Finite Group Databasebeta

← #14426293

Subgroups — #14426293

Order 1920.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6817C211yesmaximal, characteristic, complemented
2C3×#65811C211yesmaximal, characteristic, complemented
3C2×#91311C212yesmaximal, complemented
4C2×C5×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5#11748365C212yesmaximal, complemented
6#11747104C211yesmaximal, characteristic, complemented
7#11747085C211yesmaximal, characteristic, complemented
8#11747084C212yesmaximal, complemented
9#11747083C211yesmaximal, characteristic, complemented
10#11747044C212yesmaximal, complemented
11#11747043C211yesmaximal, characteristic, complemented
12#10610752—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C2×C3×#709C2212yes—
15C2×C3×C42×C5C2211yescharacteristic
16C4×#1220—21no—
17C4×#1220—21no—
18#91815C2211yescharacteristic
19C4×D60C2211yescharacteristic
20C4×D60C2211yescharacteristic
21#91761—21no—
22C42×D15—21no—
23C4×Dic30C2211yescharacteristic
24C4×Dic30C2211yescharacteristic
25C22×C5×C3⋊C8C2212yes—
26C3×#4076—21no—
27C3×#4076—21no—
28C3×#4021C2212yes—
29C3×#4020C2211yescharacteristic
30C3×C4×D20—21no—
31C3×C4×D20—21no—
32C3×#4015C2211yescharacteristic
33C2×#6662C2212yes—
34C2×#6658C2212yes—
35#91311—28no—
36C2×#6657C2212yes—
37C2×#6653C2212yes—
38C5×C4⋊(C3⋊C8)C2211yescharacteristic
39C5×C4⋊(C3⋊C8)C2212yes—
40C5×C4⋊(C3⋊C8)C2211yescharacteristic
41#90971C2212yes—
42#90971C2212yes—
43#90965C2212yes—
44#90965C2212yes—
45#83296—51nomaximal, Hall
46C2×#4050—32no—
47#65811D311yescharacteristic, complemented
48C4×#709—31no—
49C2×C5×C4⋊C8—31no—
50#65402—32no—
51#65138—31no—
52#65137—31no—
53#65134—32no—
54#65133—32no—
55#65132—31no—
56#65130—31no—
57C22×C3×C4×C5D411yescharacteristic
58C22×C3×C4×C5D411yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—21no—
61#6817—22no—
62#6817D414yescomplemented
63C2×D60C2311yescharacteristic
64C2×C4×D15—21no—
65C2×C4×D15—21no—
66C2×C4×D15—21no—
67C2×Dic30C2311yescharacteristic
68C2×C3×C22⋊D5—22no—
69C3×#709—24no—
70C3×#709—24no—
71C2×C3×D20—21no—
72C2×C3×D20—21no—
73C2×C3×D20C2312yes—
74C2×C3×C4×D5—22no—
75C2×C3×Dic10C2312yes—
76C3×C42×C5D411yescharacteristic
77C3×C42×C5D411yescharacteristic
78C3×C42×C5C2311yescharacteristic
79C3×C42×C5C2311yescharacteristic
80#6719—21no—
81#6714—21no—
82#6714—21no—
83C4⋊Dic15C2311yescharacteristic
84C4⋊Dic15C2311yescharacteristic
85#6712—21no—
86#6712—21no—
87C4×Dic15—21no—
88C4×Dic15—21no—
89C4×Dic15—21no—
90C2×C5×C3⋊C8—24no—
91C2×C5×C3⋊C8C2312yes—
92C2×C5×C3⋊C8C2312yes—
93C3×#685—22no—
94C3×#685—22no—
95C3×C4⋊Dic5—21no—
96C3×C4⋊Dic5—21no—
97#6662—24no—
98#6658—24no—
99#6657—24no—
100#6653—24no—
101C3×#532—51no—
102C2×#950—52no—
103C4×#235—51no—
104#5233—52no—
