Finite Group Databasebeta

← #14438688

Subgroups — #14438688

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756591C211yesmaximal, characteristic
2C2×#91384C211yesmaximal, characteristic, complemented
3C5×#5232C211yesmaximal, characteristic, complemented
4C3×#65521C211yesmaximal, characteristic, complemented
5#11749336C212yesmaximal, complemented
6#11749190C212yesmaximal, complemented
7#11749090C212yesmaximal, complemented
8#11748944C212yesmaximal, complemented
9#11748413C211yesmaximal, characteristic, complemented
10#11748394C211yesmaximal, characteristic, complemented
11#11748328C211yesmaximal, characteristic, complemented
12#10610610—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C2×#6817C2211yescharacteristic
15#91836C2211yescharacteristic
16#91833C2211yescharacteristic
17#91829C2211yescharacteristic
18#91828—21no—
19#91828—21no—
20#91825C2211yescharacteristic
21C4×#1220—21no—
22C4×#1220—21no—
23C2×C4×Dic15C2211yescharacteristic
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×#208C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×#681C2211yescharacteristic
28C2×Dic3⋊Dic5—21no—
29C2×#6668—21no—
30#91384—21no—
31#91384—21no—
32#91384C2212yes—
33#91384C2212yes—
34C5×#835C2212yes—
35C5×#832C2212yes—
36C3×#3967C2212yes—
37C3×#3964C2212yes—
38#90975C2212yes—
39#90974C2212yes—
40#90969C2212yes—
41#90968C2212yes—
42#90954C2212yes—
43#90951C2212yes—
44#78879—51nomaximal, Hall
45#66166—31no—
46C2×#4036—31no—
47C5×#417—31no—
48#65521D311yescharacteristic, complemented
49#65422—31no—
50M16×Dic5—31no—
51#65255—31no—
52#65089—32no—
53#65086—32no—
54#65057—32no—
55#65053—32no—
56C3×C5×#41—22no—
57C3×C5×#41—22no—
58C2×C3×C5×Q8C2311yescharacteristic
59C2×C3×C5×D4—21no—
60C2×C3×C5×D4C2311yescharacteristic
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5C2311yescharacteristic
63C2×#1220—21no—
64C2×#1220—21no—
65C2×#1220—21no—
66C22×Dic15D411yescharacteristic
67C22×Dic15D411yescharacteristic
68#6817—22no—
69#6817—22no—
70C2×D60C2311yescharacteristic
71C2×C4×D15—21no—
72C2×Dic30C2311yescharacteristic
73C22×C5×Dic3—21no—
74C22×C3×Dic5—21no—
75#6719—21no—
76#6719—21no—
77#6714—21no—
78#6714—21no—
79#6712—21no—
80#6712—21no—
81C4×Dic15D411yescharacteristic
82C4×Dic15D411yescharacteristic
83C4×Dic15C2311yescharacteristic
84C4×Dic15C2311yescharacteristic
85C5×C4⋊Dic3—21no—
86C5×C4⋊Dic3C2312yes—
87C5×#208—22no—
88C2×C5×C3⋊C8C2312yes—
89C3×C4⋊Dic5—21no—
90C3×C4⋊Dic5C2312yes—
91C3×#681—22no—
92C2×C3×#168C2312yes—
93Dic3⋊Dic5—22no—
94Dic3⋊Dic5—22no—
95#6668—22no—
96#6668—22no—
97#5832—51no—
98C2×#890—51no—
99C3×#417—51no—
100#5232D511yescharacteristic, complemented
101#5133—51no—
102Dic3×M16—51no—
103#4966—51no—
104#4800—52no—
105#4797—52no—
106#4768—52no—
107#4764—52no—
108C2×C5×#41C2×D311yescharacteristic
109C2×#709—31no—
110C2×C5×M16—31no—
111C2×C5×C4⋊C4—31no—
112#4093—31no—
113#4090—31no—
114#4086—31no—
115#4085—61no—
116#4085—61no—
117#4082—31no—
118C4×C22⋊D5—61no—
119C4×C22⋊D5—61no—
120C2×C4⋊Dic5—61no—
121C2×C4⋊Dic5C2×D311yescharacteristic
122C2×#683—61no—
123C2×C4×Dic5—31no—
124C2×#681C2×D311yescharacteristic
125#4036—32no—
126#4036—32no—
