Finite Group Databasebeta

← #14430215

Subgroups — #14430215

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#41×Dic15C211yesmaximal, characteristic
2#11756276C211yesmaximal, characteristic, complemented
3C3×C5×#416C211yesmaximal, characteristic
4#11751334C211yesmaximal, characteristic
5#11751207C211yesmaximal, characteristic, complemented
6C2×#91098C211yesmaximal, characteristic, complemented
7#11751099C211yesmaximal, characteristic, complemented
8#11750900C211yesmaximal, characteristic, complemented
9#11750897C211yesmaximal, characteristic, complemented
10#11750894C211yesmaximal, characteristic, complemented
11#11750892C211yesmaximal, characteristic, complemented
12#11750867C211yesmaximal, characteristic, complemented
13#11750864C211yesmaximal, characteristic, complemented
14#11750862C211yesmaximal, characteristic, complemented
15#11750860C211yesmaximal, characteristic, complemented
16#10610553—31nomaximal, Hall
17C2×C3×C5×#41C2211yescharacteristic
18C22×C3×C5×C8C2211yescharacteristic
19C3×C5×#117C2211yescharacteristic
20Q8×Dic15C2211yescharacteristic
21Q8×Dic15C2211yescharacteristic
22D4×Dic15—21no—
23D4×Dic15—21no—
24D4×Dic15C2211yescharacteristic
25D4×Dic15C2211yescharacteristic
26#91815—21no—
27#91815C2211yescharacteristic
28#91815C2211yescharacteristic
29C2×C4×Dic15—21no—
30C2×C4×Dic15C2211yescharacteristic
31C22×C3⋊#168C2211yescharacteristic
32#91779C2211yescharacteristic
33#91779C2211yescharacteristic
34#91777C2211yescharacteristic
35#91777C2211yescharacteristic
36#91767—21no—
37#91767—21no—
38C4×Dic30—21no—
39C4×Dic30—21no—
40C3×C5×#103C2211yescharacteristic
41C3×C5×#103C2211yescharacteristic
42C3×C5×#102C2211yescharacteristic
43C3×C5×#102C2211yescharacteristic
44#91119C2211yescharacteristic
45#91119C2211yescharacteristic
46#91116C2211yescharacteristic
47#91116C2211yescharacteristic
48#91102C2211yescharacteristic
49#91102C2211yescharacteristic
50#91101C2211yescharacteristic
51#91101C2211yescharacteristic
52#91098C2211yescharacteristic
53#91098C2211yescharacteristic
54#91098C2211yescharacteristic
55#91098C2211yescharacteristic
56#91092C2211yescharacteristic
57#91092C2211yescharacteristic
58#91091C2211yescharacteristic
59#91091C2211yescharacteristic
60#78822—51nomaximal, Hall
61#41×Dic5—31no—
62#65851—31no—
63C5×#416D311yescharacteristic
64#65522—31no—
65#65395—31no—
66C2×#3946—31no—
67#65287—31no—
68#65088—31no—
69#65085—31no—
70#65083—31no—
71#65080—31no—
72#65055—31no—
73#65052—31no—
74#65050—31no—
75#65048—31no—
76C3×C5×#41—22no—
77C3×C5×#41Q814yescomplemented
78C2×C3×C5×Q8C2311yescharacteristic
79C2×C3×C5×D4—21no—
80C2×C3×C5×D4C2311yescharacteristic
81C22×C3×C4×C5—21no—
82C22×C3×C4×C5C2311yescharacteristic
83C22×Dic15—21no—
84C22×Dic15—21no—
85C22×Dic15D411yescharacteristic
86C22×Dic15D411yescharacteristic
87C2×Dic30—21no—
