Finite Group Databasebeta

← #14412628

Subgroups — #14412628

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×#92095C211yesmaximal, characteristic
2C2×#91796C211yesmaximal, characteristic, complemented
3#11756282C211yesmaximal, characteristic, complemented
4C3×C5×#418C211yesmaximal, characteristic, complemented
5C2×#91116C211yesmaximal, characteristic, complemented
6#11751234C211yesmaximal, characteristic, complemented
7#11751103C211yesmaximal, characteristic, complemented
8#11750873C212yesmaximal, complemented
9#11750872C212yesmaximal, complemented
10#11750871C212yesmaximal, complemented
11D15×#102C212yesmaximal, complemented
12#10610656—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C22×#1220—22no—
15#92095—24no—
16#92095C418yescomplemented
17C2×#6819—21no—
18C2×#6819C2212yes—
19C2×D4×D15—21no—
20C2×D4×D15C2212yes—
21C2×#6817C2211yescharacteristic
22C2×#6817C2211yescharacteristic
23C2×C3×C5×M16C2211yescharacteristic, complemented
24C3×C5×#117C2211yescharacteristic, complemented
25C2×C4⋊Dic15C2211yescharacteristic, complemented
26C22×C3⋊#168C2211yescharacteristic, complemented
27#91796—24no—
28C2×#6705C2212yescomplemented
29C2×C8×D15C2212yescomplemented
30#91781C2212yescomplemented
31C4⋊C4×D15C2212yescomplemented
32#91770—22no—
33C4×D60—21no—
34C4×Dic30—21no—
35C3×C5×#102C2212yescomplemented
36C3×C5×#102C2212yescomplemented
37#91116C2212yescomplemented
38#91116C2212yescomplemented
39#91105C2212yescomplemented
40#91100C2212yescomplemented
41#91094C2211yescharacteristic, complemented
42#91094C2211yescharacteristic, complemented
43#91093C2211yescharacteristic, complemented
44#91093C2211yescharacteristic, complemented
45#78925—51nomaximal, Hall
46C2×#4144—31no—
47C2×#4053—31no—
48#65857—31no—
49C5×#418D311yescharacteristic, complemented
50C2×#3964—31no—
51#65422—31no—
52#65291—31no—
53#65061—32no—
54#65060—32no—
55#65059—32no—
56D5×#102—32no—
57C24×C3×C5—21no—
58C23×D15—22no—
59C2×C3×C5×D4—21no—
60C2×C3×C5×D4—22no—
61C2×C3×C5×D4C2×C414yescomplemented
62C2×C3×C5×D4C2312yes—
63C22×C3×C4×C5C2311yescharacteristic
64C2×#1220—21no—
65C2×#1220—22no—
66C2×#1220—216no—
67C2×#1220—22no—
68C2×#1220D412yes—
69C22×Dic15—22no—
70C22×Dic15—21no—
71#6819—24no—
72#6819—28no—
73#6819C2×C418yescomplemented
74D4×D15—28no—
75D4×D15—24no—
76D4×D15C2×C418yescomplemented
77#6817—22no—
78#6817C2×C414yes—
79#6817C2×C414yescomplemented
80#6817D414yes—
81C2×D60D411yescharacteristic
82C2×D60C2311yescharacteristic
83C2×C4×D15—21no—
84C2×C4×D15—22no—
85C2×C4×D15C2312yes—
86C2×Dic30D411yescharacteristic
87C2×Dic30C2311yescharacteristic
88C3×C5×M16—22no—
89C2×C3×C5×C8C2312yes—
90C3×C5×C4⋊C4C2311yescharacteristic
91C3×C5×C4⋊C4C2311yescharacteristic
92C3×C5×C22⋊C4—21no—
93C3×C42×C5—21no—
94#6714—22no—
95C4⋊Dic15—21no—
96C4⋊Dic15C2312yes—
97#6712—22no—
98C4×Dic15—22no—
99C2×C3⋊#168—22no—
100C2×C3⋊#168C2311yescharacteristic
101C2×C3⋊#168C2311yescharacteristic
102#6705—24no—
103C8×D15—24no—
104C2×#1004—51no—
105C2×#907—51no—
106#5523—51no—
107C3×#418D511yescharacteristic, complemented