127#4036—61no—
128#4036—61no—
129C5×#103—32no—
130C5×#102—32no—
131#3967C2×D312yes—
132#3964C2×D312yes—
133#3953—32no—
134#3948—32no—
135#3947—32no—
136C8×Dic5—32no—
137#3942—32no—
138#3941—32no—
139#3142—151noSylow
140C23×C3×C5—21no—
141C23×C3×C5C2×D411yescharacteristic
142C22×D15—21no—
143C3×C5×Q8—21no—
144C3×C5×Q8—21no—
145C3×C5×D4—21no—
146C3×C5×D4—21no—
147C3×C5×D4—42no—
148C2×C3×C4×C5—21no—
149C2×C3×C4×C5—42no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5#4112yes—
152C2×C3×C4×C5#4112yes—
153C2×C3×C4×C5C2411yescharacteristic
154#1220—42no—
155#1220—42no—
156#1220—42no—
157C2×Dic15—21no—
158C2×Dic15—41no—
159C2×Dic15—41no—
160C2×Dic15C2×D411yescharacteristic
161C2×Dic15C2×D411yescharacteristic
162C2×Dic15C2×D411yescharacteristic
163C2×Dic15C2×D411yescharacteristic
164D60—21no—
165D60—21no—
166C4×D15—42no—
167Dic30—21no—
168Dic30—21no—
169C2×C5×Dic3—22no—
170C2×C5×Dic3—42no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5—42no—
173C3×#168—22no—
174C5×C3⋊C8—22no—
175C2×C3×#41C2×D511yescharacteristic
176C2×#235—51no—
177C2×C3×M16—51no—
178C2×C3×C4⋊C4—51no—
179#947—51no—
180#944—51no—
181#940—51no—
182#939—101no—
183#939—101no—
184#936—51no—
185C4×#67—101no—
186C4×#67—101no—
187C2×C4⋊Dic3—101no—
188C2×C4⋊Dic3C2×D511yescharacteristic
189C2×Dic3⋊C4—101no—
190C2×C4×Dic3—51no—
191C2×#208C2×D511yescharacteristic
192#890—52no—
193#890—52no—
194#890—101no—
195#890—101no—
196C3×#103—52no—
197C3×#102—52no—
198#835C2×D512yes—
199#832C2×D512yes—
200#821—52no—
201#816—52no—
202#815—52no—
203C8×Dic3—52no—
204#810—52no—
205#809—52no—
206C5×#41—22no—
207C5×#41—22no—
208C2×C5×Q8C22×D311yescharacteristic
209C2×C5×D4—21no—
210C2×C5×D4C22×D311yescharacteristic
211C22×C4×C5—21no—
212C22×C4×C5—61no—
213C22×C4×C5C22×D311yescharacteristic
214C2×C22⋊D5—61no—
215C2×C22⋊D5—61no—
216C2×C22⋊D5—61no—
217C22×Dic5—21no—
218C22×Dic5—31no—
219C22×Dic5—31no—
220#709—62no—
221#709—62no—
222C2×D20—31no—
223C2×C4×D5—61no—
224C2×Dic10—31no—
225C5×M16—62no—
226C2×C5×C8—32no—
227C5×C4⋊C4—32no—
228C5×C4⋊C4—61no—
229C22⋊Dic5—61no—
230C22⋊Dic5—61no—
231#685—61no—
232#685—61no—
233C4⋊Dic5—21no—
234C4⋊Dic5—62no—
235C4⋊Dic5—62no—
236C4⋊Dic5C22×D312yes—
237#683—62no—
238#683—62no—
239#683—61no—
240#683—61no—
241C4×Dic5—31no—
242C4×Dic5—31no—
243C4×Dic5—31no—
244C4×Dic5—31no—
245#681—22no—
246C2×#168C22×D312yes—
247#532—151no—
248C2×#125—151no—
249#491—152no—
250#490—152no—
251#488—152no—
252#488—152no—
253#417—31no—
254#417—51no—
255#417—151no—
256#417—151no—
257C4×M16—151no—
258C22×C3×C5—21no—
259C22×C3×C5—21no—
260C22×C3×C5—41no—
261C22×C3×C5—41no—
262C22×C3×C5—41no—
263C22×C3×C5#12412yes—
264C22×C3×C5C22×D411yescharacteristic
265C2×D15—41no—
266C2×D15—41no—
267C2×D15—41no—
268C3×C4×C5—41no—
269C3×C4×C5#12412yes—
270C3×C4×C5C2×#4111yescharacteristic
271C3×C4×C5C2×#4111yescharacteristic
272Dic15—21no—
273Dic15—21no—
274Dic15—21no—
275Dic15—21no—
276Dic15—41no—
277C3×Dic5—42no—
278C5×Dic3—42no—
279C3×#41—22no—
280C3×#41—22no—
281C2×C3×Q8C22×D511yescharacteristic
282C2×C3×D4—21no—
283C2×C3×D4C22×D511yescharacteristic
284C22×C3×C4—21no—
285C22×C3×C4—101no—
286C22×C3×C4C22×D511yescharacteristic
