Finite Group Databasebeta

← #14416739

Subgroups — #14416739

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754455C211yesmaximal, characteristic, complemented
2C2×#91434C211yesmaximal, characteristic, complemented
3C2×#91298C211yesmaximal, characteristic
4M16×Dic15C211yesmaximal, characteristic, complemented
5C5×#5133C211yesmaximal, characteristic, complemented
6C3×#65420C211yesmaximal, characteristic, complemented
7#11748472C211yesmaximal, characteristic, complemented
8#11747736C212yesmaximal, complemented
9#11747735C212yesmaximal, complemented
10#11747734C212yesmaximal, complemented
11#11747733C212yesmaximal, complemented
12#10610597—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×D5×Dic3—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C4×Dic15C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C3×#4072C2211yescharacteristic
21#67⋊Dic5—21no—
22#67⋊Dic5—21no—
23C2×#6665—21no—
24#91435C2211yescharacteristic, complemented
25#91435C2211yescharacteristic, complemented
26#91434—21no—
27#91434—21no—
28#91434C2212yes—
29#91434C2212yes—
30C4×D3×Dic5—21no—
31#91371—21no—
32#91344C2211yescharacteristic, complemented
33#91344C2211yescharacteristic, complemented
34C2×#6655C2212yes—
35#91298C418yescomplemented
36C2×#6652C2212yes—
37#91099C2212yescomplemented
38C8×Dic15C2212yescomplemented
39C5×#821C2212yescomplemented
40C5×#816C2212yescomplemented
41C3×#3953C2212yes—
42C3×#3953C2212yes—
43#90989C2212yescomplemented
44#90986C2212yescomplemented
45#78914—51nomaximal, Hall
46#41×Dic5—31no—
47C2×#4095—31no—
48C2×#4039—31no—
49C5×#417—31no—
50#65420D311yescharacteristic, complemented
51M16×Dic5—31no—
52#65253—31no—
53#65106—32no—
54#65077—32no—
55#65074—32no—
56#65045—32no—
57C23×C3×D5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15D411yescharacteristic
60C22×Dic15D411yescharacteristic
61C2×C5×#67—21no—
62C22×C5×Dic3—21no—
63C5×#235—22no—
64C5×#235C2×C414yescomplemented
65C2×C5×D12C2311yescharacteristic
66C2×C4×C5×D3—21no—
67C2×C5×Dic6C2311yescharacteristic
68C2×C3×D20—21no—
69C2×C3×D20—22no—
70C2×C3×D20C2×C414yescomplemented
71C2×C3×D20C2312yes—
72C2×D3×Dic5—21no—
73C2×D3×Dic5—21no—
74C2×D5×Dic3—22no—
75C2×D5×Dic3—22no—
76C2×D5×Dic3—44no—
77C3×C5×M16—22no—
78C2×C3×C5×C8C2312yes—
79C4×Dic15D411yescharacteristic
80C4×Dic15D411yescharacteristic
81C4×Dic15C2311yescharacteristic
82C4×Dic15C2311yescharacteristic
83#6710C2×C414yescomplemented
84C2×C3⋊#168C2312yes—
85C5×C4⋊Dic3—21no—
86C5×C4⋊Dic3C2312yes—
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5C2311yescharacteristic
89C3×C4⋊Dic5C2311yescharacteristic
90C3×C4×Dic5—21no—
91Dic3⋊Dic5—21no—
92Dic3⋊Dic5—21no—
93(C2×D3)⋊Dic5—21no—
94(C2×D3)⋊Dic5—21no—
95#6665—22no—
96#6665—22no—
97Dic3×Dic5—21no—
98Dic3×Dic5—21no—
99#6655C2×C418yescomplemented
100#6652C2×C418yescomplemented
101C2×D4×Dic3—51no—
102C2×#932—51no—
103#5522—51no—
104C3×#418—51no—
105#5133D511yescharacteristic, complemented
106Dic3×M16—51no—
107#4995—51no—
108#4785—52no—
109#4758—52no—
110#4757—52no—
111#4756—52no—
112C2×C5×#41—31no—
113C22×D20C2×D311yescharacteristic
114C22×C4×D5—61no—
115C2×C5×M16C2×D311yescharacteristic
116C2×C5×C4⋊C4—31no—
117#4095—38no—
118Q8×Dic5—31no—
119Q8×Dic5—31no—
120C2×#688—32no—
121D4×Dic5—31no—
122D4×Dic5—31no—
123D4×Dic5—61no—
124D4×Dic5—61no—
125C2×#686—32no—
126C2×#685—61no—
127#4072—61no—
128#4072C2×D311yescharacteristic
129C2×C4×Dic5—31no—
130C2×C4×Dic5—61no—
131C2×#681—31no—
132#4039—32no—
133#4039—32no—
134#4039—61no—
135#4039—61no—
136C5×#103—32no—
