Finite Group Databasebeta

← #14513640

Subgroups — #14513640

Order 1920.

726 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×SL(2,ℤ/4)C211yesmaximal, characteristic
2C2×#91899C211yesmaximal, characteristic, complemented
3A4×C22⋊Dic5C211yesmaximal, characteristic
4#11752151C211yesmaximal, characteristic
5#11751636C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊Dic5C211yesmaximal, characteristic, complemented
7#11751634C211yesmaximal, characteristic, complemented
8#10613841—31nomaximal, Hall
9C23×C5×A4C2211yescharacteristic
10C2×A4×Dic5C2211yescharacteristic
11C2×A4×Dic5C2211yescharacteristic
12C2×A4⋊Dic5C2211yescharacteristic
13C2×A4⋊Dic5C2211yescharacteristic
14C2×C5×SL(2,ℤ/4)—22no—
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C2×C5×SL(2,ℤ/4)C2211yescharacteristic
17#91899—22no—
18#91575—41nomaximal
19#85293—51nomaximal, Hall
20C22×C5×C22⋊C4—31no—
21C22×C22⋊Dic5D311yescharacteristic
22C2×#4070—31no—
23C2×#4024—31no—
24#65536—31no—
25#65501—31no—
26#65499—31no—
27#65499—31no—
28#65239—31no—
29#65238—31no—
30#65237—31no—
31#65237—31no—
32#65236—31no—
33#65235—31no—
34#65231—31no—
35C22×C5×A4—21no—
36C22×C5×A4—21no—
37C22×C5×A4Q812yescomplemented
38C22×C5×A4C2311yescharacteristic
39C22×C5×Dic3—41no—
40C2×#1204—41no—
41A4×Dic5—21no—
42A4×Dic5—21no—
43A4⋊Dic5—21no—
44A4⋊Dic5—21no—
45C5×SL(2,ℤ/4)—21no—
46C5×SL(2,ℤ/4)D412yes—
47#6719—41no—
48C3×C22⋊Dic5—41no—
49Dic3⋊Dic5—41no—
50#6669—41no—
51#6668—41no—
52C22×SL(2,ℤ/4)D511yescharacteristic
53C2×A4⋊Q8—51no—
54A4×C22⋊C4—51no—
55#5432—51no—
56C4⋊SL(2,ℤ/4)—51no—
57SL(2,ℤ/4)⋊C4—51no—
58SL(2,ℤ/4)⋊C4—51no—
59C25×C5C2×D311yescharacteristic
60C23×C4×C5—31no—
61C23×C4×C5—31no—
62C23×Dic5C2×D311yescharacteristic
63C23×Dic5C2×D311yescharacteristic
64C22×Dic10—31no—
65C2×C5×C22⋊C4—31no—
66C2×C5×C22⋊C4—31no—
67C2×C5×C22⋊C4—31no—
68C2×C5×C22⋊C4—31no—
69C2×C5×C22⋊C4—62no—
70C2×C5×C22⋊C4—62no—
71C2×C22⋊Dic5—31no—
72C2×C22⋊Dic5—31no—
73C2×C22⋊Dic5—31no—
74C2×C22⋊Dic5—31no—
75C2×C22⋊Dic5—31no—
76C2×C22⋊Dic5—31no—
77C2×C22⋊Dic5—31no—
78C2×C22⋊Dic5—31no—
79C2×C22⋊Dic5—31no—
80C2×C22⋊Dic5—31no—
81C2×C4⋊Dic5—31no—
82C2×C4⋊Dic5—31no—
83C2×C4⋊Dic5—31no—
84#4070—61no—
85#4070—61no—
86#4070—61no—
87#4070—61no—
88C2×#683—31no—
89C2×#683—31no—
90C2×#683—31no—
91C2×#683—31no—
92C2×#683—31no—
93C2×#683—31no—
94#4024—62no—
95#4024—62no—
96#3963—31no—
97#3963—31no—
98#3963—31no—
99#3963—31no—
100#3963—31no—
101#3963—31no—
