—61no—
159C4×Dic10—61no—
160C4×Dic10—61no—
161C5×#107C2×D311yescharacteristic
162C5×#107C2×D311yescharacteristic
163C5×#106C2×D311yescharacteristic
164C5×#106C2×D311yescharacteristic
165#3953—31no—
166#3953—31no—
167#3948—31no—
168#3948—31no—
169#3946—31no—
170#3946—31no—
171#3946—31no—
172#3946—31no—
173#3942—31no—
174#3942—31no—
175#3941—31no—
176#3941—31no—
177#3940—31no—
178#3940—31no—
179#3939—31no—
180#3939—31no—
181#3076—151noSylow
182C23×C3×C5C2×D411yescharacteristic
183C22×D15—21no—
184C2×C3×C4×C5—21no—
185C2×C3×C4×C5—21no—
186C2×C3×C4×C5—21no—
187C2×C3×C4×C5—21no—
188C2×C3×C4×C5C2×D411yescharacteristic
189C2×C3×C4×C5C2×Q812yes—
190C2×C3×C4×C5#4112yes—
191C2×C3×C4×C5C2411yescharacteristic
192#1220—24no—
193C2×Dic15—21no—
194C2×Dic15—22no—
195C2×Dic15—21no—
196C2×Dic15—21no—
197C2×Dic15—41no—
198C2×Dic15C2×D411yescharacteristic
199C2×Dic15C2×D411yescharacteristic
200C2×Dic15C2×D411yescharacteristic
201C2×Dic15C2×D411yescharacteristic
202D60—21no—
203D60C2×Q812yes—
204C4×D15—24no—
205C4×D15—42no—
206C4×D15—42no—
207Dic30—21no—
208Dic30—42no—
209Dic30—42no—
210Dic30C2×Q812yes—
211C3×C5×C8—21no—
212C3×C5×C8—21no—
213C3⋊#168—21no—
214C3⋊#168—21no—
215C2×#235—51no—
216C22×C3×C8C2×D511yescharacteristic
217C3×#117C2×D511yescharacteristic
218C3×#117C2×D511yescharacteristic
219#926—51no—
220C2×C4×Dic3—51no—
221C22×C3⋊C8—51no—
222#893—51no—
223#893—51no—
224#890—51no—
225#890—51no—
226#888—51no—
227#888—51no—
228#887—51no—
229#887—51no—
230#67⋊C4—101no—
231#67⋊C4—101no—
232#878—101no—
233#878—101no—
234#872—101no—
235C42×D3—101no—
236C4×Dic6—101no—
237C4×Dic6—101no—
238C3×#107C2×D511yescharacteristic
239C3×#107C2×D511yescharacteristic
240C3×#106C2×D511yescharacteristic
241C3×#106C2×D511yescharacteristic
242#821—51no—
243#821—51no—
244#816—51no—
245#816—51no—
246Dic3⋊C8—51no—
247Dic3⋊C8—51no—
248Dic3⋊C8—51no—
249Dic3⋊C8—51no—
250#810—51no—
251#810—51no—
252#809—51no—
253#809—51no—
254#808—51no—
255#808—51no—
256#807—51no—
257#807—51no—
258C22×C4×C5C22×D311yescharacteristic
259C2×C22⋊D5—61no—
260C22×Dic5—31no—
261C22×Dic5—31no—
262#709—34no—
263#709—62no—
264C2×D20—31no—
265C2×C4×D5—61no—
266C2×C4×D5—61no—
267C2×C4×D5—61no—
268C2×Dic10—31no—
269C2×Dic10—61no—
270C2×Dic10—61no—
271C2×C5×C8—22no—
272C2×C5×C8C22×D311yescharacteristic
273C2×C5×C8C22×D311yescharacteristic
274C5×C4⋊C4C22×D311yescharacteristic
275C5×C4⋊C4C22×D311yescharacteristic
276C5×C4⋊C4C22×D311yescharacteristic
277C5×C4⋊C4C22×D311yescharacteristic
278C5×C22⋊C4—21no—
279C5×C22⋊C4—21no—
280C42×C5—21no—
281C42×C5—21no—
282C22⋊Dic5—61no—
283#685—61no—
284#685—61no—
285C4⋊Dic5—31no—
286C4⋊Dic5—31no—
287C4⋊Dic5—61no—
288C4⋊Dic5—61no—
289#683—61no—
290#683—61no—
291#683—61no—
292#683—61no—