108C2×#832—51no—
109#5133—51no—
110#5002—51no—
111#4772—52no—
112#4771—52no—
113#4770—52no—
114D3×#102—52no—
115C22×C5×D4C2×D311yescharacteristic
116C22×C22⋊D5—62no—
117#4144—38no—
118#4144—64no—
119C2×#711—32no—
120C2×#711—61no—
121C2×D4×D5—32no—
122C2×D4×D5—61no—
123C2×#709—31no—
124C2×#709—31no—
125C2×C5×M16C2×D311yescharacteristic, complemented
126C5×#117C2×D311yescharacteristic, complemented
127C2×C4⋊Dic5—31no—
128C22×#168—31no—
129#4053—64no—
130C2×#676—32no—
131C2×C8×D5—32no—
132#4038—32no—
133D5×C4⋊C4—32no—
134#4027—62no—
135C4×D20—61no—
136C4×Dic10—61no—
137C5×#102C2×D312yescomplemented
138C5×#102C2×D312yescomplemented
139#3964—32no—
140#3964—32no—
141#3953—32no—
142#3948—32no—
143#3942—31no—
144#3942—31no—
145#3941—31no—
146#3941—31no—
147#2897—151noSylow
148C23×C3×C5—22no—
149C23×C3×C5—24no—
150C23×C3×C5—81no—
151C23×C3×C5—81no—
152C23×C3×C5C2×D411yescharacteristic
153C22×D15—21no—
154C22×D15—28no—
155C22×D15—82no—
156C22×D15—82no—
157C22×D15C2×D412yes—
158C3×C5×D4—24no—
159C3×C5×D4—22no—
160C3×C5×D4C22×C414yes—
161C2×C3×C4×C5—21no—
162C2×C3×C4×C5—21no—
163C2×C3×C4×C5C22×C412yes—
164C2×C3×C4×C5C2×D411yescharacteristic
165C2×C3×C4×C5C2×D412yes—
166C2×C3×C4×C5C2411yescharacteristic
167#1220—24no—
168#1220—28no—
169#1220—28no—
170#1220C22⋊C418yes—
171C2×Dic15—21no—
172C2×Dic15—28no—
173C2×Dic15—22no—
174C2×Dic15—81no—
175C2×Dic15C2×D412yes—
176D60—21no—
177D60C22⋊C414yes—
178D60C22×C412yes—
179C4×D15—24no—
180C4×D15—82no—
181C4×D15C22×C414yes—
182C4×D15C2×D414yes—
183Dic30—21no—
184Dic30C22⋊C414yes—
185Dic30C22×C412yes—
186C3×C5×C8—22no—
187C3⋊#168—21no—
188C3⋊#168—21no—
189C22×C3×D4C2×D511yescharacteristic
190C22×#67—102no—
191#1004—58no—
192#1004—104no—
193C2×#237—52no—
194C2×#237—101no—
195C2×D3×D4—52no—
196C2×D3×D4—101no—
197C2×#235—51no—
198C2×#235—51no—
199C2×C3×M16C2×D511yescharacteristic, complemented
200C3×#117C2×D511yescharacteristic, complemented
201C2×C4⋊Dic3—51no—
202C22×C3⋊C8—51no—
203#907—104no—
204C2×ΓL(1,𝔽25)—52no—
205C2×D3×C8—52no—
206#892—52no—
207D3×C4⋊C4—52no—
208#67⋊C4—102no—
209C4×D12—101no—
210C4×Dic6—101no—
211C3×#102C2×D512yescomplemented
212C3×#102C2×D512yescomplemented
213#832—52no—
214#832—52no—
215#821—52no—
216#816—52no—
217#810—51no—
218#810—51no—
219#809—51no—
220#809—51no—
221C24×C5—21no—
222C23×D5—62no—
223C2×C5×D4—21no—
224C2×C5×D4—22no—
225C2×C5×D4C4×D314yescomplemented
226C2×C5×D4C22×D312yes—
227C22×C4×C5C22×D311yescharacteristic
228C2×C22⋊D5—32no—
229C2×C22⋊D5—62no—
230C2×C22⋊D5—616no—
231C2×C22⋊D5—62no—
232C2×C22⋊D5—61no—
233C22×Dic5—62no—
234C22×Dic5—61no—
235#711—38no—
236#711—68no—
237#711—64no—
238D4×D5—38no—
239D4×D5—64no—
240D4×D5—68no—
241#709—34no—
242#709—34no—
243#709—34no—
244#709—62no—
245C2×D20—31no—
246C2×D20—31no—
247C2×C4×D5—32no—
248C2×C4×D5—61no—
249C2×C4×D5—62no—
250C2×Dic10—31no—
251C2×Dic10—31no—
252C5×M16—22no—
253C2×C5×C8C22×D312yes—
254C5×C4⋊C4C22×D311yescharacteristic
255C5×C4⋊C4C22×D311yescharacteristic
256C5×C22⋊C4—21no—
257C42×C5—21no—
258#685—62no—
259C4⋊Dic5—32no—
