Finite Group Databasebeta

← #14412671

Subgroups — #14412671

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×#92017C211yesmaximal, characteristic
2#11756282C211yesmaximal, characteristic, complemented
3C2×#91291C211yesmaximal, characteristic, complemented
4C5×#5002C211yesmaximal, characteristic, complemented
5C3×#65290C211yesmaximal, characteristic, complemented
6#11748390C211yesmaximal, characteristic, complemented
7C2×#90965C211yesmaximal, characteristic, complemented
8#11747507C212yesmaximal, complemented
9#11747506C212yesmaximal, complemented
10#11747479C212yesmaximal, complemented
11#11747478C212yesmaximal, complemented
12#10610562—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C2×C5×#235C2211yescharacteristic
15C22×C3×D20C2211yescharacteristic
16C2×D5×#67—22no—
17#92017—24no—
18#92017C418yescomplemented
19C2×#6777C2212yes—
20C2×D3×D20—21no—
21C2×#6766C2212yes—
22C2×#6765—21no—
23C3×C5×#117C2211yescharacteristic, complemented
24C2×C4⋊Dic15C2211yescharacteristic, complemented
25#91781C2212yescomplemented
26C4⋊C4×D15C2212yescomplemented
27#91770—22no—
28C4×D60—21no—
29C4×Dic30—21no—
30C22×C5×C3⋊C8C2211yescharacteristic, complemented
31C2×C3×#681C2211yescharacteristic, complemented
32C2×#6651C2212yescomplemented
33#91291—24no—
34C2×#6648C2212yescomplemented
35C5×#810C2211yescharacteristic, complemented
36C5×#810C2211yescharacteristic, complemented
37C5×#809C2211yescharacteristic, complemented
38C5×#809C2211yescharacteristic, complemented
39C3×#3941C2212yescomplemented
40C3×#3941C2212yescomplemented
41#90973C2212yescomplemented
42#90967C2212yescomplemented
43#90965C2212yescomplemented
44#90965C2212yescomplemented
45#78925—51nomaximal, Hall
46C2×#4149—31no—
47C2×#4047—31no—
48#65857—31no—
49C5×#416—31no—
50#65521—31no—
51C2×#3953—31no—
52#65290D311yescharacteristic, complemented
53#65093—32no—
54#65092—32no—
55#65059—32no—
56D5×#102—32no—
57C23×C3×D5—21no—
58C22×D3×D5—22no—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220D412yes—
61C22×Dic15—21no—
62#6817C2×C414yes—
63#6817D414yes—
64C2×D60D411yescharacteristic
65C2×C4×D15—22no—
66C2×C4×D15C2312yes—
67C2×Dic30D411yescharacteristic
68C2×C5×#67—21no—
69C5×#235—22no—
70C5×#235C2×C414yescomplemented
71C2×C5×D12C2311yescharacteristic
72C2×C4×C5×D3—21no—
73C2×C5×Dic6C2311yescharacteristic
74C2×C3×D20—21no—
75C2×C3×D20—22no—
76C2×C3×D20C2×C414yescomplemented
77C2×C3×D20C2312yes—
78D5×#67—216no—
79C2×#1202—22no—
80C2×#1201—22no—
81C2×D5×Dic3—22no—
82#6777—24no—
83#6777C2×C418yescomplemented
84D3×D20—28no—
85#6766—24no—
86#6766C2×C418yescomplemented
87#6765—28no—
88C3×C5×C4⋊C4C2311yescharacteristic
89C3×C5×C4⋊C4C2311yescharacteristic
90C3×C5×C22⋊C4—21no—
91C3×C42×C5—21no—
92#6714—22no—
93C4⋊Dic15—21no—
