Finite Group Databasebeta

← #14424209

Subgroups — #14424209

Order 1920.

1568 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#11756589C211yesmaximal, characteristic, complemented
3C2×#91838C211yesmaximal, characteristic, complemented
4C22×#6715C211yesmaximal, characteristic, complemented
5#11751339C214yesmaximal, complemented
6#11751335C211yesmaximal, characteristic, complemented
7#11751334C211yesmaximal, characteristic, complemented
8#11751318C214yesmaximal, complemented
9#11751316C211yesmaximal, characteristic, complemented
10#10615832—31nomaximal, Hall
11C3×C5×H(5,𝔽2)—24no—
12C3×C5×H(5,𝔽2)—24no—
13C2×C3×C5×#41—21no—
14C2×C3×C5×#41C2211yescharacteristic
15C2×C3×C5×#41C2211yescharacteristic
16C2×C3×C5×#41C2212yescomplemented
17C22×C3×C5×D4—21no—
18C22×C3×C5×D4—22no—
19C22×C3×C5×D4C2211yescharacteristic
20C22×C3×C5×D4C2212yescomplemented
21C22×D60C2211yescharacteristic, complemented
22#91838—24no—
23#91836—21no—
24C2×#6717C2212yescomplemented
25#91829—21no—
26#91826—22no—
27C2×#6715—24no—
28C2×#6715C2214yescomplemented
29C2×#6715C2212yescomplemented
30#91819C2214yescomplemented
31#91815C2211yescharacteristic, complemented
32C2×#6710C2211yescharacteristic, complemented
33C22×C3⋊#168C2211yescharacteristic, complemented
34#91119C2212yescomplemented
35#91116C2214yescomplemented
36#91116C2212yescomplemented
37#91114C2214yescomplemented
38#84101—51nomaximal, Hall
39C2×C5×H(5,𝔽2)D311yescharacteristic, complemented
40#66164—31no—
41C2×#4095—31no—
42C22×#686—31no—
43#65527—34no—
44#65523—31no—
45#65522—31no—
46#65506—34no—
47#65504—31no—
48C24×C3×C5—21no—
49C24×C3×C5—22no—
50C23×D15—21no—
51C3×C5×#41—28no—
52C3×C5×#41—24no—
53C3×C5×#41—24no—
54C3×C5×#41—22no—
55C3×C5×#41—22no—
56C3×C5×#41—22no—
57C3×C5×#41D414yescomplemented
58C2×C3×C5×Q8D411yescharacteristic
59C2×C3×C5×Q8C2311yescharacteristic
60C2×C3×C5×D4—216no—
61C2×C3×C5×D4—28no—
62C2×C3×C5×D4—21no—
63C2×C3×C5×D4—22no—
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4—24no—
66C2×C3×C5×D4—21no—
67C2×C3×C5×D4—22no—
68C2×C3×C5×D4—22no—
69C2×C3×C5×D4D412yes—
70C2×C3×C5×D4D411yescharacteristic
71C2×C3×C5×D4D414yescomplemented
72C2×C3×C5×D4C2312yes—
73C2×C3×C5×D4C2311yescharacteristic
74C22×C3×C4×C5—21no—
75C22×C3×C4×C5—22no—
76C22×C3×C4×C5C2312yes—
77C22×C3×C4×C5C2311yescharacteristic
78C2×#1220—24no—
79C2×D60—22no—
80C2×D60—22no—
81C2×D60—21no—
82C2×D60C2312yes—
83#6719—21no—
84#6717—24no—
85#6715—28no—
86#6715—24no—
87#6714—24no—
88C4⋊Dic15C2312yes—
89C4×Dic15—21no—
90#6710—22no—
91C2×C3⋊#168—22no—
92C2×C3⋊#168C2312yes—
93C2×C3⋊#168C2312yes—
94C2×C3×H(5,𝔽2)D511yescharacteristic, complemented
95#5830—51no—
96C2×#949—51no—
97C22×#213—51no—
98#5238—54no—
99#5234—51no—
100#5233—51no—
101#5217—54no—
102#5215—51no—
103C5×H(5,𝔽2)—24no—
104C5×H(5,𝔽2)—24no—
105C2×C5×#41—21no—
