Finite Group Databasebeta

← #14412625

Subgroups — #14412625

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×#92100C211yesmaximal, characteristic
2C2×#91796C211yesmaximal, characteristic, complemented
3#11756263C211yesmaximal, characteristic, complemented
4C3×C5×#417C211yesmaximal, characteristic, complemented
5#11751334C211yesmaximal, characteristic, complemented
6#11751232C211yesmaximal, characteristic, complemented
7C2×#91093C211yesmaximal, characteristic, complemented
8#11750905C212yesmaximal, complemented
9#11750904C212yesmaximal, complemented
10#11750871C212yesmaximal, complemented
11D15×#102C212yesmaximal, complemented
12#10610613—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14#92100—24no—
15#92100C418yescomplemented
16C2×#6821C2212yes—
17C2×D4×D15—22no—
18C2×D4×D15C2212yes—
19C2×#6817—21no—
20C2×#6817C2211yescharacteristic
21C22×D60—21no—
22C22×D60C2211yescharacteristic
23C2×C3×C5×M16C2211yescharacteristic, complemented
24C2×C3×C5×C4⋊C4C2211yescharacteristic, complemented
25C4×#1220—22no—
26#91815C2211yescharacteristic, complemented
27C22×C3⋊#168C2211yescharacteristic, complemented
28#91796—24no—
29C2×#6705C2212yescomplemented
30C2×C8×D15C2212yescomplemented
31#91782—21no—
32#91781C2212yescomplemented
33C4⋊C4×D15C2212yescomplemented
34#91776—21no—
35C3×C5×#103C2212yescomplemented
36C3×C5×#102C2212yescomplemented
37#91119C2211yescharacteristic, complemented
38#91119C2211yescharacteristic, complemented
39#91116C2211yescharacteristic, complemented
40#91116C2211yescharacteristic, complemented
41#91105C2212yescomplemented
42#91105C2212yescomplemented
43#91093C2212yescomplemented
44#91093C2212yescomplemented
45#78882—51nomaximal, Hall
46C2×#4149—31no—
47C2×#4053—31no—
48#65838—31no—
49C5×#417D311yescharacteristic, complemented
50#65522—31no—
51#65420—31no—
52C2×#3941—31no—
53#65093—32no—
54#65092—32no—
55#65059—32no—
56D5×#102—32no—
57C23×D15—21no—
58C23×D15—22no—
59C3×C5×#41—22no—
60C3×C5×#41C2×C414yescomplemented
61C2×C3×C5×Q8C2311yescharacteristic
62C2×C3×C5×D4—21no—
63C2×C3×C5×D4C2311yescharacteristic
64C22×C3×C4×C5—21no—
65C22×C3×C4×C5—21no—
66C22×C3×C4×C5C2311yescharacteristic
67C2×#1220—22no—
68C2×#1220D412yes—
69#6821—24no—
70#6821C2×C418yescomplemented
71D4×D15—216no—
72D4×D15—24no—
73D4×D15C2×C418yescomplemented
74#6817—28no—
75#6817C2×C414yes—
76#6817D414yes—
77C2×D60—22no—
78C2×D60—21no—
79C2×D60—22no—
80C2×D60—28no—
81C2×D60C2×C414yescomplemented
82C2×D60D411yescharacteristic
83C2×D60C2312yes—
84C2×C4×D15—22no—
85C2×C4×D15—22no—
86C2×C4×D15C2312yes—
87C2×Dic30D411yescharacteristic