137C5×#102—32no—
138#3953C2×D312yes—
139#3953C2×D312yes—
140#3947—32no—
141C8×Dic5—32no—
142#3940—32no—
143#3939—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×C5×D3—21no—
147C22×C3×D5—24no—
148C22×C3×D5—22no—
149C22×C3×D5—42no—
150C22×C3×D5—42no—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5#4112yes—
154C2×C3×C4×C5C2411yescharacteristic
155C2×Dic15—22no—
156C2×Dic15—41no—
157C2×Dic15C2×D411yescharacteristic
158C2×Dic15C2×D411yescharacteristic
159C2×Dic15C2×D411yescharacteristic
160C2×Dic15C2×D411yescharacteristic
161C5×#67—24no—
162C2×C5×Dic3—21no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—42no—
165C5×D12—21no—
166C5×D12C22×C412yescomplemented
167C4×C5×D3—24no—
168C5×Dic6—21no—
169C5×Dic6C22×C412yescomplemented
170C2×C3×Dic5—21no—
171C2×C3×Dic5—21no—
172C3×D20—24no—
173C3×D20—22no—
174C3×D20C22×C414yes—
175D3×Dic5—42no—
176D3×Dic5—42no—
177D5×Dic3—44no—
178D5×Dic3—44no—
179C3×C5×C8—22no—
180C3⋊#168C22×C414yescomplemented
181C22×C3×D4—51no—
182C23×Dic3—101no—
183C2×#235C2×D511yescharacteristic
184C2×C3×M16C2×D511yescharacteristic
185C3×#117—51no—
186C2×#217—101no—
187D4×Dic3—52no—
188D4×Dic3—52no—
189D4×Dic3—101no—
190D4×Dic3—101no—
191C2×#214—52no—
192#932—58no—
193C2×#213—52no—
194C2×C4⋊Dic3C2×D511yescharacteristic
195C2×C4×Dic3—51no—
196C2×#208—51no—
197#893—51no—
198#893—51no—
199#887—51no—
200#887—51no—
201#67⋊C4—101no—
202#67⋊C4—101no—
203#872—101no—
204C42×D3—101no—
205C3×#102—52no—
206C3×#102—52no—
207#821C2×D512yescomplemented
208#816C2×D512yescomplemented
209#815—52no—
210C8×Dic3—52no—
211#808—52no—
212#807—52no—
213C23×D5—21no—
214C5×#41—34no—
215C5×#41—62no—
216C2×C5×Q8—31no—
217C2×C5×D4—31no—
218C2×C5×D4—61no—
219C22×C4×C5—61no—
220C22×C4×C5—61no—
221C22×C4×C5C22×D311yescharacteristic
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5—61no—
225C22×Dic5—61no—
226C2×D20—21no—
227C2×D20—22no—
228C2×D20C4×D314yescomplemented
229C2×D20C22×D312yes—
230C2×C4×D5—62no—
231C2×C4×D5—62no—
232C2×C4×D5—124no—
233C5×M16—22no—
234C2×C5×C8C22×D312yes—
235C5×C4⋊C4—32no—
236C5×C4⋊C4—61no—
237C22⋊Dic5—21no—
238C22⋊Dic5—61no—
239C22⋊Dic5—61no—
240#688—38no—
241#686—38no—
242#685—62no—
243#685—62no—
244C4⋊Dic5—61no—
245C4⋊Dic5—61no—
246C4⋊Dic5C22×D311yescharacteristic
247C4⋊Dic5C22×D311yescharacteristic
248C4×Dic5—21no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—31no—
252C4×Dic5—31no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255#681—34no—
256C2×#168—32no—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—151no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—151no—
265#418—51no—
266#417—31no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×C5×D3—22no—
273C2×C5×D3—41no—
274C2×C5×D3—41no—
275C2×C3×D5—24no—
276C2×C3×D5—48no—
277C2×C3×D5—42no—
278C2×C3×D5—42no—
279C3×C4×C5C4×D412yes—
280C3×C4×C5C23×C411yescharacteristic
281C3×C4×C5C2×#4111yescharacteristic
282Dic15—21no—
283Dic15—21no—
284Dic15—21no—
285Dic15AGL(1,ℤ/8)12yescomplemented
286C3×Dic5—41no—
287C3×Dic5—41no—
288C5×Dic3—22no—
289C5×Dic3—42no—
290C24×C3—101no—
291C2×C3×D4—54no—
292C2×C3×D4—52no—
293C2×C3×D4—101no—
294C2×C3×D4—102no—
295C22×C3×C4C22×D511yescharacteristic
296C2×#67—21no—
297C22×Dic3—21no—
298C22×Dic3—51no—
299C22×Dic3—51no—
300C22×Dic3—102no—
301C22×Dic3—102no—
302C22×Dic3—204no—
303#235—22no—
304#235C4×D514yescomplemented
305C2×D12C22×D511yescharacteristic
306C2×C4×D3—21no—
307C2×C4×D3—101no—
308C2×C4×D3—101no—
309C2×Dic6C22×D511yescharacteristic
310C3×M16—22no—
311C2×C3×C8C22×D512yes—
