Finite Group Databasebeta

← #14300551

Subgroups — #14300551

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
1#11754564C211yesmaximal, characteristic
2C2×D5×#208C211yesmaximal, characteristic, complemented
3C2×#91089C211yesmaximal, characteristic, complemented
4C3×C4×#685C211yesmaximal, characteristic, complemented
5C2×#90981C211yesmaximal, characteristic, complemented
6C5×#4523C211yesmaximal, characteristic, complemented
7#11745977C211yesmaximal, characteristic, complemented
8#10599548—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C2×C3×#685—21no—
16C2×C3×C4×Dic5C2211yescharacteristic
17#91564—21no—
18#91546—21no—
19#91450C411yescharacteristic, complemented
20#91450C411yescharacteristic, complemented
21C4×#1202—21no—
22C4×#1202—21no—
23#91360C411yescharacteristic, complemented
24#91360C411yescharacteristic, complemented
25C2×#6649—21no—
26D5×#208—22no—
27D5×#208—22no—
28C2×D5×C3⋊C8—21no—
29#91089—22no—
30C3×#3963—21no—
31#90981—22no—
32#69453—51nomaximal, Hall
33C2×D5×M16—31no—
34#65858—31no—
35C2×#3957—31no—
36C4×#685D311yescharacteristic, complemented
37C2×#3937—31no—
38C5×#339—31no—
39#64752—31no—
40C23×C3×D5D411yescharacteristic
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C2×#1220—21no—
44#6817—21no—
45#6817—21no—
46#6817—21no—
47#6817—21no—
48C2×D60C2×C411yescharacteristic
49C2×C4×D15—21no—
50C2×Dic30C2×C411yescharacteristic
51C22×C3×Dic5—21no—
52C22×C3×Dic5D411yescharacteristic
53C2×C3×C4×D5—21no—
54C2×C3×C4×D5—21no—
55C2×C3×C4×D5—21no—
56C2×C3×C4×D5—21no—
57C2×C3×C4×D5C2×C411yescharacteristic
58C2×C3×C4×D5C2×C411yescharacteristic
59C2×C3×C4×D5D411yescharacteristic
60C2×C3×C4×D5D411yescharacteristic
61C2×#1202—21no—
62C2×#1202—21no—
63C3×C42×C5—21no—
64C3×C42×C5—21no—
65#6710—21no—
66#6710D412yescomplemented
67C2×C3⋊#168—21no—
68C5×#217—21no—
69C5×C4⋊Dic3C2×C411yescharacteristic
70C5×C4⋊Dic3C2×C411yescharacteristic
71C4×C5×Dic3—21no—
72C5×#208—21no—
73C5×#208D412yescomplemented
74C2×C5×C3⋊C8—21no—
75C3×#685—24no—
76C3×C4×Dic5—21no—
77C3×C4×Dic5—21no—
78Dic3⋊Dic5—21no—
79Dic3⋊Dic5—21no—
80#6667—21no—
81#6667—21no—
82#6665—21no—
83#6665—21no—
84#6649—24no—
85D5×C3⋊C8—24no—
86#5791—51no—
87C22×#208—51no—
88C3×C4×C22⋊C4—51no—
89C2×#805—51no—
90C2×#805—51no—
91#4523—51no—
92#4523D511yescharacteristic, complemented
93C2×#709—31no—
94C22×C4×D5C2×D311yescharacteristic
95C2×C5×M16—31no—
96C5×#117—31no—
97C2×C42×C5C2×D311yescharacteristic
98C2×#685—21no—
99C2×C4×Dic5C2×D311yescharacteristic
100C2×#681—31no—
101D5×M16—62no—
102D5×M16—62no—
103C2×#676—61no—
104C2×C8×D5—61no—
105#4043—31no—
106#4043—31no—
107#4041—31no—
108#4041—31no—
109#4032—61no—
110#4028—61no—
111C4×D20—61no—
112C4×D20—61no—
113#3963—21no—
114#3957—62no—
115#3937—62no—
116#1871—151noSylow
117C23×C3×C5C2×D411yescharacteristic
118C22×D15—21no—
119C22×C3×D5—21no—
120C22×C3×D5—21no—
121C22×C3×D5—22no—
122C22×C3×D5—41no—
123C22×C3×D5C22⋊C411yescharacteristic
124C22×C3×D5C22⋊C411yescharacteristic
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—41no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5C22×C411yescharacteristic
129C2×C3×C4×C5C2×D411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5#4111yescharacteristic
133C2×C3×C4×C5#4111yescharacteristic
134#1220—41no—
135#1220—41no—