106C2×C5×#41C2×D311yescharacteristic
107C2×C5×#41C2×D311yescharacteristic
108C2×C5×#41C2×D312yescomplemented
109C22×C5×D4—21no—
110C22×C5×D4—22no—
111C22×C5×D4C2×D311yescharacteristic
112C22×C5×D4C2×D312yescomplemented
113C22×D20—31no—
114#4095—64no—
115#4093—61no—
116C2×#688—32no—
117#4086—61no—
118#4083—62no—
119C2×#686—34no—
120C2×#686—32no—
121C2×#686—64no—
122#4076—34no—
123#4072—31no—
124C2×#681—31no—
125C22×#168—31no—
126#3967—32no—
127#3964—34no—
128#3964—32no—
129C22⋊#168—34no—
130#3008—151noSylow
131C23×C3×C5—28no—
132C23×C3×C5—21no—
133C23×C3×C5—22no—
134C23×C3×C5—24no—
135C23×C3×C5—82no—
136C23×C3×C5—84no—
137C23×C3×C5C2×D412yes—
138C23×C3×C5C2×D411yescharacteristic
139C22×D15—22no—
140C22×D15—42no—
141C22×D15—82no—
142C3×C5×Q8—22no—
143C3×C5×Q8—21no—
144C3×C5×Q8C2×D412yes—
145C3×C5×D4—28no—
146C3×C5×D4—28no—
147C3×C5×D4—24no—
148C3×C5×D4—24no—
149C3×C5×D4—22no—
150C3×C5×D4—24no—
151C3×C5×D4—22no—
152C3×C5×D4—21no—
153C3×C5×D4C2×D412yes—
154C3×C5×D4C2×D414yes—
155C2×C3×C4×C5—28no—
156C2×C3×C4×C5—21no—
157C2×C3×C4×C5—24no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—22no—
160C2×C3×C4×C5C2×D412yes—
161C2×C3×C4×C5C2×D411yescharacteristic
162C2×C3×C4×C5C2×D412yes—
163C2×C3×C4×C5C2×D412yes—
164C2×C3×C4×C5C2411yescharacteristic
165#1220—84no—
166C2×Dic15—22no—
167D60—22no—
168D60—22no—
169D60—42no—
170C3⋊#168—22no—
171C3⋊#168—22no—
172C3×H(5,𝔽2)—24no—
173C3×H(5,𝔽2)—24no—
174C2×C3×#41—21no—
175C2×C3×#41C2×D511yescharacteristic
176C2×C3×#41C2×D511yescharacteristic
177C2×C3×#41C2×D512yescomplemented
178C22×C3×D4—21no—
179C22×C3×D4—22no—
180C22×C3×D4C2×D511yescharacteristic
181C22×C3×D4C2×D512yescomplemented
182C22×D12—51no—
183#949—104no—
184#947—101no—
185C2×#215—52no—
186#940—101no—
187#937—102no—
188C2×#213—52no—
189C2×#213—54no—
190C2×#213—104no—
191#930—54no—
192#926—51no—
193C2×#208—51no—
194C22×C3⋊C8—51no—
195#835—52no—
196#832—54no—
197#832—52no—
198#830—54no—
199C24×C5—21no—
200C24×C5—22no—
201C23×D5—61no—
202C5×#41—28no—
203C5×#41—24no—
204C5×#41—24no—
205C5×#41—22no—
206C5×#41—22no—
207C5×#41—22no—
208C5×#41#6714yescomplemented
209C2×C5×Q8#6711yescharacteristic
210C2×C5×Q8C22×D311yescharacteristic
211C2×C5×D4—216no—
212C2×C5×D4—28no—
213C2×C5×D4—21no—
214C2×C5×D4—22no—
215C2×C5×D4—24no—
216C2×C5×D4—24no—
217C2×C5×D4—21no—
218C2×C5×D4—22no—
219C2×C5×D4—22no—
220C2×C5×D4#6712yes—
221C2×C5×D4#6711yescharacteristic
222C2×C5×D4#6714yescomplemented
223C2×C5×D4C22×D312yes—
224C2×C5×D4C22×D311yescharacteristic
225C22×C4×C5—21no—
226C22×C4×C5—22no—
227C22×C4×C5C22×D312yes—
228C22×C4×C5C22×D311yescharacteristic
229C2×C22⋊D5—64no—
230C2×D20—32no—
231C2×D20—62no—
232C2×D20—62no—
233C2×D20—61no—
234C22⋊Dic5—61no—
235#688—64no—
236#686—68no—
237#686—64no—
238#685—64no—
239C4⋊Dic5—32no—
240C4×Dic5—61no—
241#681—62no—
242C2×#168—32no—