287C2×#67—101no—
288C2×#67—101no—
289C2×#67—101no—
290C22×Dic3—21no—
291C22×Dic3—51no—
292C22×Dic3—51no—
293#235—102no—
294#235—102no—
295C2×D12—51no—
296C2×C4×D3—101no—
297C2×Dic6—51no—
298C3×M16—102no—
299C2×C3×C8—52no—
300C3×C4⋊C4—52no—
301C3×C4⋊C4—101no—
302#217—101no—
303#217—101no—
304(C2×D3)⋊C4—101no—
305(C2×D3)⋊C4—101no—
306C4⋊Dic3—21no—
307C4⋊Dic3—102no—
308C4⋊Dic3—102no—
309C4⋊Dic3C22×D512yes—
310Dic3⋊C4—102no—
311Dic3⋊C4—102no—
312Dic3⋊C4—101no—
313Dic3⋊C4—101no—
314C4×Dic3—51no—
315C4×Dic3—51no—
316C4×Dic3—51no—
317C4×Dic3—51no—
318#208—22no—
319C2×C3⋊C8C22×D512yes—
320C23×C5—21no—
321C23×C5D3×D411yescharacteristic
322C22×D5—61no—
323C5×Q8—21no—
324C5×Q8—21no—
325C5×D4—21no—
326C5×D4—21no—
327C5×D4—42no—
328C2×C4×C5—21no—
329C2×C4×C5—42no—
330C2×C4×C5—62no—
331C2×C4×C5—122no—
332C2×C4×C5#23512yes—
333C2×C4×C5D3×D411yescharacteristic
334C2×C4×C5#23712yes—
335C2×C4×C5C23×D311yescharacteristic
336C22⋊D5—122no—
337C22⋊D5—122no—
338C22⋊D5—122no—
339C2×Dic5—22no—
340C2×Dic5—31no—
341C2×Dic5—31no—
342C2×Dic5—31no—
343C2×Dic5—31no—
344C2×Dic5—42no—
345C2×Dic5—61no—
346C2×Dic5—121no—
347C2×Dic5—121no—
348D20—61no—
349D20—61no—
350C4×D5—122no—
351Dic10—61no—
352Dic10—61no—
353C5×C8—62no—
354#168—22no—
355C2×#41—151no—
356C2×#41D3×D511yescharacteristic
357C2×M16—31no—
358C2×M16—51no—
359C4⋊Q8—151no—
360#127—151no—
361#125—152no—
362#125—152no—
363#125—301no—
364#125—301no—
365#124—151no—
366#124—151no—
367#121—301no—
368#121—301no—
369C4×D4—301no—
370C4×D4—301no—
371C2×C4⋊C4—31no—
372C2×C4⋊C4—51no—
373C2×C4⋊C4—301no—
374C2×C4⋊C4—301no—
375C2×C42—151no—
376#103—32no—
377#103—52no—
378#103—152no—
379#103—152no—
380#102—32no—
381#102—52no—
382#102—152no—
383#102—152no—
384#97—152no—
385C4×C8—152no—
386C2×C3×C5—22no—
387C2×C3×C5—81no—
388C2×C3×C5C2×#12411yescharacteristic
389C2×C3×C5#57511yescharacteristic
390C2×C3×C5#57511yescharacteristic
391D15—81no—
392C23×C3—21no—
393C23×C3D4×D511yescharacteristic
394C22×D3—101no—
395C3×Q8—21no—
396C3×Q8—21no—
397C3×D4—21no—
398C3×D4—21no—
399C3×D4—42no—
400C2×C3×C4—21no—
401C2×C3×C4—42no—
402C2×C3×C4—102no—
403C2×C3×C4—202no—
404C2×C3×C4#70912yes—
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4#71112yes—
407C2×C3×C4C23×D511yescharacteristic
408#67—202no—
409#67—202no—
410#67—202no—
411C2×Dic3—22no—
412C2×Dic3—42no—
413C2×Dic3—51no—
414C2×Dic3—51no—
415C2×Dic3—51no—
416C2×Dic3—51no—
417C2×Dic3—101no—
418C2×Dic3—201no—
419C2×Dic3—201no—
420D12—101no—
421D12—101no—
422C4×D3—202no—
423Dic6—101no—
424Dic6—101no—
425C3×C8—102no—
426C3⋊C8—22no—
427C22×C5—21no—
428C22×C5—21no—
429C22×C5—41no—
430C22×C5—41no—
431C22×C5—41no—
432C22×C5#88512yes—
433C22×C5C2×D3×D411yescharacteristic
434C2×D5—121no—
435C2×D5—121no—
436C2×D5—121no—
437C4×C5—41no—
438C4×C5—122no—
439C4×C5#88512yes—
440C4×C5C2×#23511yescharacteristic
441C4×C5C2×#23711yescharacteristic
442Dic5—42no—
443Dic5—61no—
444Dic5—61no—
445Dic5—61no—
446Dic5—61no—
447Dic5—121no—
448#41—22no—
449#41—22no—
450#41—302no—
451#41—302no—
452C2×Q8—151no—
453C2×Q8C2×D3×D511yescharacteristic
454C2×D4—21no—
455C2×D4—151no—
456C2×D4—301no—
457C2×D4—301no—
458C2×D4—301no—
