Finite Group Databasebeta

← #14426238

Subgroups — #14426238

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
1C3×#66050C211yesmaximal, characteristic, complemented
2C2×#91490C212yesmaximal, complemented
3C4×#6772C211yesmaximal, characteristic, complemented
4#11751176C211yesmaximal, characteristic, complemented
5C5×#5072C211yesmaximal, characteristic, complemented
6#11749002C212yesmaximal, complemented
7#11748978C212yesmaximal, complemented
8#11748961C212yesmaximal, complemented
9#11747104C211yesmaximal, characteristic, complemented
10#11747070C211yesmaximal, characteristic, complemented
11D5×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
12#10612062—31nomaximal, Hall
13C2×C3×#711C2212yes—
14C2×#6772C2211yescharacteristic
15C3×C5×#127C2211yescharacteristic
16C2×#6716C2212yes—
17C4×Dic30C2211yescharacteristic
18C2×C5×#213C2212yes—
19C4×C5×D12C2211yescharacteristic
20C3×#4084—21no—
21C3×#4084—21no—
22C3×#4082C2212yes—
23C3×D4×Dic5—21no—
24C3×D4×Dic5—21no—
25C3×C42×D5C2211yescharacteristic
26C3×#4015C2211yescharacteristic
27#91490—28no—
28#91440—21no—
29C4×#1201—21no—
30Dic5×D12C2211yescharacteristic
31#91398—21no—
32C4×D3×Dic5—21no—
33#91358C2211yescharacteristic
34#91333C2211yescharacteristic
35C2×#6662C2212yes—
36C2×#6656C2212yes—
37C2×D5×C3⋊C8C2212yes—
38#91116C2212yes—
39C4⋊(C3⋊#168)C2211yescharacteristic
40C5×#832C2212yes—
41C5×C4⋊(C3⋊C8)C2211yescharacteristic
42#90982C2212yes—
43#90971C2212yes—
44#90970C2212yes—
45#83296—51nomaximal, Hall
46C2×#4058—32no—
47#66050D311yescharacteristic, complemented
48C4×#711—31no—
49C5×#459—31no—
50#65442—32no—
51#65364—31no—
52#65308—31no—
53#65138—31no—
54D5×C4⋊C8—31no—
55#65061—32no—
56#65056—32no—
57C2×C3×C5×D4—21no—
58C2×C3×C5×D4—21no—
59C2×C3×C5×D4C2312yes—
60C2×Dic30C2311yescharacteristic
61C2×C5×D12C2311yescharacteristic
62C2×C4×C5×D3—21no—
63C2×C3×C22⋊D5—22no—
64C22×C3×Dic5—22no—
65C3×#711—24no—
66C3×#711—24no—
67C2×C3×C4×D5D411yescharacteristic
68C2×C3×C4×D5D411yescharacteristic
69C2×C3×C4×D5C2311yescharacteristic
70C2×C3×Dic10C2312yes—
71C2×#1201—21no—
72C2×D3×Dic5—21no—
73C2×D3×Dic5—21no—
74#6772—22no—
75#6772D414yescomplemented
76C3×C42×C5C2311yescharacteristic
77#6716—24no—
78C4⋊Dic15C2311yescharacteristic
79#6712—21no—
80C4×Dic15—21no—
81C2×C3⋊#168C2312yes—
82C5×#213—24no—
83C5×(C2×D3)⋊C4—21no—
84C5×C4⋊Dic3C2311yescharacteristic
85C2×C5×C3⋊C8C2312yes—
86C3×C22⋊Dic5—22no—
87C3×C22⋊Dic5—22no—
88C3×C4⋊Dic5—21no—
89C3×C4⋊Dic5—21no—
90C3×C4×Dic5D411yescharacteristic
91C3×C4×Dic5D411yescharacteristic