88C3×C5×M16—22no—
89C2×C3×C5×C8C2312yes—
90C3×C5×C4⋊C4—21no—
91C3×C5×C4⋊C4C2312yes—
92#6719—21no—
93#6714—22no—
94C4⋊Dic15C2311yescharacteristic
95C4⋊Dic15C2311yescharacteristic
96#6712—22no—
97C4×Dic15—21no—
98C4×Dic15—22no—
99C2×C3⋊#168—22no—
100C2×C3⋊#168C2311yescharacteristic
101C2×C3⋊#168C2311yescharacteristic
102#6705—24no—
103C8×D15—24no—
104C2×#1009—51no—
105C2×#907—51no—
106#5504—51no—
107C3×#417D511yescharacteristic, complemented
108#5233—51no—
109#5131—51no—
110C2×#809—51no—
111#4804—52no—
112#4803—52no—
113#4770—52no—
114D3×#102—52no—
115C2×C5×#41C2×D311yescharacteristic
116#4149—38no—
117#4149—64no—
118C2×#713—32no—
119C2×D4×D5—32no—
120C2×D4×D5—62no—
121C2×#709—31no—
122C2×#709—61no—
123C22×D20—31no—
124C22×D20—61no—
125C2×C5×M16C2×D311yescharacteristic, complemented
126C2×C5×C4⋊C4C2×D311yescharacteristic, complemented
127C4×C22⋊D5—62no—
128#4072—31no—
129C22×#168—31no—
130#4053—64no—
131C2×#676—32no—
132C2×C8×D5—32no—
133#4039—61no—
134#4038—32no—
135D5×C4⋊C4—32no—
136#4033—61no—
137C5×#103C2×D312yescomplemented
138C5×#102C2×D312yescomplemented
139#3967—31no—
140#3967—31no—
141#3964—31no—
142#3964—31no—
143#3953—32no—
144#3953—32no—
145#3941—32no—
146#3941—32no—
147#2897—151noSylow
148C23×C3×C5—21no—
149C23×C3×C5C2×D411yescharacteristic
150C22×D15—22no—
151C22×D15—24no—
152C22×D15—28no—
153C22×D15—82no—
154C22×D15—82no—
155C22×D15—81no—
156C22×D15—81no—
157C22×D15C2×D412yes—
158C3×C5×Q8—21no—
159C3×C5×Q8C22×C412yes—
160C3×C5×D4—24no—
161C3×C5×D4—21no—
162C3×C5×D4C22×C412yes—
163C2×C3×C4×C5—21no—
164C2×C3×C4×C5—24no—
165C2×C3×C4×C5—22no—
166C2×C3×C4×C5—81no—
167C2×C3×C4×C5C22×C412yes—
168C2×C3×C4×C5C2×D411yescharacteristic
169C2×C3×C4×C5C2×D412yes—
170C2×C3×C4×C5C2411yescharacteristic
171#1220—28no—
172#1220C22⋊C418yes—
173C2×Dic15—21no—
174C2×Dic15—21no—
175C2×Dic15C2×D412yes—
176D60—28no—
177D60—24no—
178D60—22no—
179D60C22⋊C414yes—
180D60C22×C414yes—
181C4×D15—28no—
182C4×D15—82no—
183C4×D15C22×C414yes—
184C4×D15C2×D414yes—
185Dic30C22⋊C414yes—
186C3×C5×C8—22no—
187C3⋊#168—21no—
188C3⋊#168—21no—
189C2×C3×#41C2×D511yescharacteristic
190#1009—58no—
191#1009—104no—
192C2×#239—52no—
193C2×D3×D4—52no—
194C2×D3×D4—102no—
195C2×#235—51no—
196C2×#235—101no—
197C22×D12—51no—
198C22×D12—101no—
199C2×C3×M16C2×D511yescharacteristic, complemented
200C2×C3×C4⋊C4C2×D511yescharacteristic, complemented
201C4×#67—102no—
202#926—51no—
203C22×C3⋊C8—51no—
204#907—104no—
205C2×ΓL(1,𝔽25)—52no—
206C2×D3×C8—52no—
207#893—101no—
208#892—52no—
209D3×C4⋊C4—52no—
210#887—101no—
211C3×#103C2×D512yescomplemented