102#2437—151noSylow
103C23×C3×C5—41no—
104C2×C5×A4—22no—
105C2×C5×A4C2×D411yescharacteristic
106C2×C5×A4C2×Q811yescharacteristic
107C2×C5×A4#4111yescharacteristic
108C2×Dic15—41no—
109C2×Dic15—41no—
110C2×C5×Dic3—41no—
111C2×C5×Dic3—41no—
112C2×C5×Dic3—82no—
113C2×C3×Dic5—41no—
114C2×C3×Dic5—41no—
115#1204—82no—
116C23×A4C2×D511yescharacteristic
117C2×C4×A4—51no—
118C2×C4×A4—51no—
119C2×SL(2,ℤ/4)—22no—
120C2×SL(2,ℤ/4)—51no—
121C2×SL(2,ℤ/4)—51no—
122C2×SL(2,ℤ/4)C2×D511yescharacteristic
123C2×SL(2,ℤ/4)C2×D511yescharacteristic
124A4⋊Q8—102no—
125#878—201no—
126C24×C5—21no—
127C24×C5—21no—
128C24×C5—31no—
129C24×C5—31no—
130C24×C5—32no—
131C24×C5—61no—
132C24×C5—61no—
133C24×C5Dic612yescomplemented
134C24×C5C22×D311yescharacteristic
135C22×C4×C5—31no—
136C22×C4×C5—31no—
137C22×C4×C5—31no—
138C22×C4×C5—31no—
139C22×C4×C5—62no—
140C22×C4×C5—62no—
141C22×C4×C5—62no—
142C22×C4×C5—61no—
143C22×C4×C5—61no—
144C22×C4×C5—121no—
145C22×C4×C5—121no—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5—31no—
149C22×Dic5—31no—
150C22×Dic5—31no—
151C22×Dic5—31no—
152C22×Dic5—31no—
153C22×Dic5—31no—
154C22×Dic5—31no—
155C22×Dic5—31no—
156C22×Dic5—61no—
157C22×Dic5—61no—
158C2×Dic10—61no—
159C2×Dic10—62no—
160C2×Dic10—61no—
161C2×Dic10—121no—
162C5×C22⋊C4—32no—
163C5×C22⋊C4—32no—
164C5×C22⋊C4—62no—
165C5×C22⋊C4—62no—
166C5×C22⋊C4—61no—
167C5×C22⋊C4—61no—
168C22⋊Dic5—31no—
169C22⋊Dic5—31no—
170C22⋊Dic5—31no—
171C22⋊Dic5—61no—
172C22⋊Dic5—61no—
173C22⋊Dic5—62no—
174C22⋊Dic5—61no—
175C22⋊Dic5—61no—
176C22⋊Dic5—61no—
177C22⋊Dic5—61no—
178C22⋊Dic5—61no—
179C22⋊Dic5—121no—
180C22⋊Dic5—121no—
181C22⋊Dic5S411yescharacteristic
182C4⋊Dic5—61no—
183C4⋊Dic5—62no—
184C4⋊Dic5—61no—
185C4⋊Dic5—121no—
186#683—61no—
187#683—61no—
188#683—62no—
189#683—62no—
190#683—61no—
191#683—61no—
192#683—121no—
193#683—121no—
194C2×C22⋊Q8—151no—
195C2×C22⋊Q8—151no—
196C22×C22⋊C4—31no—
197C22×C22⋊C4—51no—
198#399—151no—
199#396—151no—
200#396—151no—
201#393—151no—
202#385—151no—
203#385—151no—
204#385—151no—
205#385—151no—
206#380—151no—
207#380—151no—
208#379—151no—
209C22×C3×C5—42no—
210C22×C3×C5—41no—
211C22×C3×C5—81no—
212C22×C3×C5—81no—
213C5×A4C22⋊Q811yescharacteristic, complemented
214Dic15—81no—
215Dic15—81no—
216C3×Dic5—81no—
217C3×Dic5—81no—
218C5×Dic3—42no—
219C5×Dic3—81no—
220C22×A4—21no—
221C22×A4—21no—
222C22×A4Dic1012yescomplemented
223C22×A4C22×D511yescharacteristic
224C22×Dic3—41no—
225C2×Dic6—201no—