92C3×C4×Dic5C2311yescharacteristic
93#6669—21no—
94(C2×D3)⋊Dic5—21no—
95#6665—21no—
96Dic3×Dic5—21no—
97Dic3×Dic5—21no—
98#6662—24no—
99#6656—24no—
100D5×C3⋊C8—24no—
101C3×#532—51no—
102C2×#950—52no—
103C4×#235—51no—
104#5233—52no—
105#5090—51no—
106#5083—51no—
107#5075—51no—
108#5072D511yescharacteristic, complemented
109#5057—52no—
110#5054—52no—
111C2×C4⋊(C3⋊C8)—51no—
112C2×#711—31no—
113C2×#711C2×D312yes—
114C2×C5×D8—32no—
115C5×#127C2×D311yescharacteristic
116C4×C5×D4—31no—
117#4084—21no—
118#4084—21no—
119#4082C2×D312yes—
120D4×Dic5—21no—
121D4×Dic5—21no—
122D4×Dic5—31no—
123C2×#687—32no—
124C2×#687—32no—
125C4×C22⋊D5—61no—
126C2×C4×Dic5—61no—
127#4058—68no—
128C2×Dic20—32no—
129C2×C8×D5—32no—
130#4038—31no—
131#4033—31no—
132#4027—61no—
133#4023—61no—
134C42×D5C2×D311yescharacteristic
135#4015C2×D311yescharacteristic
136C4×Dic10—31no—
137C5×C4⋊C8—31no—
138C5×#102—32no—
139#3964—32no—
140#3948—32no—
141#3946—32no—
142#3942—32no—
143C4⋊#168—31no—
144#3035—151noSylow
145C23×C3×C5—22no—
146C22×C5×D3—21no—
147C22×C3×D5C2×D411yescharacteristic
148C3×C5×D4—22no—
149C3×C5×D4—22no—
150C3×C5×D4—42no—
151C3×C5×D4—42no—
152C2×C3×C4×C5C2×D411yescharacteristic
153C2×C3×C4×C5C2×D411yescharacteristic
154C2×C3×C4×C5C2411yescharacteristic
155C2×Dic15—21no—
156C2×Dic15—21no—
157Dic30—21no—
158Dic30C2×D412yes—
159C2×C5×Dic3—21no—
160C5×D12—21no—
161C5×D12C2×D412yes—
162C4×C5×D3—42no—
163C3×C22⋊D5—44no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—44no—
166C2×C3×Dic5C2×D411yescharacteristic
167C2×C3×Dic5C2×D411yescharacteristic
168C2×C3×Dic5C2×D411yescharacteristic
169C3×C4×D5—22no—
170C3×C4×D5—41no—
171C3×C4×D5C2×D412yes—
172C3×C4×D5#4112yes—
173C3×Dic10—22no—
174C3×Dic10—22no—
175#1201—24no—
176D3×Dic5—24no—
177D3×Dic5—42no—
178C3⋊#168—22no—
179C5×C3⋊C8—22no—
180C2×C3×#41—52no—
181C2×#235—51no—
182C3×C4⋊Q8—51no—
183C3×#127C2×D511yescharacteristic
184C3×#124—52no—
185C3×#121—101no—
186C3×#121—101no—
187C3×C4×D4—101no—
188C3×C4×D4—101no—
189C2×C3×C42—51no—
190#950—108no—
191C2×#216—52no—
192C2×#215—52no—
193C2×#214—52no—
194C2×#213C2×D512yes—
195C4×#67—101no—
196C4×#67—101no—
197#926—51no—
198C22×C3⋊C8—52no—
199C4×D12—51no—
200C4×D12C2×D511yescharacteristic
201#872—101no—
202C42×D3—101no—
203C4×Dic6—51no—
204C4×Dic6—51no—
205#835—52no—
206#835—52no—
207#832—52no—
208#832C2×D512yes—
209C4⋊(C3⋊C8)—51no—
210C4⋊(C3⋊C8)—52no—
211C4⋊(C3⋊C8)C2×D511yescharacteristic
212C2×C5×D4—21no—
213C2×C5×D4—21no—
214C2×C5×D4—31no—
215C2×C5×D4C22×D312yes—
216C22×C4×C5—61no—
217C2×C22⋊D5—22no—
