Finite Group Databasebeta

← #14412672

Subgroups — #14412672

Order 1920.

1556 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92019C211yesmaximal, characteristic
2C3×#65857C211yesmaximal, characteristic, complemented
3C2×#91283C211yesmaximal, characteristic, complemented
4#11751102C211yesmaximal, characteristic, complemented
5C5×#5002C211yesmaximal, characteristic, complemented
6C2×#90968C211yesmaximal, characteristic, complemented
7#11748384C211yesmaximal, characteristic, complemented
8#11747501C212yesmaximal, complemented
9#11747500C212yesmaximal, complemented
10#11747475C212yesmaximal, complemented
11D5×#809C212yesmaximal, complemented
12#10610562—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C2×#6790—22no—
17#92019—24no—
18#92019C418yescomplemented
19C2×D3×D20—21no—
20C2×D5×D12C2212yes—
21C2×#6773C2212yes—
22C2×#6769—21no—
23C3×C5×#117C2211yescharacteristic, complemented
24C2×#6710C2211yescharacteristic, complemented
25C22×C5×C3⋊C8C2211yescharacteristic, complemented
26C2×C3×C4⋊Dic5C2211yescharacteristic, complemented
27C3×#4038C2212yescomplemented
28C3×D5×C4⋊C4C2212yescomplemented
29C3×#4027—22no—
30C3×C4×D20—21no—
31C3×C4×Dic10—21no—
32C2×#6649C2212yescomplemented
33#91283—24no—
34C2×D5×C3⋊C8C2212yescomplemented
35#91093C2212yescomplemented
36#91093C2212yescomplemented
37C5×#810C2211yescharacteristic, complemented
38C5×#810C2211yescharacteristic, complemented
39C5×#809C2211yescharacteristic, complemented
40C5×#809C2211yescharacteristic, complemented
41#90972C2212yescomplemented
42#90968C2212yescomplemented
43#90968C2212yescomplemented
44#90966C2212yescomplemented
45#82563—51nomaximal, Hall
46C2×#4149—31no—
47C2×#4047—31no—
48#65857D311yescharacteristic, complemented
49C5×#416—31no—
50#65521—31no—
51C2×#3953—31no—
52#65290—31no—
53#65093—32no—
54#65092—32no—
55#65059—32no—
56D5×#102—32no—
57C23×D15—21no—
58C22×D3×D5—22no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×D60—21no—
61C2×D60—22no—
62C2×D60C2×C414yescomplemented
63C2×D60C2312yes—
64C2×C5×#67—21no—
65C5×#235—22no—
66C5×#235C2×C414yescomplemented
67C2×C5×D12C2311yescharacteristic
68C2×C4×C5×D3—21no—
69C2×C5×Dic6C2311yescharacteristic
70C2×C3×C22⋊D5D412yes—
71C22×C3×Dic5—21no—
72C3×#709C2×C414yes—
73C3×#709D414yes—
74C2×C3×D20D411yescharacteristic
75C2×C3×C4×D5—22no—
76C2×C3×C4×D5C2312yes—
77C2×C3×Dic10D411yescharacteristic
78#6790—216no—
79C2×#1203—22no—
80C2×#1202—22no—
81C2×#1200—22no—
82D3×D20—28no—
83D5×D12—24no—
84D5×D12C2×C418yescomplemented
85#6773—24no—
86#6773C2×C418yescomplemented
87#6769—28no—