226C4×A4—101no—
227C4×A4—101no—
228SL(2,ℤ/4)—21no—
229SL(2,ℤ/4)—101no—
230SL(2,ℤ/4)—101no—
231SL(2,ℤ/4)C22⋊D512yes—
232C3×C22⋊C4—201no—
233#217—201no—
234C4⋊Dic3—201no—
235Dic3⋊C4—201no—
236Dic3⋊C4—201no—
237C23×C5—22no—
238C23×C5—31no—
239C23×C5—31no—
240C23×C5—31no—
241C23×C5—31no—
242C23×C5—32no—
243C23×C5—32no—
244C23×C5—32no—
245C23×C5—31no—
246C23×C5—61no—
247C23×C5—61no—
248C23×C5—61no—
249C23×C5—61no—
250C23×C5—61no—
251C23×C5—62no—
252C23×C5—61no—
253C23×C5—61no—
254C23×C5—61no—
255C23×C5—62no—
256C23×C5—62no—
257C23×C5—61no—
258C23×C5—122no—
259C23×C5C2×Dic611yescharacteristic
260C23×C5D3×D411yescharacteristic
261C23×C5#23711yescharacteristic
262C23×C5C2×S411yescharacteristic
263C2×C4×C5—32no—
264C2×C4×C5—32no—
265C2×C4×C5—61no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5—61no—
273C2×C4×C5—61no—
274C2×C4×C5—61no—
275C2×C4×C5—122no—
276C2×C4×C5—124no—
277C2×C4×C5—122no—
278C2×Dic5—31no—
279C2×Dic5—31no—
280C2×Dic5—61no—
281C2×Dic5—61no—
282C2×Dic5—61no—
283C2×Dic5—61no—
284C2×Dic5—61no—
285C2×Dic5—61no—
286C2×Dic5—61no—
287C2×Dic5—61no—
288C2×Dic5—61no—
289C2×Dic5—61no—
290C2×Dic5—61no—
291C2×Dic5—61no—
292C2×Dic5—61no—
293C2×Dic5—61no—
294C2×Dic5—61no—
295C2×Dic5—61no—
296C2×Dic5—61no—
297C2×Dic5—61no—
298C2×Dic5—61no—
299C2×Dic5—61no—
300C2×Dic5C2×S411yescharacteristic
301C2×Dic5C2×S411yescharacteristic
302Dic10—122no—
303Dic10—122no—
304C25D3×D511yescharacteristic
305C22×Q8—151no—
306C23×C4—31no—
307C23×C4—31no—
308C23×C4—51no—
309C23×C4—51no—
310C22⋊Q8—302no—
311C22⋊Q8—302no—
312C22⋊Q8—301no—
313C22⋊Q8—301no—
314C22⋊Q8—301no—
315C22⋊Q8—301no—
316C2×C4⋊C4—151no—
317C2×C4⋊C4—151no—
318C2×C4⋊C4—151no—
319C2×C4⋊C4—151no—
320C2×C4⋊C4—151no—
321C2×C4⋊C4—151no—
322C2×C4⋊C4—151no—
323C2×C4⋊C4—151no—
324C2×C4⋊C4—151no—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—62no—
330C2×C22⋊C4—62no—
331C2×C22⋊C4—151no—
332C2×C22⋊C4—151no—
333C2×C22⋊C4—151no—
334C2×C22⋊C4—151no—
335C2×C22⋊C4—151no—
336C2×C22⋊C4—151no—
337C2×C22⋊C4—151no—
338C2×C22⋊C4—151no—
339C2×C22⋊C4—151no—
340C2×C22⋊C4—151no—
341#95—151no—
342#95—151no—
343#95—151no—
344#95—151no—
345#95—151no—
346#95—151no—
347C2×C3×C5—41no—
348C2×C3×C5—41no—
349C2×C3×C5—41no—
350C2×C3×C5—82no—
351C23×C3—41no—
352C2×A4—22no—
353C2×A4C2×Dic1011yescharacteristic
354C2×A4#70911yescharacteristic
355C2×A4C2×C22⋊D511yescharacteristic
356C2×C3×C4—201no—
357C2×C3×C4—201no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—82no—
