Finite Group Databasebeta

← #14300567

Subgroups — #14300567

Order 1920.

752 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65858C211yesmaximal, characteristic
2C2×D5×#208C211yesmaximal, characteristic, complemented
3C4×#6665C211yesmaximal, characteristic, complemented
4C2×#90977C211yesmaximal, characteristic, complemented
5C2×#90976C211yesmaximal, characteristic, complemented
6#11746353C211yesmaximal, characteristic, complemented
7C5×#4532C211yesmaximal, characteristic, complemented
8#10599548—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×C4×Dic15C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C2×C4×C5×Dic3C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C3×#4043C411yescharacteristic, complemented
17C3×#4043C411yescharacteristic, complemented
18C3×#4041C411yescharacteristic, complemented
19C3×#4041C411yescharacteristic, complemented
20C3×#4032—21no—
21C3×#4028—21no—
22C3×C4×D20—21no—
23C3×C4×D20—21no—
24C2×#6665—21no—
25C2×#6649—21no—
26D5×#208—22no—
27D5×#208—22no—
28C2×D5×C3⋊C8—21no—
29#90994—21no—
30#90977—22no—
31#90976—22no—
32#71713—51nomaximal, Hall
33C2×D5×M16—31no—
34#65858D311yescharacteristic
35C2×#3957—31no—
36C4×#685—31no—
37C2×#3937—31no—
38C5×#339—31no—
39#64752—31no—
40C23×C3×D5D411yescharacteristic
41C22×C3×C4×C5C2311yescharacteristic
42C22×Dic15—21no—
43C22×C5×Dic3—21no—
44C2×C3×C22⋊D5—21no—
45C22×C3×Dic5D411yescharacteristic
46C3×#709—21no—
47C3×#709—21no—
48C3×#709—21no—
49C3×#709—21no—
50C2×C3×D20—21no—
51C2×C3×D20—21no—
52C2×C3×D20C2×C411yescharacteristic
53C2×C3×C4×D5—21no—
54C2×C3×C4×D5—21no—
55C2×C3×C4×D5—21no—
56C2×C3×C4×D5—21no—
57C2×C3×C4×D5—21no—
58C2×C3×C4×D5C2×C411yescharacteristic
59C2×C3×C4×D5C2×C411yescharacteristic
60C2×C3×C4×D5D411yescharacteristic
61C2×C3×C4×D5D411yescharacteristic
62C2×C3×Dic10C2×C411yescharacteristic
63C3×C5×C4⋊C4C2×C411yescharacteristic
64C3×C5×C4⋊C4C2×C411yescharacteristic
65C3×C5×C22⋊C4—21no—
66C3×C42×C5—21no—
67C4×Dic15—21no—
68C4×Dic15—21no—
69#6710—21no—
70#6710D412yescomplemented
71C2×C3⋊#168—21no—
72C4×C5×Dic3—21no—
73C4×C5×Dic3—21no—
74C5×#208—21no—
75C5×#208D412yescomplemented
76C2×C5×C3⋊C8—21no—
77C3×#685—21no—
78C3×#685—21no—
79C3×#685—21no—
80C3×#685—21no—
81C3×C4⋊Dic5—21no—
82C3×C4⋊Dic5—21no—
83#6665—24no—
84#6649—24no—
85D5×C3⋊C8—24no—
86C3×#525—51no—
87C22×#208—51no—
88C2×#837—51no—
89C2×#837—51no—
90C4×#217—51no—
91#4532—51no—
92#4532D511yescharacteristic, complemented
93C2×#709C2×D311yescharacteristic
94C22×C4×D5C2×D311yescharacteristic
95C2×C5×M16—31no—
96C5×#117C2×D311yescharacteristic
97C2×C42×C5—31no—
98C2×#685—61no—
99C2×C4×Dic5—31no—
100C2×#681—31no—
101D5×M16—62no—
102D5×M16—62no—
103C2×#676—61no—
104C2×C8×D5—61no—
105#4043Dic311yescharacteristic, complemented
106#4043Dic311yescharacteristic, complemented
107#4041Dic311yescharacteristic, complemented
108#4041Dic311yescharacteristic, complemented
109#4032—21no—
110#4028—21no—
111C4×D20—21no—