312C3×C4⋊C4—51no—
313C3×C4⋊C4—51no—
314C3×C22⋊C4—101no—
315C3×C42—101no—
316#217—102no—
317#217—102no—
318#214—58no—
319#213—58no—
320(C2×D3)⋊C4—101no—
321(C2×D3)⋊C4—101no—
322C4⋊Dic3—21no—
323C4⋊Dic3C22×D512yes—
324Dic3⋊C4—101no—
325Dic3⋊C4—101no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328C4×Dic3—51no—
329C4×Dic3—51no—
330C4×Dic3—101no—
331C4×Dic3—101no—
332#208—54no—
333C2×C3⋊C8—52no—
334C23×C5—61no—
335C23×C5D3×D411yescharacteristic
336C22×D5—24no—
337C22×D5—22no—
338C22×D5—42no—
339C22×D5—42no—
340C5×Q8—32no—
341C5×Q8—61no—
342C5×D4—32no—
343C5×D4—64no—
344C5×D4—61no—
345C2×C4×C5—61no—
346C2×C4×C5—64no—
347C2×C4×C5—62no—
348C2×C4×C5—122no—
349C2×C4×C5C2×C4×D312yes—
350C2×C4×C5D3×D411yescharacteristic
351C2×C4×C5#23712yes—
352C2×C4×C5C23×D311yescharacteristic
353C2×Dic5—21no—
354C2×Dic5—21no—
355C2×Dic5—31no—
356C2×Dic5—31no—
357C2×Dic5—31no—
358C2×Dic5—31no—
359C2×Dic5—62no—
360C2×Dic5—122no—
361C2×Dic5—122no—
362C2×Dic5—121no—
363D20—24no—
364D20—22no—
365D20C2×C4×D314yes—
366C4×D5—124no—
367C4×D5—124no—
368C5×C8—22no—
369#168—34no—
370C2×#41—31no—
371C22×D4—51no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—151no—
377C2×M16D3×D511yescharacteristic
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—152no—
381C4×D4—152no—
382C4×D4—151no—
383C4×D4—151no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—51no—
389#117—301no—
390C2×C4⋊C4—31no—
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107—152no—
395#106—152no—
396#103—32no—
397#102—32no—
398#102—52no—
399#102—52no—
400#97—152no—
401C4×C8—152no—
402C2×C3×C5—22no—
403C2×C3×C5C2×C4×D411yescharacteristic
404C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C3×C5#57511yescharacteristic
406C3×D5—44no—
407C5×D3—42no—
408C23×C3—102no—
409C23×C3—104no—
410C23×C3—202no—
411C23×C3—202no—
412C23×C3D4×D511yescharacteristic
413C22×D3—21no—
414C3×D4—54no—
415C3×D4—102no—
416C3×D4—104no—
417C2×C3×C4—101no—
418C2×C3×C4—101no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4D4×D511yescharacteristic
421C2×C3×C4#71112yes—
422C2×C3×C4C23×D511yescharacteristic
423#67—24no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—42no—
427C2×Dic3—51no—
428C2×Dic3—51no—
429C2×Dic3—51no—
430C2×Dic3—51no—
431C2×Dic3—102no—
432C2×Dic3—204no—
433C2×Dic3—204no—
434C2×Dic3—201no—
435D12—21no—
436D12C2×C4×D512yescomplemented
437C4×D3—24no—
438C4×D3—202no—
439C4×D3—202no—
440Dic6—21no—
441Dic6C2×C4×D512yescomplemented
442C3×C8—22no—
443C3⋊C8—54no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—62no—
447C22×C5—121no—
448C22×C5—121no—
449C22×C5#67⋊C412yes—
450C22×C5C2×D3×D411yescharacteristic
451C2×D5—24no—
452C2×D5—48no—
453C2×D5—42no—
454C2×D5—42no—
455C4×C5—62no—
456C4×C5—122no—
457C4×C5#67⋊C412yes—
458C4×C5C22×C4×D311yescharacteristic
459C4×C5C2×#23711yescharacteristic
460Dic5—32no—
461Dic5—41no—
462Dic5—41no—
463Dic5—61no—
464Dic5—61no—
465Dic5—61no—
466C24—101no—
467#41—34no—
468#41—62no—
469C2×Q8—31no—
470C2×D4—31no—
471C2×D4—54no—
472C2×D4—52no—
473C2×D4—61no—
474C2×D4—101no—
475C2×D4—102no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—151no—
479C22×C4—151no—
480C22×C4—302no—
481C22×C4—302no—
482C22×C4—301no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—22no—
489M16—154no—
490C2×C8—152no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—32no—
493C4⋊C4—51no—
494C4⋊C4—51no—
495C4⋊C4—61no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—101no—
499C22⋊C4—302no—
500C22⋊C4—302no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—101no—