243C2×#168—32no—
244C2×#168—62no—
245C2×H(5,𝔽2)D1511yescharacteristic, complemented
246C2×AGL(1,ℤ/8)—151no—
247C22×D8—151no—
248#535—151no—
249#449—154no—
250#447—154no—
251#418—151no—
252#416—151no—
253#408—151no—
254C22×C3×C5—22no—
255C22×C3×C5—22no—
256C22×C3×C5—24no—
257C22×C3×C5—22no—
258C22×C3×C5—24no—
259C22×C3×C5—44no—
260C22×C3×C5—42no—
261C22×C3×C5—84no—
262C22×C3×C5—88no—
263C22×C3×C5#12012yes—
264C22×C3×C5C22×D411yescharacteristic
265C2×D15—42no—
266C2×D15—44no—
267C2×D15—84no—
268C3×C4×C5—22no—
269C3×C4×C5—22no—
270C3×C4×C5#12012yes—
271C3×C4×C5C22×D411yescharacteristic
272C3×C4×C5C22×D411yescharacteristic
273Dic15—42no—
274C24×C3—21no—
275C24×C3—22no—
276C23×D3—101no—
277C3×#41—28no—
278C3×#41—24no—
279C3×#41—24no—
280C3×#41—22no—
281C3×#41—22no—
282C3×#41—22no—
283C3×#41C22⋊D514yescomplemented
284C2×C3×Q8C22⋊D511yescharacteristic
285C2×C3×Q8C22×D511yescharacteristic
286C2×C3×D4—216no—
287C2×C3×D4—28no—
288C2×C3×D4—21no—
289C2×C3×D4—22no—
290C2×C3×D4—24no—
291C2×C3×D4—24no—
292C2×C3×D4—21no—
293C2×C3×D4—22no—
294C2×C3×D4—22no—
295C2×C3×D4C22⋊D512yes—
296C2×C3×D4C22⋊D511yescharacteristic
297C2×C3×D4C22⋊D514yescomplemented
298C2×C3×D4C22×D512yes—
299C2×C3×D4C22×D511yescharacteristic
300C22×C3×C4—21no—
301C22×C3×C4—22no—
302C22×C3×C4C22×D512yes—
303C22×C3×C4C22×D511yescharacteristic
304C2×#67—104no—
305C2×D12—52no—
306C2×D12—102no—
307C2×D12—102no—
308C2×D12—101no—
309#217—101no—
310#215—104no—
311#213—108no—
312#213—104no—
313(C2×D3)⋊C4—104no—
314C4⋊Dic3—52no—
315C4×Dic3—101no—
316#208—102no—
317C2×C3⋊C8—52no—
318C2×C3⋊C8—52no—
319C2×C3⋊C8—102no—
320C23×C5—28no—
321C23×C5—21no—
322C23×C5—22no—
323C23×C5—24no—
324C23×C5—82no—
325C23×C5—84no—
326C23×C5C2×#6712yes—
327C23×C5C2×#6711yescharacteristic
328C22×D5—62no—
329C22×D5—122no—
330C22×D5—242no—
331C5×Q8—22no—
332C5×Q8—21no—
333C5×Q8C2×#6712yes—
334C5×D4—28no—
335C5×D4—28no—
336C5×D4—24no—
337C5×D4—24no—
338C5×D4—22no—
339C5×D4—24no—
340C5×D4—22no—
341C5×D4—21no—
342C5×D4C2×#6712yes—
343C5×D4C2×#6714yes—
344C2×C4×C5—28no—
345C2×C4×C5—21no—
346C2×C4×C5—24no—
347C2×C4×C5—22no—
348C2×C4×C5—22no—
349C2×C4×C5C2×#6712yes—
350C2×C4×C5C2×#6711yescharacteristic
351C2×C4×C5C2×#6712yes—
352C2×C4×C5C2×#6712yes—
353C2×C4×C5C23×D311yescharacteristic
354C22⋊D5—244no—
355C2×Dic5—62no—
356D20—62no—
357D20—62no—
358D20—122no—
359#168—62no—
360#168—62no—
361H(5,𝔽2)—24no—
362H(5,𝔽2)—24no—
363C2×#41—21no—
364C2×#41C2×D1511yescharacteristic
365C2×#41C2×D1511yescharacteristic
366C2×#41C2×D1512yescomplemented
367C22×D4—21no—
368C22×D4—22no—
369C22×D4—151no—
370C22×D4C2×D1511yescharacteristic
371C22×D4C2×D1512yescomplemented
372AGL(1,ℤ/8)—304no—
373C2×SD16—152no—
374C2×D8—154no—
375C2×D8—152no—
376C2×D8—304no—
377C2×M16—151no—
378C22×C8—151no—
379#127—301no—
380#124—301no—
381#121—154no—
382#120—302no—