94C4⋊Dic15C2312yes—
95#6712—22no—
96C4×Dic15—22no—
97C2×C5×C3⋊C8—22no—
98C2×C5×C3⋊C8C2311yescharacteristic
99C2×C5×C3⋊C8C2311yescharacteristic
100C3×#681—22no—
101C2×C3×#168C2312yes—
102#6651—24no—
103#6648—24no—
104C2×#1004—51no—
105C2×#907—51no—
106#5523—51no—
107C3×#418—51no—
108C2×#832—51no—
109#5133—51no—
110#5002D511yescharacteristic, complemented
111#4772—52no—
112#4771—52no—
113#4770—52no—
114D3×#102—52no—
115C2×C5×#41—31no—
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—61no—
124C22×D20C2×D311yescharacteristic
125C22×C5×C8—31no—
126C5×#117C2×D311yescharacteristic, complemented
127C2×C4⋊Dic5—31no—
128C2×#681C2×D311yescharacteristic, complemented
129#4047—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×#103—31no—
138C5×#103—31no—
139C5×#102—31no—
140C5×#102—31no—
141#3967—32no—
142#3964—32no—
143#3953—32no—
144#3953—32no—
145#3941C2×D312yescomplemented
146#3941C2×D312yescomplemented
147#2897—151noSylow
148C23×C3×C5C2×D411yescharacteristic
149C22×D15C2×D412yes—
150C22×C5×D3—21no—
151C22×C3×D5—22no—
152C22×C3×D5—24no—
153C22×C3×D5—81no—
154C22×C3×D5—81no—
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
164#1220C22⋊C418yes—
165C2×Dic15—22no—
166C2×Dic15—81no—
167C2×Dic15C2×D412yes—
168D60C22⋊C414yes—
169C4×D15—82no—
170C4×D15C22×C414yes—
171C4×D15C2×D414yes—
172Dic30C22⋊C414yes—
173C5×#67—24no—
174C2×C5×Dic3—21no—
175C5×D12—21no—
176C5×D12C22×C412yes—
177C4×C5×D3—24no—
178C5×Dic6—21no—
179C5×Dic6C22×C412yes—
180C3×D20—24no—
181C3×D20—22no—
182C3×D20C22×C414yes—
183#1202—28no—
184#1201—28no—
185D5×Dic3—28no—
186C3×#168—22no—
187C5×C3⋊C8—21no—
188C5×C3⋊C8—21no—
189C22×C3×D4—51no—
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×#235C2×D511yescharacteristic
199C2×C3×M16—51no—
200C3×#117C2×D511yescharacteristic, complemented
201C2×C4⋊Dic3—51no—
202C22×C3⋊C8C2×D511yescharacteristic, complemented
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×#102—52no—
212C3×#102—52no—
213#832—52no—
214#832—52no—
215#821—52no—
216#816—52no—
217#810C2×D511yescharacteristic, complemented
218#810C2×D511yescharacteristic, complemented
219#809C2×D511yescharacteristic, complemented
220#809C2×D511yescharacteristic, complemented
221C23×D5—21no—
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—32no—
231C2×C22⋊D5—62no—
232C22×Dic5—61no—
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—21no—
242C2×D20—22no—
243C2×D20—31no—
244C2×D20—68no—
245C2×D20—62no—
246C2×D20C4×D314yescomplemented
247C2×D20C22×D312yes—
248C2×C4×D5—32no—
249C2×C4×D5—62no—
250C2×C4×D5—62no—
251C2×Dic10—31no—
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—62no—
260C4⋊Dic5—32no—
261C4⋊Dic5—61no—
262#683—62no—
263C4×Dic5—62no—
264#681—22no—
265C2×#168C22×D312yes—
266#676—64no—
267C8×D5—64no—