504C42—151no—
505C42—151no—
506C42—151no—
507C42—151no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—104no—
514C22×C3—208no—
515C22×C3—202no—
516C22×C3—202no—
517C22×C3#402712yes—
518C22×C3C2×D4×D511yescharacteristic
519C2×D3—22no—
520C2×D3—41no—
521C2×D3—41no—
522C3×C4—201no—
523C3×C4—201no—
524C3×C4#402712yes—
525C3×C4C22×C4×D511yescharacteristic
526C3×C4C2×#71111yescharacteristic
527Dic3—22no—
528Dic3—42no—
529Dic3—52no—
530Dic3—101no—
531Dic3—101no—
532Dic3—101no—
533C2×C5—22no—
534C2×C5—122no—
535C2×C5C2×#67⋊C411yescharacteristic
536C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
537C2×C5#577311yescharacteristic
538D5—44no—
539C23—61no—
540C23—104no—
541C23—102no—
542C23—202no—
543C23—202no—
544C23#677711yescharacteristic
545Q8—32no—
546Q8—61no—
547D4—32no—
548D4—54no—
549D4—64no—
550D4—61no—
551D4—104no—
552D4—102no—
553C2×C4—61no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—101no—
557C2×C4—101no—
558C2×C4—122no—
559C2×C4—151no—
560C2×C4—151no—
561C2×C4—151no—
562C2×C4—151no—
563C2×C4—302no—
564C2×C4—604no—
565C2×C4—604no—
566C2×C4—602no—
567C2×C4—602no—
568C2×C4—601no—
569C2×C4C4×D3×D512yes—
570C2×C4#677711yescharacteristic
571C2×C4#678012yes—
572C2×C4C22×D3×D511yescharacteristic
573C8—22no—
574C8—154no—
575C2×C3—22no—
576C2×C3—204no—
577C2×C3C2×#402711yescharacteristic
578C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
579C2×C3#6611111yescharacteristic
580D3—42no—
581C5#7891411yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—62no—
585C22—104no—
586C22—121no—
587C22—121no—
588C22—208no—
589C22—202no—
590C22—202no—
591C22#9144212yes—
592C22C2×#677711yescharacteristic
593C4—62no—
594C4—122no—
595C4—152no—
596C4—201no—
597C4—201no—
598C4—301no—
599C4—301no—
600C4—301no—
601C4#9144212yes—
602C4C2×C4×D3×D511yescharacteristic
603C4C2×#678011yescharacteristic
604C3#1061059711yescharacteristic, complemented, Sylow
605C2—22no—
606C2—122no—
607C2—204no—
608C2#1175403311yescharacteristic
609C2#1175403611yescharacteristic
610C2C2×#9144211yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#91298C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
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
1. #11754455C2semidirect
2. C2×#91434C2semidirect
4. M16×Dic15C2semidirect
5. C5×#5133C2semidirect
6. C3×#65420C2semidirect
7. #11748472C2semidirect
8. #11747736C2semidirect
9. #11747735C2semidirect
10. #11747734C2semidirect
11. #11747733C2semidirect
24. #91435C22semidirect
25. #91435C22semidirect
32. #91344C22semidirect
33. #91344C22semidirect
35. #91298C4semidirect
37. #91099C22semidirect
38. C8×Dic15C22semidirect
39. C5×#821C22semidirect
40. C5×#816C22semidirect
43. #90989C22semidirect
44. #90986C22semidirect
50. #65420D3semidirect
64. C5×#235C2×C4semidirect
70. C2×C3×D20C2×C4semidirect
83. #6710C2×C4semidirect
99. #6655C2×C4semidirect
100. #6652C2×C4semidirect
105. #5133D5semidirect
166. C5×D12C22×C4semidirect
169. C5×Dic6C22×C4semidirect
180. C3⋊#168C22×C4semidirect
207. #821C2×D5semidirect
208. #816C2×D5semidirect
228. C2×D20C4×D3semidirect
285. Dic15AGL(1,ℤ/8)semidirect
304. #235C4×D5semidirect
436. D12C2×C4×D5semidirect
441. Dic6C2×C4×D5semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
581. C5#78914semidirect
604. C3#10610597semidirect

Maximal subgroups

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

subgroupindex
#117544552
C2×#914342
C2×#912982
M16×Dic152
C5×#51332
C3×#654202
#117484722
#117477362
#117477352
#117477342
#117477332
#106105973
#789145