383#117—151no—
384#103—152no—
385#102—152no—
386#102—154no—
387C22⋊C8—154no—
388C2×C3×C5—22no—
389C2×C3×C5—42no—
390C2×C3×C5—44no—
391C2×C3×C5—42no—
392C2×C3×C5C2×#12011yescharacteristic
393C2×C3×C5#57612yes—
394D15—82no—
395C23×C3—28no—
396C23×C3—21no—
397C23×C3—22no—
398C23×C3—24no—
399C23×C3—82no—
400C23×C3—84no—
401C23×C3C2×C22⋊D512yes—
402C23×C3C2×C22⋊D511yescharacteristic
403C22×D3—102no—
404C22×D3—202no—
405C22×D3—402no—
406C3×Q8—22no—
407C3×Q8—21no—
408C3×Q8C2×C22⋊D512yes—
409C3×D4—28no—
410C3×D4—28no—
411C3×D4—24no—
412C3×D4—24no—
413C3×D4—22no—
414C3×D4—24no—
415C3×D4—22no—
416C3×D4—21no—
417C3×D4C2×C22⋊D512yes—
418C3×D4C2×C22⋊D514yes—
419C2×C3×C4—28no—
420C2×C3×C4—21no—
421C2×C3×C4—24no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4C2×C22⋊D512yes—
425C2×C3×C4C2×C22⋊D511yescharacteristic
426C2×C3×C4C2×C22⋊D512yes—
427C2×C3×C4C2×C22⋊D512yes—
428C2×C3×C4C23×D511yescharacteristic
429#67—404no—
430C2×Dic3—102no—
431D12—102no—
432D12—102no—
433D12—202no—
434C3⋊C8—102no—
435C3⋊C8—102no—
436C22×C5—22no—
437C22×C5—24no—
438C22×C5—22no—
439C22×C5—24no—
440C22×C5—22no—
441C22×C5—44no—
442C22×C5—42no—
443C22×C5—84no—
444C22×C5—88no—
445C22×C5#95312yes—
446C22×C5C22×#6711yescharacteristic
447C2×D5—124no—
448C2×D5—122no—
449C2×D5—244no—
450C4×C5—22no—
451C4×C5—22no—
452C4×C5#95312yes—
453C4×C5C22×#6711yescharacteristic
454C4×C5C22×#6711yescharacteristic
455Dic5—122no—
456C24—21no—
457C24—22no—
458C24—301no—
459#41—28no—
460#41—24no—
461#41—24no—
462#41—22no—
463#41—22no—
464#41—22no—
465#41#122014yescomplemented
466C2×Q8#122011yescharacteristic
467C2×Q8C22×D1511yescharacteristic
468C2×D4—216no—
469C2×D4—28no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—152no—
478C2×D4—304no—
479C2×D4—302no—
480C2×D4—302no—
481C2×D4—301no—
482C2×D4#122012yes—
483C2×D4#122011yescharacteristic
484C2×D4#122014yescomplemented
485C2×D4C22×D1512yes—
486C2×D4C22×D1511yescharacteristic
487C22×C4—21no—
488C22×C4—22no—
489C22×C4C22×D1512yes—
490C22×C4C22×D1511yescharacteristic
491SD16—304no—
492D8—308no—
493D8—304no—
494M16—302no—
495C2×C8—152no—
496C2×C8—152no—
497C2×C8—302no—
498C4⋊C4—152no—
499C22⋊C4—301no—
500C22⋊C4—304no—
501C42—301no—
502C3×C5#300811yescharacteristic, complemented, Hall
503C22×C3—22no—
504C22×C3—24no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—22no—
508C22×C3—44no—
509C22×C3—42no—
510C22×C3—84no—
511C22×C3—88no—
512C22×C3#409912yes—
513C22×C3C22×C22⋊D511yescharacteristic
514C2×D3—204no—
515C2×D3—202no—
516C2×D3—404no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4#409912yes—
520C3×C4C22×C22⋊D511yescharacteristic
521C3×C4C22×C22⋊D511yescharacteristic
522Dic3—202no—
523C2×C5—22no—
524C2×C5—42no—
525C2×C5—44no—
526C2×C5—42no—
527C2×C5#583512yes—
528C2×C5C2×#95311yescharacteristic
529D5—242no—
530C23—28no—
531C23—21no—
