Finite Group Databasebeta

← #14412767

Subgroups — #14412767

Order 1920.

1452 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92000C211yesmaximal, characteristic
2C2×#91796C211yesmaximal, characteristic, complemented
3#11754496C211yesmaximal, characteristic, complemented
4C5×#5120C211yesmaximal, characteristic, complemented
5C3×#65422C211yesmaximal, characteristic, complemented
6C2×#90972C211yesmaximal, characteristic, complemented
7#11748364C211yesmaximal, characteristic, complemented
8#11748142C212yesmaximal, complemented
9#11748141C212yesmaximal, complemented
10#11747644C212yesmaximal, complemented
11#11747609C212yesmaximal, complemented
12#10610688—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C2×#6779—22no—
17C2×#6773—21no—
18C2×D15⋊Q8C2212yes—
19#92000—24no—
20#92000C418yescomplemented
21C2×#6766C2212yes—
22C2×D5×Dic6—21no—
23C2×C3×C5×M16C2211yescharacteristic
24C22×C3⋊#168C2211yescharacteristic
25#91796—24no—
26C2×#6705C2212yes—
27C2×C8×D15C2212yes—
28C5×#926C2211yescharacteristic
29C2×C3×C4⋊Dic5C2211yescharacteristic
30#91552—22no—
31#91437C2212yes—
32#91428—21no—
33#91347C2212yes—
34#91340—21no—
35C5×#816C2212yes—
36C5×#816C2212yes—
37C3×#3953C2212yes—
38C3×#3948C2212yes—
39#90972C2212yes—
40#90972C2212yes—
41#90970C2211yescharacteristic
42#90970C2211yescharacteristic
43#90964C2211yescharacteristic
44#90964C2211yescharacteristic
45#78883—51nomaximal, Hall
46C2×#4147—31no—
47C2×#4053—31no—
48#65857—31no—
49C5×#419—31no—
50C2×#3967—31no—
51#65422D311yescharacteristic, complemented
52#65291—31no—
53#65093—32no—
54#65092—32no—
55#65091—32no—
56D5×#103—32no—
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220D412yes—
59#6817C2×C414yes—
60#6817D414yes—
61C2×D60D411yescharacteristic
62C2×C4×D15C2312yes—
63C2×Dic30D411yescharacteristic
64C22×C5×Dic3—21no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6—22no—
67C2×C5×Dic6C2×C414yescomplemented
68C2×C5×Dic6C2312yes—
69C2×C3×C22⋊D5—21no—
70C22×C3×Dic5—21no—
71C3×#709—22no—
72C3×#709C2×C414yescomplemented
73C2×C3×D20C2311yescharacteristic
74C2×C3×C4×D5—21no—
75C2×C3×Dic10C2311yescharacteristic
76C2×#1204—22no—
77C2×#1202—22no—
78C2×#1200—22no—
79C2×#1200—22no—
80#6779—216no—
81C2×D5×Dic3—22no—
82#6773—28no—
83D15⋊Q8—24no—
84D15⋊Q8C2×C418yescomplemented
85#6766—24no—
86#6766C2×C418yescomplemented
87D5×Dic6—28no—
88C3×C5×M16—22no—
89C2×C3×C5×C8C2312yes—
90C2×C3⋊#168—22no—
91C2×C3⋊#168C2311yescharacteristic
92C2×C3⋊#168C2311yescharacteristic
93#6705—24no—
94C8×D15—24no—
95C5×#217—21no—
96C5×C4⋊Dic3C2311yescharacteristic
97C5×C4⋊Dic3C2311yescharacteristic
98C4×C5×Dic3—21no—
99C3×C4⋊Dic5—21no—
100C3×C4⋊Dic5C2312yes—
101#6669—22no—