88C3×C5×C4⋊C4C2311yescharacteristic
89C3×C5×C4⋊C4C2311yescharacteristic
90C3×C5×C22⋊C4—21no—
91C3×C42×C5—21no—
92#6710—22no—
93C2×C3⋊#168C2312yes—
94C2×C5×C3⋊C8—22no—
95C2×C5×C3⋊C8C2311yescharacteristic
96C2×C5×C3⋊C8C2311yescharacteristic
97C3×#685—22no—
98C3×C4⋊Dic5—21no—
99C3×C4⋊Dic5C2312yes—
100C3×#683—22no—
101C3×C4×Dic5—22no—
102#6649—24no—
103D5×C3⋊C8—24no—
104C2×#1009—51no—
105C3×#520—51no—
106C2×#948—51no—
107#5002—52no—
108#5002D511yescharacteristic, complemented
109#5001—51no—
110#5001—52no—
111#4966—51no—
112#4966—52no—
113C2×#809—51no—
114C2×#809—52no—
115C2×C5×#41—31no—
116#4149—38no—
117#4149—64no—
118C2×#713—32no—
119C2×D4×D5—32no—
120C2×D4×D5—62no—
121C2×#709—61no—
122C2×#709C2×D311yescharacteristic
123C22×D20—31no—
124C22×D20—61no—
125C22×C5×C8—31no—
126C5×#117C2×D311yescharacteristic, complemented
127C2×C4⋊Dic5C2×D311yescharacteristic, complemented
128C2×#681—31no—
129#4047—64no—
130C2×#676—32no—
131C2×C8×D5—32no—
132#4038C2×D312yescomplemented
133D5×C4⋊C4C2×D312yescomplemented
134#4027—22no—
135C4×D20—21no—
136C4×Dic10—21no—
137C5×#103—31no—
138C5×#103—31no—
139C5×#102—31no—
140C5×#102—31no—
141#3967—32no—
142#3964—32no—
143#3953—32no—
144#3953—32no—
145#3941—32no—
146#3941—32no—
147#2897—151noSylow
148C23×C3×C5C2×D411yescharacteristic
149C22×D15—22no—
150C22×D15—24no—
151C22×D15—81no—
152C22×D15—81no—
153C22×C5×D3—21no—
154C22×C3×D5C2×D412yes—
155C2×D3×D5—28no—
156C2×D3×D5—82no—
157C2×D3×D5—82no—
158C2×C3×C4×C5—21no—
159C2×C3×C4×C5—21no—
160C2×C3×C4×C5C22×C412yes—
161C2×C3×C4×C5C2×D411yescharacteristic
162C2×C3×C4×C5C2×D412yes—
163C2×C3×C4×C5C2411yescharacteristic
164D60—24no—
165D60—22no—
166D60C22×C414yes—
167C5×#67—24no—
168C2×C5×Dic3—21no—
169C5×D12—21no—
170C5×D12C22×C412yes—
171C4×C5×D3—24no—
172C5×Dic6—21no—
173C5×Dic6C22×C412yes—
174C3×C22⋊D5C22⋊C418yes—
175C2×C3×Dic5—22no—
176C2×C3×Dic5—81no—
177C2×C3×Dic5C2×D412yes—
178C3×D20C22⋊C414yes—
179C3×C4×D5—82no—
180C3×C4×D5C22×C414yes—
181C3×C4×D5C2×D414yes—
182C3×Dic10C22⋊C414yes—
183#1203—28no—
184#1202—28no—
185#1200—28no—
186C3⋊#168—22no—
187C5×C3⋊C8—21no—
188C5×C3⋊C8—21no—
189C2×C3×#41—51no—
190#1009—58no—
191#1009—104no—
192C2×#239—101no—
193C2×D3×D4—101no—
194C2×D3×D4—102no—
195C2×#235—52no—
196C2×#235C2×D511yescharacteristic
197C22×D12—51no—
198C22×D12—52no—
199C3×C4×Q8—101no—
200C3×C4×D4—101no—
201C3×C4×D4—102no—
202C3×#117—52no—
203C3×#117C2×D511yescharacteristic, complemented
204C2×C3×C4⋊C4—51no—
205C2×C3×C4⋊C4—52no—
206#948—104no—
207C2×#208—51no—
208C2×#208—52no—
209C22×C3⋊C8—52no—
210C22×C3⋊C8C2×D511yescharacteristic, complemented