532C23—22no—
533C23—24no—
534C23—82no—
535C23—84no—
536C23—302no—
537C23—602no—
538C23—1202no—
539C23C2×#122012yes—
540C23C2×#122011yescharacteristic
541Q8—22no—
542Q8—21no—
543Q8C2×#122012yes—
544D4—28no—
545D4—28no—
546D4—24no—
547D4—24no—
548D4—22no—
549D4—24no—
550D4—22no—
551D4—21no—
552D4—302no—
553D4—302no—
554D4—602no—
555D4—1204no—
556D4C2×#122012yes—
557D4C2×#122014yes—
558C2×C4—28no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—302no—
564C2×C4C2×#122012yes—
565C2×C4C2×#122011yescharacteristic
566C2×C4C2×#122012yes—
567C2×C4C2×#122012yes—
568C2×C4C23×D1511yescharacteristic
569C8—302no—
570C8—302no—
571C2×C3—22no—
572C2×C3—42no—
573C2×C3—44no—
574C2×C3—42no—
575C2×C3#6616912yes—
576C2×C3C2×#409911yescharacteristic
577D3—402no—
578C5#8410111yescharacteristic, complemented, Sylow
579C22—22no—
580C22—24no—
581C22—22no—
582C22—24no—
583C22—22no—
584C22—44no—
585C22—42no—
586C22—84no—
587C22—88no—
588C22—604no—
589C22—602no—
590C22—1204no—
591C22#9184212yes—
592C22C22×#122011yescharacteristic
593C4—22no—
594C4—22no—
595C4—602no—
596C4#9184212yes—
597C4C22×#122011yescharacteristic
598C4C22×#122011yescharacteristic
599C3#1061583211yescharacteristic, complemented, Sylow
600C2—22no—
601C2—42no—
602C2—44no—
603C2—42no—
604C2—1202no—
605C2#1175659412yes—
606C2C2×#9184211yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×H(5,𝔽2)C2
socleC22×C3×C5C22×D4
hypercenterC2×H(5,𝔽2)D15
hypocenterC3×C5#3008
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×H(5,𝔽2)C2semidirect
2. #11756589C2semidirect
3. C2×#91838C2semidirect
4. C22×#6715C2semidirect
5. #11751339C2semidirect
6. #11751335C2semidirect
7. #11751334C2semidirect
8. #11751318C2semidirect
9. #11751316C2semidirect
16. C2×C3×C5×#41C22semidirect
20. C22×C3×C5×D4C22semidirect
21. C22×D60C22semidirect
24. C2×#6717C22semidirect
28. C2×#6715C22semidirect
29. C2×#6715C22semidirect
30. #91819C22semidirect
31. #91815C22semidirect
32. C2×#6710C22semidirect
33. C22×C3⋊#168C22semidirect
34. #91119C22semidirect
35. #91116C22semidirect
36. #91116C22semidirect
37. #91114C22semidirect
39. C2×C5×H(5,𝔽2)D3semidirect
57. C3×C5×#41D4semidirect
71. C2×C3×C5×D4D4semidirect
94. C2×C3×H(5,𝔽2)D5semidirect
108. C2×C5×#41C2×D3semidirect
112. C22×C5×D4C2×D3semidirect
177. C2×C3×#41C2×D5semidirect
181. C22×C3×D4C2×D5semidirect
208. C5×#41#67semidirect
222. C2×C5×D4#67semidirect
245. C2×H(5,𝔽2)D15semidirect
283. C3×#41C22⋊D5semidirect
297. C2×C3×D4C22⋊D5semidirect
366. C2×#41C2×D15semidirect
371. C22×D4C2×D15semidirect
465. #41#1220semidirect
484. C2×D4#1220semidirect
502. C3×C5#3008semidirect
578. C5#84101semidirect
599. C3#10615832semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×H(5,𝔽2)2
#117565892
C2×#918382
C22×#67152
#117513392
#117513352
#117513342
#117513182
#117513162
#106158323
#841015