88C2×Dic30—21no—
89C2×C3×C5×C8—22no—
90C2×C3×C5×C8C2311yescharacteristic
91C2×C3×C5×C8C2311yescharacteristic
92C3×C5×C4⋊C4C2311yescharacteristic
93C3×C5×C4⋊C4C2311yescharacteristic
94C3×C5×C22⋊C4—21no—
95C3×C42×C5—21no—
96#6719—21no—
97#6719—21no—
98#6719—21no—
99#6719—21no—
100C4⋊Dic15—21no—
101C4⋊Dic15—21no—
102C4⋊Dic15—21no—
103C4⋊Dic15—21no—
104C4⋊Dic15C2311yescharacteristic
105C4⋊Dic15C2311yescharacteristic
106C4⋊Dic15C2311yescharacteristic
107C4⋊Dic15C2311yescharacteristic
108#6712—21no—
109#6712—21no—
110#6712—21no—
111#6712—21no—
112C4×Dic15—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
123Dic3×#41—51no—
124#5517—51no—
125C3×#416D511yescharacteristic
126#5233—51no—
127#5106—51no—
128C2×Dic3⋊C8—51no—
129#4998—51no—
130#4799—51no—
131#4796—51no—
132#4793—51no—
133#4791—51no—
134#4766—51no—
135#4763—51no—
136#4761—51no—
137#4759—51no—
138C2×C5×#41C2×D311yescharacteristic
139C22×C5×C8C2×D311yescharacteristic
140C5×#117C2×D311yescharacteristic
141Q8×Dic5—31no—
142Q8×Dic5—31no—
143D4×Dic5—31no—
144D4×Dic5—31no—
145D4×Dic5—61no—
146D4×Dic5—61no—
147#4072—31no—
148#4072—31no—
149#4072—61no—
150C2×C4×Dic5—31no—
151C2×C4×Dic5—61no—
152C22×#168—31no—
153#4036—31no—
154#4036—31no—
155#4034—31no—
156#4034—31no—
157#4024—61no—
158#4024—61no—
159C4×Dic10—61no—
160C4×Dic10—61no—
161C5×#103C2×D311yescharacteristic
162C5×#103C2×D311yescharacteristic
163C5×#102C2×D311yescharacteristic
164C5×#102C2×D311yescharacteristic
165#3967—31no—
166#3967—31no—
167#3964—31no—
168#3964—31no—
169#3950—31no—
170#3950—31no—
171#3949—31no—
172#3949—31no—
173#3946—31no—
174#3946—31no—
175#3946—31no—
176#3946—31no—
177#3940—31no—
178#3940—31no—
179#3939—31no—
180#3939—31no—
181#3076—151noSylow
182C23×C3×C5—21no—
183C23×C3×C5C2×D411yescharacteristic
184C3×C5×Q8—21no—
185C3×C5×Q8C2×Q812yes—
186C3×C5×D4—24no—
187C3×C5×D4—21no—
188C3×C5×D4C2×Q812yes—
189C2×C3×C4×C5—21no—
190C2×C3×C4×C5—24no—
191C2×C3×C4×C5—21no—
192C2×C3×C4×C5—21no—
193C2×C3×C4×C5C2×D411yescharacteristic
194C2×C3×C4×C5C2×Q812yes—
195C2×C3×C4×C5#4112yes—
196C2×C3×C4×C5C2411yescharacteristic
197C2×Dic15—22no—
198C2×Dic15—21no—
199C2×Dic15—21no—
200C2×Dic15—21no—
201C2×Dic15—21no—
202C2×Dic15—41no—
203C2×Dic15—42no—
204C2×Dic15—42no—
205C2×Dic15C2×D411yescharacteristic
206C2×Dic15C2×D411yescharacteristic
207C2×Dic15C2×D411yescharacteristic
208C2×Dic15C2×D411yescharacteristic
209Dic30—42no—
210Dic30—42no—
211C3×C5×C8—21no—
212C3×C5×C8—21no—
213C3⋊#168—21no—
214C3⋊#168—21no—
215C2×C3×#41C2×D511yescharacteristic
216C22×C3×C8C2×D511yescharacteristic
217C3×#117C2×D511yescharacteristic
218Q8×Dic3—51no—
219Q8×Dic3—51no—
220D4×Dic3—51no—
221D4×Dic3—51no—
222D4×Dic3—101no—
223D4×Dic3—101no—
224#926—51no—
225#926—51no—
226#926—101no—
227C2×C4×Dic3—51no—
228C2×C4×Dic3—101no—
229C22×C3⋊C8—51no—
230#890—51no—
231#890—51no—