212C3×#102C2×D512yescomplemented
213#835—51no—
214#835—51no—
215#832—51no—
216#832—51no—
217#821—52no—
218#821—52no—
219#809—52no—
220#809—52no—
221C23×D5—62no—
222C23×D5—61no—
223C5×#41—22no—
224C5×#41C4×D314yescomplemented
225C2×C5×Q8C22×D311yescharacteristic
226C2×C5×D4—21no—
227C2×C5×D4C22×D311yescharacteristic
228C22×C4×C5—21no—
229C22×C4×C5—21no—
230C22×C4×C5C22×D311yescharacteristic
231C2×C22⋊D5—32no—
232C2×C22⋊D5—62no—
233#713—38no—
234#713—64no—
235D4×D5—38no—
236D4×D5—64no—
237D4×D5—616no—
238#709—34no—
239#709—34no—
240#709—68no—
241C2×D20—31no—
242C2×D20—34no—
243C2×D20—32no—
244C2×D20—68no—
245C2×D20—62no—
246C2×D20—61no—
247C2×D20—62no—
248C2×C4×D5—32no—
249C2×C4×D5—62no—
250C2×C4×D5—62no—
251C2×Dic10—31no—
252C5×M16—22no—
253C2×C5×C8C22×D312yes—
254C5×C4⋊C4—21no—
255C5×C4⋊C4C22×D312yes—
256C22⋊Dic5—61no—
257#685—62no—
258C4⋊Dic5—31no—
259C4⋊Dic5—31no—
260#683—62no—
261C4×Dic5—62no—
262C4×Dic5—61no—
263C2×#168—31no—
264C2×#168—31no—
265C2×#168—62no—
266#676—64no—
267C8×D5—64no—
268C2×H(5,𝔽2)—151no—
269C2×#131—151no—
270#520—151no—
271#418—151no—
272#418—152no—
273#417—152no—
274#417D1511yescharacteristic, complemented
275#416—151no—
276#416—152no—
277C2×#102—151no—
278C2×#102—152no—
279C22×C3×C5—21no—
280C22×C3×C5—21no—
281C22×C3×C5—22no—
282C22×C3×C5—41no—
283C22×C3×C5—41no—
284C22×C3×C5C2×C22⋊C412yes—
285C22×C3×C5C22×D411yescharacteristic
286C2×D15—22no—
287C2×D15—22no—
288C2×D15—24no—
289C2×D15—42no—
290C2×D15—42no—
291C2×D15—88no—
292C2×D15—84no—
293C2×D15C2×C22⋊C414yes—
294C3×C4×C5—22no—
295C3×C4×C5—42no—
296C3×C4×C5C2×C22⋊C412yes—
297C3×C4×C5C23×C411yescharacteristic
298C3×C4×C5C22×D411yescharacteristic
299Dic15—41no—
300Dic15—41no—
301Dic15C2×C22⋊C414yes—
302C23×D3—102no—
303C23×D3—101no—
304C3×#41—22no—
305C3×#41C4×D514yescomplemented
306C2×C3×Q8C22×D511yescharacteristic
307C2×C3×D4—21no—
308C2×C3×D4C22×D511yescharacteristic
309C22×C3×C4—21no—
310C22×C3×C4—21no—
311C22×C3×C4C22×D511yescharacteristic
312C2×#67—52no—
313C2×#67—102no—
314#239—58no—
315#239—104no—
316D3×D4—58no—
317D3×D4—104no—
318D3×D4—1016no—
319#235—54no—
320#235—54no—
321#235—108no—
322C2×D12—51no—
323C2×D12—54no—
324C2×D12—52no—
325C2×D12—108no—
326C2×D12—102no—
327C2×D12—101no—
328C2×D12—102no—
329C2×C4×D3—52no—
330C2×C4×D3—102no—
331C2×C4×D3—102no—
332C2×Dic6—51no—
333C3×M16—22no—
334C2×C3×C8C22×D512yes—
335C3×C4⋊C4—21no—
336C3×C4⋊C4C22×D512yes—
337#217—101no—
338(C2×D3)⋊C4—102no—
339C4⋊Dic3—51no—
340C4⋊Dic3—51no—
341Dic3⋊C4—102no—
342C4×Dic3—102no—
343C4×Dic3—101no—
344C2×C3⋊C8—51no—
345C2×C3⋊C8—51no—