211#810—52no—
212#810C2×D511yescharacteristic, complemented
213#810C2×D511yescharacteristic, complemented
214#809—52no—
215#809—52no—
216#809—52no—
217#809—52no—
218#809—52no—
219#809C2×D511yescharacteristic, complemented
220#809C2×D511yescharacteristic, complemented
221C23×D5—61no—
222C23×D5—62no—
223C5×#41—34no—
224C5×#41—62no—
225C2×C5×Q8—31no—
226C2×C5×D4—31no—
227C2×C5×D4—61no—
228C22×C4×C5—61no—
229C22×C4×C5C22×D311yescharacteristic
230C2×C22⋊D5—62no—
231C2×C22⋊D5#6712yes—
232C22×Dic5—21no—
233#713—38no—
234#713—64no—
235D4×D5—38no—
236D4×D5—616no—
237D4×D5—64no—
238#709—68no—
239#709C4×D314yes—
240#709D1214yes—
241C2×D20—34no—
242C2×D20—32no—
243C2×D20—62no—
244C2×D20—62no—
245C2×D20—61no—
246C2×D20—68no—
247C2×D20#6711yescharacteristic
248C2×C4×D5—22no—
249C2×C4×D5—62no—
250C2×C4×D5C22×D312yes—
251C2×Dic10#6711yescharacteristic
252C2×C5×C8—31no—
253C2×C5×C8—31no—
254C2×C5×C8—62no—
255C5×C4⋊C4C22×D311yescharacteristic
256C5×C4⋊C4C22×D311yescharacteristic
257C5×C22⋊C4—21no—
258C42×C5—21no—
259#685—22no—
260C4⋊Dic5—21no—
261C4⋊Dic5C22×D312yes—
262#683—22no—
263C4×Dic5—22no—
264#681—62no—
265C2×#168—32no—
266#676—64no—
267C8×D5—64no—
268C2×H(5,𝔽2)—151no—
269C2×#131—151no—
270#520—51no—
271#418—152no—
272#418—151no—
273#417—152no—
274#417—151no—
275#416—31no—
276#416—152no—
277C2×#102—152no—
278C2×#102—151no—
279C22×C3×C5—21no—
280C22×C3×C5—21no—
281C22×C3×C5C2×C22⋊C412yes—
282C22×C3×C5C22×D411yescharacteristic
283C2×D15—24no—
284C2×D15—42no—
285C2×D15—42no—
286C2×D15—84no—
287C2×C5×D3—22no—
288C2×C5×D3—41no—
289C2×C5×D3—41no—
290C2×C3×D5—22no—
291C2×C3×D5—22no—
292C2×C3×D5C2×C22⋊C414yes—
293D3×D5—88no—
294C3×C4×C5—41no—
295C3×C4×C5—41no—
296C3×C4×C5C2×C22⋊C412yes—
297C3×C4×C5C23×C411yescharacteristic
298C3×C4×C5C22×D411yescharacteristic
299C3×Dic5—42no—
300C3×Dic5C2×C22⋊C414yes—
301C5×Dic3—22no—
302C23×D3—102no—
303C23×D3—101no—
304C3×#41—54no—
305C3×#41—54no—
306C2×C3×Q8—51no—
307C2×C3×D4—52no—
308C2×C3×D4—51no—
309C22×C3×C4—52no—
310C22×C3×C4—102no—
311C22×C3×C4—101no—
312C22×C3×C4C22×D511yescharacteristic
313C2×#67—21no—
314C2×#67—102no—
315#239—108no—
316D3×D4—108no—
317D3×D4—1016no—
318#235—22no—
319#235—58no—
320#235—104no—
321#235C4×D514yescomplemented
322C2×D12—54no—
323C2×D12—52no—
324C2×D12—58no—
325C2×D12—104no—
326C2×D12—102no—
327C2×D12—101no—
328C2×D12—102no—
329C2×D12C22×D511yescharacteristic
330C2×C4×D3—21no—
331C2×C4×D3—102no—
332C2×Dic6C22×D511yescharacteristic
333C3×C4⋊C4—52no—
334C3×C4⋊C4—102no—
335C3×C4⋊C4—101no—
336C3×C4⋊C4C22×D511yescharacteristic