105#5090—51no—
106#5083—51no—
107#5075—51no—
108#5072—51no—
109#5057—52no—
110#5054—52no—
111C2×C4⋊(C3⋊C8)D511yescharacteristic, complemented
112C2×#709—31no—
113C2×#709C2×D312yes—
114C22×C5×C8—32no—
115C2×C42×C5C2×D311yescharacteristic
116#4076—21no—
117#4076—21no—
118C4×C22⋊D5—61no—
119C4×C22⋊D5—61no—
120#4072—31no—
121C2×Dic20—32no—
122#4050—68no—
123C2×D40—32no—
124C2×#677—32no—
125C2×#677—32no—
126#4021C2×D312yes—
127#4020C2×D311yescharacteristic
128C4×D20—21no—
129C4×D20—21no—
130C4×D20—31no—
131C4×D20—31no—
132#4018—61no—
133C42×D5—61no—
134#4015C2×D311yescharacteristic
135C4×Dic10—31no—
136C4×Dic10—31no—
137C5×C4⋊C8—31no—
138C5×C4⋊C8—31no—
139C5×C4⋊C8—32no—
140#3953—32no—
141#3953—32no—
142#3948—32no—
143#3948—32no—
144#3035—151noSylow
145C23×C3×C5C2×D411yescharacteristic
146C22×D15—21no—
147C22×C3×D5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—41no—
150C2×C3×C4×C5C2×D411yescharacteristic
151C2×C3×C4×C5C2×D411yescharacteristic
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D411yescharacteristic
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5C2×D411yescharacteristic
156C2×C3×C4×C5#4112yes—
157C2×C3×C4×C5C2411yescharacteristic
158#1220—24no—
159C2×Dic15—21no—
160C2×Dic15—21no—
161C2×Dic15—21no—
162D60—21no—
163D60C2×D412yes—
164C4×D15—24no—
165C4×D15—42no—
166C4×D15—42no—
167Dic30—21no—
168Dic30C2×D412yes—
169C3×C22⋊D5—44no—
170C2×C3×Dic5—22no—
171C3×D20—22no—
172C3×D20—22no—
173C3×D20—42no—
174C3×D20—42no—
175C3×C4×D5—44no—
176C3×Dic10—22no—
177C3×Dic10—22no—
178C5×C3⋊C8—22no—
179C5×C3⋊C8—22no—
180C2×C3×#41—52no—
181C2×#235—51no—
182C3×C4⋊Q8—51no—
183C3×#127—51no—
184C3×#124—52no—
185C3×#121—101no—
186C3×#121—101no—
187C3×C4×D4—101no—
188C3×C4×D4—101no—
189C2×C3×C42C2×D511yescharacteristic
190#950—108no—
191C2×#216—52no—
192C2×#215—52no—
193C2×#214—52no—
194C2×#213—52no—
195C4×#67—101no—
196C4×#67—101no—
197#926—51no—
198C22×C3⋊C8C2×D512yes—
199C4×D12—51no—
200C4×D12—51no—
201#872—101no—
202C42×D3—101no—
203C4×Dic6—51no—
204C4×Dic6—51no—
205#835—52no—
206#835—52no—
207#832—52no—
208#832—52no—
209C4⋊(C3⋊C8)C2×D512yes—
210C4⋊(C3⋊C8)C2×D511yescharacteristic
211C4⋊(C3⋊C8)C2×D511yescharacteristic
212C22×C4×C5#6711yescharacteristic
213C22×C4×C5#6711yescharacteristic
214C22×C4×C5C22×D311yescharacteristic
215C2×C22⋊D5—22no—
216C2×C22⋊D5—61no—
217#709—24no—
218#709—24no—
219#709—34no—
220#709—62no—
221C2×D20—21no—
222C2×D20—21no—
223C2×D20—31no—
224C2×D20C22×D312yes—
225C2×C4×D5—22no—
226C2×C4×D5—61no—
227C2×C4×D5—61no—
228C2×C4×D5—61no—
229C2×Dic10—31no—
230C2×Dic10C22×D312yes—
231C2×C5×C8—32no—
232C2×C5×C8—32no—
233C2×C5×C8—64no—
234C42×C5#6711yescharacteristic
235C42×C5#6711yescharacteristic
236C42×C5C22×D311yescharacteristic