346C2×C3⋊C8—102no—
347ΓL(1,𝔽25)—104no—
348D3×C8—104no—
349C23×C5—21no—
350C23×C5D3×D411yescharacteristic
351C22×D5—32no—
352C22×D5—68no—
353C22×D5—62no—
354C22×D5—64no—
355C22×D5—242no—
356C22×D5—242no—
357C22×D5—241no—
358C22×D5—241no—
359C5×Q8—21no—
360C5×Q8C2×C4×D312yes—
361C5×D4—24no—
362C5×D4—21no—
363C5×D4C2×C4×D312yes—
364C2×C4×C5—21no—
365C2×C4×C5—24no—
366C2×C4×C5—22no—
367C2×C4×C5—81no—
368C2×C4×C5C2×C4×D312yes—
369C2×C4×C5D3×D411yescharacteristic
370C2×C4×C5D3×D412yes—
371C2×C4×C5C23×D311yescharacteristic
372C22⋊D5—38no—
373C22⋊D5—68no—
374C2×Dic5—32no—
375C2×Dic5—61no—
376C2×Dic5—61no—
377D20—34no—
378D20—34no—
379D20—68no—
380D20—64no—
381D20—62no—
382C4×D5—34no—
383C4×D5—34no—
384C4×D5—68no—
385C4×D5—242no—
386Dic10—34no—
387C5×C8—22no—
388#168—61no—
389#168—61no—
390H(5,𝔽2)—158no—
391H(5,𝔽2)—304no—
392C2×#41—151no—
393C2×#41—152no—
394C2×#41—301no—
395C2×#41C2×D1511yescharacteristic
396C22×D4—152no—
397C22×D4—151no—
398C22×D4—301no—
399C22×D4—302no—
400#131—304no—
401C2×M16—152no—
402C2×M16C2×D1511yescharacteristic, complemented
403C22×C8—151no—
404C22×C8—152no—
405C4×Q8—301no—
406C4×D4—301no—
407C4×D4—302no—
408#117—151no—
409#117—152no—
410C2×C4⋊C4—152no—
411C2×C4⋊C4C2×D1511yescharacteristic, complemented
412#103—151no—
413#103—151no—
414#103C2×D1512yescomplemented
415#102—151no—
416#102—151no—
417#102—152no—
418#102—152no—
419#102—152no—
420#102—152no—
421#102C2×D1512yescomplemented
422C2×C3×C5—22no—
423C2×C3×C5—42no—
424C2×C3×C5C22×C22⋊C411yescharacteristic
425C2×C3×C5#57611yescharacteristic
426C2×C3×C5#57711yescharacteristic
427D15—24no—
428D15—44no—
429C23×C3—21no—
430C23×C3D4×D511yescharacteristic
431C22×D3—52no—
432C22×D3—108no—
433C22×D3—102no—
434C22×D3—104no—
435C22×D3—402no—
436C22×D3—402no—
437C22×D3—401no—
438C22×D3—401no—
439C3×Q8—21no—
440C3×Q8C2×C4×D512yes—
441C3×D4—24no—
442C3×D4—21no—
443C3×D4C2×C4×D512yes—
444C2×C3×C4—21no—
445C2×C3×C4—24no—
446C2×C3×C4—22no—
447C2×C3×C4—81no—
448C2×C3×C4C2×C4×D512yes—
449C2×C3×C4D4×D512yes—
450C2×C3×C4D4×D511yescharacteristic
451C2×C3×C4C23×D511yescharacteristic
452#67—58no—
453#67—108no—
454C2×Dic3—52no—
455C2×Dic3—101no—
456C2×Dic3—101no—
457D12—54no—
458D12—54no—
459D12—108no—
460D12—102no—
461D12—104no—
462C4×D3—54no—
463C4×D3—54no—
464C4×D3—108no—
465C4×D3—402no—
466Dic6—54no—
467C3×C8—22no—
468C3⋊C8—101no—
469C3⋊C8—101no—
470C22×C5—21no—
471C22×C5—21no—
472C22×C5—22no—
473C22×C5—41no—
474C22×C5—41no—
475C22×C5D3×C22⋊C412yes—
476C22×C5C2×D3×D411yescharacteristic
477C2×D5—34no—
478C2×D5—62no—
479C2×D5—62no—
480C2×D5—64no—