293#683—61no—
294#683—61no—
295C4×Dic5—31no—
296C4×Dic5—31no—
297C4×Dic5—31no—
298C4×Dic5—31no—
299C4×Dic5—61no—
300C4×Dic5—61no—
301C4×Dic5—61no—
302C2×#168—31no—
303C2×#168—31no—
304C2×#168—62no—
305#533—151no—
306C4×#41—151no—
307#479—151no—
308#479—151no—
309#478—151no—
310#478—151no—
311#477—151no—
312#476—151no—
313#475—151no—
314#474—151no—
315#427—151no—
316#427D1511yescharacteristic, complemented
317C2×C4⋊C8—151no—
318#416—151no—
319#416—151no—
320C22×C3×C5—21no—
321C22×C3×C5—21no—
322C22×C3×C5C22⋊Q812yes—
323C22×C3×C5C22×D411yescharacteristic
324C2×D15—22no—
325C2×D15—41no—
326C2×D15—41no—
327C3×C4×C5—41no—
328C3×C4×C5—41no—
329C3×C4×C5—41no—
330C3×C4×C5—41no—
331C3×C4×C5C22⋊Q812yes—
332C3×C4×C5C22×Q811yescharacteristic
333C3×C4×C5C2×#4111yescharacteristic
334Dic15—22no—
335Dic15—21no—
336Dic15—21no—
337Dic15—21no—
338Dic15—41no—
339Dic15—41no—
340Dic15#13512yes—
341C22×C3×C4C22×D511yescharacteristic
342C2×#67—101no—
343C22×Dic3—51no—
344C22×Dic3—51no—
345#235—54no—
346#235—102no—
347C2×D12—51no—
348C2×C4×D3—101no—
349C2×C4×D3—101no—
350C2×C4×D3—101no—
351C2×Dic6—51no—
352C2×Dic6—101no—
353C2×Dic6—101no—
354C2×C3×C8—22no—
355C2×C3×C8C22×D511yescharacteristic
356C2×C3×C8C22×D511yescharacteristic
357C3×C4⋊C4C22×D511yescharacteristic
358C3×C4⋊C4C22×D511yescharacteristic
359C3×C4⋊C4C22×D511yescharacteristic
360C3×C4⋊C4C22×D511yescharacteristic
361C3×C22⋊C4—21no—
362C3×C22⋊C4—21no—
363C3×C42—21no—
364C3×C42—21no—
365#217—101no—
366(C2×D3)⋊C4—101no—
367(C2×D3)⋊C4—101no—
368C4⋊Dic3—51no—
369C4⋊Dic3—51no—
370C4⋊Dic3—101no—
371C4⋊Dic3—101no—
372Dic3⋊C4—101no—
373Dic3⋊C4—101no—
374Dic3⋊C4—101no—
375Dic3⋊C4—101no—
376Dic3⋊C4—101no—
377Dic3⋊C4—101no—
378C4×Dic3—51no—
379C4×Dic3—51no—
380C4×Dic3—51no—
381C4×Dic3—51no—
382C4×Dic3—101no—
383C4×Dic3—101no—
384C4×Dic3—101no—
385C2×C3⋊C8—51no—
386C2×C3⋊C8—51no—
387C2×C3⋊C8—102no—
388C23×C5D3×D411yescharacteristic
389C22×D5—61no—
390C2×C4×C5—21no—
391C2×C4×C5—21no—
392C2×C4×C5—21no—
393C2×C4×C5—21no—
394C2×C4×C5#23512yes—
395C2×C4×C5D3×D411yescharacteristic
396C2×C4×C5D3×Q812yes—
397C2×C4×C5C23×D311yescharacteristic
398C22⋊D5—64no—
399C2×Dic5—31no—
400C2×Dic5—31no—
401C2×Dic5—31no—
402C2×Dic5—31no—
403C2×Dic5—62no—
404C2×Dic5—61no—
405C2×Dic5—61no—
406C2×Dic5—61no—
407C2×Dic5—121no—
408D20—32no—
409D20—61no—
410C4×D5—64no—
411C4×D5—122no—
412C4×D5—122no—
413Dic10—32no—
414Dic10—61no—
415Dic10—122no—
416Dic10—122no—
417C5×C8—21no—
418C5×C8—21no—
419#168—61no—
420#168—61no—
421C2×#41—151no—
422C22×C8—151no—
423C22×C8C2×D1511yescharacteristic
424C4⋊Q8—151no—
425C4⋊Q8—151no—
426#125—151no—
427#125—151no—
428C22⋊Q8—301no—
429C22⋊Q8—301no—
430C4×Q8—151no—
431C4×Q8—151no—
432C4×Q8—301no—
433C4×Q8—301no—