232#888—51no—
233#888—51no—
234#878—101no—
235#878—101no—
236C4×Dic6—101no—
237C4×Dic6—101no—
238C3×#103C2×D511yescharacteristic
239C3×#103C2×D511yescharacteristic
240C3×#102C2×D511yescharacteristic
241C3×#102C2×D511yescharacteristic
242#835—51no—
243#835—51no—
244#832—51no—
245#832—51no—
246#818—51no—
247#818—51no—
248#817—51no—
249#817—51no—
250Dic3⋊C8—51no—
251Dic3⋊C8—51no—
252Dic3⋊C8—51no—
253Dic3⋊C8—51no—
254#808—51no—
255#808—51no—
256#807—51no—
257#807—51no—
258C5×#41—22no—
259C5×#41Dic614yescomplemented
260C2×C5×Q8C22×D311yescharacteristic
261C2×C5×D4—21no—
262C2×C5×D4C22×D311yescharacteristic
263C22×C4×C5—21no—
264C22×C4×C5C22×D311yescharacteristic
265C22×Dic5—31no—
266C22×Dic5—31no—
267C22×Dic5—61no—
268C22×Dic5—61no—
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×C22⋊C4—21no—
277C42×C5—21no—
278C22⋊Dic5—61no—
279C22⋊Dic5—61no—
280C22⋊Dic5—61no—
281C22⋊Dic5—61no—
282C4⋊Dic5—31no—
283C4⋊Dic5—31no—
284C4⋊Dic5—31no—
285C4⋊Dic5—31no—
286C4⋊Dic5—61no—
287C4⋊Dic5—61no—
288C4⋊Dic5—61no—
289C4⋊Dic5—61no—
290#683—61no—
291#683—61no—
292#683—61no—
293#683—61no—
294C4×Dic5—31no—
295C4×Dic5—31no—
296C4×Dic5—31no—
297C4×Dic5—31no—
298C4×Dic5—61no—
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#427—151no—
317C2×C4⋊C8—151no—
318#416—151no—
319#416D1511yescharacteristic
320C22×C3×C5—21no—
321C22×C3×C5—21no—
322C22×C3×C5—22no—
323C22×C3×C5—41no—
324C22×C3×C5—41no—
325C22×C3×C5C22⋊Q812yes—
326C22×C3×C5C22×D411yescharacteristic
327C3×C4×C5—22no—
328C3×C4×C5—41no—
329C3×C4×C5—41no—
330C3×C4×C5C22⋊Q812yes—
331C3×C4×C5C22×Q811yescharacteristic
332C3×C4×C5C2×#4111yescharacteristic
333Dic15—21no—
334Dic15—21no—
335Dic15—21no—
336Dic15—41no—
337Dic15—41no—
338Dic15—41no—
339Dic15—41no—
340Dic15#13512yes—
341C3×#41—22no—
342C3×#41Dic1014yescomplemented
343C2×C3×Q8C22×D511yescharacteristic
344C2×C3×D4—21no—
345C2×C3×D4C22×D511yescharacteristic
346C22×C3×C4—21no—
347C22×C3×C4C22×D511yescharacteristic
348C22×Dic3—51no—
349C22×Dic3—51no—
350C22×Dic3—101no—
351C22×Dic3—101no—
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×C22⋊C4—21no—
360C3×C42—21no—
361#217—101no—
362#217—101no—
363#217—101no—
364#217—101no—
365C4⋊Dic3—51no—
366C4⋊Dic3—51no—
367C4⋊Dic3—51no—
368C4⋊Dic3—51no—
369C4⋊Dic3—101no—
370C4⋊Dic3—101no—
371C4⋊Dic3—101no—
372C4⋊Dic3—101no—
373Dic3⋊C4—101no—
374Dic3⋊C4—101no—
375Dic3⋊C4—101no—
376Dic3⋊C4—101no—
377C4×Dic3—51no—
378C4×Dic3—51no—
379C4×Dic3—51no—
380C4×Dic3—51no—
381C4×Dic3—101no—
382C4×Dic3—101no—
383C4×Dic3—101no—
384C4×Dic3—101no—
385C2×C3⋊C8—51no—
386C2×C3⋊C8—51no—
387C2×C3⋊C8—102no—
388C23×C5—21no—
389C23×C5D3×D411yescharacteristic
390C5×Q8—21no—
391C5×Q8C2×Dic612yes—
392C5×D4—24no—
393C5×D4—21no—
394C5×D4C2×Dic612yes—
395C2×C4×C5—21no—
396C2×C4×C5—24no—
397C2×C4×C5—21no—
398C2×C4×C5—21no—
399C2×C4×C5C2×Dic612yes—
