Finite Group Databasebeta

← #14416737

Subgroups — #14416737

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#11754436C211yesmaximal, characteristic, complemented
2C2×Dic3×D20C211yesmaximal, characteristic, complemented
3C2×#91312C211yesmaximal, characteristic
4#11751234C211yesmaximal, characteristic, complemented
5C5×Dic3×M16C211yesmaximal, characteristic, complemented
6C3×#65420C211yesmaximal, characteristic, complemented
7#11748455C211yesmaximal, characteristic, complemented
8#11747742C212yesmaximal, complemented
9#11747741C212yesmaximal, complemented
10#11747740C212yesmaximal, complemented
11#11747739C212yesmaximal, complemented
12C4×#4054—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×D5×Dic3—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C4⋊Dic15C2211yescharacteristic
18C2×C4×C5×Dic3C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C3×#4072C2211yescharacteristic
21#91556—21no—
22#91556—21no—
23C2×#6665—21no—
24#91428C2211yescharacteristic, complemented
25#91428C2211yescharacteristic, complemented
26Dic3×D20—21no—
27Dic3×D20—21no—
28Dic3×D20C2212yes—
29Dic3×D20C2212yes—
30C4×#1200—21no—
31#91386—21no—
32#91340C2211yescharacteristic, complemented
33#91340C2211yescharacteristic, complemented
34#91312C418yescomplemented
35C2×#6657C2212yes—
36C2×#6653C2212yes—
37#91105C2212yescomplemented
38#91100C2212yescomplemented
39C5×#815C2212yescomplemented
40C5×C8×Dic3C2212yescomplemented
41C3×#3953C2212yes—
42C3×#3953C2212yes—
43#90990C2212yescomplemented
44#90987C2212yescomplemented
45#78914—51nomaximal, Hall
46C2×#4054—31no—
47C4×#709—31no—
48C2×C4×D20—31no—
49C4×C5×M16—31no—
50#65422—31no—
51#65420D311yescharacteristic, complemented
52#65410—31no—
53#65000—32no—
54#64999—32no—
55C4×D40—32no—
56C4×#677—32no—
57C23×C3×D5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C2×#1220—21no—
60C22×Dic15—21no—
61#6817—22no—
62#6817C2×C414yescomplemented
63C2×D60C2311yescharacteristic
64C2×C4×D15—21no—
65C2×Dic30C2311yescharacteristic
66C22×C5×Dic3D411yescharacteristic
67C22×C5×Dic3D411yescharacteristic
68C2×C3×D20—21no—
69C2×C3×D20—22no—
70C2×C3×D20C2×C414yescomplemented
71C2×C3×D20C2312yes—
72C2×#1200—21no—
73C2×#1200—21no—
74C2×D5×Dic3—22no—
75C2×D5×Dic3—22no—
76C2×D5×Dic3—44no—
77C3×C5×M16—22no—
78C2×C3×C5×C8C2312yes—
79C4⋊Dic15—21no—
80C4⋊Dic15C2312yes—
81C4×C5×Dic3D411yescharacteristic
82C4×C5×Dic3D411yescharacteristic
83C4×C5×Dic3C2311yescharacteristic
84C4×C5×Dic3C2311yescharacteristic
85C5×#208C2×C414yescomplemented
86C2×C5×C3⋊C8C2312yes—
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5C2311yescharacteristic
89C3×C4⋊Dic5C2311yescharacteristic
90C3×C4×Dic5—21no—
91#6669—21no—
92#6669—21no—
93#6667—21no—
94#6667—21no—
95#6665—22no—
96#6665—22no—
97Dic3×Dic5—21no—
98Dic3×Dic5—21no—
99#6657C2×C418yescomplemented
100#6653C2×C418yescomplemented
101C2×D4×Dic3—51no—
102C2×#932—51no—
103#5522—51no—
104C3×#418—51no—
105#5133—51no—
106Dic3×M16D511yescharacteristic, complemented
107#4995—51no—
108#4785—52no—
109#4758—52no—
110#4757—52no—
111#4756—52no—
112C2×#709—31no—
113C22×D20C2×D311yescharacteristic
114C22×C4×D5—61no—
115C2×C5×M16—31no—
116C2×C5×M16C2×D311yescharacteristic
117C2×C42×C5—31no—
118C4×C22⋊D5—61no—
119C4×C22⋊D5—61no—
120C2×#685—61no—