337C3×C4⋊C4C22×D511yescharacteristic
338C3×C22⋊C4—21no—
339C3×C22⋊C4—102no—
340C3×C42—21no—
341C3×C42—102no—
342#208—104no—
343#208—102no—
344C2×C3⋊C8—22no—
345C2×C3⋊C8—52no—
346C2×C3⋊C8—104no—
347C2×C3⋊C8C22×D511yescharacteristic
348C2×C3⋊C8C22×D511yescharacteristic
349C23×C5—61no—
350C23×C5C2×#6711yescharacteristic
351C22×D5—62no—
352C22×D5—64no—
353C22×D5—68no—
354C22×D5—242no—
355C22×D5—242no—
356C22×D5—241no—
357C22×D5—241no—
358C22×D5C2×#6712yes—
359C5×Q8—32no—
360C5×Q8—61no—
361C5×D4—32no—
362C5×D4—64no—
363C5×D4—61no—
364C2×C4×C5—21no—
365C2×C4×C5—21no—
366C2×C4×C5—61no—
367C2×C4×C5—64no—
368C2×C4×C5C2×C4×D312yes—
369C2×C4×C5C2×D1212yes—
370C2×C4×C5C2×#6711yescharacteristic
371C2×C4×C5C23×D311yescharacteristic
372C22⋊D5—68no—
373C22⋊D5(C2×D3)⋊C418yes—
374C2×Dic5—22no—
375C2×Dic5—81no—
376C2×Dic5C2×#6712yes—
377D20—34no—
378D20—68no—
379D20—64no—
380D20—62no—
381D20(C2×D3)⋊C414yes—
382C4×D5—68no—
383C4×D5—82no—
384C4×D5C2×C4×D314yes—
385C4×D5C2×D1214yes—
386Dic10(C2×D3)⋊C414yes—
387C5×C8—61no—
388C5×C8—61no—
389#168—62no—
390H(5,𝔽2)—158no—
391H(5,𝔽2)—304no—
392C2×#41—31no—
393C2×#41—51no—
394C2×#41—152no—
395C2×#41—301no—
396C22×D4—152no—
397C22×D4—151no—
398C22×D4—301no—
399C22×D4—302no—
400#131—304no—
401C2×M16—152no—
402C2×M16—151no—
403C22×C8—31no—
404C22×C8—152no—
405C4×Q8—101no—
406C4×D4—101no—
407C4×D4—102no—
408#117—52no—
409#117D3×D511yescharacteristic, complemented
410C2×C4⋊C4—51no—
411C2×C4⋊C4—52no—
412#103—31no—
413#103—31no—
414#103—152no—
415#102—31no—
416#102—31no—
417#102—152no—
418#102—152no—
419#102—152no—
420#102—152no—
421#102—152no—
422C2×C3×C5—22no—
423C2×C3×C5C22×C22⋊C411yescharacteristic
424C2×C3×C5#57611yescharacteristic
425C2×C3×C5#57711yescharacteristic
426D15—44no—
427C3×D5—24no—
428C5×D3—42no—
429C23×C3—52no—
430C23×C3D4×D511yescharacteristic
431C22×D3—21no—
432C22×D3—108no—
433C22×D3—102no—
434C22×D3—104no—
435C22×D3—402no—
436C22×D3—402no—
437C22×D3—401no—
438C22×D3—401no—
439C3×Q8—54no—
440C3×D4—58no—
441C3×D4—54no—
442C2×C3×C4—21no—
443C2×C3×C4—21no—
444C2×C3×C4—52no—
445C2×C3×C4—54no—
446C2×C3×C4—54no—
447C2×C3×C4—102no—
448C2×C3×C4—402no—
449C2×C3×C4—401no—
450C2×C3×C4C2×C4×D512yes—
451C2×C3×C4D4×D512yes—
452C2×C3×C4D4×D511yescharacteristic
453C2×C3×C4C23×D511yescharacteristic
454#67—24no—
455#67—108no—
456C2×Dic3—21no—
457D12—21no—
458D12—54no—
459D12—108no—
460D12—102no—
461D12—104no—
462D12C2×C4×D512yes—
463C4×D3—24no—
464C4×D3—108no—
465Dic6—21no—
466Dic6C2×C4×D512yes—
467C3⋊C8—21no—
468C3⋊C8—21no—
469C3⋊C8—102no—
470C22×C5—21no—
471C22×C5—21no—