481C2×D5—122no—
482C2×D5—122no—
483C2×D5—248no—
484C2×D5—244no—
485C4×C5—22no—
486C4×C5—42no—
487C4×C5D3×C22⋊C412yes—
488C4×C5C22×C4×D311yescharacteristic
489C4×C5C2×D3×D411yescharacteristic
490Dic5—34no—
491Dic5—121no—
492Dic5—121no—
493C24—302no—
494C24—301no—
495#41—22no—
496#41—154no—
497#41—154no—
498#41—158no—
499#41—308no—
500#41—304no—
501#41C4×D1514yescomplemented
502C2×Q8—151no—
503C2×Q8C22×D1511yescharacteristic
504C2×D4—21no—
505C2×D4—152no—
506C2×D4—158no—
507C2×D4—154no—
508C2×D4—151no—
509C2×D4—152no—
510C2×D4—308no—
511C2×D4—302no—
512C2×D4—304no—
513C2×D4—302no—
514C2×D4—3016no—
515C2×D4—301no—
516C2×D4—302no—
517C2×D4C22×D1511yescharacteristic
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—152no—
521C22×C4—302no—
522C22×C4—302no—
523C22×C4C22×D1511yescharacteristic
524M16—22no—
525M16—304no—
526C2×C8—151no—
527C2×C8—151no—
528C2×C8—304no—
529C2×C8—302no—
530C2×C8C22×D1512yes—
531C4⋊C4—21no—
532C4⋊C4—151no—
533C4⋊C4—151no—
534C4⋊C4—302no—
535C4⋊C4C22×D1512yes—
536C22⋊C4—302no—
537C22⋊C4—301no—
538C42—302no—
539C42—301no—
540C3×C5#289711yescharacteristic, complemented, Hall
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—22no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3D5×C22⋊C412yes—
547C22×C3C2×D4×D511yescharacteristic
548C2×D3—54no—
549C2×D3—102no—
550C2×D3—102no—
551C2×D3—104no—
552C2×D3—202no—
553C2×D3—202no—
554C2×D3—408no—
555C2×D3—404no—
556C3×C4—22no—
557C3×C4—42no—
558C3×C4D5×C22⋊C412yes—
559C3×C4C22×C4×D511yescharacteristic
560C3×C4C2×D4×D511yescharacteristic
561Dic3—54no—
562Dic3—201no—
563Dic3—201no—
564C2×C5—22no—
565C2×C5—42no—
566C2×C5C2×D3×C22⋊C411yescharacteristic
567C2×C5#577411yescharacteristic
568C2×C5#577811yescharacteristic
569D5—64no—
570D5—124no—
571C23—21no—
572C23—152no—
573C23—308no—
574C23—302no—
575C23—304no—
576C23—1201no—
577C23—1201no—
578C23—1202no—
579C23—1202no—
580C23D4×D1511yescharacteristic
581Q8—21no—
582Q8—154no—
583Q8C2×C4×D1512yes—
584D4—24no—
585D4—21no—
586D4—158no—
587D4—154no—
588D4—154no—
589D4—308no—
590D4—308no—
591D4—304no—
592D4—302no—
593D4C2×C4×D1512yes—
594C2×C4—21no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—81no—
598C2×C4—152no—
599C2×C4—154no—
600C2×C4—154no—
601C2×C4—308no—
602C2×C4—301no—
603C2×C4—301no—
604C2×C4—1202no—
605C2×C4C2×C4×D1512yes—
606C2×C4D4×D1511yescharacteristic
607C2×C4D4×D1512yes—
608C2×C4C23×D1511yescharacteristic
609C8—22no—
610C8—301no—
611C8—301no—
612C2×C3—22no—
613C2×C3—42no—
614C2×C3C2×D5×C22⋊C411yescharacteristic
615C2×C3#6610811yescharacteristic
616C2×C3#6611211yescharacteristic