136C2×Dic15—21no—
137D60—21no—
138D60—21no—
139C4×D15—41no—
140C4×D15—41no—
141Dic30—21no—
142Dic30—21no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—41no—
149C2×C3×Dic5—41no—
150C2×C3×Dic5C22⋊C411yescharacteristic
151C2×C3×Dic5C22⋊C411yescharacteristic
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—21no—
159C3×C4×D5—21no—
160#1202—42no—
161#1202—42no—
162C3⋊#168—22no—
163C5×C3⋊C8—22no—
164C23×C3×C4—51no—
165C2×#235—51no—
166C2×C3×C22⋊C4—101no—
167C2×C3×C42—51no—
168C2×C3×C42C2×D511yescharacteristic
169#953—101no—
170#930—51no—
171#930—51no—
172#929—101no—
173C4×#67—101no—
174C4×#67—101no—
175#926C2×D511yescharacteristic
176#924—51no—
177#924—51no—
178C2×#208—51no—
179C2×#208—101no—
180C2×#208—102no—
181C2×#208—102no—
182C2×#208C2×D511yescharacteristic
183C22×C3⋊C8—101no—
184C3×#95—101no—
185#805—102no—
186#805—102no—
187C23×D5#6711yescharacteristic
188C22×C4×C5—21no—
189C22×C4×C5C22×D311yescharacteristic
190C2×C22⋊D5—61no—
191C22×Dic5—21no—
192C22×Dic5#6711yescharacteristic
193#709—61no—
194#709—61no—
195#709—61no—
196#709—61no—
197C2×D20—31no—
198C2×D20—61no—
199C2×D20—61no—
200C2×C4×D5—21no—
201C2×C4×D5—21no—
202C2×C4×D5—21no—
203C2×C4×D5—21no—
204C2×C4×D5—61no—
205C2×C4×D5C4×D311yescharacteristic
206C2×C4×D5C4×D311yescharacteristic
207C2×C4×D5D1211yescharacteristic
208C2×C4×D5D1211yescharacteristic
209C2×Dic10—31no—
210C5×M16—32no—
211C5×M16—61no—
212C2×C5×C8—61no—
213C5×C4⋊C4—31no—
214C5×C4⋊C4—31no—
215C5×C22⋊C4—61no—
216C42×C5—21no—
217C42×C5—21no—
218C42×C5—61no—
219#685—24no—
220#685—61no—
221#685—61no—
222#685—61no—
223#685—61no—
224C4⋊Dic5—61no—
225C4⋊Dic5—61no—
226C4×Dic5—21no—
227C4×Dic5—21no—
228#681—32no—
229#681—61no—
230C2×#168—61no—
231#676—64no—
232C8×D5—64no—
233C22×M16—151no—
234#525—151no—
235C2×#104—151no—
236C2×#104—151no—
237C4×C22⋊C4—51no—
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×D15—41no—
246C2×D15—41no—
247C2×D15—41no—
248C2×C3×D5—21no—
249C2×C3×D5—21no—
250C2×C3×D5—21no—
251C2×C3×D5—21no—
252C2×C3×D5—21no—
253C2×C3×D5—42no—
254C2×C3×D5—42no—
255C2×C3×D5#10412yescomplemented
256C3×C4×C5—42no—
257C3×C4×C5C4×D411yescharacteristic
258C3×C4×C5C4×D411yescharacteristic
259C3×C4×C5#12011yescharacteristic
260C3×C4×C5#12311yescharacteristic
261Dic15—41no—
262C3×Dic5—21no—
263C3×Dic5—21no—
264C3×Dic5—42no—
265C5×Dic3—41no—
266C5×Dic3—41no—
267C24×C3—51no—
268C22×C3×C4—21no—
269C22×C3×C4—51no—
270C22×C3×C4—51no—
271C22×C3×C4—51no—
272C22×C3×C4—51no—
273C22×C3×C4—51no—
274C22×C3×C4—101no—
275C22×C3×C4—101no—
276C22×C3×C4—101no—
277C22×C3×C4—101no—
278C22×C3×C4—101no—
279C22×C3×C4C22×D511yescharacteristic
280C2×#67—101no—
281C2×#67—101no—
282C2×#67—101no—
283#235—101no—
284#235—101no—
285#235—101no—
286#235—101no—
287C2×D12—51no—
288C2×C4×D3—101no—
289C2×Dic6—51no—
290C3×C22⋊C4—104no—
291C3×C42—21no—
292C3×C42—21no—
293C3×C42—101no—
294C3×C42—101no—
295#217—21no—
296#217—101no—
297#217—101no—
298(C2×D3)⋊C4—101no—
299(C2×D3)⋊C4—101no—
300C4⋊Dic3C4×D511yescharacteristic
301C4⋊Dic3C4×D511yescharacteristic
302Dic3⋊C4—101no—
303Dic3⋊C4—101no—
304C4×Dic3—21no—
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—41no—
318C22×D5—61no—
319C22×D5(C2×D3)⋊C411yescharacteristic
320C22×D5(C2×D3)⋊C411yescharacteristic
321C2×C4×C5—22no—
