Finite Group Databasebeta

← #14416719

Subgroups — #14416719

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
1C5×#5522C211yesmaximal, characteristic, complemented
2C2×Dic3×D20C211yesmaximal, characteristic, complemented
3C2×#91298C211yesmaximal, characteristic
4#11751096C211yesmaximal, characteristic, complemented
5C3×#65290C211yesmaximal, characteristic, complemented
6#11748390C211yesmaximal, characteristic, complemented
7Dic3×#681C211yesmaximal, characteristic, complemented
8#11747222C212yesmaximal, complemented
9#11747221C212yesmaximal, complemented
10#11747220C212yesmaximal, complemented
11#11747219C212yesmaximal, complemented
12C4×#4095—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C22×D5×Dic3—21no—
16C3×C5×#117C2211yescharacteristic
17C2×C4⋊Dic15C2211yescharacteristic
18C2×#6710C2211yescharacteristic
19C2×C4×C5×Dic3C2211yescharacteristic
20C5×#893C2211yescharacteristic, complemented
21C5×#893C2211yescharacteristic, complemented
22C5×#887C2211yescharacteristic, complemented
23C5×#887C2211yescharacteristic, complemented
24C5×#67⋊C4—21no—
25C5×#67⋊C4—21no—
26C5×#872—21no—
27C42×C5×D3—21no—
28C2×C3×#681C2211yescharacteristic
29C2×#6665—21no—
30Dic3×D20—21no—
31Dic3×D20—21no—
32Dic3×D20C2212yes—
33Dic3×D20C2212yes—
34C2×#6655C2212yes—
35#91298C418yescomplemented
36C2×#6652C2212yes—
37#91092C2212yescomplemented
38#91091C2212yescomplemented
39C3×#3941C2212yes—
40C3×#3941C2212yes—
41#90973C2212yescomplemented
42#90967C2212yescomplemented
43#90953C2212yescomplemented
44Dic3×#168C2212yescomplemented
45#78914—51nomaximal, Hall
46C4×C5×#41—31no—
47C2×#4095—31no—
48C2×C4×D20—31no—
49#65521—31no—
50#65315—32no—
51C4×#688—32no—
52#65303—32no—
53C4×#686—32no—
54#65290D311yescharacteristic, complemented
55#65284—31no—
56C4×#681—31no—
57C23×C3×D5—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C22×Dic15—21no—
60C2×C5×#67—21no—
61C22×C5×Dic3D411yescharacteristic
62C22×C5×Dic3D411yescharacteristic
63C5×#235—22no—
64C5×#235C2×C414yescomplemented
65C2×C5×D12C2311yescharacteristic
66C2×C4×C5×D3—21no—
67C2×C4×C5×D3—21no—
68C2×C4×C5×D3—21no—
69C2×C5×Dic6C2311yescharacteristic
70C2×C3×D20—21no—
71C2×C3×D20—22no—
72C2×C3×D20C2×C414yescomplemented
73C2×C3×D20C2312yes—
74C2×D5×Dic3—22no—
75C2×D5×Dic3—22no—
76C2×D5×Dic3—44no—
77C3×C5×C4⋊C4C2311yescharacteristic
78C3×C5×C4⋊C4C2311yescharacteristic
79C3×C5×C22⋊C4—21no—
80C3×C42×C5—21no—
81C4⋊Dic15—21no—
82C4⋊Dic15C2312yes—
83#6710C2×C414yescomplemented
84C2×C3⋊#168C2312yes—
85C5×(C2×D3)⋊C4—21no—
86C5×(C2×D3)⋊C4—21no—
87C5×Dic3⋊C4—21no—
88C5×Dic3⋊C4—21no—
89C4×C5×Dic3—21no—
90C4×C5×Dic3—21no—
91C4×C5×Dic3D411yescharacteristic
92C4×C5×Dic3D411yescharacteristic
93C4×C5×Dic3C2311yescharacteristic
94C4×C5×Dic3C2311yescharacteristic
95C3×#681—22no—
96C2×C3×#168C2312yes—
97#6665—22no—
98#6665—22no—
99#6655C2×C418yescomplemented
100#6652C2×C418yescomplemented
101C2×D4×Dic3—51no—
102C2×#932—51no—
103#5522D511yescharacteristic, complemented
104C3×#418—51no—
105#5133—51no—
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—
115C4×C5×Q8—31no—
116C4×C5×Q8—31no—
117C4×C5×D4—31no—
118C4×C5×D4—31no—
119C4×C5×D4—61no—