434C4×D4—151no—
435C4×D4—151no—
436C4×D4—301no—
437C4×D4—301no—
438#117—151no—
439#117—301no—
440#117C2×D1511yescharacteristic
441#117C2×D1511yescharacteristic
442C2×C42—151no—
443C2×C42—301no—
444#107—151no—
445#107—151no—
446#107C2×D1511yescharacteristic
447#107C2×D1511yescharacteristic
448#106—151no—
449#106—151no—
450#106C2×D1511yescharacteristic
451#106C2×D1511yescharacteristic
452C4⋊C8—151no—
453C4⋊C8—151no—
454C4⋊C8—151no—
455C4⋊C8—151no—
456#103—151no—
457#103—151no—
458#103—151no—
459#103—151no—
460#102—151no—
461#102—151no—
462#102—151no—
463#102—151no—
464C2×C3×C5—22no—
465C2×C3×C5C2×C22⋊Q811yescharacteristic
466C2×C3×C5C2×#13511yescharacteristic
467C2×C3×C5#57511yescharacteristic
468D15—42no—
469C23×C3D4×D511yescharacteristic
470C22×D3—101no—
471C2×C3×C4—21no—
472C2×C3×C4—21no—
473C2×C3×C4—21no—
474C2×C3×C4—21no—
475C2×C3×C4#70912yes—
476C2×C3×C4D4×D511yescharacteristic
477C2×C3×C4Q8×D512yes—
478C2×C3×C4C23×D511yescharacteristic
479#67—104no—
480C2×Dic3—51no—
481C2×Dic3—51no—
482C2×Dic3—51no—
483C2×Dic3—51no—
484C2×Dic3—102no—
485C2×Dic3—101no—
486C2×Dic3—101no—
487C2×Dic3—101no—
488C2×Dic3—201no—
489D12—52no—
490D12—101no—
491C4×D3—104no—
492C4×D3—202no—
493C4×D3—202no—
494Dic6—52no—
495Dic6—101no—
496Dic6—202no—
497Dic6—202no—
498C3×C8—21no—
499C3×C8—21no—
500C3⋊C8—101no—
501C3⋊C8—101no—
502C22×C5—21no—
503C22×C5—21no—
504C22×C5#89612yes—
505C22×C5C2×D3×D411yescharacteristic
506C2×D5—62no—
507C2×D5—121no—
508C2×D5—121no—
509C4×C5—41no—
510C4×C5—41no—
511C4×C5—41no—
512C4×C5—41no—
513C4×C5#89612yes—
514C4×C5C2×#23511yescharacteristic
515C4×C5C2×D3×Q811yescharacteristic
516Dic5—32no—
517Dic5—62no—
518Dic5—61no—
519Dic5—61no—
520Dic5—61no—
521Dic5—121no—
522Dic5—121no—
523#41—154no—
524#41—302no—
525C2×Q8—151no—
526C2×Q8—301no—
527C2×Q8—301no—
528C2×D4—151no—
529C2×D4—301no—
530C22×C4—151no—
531C22×C4—151no—
532C22×C4—301no—
533C22×C4—301no—
534C22×C4—301no—
535C22×C4C22×D1511yescharacteristic
536C2×C8—22no—
537C2×C8—151no—
538C2×C8—151no—
539C2×C8—302no—
540C2×C8C22×D1511yescharacteristic
541C2×C8C22×D1511yescharacteristic
542C4⋊C4—151no—
543C4⋊C4—151no—
544C4⋊C4—301no—
545C4⋊C4—301no—
546C4⋊C4—301no—
547C4⋊C4—301no—
548C4⋊C4—301no—
549C4⋊C4—301no—
550C4⋊C4—301no—
551C4⋊C4—301no—
552C4⋊C4C22×D1511yescharacteristic
553C4⋊C4C22×D1511yescharacteristic
554C4⋊C4C22×D1511yescharacteristic
555C4⋊C4C22×D1511yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—21no—
558C22⋊C4—301no—
559C22⋊C4—301no—
560C22⋊C4—301no—
561C42—21no—
562C42—21no—
563C42—151no—
564C42—151no—
565C42—151no—
566C42—151no—
567C42—301no—
568C42—301no—
569C42—301no—
570C3×C5#307611yescharacteristic, complemented, Hall
571C22×C3—21no—
572C22×C3—21no—
573C22×C3#404212yes—
