Finite Group Databasebeta

← #14438619

Subgroups — #14438619

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91777C211yesmaximal, characteristic, complemented
2#11754586C211yesmaximal, characteristic
3C5×#4966C211yesmaximal, characteristic, complemented
4C3×#65255C211yesmaximal, characteristic, complemented
5#11748390C211yesmaximal, characteristic, complemented
6#11748369C211yesmaximal, characteristic, complemented
7#11748328C211yesmaximal, characteristic, complemented
8#11747428C212yesmaximal, complemented
9#11747407C212yesmaximal, complemented
10#11747388C212yesmaximal, complemented
11#11747384C212yesmaximal, complemented
12#10610609—31nomaximal, Hall
13C22×Dic30—21no—
14C2×C5×#235C2211yescharacteristic
15C2×C3×#709C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C4⋊Dic15C2211yescharacteristic
18C2×#6712—21no—
19C2×C4×Dic15C2211yescharacteristic
20#91777—21no—
21#91777—21no—
22#91777C2212yes—
23#91777C2212yes—
24C2×C5×#208C2211yescharacteristic
25C2×C3×#681C2211yescharacteristic
26#91571—21no—
27#91571—21no—
28#91463C2211yescharacteristic
29C4×#1201—21no—
30C4×#1201—21no—
31#91385C2211yescharacteristic
32#91370C2211yescharacteristic
33#91368C2211yescharacteristic
34C5×#810C2212yes—
35C5×#809C2212yes—
36C3×#3942C2212yes—
37C3×#3941C2212yes—
38#90973C2212yes—
39#90971C2212yes—
40#90967C2212yes—
41#90965C2212yes—
42#90954C2212yes—
43#90951C2212yes—
44#78878—51nomaximal, Hall
45#66166—31no—
46C2×#4034—31no—
47C5×#417—31no—
48#65521—31no—
49#65422—31no—
50M16×Dic5—31no—
51#65255D311yescharacteristic, complemented
52#65084—32no—
53#65081—32no—
54#65054—32no—
55#65049—32no—
56C22×C3×C4×C5—21no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15—21no—
59C22×Dic15D411yescharacteristic
60C22×Dic15D411yescharacteristic
61C2×Dic30—22no—
62C2×Dic30—22no—
63C2×Dic30—44no—
64C2×C5×#67—21no—
65C5×#235—22no—
66C5×#235—22no—
67C2×C5×D12C2311yescharacteristic
68C2×C4×C5×D3—21no—
69C2×C5×Dic6C2311yescharacteristic
70C2×C3×C22⋊D5—21no—
71C3×#709—22no—
72C3×#709—22no—
73C2×C3×D20C2311yescharacteristic
74C2×C3×C4×D5—21no—
75C2×C3×Dic10C2311yescharacteristic
76C2×#1201—21no—
77C2×#1201—21no—
78C3×C5×C4⋊C4—21no—
79C3×C5×C4⋊C4C2312yes—
80C4⋊Dic15—21no—
81C4⋊Dic15C2312yes—
82#6712—22no—
83#6712—22no—
84C4×Dic15D411yescharacteristic
85C4×Dic15D411yescharacteristic
86C4×Dic15C2311yescharacteristic
87C4×Dic15C2311yescharacteristic
88C5×#208—22no—
89C2×C5×C3⋊C8C2312yes—
90C3×#681—22no—
91C2×C3×#168C2312yes—
92#6669—21no—
93#6669—21no—
94(C2×D3)⋊Dic5—21no—
95(C2×D3)⋊Dic5—21no—
96#6665—21no—
97#6665—21no—
98#5832—51no—
99C2×#888—51no—
100C3×#417—51no—
101#5232—51no—
102#5133—51no—
103Dic3×M16—51no—
104#4966D511yescharacteristic, complemented
105#4794—52no—
106#4792—52no—
107#4765—52no—
108#4760—52no—
109C2×C5×#41—31no—
110C2×#709C2×D311yescharacteristic
111C22×Dic10—61no—
112C2×C5×M16—31no—
113C2×C5×C4⋊C4C2×D311yescharacteristic
114#4093—31no—
115#4090—31no—
116#4086—31no—
117#4085—61no—
118#4085—61no—
119#4082—31no—
120C4×C22⋊D5—61no—
121C4×C22⋊D5—61no—
122C2×C4⋊Dic5—31no—
123C2×#683—61no—
124C2×C4×Dic5—31no—
125C2×#681C2×D311yescharacteristic
126#4034—32no—
127#4034—32no—
128#4034—61no—
129#4034—61no—
130C5×#103—32no—
131C5×#102—32no—
132#3967—32no—
133#3964—32no—
