Finite Group Databasebeta

← #14416736

Subgroups — #14416736

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#11754444C211yesmaximal, characteristic, complemented
2C2×#91435C211yesmaximal, characteristic, complemented
3C2×#91299C211yesmaximal, characteristic
4M16×Dic15C211yesmaximal, characteristic, complemented
5C5×#5131C211yesmaximal, characteristic, complemented
6C3×#65422C211yesmaximal, characteristic, complemented
7#11748470C211yesmaximal, characteristic, complemented
8#11747738C212yesmaximal, complemented
9#11747737C212yesmaximal, complemented
10#11747734C212yesmaximal, complemented
11#11747733C212yesmaximal, complemented
12#10610645—31nomaximal, Hall
13C22×C5×D12C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C22×D3×Dic5—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C4×Dic15C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C5×#926C2211yescharacteristic
20C2×C3×C4⋊Dic5C2211yescharacteristic
21#91559—21no—
22#91559—21no—
23C2×(C2×D3)⋊Dic5—21no—
24#91435—21no—
25#91435—21no—
26#91435C2212yes—
27#91435C2212yes—
28#91434C2211yescharacteristic, complemented
29#91434C2211yescharacteristic, complemented
30C4×D5×Dic3—21no—
31#91357—21no—
32#91331C2211yescharacteristic, complemented
33#91331C2211yescharacteristic, complemented
34C2×#6656C2212yes—
35#91299C418yescomplemented
36C2×#6652C2212yes—
37#91099C2212yescomplemented
38C8×Dic15C2212yescomplemented
39C5×#821C2212yes—
40C5×#821C2212yes—
41C3×#3953C2212yescomplemented
42C3×#3948C2212yescomplemented
43#90989C2212yescomplemented
44#90986C2212yescomplemented
45#78866—51nomaximal, Hall
46C2×D4×Dic5—31no—
47C2×#4078—31no—
48#65856—31no—
49C5×#418—31no—
50#65422D311yescharacteristic, complemented
51M16×Dic5—31no—
52#65284—31no—
53#65074—32no—
54#65047—32no—
55#65046—32no—
56#65045—32no—
57C23×C5×D3—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15D411yescharacteristic
60C22×Dic15D411yescharacteristic
61C2×C5×D12—21no—
62C2×C5×D12—22no—
63C2×C5×D12C2×C414yescomplemented
64C2×C5×D12C2312yes—
65C2×C3×C22⋊D5—21no—
66C22×C3×Dic5—21no—
67C3×#709—22no—
68C3×#709C2×C414yescomplemented
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72C2×D3×Dic5—22no—
73C2×D3×Dic5—22no—
74C2×D3×Dic5—44no—
75C2×D5×Dic3—21no—
76C2×D5×Dic3—21no—
77C3×C5×M16—22no—
78C2×C3×C5×C8C2312yes—
79C4×Dic15D411yescharacteristic
80C4×Dic15D411yescharacteristic
81C4×Dic15C2311yescharacteristic
82C4×Dic15C2311yescharacteristic
83#6710C2×C414yescomplemented
84C2×C3⋊#168C2312yes—
85C5×#217—21no—
86C5×C4⋊Dic3C2311yescharacteristic
87C5×C4⋊Dic3C2311yescharacteristic
88C4×C5×Dic3—21no—
89C3×C4⋊Dic5—21no—
90C3×C4⋊Dic5C2312yes—
91#6668—21no—
92#6668—21no—
93(C2×D3)⋊Dic5—22no—
94(C2×D3)⋊Dic5—22no—
95#6665—21no—
96#6665—21no—
97Dic3×Dic5—21no—
98Dic3×Dic5—21no—
99#6656C2×C418yescomplemented
100#6652C2×C418yescomplemented
101Dic3×#41—51no—
102C2×#949—51no—
103C2×#893—51no—
104C3×#417—51no—
105#5131D511yescharacteristic, complemented
106Dic3×M16—51no—
107#4964—51no—
108#4817—52no—
109#4788—52no—
110#4785—52no—
111#4756—52no—
112C22×C5×D4—31no—
113C23×Dic5—61no—
114C2×#709C2×D311yescharacteristic
115C2×C5×M16C2×D311yescharacteristic
116C5×#117—31no—
117C2×C22⋊Dic5—61no—
118D4×Dic5—32no—