617D3—104no—
618D3—204no—
619C5#7888211yescharacteristic, complemented, Sylow
620C22—21no—
621C22—21no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—154no—
626C22—302no—
627C22—302no—
628C22—304no—
629C22—602no—
630C22—602no—
631C22—1204no—
632C22—1208no—
633C22C22⋊C4×D1512yes—
634C22C2×D4×D1511yescharacteristic
635C4—22no—
636C4—42no—
637C4—154no—
638C4—601no—
639C4—601no—
640C4C22⋊C4×D1512yes—
641C4C22×C4×D1511yescharacteristic
642C4C2×D4×D1511yescharacteristic
643C3#1061061311yescharacteristic, complemented, Sylow
644C2—22no—
645C2—42no—
646C2—304no—
647C2—604no—
648C2C2×C22⋊C4×D1511yescharacteristic
649C2#1175653311yescharacteristic
650C2#1175653711yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#417C2
socleC22×C3×C5C22×D4
prime radicalC2×#92100C2
cosocleC2×C3×C4×C5C24
hypercenter#417D15
hypocenterC3×C5#2897
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#91796C2semidirect
3. #11756263C2semidirect
4. C3×C5×#417C2semidirect
5. #11751334C2semidirect
6. #11751232C2semidirect
7. C2×#91093C2semidirect
8. #11750905C2semidirect
9. #11750904C2semidirect
10. #11750871C2semidirect
11. D15×#102C2semidirect
15. #92100C4semidirect
23. C2×C3×C5×M16C22semidirect
24. C2×C3×C5×C4⋊C4C22semidirect
26. #91815C22semidirect
27. C22×C3⋊#168C22semidirect
29. C2×#6705C22semidirect
30. C2×C8×D15C22semidirect
32. #91781C22semidirect
33. C4⋊C4×D15C22semidirect
35. C3×C5×#103C22semidirect
36. C3×C5×#102C22semidirect
37. #91119C22semidirect
38. #91119C22semidirect
39. #91116C22semidirect
40. #91116C22semidirect
41. #91105C22semidirect
42. #91105C22semidirect
43. #91093C22semidirect
44. #91093C22semidirect
49. C5×#417D3semidirect
60. C3×C5×#41C2×C4semidirect
70. #6821C2×C4semidirect
73. D4×D15C2×C4semidirect
81. C2×D60C2×C4semidirect
107. C3×#417D5semidirect
125. C2×C5×M16C2×D3semidirect
126. C2×C5×C4⋊C4C2×D3semidirect
137. C5×#103C2×D3semidirect
138. C5×#102C2×D3semidirect
199. C2×C3×M16C2×D5semidirect
200. C2×C3×C4⋊C4C2×D5semidirect
211. C3×#103C2×D5semidirect
212. C3×#102C2×D5semidirect
224. C5×#41C4×D3semidirect
274. #417D15semidirect
305. C3×#41C4×D5semidirect
402. C2×M16C2×D15semidirect
411. C2×C4⋊C4C2×D15semidirect
414. #103C2×D15semidirect
421. #102C2×D15semidirect
501. #41C4×D15semidirect
540. C3×C5#2897semidirect
619. C5#78882semidirect
643. C3#10610613semidirect

Maximal subgroups

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

subgroupindex
C2×#921002
C2×#917962
#117562632
C3×C5×#4172
#117513342
#117512322
C2×#910932
#117509052
#117509042
#117508712
D15×#1022
#106106133
#788825