400C2×C4×C5D3×D411yescharacteristic
401C2×C4×C5#23712yes—
402C2×C4×C5C23×D311yescharacteristic
403C2×Dic5—31no—
404C2×Dic5—31no—
405C2×Dic5—31no—
406C2×Dic5—31no—
407C2×Dic5—62no—
408C2×Dic5—61no—
409C2×Dic5—61no—
410C2×Dic5—61no—
411C2×Dic5—61no—
412C2×Dic5—122no—
413C2×Dic5—121no—
414C2×Dic5—122no—
415Dic10—122no—
416Dic10—122no—
417C5×C8—21no—
418C5×C8—21no—
419#168—61no—
420#168—61no—
421C2×#41C2×D1511yescharacteristic
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—151no—
440#117—301no—
441#117C2×D1511yescharacteristic
442C2×C42—151no—
443C2×C42—301no—
444#107—151no—
445#107—151no—
446#107—151no—
447#107—151no—
448#106—151no—
449#106—151no—
450#106—151no—
451#106—151no—
452C4⋊C8—151no—
453C4⋊C8—151no—
454C4⋊C8—151no—
455C4⋊C8—151no—
456#103—151no—
457#103—151no—
458#103C2×D1511yescharacteristic
459#103C2×D1511yescharacteristic
460#102—151no—
461#102—151no—
462#102C2×D1511yescharacteristic
463#102C2×D1511yescharacteristic
464C2×C3×C5—22no—
465C2×C3×C5—42no—
466C2×C3×C5C2×C22⋊Q811yescharacteristic
467C2×C3×C5C2×#13511yescharacteristic
468C2×C3×C5#57511yescharacteristic
469C23×C3—21no—
470C23×C3D4×D511yescharacteristic
471C3×Q8—21no—
472C3×Q8C2×Dic1012yes—
473C3×D4—24no—
474C3×D4—21no—
475C3×D4C2×Dic1012yes—
476C2×C3×C4—21no—
477C2×C3×C4—24no—
478C2×C3×C4—21no—
479C2×C3×C4—21no—
480C2×C3×C4C2×Dic1012yes—
481C2×C3×C4D4×D511yescharacteristic
482C2×C3×C4#71112yes—
483C2×C3×C4C23×D511yescharacteristic
484C2×Dic3—51no—
485C2×Dic3—51no—
486C2×Dic3—51no—
487C2×Dic3—51no—
488C2×Dic3—102no—
489C2×Dic3—101no—
490C2×Dic3—101no—
491C2×Dic3—101no—
492C2×Dic3—101no—
493C2×Dic3—202no—
494C2×Dic3—201no—
495C2×Dic3—202no—
496Dic6—202no—
497Dic6—202no—
498C3×C8—21no—
499C3×C8—21no—
500C3⋊C8—101no—
501C3⋊C8—101no—
502C22×C5—21no—
503C22×C5—21no—
504C22×C5—22no—
505C22×C5—41no—
506C22×C5—41no—
507C22×C5#87812yes—
508C22×C5C2×D3×D411yescharacteristic
509C4×C5—22no—
510C4×C5—41no—
511C4×C5—41no—
512C4×C5#87812yes—
513C4×C5C22×Dic611yescharacteristic
514C4×C5C2×#23711yescharacteristic
515Dic5—32no—
516Dic5—61no—
517Dic5—61no—
518Dic5—61no—
519Dic5—121no—
520Dic5—121no—
521Dic5—121no—
522Dic5—121no—
523#41—22no—
524#41Dic3014yescomplemented
525C2×Q8—301no—
526C2×Q8—301no—
527C2×Q8C22×D1511yescharacteristic
528C2×D4—21no—
529C2×D4C22×D1511yescharacteristic
530C22×C4—21no—
531C22×C4—151no—
532C22×C4—151no—
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—151no—
545C4⋊C4—151no—
546C4⋊C4—301no—
547C4⋊C4—301no—
548C4⋊C4—301no—
549C4⋊C4—301no—
550C4⋊C4—301no—
551C4⋊C4—301no—
552C4⋊C4—301no—
553C4⋊C4—301no—
554C4⋊C4C22×D1511yescharacteristic
555C4⋊C4C22×D1511yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—301no—
558C22⋊C4—301no—
559C22⋊C4—301no—
560C22⋊C4—301no—
561C42—21no—
562C42—151no—
563C42—151no—
564C42—151no—
565C42—151no—
566C42—301no—
567C42—301no—
568C42—301no—