119D4×Dic5—32no—
120D4×Dic5—61no—
121D4×Dic5—61no—
122C2×#687—32no—
123#4078—38no—
124C2×#686—32no—
125C2×C4⋊Dic5C2×D311yescharacteristic
126C2×C4×Dic5—31no—
127C2×#681—31no—
128#4039—31no—
129#4039—31no—
130#4033—31no—
131#4033—31no—
132#4027—61no—
133#4027—61no—
134#4018—61no—
135C42×D5—61no—
136C5×#102—32no—
137C5×#102—32no—
138#3953C2×D312yescomplemented
139#3948C2×D312yescomplemented
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—24no—
147C22×C5×D3—22no—
148C22×C5×D3—42no—
149C22×C5×D3—42no—
150C22×C3×D5—21no—
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
161C2×C5×Dic3—21no—
162C2×C5×Dic3—21no—
163C5×D12—24no—
164C5×D12—22no—
165C5×D12C22×C414yes—
166C3×C22⋊D5—24no—
167C2×C3×Dic5—21no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—42no—
170C3×D20—21no—
171C3×D20C22×C412yescomplemented
172C3×C4×D5—24no—
173C3×Dic10—21no—
174C3×Dic10C22×C412yescomplemented
175D3×Dic5—44no—
176D3×Dic5—44no—
177D5×Dic3—42no—
178D5×Dic3—42no—
179C3×C5×C8—22no—
180C3⋊#168C22×C414yescomplemented
181C2×C3×#41—51no—
182C22×D12C2×D511yescharacteristic
183C22×C4×D3—101no—
184C2×C3×M16C2×D511yescharacteristic
185C2×C3×C4⋊C4—51no—
186#949—58no—
187Q8×Dic3—51no—
188Q8×Dic3—51no—
189C2×#215—52no—
190D4×Dic3—51no—
191D4×Dic3—51no—
192D4×Dic3—101no—
193D4×Dic3—101no—
194C2×#213—52no—
195C2×(C2×D3)⋊C4—101no—
196#926—101no—
197#926C2×D511yescharacteristic
198C2×C4×Dic3—51no—
199C2×C4×Dic3—101no—
200C2×#208—51no—
201#893—52no—
202#893—52no—
203#893—101no—
204#893—101no—
205C3×#103—52no—
206C3×#102—52no—
207#821C2×D512yes—
208#821C2×D512yes—
209#815—52no—
210C8×Dic3—52no—
211#808—52no—
212#807—52no—
213C24×C5—61no—
214C2×C5×D4—34no—
215C2×C5×D4—32no—
216C2×C5×D4—61no—
217C2×C5×D4—62no—
218C22×C4×C5C22×D311yescharacteristic
219C2×C22⋊D5—21no—
220C22×Dic5—21no—
221C22×Dic5—31no—
222C22×Dic5—31no—
223C22×Dic5—62no—
224C22×Dic5—62no—
225C22×Dic5—124no—
226#709—22no—
227#709C4×D314yescomplemented
228C2×D20C22×D311yescharacteristic
229C2×C4×D5—21no—
230C2×C4×D5—61no—
231C2×C4×D5—61no—
232C2×Dic10C22×D311yescharacteristic
233C5×M16—22no—
234C2×C5×C8C22×D312yes—
235C5×C4⋊C4—31no—
236C5×C4⋊C4—31no—
237C5×C22⋊C4—61no—
238C42×C5—61no—
239C22⋊Dic5—62no—
240C22⋊Dic5—62no—
241#687—38no—
242#686—38no—
243#685—61no—
244#685—61no—
245C4⋊Dic5—21no—
246C4⋊Dic5C22×D312yes—
247#683—61no—
248#683—61no—
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—31no—
266#417—51no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×C5×D3—24no—
273C2×C5×D3—48no—
274C2×C5×D3—42no—
275C2×C5×D3—42no—
276C2×C3×D5—22no—
277C2×C3×D5—41no—
278C2×C3×D5—41no—
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—22no—
287C3×Dic5—42no—
288C5×Dic3—41no—
289C5×Dic3—41no—
290C23×D3—21no—
291C3×#41—54no—
292C3×#41—102no—
293C2×C3×Q8—51no—
294C2×C3×D4—51no—
295C2×C3×D4—101no—
296C22×C3×C4—101no—
297C22×C3×C4—101no—
298C22×C3×C4C22×D511yescharacteristic
299C22×Dic3—51no—
300C22×Dic3—51no—
301C22×Dic3—101no—
302C22×Dic3—101no—
303C2×D12—21no—
304C2×D12—22no—
305C2×D12C4×D514yescomplemented
306C2×D12C22×D512yes—
307C2×C4×D3—102no—
308C2×C4×D3—102no—
309C2×C4×D3—204no—
310C3×M16—22no—
311C2×C3×C8C22×D512yes—
312C3×C4⋊C4—52no—
313C3×C4⋊C4—101no—
314#217—21no—
315#217—101no—