102#6667—22no—
103Dic3×Dic5—22no—
104C2×#1010—51no—
105C2×#907—51no—
106#5504—51no—
107C3×#417—51no—
108#5233—51no—
109#5120D511yescharacteristic, complemented
110C2×#810—51no—
111#4802—52no—
112D3×#103—52no—
113#4772—52no—
114#4771—52no—
115C22×C5×Q8—31no—
116#4147—38no—
117#4147—64no—
118C2×#713—32no—
119C2×#713—61no—
120C2×Q8×D5—32no—
121C2×Q8×D5—61no—
122C2×#709—31no—
123C2×#709—62no—
124C2×#709C2×D311yescharacteristic
125C2×C5×M16C2×D311yescharacteristic
126C5×#117—31no—
127C2×C4⋊Dic5C2×D311yescharacteristic
128C22×#168—31no—
129#4053—64no—
130C2×#676—32no—
131C2×C8×D5—32no—
132#4038—32no—
133D5×C4⋊C4—32no—
134#4027—62no—
135C4×D20—61no—
136C4×Dic10—61no—
137C5×#103—32no—
138C5×#103—32no—
139#3967—32no—
140#3967—32no—
141#3953C2×D312yes—
142#3948C2×D312yes—
143#3942—31no—
144#3942—31no—
145#3941—31no—
146#3941—31no—
147#2898—151noSylow
148C23×C3×C5C2×D411yescharacteristic
149C22×D15C2×D412yes—
150C22×C3×D5—21no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D412yes—
154C2×C3×C4×C5C2411yescharacteristic
155#1220C22⋊C418yes—
156C2×Dic15C2×D412yes—
157D60C22⋊C414yes—
158C4×D15C22×C414yes—
159C4×D15C2×D414yes—
160Dic30C22⋊C414yes—
161C2×C5×Dic3—24no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—21no—
164C2×C5×Dic3—21no—
165C5×Dic6—24no—
166C5×Dic6—22no—
167C5×Dic6C22×C414yes—
168C3×C22⋊D5—24no—
169C2×C3×Dic5—21no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5—81no—
172C3×D20—21no—
173C3×D20C22×C412yes—
174C3×C4×D5—24no—
175C3×Dic10—21no—
176C3×Dic10C22×C412yes—
177#1204—28no—
178#1202—28no—
179#1200—28no—
180#1200—82no—
181D5×Dic3—28no—
182C3×C5×C8—22no—
183C3⋊#168—21no—
184C3⋊#168—21no—
185C2×C3×#41—51no—
186#1010—58no—
187#1010—104no—
188C2×D3×Q8—52no—
189C2×#237—52no—
190C2×#237—102no—
191C2×#235—51no—
192C2×#235—101no—
193C22×Dic6—101no—
194C22×Dic6C2×D511yescharacteristic
195C2×C3×M16C2×D511yescharacteristic
196C2×C3×C4⋊C4—51no—
197C4×#67—102no—
198#926C2×D511yescharacteristic
199C22×C3⋊C8—51no—
200#907—104no—
201C2×ΓL(1,𝔽25)—52no—
202C2×D3×C8—52no—
203#893—101no—
204#892—52no—
205D3×C4⋊C4—52no—
206#887—101no—
207C3×#103—52no—
208C3×#102—52no—
209#835—51no—
210#835—51no—
211#832—51no—
212#832—51no—
213#816C2×D512yes—
214#816C2×D512yes—
215#810—52no—
216#810—52no—
217C2×C5×Q8—34no—
218C2×C5×Q8—32no—
219C2×C5×Q8—61no—
220C2×C5×Q8—62no—
221C22×C4×C5—61no—
222C22×C4×C5C22×D311yescharacteristic
223C2×C22⋊D5—21no—
224C2×C22⋊D5—32no—
225C22×Dic5—21no—
226#713—38no—
227#713—68no—
228#713—64no—
229Q8×D5—38no—
230Q8×D5—64no—
231Q8×D5—68no—
232#709—22no—
233#709—34no—
234#709—34no—
235#709—616no—
236#709C4×D314yescomplemented
237C2×D20—31no—
238C2×D20—62no—
239C2×D20C22×D311yescharacteristic