260C4⋊Dic5—61no—
261#683—62no—
262C4×Dic5—62no—
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—152no—
272#418D1511yescharacteristic, complemented
273#417—151no—
274#417—152no—
275#416—151no—
276#416—152no—
277C2×#102—151no—
278C2×#102—152no—
279C22×C3×C5—21no—
280C22×C3×C5—21no—
281C22×C3×C5—24no—
282C22×C3×C5—42no—
283C22×C3×C5—42no—
284C22×C3×C5—84no—
285C22×C3×C5C2×C22⋊C412yes—
286C22×C3×C5C22×D411yescharacteristic
287C2×D15—22no—
288C2×D15—22no—
289C2×D15—22no—
290C2×D15—41no—
291C2×D15—41no—
292C2×D15—88no—
293C2×D15C2×C22⋊C414yes—
294C3×C4×C5—41no—
295C3×C4×C5—41no—
296C3×C4×C5C2×C22⋊C412yes—
297C3×C4×C5C23×C411yescharacteristic
298C3×C4×C5C22×D411yescharacteristic
299Dic15—22no—
300Dic15—42no—
301Dic15C2×C22⋊C414yes—
302C24×C3—21no—
303C23×D3—102no—
304C2×C3×D4—21no—
305C2×C3×D4—22no—
306C2×C3×D4C4×D514yescomplemented
307C2×C3×D4C22×D512yes—
308C22×C3×C4C22×D511yescharacteristic
309C2×#67—52no—
310C2×#67—102no—
311C2×#67—1016no—
312C2×#67—102no—
313C2×#67—101no—
314C22×Dic3—102no—
315C22×Dic3—101no—
316#237—58no—
317#237—108no—
318#237—104no—
319D3×D4—58no—
320D3×D4—104no—
321D3×D4—108no—
322#235—54no—
323#235—54no—
324#235—54no—
325#235—102no—
326C2×D12—51no—
327C2×D12—51no—
328C2×C4×D3—52no—
329C2×C4×D3—101no—
330C2×C4×D3—102no—
331C2×Dic6—51no—
332C2×Dic6—51no—
333C3×M16—22no—
334C2×C3×C8C22×D512yes—
335C3×C4⋊C4C22×D511yescharacteristic
336C3×C4⋊C4C22×D511yescharacteristic
337C3×C22⋊C4—21no—
338C3×C42—21no—
339(C2×D3)⋊C4—102no—
340C4⋊Dic3—52no—
341C4⋊Dic3—101no—
342Dic3⋊C4—102no—
343C4×Dic3—102no—
344C2×C3⋊C8—51no—
345C2×C3⋊C8—51no—
346C2×C3⋊C8—102no—
347ΓL(1,𝔽25)—104no—
348D3×C8—104no—
349C23×C5—22no—
350C23×C5—24no—
351C23×C5—81no—
352C23×C5—81no—
353C23×C5D3×D411yescharacteristic
354C22×D5—32no—
355C22×D5—68no—
356C22×D5—61no—
357C22×D5—242no—
358C22×D5—242no—
359C5×D4—24no—
360C5×D4—22no—
361C5×D4C2×C4×D314yes—
362C2×C4×C5—21no—
363C2×C4×C5—21no—
364C2×C4×C5C2×C4×D312yes—
365C2×C4×C5D3×D411yescharacteristic
366C2×C4×C5D3×D412yes—
367C2×C4×C5C23×D311yescharacteristic
368C22⋊D5—38no—
369C22⋊D5—68no—
370C22⋊D5—68no—
371C22⋊D5—64no—
372C2×Dic5—32no—
373C2×Dic5—68no—
374C2×Dic5—61no—
375C2×Dic5—62no—
376C2×Dic5—241no—
377D20—34no—
378D20—32no—
379D20—61no—
380C4×D5—34no—
381C4×D5—34no—
382C4×D5—64no—
383C4×D5—242no—
384Dic10—34no—
385Dic10—32no—
386Dic10—61no—
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—151no—
395C2×#41—301no—
396C22×D4—152no—
397C22×D4—301no—
398C22×D4—302no—
399C22×D4C2×D1511yescharacteristic
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—152no—
409#117C2×D1511yescharacteristic, complemented
410C2×C4⋊C4—151no—
411C2×C4⋊C4—152no—
412#103—152no—
413#103—151no—
414#103—151no—
415#102—152no—
416#102—152no—
417#102—152no—
418#102—151no—
419#102—151no—
420#102C2×D1512yescomplemented
421#102C2×D1512yescomplemented
422C2×C3×C5—22no—
423C2×C3×C5—44no—
424C2×C3×C5C22×C22⋊C411yescharacteristic
425C2×C3×C5#57611yescharacteristic
426C2×C3×C5#57711yescharacteristic
427D15—24no—
428D15—42no—
429C23×C3—22no—