237C42×C5C22×D311yescharacteristic
238C22⋊Dic5—61no—
239#685—22no—
240#685—22no—
241#685—61no—
242#685—61no—
243C4⋊Dic5—21no—
244C4⋊Dic5—21no—
245C4⋊Dic5—31no—
246C4⋊Dic5—31no—
247#683—61no—
248#683—61no—
249C4×Dic5—61no—
250C4×Dic5—61no—
251C4×Dic5—61no—
252Dic20—64no—
253D40—64no—
254#677—64no—
255#677—64no—
256C2×#135—152no—
257#532—51no—
258C4×#41—151no—
259#464—152no—
260#463—152no—
261#462—151no—
262#461—151no—
263#460—151no—
264#459—151no—
265C2×C4⋊C8—31no—
266#416—152no—
267C22×C3×C5—21no—
268C22×C3×C5—21no—
269C22×C3×C5#12112yes—
270C22×C3×C5C22×D411yescharacteristic
271C2×D15—22no—
272C2×D15—41no—
273C2×D15—41no—
274C2×C3×D5—42no—
275C2×C3×D5—42no—
276C2×C3×D5—42no—
277C3×C4×C5—21no—
278C3×C4×C5—21no—
279C3×C4×C5—21no—
280C3×C4×C5#12112yes—
281C3×C4×C5#13512yescomplemented
282C3×C4×C5C22×D411yescharacteristic
283C3×C4×C5C2×#4111yescharacteristic
284Dic15—22no—
285Dic15—41no—
286Dic15—41no—
287C3×Dic5—42no—
288C3×#41—104no—
289C3×#41—104no—
290C2×C3×Q8—52no—
291C2×C3×D4—52no—
292C2×C3×D4—102no—
293C2×C3×D4—101no—
294C2×C3×D4—101no—
295C22×C3×C4—102no—
296C22×C3×C4D2011yescharacteristic
297C22×C3×C4D2011yescharacteristic
298C22×C3×C4C22×D511yescharacteristic
299C2×#67—101no—
300#235—54no—
301#235—102no—
302C2×D12—51no—
303C2×C4×D3—101no—
304C2×C4×D3—101no—
305C2×C4×D3—101no—
306C2×Dic6—51no—
307C3×C4⋊C4—101no—
308C3×C4⋊C4—101no—
309C3×C22⋊C4—102no—
310C3×C22⋊C4—102no—
311C3×C42D2011yescharacteristic
312C3×C42D2011yescharacteristic
313C3×C42C22×D511yescharacteristic
314C3×C42C22×D511yescharacteristic
315#217—101no—
316#216—104no—
317#215—104no—
318#214—104no—
319#213—104no—
320(C2×D3)⋊C4—101no—
321(C2×D3)⋊C4—101no—
322C4⋊Dic3—51no—
323C4⋊Dic3—51no—
324Dic3⋊C4—101no—
325Dic3⋊C4—101no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3—101no—
329C2×C3⋊C8—24no—
330C2×C3⋊C8C22×D512yes—
331C2×C3⋊C8C22×D512yes—
332C23×C5C2×#6711yescharacteristic
333C22×D5—22no—
334C22×D5—61no—
335C2×C4×C5—22no—
336C2×C4×C5—41no—
337C2×C4×C5D3×D412yes—
338C2×C4×C5#23712yes—
339C2×C4×C5C2×#6711yescharacteristic
340C2×C4×C5C2×#6711yescharacteristic
341C2×C4×C5C2×#6711yescharacteristic
342C2×C4×C5C2×#6711yescharacteristic
343C2×C4×C5C2×#6711yescharacteristic
344C2×C4×C5C23×D311yescharacteristic
345C22⋊D5—44no—
346C22⋊D5—64no—
347C2×Dic5—22no—
348C2×Dic5—61no—
349C2×Dic5—61no—
350C2×Dic5—61no—
351D20—22no—
352D20—22no—
353D20—32no—
354D20—42no—
355D20—42no—
356D20—61no—
357C4×D5—44no—
358C4×D5—64no—
359C4×D5—122no—
360C4×D5—122no—
361Dic10—22no—
362Dic10—22no—
363Dic10—32no—
364Dic10—61no—
365C5×C8—62no—
366C5×C8—62no—
367C2×#41—52no—
368C2×#41—151no—
369#135—308no—
370C2×Q16—152no—
371C2×SD16—152no—
372C2×SD16—152no—
373C2×D8—152no—
374C22×C8—32no—
375C4⋊Q8—51no—