218C2×C22⋊D5—61no—
219C22×Dic5—22no—
220C22×Dic5—61no—
221C22×Dic5—61no—
222#711—24no—
223#711—24no—
224#711—34no—
225#711—62no—
226C2×C4×D5#6711yescharacteristic
227C2×C4×D5#6711yescharacteristic
228C2×C4×D5C22×D311yescharacteristic
229C2×Dic10—31no—
230C2×Dic10C22×D312yes—
231C5×D8—64no—
232C2×C5×C8—32no—
233C5×C4⋊C4—31no—
234C5×C22⋊C4—61no—
235C42×C5C22×D311yescharacteristic
236C22⋊Dic5—22no—
237C22⋊Dic5—22no—
238C22⋊Dic5—61no—
239#687—64no—
240#687—64no—
241#685—61no—
242C4⋊Dic5—21no—
243C4⋊Dic5—21no—
244C4⋊Dic5—31no—
245#683—61no—
246#683—61no—
247C4×Dic5—61no—
248C4×Dic5—61no—
249C4×Dic5—61no—
250C4×Dic5#6711yescharacteristic
251C4×Dic5#6711yescharacteristic
252C4×Dic5C22×D311yescharacteristic
253C2×#168—32no—
254Dic20—64no—
255C8×D5—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—31no—
265C2×C4⋊C8—151no—
266#416—152no—
267C22×C3×C5—42no—
268C22×C3×C5—42no—
269C22×C3×C5—42no—
270C22×C3×C5C22×D411yescharacteristic
271C2×C5×D3—22no—
272C2×C5×D3—41no—
273C2×C5×D3—41no—
274C2×C3×D5—21no—
275C2×C3×D5—21no—
276C2×C3×D5#12112yes—
277C3×C4×C5—21no—
278C3×C4×C5#13512yescomplemented
279C3×C4×C5C22×D411yescharacteristic
280C3×C4×C5C2×#4111yescharacteristic
281Dic15—22no—
282Dic15—41no—
283C3×Dic5—21no—
284C3×Dic5—21no—
285C3×Dic5—42no—
286C3×Dic5#12112yes—
287C5×Dic3—41no—
288C3×#41—104no—
289C3×#41—104no—
290C2×C3×Q8—52no—
291C2×C3×D4—21no—
292C2×C3×D4—21no—
293C2×C3×D4—102no—
294C2×C3×D4C22×D512yes—
295C22×C3×C4—51no—
296C22×C3×C4—51no—
297C22×C3×C4—51no—
298C22×C3×C4—102no—
299C2×#67—101no—
300#235—54no—
301#235—102no—
302C2×D12C22×D511yescharacteristic
303C2×C4×D3—21no—
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×C42—51no—
312C3×C42—51no—
313C3×C42—51no—
314C3×C42C22×D511yescharacteristic
315#217—101no—
316#216—104no—
317#215—104no—
318#214—104no—
319#213—24no—
320(C2×D3)⋊C4—21no—
321(C2×D3)⋊C4—101no—
322C4⋊Dic3—51no—
323C4⋊Dic3C22×D511yescharacteristic
324Dic3⋊C4—101no—
325Dic3⋊C4—101no—
326C4×Dic3—101no—
327C4×Dic3—101no—
328C4×Dic3—101no—
329C2×C3⋊C8—52no—
330C2×C3⋊C8—104no—
331C2×C3⋊C8C22×D512yes—
332C23×C5—22no—
333C23×C5—61no—
334C22×D5C2×#6711yescharacteristic
335C5×D4—22no—
336C5×D4—22no—
337C5×D4—32no—
338C5×D4—42no—
339C5×D4—42no—
340C5×D4—61no—
341C2×C4×C5—61no—
342C2×C4×C5—122no—
343C2×C4×C5C2×#6711yescharacteristic
344C2×C4×C5C2×#6711yescharacteristic
345C2×C4×C5C23×D311yescharacteristic
346C22⋊D5—44no—
347C22⋊D5—64no—
348C2×Dic5—22no—
349C2×Dic5—44no—
350C2×Dic5—61no—
351C2×Dic5—64no—
352C2×Dic5—61no—
353C2×Dic5—122no—
354C2×Dic5C2×#6711yescharacteristic