268C2×H(5,𝔽2)—151no—
269C2×#131—151no—
270#520—151no—
271#418—51no—
272#418—152no—
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—22no—
284C2×D15—22no—
285C2×D15C2×C22⋊C414yes—
286C2×C5×D3—22no—
287C2×C5×D3—41no—
288C2×C5×D3—41no—
289C2×C3×D5—24no—
290C2×C3×D5—42no—
291C2×C3×D5—42no—
292C2×C3×D5—84no—
293D3×D5—88no—
294C3×C4×C5—41no—
295C3×C4×C5—41no—
296C3×C4×C5C2×C22⋊C412yes—
297C3×C4×C5C23×C411yescharacteristic
298C3×C4×C5C22×D411yescharacteristic
299Dic15—42no—
300Dic15C2×C22⋊C414yes—
301C5×Dic3—22no—
302C24×C3—101no—
303C23×D3—102no—
304C2×C3×D4—54no—
305C2×C3×D4—52no—
306C2×C3×D4—101no—
307C2×C3×D4—102no—
308C22×C3×C4C22×D511yescharacteristic
309C2×#67—21no—
310C2×#67—52no—
311C2×#67—102no—
312C2×#67—102no—
313C2×#67—1016no—
314C22×Dic3—102no—
315C22×Dic3—101no—
316#237—58no—
317#237—104no—
318#237—108no—
319D3×D4—58no—
320D3×D4—108no—
321D3×D4—104no—
322#235—22no—
323#235—54no—
324#235—54no—
325#235C4×D514yescomplemented
326C2×D12—51no—
327C2×D12C22×D511yescharacteristic
328C2×C4×D3—21no—
329C2×C4×D3—52no—
330C2×C4×D3—102no—
331C2×Dic6—51no—
332C2×Dic6C22×D511yescharacteristic
333C3×M16—102no—
334C2×C3×C8—52no—
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—22no—
345C2×C3⋊C8C22×D511yescharacteristic
346C2×C3⋊C8C22×D511yescharacteristic
347ΓL(1,𝔽25)—104no—
348D3×C8—104no—
349C23×C5—61no—
350C23×C5D3×D411yescharacteristic
351C22×D5—22no—
352C22×D5—24no—
353C22×D5—32no—
354C22×D5—68no—
355C22×D5—81no—
356C22×D5—81no—
357C22×D5—242no—
358C22×D5—242no—
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×C5D3×D412yes—
370C2×C4×C5D3×D411yescharacteristic
371C2×C4×C5C23×D311yescharacteristic
372C22⋊D5—38no—
373C22⋊D5—68no—
374C2×Dic5—32no—
375C2×Dic5—62no—
376C2×Dic5—241no—
377D20—24no—
378D20—22no—
379D20—34no—
380D20—68no—
381D20C2×C4×D314yes—
382C4×D5—34no—
383C4×D5—34no—
384C4×D5—68no—
385C4×D5—242no—
386Dic10—34no—
387C5×C8—61no—
388C5×C8—61no—
389#168—22no—
390H(5,𝔽2)—158no—
391H(5,𝔽2)—304no—
392C2×#41—31no—
393C2×#41—151no—
394C2×#41—152no—
395C2×#41—301no—
396C22×D4—51no—
397C22×D4—152no—
398C22×D4—301no—
399C22×D4—302no—
400#131—304no—
401C2×M16—51no—
402C2×M16—152no—
403C22×C8—31no—
404C22×C8—152no—
405C4×Q8—301no—
406C4×D4—301no—
407C4×D4—302no—
408#117—152no—
409#117D3×D511yescharacteristic, complemented
410C2×C4⋊C4—151no—
411C2×C4⋊C4—152no—
412#103—31no—
413#103—31no—
414#103—152no—
415#102—31no—
416#102—31no—
417#102—52no—
418#102—52no—
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—24no—
427C3×D5—44no—
428C5×D3—42no—
429C23×C3—102no—
430C23×C3—104no—
431C23×C3—401no—
432C23×C3—401no—
433C23×C3D4×D511yescharacteristic
434C22×D3—21no—
435C22×D3—52no—
436C22×D3—108no—
437C22×D3—402no—
438C22×D3—402no—
439C3×D4—54no—
440C3×D4—102no—
441C3×D4—104no—