376#127—51no—
377#124—52no—
378#121—101no—
379#121—101no—
380C4×Q8—151no—
381C4×Q8—151no—
382C4×D4—101no—
383C4×D4—101no—
384C4×D4—151no—
385C4×D4—151no—
386C4×D4—301no—
387C4×D4—301no—
388#117—151no—
389#117—301no—
390C2×C42—301no—
391C2×C42D3×D511yescharacteristic
392C4⋊C8—31no—
393C4⋊C8—31no—
394C4⋊C8—32no—
395#103—152no—
396#103—152no—
397#102—152no—
398#102—152no—
399C2×C3×C5—22no—
400C2×C3×C5C2×#12111yescharacteristic
401C2×C3×C5C2×#13511yescharacteristic
402C2×C3×C5#57511yescharacteristic
403D15—42no—
404C3×D5—82no—
405C23×C3—102no—
406C23×C3C2×D2011yescharacteristic
407C22×D3—101no—
408C3×Q8—102no—
409C3×Q8—102no—
410C3×D4—102no—
411C3×D4—102no—
412C3×D4—202no—
413C3×D4—204no—
414C3×D4—202no—
415C2×C3×C4—22no—
416C2×C3×C4—41no—
417C2×C3×C4—102no—
418C2×C3×C4—204no—
419C2×C3×C4C2×D2011yescharacteristic
420C2×C3×C4C2×D2011yescharacteristic
421C2×C3×C4C2×D2011yescharacteristic
422C2×C3×C4C2×D2011yescharacteristic
423C2×C3×C4C2×D2011yescharacteristic
424C2×C3×C4D4×D512yes—
425C2×C3×C4#71312yes—
426C2×C3×C4C23×D511yescharacteristic
427#67—104no—
428C2×Dic3—101no—
429C2×Dic3—101no—
430C2×Dic3—101no—
431D12—52no—
432D12—101no—
433C4×D3—104no—
434C4×D3—202no—
435C4×D3—202no—
436Dic6—52no—
437Dic6—101no—
438C3⋊C8—22no—
439C3⋊C8—22no—
440C22×C5—21no—
441C22×C5—21no—
442C22×C5#93812yes—
443C22×C5C22×#6711yescharacteristic
444C2×D5—42no—
445C2×D5—42no—
446C2×D5—42no—
447C2×D5—62no—
448C2×D5—121no—
449C2×D5—121no—
450C4×C5—21no—
451C4×C5—21no—
452C4×C5—21no—
453C4×C5#93812yes—
454C4×C5#95012yescomplemented
455C4×C5C2×D3×D411yescharacteristic
456C4×C5C2×#23711yescharacteristic
457Dic5—42no—
458Dic5—62no—
459Dic5—121no—
460Dic5—121no—
461#41—104no—
462#41—104no—
463#41—154no—
464#41—302no—
465C2×Q8—52no—
466C2×Q8—151no—
467C2×D4—52no—
468C2×D4—102no—
469C2×D4—101no—
470C2×D4—101no—
471C2×D4—151no—
472C2×D4—301no—
473C22×C4—102no—
474C22×C4—301no—
475C22×C4—301no—
476C22×C4—301no—
477C22×C4#120211yescharacteristic
478C22×C4#120211yescharacteristic
479C22×C4C2×D3×D511yescharacteristic
480Q16—304no—
481SD16—304no—
482SD16—304no—
483D8—304no—
484C2×C8—32no—
485C2×C8—32no—
486C2×C8—64no—
487C4⋊C4—101no—
488C4⋊C4—101no—
489C4⋊C4—151no—
490C4⋊C4—151no—
491C4⋊C4—301no—
492C4⋊C4—301no—
493C22⋊C4—102no—
494C22⋊C4—102no—
495C22⋊C4—301no—
496C22⋊C4—301no—
497C22⋊C4—301no—
498C42—301no—
499C42—301no—
500C42—301no—
501C42#120211yescharacteristic
502C42#120211yescharacteristic
503C42C2×D3×D511yescharacteristic
504C42C2×D3×D511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—202no—
509C22×C3—202no—
510C22×C3—202no—
511C22×C3#404112yes—
512C22×C3C22×D2011yescharacteristic
513C2×D3—102no—
514C2×D3—201no—
515C2×D3—201no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—202no—