361C2×Dic3—201no—
362C2×Dic3—201no—
363Dic6—402no—
364C22×C5—21no—
365C22×C5—21no—
366C22×C5—31no—
367C22×C5—31no—
368C22×C5—31no—
369C22×C5—31no—
370C22×C5—31no—
371C22×C5—31no—
372C22×C5—31no—
373C22×C5—31no—
374C22×C5—31no—
375C22×C5—32no—
376C22×C5—62no—
377C22×C5—62no—
378C22×C5—62no—
379C22×C5—62no—
380C22×C5—61no—
381C22×C5—62no—
382C22×C5—62no—
383C22×C5—61no—
384C22×C5—61no—
385C22×C5—121no—
386C22×C5—121no—
387C22×C5#87811yescharacteristic, complemented
388C22×C5A4⋊Q812yescomplemented
389C22×C5C22×S411yescharacteristic
390C4×C5—62no—
391C4×C5—62no—
392C4×C5—121no—
393C4×C5—121no—
394Dic5—21no—
395Dic5—21no—
396Dic5—61no—
397Dic5—61no—
398Dic5—121no—
399Dic5—121no—
400Dic5—121no—
401Dic5—121no—
402C24—21no—
403C24—21no—
404C24—31no—
405C24—31no—
406C24—32no—
407C24—61no—
408C24—61no—
409C24#120412yescomplemented
410C24C2×D3×D511yescharacteristic
411C2×Q8—302no—
412C2×Q8—301no—
413C2×Q8—301no—
414C2×Q8—601no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—61no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—121no—
427C22×C4—121no—
428C22×C4—151no—
429C22×C4—151no—
430C22×C4—151no—
431C22×C4—151no—
432C22×C4—151no—
433C22×C4—151no—
434C22×C4—151no—
435C22×C4—151no—
436C22×C4—301no—
437C22×C4—301no—
438C4⋊C4—302no—
439C4⋊C4—302no—
440C4⋊C4—302no—
441C4⋊C4—301no—
442C4⋊C4—301no—
443C4⋊C4—301no—
444C4⋊C4—301no—
445C4⋊C4—301no—
446C4⋊C4—301no—
447C4⋊C4—601no—
448C4⋊C4—601no—
449C4⋊C4—601no—
450C22⋊C4—32no—
451C22⋊C4—32no—
452C22⋊C4—51no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—61no—
456C22⋊C4—61no—
457C22⋊C4—151no—
458C22⋊C4—151no—
459C22⋊C4—151no—
460C22⋊C4—301no—
461C22⋊C4—301no—
462C22⋊C4—301no—
463C22⋊C4—301no—
464C22⋊C4—302no—
465C22⋊C4—301no—
466C22⋊C4—301no—
467C22⋊C4—301no—
468C22⋊C4—601no—
469C22⋊C4—601no—
470C3×C5—41noHall
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—81no—
474C22×C3—81no—
475A4#407011yescharacteristic, complemented
476C3×C4—401no—
477C3×C4—401no—
478Dic3—42no—
479Dic3—81no—
480Dic3—401no—
481Dic3—401no—
482C2×C5—22no—
483C2×C5—31no—
484C2×C5—31no—
485C2×C5—31no—
486C2×C5—31no—
487C2×C5—62no—
488C2×C5C2×A4⋊Q811yescharacteristic
489C2×C5D4×S411yescharacteristic
490C2×C5#591411yescharacteristic
491C23—22no—
492C23—31no—
493C23—31no—
494C23—31no—
495C23—31no—
496C23—32no—
497C23—31no—
498C23—32no—
499C23—32no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—62no—
506C23—61no—
507C23—61no—