121#4072C2×D311yescharacteristic
122C2×C4⋊Dic5—31no—
123#4054—38no—
124C2×D40—32no—
125C2×#677—32no—
126C4×D20—32no—
127C4×D20—32no—
128C4×D20—31no—
129C4×D20—31no—
130C4×D20—61no—
131C4×D20—61no—
132#4018—61no—
133C42×D5—61no—
134C4×Dic10—31no—
135C4×Dic10—31no—
136C5×#97—32no—
137C4×C5×C8—32no—
138#3953—32no—
139#3953C2×D312yes—
140#3953C2×D312yes—
141#3950—32no—
142#3949—32no—
143#3948—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—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
155#1220—24no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15—42no—
159D60—21no—
160D60C22×C412yescomplemented
161C4×D15—24no—
162Dic30—21no—
163Dic30C22×C412yescomplemented
164C2×C5×Dic3—22no—
165C2×C5×Dic3—41no—
166C2×C5×Dic3C2×D411yescharacteristic
167C2×C5×Dic3C2×D411yescharacteristic
168C2×C5×Dic3C2×D411yescharacteristic
169C2×C5×Dic3C2×D411yescharacteristic
170C2×C3×Dic5—21no—
171C2×C3×Dic5—21no—
172C3×D20—24no—
173C3×D20—22no—
174C3×D20C22×C414yes—
175#1200—42no—
176#1200—42no—
177D5×Dic3—44no—
178D5×Dic3—44no—
179C3×C5×C8—22no—
180C5×C3⋊C8C22×C414yescomplemented
181C22×C3×D4—51no—
182C23×Dic3—101no—
183C2×#235—51no—
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⋊Dic3—51no—
195C2×C4×Dic3C2×D511yescharacteristic
196C2×#208C2×D511yescharacteristic
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#821—52no—
208#816—52no—
209#815C2×D512yescomplemented
210C8×Dic3C2×D512yescomplemented
211#808—52no—
212#807—52no—
213C23×D5—21no—
214C22×C4×C5—31no—
215C22×C4×C5—31no—
216C22×C4×C5C22×D311yescharacteristic
217C2×C22⋊D5—61no—
218C22×Dic5—61no—
219#709—34no—
220#709—62no—
221C2×D20—21no—
222C2×D20—22no—
223C2×D20—31no—
224C2×D20C4×D314yescomplemented
225C2×D20C22×D312yes—
226C2×C4×D5—61no—
227C2×C4×D5—61no—
228C2×C4×D5—61no—
229C2×C4×D5—62no—
230C2×C4×D5—62no—
231C2×C4×D5—124no—
232C2×Dic10—31no—
233C5×M16—22no—
234C5×M16—34no—
235C2×C5×C8—32no—
236C2×C5×C8C22×D312yes—
237C42×C5—31no—
238C42×C5—31no—
239C42×C5—31no—
240C42×C5—31no—
241C22⋊Dic5—21no—
242#685—61no—
243#685—61no—
244#685—62no—
245#685—62no—
246C4⋊Dic5—32no—
247C4⋊Dic5—61no—
248C4⋊Dic5C22×D311yescharacteristic
249C4⋊Dic5C22×D311yescharacteristic
250#683—61no—
251#683—61no—
252C4×Dic5—21no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255D40—38no—
256#677—38no—
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—151no—
267C4×M16—31no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×D15—22no—
273C2×D15—41no—
274C2×D15—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—22no—
283Dic15—42no—
284C3×Dic5—41no—
285C3×Dic5—41no—
286C5×Dic3—21no—
287C5×Dic3—21no—
288C5×Dic3—21no—
289C5×Dic3AGL(1,ℤ/8)12yescomplemented
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—101no—
297C22×Dic3—101no—
298C22×Dic3—102no—
299C22×Dic3—102no—
300C22×Dic3—204no—
301C22×Dic3D2011yescharacteristic
302C22×Dic3D2011yescharacteristic
303#235—54no—
304#235—102no—
305C2×D12—51no—
306C2×C4×D3—101no—
307C2×C4×D3—101no—
308C2×C4×D3—101no—
309C2×Dic6—51no—
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—52no—
323C4⋊Dic3—101no—
324Dic3⋊C4—101no—
325Dic3⋊C4—101no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3D2011yescharacteristic
329C4×Dic3D2011yescharacteristic
330C4×Dic3C22×D511yescharacteristic
331C4×Dic3C22×D511yescharacteristic
