Finite Group Databasebeta

← #14416722

Subgroups — #14416722

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
1C3×#65856C211yesmaximal, characteristic, complemented
2C2×#91418C211yesmaximal, characteristic, complemented
3C2×#91301C211yesmaximal, characteristic
4#11751102C211yesmaximal, characteristic, complemented
5C5×#4995C211yesmaximal, characteristic, complemented
6#11748368C211yesmaximal, characteristic, complemented
7Dic5×#208C211yesmaximal, characteristic, complemented
8#11747469C212yesmaximal, complemented
9#11747225C212yesmaximal, complemented
10#11747224C212yesmaximal, complemented
11#11747223C212yesmaximal, complemented
12#10610930—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C22×#1200—21no—
16C3×C5×#117C2211yescharacteristic
17C2×#6710C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C2×C3×C4×Dic5C2211yescharacteristic
21C3×#4039C2211yescharacteristic, complemented
22C3×#4039C2211yescharacteristic, complemented
23C3×#4033C2211yescharacteristic, complemented
24C3×#4033C2211yescharacteristic, complemented
25C3×#4027—21no—
26C3×#4027—21no—
27C3×#4018—21no—
28C3×C42×D5—21no—
29C2×#6667—21no—
30#91418—21no—
31#91418—21no—
32#91418C2212yes—
33#91418C2212yes—
34C2×#6658C2212yes—
35#91301C418yescomplemented
36C2×#6653C2212yes—
37#91093C2212yes—
38#91093C2212yes—
39C5×#808C2212yescomplemented
40C5×#807C2212yescomplemented
41#90970C2212yescomplemented
42#90964C2212yescomplemented
43#90952C2212yescomplemented
44Dic5×C3⋊C8C2212yescomplemented
45C4×#949—51nomaximal, Hall
46C2×#4054—31no—
47#65856D311yescharacteristic, complemented
48C2×#4039—31no—
49C5×#428—31no—
50M16×Dic5—31no—
51#65290—31no—
52#65255—31no—
53#65178—32no—
54#65161—32no—
55#65158—32no—
56#65141—32no—
57C23×D15—21no—
58C22×C3×C4×C5C2311yescharacteristic
59C2×D60—21no—
60C2×D60—22no—
61C2×D60C2×C414yescomplemented
62C2×D60C2312yes—
63C22×C5×Dic3—21no—
64C2×C3×C22⋊D5—21no—
65C22×C3×Dic5D411yescharacteristic
66C22×C3×Dic5D411yescharacteristic
67C3×#709—22no—
68C3×#709C2×C414yescomplemented
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×C4×D5—21no—
72C2×C3×C4×D5—21no—
73C2×C3×Dic10C2311yescharacteristic
74C2×#1200—22no—
75C2×#1200—22no—
76C2×#1200—44no—
77C3×C5×C4⋊C4C2311yescharacteristic
78C3×C5×C4⋊C4C2311yescharacteristic
79C3×C5×C22⋊C4—21no—
80C3×C42×C5—21no—
81#6710—22no—
82C2×C3⋊#168C2312yes—
83C5×C4⋊Dic3—21no—
84C5×C4⋊Dic3C2312yes—
85C5×#208C2×C414yescomplemented
86C2×C5×C3⋊C8C2312yes—
87C3×#685—21no—
88C3×#685—21no—
89C3×#683—21no—
90C3×#683—21no—
91C3×C4×Dic5—21no—
92C3×C4×Dic5—21no—
93C3×C4×Dic5D411yescharacteristic
94C3×C4×Dic5D411yescharacteristic
95C3×C4×Dic5C2311yescharacteristic
96C3×C4×Dic5C2311yescharacteristic
97#6667—22no—
98#6667—22no—
99#6658C2×C418yescomplemented
100#6653C2×C418yescomplemented
101C3×C4×#41—51no—
102C2×#949—51no—
103C2×C4×D12—51no—
104#5232—51no—