520C3×C4#404112yes—
521C3×C4#405012yescomplemented
522C3×C4C2×D4×D511yescharacteristic
523C3×C4C2×#71311yescharacteristic
524Dic3—102no—
525Dic3—201no—
526Dic3—201no—
527C2×C5—22no—
528C2×C5C2×#93811yescharacteristic
529C2×C5C2×#95011yescharacteristic
530C2×C5#582211yescharacteristic
531D5—82no—
532D5—122no—
533C23—102no—
534C23—301no—
535C23C2×#120211yescharacteristic
536Q8—102no—
537Q8—102no—
538Q8—152no—
539Q8—301no—
540D4—102no—
541D4—102no—
542D4—152no—
543D4—202no—
544D4—204no—
545D4—202no—
546D4—304no—
547D4—301no—
548C2×C4—22no—
549C2×C4—41no—
550C2×C4—102no—
551C2×C4—204no—
552C2×C4—301no—
553C2×C4—304no—
554C2×C4—301no—
555C2×C4—301no—
556C2×C4—602no—
557C2×C4—602no—
558C2×C4#676612yes—
559C2×C4#677712yes—
560C2×C4C2×#120211yescharacteristic
561C2×C4C2×#120211yescharacteristic
562C2×C4C2×#120211yescharacteristic
563C2×C4C2×#120211yescharacteristic
564C2×C4C2×#120211yescharacteristic
565C2×C4C22×D3×D511yescharacteristic
566C8—62no—
567C8—62no—
568C2×C3—22no—
569C2×C3—402no—
570C2×C3C2×#404111yescharacteristic
571C2×C3C2×#405011yescharacteristic
572C2×C3#6608211yescharacteristic
573D3—202no—
574C5#8329611yescharacteristic, complemented, Sylow
575C22—21no—
576C22—21no—
577C22—202no—
578C22—202no—
579C22—202no—
580C22—302no—
581C22—601no—
582C22—601no—
583C22#9146512yes—
584C22C22×#120211yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—202no—
589C4—302no—
590C4—601no—
591C4—601no—
592C4#9131112yescomplemented
593C4#9146512yes—
594C4C2×#676611yescharacteristic
595C4C2×#677711yescharacteristic
596C3#1061075211yescharacteristic, complemented, Sylow
597C2—22no—
598C2—402no—
599C2—602no—
600C2C2×#9131111yescharacteristic
601C2#1175415411yescharacteristic
602C2C2×#9146511yescharacteristic
subgroupG/H
centerC2×C4C2×#1202
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5C22×D4
hypercenterC2×C42D3×D5
hypocenterC3×C5#3035
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C4×#6817C2semidirect
2. C3×#65811C2semidirect
3. C2×#91311C2semidirect
4. C2×C5×C4⋊(C3⋊C8)C2semidirect
5. #11748365C2semidirect
6. #11747104C2semidirect
7. #11747085C2semidirect
8. #11747084C2semidirect
9. #11747083C2semidirect
10. #11747044C2semidirect
11. #11747043C2semidirect
47. #65811D3semidirect
62. #6817D4semidirect
111. C2×C4⋊(C3⋊C8)D5semidirect
281. C3×C4×C5#135semidirect
454. C4×C5#950semidirect
505. C3×C5#3035semidirect
521. C3×C4#4050semidirect
574. C5#83296semidirect
592. C4#91311semidirect
596. C3#10610752semidirect

Maximal subgroups

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

subgroupindex
C4×#68172
C3×#658112
C2×#913112
C2×C5×C4⋊(C3⋊C8)2
#117483652
#117471042
#117470852
#117470842
#117470832
#117470442
#117470432
#106107523
#832965