332#208C4×D514yescomplemented
333C2×C3⋊C8C22×D512yes—
334C23×C5D3×D411yescharacteristic
335C22×D5—24no—
336C22×D5—22no—
337C22×D5—42no—
338C22×D5—42no—
339C22×D5—61no—
340C2×C4×C5—31no—
341C2×C4×C5—31no—
342C2×C4×C5—31no—
343C2×C4×C5—31no—
344C2×C4×C5—62no—
345C2×C4×C5—121no—
346C2×C4×C5C2×C4×D312yes—
347C2×C4×C5D3×D411yescharacteristic
348C2×C4×C5#23712yes—
349C2×C4×C5C23×D311yescharacteristic
350C22⋊D5—64no—
351C2×Dic5—21no—
352C2×Dic5—21no—
353C2×Dic5—61no—
354C2×Dic5—62no—
355C2×Dic5—122no—
356D20—24no—
357D20—22no—
358D20—32no—
359D20—61no—
360D20C2×C4×D314yes—
361C4×D5—64no—
362C4×D5—122no—
363C4×D5—122no—
364C4×D5—124no—
365C4×D5—124no—
366Dic10—32no—
367Dic10—61no—
368C5×C8—22no—
369C5×C8—34no—
370C2×#41—151no—
371C22×D4—51no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—31no—
377C2×M16D3×D511yescharacteristic
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—151no—
381C4×D4—151no—
382C4×D4—152no—
383C4×D4—152no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—51no—
389#117—301no—
390C2×C4⋊C4—151no—
391C2×C22⋊C4—301no—
392C2×C42—31no—
393C2×C42—301no—
394#107—152no—
395#106—152no—
396#103—152no—
397#102—52no—
398#102—52no—
399#102—152no—
400#97—32no—
401C4×C8—32no—
402C2×C3×C5—22no—
403C2×C3×C5C2×C4×D411yescharacteristic
404C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C3×C5#57511yescharacteristic
406D15—42no—
407C3×D5—44no—
408C23×C3—102no—
409C23×C3—104no—
410C23×C3—202no—
411C23×C3—202no—
412C23×C3C2×D2011yescharacteristic
413C22×D3—101no—
414C3×D4—54no—
415C3×D4—102no—
416C3×D4—104no—
417C2×C3×C4—101no—
418C2×C3×C4—101no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4C2×D2011yescharacteristic
421C2×C3×C4#70912yes—
422C2×C3×C4C23×D511yescharacteristic
423#67—104no—
424C2×Dic3—22no—
425C2×Dic3—41no—
426C2×Dic3—101no—
427C2×Dic3—102no—
428C2×Dic3—204no—
429C2×Dic3—202no—
430C2×Dic3—204no—
431C2×Dic3C2×D2011yescharacteristic
432C2×Dic3C2×D2011yescharacteristic
433C2×Dic3C2×D2011yescharacteristic
434C2×Dic3C2×D2011yescharacteristic
435D12—52no—
436D12—101no—
437C4×D3—104no—
438C4×D3—202no—
439C4×D3—202no—
440Dic6—52no—
441Dic6—101no—
442C3×C8—22no—
443C3⋊C8C2×C4×D514yescomplemented
444C22×C5—21no—
445C22×C5—21no—
446C22×C5#67⋊C412yes—
447C22×C5C2×D3×D411yescharacteristic
448C2×D5—24no—
449C2×D5—48no—
450C2×D5—42no—
451C2×D5—42no—
452C2×D5—62no—
453C2×D5—121no—
454C2×D5—121no—
455C4×C5—32no—
456C4×C5—61no—
457C4×C5—61no—
458C4×C5—61no—
459C4×C5#67⋊C412yes—
460C4×C5C22×C4×D311yescharacteristic
461C4×C5C2×#23711yescharacteristic
462Dic5—41no—
463Dic5—41no—
464Dic5—62no—
465Dic5—122no—
466C24—101no—
467#41—154no—
468#41—302no—
469C2×Q8—151no—
470C2×D4—54no—
471C2×D4—52no—
472C2×D4—101no—
473C2×D4—102no—
474C2×D4—151no—
475C2×D4—301no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—301no—
479C22×C4—301no—
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—34no—
490C2×C8—32no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—51no—
493C4⋊C4—51no—
494C4⋊C4—152no—
495C4⋊C4—301no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—101no—
499C22⋊C4—301no—
500C22⋊C4—301no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—31no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—101no—
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×C3C4×D2012yes—