569C42—301no—
570C3×C5#307611yescharacteristic, complemented, Hall
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—22no—
574C22×C3—41no—
575C22×C3—41no—
576C22×C3#402412yes—
577C22×C3C2×D4×D511yescharacteristic
578C3×C4—22no—
579C3×C4—41no—
580C3×C4—41no—
581C3×C4#402412yes—
582C3×C4C22×Dic1011yescharacteristic
583C3×C4C2×#71111yescharacteristic
584Dic3—52no—
585Dic3—101no—
586Dic3—101no—
587Dic3—101no—
588Dic3—201no—
589Dic3—201no—
590Dic3—201no—
591Dic3—201no—
592C2×C5—22no—
593C2×C5—42no—
594C2×C5C2×#87811yescharacteristic
595C2×C5D3×#13511yescharacteristic
596C2×C5#576811yescharacteristic
597C23—21no—
598C23D4×D1511yescharacteristic
599Q8—21no—
600Q8—602no—
601Q8—602no—
602Q8C2×Dic3012yes—
603D4—24no—
604D4—21no—
605D4C2×Dic3012yes—
606C2×C4—21no—
607C2×C4—24no—
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—301no—
617C2×C4—301no—
618C2×C4—301no—
619C2×C4—602no—
620C2×C4—601no—
621C2×C4—602no—
622C2×C4C2×Dic3012yes—
623C2×C4D4×D1511yescharacteristic
624C2×C4#681912yes—
625C2×C4C23×D1511yescharacteristic
626C8—21no—
627C8—21no—
628C8—301no—
629C8—301no—
630C2×C3—22no—
631C2×C3—42no—
632C2×C3C2×#402411yescharacteristic
633C2×C3D5×#13511yescharacteristic
634C2×C3#6610211yescharacteristic
635C5#7882211yescharacteristic, complemented, Sylow
636C22—21no—
637C22—21no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22#9176712yes—
642C22C2×D4×D1511yescharacteristic
643C4—22no—
644C4—41no—
645C4—41no—
646C4—152no—
647C4—301no—
648C4—301no—
649C4—301no—
650C4—601no—
651C4—601no—
652C4—601no—
653C4—601no—
654C4#9176712yes—
655C4C22×Dic3011yescharacteristic
656C4C2×#681911yescharacteristic
657C3#1061055311yescharacteristic, complemented, Sylow
658C2—22no—
659C2—42no—
660C2C2×#9176711yescharacteristic
661C2D15×#13511yescharacteristic
662C2#1175652711yescharacteristic
subgroupG/H
centerC2×C4D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#416C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×#41C22
hypercenter#416D15
hypocenterC3×C5#3076
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #11756276C2semidirect
5. #11751207C2semidirect
6. C2×#91098C2semidirect
7. #11751099C2semidirect
8. #11750900C2semidirect
9. #11750897C2semidirect
10. #11750894C2semidirect
11. #11750892C2semidirect
12. #11750867C2semidirect
13. #11750864C2semidirect
14. #11750862C2semidirect
15. #11750860C2semidirect
77. C3×C5×#41Q8semidirect
259. C5×#41Dic6semidirect
342. C3×#41Dic10semidirect
524. #41Dic30semidirect
570. C3×C5#3076semidirect
635. C5#78822semidirect
657. C3#10610553semidirect

Maximal subgroups

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

subgroupindex
#41×Dic152
#117562762
C3×C5×#4162
#117513342
#117512072
C2×#910982
#117510992
#117509002
#117508972
#117508942
#117508922
#117508672
#117508642
#117508622
#117508602
#106105533
#788225