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—24no—
449#67—58no—
450#67—108no—
451#67—108no—
452C2×Dic3—21no—
453C2×Dic3—52no—
454C2×Dic3—108no—
455C2×Dic3—102no—
456C2×Dic3—401no—
457D12—21no—
458D12—54no—
459D12C2×C4×D512yes—
460C4×D3—24no—
461C4×D3—54no—
462C4×D3—54no—
463C4×D3—402no—
464Dic6—21no—
465Dic6—54no—
466Dic6C2×C4×D512yes—
467C3×C8—102no—
468C3⋊C8—21no—
469C3⋊C8—21no—
470C22×C5—21no—
471C22×C5—21no—
472C22×C5—62no—
473C22×C5—121no—
474C22×C5—121no—
475C22×C5D3×C22⋊C412yes—
476C22×C5C2×D3×D411yescharacteristic
477C2×D5—24no—
478C2×D5—34no—
479C2×D5—42no—
480C2×D5—42no—
481C2×D5—62no—
482C2×D5—62no—
483C2×D5—84no—
484C2×D5—248no—
485C4×C5—41no—
486C4×C5—41no—
487C4×C5—62no—
488C4×C5D3×C22⋊C412yes—
489C4×C5C22×C4×D311yescharacteristic
490C4×C5C2×D3×D411yescharacteristic
491Dic5—34no—
492Dic5—122no—
493C24—101no—
494C24—302no—
495#41—34no—
496#41—62no—
497#41—154no—
498#41—154no—
499#41—158no—
500#41—308no—
501#41—304no—
502C2×Q8—31no—
503C2×Q8—151no—
504C2×D4—31no—
505C2×D4—54no—
506C2×D4—52no—
507C2×D4—61no—
508C2×D4—102no—
509C2×D4—101no—
510C2×D4—158no—
511C2×D4—152no—
512C2×D4—151no—
513C2×D4—308no—
514C2×D4—302no—
515C2×D4—304no—
516C2×D4—302no—
517C2×D4—3016no—
518C22×C4—61no—
519C22×C4—152no—
520C22×C4—302no—
521C22×C4—302no—
522C22×C4—301no—
523C22×C4C2×D3×D511yescharacteristic
524M16—102no—
525M16—304no—
526C2×C8—31no—
527C2×C8—31no—
528C2×C8—52no—
529C2×C8—62no—
530C2×C8—304no—
531C4⋊C4—152no—
532C4⋊C4—302no—
533C4⋊C4—301no—
534C4⋊C4C2×D3×D511yescharacteristic
535C4⋊C4C2×D3×D511yescharacteristic
536C22⋊C4—21no—
537C22⋊C4—302no—
538C42—21no—
539C42—302no—
540C3×C5#289711yescharacteristic, complemented, Hall
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—104no—
544C22×C3—202no—
545C22×C3—202no—
546C22×C3—404no—
547C22×C3D5×C22⋊C412yes—
548C22×C3C2×D4×D511yescharacteristic
549C2×D3—22no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—54no—
553C2×D3—102no—
554C2×D3—102no—
555C2×D3—408no—
556C3×C4—41no—
557C3×C4—41no—
558C3×C4D5×C22⋊C412yes—
559C3×C4C22×C4×D511yescharacteristic
560C3×C4C2×D4×D511yescharacteristic
561Dic3—22no—
562Dic3—54no—
563Dic3—202no—
564C2×C5—22no—
565C2×C5—122no—
566C2×C5C2×D3×C22⋊C411yescharacteristic
567C2×C5#577411yescharacteristic
568C2×C5#577511yescharacteristic
569D5—44no—
570D5—64no—
571C23—61no—
572C23—102no—
573C23—104no—
574C23—152no—
575C23—308no—
576C23—401no—
577C23—401no—
578C23—1202no—
579C23—1202no—
580C23#679011yescharacteristic
581Q8—32no—
582Q8—61no—
583Q8—154no—
584D4—32no—
585D4—54no—
586D4—64no—
587D4—61no—
588D4—104no—
589D4—102no—
590D4—158no—
591D4—154no—
592D4—308no—
593D4—308no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—61no—
597C2×C4—64no—
598C2×C4—152no—
599C2×C4—154no—