240C2×C4×D5—21no—
241C2×C4×D5—32no—
242C2×C4×D5—62no—
243C2×C4×D5—62no—
244C2×C4×D5—62no—
245C2×Dic10—31no—
246C2×Dic10—62no—
247C2×Dic10C22×D311yescharacteristic
248C5×M16—22no—
249C2×C5×C8C22×D312yes—
250C5×C4⋊C4—31no—
251C5×C4⋊C4—31no—
252C5×C22⋊C4—61no—
253C42×C5—61no—
254#685—62no—
255C4⋊Dic5—21no—
256C4⋊Dic5C22×D312yes—
257#683—62no—
258C4×Dic5—62no—
259C2×#168—31no—
260C2×#168—31no—
261C2×#168—62no—
262#676—64no—
263C8×D5—64no—
264C2×2−1+4—151no—
265C2×#131—151no—
266#520—151no—
267#419—31no—
268#419—152no—
269#417—51no—
270#417—152no—
271#416—152no—
272#416—151no—
273C2×#103—152no—
274C2×#103—151no—
275C22×C3×C5—21no—
276C22×C3×C5—21no—
277C22×C3×C5C2×C22⋊C412yes—
278C22×C3×C5C22×D411yescharacteristic
279C2×D15—22no—
280C2×D15—22no—
281C2×D15C2×C22⋊C414yes—
282C2×C3×D5—22no—
283C2×C3×D5—41no—
284C2×C3×D5—41no—
285C3×C4×C5C2×C22⋊C412yes—
286C3×C4×C5C23×C411yescharacteristic
287C3×C4×C5C22×D411yescharacteristic
288Dic15C2×C22⋊C414yes—
289C3×Dic5—22no—
290C3×Dic5—42no—
291C5×Dic3—24no—
292C5×Dic3—41no—
293C5×Dic3—41no—
294C3×#41—54no—
295C3×#41—102no—
296C2×C3×Q8—51no—
297C2×C3×D4—51no—
298C2×C3×D4—101no—
299C22×C3×C4—101no—
300C22×C3×C4—101no—
301C22×C3×C4C22×D511yescharacteristic
302C2×#67—52no—
303C2×#67—102no—
304C22×Dic3—21no—
305C22×Dic3—102no—
306D3×Q8—58no—
307D3×Q8—104no—
308#237—58no—
309#237—104no—
310#237—1016no—
311#235—54no—
312#235—54no—
313#235—108no—
314C2×D12—51no—
315C2×C4×D3—52no—
316C2×C4×D3—102no—
317C2×C4×D3—102no—
318C2×Dic6—21no—
319C2×Dic6—22no—
320C2×Dic6—51no—
321C2×Dic6—108no—
322C2×Dic6—102no—
323C2×Dic6C4×D514yescomplemented
324C2×Dic6C22×D512yes—
325C3×M16—22no—
326C2×C3×C8C22×D512yes—
327C3×C4⋊C4—52no—
328C3×C4⋊C4—101no—
329#217—21no—
330(C2×D3)⋊C4—102no—
331C4⋊Dic3C22×D511yescharacteristic
332C4⋊Dic3C22×D511yescharacteristic
333Dic3⋊C4—102no—
334C4×Dic3—21no—
335C4×Dic3—102no—
336C2×C3⋊C8—51no—
337C2×C3⋊C8—51no—
338C2×C3⋊C8—102no—
339ΓL(1,𝔽25)—104no—
340D3×C8—104no—
341C23×C5D3×D411yescharacteristic
342C22×D5—21no—
343C22×D5—32no—
344C5×Q8—34no—
345C5×Q8—64no—
346C5×Q8—62no—
347C2×C4×C5—64no—
348C2×C4×C5—62no—
349C2×C4×C5—61no—
350C2×C4×C5—61no—
351C2×C4×C5C2×C4×D312yes—
352C2×C4×C5D3×D411yescharacteristic
353C2×C4×C5D3×D412yes—
354C2×C4×C5C23×D311yescharacteristic
355C22⋊D5—24no—
356C22⋊D5—38no—
357C2×Dic5—21no—
358C2×Dic5—22no—
359C2×Dic5—32no—
360C2×Dic5—81no—
361D20—21no—
362D20—34no—
363D20—68no—
364D20C2×C4×D312yes—
365C4×D5—24no—
366C4×D5—34no—
367C4×D5—34no—
368C4×D5—68no—
369C4×D5—68no—
370C4×D5—242no—
371Dic10—21no—
372Dic10—34no—
373Dic10—68no—
374Dic10C2×C4×D312yes—
375C5×C8—22no—
376#168—61no—
377#168—61no—
3782−1+4—158no—
3792−1+4—304no—