105#5026—52no—
106C4×#215—52no—
107#5014—52no—
108C4×#213—52no—
109#5001—51no—
110#4995D511yescharacteristic, complemented
111C4×#208—51no—
112C2×#709C2×D311yescharacteristic
113C22×D20—31no—
114C22×C4×D5—61no—
115C2×C5×M16—31no—
116C5×#117C2×D311yescharacteristic
117C2×C5×C4⋊C4—31no—
118C2×#685—61no—
119C2×C4×Dic5C2×D311yescharacteristic
120C2×#681—31no—
121#4054—38no—
122C2×D40—32no—
123C2×#677—32no—
124#4039—32no—
125#4039—32no—
126#4039—61no—
127#4039—61no—
128#4039C2×D311yescharacteristic, complemented
129#4039C2×D311yescharacteristic, complemented
130#4033C2×D311yescharacteristic, complemented
131#4033C2×D311yescharacteristic, complemented
132#4027—21no—
133#4027—21no—
134#4018—21no—
135C42×D5—21no—
136C5×#107—32no—
137C5×#106—32no—
138#3947—32no—
139C8×Dic5—32no—
140#3942—32no—
141#3941—32no—
142#3941—32no—
143#3941—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—24no—
147C22×D15—22no—
148C22×D15—42no—
149C22×D15—42no—
150C22×C3×D5—21no—
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
157D60—24no—
158D60—22no—
159D60C22×C414yes—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—42no—
162C3×C22⋊D5—24no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—41no—
166C2×C3×Dic5C2×D411yescharacteristic
167C2×C3×Dic5C2×D411yescharacteristic
168C2×C3×Dic5C2×D411yescharacteristic
169C2×C3×Dic5C2×D411yescharacteristic
170C3×D20—21no—
171C3×D20C22×C412yescomplemented
172C3×C4×D5—24no—
173C3×C4×D5—42no—
174C3×C4×D5—42no—
175C3×Dic10—21no—
176C3×Dic10C22×C412yescomplemented
177#1200—44no—
178#1200—44no—
179C3⋊#168—22no—
180C5×C3⋊C8C22×C414yescomplemented
181C2×C3×#41—51no—
182C22×D12—51no—
183C22×C4×D3—101no—
184C3×C4×Q8—51no—
185C3×C4×Q8—51no—
186C3×C4×D4—51no—
187C3×C4×D4—51no—
188C3×C4×D4—101no—
189C3×C4×D4—101no—
190C3×#117—101no—
191C3×#117C2×D511yescharacteristic
192C2×C3×C42—51no—
193C2×C3×C42—101no—
194#949—58no—
195C2×#215—52no—
196C2×#213—52no—
197C2×(C2×D3)⋊C4—101no—
198C2×C4⋊Dic3C2×D511yescharacteristic
199C2×#208—51no—
200C2×#208C2×D511yescharacteristic
201C4×D12—52no—
202C4×D12—52no—
203C4×D12—101no—
204C4×D12—101no—
205#835—52no—
206#832—52no—
207#809—52no—
208#809—52no—
209#808C2×D512yescomplemented
210#807C2×D512yescomplemented
211#803—52no—
212C4×C3⋊C8—52no—
213C23×D5—61no—
214C22×C4×C5—61no—
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—21no—
217C22×Dic5#6711yescharacteristic
218C22×Dic5#6711yescharacteristic
219#709—22no—
220#709C4×D314yescomplemented
221C2×D20—34no—
222C2×D20—32no—
223C2×D20—62no—
224C2×D20—61no—
225C2×D20C22×D311yescharacteristic
226C2×C4×D5—21no—
227C2×C4×D5—21no—
228C2×C4×D5—21no—
229C2×C4×D5—62no—
230C2×C4×D5—62no—
231C2×C4×D5—124no—
232C2×Dic10C22×D311yescharacteristic
233C5×M16—34no—
234C2×C5×C8—32no—
235C5×C4⋊C4—32no—
236C5×C4⋊C4—61no—
237C5×C4⋊C4C22×D311yescharacteristic
238C5×C4⋊C4C22×D311yescharacteristic