322C2×C4×C5—41no—
323C2×C4×C5—41no—
324C2×C4×C5—61no—
325C2×C4×C5—61no—
326C2×C4×C5C2×C4×D311yescharacteristic
327C2×C4×C5C2×D1211yescharacteristic
328C2×C4×C5D3×D411yescharacteristic
329C2×C4×C5D3×D411yescharacteristic
330C2×C4×C5#23711yescharacteristic
331C2×C4×C5#23711yescharacteristic
332C22⋊D5—121no—
333C22⋊D5—121no—
334C2×Dic5—21no—
335C2×Dic5—21no—
336C2×Dic5—22no—
337C2×Dic5—41no—
338C2×Dic5—41no—
339C2×Dic5—61no—
340C2×Dic5(C2×D3)⋊C411yescharacteristic
341C2×Dic5(C2×D3)⋊C411yescharacteristic
342D20—61no—
343D20—61no—
344D20—122no—
345D20—122no—
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—121no—
355C4×D5—121no—
356Dic10—61no—
357Dic10—61no—
358C5×C8—62no—
359#168—62no—
360C2×#41—151no—
361C23×C4—51no—
362C2×M16—31no—
363C2×M16—151no—
364C2×M16—301no—
365C2×M16—302no—
366C2×M16—302no—
367C22×C8—301no—
368#123—301no—
369C22⋊Q8—151no—
370C22⋊Q8—151no—
371#121—151no—
372#121—151no—
373#120—301no—
374C4×D4—301no—
375C4×D4—301no—
376#117—31no—
377C2×C22⋊C4—101no—
378C2×C42—51no—
379C2×C42D3×D511yescharacteristic
380#104—302no—
381#104—302no—
382#95—101no—
383C2×C3×C5—21no—
384C2×C3×C5—21no—
385C2×C3×C5#38611yescharacteristic
386C2×C3×C5C2×#10411yescharacteristic
387C2×C3×C5#42111yescharacteristic
388D15—81no—
389C3×D5—22no—
390C3×D5—41no—
391C23×C3—51no—
392C23×C3—51no—
393C23×C3—102no—
394C23×C3—101no—
395C23×C3—101no—
396C23×C3—201no—
397C23×C3D4×D511yescharacteristic
398C22×D3—101no—
399C2×C3×C4—22no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—51no—
403C2×C3×C4—51no—
404C2×C3×C4—101no—
405C2×C3×C4—101no—
406C2×C3×C4—101no—
407C2×C3×C4—101no—
408C2×C3×C4—101no—
409C2×C3×C4—101no—
410C2×C3×C4—101no—
411C2×C3×C4—101no—
412C2×C3×C4—101no—
413C2×C3×C4—101no—
414C2×C3×C4—102no—
415C2×C3×C4—201no—
416C2×C3×C4—201no—
417C2×C3×C4C2×C4×D511yescharacteristic
418C2×C3×C4C2×D2011yescharacteristic
419C2×C3×C4#70911yescharacteristic
420C2×C3×C4D4×D511yescharacteristic
421C2×C3×C4D4×D511yescharacteristic
422C2×C3×C4#71311yescharacteristic
423#67—202no—
424#67—202no—
425#67—201no—
426#67—201no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—101no—
430D12—101no—
431D12—101no—
432C4×D3—201no—
433C4×D3—201no—
434Dic6—101no—
435Dic6—101no—
436C3⋊C8—22no—
437C3⋊C8—102no—
438C22×C5—21no—
439C22×C5—21no—
440C22×C5C2×(C2×D3)⋊C411yescharacteristic
441C22×C5#93911yescharacteristic
442C22×C5#93911yescharacteristic
443C2×D5—21no—
444C2×D5—21no—
445C2×D5—21no—
446C2×D5—21no—
447C2×D5—21no—
448C2×D5—42no—
449C2×D5—42no—
450C2×D5—121no—
451C2×D5—121no—
452C2×D5—121no—
453C2×D5#80512yescomplemented
454C4×C5—42no—
455C4×C5—121no—
456C4×C5—121no—
457C4×C5#67⋊C411yescharacteristic
458C4×C5#67⋊C411yescharacteristic
459C4×C5#88211yescharacteristic
460C4×C5#88611yescharacteristic
461Dic5—21no—
462Dic5—21no—
463Dic5—42no—
464Dic5—121no—
465C24—51no—
466#41—301no—
467#41—301no—
468#41—301no—
469#41—301no—
470C2×Q8—151no—
471C2×D4—151no—
472C2×D4—301no—
473C2×D4—301no—
474C2×D4—301no—
475C22×C4—21no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—51no—
480C22×C4—51no—
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—31no—
497C4⋊C4—31no—
498C4⋊C4—301no—
499C4⋊C4—301no—
500C22⋊C4—61no—
501C22⋊C4—104no—
502C22⋊C4—301no—
503C22⋊C4—301no—
504C22⋊C4—301no—
505C22⋊C4—301no—
506C42—21no—