112C4×D20—21no—
113#3963—61no—
114#3957—62no—
115#3937—62no—
116#1871—151noSylow
117C23×C3×C5C2×D411yescharacteristic
118C22×C3×D5—21no—
119C22×C3×D5—21no—
120C22×C3×D5—22no—
121C22×C3×D5—21no—
122C22×C3×D5—41no—
123C22×C3×D5C22⋊C411yescharacteristic
124C22×C3×D5C22⋊C411yescharacteristic
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5C22×C411yescharacteristic
128C2×C3×C4×C5C2×D411yescharacteristic
129C2×C3×C4×C5C2×D411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5#4111yescharacteristic
132C2×C3×C4×C5#4111yescharacteristic
133C2×Dic15—22no—
134C2×Dic15—41no—
135C2×Dic15—41no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—41no—
138C2×C5×Dic3—41no—
139C3×C22⋊D5—41no—
140C3×C22⋊D5—41no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—21no—
143C2×C3×Dic5—21no—
144C2×C3×Dic5C22⋊C411yescharacteristic
145C2×C3×Dic5C22⋊C411yescharacteristic
146C3×D20—21no—
147C3×D20—21no—
148C3×D20—42no—
149C3×D20—42no—
150C3×C4×D5—21no—
151C3×C4×D5—21no—
152C3×C4×D5—21no—
153C3×C4×D5—21no—
154C3×C4×D5—21no—
155C3×C4×D5—21no—
156C3×C4×D5—21no—
157C3×C4×D5—21no—
158C3×C4×D5—41no—
159C3×C4×D5—41no—
160C3×Dic10—21no—
161C3×Dic10—21no—
162C3⋊#168—22no—
163C5×C3⋊C8—22no—
164C2×C3×#41—51no—
165C23×C3×C4—51no—
166C3×#123—101no—
167C3×C22⋊Q8—51no—
168C3×C22⋊Q8—51no—
169C3×#121—51no—
170C3×#121—51no—
171C3×#120—101no—
172C3×C4×D4—101no—
173C3×C4×D4—101no—
174C3×#117C2×D511yescharacteristic
175C2×#217—101no—
176C2×C4×Dic3—51no—
177C2×C4×Dic3C2×D511yescharacteristic
178C2×#208—51no—
179C2×#208—101no—
180C2×#208—102no—
181C2×#208—102no—
182C2×#208C2×D511yescharacteristic
183C22×C3⋊C8—101no—
184#837—102no—
185#837—102no—
186C3⋊#95—101no—
187C23×D5#6711yescharacteristic
188C22×C4×C5—61no—
189C22×C4×C5C22×D311yescharacteristic
190C2×C22⋊D5—21no—
191C22×Dic5—61no—
192C22×Dic5#6711yescharacteristic
193#709—21no—
194#709—21no—
195#709—21no—
196#709—21no—
197C2×D20—21no—
198C2×D20—21no—
199C2×D20C2×Dic311yescharacteristic
200C2×C4×D5—21no—
201C2×C4×D5—21no—
202C2×C4×D5—21no—
203C2×C4×D5—21no—
204C2×C4×D5—21no—
205C2×C4×D5C2×Dic311yescharacteristic
206C2×C4×D5C2×Dic311yescharacteristic
207C2×C4×D5#6711yescharacteristic
208C2×C4×D5#6711yescharacteristic
209C2×Dic10C2×Dic311yescharacteristic
210C5×M16—32no—
211C5×M16—61no—
212C2×C5×C8—61no—
213C5×C4⋊C4C2×Dic311yescharacteristic
214C5×C4⋊C4C2×Dic311yescharacteristic
215C5×C22⋊C4—21no—
216C42×C5—21no—
217C42×C5—61no—
218C42×C5—61no—
219#685—21no—
220#685—21no—
221#685—21no—
222#685—21no—
223#685—64no—
224C4⋊Dic5—21no—
225C4⋊Dic5—21no—
226C4×Dic5—61no—
227C4×Dic5—61no—
228#681—32no—
229#681—61no—
230C2×#168—61no—
231#676—64no—
232C8×D5—64no—
233C22×M16—151no—
234#525—51no—
235C2×#104—151no—
236C2×#104—151no—
237C4×C22⋊C4—151no—
238#339—31no—
239#339—151no—
240C22×C3×C5—21no—
241C22×C3×C5—21no—
242C22×C3×C5C2×C22⋊C411yescharacteristic
243C22×C3×C5#12111yescharacteristic
244C22×C3×C5#12111yescharacteristic
245C2×C3×D5—21no—
246C2×C3×D5—21no—