239C5×C22⋊C4—21no—
240C42×C5—21no—
241#685—21no—
242#685—21no—
243#685—62no—
244#685—62no—
245#683—21no—
246#683—21no—
247C4×Dic5—21no—
248C4×Dic5—21no—
249C4×Dic5#6711yescharacteristic
250C4×Dic5#6711yescharacteristic
251C4×Dic5C22×D311yescharacteristic
252C4×Dic5C22×D311yescharacteristic
253#681—62no—
254C2×#168—32no—
255D40—38no—
256#677—38no—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—51no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—31no—
265#418—151no—
266#417—151no—
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5C4×D412yes—
271C22×C3×C5C22×D411yescharacteristic
272C2×D15—24no—
273C2×D15—48no—
274C2×D15—42no—
275C2×D15—42no—
276C2×C3×D5—22no—
277C2×C3×D5—41no—
278C2×C3×D5—41no—
279C3×C4×C5—41no—
280C3×C4×C5—41no—
281C3×C4×C5C4×D412yes—
282C3×C4×C5C23×C411yescharacteristic
283C3×C4×C5C2×#4111yescharacteristic
284C3×Dic5—22no—
285C3×Dic5—21no—
286C3×Dic5—21no—
287C3×Dic5—21no—
288C3×Dic5AGL(1,ℤ/8)12yescomplemented
289C5×Dic3—42no—
290C23×D3—101no—
291C3×#41—54no—
292C3×#41—102no—
293C2×C3×Q8—51no—
294C2×C3×D4—51no—
295C2×C3×D4—101no—
296C22×C3×C4—51no—
297C22×C3×C4—51no—
298C22×C3×C4—101no—
299C22×C3×C4—101no—
300C22×C3×C4—101no—
301C22×C3×C4C22×D511yescharacteristic
302C22×Dic3—21no—
303C2×D12—54no—
304C2×D12—52no—
305C2×D12—101no—
306C2×D12—102no—
307C2×C4×D3—102no—
308C2×C4×D3—102no—
309C2×C4×D3—204no—
310C3×C4⋊C4—101no—
311C3×C4⋊C4—101no—
312C3×C4⋊C4C22×D511yescharacteristic
313C3×C4⋊C4C22×D511yescharacteristic
314C3×C22⋊C4—21no—
315C3×C22⋊C4—101no—
316C3×C22⋊C4—101no—
317C3×C42—21no—
318C3×C42—51no—
319C3×C42—51no—
320C3×C42—51no—
321C3×C42—51no—
322C3×C42—101no—
323C3×C42—101no—
324#215—58no—
325#213—58no—
326(C2×D3)⋊C4—102no—
327(C2×D3)⋊C4—102no—
328C4⋊Dic3—21no—
329C4⋊Dic3C22×D512yes—
330#208—102no—
331#208C4×D514yescomplemented
332C2×C3⋊C8—52no—
333C2×C3⋊C8C22×D512yes—
334C23×C5C2×#6711yescharacteristic
335C22×D5—21no—
336C22×D5—64no—
337C22×D5—62no—
338C22×D5—122no—
339C22×D5—122no—
340C2×C4×C5—21no—
341C2×C4×C5—21no—
342C2×C4×C5—62no—
343C2×C4×C5—122no—
344C2×C4×C5C2×C4×D312yes—
345C2×C4×C5#23512yes—
346C2×C4×C5C2×#6711yescharacteristic
347C2×C4×C5C23×D311yescharacteristic
348C22⋊D5—24no—
349C2×Dic5—21no—
350C2×Dic5—22no—
351C2×Dic5—41no—
352C2×Dic5C2×#6711yescharacteristic
353C2×Dic5C2×#6711yescharacteristic
354C2×Dic5C2×#6711yescharacteristic
355C2×Dic5C2×#6711yescharacteristic
356D20—21no—
357D20—34no—
358D20—64no—
359D20—62no—
360D20C2×C4×D312yescomplemented
361C4×D5—24no—
362C4×D5—42no—
363C4×D5—42no—
364C4×D5—124no—
365C4×D5—124no—
366Dic10—21no—
367Dic10C2×C4×D312yescomplemented
368C5×C8—34no—
369#168—62no—
370C2×#41—51no—
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—31no—
377C2×M16—151no—
378C4×Q8—51no—
379C4×Q8—51no—
380C4×D4—51no—
381C4×D4—51no—
382C4×D4—101no—
383C4×D4—101no—
384C4×D4—152no—
385C4×D4—152no—
386C4×D4—301no—