508C23—61no—
509C23—62no—
510C23—62no—
511C23—61no—
512C23—122no—
513C23#677911yescharacteristic
514C23C2×#120411yescharacteristic
515C23D3×C22⋊D511yescharacteristic
516C23D5×S411yescharacteristic
517Q8—602no—
518Q8—602no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—51no—
522C2×C4—51no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—122no—
534C2×C4—124no—
535C2×C4—122no—
536C2×C4—151no—
537C2×C4—151no—
538C2×C4—301no—
539C2×C4—301no—
540C2×C4—301no—
541C2×C4—301no—
542C2×C4—301no—
543C2×C4—301no—
544C2×C4—301no—
545C2×C4—301no—
546C2×C4—301no—
547C2×C4—301no—
548C2×C4—301no—
549C2×C4—301no—
550C2×C4—301no—
551C2×C4—301no—
552C2×C4—301no—
553C2×C4—301no—
554C2×C4—301no—
555C2×C4—301no—
556C2×C4—301no—
557C2×C4—301no—
558C2×C3—41no—
559C2×C3—41no—
560C2×C3—41no—
561C2×C3—82no—
562C5#8529311yescharacteristic, complemented, Sylow
563C22—21no—
564C22—21no—
565C22—31no—
566C22—31no—
567C22—31no—
568C22—31no—
569C22—31no—
570C22—31no—
571C22—31no—
572C22—31no—
573C22—31no—
574C22—32no—
575C22—62no—
576C22—62no—
577C22—62no—
578C22—62no—
579C22—61no—
580C22—62no—
581C22—62no—
582C22—61no—
583C22—61no—
584C22—121no—
585C22—121no—
586C22#9157511yescharacteristic, complemented
587C22#9189912yescomplemented
588C22C2×D5×S411yescharacteristic
589C4—62no—
590C4—62no—
591C4—101no—
592C4—101no—
593C4—121no—
594C4—121no—
595C4—301no—
596C4—301no—
597C4—601no—
598C4—601no—
599C4—601no—
600C4—601no—
601C3—41noSylow
602C2—22no—
603C2—31no—
604C2—31no—
605C2—31no—
606C2—31no—
607C2—62no—
608C2C2×#9189911yescharacteristic
609C2#1175679311yescharacteristic
610C2S4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC22×C5×A4C23
FittingC25×C5C2×D3
socleC24×C5C22×D3
prime radicalC23×C5×A4C22
hypercenterC23D5×S4
hypocenterC5×A4C22⋊Q8
LCSG ▷ C22×C5×A4 ▷ C5×A4
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#91899C2semidirect
5. #11751636C2semidirect
6. SL(2,ℤ/4)⋊Dic5C2semidirect
7. #11751634C2semidirect
37. C22×C5×A4Q8semidirect
133. C24×C5Dic6semidirect
213. C5×A4C22⋊Q8semidirect
222. C22×A4Dic10semidirect
387. C22×C5#878semidirect
388. C22×C5A4⋊Q8semidirect
409. C24#1204semidirect
475. A4#4070semidirect
562. C5#85293semidirect
586. C22#91575semidirect
587. C22#91899semidirect

Maximal subgroups

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

subgroupindex
C22×C5×SL(2,ℤ/4)2
C2×#918992
A4×C22⋊Dic52
#117521512
#117516362
SL(2,ℤ/4)⋊Dic52
#117516342
#106138413
#915754
#852935