120C4×C5×D4—61no—
121C5×#117—61no—
122C5×#117C2×D311yescharacteristic
123C2×C42×C5—31no—
124C2×C42×C5—61no—
125#4095—38no—
126C2×#688—32no—
127C2×#686—32no—
128C2×#685—61no—
129C2×C4⋊Dic5—31no—
130C2×#681—31no—
131C2×#681C2×D311yescharacteristic
132C4×D20—32no—
133C4×D20—32no—
134C4×D20—61no—
135C4×D20—61no—
136#3967—32no—
137#3964—32no—
138#3941C2×D312yes—
139#3941C2×D312yes—
140#3940—32no—
141#3939—32no—
142#3935—32no—
143C4×#168—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×C5—21no—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5C22×C412yes—
154C2×C3×C4×C5C2×D411yescharacteristic
155C2×C3×C4×C5#4112yes—
156C2×C3×C4×C5C2411yescharacteristic
157C2×Dic15—22no—
158C2×Dic15—42no—
159C5×#67—24no—
160C2×C5×Dic3—21no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—41no—
163C2×C5×Dic3C2×D411yescharacteristic
164C2×C5×Dic3C2×D411yescharacteristic
165C2×C5×Dic3C2×D411yescharacteristic
166C2×C5×Dic3C2×D411yescharacteristic
167C5×D12—21no—
168C5×D12C22×C412yescomplemented
169C4×C5×D3—24no—
170C4×C5×D3—42no—
171C4×C5×D3—42no—
172C5×Dic6—21no—
173C5×Dic6C22×C412yescomplemented
174C3×D20—24no—
175C3×D20—22no—
176C3×D20C22×C414yes—
177D5×Dic3—44no—
178D5×Dic3—44no—
179C3⋊#168C22×C414yescomplemented
180C3×#168—22no—
181C22×C3×D4—51no—
182C23×Dic3—101no—
183C2×#235C2×D511yescharacteristic
184C2×C3×M16—51no—
185C3×#117C2×D511yescharacteristic
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×#208—51no—
197#893C2×D511yescharacteristic, complemented
198#893C2×D511yescharacteristic, complemented
199#887C2×D511yescharacteristic, complemented
200#887C2×D511yescharacteristic, complemented
201#67⋊C4—21no—
202#67⋊C4—21no—
203#872—21no—
204C42×D3—21no—
205C3×#102—52no—
206C3×#102—52no—
207#821—52no—
208#816—52no—
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—31no—
220C22×C4×C5—31no—
221C22×C4×C5—61no—
222C22×C4×C5—61no—
223C22×C4×C5—61no—
224C22×C4×C5C22×D311yescharacteristic
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×C4⋊C4—61no—
234C5×C4⋊C4—61no—
235C5×C4⋊C4C22×D311yescharacteristic
236C5×C4⋊C4C22×D311yescharacteristic
237C5×C22⋊C4—21no—
238C5×C22⋊C4—61no—
239C5×C22⋊C4—61no—
240C42×C5—21no—
241C42×C5—31no—
242C42×C5—31no—
243C42×C5—31no—
244C42×C5—31no—
245C42×C5—61no—
246C42×C5—61no—
247#688—38no—
248#686—38no—
249#685—62no—
250#685—62no—
251C4⋊Dic5—32no—
252C4⋊Dic5—61no—
253#681—22no—
254#681—34no—
255C2×#168—32no—
256C2×#168C22×D312yes—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—31no—
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—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×C5—41no—
280C3×C4×C5—41no—
281C3×C4×C5C4×D412yes—
282C3×C4×C5C23×C411yescharacteristic
283C3×C4×C5C2×#4111yescharacteristic
284Dic15—42no—
285C5×Dic3—22no—
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—102no—
294C2×C3×D4—101no—
295C22×C3×C4C22×D511yescharacteristic
296C2×#67—21no—
297C22×Dic3—101no—
298C22×Dic3—102no—
299C22×Dic3—102no—
300C22×Dic3—204no—
301C22×Dic3C22⋊D511yescharacteristic
302C22×Dic3C22⋊D511yescharacteristic
303#235—22no—
304#235C4×D514yescomplemented
305C2×D12C22×D511yescharacteristic
306C2×C4×D3—21no—