380C2×#41—51no—
381C2×#41—151no—
382C2×#41—152no—
383C2×#41—302no—
384C2×#41—301no—
385C22×Q8—31no—
386C22×Q8—152no—
387C22×Q8—301no—
388#131—304no—
389C2×M16—152no—
390C2×M16D3×D511yescharacteristic
391C22×C8—151no—
392C22×C8—152no—
393C4×Q8—301no—
394C4×D4—301no—
395C4×D4—302no—
396#117—31no—
397#117—152no—
398C2×C4⋊C4—51no—
399C2×C4⋊C4—152no—
400#103—32no—
401#103—32no—
402#103—52no—
403#103—151no—
404#103—151no—
405#103—152no—
406#103—152no—
407#102—52no—
408#102—151no—
409#102—151no—
410C2×C3×C5—22no—
411C2×C3×C5C22×C22⋊C411yescharacteristic
412C2×C3×C5#57711yescharacteristic
413C2×C3×C5#57811yescharacteristic
414D15—24no—
415C3×D5—42no—
416C23×C3—101no—
417C23×C3D4×D511yescharacteristic
418C22×D3—52no—
419C3×Q8—52no—
420C3×Q8—101no—
421C3×D4—52no—
422C3×D4—104no—
423C3×D4—101no—
424C2×C3×C4—101no—
425C2×C3×C4—104no—
426C2×C3×C4—102no—
427C2×C3×C4—401no—
428C2×C3×C4C2×C4×D512yes—
429C2×C3×C4D4×D511yescharacteristic
430C2×C3×C4D4×D512yes—
431C2×C3×C4C23×D511yescharacteristic
432#67—58no—
433#67—108no—
434C2×Dic3—22no—
435C2×Dic3—24no—
436C2×Dic3—21no—
437C2×Dic3—21no—
438C2×Dic3—52no—
439C2×Dic3—108no—
440D12—54no—
441C4×D3—54no—
442C4×D3—54no—
443C4×D3—108no—
444C4×D3—402no—
445Dic6—22no—
446Dic6—24no—
447Dic6—54no—
448Dic6—108no—
449Dic6C2×C4×D514yes—
450C3×C8—22no—
451C3⋊C8—101no—
452C3⋊C8—101no—
453C22×C5—21no—
454C22×C5—21no—
455C22×C5D3×C22⋊C412yes—
456C22×C5C2×D3×D411yescharacteristic
457C2×D5—22no—
458C2×D5—34no—
459C2×D5—41no—
460C2×D5—41no—
461C2×D5—62no—
462C2×D5—62no—
463C4×C5—64no—
464C4×C5—121no—
465C4×C5—121no—
466C4×C5D3×C22⋊C412yes—
467C4×C5C22×C4×D311yescharacteristic
468C4×C5C2×D3×D411yescharacteristic
469Dic5—22no—
470Dic5—34no—
471Dic5—42no—
472#41—54no—
473#41—102no—
474#41—154no—
475#41—154no—
476#41—158no—
477#41—3016no—
478#41—308no—
479#41—304no—
480C2×Q8—34no—
481C2×Q8—32no—
482C2×Q8—51no—
483C2×Q8—61no—
484C2×Q8—62no—
485C2×Q8—151no—
486C2×Q8—158no—
487C2×Q8—302no—
488C2×Q8—308no—
489C2×Q8—304no—
490C2×D4—51no—
491C2×D4—101no—
492C2×D4—152no—
493C2×D4—151no—
494C2×D4—302no—
495C22×C4—61no—
496C22×C4—101no—
497C22×C4—101no—
498C22×C4—152no—
499C22×C4—302no—
500C22×C4—302no—
501C22×C4—302no—
502C22×C4C2×D3×D511yescharacteristic
503M16—22no—
504M16—304no—
505C2×C8—151no—
506C2×C8—151no—
507C2×C8—304no—
508C2×C8—302no—
509C2×C8C2×D3×D512yes—
510C4⋊C4—31no—
511C4⋊C4—31no—
512C4⋊C4—52no—
513C4⋊C4—101no—
514C4⋊C4—302no—
515C22⋊C4—61no—
516C22⋊C4—302no—
517C42—61no—
518C42—302no—
519C3×C5#289811yescharacteristic, complemented, Hall
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—102no—
523C22×C3—201no—
524C22×C3—201no—
525C22×C3D5×C22⋊C412yes—
526C22×C3C2×D4×D511yescharacteristic