518C22×C3C22×D2011yescharacteristic
519C2×D3—102no—
520C2×D3—201no—
521C2×D3—201no—
522C3×C4—201no—
523C3×C4—201no—
524C3×C4C4×D2012yes—
525C3×C4C22×C4×D511yescharacteristic
526C3×C4C2×#70911yescharacteristic
527Dic3—21no—
528Dic3—21no—
529Dic3—21no—
530Dic3—102no—
531Dic3—202no—
532Dic3#405412yescomplemented
533C2×C5—22no—
534C2×C5C2×#67⋊C411yescharacteristic
535C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
536C2×C5#577311yescharacteristic
537D5—44no—
538D5—122no—
539C23—104no—
540C23—102no—
541C23—202no—
542C23—202no—
543C23—301no—
544C23D3×D2011yescharacteristic
545Q8—152no—
546Q8—301no—
547D4—54no—
548D4—104no—
549D4—102no—
550D4—152no—
551D4—304no—
552D4—301no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—62no—
558C2×C4—101no—
559C2×C4—101no—
560C2×C4—121no—
561C2×C4—301no—
562C2×C4—304no—
563C2×C4—302no—
564C2×C4—602no—
565C2×C4—604no—
566C2×C4—602no—
567C2×C4—604no—
568C2×C4—602no—
569C2×C4C4×D3×D512yes—
570C2×C4D3×D2011yescharacteristic
571C2×C4#677912yes—
572C2×C4C22×D3×D511yescharacteristic
573C8—22no—
574C8—34no—
575C2×C3—22no—
576C2×C3—204no—
577C2×C3C2×C4×D2011yescharacteristic
578C2×C3C2×#405411yescharacteristic
579C2×C3#6608211yescharacteristic
580D3—202no—
581C5#7891411yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—104no—
585C22—208no—
586C22—202no—
587C22—202no—
588C22—302no—
589C22—601no—
590C22—601no—
591C22#9139512yes—
592C22C2×D3×D2011yescharacteristic
593C4—32no—
594C4—61no—
595C4—61no—
596C4—61no—
597C4—201no—
598C4—201no—
599C4—302no—
600C4—602no—
601C4#9139512yes—
602C4C2×C4×D3×D511yescharacteristic
603C4C2×#677911yescharacteristic
604C3C4×#405411yescharacteristic, complemented, Sylow
605C2—22no—
606C2—204no—
607C2—602no—
608C2D3×#405411yescharacteristic
609C2#1175402911yescharacteristic
610C2C2×#9139511yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#91312C2
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. #11754436C2semidirect
2. C2×Dic3×D20C2semidirect
4. #11751234C2semidirect
5. C5×Dic3×M16C2semidirect
6. C3×#65420C2semidirect
7. #11748455C2semidirect
8. #11747742C2semidirect
9. #11747741C2semidirect
10. #11747740C2semidirect
11. #11747739C2semidirect
24. #91428C22semidirect
25. #91428C22semidirect
32. #91340C22semidirect
33. #91340C22semidirect
34. #91312C4semidirect
37. #91105C22semidirect
38. #91100C22semidirect
39. C5×#815C22semidirect
40. C5×C8×Dic3C22semidirect
43. #90990C22semidirect
44. #90987C22semidirect
51. #65420D3semidirect
62. #6817C2×C4semidirect
70. C2×C3×D20C2×C4semidirect
85. C5×#208C2×C4semidirect
99. #6657C2×C4semidirect
100. #6653C2×C4semidirect
106. Dic3×M16D5semidirect
160. D60C22×C4semidirect
163. Dic30C22×C4semidirect
180. C5×C3⋊C8C22×C4semidirect
209. #815C2×D5semidirect
210. C8×Dic3C2×D5semidirect
224. C2×D20C4×D3semidirect
289. C5×Dic3AGL(1,ℤ/8)semidirect
332. #208C4×D5semidirect
443. C3⋊C8C2×C4×D5semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
532. Dic3#4054semidirect
581. C5#78914semidirect
604. C3C4×#4054semidirect

Maximal subgroups

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

subgroupindex
#117544362
C2×Dic3×D202
C2×#913122
#117512342
C5×Dic3×M162
C3×#654202
#117484552
#117477422
#117477412
#117477402
#117477392
C4×#40543
#789145