507C42—21no—
508C42—61no—
509C42—101no—
510C42—101no—
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×C3D5×C22⋊C411yescharacteristic
523C22×C3#403011yescharacteristic
524C22×C3#404111yescharacteristic
525C2×D3—201no—
526C2×D3—201no—
527C2×D3—201no—
528C3×C4—42no—
529C3×C4—101no—
530C3×C4—101no—
531C3×C4—202no—
532C3×C4C4×D2011yescharacteristic
533C3×C4#402811yescharacteristic
534C3×C4#403911yescharacteristic
535C3×C4#404011yescharacteristic
536Dic3—41no—
537Dic3—41no—
538Dic3—201no—
539C2×C5—21no—
540C2×C5—21no—
541C2×C5C2×#80511yescharacteristic
542C2×C5#495811yescharacteristic
543C2×C5#500511yescharacteristic
544D5—22no—
545D5—41no—
546D5—241no—
547C23—51no—
548C23—51no—
549C23—102no—
550C23—101no—
551C23—101no—
552C23—201no—
553C23—301no—
554C23D5×#6711yescharacteristic
555Q8—301no—
556Q8—301no—
557D4—301no—
558D4—301no—
559D4—602no—
560D4—602no—
561D4—601no—
562D4—601no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—51no—
567C2×C4—51no—
568C2×C4—61no—
569C2×C4—61no—
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—101no—
580C2×C4—102no—
581C2×C4—201no—
582C2×C4—201no—
583C2×C4—301no—
584C2×C4—601no—
585C2×C4—601no—
586C2×C4#676611yescharacteristic
587C2×C4C4×D3×D511yescharacteristic
588C2×C4D5×D1211yescharacteristic
589C2×C4D3×D2011yescharacteristic
590C2×C4#677911yescharacteristic
591C2×C4#679011yescharacteristic
592C8—62no—
593C8—302no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—102no—
597C2×C3—201no—
598C2×C3#6495411yescharacteristic
599C2×C3D5×#10411yescharacteristic
600C2×C3#6511011yescharacteristic
601D3—401no—
602C5#6945311yescharacteristic, complemented, Sylow
603C22—21no—
604C22—21no—
605C22—52no—
606C22—101no—
607C22—101no—
608C22—101no—
609C22—101no—
610C22—101no—
611C22—202no—
612C22—202no—
613C22—601no—
614C22—601no—
615C22—601no—
616C22#9140911yescharacteristic
617C22D5×(C2×D3)⋊C411yescharacteristic
618C22#9155011yescharacteristic
619C4—42no—
620C4—101no—
621C4—101no—
622C4—121no—
623C4—121no—
624C4—202no—
625C4—601no—
626C4#9139511yescharacteristic
627C4#9139611yescharacteristic
628C4#9141311yescharacteristic
629C4#9147611yescharacteristic
630C3#1059954811yescharacteristic, complemented, Sylow
631C2—21no—
632C2—21no—
633C2—102no—
634C2—201no—
635C2—1201no—
636C2D5×#80511yescharacteristic
637C2#1174710911yescharacteristic
638C2#1174887211yescharacteristic
subgroupG/H
centerC2×C4D5×D12
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5C2×C22⋊C4
prime radical#11754564C2
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1871
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×D5×#208C2semidirect
3. C2×#91089C2semidirect
4. C3×C4×#685C2semidirect
5. C2×#90981C2semidirect
6. C5×#4523C2semidirect
7. #11745977C2semidirect
19. #91450C4semidirect
20. #91450C4semidirect
23. #91360C4semidirect
24. #91360C4semidirect
36. C4×#685D3semidirect
66. #6710D4semidirect
73. C5×#208D4semidirect
92. #4523D5semidirect
255. C2×C3×D5#104semidirect
309. #208D20semidirect
453. C2×D5#805semidirect
511. C3×C5#1871semidirect
602. C5#69453semidirect
630. C3#10599548semidirect

Maximal subgroups

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

subgroupindex
#117545642
C2×D5×#2082
C2×#910892
C3×C4×#6852
C2×#909812
C5×#45232
#117459772
#105995483
#694535