307C2×C4×D3—21no—
308C2×C4×D3—21no—
309C2×Dic6C22×D511yescharacteristic
310C3×M16—102no—
311C2×C3×C8—52no—
312C3×C4⋊C4C22×D511yescharacteristic
313C3×C4⋊C4C22×D511yescharacteristic
314C3×C22⋊C4—21no—
315C3×C42—21no—
316#217—102no—
317#217—102no—
318#214—58no—
319#213—58no—
320(C2×D3)⋊C4—21no—
321(C2×D3)⋊C4—21no—
322C4⋊Dic3—52no—
323C4⋊Dic3—101no—
324Dic3⋊C4—21no—
325Dic3⋊C4—21no—
326C4×Dic3—21no—
327C4×Dic3—21no—
328C4×Dic3C22⋊D511yescharacteristic
329C4×Dic3C22⋊D511yescharacteristic
330C4×Dic3C22×D511yescharacteristic
331C4×Dic3C22×D511yescharacteristic
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—21no—
346C2×C4×C5—21no—
347C2×C4×C5—31no—
348C2×C4×C5—31no—
349C2×C4×C5—31no—
350C2×C4×C5—31no—
351C2×C4×C5—62no—
352C2×C4×C5—61no—
353C2×C4×C5—64no—
354C2×C4×C5—122no—
355C2×C4×C5—122no—
356C2×C4×C5—121no—
357C2×C4×C5C2×C4×D312yes—
358C2×C4×C5D3×D411yescharacteristic
359C2×C4×C5#23712yes—
360C2×C4×C5C23×D311yescharacteristic
361C2×Dic5—62no—
362C2×Dic5—122no—
363D20—24no—
364D20—22no—
365D20C2×C4×D314yes—
366C4×D5—124no—
367C4×D5—124no—
368#168—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—51no—
377C2×M16—151no—
378C4×Q8—31no—
379C4×Q8—31no—
380C4×D4—31no—
381C4×D4—31no—
382C4×D4—61no—
383C4×D4—61no—
384C4×D4—152no—
385C4×D4—152no—
386C4×D4—301no—
387C4×D4—301no—
388#117—61no—
389#117D3×D511yescharacteristic
390C2×C4⋊C4—151no—
391C2×C22⋊C4—301no—
392C2×C42—31no—
393C2×C42—61no—
394#107—152no—
395#106—152no—
396#103—152no—
397#102—52no—
398#102—52no—
399#102—152no—
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×C3C2×C22⋊D511yescharacteristic
413C22×D3—21no—
414C3×D4—54no—
415C3×D4—102no—
416C3×D4—104no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4#70912yes—
421C2×C3×C4C2×C22⋊D511yescharacteristic
422C2×C3×C4C23×D511yescharacteristic
423#67—24no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—41no—
427C2×Dic3—102no—
428C2×Dic3—204no—
429C2×Dic3—202no—
430C2×Dic3—204no—
431C2×Dic3C2×C22⋊D511yescharacteristic
432C2×Dic3C2×C22⋊D511yescharacteristic
433C2×Dic3C2×C22⋊D511yescharacteristic
434C2×Dic3C2×C22⋊D511yescharacteristic
435D12—21no—
436D12C2×C4×D512yescomplemented
437C4×D3—24no—
438C4×D3—42no—
439C4×D3—42no—
440Dic6—21no—
441Dic6C2×C4×D512yescomplemented
442C3×C8—102no—
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—32no—
456C4×C5—41no—
457C4×C5—41no—
458C4×C5—62no—
459C4×C5—61no—
460C4×C5—61no—
461C4×C5—61no—
462C4×C5#67⋊C412yes—
463C4×C5C22×C4×D311yescharacteristic
464C4×C5C2×#23711yescharacteristic
465Dic5—122no—
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—102no—
475C2×D4—101no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—302no—
482C22×C4—302no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—102no—
489M16—154no—
490C2×C8—52no—
491C2×C8—152no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—152no—
495C4⋊C4—301no—
496C4⋊C4C2×D3×D511yescharacteristic
497C4⋊C4C2×D3×D511yescharacteristic
498C22⋊C4—21no—
499C22⋊C4—61no—
500C22⋊C4—61no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—21no—
504C42—31no—
505C42—31no—
506C42—31no—
507C42—31no—
508C42—61no—
509C42—61no—