527C2×D3—54no—
528C2×D3—102no—
529C2×D3—102no—
530C3×C4—102no—
531C3×C4—202no—
532C3×C4D5×C22⋊C412yes—
533C3×C4C22×C4×D511yescharacteristic
534C3×C4C2×D4×D511yescharacteristic
535Dic3—24no—
536Dic3—41no—
537Dic3—41no—
538Dic3—54no—
539C2×C5—22no—
540C2×C5C2×D3×C22⋊C411yescharacteristic
541C2×C5#577511yescharacteristic
542C2×C5#577911yescharacteristic
543D5—42no—
544D5—64no—
545C23—101no—
546C23—152no—
547C23#677711yescharacteristic
548Q8—34no—
549Q8—52no—
550Q8—64no—
551Q8—62no—
552Q8—101no—
553Q8—154no—
554Q8—308no—
555D4—52no—
556D4—104no—
557D4—101no—
558D4—158no—
559D4—154no—
560D4—308no—
561C2×C4—64no—
562C2×C4—62no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—101no—
566C2×C4—104no—
567C2×C4—102no—
568C2×C4—152no—
569C2×C4—154no—
570C2×C4—154no—
571C2×C4—308no—
572C2×C4—308no—
573C2×C4—401no—
574C2×C4—1202no—
575C2×C4C4×D3×D512yes—
576C2×C4#677711yescharacteristic
577C2×C4#679012yes—
578C2×C4C22×D3×D511yescharacteristic
579C8—22no—
580C8—301no—
581C8—301no—
582C2×C3—22no—
583C2×C3—202no—
584C2×C3C2×D5×C22⋊C411yescharacteristic
585C2×C3#6611211yescharacteristic
586C2×C3#6611311yescharacteristic
587D3—104no—
588C5#7888311yescharacteristic, complemented, Sylow
589C22—21no—
590C22—21no—
591C22—102no—
592C22—154no—
593C22—201no—
594C22—201no—
595C22—302no—
596C22—302no—
597C22#9147312yes—
598C22C2×#677711yescharacteristic
599C4—64no—
600C4—102no—
601C4—121no—
602C4—121no—
603C4—154no—
604C4—202no—
605C4#9147312yes—
606C4C2×C4×D3×D511yescharacteristic
607C4C2×#679011yescharacteristic
608C3#1061068811yescharacteristic, complemented, Sylow
609C2—22no—
610C2—202no—
611C2—304no—
612C2#1175403711yescharacteristic
613C2#1175404311yescharacteristic
614C2C2×#9147311yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#92000C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5#2898
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#91796C2semidirect
3. #11754496C2semidirect
4. C5×#5120C2semidirect
5. C3×#65422C2semidirect
6. C2×#90972C2semidirect
7. #11748364C2semidirect
8. #11748142C2semidirect
9. #11748141C2semidirect
10. #11747644C2semidirect
11. #11747609C2semidirect
20. #92000C4semidirect
51. #65422D3semidirect
67. C2×C5×Dic6C2×C4semidirect
72. C3×#709C2×C4semidirect
84. D15⋊Q8C2×C4semidirect
86. #6766C2×C4semidirect
109. #5120D5semidirect
236. #709C4×D3semidirect
323. C2×Dic6C4×D5semidirect
519. C3×C5#2898semidirect
588. C5#78883semidirect
608. C3#10610688semidirect

Maximal subgroups

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

subgroupindex
C2×#920002
C2×#917962
#117544962
C5×#51202
C3×#654222
C2×#909722
#117483642
#117481422
#117481412
#117476442
#117476092
#106106883
#788835