387C4×D4—301no—
388#117—101no—
389#117D3×D511yescharacteristic
390C2×C4⋊C4—31no—
391C2×C22⋊C4—301no—
392C2×C42—51no—
393C2×C42—101no—
394#107—32no—
395#106—32no—
396#103—152no—
397#102—152no—
398#102—152no—
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
406D15—44no—
407C3×D5—42no—
408C23×C3—101no—
409C23×C3D4×D511yescharacteristic
410C22×D3—102no—
411C22×D3—104no—
412C22×D3—202no—
413C22×D3—202no—
414C3×Q8—52no—
415C3×Q8—101no—
416C3×D4—52no—
417C3×D4—104no—
418C3×D4—101no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—51no—
422C2×C3×C4—51no—
423C2×C3×C4—51no—
424C2×C3×C4—51no—
425C2×C3×C4—102no—
426C2×C3×C4—101no—
427C2×C3×C4—104no—
428C2×C3×C4—202no—
429C2×C3×C4—202no—
430C2×C3×C4—201no—
431C2×C3×C4C2×C4×D512yes—
432C2×C3×C4D4×D511yescharacteristic
433C2×C3×C4#71312yes—
434C2×C3×C4C23×D511yescharacteristic
435C2×Dic3—22no—
436C2×Dic3—42no—
437D12—54no—
438D12—102no—
439D12—104no—
440C4×D3—204no—
441C4×D3—204no—
442C3⋊C8—102no—
443C3⋊C8C2×C4×D514yescomplemented
444C22×C5—21no—
445C22×C5—21no—
446C22×C5C4×#6712yes—
447C22×C5C22×#6711yescharacteristic
448C2×D5—22no—
449C2×D5—41no—
450C2×D5—41no—
451C2×D5—64no—
452C2×D5—128no—
453C2×D5—122no—
454C2×D5—122no—
455C4×C5—41no—
456C4×C5—41no—
457C4×C5—122no—
458C4×C5C4×#6712yes—
459C4×C5C22×C4×D311yescharacteristic
460C4×C5C2×#23511yescharacteristic
461Dic5—22no—
462Dic5—21no—
463Dic5—21no—
464Dic5—21no—
465Dic5#94912yescomplemented
466C24—301no—
467#41—54no—
468#41—102no—
469C2×Q8—51no—
470C2×D4—51no—
471C2×D4—101no—
472C2×D4—154no—
473C2×D4—152no—
474C2×D4—302no—
475C2×D4—301no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—61no—
479C22×C4—101no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4—302no—
483C22×C4—302no—
484C22×C4—604no—
485C22×C4C2×D3×D511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—34no—
489M16—302no—
490C2×C8—32no—
491C2×C8—152no—
492C4⋊C4—32no—
493C4⋊C4—61no—
494C4⋊C4—101no—
495C4⋊C4—101no—
496C4⋊C4C2×D3×D511yescharacteristic
497C4⋊C4C2×D3×D511yescharacteristic
498C22⋊C4—21no—
499C22⋊C4—101no—
500C22⋊C4—101no—
501C22⋊C4—302no—
502C22⋊C4—302no—
503C42—21no—
504C42—51no—
505C42—51no—
506C42—51no—
507C42—51no—
508C42—101no—
509C42—101no—
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#403912yes—
517C22×C3C2×D4×D511yescharacteristic
518C2×D3—104no—
519C2×D3—208no—
520C2×D3—202no—
521C2×D3—202no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—52no—
525C3×C4—102no—
526C3×C4—101no—
527C3×C4—101no—
528C3×C4—101no—
529C3×C4#403912yes—
530C3×C4C22×C4×D511yescharacteristic
531C3×C4C2×#71311yescharacteristic
532Dic3—42no—
533C2×C5—22no—
534C2×C5C2×C4×#6711yescharacteristic
535C2×C5C2×#94911yescharacteristic
536C2×C5#582211yescharacteristic
537D5—42no—
538D5—124no—
539C23—101no—
540C23—304no—
541C23—302no—
542C23—602no—
543C23—602no—
544C23D5×#6711yescharacteristic
545Q8—52no—