247C2×C3×D5—21no—
248C2×C3×D5—21no—
249C2×C3×D5—21no—
250C2×C3×D5—42no—
251C2×C3×D5—42no—
252C2×C3×D5—41no—
253C2×C3×D5—41no—
254C2×C3×D5—41no—
255C2×C3×D5#10412yescomplemented
256C3×C4×C5—41no—
257C3×C4×C5—41no—
258C3×C4×C5C4×D411yescharacteristic
259C3×C4×C5C4×D411yescharacteristic
260C3×C4×C5#12011yescharacteristic
261C3×C4×C5#12311yescharacteristic
262Dic15—42no—
263C3×Dic5—21no—
264C3×Dic5—21no—
265C3×Dic5—41no—
266C5×Dic3—42no—
267C24×C3—51no—
268C3×#41—101no—
269C3×#41—101no—
270C3×#41—101no—
271C3×#41—101no—
272C2×C3×Q8—51no—
273C2×C3×D4—51no—
274C2×C3×D4—101no—
275C2×C3×D4—101no—
276C2×C3×D4—101no—
277C22×C3×C4—51no—
278C22×C3×C4—51no—
279C22×C3×C4—51no—
280C22×C3×C4—51no—
281C22×C3×C4—51no—
282C22×C3×C4—101no—
283C22×C3×C4—101no—
284C22×C3×C4—101no—
285C22×C3×C4—101no—
286C22×C3×C4—101no—
287C22×C3×C4C22×D511yescharacteristic
288C22×Dic3—21no—
289C22×Dic3—101no—
290C3×C4⋊C4—101no—
291C3×C4⋊C4—101no—
292C3×C4⋊C4C4×D511yescharacteristic
293C3×C4⋊C4C4×D511yescharacteristic
294C3×C22⋊C4—21no—
295C3×C22⋊C4—101no—
296C3×C22⋊C4—101no—
297C3×C22⋊C4—101no—
298C3×C22⋊C4—101no—
299C3×C42—21no—
300#217—104no—
301C4×Dic3—21no—
302C4×Dic3—21no—
303C4×Dic3—101no—
304C4×Dic3—101no—
305#208—21no—
306#208—52no—
307#208—104no—
308#208—101no—
309#208D2012yescomplemented
310C2×C3⋊C8—21no—
311C2×C3⋊C8—104no—
312C2×C3⋊C8—101no—
313C23×C5C2×#6711yescharacteristic
314C22×D5—21no—
315C22×D5—21no—
316C22×D5—22no—
317C22×D5—21no—
318C22×D5—41no—
319C22×D5#21711yescharacteristic
320C22×D5#21711yescharacteristic
321C2×C4×C5—21no—
322C2×C4×C5—21no—
323C2×C4×C5—62no—
324C2×C4×C5—121no—
325C2×C4×C5—121no—
326C2×C4×C5D3×D411yescharacteristic
327C2×C4×C5D3×D411yescharacteristic
328C2×C4×C5#23711yescharacteristic
329C2×C4×C5#23711yescharacteristic
330C2×C4×C5C22×Dic311yescharacteristic
331C2×C4×C5C2×#6711yescharacteristic
332C22⋊D5—41no—
333C22⋊D5—41no—
334C2×Dic5—21no—
335C2×Dic5—21no—
336C2×Dic5—21no—
337C2×Dic5—62no—
338C2×Dic5—121no—
339C2×Dic5—121no—
340C2×Dic5#21711yescharacteristic
341C2×Dic5#21711yescharacteristic
342D20—21no—
343D20—21no—
344D20—42no—
345D20—42no—
346C4×D5—21no—
347C4×D5—21no—
348C4×D5—21no—
349C4×D5—21no—
350C4×D5—21no—
351C4×D5—21no—
352C4×D5—21no—
353C4×D5—21no—
354C4×D5—41no—
355C4×D5—41no—
356Dic10—21no—
357Dic10—21no—
358C5×C8—62no—
359#168—62no—
360C2×#41—51no—
361C23×C4—51no—
362C2×M16—31no—
363C2×M16—151no—
364C2×M16—301no—
365C2×M16—302no—
366C2×M16—302no—
367C22×C8—301no—
368#123—101no—
369C22⋊Q8—51no—
370C22⋊Q8—51no—
371#121—51no—
372#121—51no—
373#120—101no—
374C4×D4—101no—
375C4×D4—101no—
376#117D3×D511yescharacteristic
377C2×C22⋊C4—301no—
378C2×C42—31no—
379C2×C42—151no—
380#104—302no—
381#104—302no—
382#95—301no—
383C2×C3×C5—21no—
384C2×C3×C5—21no—
385C2×C3×C5#38611yescharacteristic
386C2×C3×C5C2×#10411yescharacteristic
387C2×C3×C5#42111yescharacteristic
388C3×D5—22no—
389C3×D5—41no—
390C3×D5—81no—
391C23×C3—51no—
392C23×C3—51no—