134#3953—32no—
135#3948—32no—
136#3947—32no—
137C8×Dic5—32no—
138#3942C2×D312yes—
139#3941C2×D312yes—
140#3141—151noSylow
141C23×C3×C5C2×D411yescharacteristic
142C22×C5×D3—21no—
143C22×C3×D5—21no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—42no—
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5#4112yes—
148C2×C3×C4×C5#4112yes—
149C2×C3×C4×C5C2411yescharacteristic
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—42no—
154C2×Dic15C2×D411yescharacteristic
155C2×Dic15C2×D411yescharacteristic
156C2×Dic15C2×D411yescharacteristic
157C2×Dic15C2×D411yescharacteristic
158Dic30—44no—
159Dic30—44no—
160C5×#67—42no—
161C2×C5×Dic3—21no—
162C5×D12—21no—
163C5×D12—21no—
164C4×C5×D3—42no—
165C5×Dic6—21no—
166C5×Dic6—21no—
167C3×C22⋊D5—42no—
168C2×C3×Dic5—21no—
169C3×D20—21no—
170C3×D20—21no—
171C3×C4×D5—42no—
172C3×Dic10—21no—
173C3×Dic10—21no—
174#1201—42no—
175#1201—42no—
176C3×#168—22no—
177C5×C3⋊C8—22no—
178C2×C3×#41—51no—
179C2×#235C2×D511yescharacteristic
180C22×Dic6—101no—
181C2×C3×M16—51no—
182C2×C3×C4⋊C4C2×D511yescharacteristic
183#947—51no—
184#944—51no—
185#940—51no—
186#939—101no—
187#939—101no—
188#936—51no—
189C4×#67—101no—
190C4×#67—101no—
191C2×C4⋊Dic3—51no—
192C2×Dic3⋊C4—101no—
193C2×C4×Dic3—51no—
194C2×#208C2×D511yescharacteristic
195#888—52no—
196#888—52no—
197#888—101no—
198#888—101no—
199C3×#103—52no—
200C3×#102—52no—
201#835—52no—
202#832—52no—
203#821—52no—
204#816—52no—
205#815—52no—
206C8×Dic3—52no—
207#810C2×D512yes—
208#809C2×D512yes—
209C5×#41—62no—
210C5×#41—62no—
211C2×C5×Q8—31no—
212C2×C5×D4—31no—
213C2×C5×D4—61no—
214C22×C4×C5—21no—
215C22×C4×C5—61no—
216C22×C4×C5C22×D311yescharacteristic
217C2×C22⋊D5—21no—
218C2×C22⋊D5—61no—
219C2×C22⋊D5—61no—
220C22×Dic5—31no—
221C22×Dic5—31no—
222C22×Dic5—61no—
223#709—22no—
224#709—22no—
225C2×D20C22×D311yescharacteristic
226C2×C4×D5—21no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C2×Dic10—124no—
230C2×Dic10C22×D311yescharacteristic
231C5×M16—62no—
232C2×C5×C8—32no—
233C5×C4⋊C4—21no—
234C5×C4⋊C4C22×D312yes—
235C22⋊Dic5—61no—
236C22⋊Dic5—61no—
237#685—61no—
238#685—61no—
239C4⋊Dic5—32no—
240C4⋊Dic5—61no—
241#683—61no—
242#683—61no—
243#683—62no—
244#683—62no—
245C4×Dic5—31no—
246C4×Dic5—31no—
247C4×Dic5—31no—
248C4×Dic5—31no—
249#681—22no—
250C2×#168C22×D312yes—
251#532—151no—
252C2×C4⋊Q8—151no—
253#489—152no—
254#489—152no—
255#487—152no—
256#486—152no—
257#417—31no—
258#417—51no—
259#417—151no—
260#417—151no—
261C4×M16—151no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5#12412yes—
265C22×C3×C5C22×D411yescharacteristic
266C2×C5×D3—41no—
267C2×C5×D3—41no—
268C2×C5×D3—41no—
269C2×C3×D5—41no—
270C2×C3×D5—41no—
271C2×C3×D5—41no—
272C3×C4×C5—42no—
273C3×C4×C5#12412yes—
274C3×C4×C5C2×#4111yescharacteristic
275C3×C4×C5C2×#4111yescharacteristic
276Dic15—21no—
277Dic15—21no—
278Dic15—42no—
279Dic15AGL(1,ℤ/8)12yes—
280Dic15#13712yes—
281C3×Dic5—41no—
282C5×Dic3—41no—
283C3×#41—102no—
284C3×#41—102no—
285C2×C3×Q8—51no—
286C2×C3×D4—51no—
287C2×C3×D4—101no—
288C22×C3×C4—21no—
289C22×C3×C4—101no—
290C22×C3×C4C22×D511yescharacteristic
291C2×#67—21no—
292C2×#67—101no—
293C2×#67—101no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296C22×Dic3—101no—
297#235—22no—
298#235—22no—