574C22×C3C2×D4×D511yescharacteristic
575C2×D3—102no—
576C2×D3—201no—
577C2×D3—201no—
578C3×C4—41no—
579C3×C4—41no—
580C3×C4—41no—
581C3×C4—41no—
582C3×C4#404212yes—
583C3×C4C2×#70911yescharacteristic
584C3×C4C2×Q8×D511yescharacteristic
585Dic3—52no—
586Dic3—102no—
587Dic3—101no—
588Dic3—101no—
589Dic3—101no—
590Dic3—201no—
591Dic3—201no—
592C2×C5—22no—
593C2×C5C2×#89611yescharacteristic
594C2×C5D3×#13511yescharacteristic
595C2×C5#576811yescharacteristic
596D5—122no—
597C23—301no—
598C23D4×D1511yescharacteristic
599Q8—152no—
600Q8—301no—
601Q8—602no—
602Q8—602no—
603D4—152no—
604D4—304no—
605D4—301no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—151no—
611C2×C4—151no—
612C2×C4—151no—
613C2×C4—151no—
614C2×C4—302no—
615C2×C4—301no—
616C2×C4—304no—
617C2×C4—301no—
618C2×C4—301no—
619C2×C4—602no—
620C2×C4—602no—
621C2×C4—601no—
622C2×C4#681712yes—
623C2×C4D4×D1511yescharacteristic
624C2×C4Q8×D1512yes—
625C2×C4C23×D1511yescharacteristic
626C8—21no—
627C8—21no—
628C8—301no—
629C8—301no—
630C2×C3—22no—
631C2×C3C2×#404211yescharacteristic
632C2×C3D5×#13511yescharacteristic
633C2×C3#6610211yescharacteristic
634D3—202no—
635C5#7919611yescharacteristic, complemented, Sylow
636C22—21no—
637C22—21no—
638C22—302no—
639C22—601no—
640C22—601no—
641C22#9178512yes—
642C22C2×D4×D1511yescharacteristic
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—152no—
648C4—302no—
649C4—301no—
650C4—301no—
651C4—301no—
652C4—601no—
653C4—601no—
654C4#9178512yes—
655C4C2×#681711yescharacteristic
656C4C2×Q8×D1511yescharacteristic
657C3#1061092711yescharacteristic, complemented, Sylow
658C2—22no—
659C2—602no—
660C2C2×#9178511yescharacteristic
661C2D15×#13511yescharacteristic
662C2#1175652711yescharacteristic
subgroupG/H
centerC2×C4D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#427C2
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
hypercenter#427D15
hypocenterC3×C5#3076
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
2. #11756276C2semidirect
3. C3×C5×#427C2semidirect
5. C2×#91098C2semidirect
7. #11751099C2semidirect
8. #11750992C2semidirect
9. #11750991C2semidirect
10. #11750978C2semidirect
11. #11750976C2semidirect
12. #11750972C2semidirect
13. #11750971C2semidirect
14. #11750958C2semidirect
15. #11750955C2semidirect
63. C5×#427D3semidirect
81. #6817Q8semidirect
125. C3×#427D5semidirect
316. #427D15semidirect
570. C3×C5#3076semidirect
635. C5#79196semidirect
657. C3#10610927semidirect

Maximal subgroups

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

subgroupindex
#117562812
#117562762
C3×C5×#4272
#117512142
C2×#910982
#117511032
#117510992
#117509922
#117509912
#117509782
#117509762
#117509722
#117509712
#117509582
#117509552
#106109273
#791965