472C22×C5—62no—
473C22×C5—121no—
474C22×C5—121no—
475C22×C5C2×(C2×D3)⋊C412yes—
476C22×C5C22×#6711yescharacteristic
477C2×D5—22no—
478C2×D5—22no—
479C2×D5—64no—
480C2×D5—122no—
481C2×D5—122no—
482C2×D5—248no—
483C2×D5—244no—
484C2×D5C2×(C2×D3)⋊C414yes—
485C4×C5—41no—
486C4×C5—41no—
487C4×C5—62no—
488C4×C5C2×(C2×D3)⋊C412yes—
489C4×C5C22×C4×D311yescharacteristic
490C4×C5C22×D1211yescharacteristic
491Dic5—42no—
492Dic5C2×(C2×D3)⋊C414yes—
493C24—302no—
494C24—301no—
495#41—34no—
496#41—54no—
497#41—54no—
498#41—62no—
499#41—158no—
500#41—304no—
501#41—308no—
502C2×Q8—31no—
503C2×Q8—51no—
504C2×D4—31no—
505C2×D4—52no—
506C2×D4—51no—
507C2×D4—61no—
508C2×D4—158no—
509C2×D4—154no—
510C2×D4—152no—
511C2×D4—304no—
512C2×D4—302no—
513C2×D4—308no—
514C2×D4—302no—
515C2×D4—3016no—
516C2×D4—302no—
517C2×D4—301no—
518C22×C4—52no—
519C22×C4—61no—
520C22×C4—102no—
521C22×C4—101no—
522C22×C4—302no—
523C22×C4C2×D3×D511yescharacteristic
524M16—304no—
525M16—302no—
526C2×C8—31no—
527C2×C8—31no—
528C2×C8—62no—
529C2×C8—152no—
530C2×C8—304no—
531C4⋊C4—52no—
532C4⋊C4—102no—
533C4⋊C4—101no—
534C4⋊C4C2×D3×D511yescharacteristic
535C4⋊C4C2×D3×D511yescharacteristic
536C22⋊C4—21no—
537C22⋊C4—102no—
538C42—21no—
539C42—102no—
540C3×C5#289711yescharacteristic, complemented, Hall
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—54no—
544C22×C3—102no—
545C22×C3—102no—
546C22×C3D5×C22⋊C412yes—
547C22×C3C2×D4×D511yescharacteristic
548C2×D3—22no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—104no—
552C2×D3—202no—
553C2×D3—202no—
554C2×D3—408no—
555C2×D3—404no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4—54no—
559C3×C4—202no—
560C3×C4D5×C22⋊C412yes—
561C3×C4C22×C4×D511yescharacteristic
562C3×C4C2×D4×D511yescharacteristic
563Dic3—22no—
564C2×C5—22no—
565C2×C5—122no—
566C2×C5C22×(C2×D3)⋊C411yescharacteristic
567C2×C5#583511yescharacteristic
568C2×C5#583711yescharacteristic
569D5—24no—
570D5—124no—
571C23—52no—
572C23—61no—
573C23—308no—
574C23—302no—
575C23—304no—
576C23—1201no—
577C23—1201no—
578C23—1202no—
579C23—1202no—
580C23D5×#6711yescharacteristic
581Q8—32no—
582Q8—54no—
583Q8—61no—
584D4—32no—
585D4—58no—
586D4—54no—
587D4—64no—
588D4—61no—
589D4—154no—
590D4—308no—
591D4—308no—
592D4—304no—
593D4—302no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—52no—
597C2×C4—54no—
598C2×C4—54no—
599C2×C4—61no—
600C2×C4—64no—
601C2×C4—102no—
602C2×C4—308no—
603C2×C4—402no—
604C2×C4—401no—
605C2×C4C4×D3×D512yes—
606C2×C4D5×D1212yes—
607C2×C4D5×#6711yescharacteristic
608C2×C4C22×D3×D511yescharacteristic