546Q8—101no—
547D4—52no—
548D4—104no—
549D4—101no—
550D4—154no—
551D4—304no—
552D4—302no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—51no—
556C2×C4—51no—
557C2×C4—51no—
558C2×C4—51no—
559C2×C4—62no—
560C2×C4—102no—
561C2×C4—101no—
562C2×C4—104no—
563C2×C4—122no—
564C2×C4—202no—
565C2×C4—202no—
566C2×C4—201no—
567C2×C4—604no—
568C2×C4—604no—
569C2×C4#677312yes—
570C2×C4C4×D3×D512yes—
571C2×C4D5×#6711yescharacteristic
572C2×C4C22×D3×D511yescharacteristic
573C8—34no—
574C8—302no—
575C2×C3—22no—
576C2×C3—202no—
577C2×C3C2×#403911yescharacteristic
578C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
579C2×C3#6611111yescharacteristic
580D3—204no—
581C5C4×#94911yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—102no—
585C22—201no—
586C22—201no—
587C22—304no—
588C22—608no—
589C22—602no—
590C22—602no—
591C22#9144412yes—
592C22C2×D5×#6711yescharacteristic
593C4—41no—
594C4—41no—
595C4—52no—
596C4—102no—
597C4—101no—
598C4—101no—
599C4—101no—
600C4—122no—
601C4#9144412yes—
602C4C2×#677311yescharacteristic
603C4C2×C4×D3×D511yescharacteristic
604C3#1061093011yescharacteristic, complemented, Sylow
605C2—22no—
606C2—202no—
607C2—604no—
608C2C2×#9144411yescharacteristic
609C2D5×#94911yescharacteristic
610C2#1175534411yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91301C2
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. C3×#65856C2semidirect
2. C2×#91418C2semidirect
4. #11751102C2semidirect
5. C5×#4995C2semidirect
6. #11748368C2semidirect
7. Dic5×#208C2semidirect
8. #11747469C2semidirect
9. #11747225C2semidirect
10. #11747224C2semidirect
11. #11747223C2semidirect
21. C3×#4039C22semidirect
22. C3×#4039C22semidirect
23. C3×#4033C22semidirect
24. C3×#4033C22semidirect
35. #91301C4semidirect
39. C5×#808C22semidirect
40. C5×#807C22semidirect
41. #90970C22semidirect
42. #90964C22semidirect
43. #90952C22semidirect
44. Dic5×C3⋊C8C22semidirect
47. #65856D3semidirect
61. C2×D60C2×C4semidirect
68. C3×#709C2×C4semidirect
85. C5×#208C2×C4semidirect
99. #6658C2×C4semidirect
100. #6653C2×C4semidirect
110. #4995D5semidirect
128. #4039C2×D3semidirect
129. #4039C2×D3semidirect
130. #4033C2×D3semidirect
131. #4033C2×D3semidirect
171. C3×D20C22×C4semidirect
176. C3×Dic10C22×C4semidirect
180. C5×C3⋊C8C22×C4semidirect
209. #808C2×D5semidirect
210. #807C2×D5semidirect
220. #709C4×D3semidirect
288. C3×Dic5AGL(1,ℤ/8)semidirect
331. #208C4×D5semidirect
360. D20C2×C4×D3semidirect
367. Dic10C2×C4×D3semidirect
443. C3⋊C8C2×C4×D5semidirect
465. Dic5#949semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
581. C5C4×#949semidirect
604. C3#10610930semidirect

Maximal subgroups

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

subgroupindex
C3×#658562
C2×#914182
C2×#913012
#117511022
C5×#49952
#117483682
Dic5×#2082
#117474692
#117472252
#117472242
#117472232
#106109303
C4×#9495