459C2×D4C2×D3×D511yescharacteristic
460C22×C4—21no—
461C22×C4—61no—
462C22×C4—101no—
463C22×C4—151no—
464C22×C4—151no—
465C22×C4—301no—
466C22×C4C2×D3×D511yescharacteristic
467M16—62no—
468M16—102no—
469C2×C8—32no—
470C2×C8—52no—
471C4⋊C4—32no—
472C4⋊C4—52no—
473C4⋊C4—61no—
474C4⋊C4—101no—
475C4⋊C4—301no—
476C4⋊C4—301no—
477C4⋊C4—302no—
478C4⋊C4—302no—
479C4⋊C4—302no—
480C4⋊C4—302no—
481C22⋊C4—301no—
482C22⋊C4—301no—
483C22⋊C4—301no—
484C22⋊C4—301no—
485C42—151no—
486C42—151no—
487C42—151no—
488C42—151no—
489C3×C5#314211yescharacteristic, complemented, Hall
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3#403112yes—
496C22×C3C2×D4×D511yescharacteristic
497C2×D3—201no—
498C2×D3—201no—
499C2×D3—201no—
500C3×C4—41no—
501C3×C4—202no—
502C3×C4#403112yes—
503C3×C4C2×#70911yescharacteristic
504C3×C4C2×#71111yescharacteristic
505Dic3—42no—
506Dic3—101no—
507Dic3—101no—
508Dic3—101no—
509Dic3—101no—
510Dic3—201no—
511C2×C5—22no—
512C2×C5—81no—
513C2×C5C2×#88511yescharacteristic
514C2×C5#577311yescharacteristic
515C2×C5#577711yescharacteristic
516D5—241no—
517C23—21no—
518C23—301no—
519C23#679011yescharacteristic
520Q8—21no—
521Q8—21no—
522Q8—301no—
523Q8—301no—
524D4—21no—
525D4—21no—
526D4—42no—
527D4—301no—
528D4—301no—
529D4—602no—
530D4—602no—
531D4—602no—
532C2×C4—21no—
533C2×C4—42no—
534C2×C4—62no—
535C2×C4—102no—
536C2×C4—122no—
537C2×C4—151no—
538C2×C4—151no—
539C2×C4—151no—
540C2×C4—151no—
541C2×C4—202no—
542C2×C4—301no—
543C2×C4—601no—
544C2×C4—601no—
545C2×C4—602no—
546C2×C4#677912yes—
547C2×C4#678212yes—
548C2×C4#679011yescharacteristic
549C2×C4C22×D3×D511yescharacteristic
550C8—62no—
551C8—102no—
552C2×C3—22no—
553C2×C3—81no—
554C2×C3C2×#403111yescharacteristic
555C2×C3#6610711yescharacteristic
556C2×C3#6611111yescharacteristic
557D3—401no—
558C5#7887911yescharacteristic, complemented, Sylow
559C22—21no—
560C22—21no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—601no—
565C22—601no—
566C22—601no—
567C22#9152612yes—
568C22C2×#679011yescharacteristic
569C4—41no—
570C4—122no—
571C4—202no—
572C4—301no—
573C4—301no—
574C4—301no—
575C4—301no—
576C4—601no—
577C4#9152612yes—
578C4C2×#677911yescharacteristic
579C4C2×#678211yescharacteristic
580C3#1061061011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—81no—
583C2—1201no—
584C2#1175534911yescharacteristic
585C2#1175535211yescharacteristic
586C2C2×#9152611yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radical#11756591C2
hypercenterC2×#41D3×D5
hypocenterC3×C5#3142
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×#91384C2semidirect
3. C5×#5232C2semidirect
4. C3×#65521C2semidirect
5. #11749336C2semidirect
6. #11749190C2semidirect
7. #11749090C2semidirect
8. #11748944C2semidirect
9. #11748413C2semidirect
10. #11748394C2semidirect
11. #11748328C2semidirect
48. #65521D3semidirect
100. #5232D5semidirect
489. C3×C5#3142semidirect
558. C5#78879semidirect
580. C3#10610610semidirect

Maximal subgroups

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

subgroupindex
#117565912
C2×#913842
C5×#52322
C3×#655212
#117493362
#117491902
#117490902
#117489442
#117484132
#117483942
#117483282
#106106103
#788795