609C8—61no—
610C8—61no—
611C8—302no—
612C2×C3—22no—
613C2×C3—104no—
614C2×C3C2×D5×C22⋊C411yescharacteristic
615C2×C3#6610311yescharacteristic
616C2×C3#6610411yescharacteristic
617D3—42no—
618D3—204no—
619C5#8256311yescharacteristic, complemented, Sylow
620C22—21no—
621C22—21no—
622C22—54no—
623C22—62no—
624C22—102no—
625C22—102no—
626C22—121no—
627C22—121no—
628C22—304no—
629C22—602no—
630C22—602no—
631C22—1204no—
632C22—1208no—
633C22D5×(C2×D3)⋊C412yes—
634C22C2×D5×#6711yescharacteristic
635C4—41no—
636C4—41no—
637C4—54no—
638C4—62no—
639C4—202no—
640C4D5×(C2×D3)⋊C412yes—
641C4C2×C4×D3×D511yescharacteristic
642C4C2×D5×D1211yescharacteristic
643C3#1061056211yescharacteristic, complemented, Sylow
644C2—22no—
645C2—104no—
646C2—122no—
647C2—604no—
648C2C2×D5×(C2×D3)⋊C411yescharacteristic
649C2#1175531811yescharacteristic
650C2#1175531911yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#92019C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2897
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
2. C3×#65857C2semidirect
3. C2×#91283C2semidirect
4. #11751102C2semidirect
5. C5×#5002C2semidirect
6. C2×#90968C2semidirect
7. #11748384C2semidirect
8. #11747501C2semidirect
9. #11747500C2semidirect
10. #11747475C2semidirect
11. D5×#809C2semidirect
18. #92019C4semidirect
23. C3×C5×#117C22semidirect
24. C2×#6710C22semidirect
25. C22×C5×C3⋊C8C22semidirect
26. C2×C3×C4⋊Dic5C22semidirect
27. C3×#4038C22semidirect
28. C3×D5×C4⋊C4C22semidirect
32. C2×#6649C22semidirect
34. C2×D5×C3⋊C8C22semidirect
35. #91093C22semidirect
36. #91093C22semidirect
37. C5×#810C22semidirect
38. C5×#810C22semidirect
39. C5×#809C22semidirect
40. C5×#809C22semidirect
41. #90972C22semidirect
42. #90968C22semidirect
43. #90968C22semidirect
44. #90966C22semidirect
48. #65857D3semidirect
62. C2×D60C2×C4semidirect
66. C5×#235C2×C4semidirect
84. D5×D12C2×C4semidirect
86. #6773C2×C4semidirect
108. #5002D5semidirect
126. C5×#117C2×D3semidirect
127. C2×C4⋊Dic5C2×D3semidirect
132. #4038C2×D3semidirect
133. D5×C4⋊C4C2×D3semidirect
203. C3×#117C2×D5semidirect
210. C22×C3⋊C8C2×D5semidirect
212. #810C2×D5semidirect
213. #810C2×D5semidirect
219. #809C2×D5semidirect
220. #809C2×D5semidirect
321. #235C4×D5semidirect
409. #117D3×D5semidirect
540. C3×C5#2897semidirect
619. C5#82563semidirect
643. C3#10610562semidirect

Maximal subgroups

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

subgroupindex
C2×#920192
C3×#658572
C2×#912832
#117511022
C5×#50022
C2×#909682
#117483842
#117475012
#117475002
#117474752
D5×#8092
#106105623
#825635