299C2×D12C22×D511yescharacteristic
300C2×C4×D3—21no—
301C2×Dic6—102no—
302C2×Dic6—102no—
303C2×Dic6—204no—
304C2×Dic6C22×D511yescharacteristic
305C3×M16—102no—
306C2×C3×C8—52no—
307C3×C4⋊C4—21no—
308C3×C4⋊C4C22×D512yes—
309#217—101no—
310#217—101no—
311(C2×D3)⋊C4—101no—
312(C2×D3)⋊C4—101no—
313C4⋊Dic3—52no—
314C4⋊Dic3—101no—
315Dic3⋊C4—101no—
316Dic3⋊C4—101no—
317Dic3⋊C4—102no—
318Dic3⋊C4—102no—
319C4×Dic3—51no—
320C4×Dic3—51no—
321C4×Dic3—51no—
322C4×Dic3—51no—
323#208—22no—
324C2×C3⋊C8C22×D512yes—
325C23×C5—61no—
326C23×C5D3×D411yescharacteristic
327C22×D5—21no—
328C5×Q8—61no—
329C5×Q8—61no—
330C5×D4—61no—
331C5×D4—61no—
332C5×D4—122no—
333C2×C4×C5—22no—
334C2×C4×C5—42no—
335C2×C4×C5—61no—
336C2×C4×C5—122no—
337C2×C4×C5#23512yes—
338C2×C4×C5D3×D411yescharacteristic
339C2×C4×C5#23712yes—
340C2×C4×C5C23×D311yescharacteristic
341C22⋊D5—42no—
342C22⋊D5—122no—
343C22⋊D5—122no—
344C2×Dic5—21no—
345C2×Dic5—31no—
346C2×Dic5—31no—
347C2×Dic5—31no—
348C2×Dic5—31no—
349C2×Dic5—62no—
350C2×Dic5—62no—
351C2×Dic5—62no—
352C2×Dic5—122no—
353D20—21no—
354D20—21no—
355C4×D5—42no—
356Dic10—21no—
357Dic10—21no—
358Dic10—124no—
359Dic10—124no—
360C5×C8—62no—
361#168—22no—
362C2×#41—31no—
363C2×#41—51no—
364C22×Q8—301no—
365C2×M16—31no—
366C2×M16—51no—
367C4⋊Q8—152no—
368C4⋊Q8—152no—
369C4⋊Q8—151no—
370C4⋊Q8—301no—
371C4⋊Q8—301no—
372#127—151no—
373#124—151no—
374#124—151no—
375#121—301no—
376#121—301no—
377C4×D4—301no—
378C4×D4—301no—
379C2×C4⋊C4—151no—
380C2×C4⋊C4—301no—
381C2×C4⋊C4D3×D511yescharacteristic
382C2×C42—151no—
383#103—32no—
384#103—52no—
385#103—152no—
386#103—152no—
387#102—32no—
388#102—52no—
389#102—152no—
390#102—152no—
391#97—152no—
392C4×C8—152no—
393C2×C3×C5—22no—
394C2×C3×C5C2×#12411yescharacteristic
395C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C3×C5C2×#13711yescharacteristic
397C3×D5—81no—
398C5×D3—81no—
399C23×C3—101no—
400C23×C3D4×D511yescharacteristic
401C22×D3—21no—
402C3×Q8—101no—
403C3×Q8—101no—
404C3×D4—101no—
405C3×D4—101no—
406C3×D4—202no—
407C2×C3×C4—22no—
408C2×C3×C4—42no—
409C2×C3×C4—101no—
410C2×C3×C4—202no—
411C2×C3×C4#70912yes—
412C2×C3×C4D4×D511yescharacteristic
413C2×C3×C4#71112yes—
414C2×C3×C4C23×D511yescharacteristic
415#67—42no—
416#67—202no—
417#67—202no—
418C2×Dic3—21no—
419C2×Dic3—51no—
420C2×Dic3—51no—
421C2×Dic3—51no—
422C2×Dic3—51no—
423C2×Dic3—102no—
424C2×Dic3—102no—
425C2×Dic3—102no—
426C2×Dic3—202no—
427D12—21no—
428D12—21no—
429C4×D3—42no—
430Dic6—21no—
431Dic6—21no—
432Dic6—204no—
433Dic6—204no—
434C3×C8—102no—
435C3⋊C8—22no—
436C22×C5—21no—
437C22×C5—21no—
438C22×C5—121no—
439C22×C5—121no—
440C22×C5—121no—
441C22×C5#88512yes—
442C22×C5C2×D3×D411yescharacteristic
443C2×D5—41no—
444C2×D5—41no—
445C2×D5—41no—
446C4×C5—42no—
447C4×C5—121no—
448C4×C5#88512yes—
449C4×C5C2×#23511yescharacteristic
450C4×C5C2×#23711yescharacteristic
451Dic5—32no—
452Dic5—32no—
453Dic5—41no—
454Dic5—61no—
455Dic5—61no—
456Dic5—122no—
457#41—62no—
458#41—62no—
459#41—102no—
460#41—102no—
461C2×Q8—31no—
462C2×Q8—51no—
463C2×Q8—302no—
464C2×Q8—302no—
465C2×Q8—604no—
466C2×D4—31no—
467C2×D4—51no—
468C2×D4—61no—
469C2×D4—101no—
470C2×D4—301no—
471C2×D4—301no—
472C22×C4—21no—
473C22×C4—61no—
474C22×C4—101no—
475C22×C4—151no—