430C23×C3—24no—
431C23×C3—81no—
432C23×C3—81no—
433C23×C3D4×D511yescharacteristic
434C22×D3—52no—
435C22×D3—108no—
436C22×D3—101no—
437C22×D3—402no—
438C22×D3—402no—
439C3×D4—22no—
440C3×D4—24no—
441C3×D4C2×C4×D514yes—
442C2×C3×C4—21no—
443C2×C3×C4—21no—
444C2×C3×C4C2×C4×D512yes—
445C2×C3×C4D4×D512yes—
446C2×C3×C4D4×D511yescharacteristic
447C2×C3×C4C23×D511yescharacteristic
448#67—58no—
449#67—108no—
450#67—108no—
451#67—104no—
452C2×Dic3—52no—
453C2×Dic3—108no—
454C2×Dic3—101no—
455C2×Dic3—102no—
456C2×Dic3—401no—
457D12—52no—
458D12—54no—
459D12—101no—
460C4×D3—54no—
461C4×D3—54no—
462C4×D3—104no—
463C4×D3—402no—
464Dic6—54no—
465Dic6—52no—
466Dic6—101no—
467C3×C8—22no—
468C3⋊C8—101no—
469C3⋊C8—101no—
470C22×C5—21no—
471C22×C5—21no—
472C22×C5—24no—
473C22×C5—42no—
474C22×C5—42no—
475C22×C5—84no—
476C22×C5D3×C22⋊C412yes—
477C22×C5C2×D3×D411yescharacteristic
478C2×D5—34no—
479C2×D5—62no—
480C2×D5—62no—
481C2×D5—62no—
482C2×D5—121no—
483C2×D5—121no—
484C2×D5—248no—
485C4×C5—41no—
486C4×C5—41no—
487C4×C5D3×C22⋊C412yes—
488C4×C5C22×C4×D311yescharacteristic
489C4×C5C2×D3×D411yescharacteristic
490Dic5—34no—
491Dic5—62no—
492Dic5—122no—
493C24—21no—
494C24—302no—
495#41—154no—
496#41—154no—
497#41—158no—
498#41—154no—
499#41—304no—
500#41—308no—
501#41—302no—
502C2×Q8—151no—
503C2×Q8—151no—
504C2×D4—21no—
505C2×D4—22no—
506C2×D4—152no—
507C2×D4—158no—
508C2×D4—151no—
509C2×D4—151no—
510C2×D4—304no—
511C2×D4—302no—
512C2×D4—308no—
513C2×D4—3016no—
514C2×D4—302no—
515C2×D4—301no—
516C2×D4C4×D1514yescomplemented
517C2×D4C22×D1512yes—
518C22×C4—152no—
519C22×C4—302no—
520C22×C4—302no—
521C22×C4—301no—
522C22×C4—301no—
523C22×C4C22×D1511yescharacteristic
524M16—22no—
525M16—304no—
526C2×C8—151no—
527C2×C8—151no—
528C2×C8—304no—
529C2×C8—302no—
530C2×C8C22×D1512yes—
531C4⋊C4—152no—
532C4⋊C4—302no—
533C4⋊C4—301no—
534C4⋊C4C22×D1511yescharacteristic
535C4⋊C4C22×D1511yescharacteristic
536C22⋊C4—21no—
537C22⋊C4—302no—
538C42—21no—
539C42—302no—
540C3×C5#289711yescharacteristic, complemented, Hall
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—24no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—84no—
547C22×C3D5×C22⋊C412yes—
548C22×C3C2×D4×D511yescharacteristic
549C2×D3—54no—
550C2×D3—102no—
551C2×D3—102no—
552C2×D3—102no—
553C2×D3—201no—
554C2×D3—201no—
555C2×D3—408no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4D5×C22⋊C412yes—
559C3×C4C22×C4×D511yescharacteristic
560C3×C4C2×D4×D511yescharacteristic
561Dic3—54no—
562Dic3—102no—
563Dic3—202no—
564C2×C5—22no—
565C2×C5—44no—
566C2×C5C2×D3×C22⋊C411yescharacteristic
567C2×C5#577411yescharacteristic
568C2×C5#577511yescharacteristic
569D5—64no—
570D5—122no—
571C23—22no—
572C23—24no—
573C23—81no—
574C23—81no—
575C23—152no—
576C23—308no—
577C23—301no—
578C23—1202no—
579C23—1202no—
580C23D4×D1511yescharacteristic
581Q8—154no—
582Q8—152no—
583Q8—301no—
584D4—24no—
585D4—22no—
586D4—158no—
587D4—154no—
588D4—152no—
589D4—308no—
590D4—308no—
591D4—304no—
592D4—301no—