355C2×Dic5C2×#6711yescharacteristic
356C2×Dic5C2×#6711yescharacteristic
357C4×D5—22no—
358C4×D5—41no—
359C4×D5D3×D412yes—
360C4×D5#23712yes—
361Dic10—22no—
362Dic10—22no—
363Dic10—32no—
364Dic10—61no—
365C5×C8—62no—
366#168—62no—
367C2×#41—52no—
368C2×#41—151no—
369#135—308no—
370C2×Q16—152no—
371C2×SD16—152no—
372C2×SD16—152no—
373C2×D8—32no—
374C22×C8—152no—
375C4⋊Q8—51no—
376#127D3×D511yescharacteristic
377#124—52no—
378#121—101no—
379#121—101no—
380C4×Q8—151no—
381C4×Q8—151no—
382C4×D4—31no—
383C4×D4—101no—
384C4×D4—101no—
385C4×D4—151no—
386C4×D4—301no—
387C4×D4—301no—
388#117—151no—
389#117—301no—
390C2×C42—51no—
391C2×C42—301no—
392C4⋊C8—31no—
393C4⋊C8—152no—
394C4⋊C8—151no—
395#103—152no—
396#103—152no—
397#102—32no—
398#102—152no—
399C2×C3×C5—82no—
400C2×C3×C5C2×#12111yescharacteristic
401C2×C3×C5C2×#13511yescharacteristic
402C2×C3×C5#57511yescharacteristic
403C3×D5—22no—
404C5×D3—42no—
405C23×C3—22no—
406C23×C3—51no—
407C22×D3—21no—
408C3×Q8—102no—
409C3×Q8—102no—
410C3×D4—22no—
411C3×D4—22no—
412C3×D4—42no—
413C3×D4—42no—
414C3×D4—204no—
415C2×C3×C4—51no—
416C2×C3×C4—52no—
417C2×C3×C4—52no—
418C2×C3×C4—51no—
419C2×C3×C4—51no—
420C2×C3×C4—102no—
421C2×C3×C4—102no—
422C2×C3×C4—201no—
423C2×C3×C4—204no—
424C2×C3×C4D4×D511yescharacteristic
425C2×C3×C4D4×D511yescharacteristic
426C2×C3×C4C23×D511yescharacteristic
427#67—104no—
428C2×Dic3—21no—
429C2×Dic3—101no—
430C2×Dic3—101no—
431D12—21no—
432D12D4×D512yes—
433C4×D3—42no—
434C4×D3—104no—
435C4×D3—202no—
436Dic6—52no—
437Dic6—101no—
438C3⋊C8—22no—
439C3⋊C8—102no—
440C22×C5—42no—
441C22×C5—42no—
442C22×C5—42no—
443C22×C5—62no—
444C22×C5—121no—
445C22×C5—121no—
446C22×C5C22×#6711yescharacteristic
447C2×D5—21no—
448C2×D5—21no—
449C2×D5#93812yes—
450C4×C5—21no—
451C4×C5—121no—
452C4×C5#95012yescomplemented
453C4×C5C2×D3×D411yescharacteristic
454C4×C5C2×#23711yescharacteristic
455Dic5—21no—
456Dic5—21no—
457Dic5—42no—
458Dic5—62no—
459Dic5—121no—
460Dic5#93812yes—
461#41—104no—
462#41—104no—
463#41—154no—
464#41—302no—
465C2×Q8—52no—
466C2×Q8—151no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—31no—
470C2×D4—102no—
471C2×D4—301no—
472C2×D4C2×D3×D512yes—
473C22×C4—51no—
474C22×C4—51no—
475C22×C4—51no—
476C22×C4—61no—
477C22×C4—102no—
478C22×C4—301no—
479C22×C4—301no—
480Q16—304no—
481SD16—304no—
482SD16—304no—
483D8—64no—
484C2×C8—32no—
485C2×C8—152no—
486C2×C8—304no—
487C4⋊C4—31no—
488C4⋊C4—101no—
489C4⋊C4—101no—
490C4⋊C4—151no—
491C4⋊C4—301no—
492C4⋊C4—301no—
493C22⋊C4—61no—
494C22⋊C4—102no—