476C22×C4—151no—
477C22×C4—301no—
478C22×C4C2×D3×D511yescharacteristic
479M16—62no—
480M16—102no—
481C2×C8—32no—
482C2×C8—52no—
483C4⋊C4—21no—
484C4⋊C4—152no—
485C4⋊C4—302no—
486C4⋊C4—302no—
487C4⋊C4—301no—
488C4⋊C4—301no—
489C4⋊C4—301no—
490C4⋊C4C2×D3×D512yes—
491C22⋊C4—301no—
492C22⋊C4—301no—
493C22⋊C4—301no—
494C22⋊C4—301no—
495C42—151no—
496C42—151no—
497C42—151no—
498C42—151no—
499C3×C5#314111yescharacteristic, complemented, Hall
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—201no—
503C22×C3—201no—
504C22×C3—201no—
505C22×C3#403112yes—
506C22×C3C2×D4×D511yescharacteristic
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—41no—
510C3×C4—42no—
511C3×C4—201no—
512C3×C4#403112yes—
513C3×C4C2×#70911yescharacteristic
514C3×C4C2×#71111yescharacteristic
515Dic3—41no—
516Dic3—52no—
517Dic3—52no—
518Dic3—101no—
519Dic3—101no—
520Dic3—202no—
521C2×C5—22no—
522C2×C5—241no—
523C2×C5C2×#88511yescharacteristic
524C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
525C2×C5D3×#13711yescharacteristic
526D5—81no—
527C23—61no—
528C23—101no—
529C23#679011yescharacteristic
530Q8—61no—
531Q8—61no—
532Q8—101no—
533Q8—101no—
534Q8—604no—
535Q8—604no—
536D4—61no—
537D4—61no—
538D4—101no—
539D4—101no—
540D4—122no—
541D4—202no—
542D4—602no—
543D4—602no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—61no—
547C2×C4—101no—
548C2×C4—122no—
549C2×C4—151no—
550C2×C4—151no—
551C2×C4—151no—
552C2×C4—151no—
553C2×C4—202no—
554C2×C4—302no—
555C2×C4—302no—
556C2×C4—302no—
557C2×C4—602no—
558C2×C4#676512yes—
559C2×C4#677212yes—
560C2×C4#679011yescharacteristic
561C2×C4C22×D3×D511yescharacteristic
562C8—62no—
563C8—102no—
564C2×C3—22no—
565C2×C3—401no—
566C2×C3C2×#403111yescharacteristic
567C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
568C2×C3D5×#13711yescharacteristic
569D3—81no—
570C5#7887811yescharacteristic, complemented, Sylow
571C22—21no—
572C22—21no—
573C22—121no—
574C22—121no—
575C22—121no—
576C22—201no—
577C22—201no—
578C22—201no—
579C22#9146212yes—
580C22C2×#679011yescharacteristic
581C4—42no—
582C4—121no—
583C4—152no—
584C4—152no—
585C4—201no—
586C4—301no—
587C4—301no—
588C4—602no—
589C4#9146212yes—
590C4C2×#676511yescharacteristic
591C4C2×#677211yescharacteristic
592C3#1061060911yescharacteristic, complemented, Sylow
593C2—22no—
594C2—241no—
595C2—401no—
596C2C2×#9146211yescharacteristic
597C2#1175470111yescharacteristic
598C2#1175509511yescharacteristic
subgroupG/H
centerC22C2×#6790
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radical#11754586C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#3141
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91777C2semidirect
3. C5×#4966C2semidirect
4. C3×#65255C2semidirect
5. #11748390C2semidirect
6. #11748369C2semidirect
7. #11748328C2semidirect
8. #11747428C2semidirect
9. #11747407C2semidirect
10. #11747388C2semidirect
11. #11747384C2semidirect
51. #65255D3semidirect
104. #4966D5semidirect
499. C3×C5#3141semidirect
570. C5#78878semidirect
592. C3#10610609semidirect

Maximal subgroups

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

subgroupindex
C2×#917772
#117545862
C5×#49662
C3×#652552
#117483902
#117483692
#117483282
#117474282
#117474072
#117473882
#117473842
#106106093
#788785