316#217—101no—
317#215—58no—
318#213—58no—
319(C2×D3)⋊C4—102no—
320(C2×D3)⋊C4—102no—
321C4⋊Dic3—101no—
322C4⋊Dic3—101no—
323C4⋊Dic3C22×D511yescharacteristic
324C4⋊Dic3C22×D511yescharacteristic
325C4×Dic3—21no—
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—64no—
335C23×C5—62no—
336C23×C5—122no—
337C23×C5—122no—
338C23×C5D3×D411yescharacteristic
339C22×D5—21no—
340C5×D4—34no—
341C5×D4—64no—
342C5×D4—62no—
343C2×C4×C5—61no—
344C2×C4×C5—61no—
345C2×C4×C5C2×C4×D312yes—
346C2×C4×C5D3×D411yescharacteristic
347C2×C4×C5#23712yes—
348C2×C4×C5C23×D311yescharacteristic
349C22⋊D5—24no—
350C2×Dic5—21no—
351C2×Dic5—22no—
352C2×Dic5—31no—
353C2×Dic5—31no—
354C2×Dic5—31no—
355C2×Dic5—31no—
356C2×Dic5—42no—
357C2×Dic5—62no—
358C2×Dic5—124no—
359C2×Dic5—124no—
360C2×Dic5—121no—
361D20—21no—
362D20C2×C4×D312yescomplemented
363C4×D5—24no—
364C4×D5—122no—
365C4×D5—122no—
366Dic10—21no—
367Dic10C2×C4×D312yescomplemented
368C5×C8—22no—
369#168—34no—
370C2×#41—51no—
371C22×D4—31no—
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—151no—
381C4×D4—151no—
382C4×D4—152no—
383C4×D4—152no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—31no—
389#117—301no—
390C2×C4⋊C4—51no—
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107—152no—
395#106—152no—
396#103—52no—
397#102—32no—
398#102—32no—
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—42no—
407C5×D3—44no—
408C23×C3—101no—
409C23×C3D4×D511yescharacteristic
410C22×D3—22no—
411C22×D3—24no—
412C22×D3—42no—
413C22×D3—42no—
414C3×Q8—52no—
415C3×Q8—101no—
416C3×D4—52no—
417C3×D4—104no—
418C3×D4—101no—
419C2×C3×C4—101no—
420C2×C3×C4—104no—
421C2×C3×C4—102no—
422C2×C3×C4—202no—
423C2×C3×C4C2×C4×D512yes—
424C2×C3×C4D4×D511yescharacteristic
425C2×C3×C4#71112yes—
426C2×C3×C4C23×D511yescharacteristic
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—51no—
430C2×Dic3—51no—
431C2×Dic3—51no—
432C2×Dic3—51no—
433C2×Dic3—102no—
434C2×Dic3—202no—
435C2×Dic3—202no—
436C2×Dic3—201no—
437D12—22no—
438D12—24no—
439D12C2×C4×D514yes—
440C4×D3—204no—
441C4×D3—204no—
442C3×C8—22no—
443C3⋊C8—54no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—64no—
447C22×C5—128no—
448C22×C5—122no—
449C22×C5—122no—
450C22×C5#67⋊C412yes—
451C22×C5C2×D3×D411yescharacteristic
452C2×D5—22no—
453C2×D5—41no—
454C2×D5—41no—
455C4×C5—121no—
456C4×C5—121no—
457C4×C5#67⋊C412yes—
458C4×C5C22×C4×D311yescharacteristic
459C4×C5C2×#23711yescharacteristic
460Dic5—22no—
461Dic5—32no—
462Dic5—42no—
463Dic5—61no—
464Dic5—61no—
465Dic5—61no—
466C24—61no—
467#41—54no—
468#41—102no—
469C2×Q8—51no—
470C2×D4—34no—
471C2×D4—32no—
472C2×D4—51no—
473C2×D4—61no—
474C2×D4—62no—
475C2×D4—101no—
476C22×C4—101no—
477C22×C4—101no—
478C22×C4—151no—
479C22×C4—151no—
480C22×C4—301no—
481C22×C4—301no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—22no—
489M16—154no—
490C2×C8—152no—
491C2×C8C2×D3×D512yes—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—52no—
495C4⋊C4—101no—
496C4⋊C4—301no—
497C4⋊C4—301no—
498C22⋊C4—61no—
499C22⋊C4—301no—
500C22⋊C4—301no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—61no—
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—102no—
514C22×C3—201no—
515C22×C3—201no—
516C22×C3#402712yes—