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×C22⋊D512yes—
518C22×C3C22×C22⋊D511yescharacteristic
519C2×D3—22no—
520C2×D3—41no—
521C2×D3—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4C4×C22⋊D512yes—
525C3×C4C22×C4×D511yescharacteristic
526C3×C4C2×#70911yescharacteristic
527Dic3—22no—
528Dic3—21no—
529Dic3—21no—
530Dic3—21no—
531Dic3—202no—
532Dic3#409512yescomplemented
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—
544C23D3×C22⋊D511yescharacteristic
545Q8—32no—
546Q8—61no—
547D4—32no—
548D4—54no—
549D4—64no—
550D4—61no—
551D4—104no—
552D4—102no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—31no—
556C2×C4—31no—
557C2×C4—31no—
558C2×C4—31no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—64no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—302no—
566C2×C4—604no—
567C2×C4—604no—
568C2×C4—602no—
569C2×C4#676512yes—
570C2×C4C4×D3×D512yes—
571C2×C4D3×C22⋊D511yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—102no—
574C8—154no—
575C2×C3—22no—
576C2×C3—204no—
577C2×C3C2×C4×C22⋊D511yescharacteristic
578C2×C3C2×#409511yescharacteristic
579C2×C3#6615611yescharacteristic
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#9144112yes—
592C22C2×D3×C22⋊D511yescharacteristic
593C4—32no—
594C4—41no—
595C4—41no—
596C4—62no—
597C4—61no—
598C4—61no—
599C4—61no—
600C4—602no—
601C4#9144112yes—
602C4C2×#676511yescharacteristic
603C4C2×C4×D3×D511yescharacteristic
604C3C4×#409511yescharacteristic, complemented, Sylow
605C2—22no—
606C2—122no—
607C2—204no—
608C2C2×#9144111yescharacteristic
609C2D3×#409511yescharacteristic
610C2#1175534511yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91298C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5C4×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5522C2semidirect
2. C2×Dic3×D20C2semidirect
4. #11751096C2semidirect
5. C3×#65290C2semidirect
6. #11748390C2semidirect
7. Dic3×#681C2semidirect
8. #11747222C2semidirect
9. #11747221C2semidirect
10. #11747220C2semidirect
11. #11747219C2semidirect
20. C5×#893C22semidirect
21. C5×#893C22semidirect
22. C5×#887C22semidirect
23. C5×#887C22semidirect
35. #91298C4semidirect
37. #91092C22semidirect
38. #91091C22semidirect
41. #90973C22semidirect
42. #90967C22semidirect
43. #90953C22semidirect
44. Dic3×#168C22semidirect
54. #65290D3semidirect
64. C5×#235C2×C4semidirect
72. C2×C3×D20C2×C4semidirect
83. #6710C2×C4semidirect
99. #6655C2×C4semidirect
100. #6652C2×C4semidirect
103. #5522D5semidirect
168. C5×D12C22×C4semidirect
173. C5×Dic6C22×C4semidirect
179. C3⋊#168C22×C4semidirect
197. #893C2×D5semidirect
198. #893C2×D5semidirect
199. #887C2×D5semidirect
200. #887C2×D5semidirect
228. C2×D20C4×D3semidirect
289. C5×Dic3AGL(1,ℤ/8)semidirect
304. #235C4×D5semidirect
436. D12C2×C4×D5semidirect
441. Dic6C2×C4×D5semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
532. Dic3#4095semidirect
581. C5#78914semidirect
604. C3C4×#4095semidirect

Maximal subgroups

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

subgroupindex
C5×#55222
C2×Dic3×D202
C2×#912982
#117510962
C3×#652902
#117483902
Dic3×#6812
#117472222
#117472212
#117472202
#117472192
C4×#40953
#789145