393C23×C3—102no—
394C23×C3—101no—
395C23×C3—101no—
396C23×C3—101no—
397C23×C3—201no—
398C23×C3D4×D511yescharacteristic
399C3×Q8—101no—
400C3×Q8—101no—
401C3×D4—101no—
402C3×D4—101no—
403C3×D4—202no—
404C3×D4—202no—
405C3×D4—201no—
406C3×D4—201no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—51no—
410C2×C3×C4—51no—
411C2×C3×C4—101no—
412C2×C3×C4—101no—
413C2×C3×C4—101no—
414C2×C3×C4—101no—
415C2×C3×C4—101no—
416C2×C3×C4—101no—
417C2×C3×C4—101no—
418C2×C3×C4—101no—
419C2×C3×C4—101no—
420C2×C3×C4—101no—
421C2×C3×C4—101no—
422C2×C3×C4—201no—
423C2×C3×C4—201no—
424C2×C3×C4C2×C4×D511yescharacteristic
425C2×C3×C4C2×D2011yescharacteristic
426C2×C3×C4#70911yescharacteristic
427C2×C3×C4D4×D511yescharacteristic
428C2×C3×C4D4×D511yescharacteristic
429C2×C3×C4#71311yescharacteristic
430C2×Dic3—22no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—102no—
434C2×Dic3—201no—
435C2×Dic3—201no—
436C3⋊C8—22no—
437C3⋊C8—102no—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5#93911yescharacteristic
441C22×C5#93911yescharacteristic
442C22×C5C2×#21711yescharacteristic
443C2×D5—21no—
444C2×D5—21no—
445C2×D5—21no—
446C2×D5—21no—
447C2×D5—21no—
448C2×D5—42no—
449C2×D5—42no—
450C2×D5—41no—
451C2×D5—41no—
452C2×D5—41no—
453C2×D5#83712yescomplemented
454C4×C5—41no—
455C4×C5—41no—
456C4×C5—122no—
457C4×C5D4×Dic311yescharacteristic
458C4×C5D4×Dic311yescharacteristic
459C4×C5#93511yescharacteristic
460C4×C5#93711yescharacteristic
461Dic5—21no—
462Dic5—21no—
463Dic5—41no—
464Dic5—122no—
465C24—51no—
466#41—101no—
467#41—101no—
468#41—101no—
469#41—101no—
470C2×Q8—51no—
471C2×D4—51no—
472C2×D4—101no—
473C2×D4—101no—
474C2×D4—101no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—51no—
480C22×C4—61no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4—101no—
484C22×C4—101no—
485C22×C4—101no—
486C22×C4—301no—
487C22×C4C2×D3×D511yescharacteristic
488M16—32no—
489M16—61no—
490M16—152no—
491M16—304no—
492M16—301no—
493C2×C8—61no—
494C2×C8—304no—
495C2×C8—301no—
496C4⋊C4—101no—
497C4⋊C4—101no—
498C4⋊C4D5×Dic311yescharacteristic
499C4⋊C4D5×Dic311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—101no—
502C22⋊C4—101no—
503C22⋊C4—101no—
504C22⋊C4—101no—
505C22⋊C4—304no—
506C42—21no—
507C42—61no—
508C42—61no—
509C42—301no—
510C42—301no—
511C3×C5#187111yescharacteristic, complemented, Hall
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—52no—
515C22×C3—101no—
516C22×C3—101no—
517C22×C3—101no—
518C22×C3—101no—
519C22×C3—101no—
520C22×C3—202no—
521C22×C3—202no—
522C22×C3—201no—
523C22×C3—201no—
524C22×C3—201no—
525C22×C3D5×C22⋊C411yescharacteristic
526C22×C3#403011yescharacteristic
527C22×C3#404111yescharacteristic
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—101no—
531C3×C4—101no—
532C3×C4—201no—
533C3×C4C4×D2011yescharacteristic
534C3×C4#402811yescharacteristic
535C3×C4#403911yescharacteristic
536C3×C4#404011yescharacteristic
537Dic3—42no—
538Dic3—202no—
539C2×C5—21no—