517C22×C3C2×D4×D511yescharacteristic
518C2×D3—24no—
519C2×D3—48no—
520C2×D3—42no—
521C2×D3—42no—
522C3×C4—102no—
523C3×C4—202no—
524C3×C4#402712yes—
525C3×C4C22×C4×D511yescharacteristic
526C3×C4C2×#71111yescharacteristic
527Dic3—41no—
528Dic3—41no—
529Dic3—52no—
530Dic3—101no—
531Dic3—101no—
532Dic3—101no—
533C2×C5—22no—
534C2×C5—124no—
535C2×C5C2×#67⋊C411yescharacteristic
536C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
537C2×C5#577711yescharacteristic
538D5—42no—
539C23—64no—
540C23—62no—
541C23—101no—
542C23—122no—
543C23—122no—
544C23#677711yescharacteristic
545Q8—52no—
546Q8—101no—
547D4—34no—
548D4—52no—
549D4—64no—
550D4—62no—
551D4—104no—
552D4—101no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—101no—
556C2×C4—104no—
557C2×C4—102no—
558C2×C4—151no—
559C2×C4—151no—
560C2×C4—151no—
561C2×C4—151no—
562C2×C4—202no—
563C2×C4—302no—
564C2×C4—602no—
565C2×C4—602no—
566C2×C4—604no—
567C2×C4—604no—
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—202no—
577C2×C3C2×#402711yescharacteristic
578C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
579C2×C3#6610711yescharacteristic
580D3—44no—
581C5#7886611yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—64no—
585C22—102no—
586C22—128no—
587C22—122no—
588C22—122no—
589C22—201no—
590C22—201no—
591C22#9144212yes—
592C22C2×#677711yescharacteristic
593C4—102no—
594C4—121no—
595C4—121no—
596C4—152no—
597C4—202no—
598C4—301no—
599C4—301no—
600C4—301no—
601C4#9144212yes—
602C4C2×C4×D3×D511yescharacteristic
603C4C2×#678011yescharacteristic
604C3#1061064511yescharacteristic, complemented, Sylow
605C2—22no—
606C2—124no—
607C2—202no—
608C2#1175403311yescharacteristic
609C2#1175403511yescharacteristic
610C2C2×#9144211yescharacteristic
subgroupG/H
centerC22C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C22×D4
prime radicalC2×#91299C2
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. #11754444C2semidirect
2. C2×#91435C2semidirect
4. M16×Dic15C2semidirect
5. C5×#5131C2semidirect
6. C3×#65422C2semidirect
7. #11748470C2semidirect
8. #11747738C2semidirect
9. #11747737C2semidirect
10. #11747734C2semidirect
11. #11747733C2semidirect
28. #91434C22semidirect
29. #91434C22semidirect
32. #91331C22semidirect
33. #91331C22semidirect
35. #91299C4semidirect
37. #91099C22semidirect
38. C8×Dic15C22semidirect
41. C3×#3953C22semidirect
42. C3×#3948C22semidirect
43. #90989C22semidirect
44. #90986C22semidirect
50. #65422D3semidirect
63. C2×C5×D12C2×C4semidirect
68. C3×#709C2×C4semidirect
83. #6710C2×C4semidirect
99. #6656C2×C4semidirect
100. #6652C2×C4semidirect
105. #5131D5semidirect
138. #3953C2×D3semidirect
139. #3948C2×D3semidirect
171. C3×D20C22×C4semidirect
174. C3×Dic10C22×C4semidirect
180. C3⋊#168C22×C4semidirect
227. #709C4×D3semidirect
285. Dic15AGL(1,ℤ/8)semidirect
305. C2×D12C4×D5semidirect
362. D20C2×C4×D3semidirect
367. Dic10C2×C4×D3semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
581. C5#78866semidirect
604. C3#10610645semidirect

Maximal subgroups

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

subgroupindex
#117544442
C2×#914352
C2×#912992
M16×Dic152
C5×#51312
C3×#654222
#117484702
#117477382
#117477372
#117477342
#117477332
#106106453
#788665