600C2×C4—154no—
601C2×C4—308no—
602C2×C4—302no—
603C2×C4—1202no—
604C2×C4—1201no—
605C2×C4C4×D3×D512yes—
606C2×C4#677712yes—
607C2×C4#679011yescharacteristic
608C2×C4C22×D3×D511yescharacteristic
609C8—61no—
610C8—61no—
611C8—102no—
612C2×C3—22no—
613C2×C3—204no—
614C2×C3C2×D5×C22⋊C411yescharacteristic
615C2×C3#6610311yescharacteristic
616C2×C3#6610411yescharacteristic
617D3—42no—
618D3—104no—
619C5#7892511yescharacteristic, complemented, Sylow
620C22—21no—
621C22—21no—
622C22—62no—
623C22—104no—
624C22—121no—
625C22—121no—
626C22—154no—
627C22—202no—
628C22—202no—
629C22—302no—
630C22—302no—
631C22—404no—
632C22—1208no—
633C22#9147312yes—
634C22C2×#679011yescharacteristic
635C4—41no—
636C4—41no—
637C4—62no—
638C4—154no—
639C4—602no—
640C4#9147312yes—
641C4C2×C4×D3×D511yescharacteristic
642C4C2×#677711yescharacteristic
643C3#1061056211yescharacteristic, complemented, Sylow
644C2—22no—
645C2—122no—
646C2—204no—
647C2—304no—
648C2C2×#9147311yescharacteristic
649C2#1175533011yescharacteristic
650C2#1175533111yescharacteristic
subgroupG/H
centerC22C2×#6790
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#92017C2
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. #11756282C2semidirect
3. C2×#91291C2semidirect
4. C5×#5002C2semidirect
5. C3×#65290C2semidirect
6. #11748390C2semidirect
7. C2×#90965C2semidirect
8. #11747507C2semidirect
9. #11747506C2semidirect
10. #11747479C2semidirect
11. #11747478C2semidirect
18. #92017C4semidirect
23. C3×C5×#117C22semidirect
24. C2×C4⋊Dic15C22semidirect
25. #91781C22semidirect
26. C4⋊C4×D15C22semidirect
30. C22×C5×C3⋊C8C22semidirect
31. C2×C3×#681C22semidirect
32. C2×#6651C22semidirect
34. C2×#6648C22semidirect
35. C5×#810C22semidirect
36. C5×#810C22semidirect
37. C5×#809C22semidirect
38. C5×#809C22semidirect
39. C3×#3941C22semidirect
40. C3×#3941C22semidirect
41. #90973C22semidirect
42. #90967C22semidirect
43. #90965C22semidirect
44. #90965C22semidirect
52. #65290D3semidirect
70. C5×#235C2×C4semidirect
76. C2×C3×D20C2×C4semidirect
83. #6777C2×C4semidirect
86. #6766C2×C4semidirect
110. #5002D5semidirect
126. C5×#117C2×D3semidirect
128. C2×#681C2×D3semidirect
145. #3941C2×D3semidirect
146. #3941C2×D3semidirect
200. C3×#117C2×D5semidirect
202. C22×C3⋊C8C2×D5semidirect
217. #810C2×D5semidirect
218. #810C2×D5semidirect
219. #809C2×D5semidirect
220. #809C2×D5semidirect
246. C2×D20C4×D3semidirect
325. #235C4×D5semidirect
409. #117D3×D5semidirect
540. C3×C5#2897semidirect
619. C5#78925semidirect
643. C3#10610562semidirect

Maximal subgroups

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

subgroupindex
C2×#920172
#117562822
C2×#912912
C5×#50022
C3×#652902
#117483902
C2×#909652
#117475072
#117475062
#117474792
#117474782
#106105623
#789255