540C2×C5—21no—
541C2×C5#522011yescharacteristic
542C2×C5C2×#83711yescharacteristic
543C2×C5#523611yescharacteristic
544D5—22no—
545D5—41no—
546D5—81no—
547C23—51no—
548C23—51no—
549C23—102no—
550C23—101no—
551C23—101no—
552C23—101no—
553C23—201no—
554C23D5×#6711yescharacteristic
555Q8—101no—
556Q8—101no—
557D4—101no—
558D4—101no—
559D4—202no—
560D4—202no—
561D4—201no—
562D4—201no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—51no—
566C2×C4—51no—
567C2×C4—62no—
568C2×C4—101no—
569C2×C4—101no—
570C2×C4—101no—
571C2×C4—101no—
572C2×C4—101no—
573C2×C4—101no—
574C2×C4—101no—
575C2×C4—101no—
576C2×C4—101no—
577C2×C4—101no—
578C2×C4—101no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—201no—
582C2×C4—201no—
583C2×C4—302no—
584C2×C4—601no—
585C2×C4—601no—
586C2×C4#676511yescharacteristic
587C2×C4#676611yescharacteristic
588C2×C4D3×D2011yescharacteristic
589C2×C4#677711yescharacteristic
590C2×C4C2×D5×Dic311yescharacteristic
591C2×C4D5×#6711yescharacteristic
592C8—62no—
593C8—302no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—102no—
597C2×C3—201no—
598C2×C3—401no—
599C2×C3#6495411yescharacteristic
600C2×C3D5×#10411yescharacteristic
601C2×C3#6511011yescharacteristic
602C5#7171311yescharacteristic, complemented, Sylow
603C22—21no—
604C22—21no—
605C22—52no—
606C22—101no—
607C22—101no—
608C22—101no—
609C22—101no—
610C22—101no—
611C22—202no—
612C22—202no—
613C22—201no—
614C22—201no—
615C22—201no—
616C22#9140811yescharacteristic
617C22#9140911yescharacteristic
618C22D5×#21711yescharacteristic
619C4—41no—
620C4—41no—
621C4—101no—
622C4—101no—
623C4—122no—
624C4—201no—
625C4—602no—
626C4#9140711yescharacteristic
627C4Dic3×D2011yescharacteristic
628C4#9143411yescharacteristic
629C4#9147411yescharacteristic
630C3#1059954811yescharacteristic, complemented, Sylow
631C2—21no—
632C2—21no—
633C2—102no—
634C2—201no—
635C2—401no—
636C2#1174889611yescharacteristic
637C2D5×#83711yescharacteristic
638C2#1174942711yescharacteristic
subgroupG/H
centerC2×C4D5×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#65858C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1871
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×D5×#208C2semidirect
3. C4×#6665C2semidirect
4. C2×#90977C2semidirect
5. C2×#90976C2semidirect
6. #11746353C2semidirect
7. C5×#4532C2semidirect
16. C3×#4043C4semidirect
17. C3×#4043C4semidirect
18. C3×#4041C4semidirect
19. C3×#4041C4semidirect
70. #6710D4semidirect
75. C5×#208D4semidirect
92. #4532D5semidirect
105. #4043Dic3semidirect
106. #4043Dic3semidirect
107. #4041Dic3semidirect
108. #4041Dic3semidirect
255. C2×C3×D5#104semidirect
309. #208D20semidirect
453. C2×D5#837semidirect
511. C3×C5#1871semidirect
602. C5#71713semidirect
630. C3#10599548semidirect

Maximal subgroups

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

subgroupindex
C3×#658582
C2×D5×#2082
C4×#66652
C2×#909772
C2×#909762
#117463532
C5×#45322
#105995483
#717135