593D4C2×C4×D1514yes—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—152no—
597C2×C4—154no—
598C2×C4—154no—
599C2×C4—308no—
600C2×C4—301no—
601C2×C4—304no—
602C2×C4—302no—
603C2×C4—1202no—
604C2×C4—1201no—
605C2×C4C2×C4×D1512yes—
606C2×C4D4×D1511yescharacteristic
607C2×C4D4×D1512yes—
608C2×C4C23×D1511yescharacteristic
609C8—22no—
610C8—301no—
611C8—301no—
612C2×C3—22no—
613C2×C3—44no—
614C2×C3C2×D5×C22⋊C411yescharacteristic
615C2×C3#6610811yescharacteristic
616C2×C3#6610911yescharacteristic
617D3—104no—
618D3—202no—
619C5#7892511yescharacteristic, complemented, Sylow
620C22—21no—
621C22—21no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—84no—
626C22—154no—
627C22—302no—
628C22—302no—
629C22—302no—
630C22—601no—
631C22—601no—
632C22—1208no—
633C22C22⋊C4×D1512yes—
634C22C2×D4×D1511yescharacteristic
635C4—41no—
636C4—41no—
637C4—154no—
638C4—302no—
639C4—602no—
640C4C22⋊C4×D1512yes—
641C4C22×C4×D1511yescharacteristic
642C4C2×D4×D1511yescharacteristic
643C3#1061065611yescharacteristic, complemented, Sylow
644C2—22no—
645C2—44no—
646C2—304no—
647C2—602no—
648C2C2×C22⋊C4×D1511yescharacteristic
649C2#1175653311yescharacteristic
650C2#1175653411yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#418C2
socleC22×C3×C5C22×D4
prime radicalC2×#92095C2
cosocleC2×C3×C4×C5C24
hypercenter#418D15
hypocenterC3×C5#2897
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#91796C2semidirect
3. #11756282C2semidirect
4. C3×C5×#418C2semidirect
5. C2×#91116C2semidirect
6. #11751234C2semidirect
7. #11751103C2semidirect
8. #11750873C2semidirect
9. #11750872C2semidirect
10. #11750871C2semidirect
11. D15×#102C2semidirect
16. #92095C4semidirect
23. C2×C3×C5×M16C22semidirect
24. C3×C5×#117C22semidirect
25. C2×C4⋊Dic15C22semidirect
26. C22×C3⋊#168C22semidirect
28. C2×#6705C22semidirect
29. C2×C8×D15C22semidirect
30. #91781C22semidirect
31. C4⋊C4×D15C22semidirect
35. C3×C5×#102C22semidirect
36. C3×C5×#102C22semidirect
37. #91116C22semidirect
38. #91116C22semidirect
39. #91105C22semidirect
40. #91100C22semidirect
41. #91094C22semidirect
42. #91094C22semidirect
43. #91093C22semidirect
44. #91093C22semidirect
49. C5×#418D3semidirect
61. C2×C3×C5×D4C2×C4semidirect
73. #6819C2×C4semidirect
76. D4×D15C2×C4semidirect
79. #6817C2×C4semidirect
107. C3×#418D5semidirect
125. C2×C5×M16C2×D3semidirect
126. C5×#117C2×D3semidirect
137. C5×#102C2×D3semidirect
138. C5×#102C2×D3semidirect
199. C2×C3×M16C2×D5semidirect
200. C3×#117C2×D5semidirect
211. C3×#102C2×D5semidirect
212. C3×#102C2×D5semidirect
225. C2×C5×D4C4×D3semidirect
272. #418D15semidirect
306. C2×C3×D4C4×D5semidirect
402. C2×M16C2×D15semidirect
409. #117C2×D15semidirect
420. #102C2×D15semidirect
421. #102C2×D15semidirect
516. C2×D4C4×D15semidirect
540. C3×C5#2897semidirect
619. C5#78925semidirect
643. C3#10610656semidirect

Maximal subgroups

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

subgroupindex
C2×#920952
C2×#917962
#117562822
C3×C5×#4182
C2×#911162
#117512342
#117511032
#117508732
#117508722
#117508712
D15×#1022
#106106563
#789255