495C22⋊C4—102no—
496C22⋊C4—301no—
497C22⋊C4—301no—
498C42—51no—
499C42—51no—
500C42—51no—
501C42—301no—
502C42—301no—
503C42—301no—
504C42C2×D3×D511yescharacteristic
505C3×C5#303511yescharacteristic, complemented, Hall
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—52no—
510C22×C3—101no—
511C22×C3—101no—
512C22×C3C2×D4×D511yescharacteristic
513C2×D3—22no—
514C2×D3—41no—
515C2×D3—41no—
516C3×C4—21no—
517C3×C4—52no—
518C3×C4—101no—
519C3×C4—101no—
520C3×C4—202no—
521C3×C4#405812yescomplemented
522C3×C4C2×D4×D511yescharacteristic
523C3×C4D5×#4111yescharacteristic
524Dic3—41no—
525Dic3—102no—
526Dic3—201no—
527C2×C5—82no—
528C2×C5—122no—
529C2×C5C2×#93811yescharacteristic
530C2×C5C2×#95011yescharacteristic
531C2×C5#582211yescharacteristic
532D5—22no—
533C23—22no—
534C23—51no—
535C23—61no—
536Q8—102no—
537Q8—102no—
538Q8—152no—
539Q8—301no—
540D4—22no—
541D4—22no—
542D4—32no—
543D4—42no—
544D4—42no—
545D4—61no—
546D4—204no—
547D4—304no—
548C2×C4—51no—
549C2×C4—52no—
550C2×C4—52no—
551C2×C4—51no—
552C2×C4—51no—
553C2×C4—61no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—122no—
557C2×C4—201no—
558C2×C4—204no—
559C2×C4—304no—
560C2×C4—301no—
561C2×C4—301no—
562C2×C4—602no—
563C2×C4D5×#6711yescharacteristic
564C2×C4D5×#6711yescharacteristic
565C2×C4C22×D3×D511yescharacteristic
566C8—62no—
567C8—302no—
568C2×C3—82no—
569C2×C3—102no—
570C2×C3D5×#12111yescharacteristic
571C2×C3C2×#405811yescharacteristic
572C2×C3#6610711yescharacteristic
573D3—42no—
574C5#8329611yescharacteristic, complemented, Sylow
575C22—42no—
576C22—42no—
577C22—42no—
578C22—52no—
579C22—62no—
580C22—101no—
581C22—101no—
582C22—121no—
583C22—121no—
584C22C2×D5×#6711yescharacteristic
585C4—21no—
586C4—52no—
587C4—101no—
588C4—101no—
589C4—121no—
590C4—202no—
591C4—302no—
592C4—601no—
593C4#9149012yescomplemented
594C4D3×D4×D511yescharacteristic
595C4D5×#23711yescharacteristic
596C3#1061206211yescharacteristic, complemented, Sylow
597C2—82no—
598C2—102no—
599C2—122no—
600C2#1175469011yescharacteristic
601C2C2×#9149011yescharacteristic
602C2D5×#93811yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#127C22
socleC22×C3×C5C22×D4
hypercenter#127D3×D5
hypocenterC3×C5#3035
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C3×#66050C2semidirect
2. C2×#91490C2semidirect
3. C4×#6772C2semidirect
4. #11751176C2semidirect
5. C5×#5072C2semidirect
6. #11749002C2semidirect
7. #11748978C2semidirect
8. #11748961C2semidirect
9. #11747104C2semidirect
10. #11747070C2semidirect
11. D5×C4⋊(C3⋊C8)C2semidirect
47. #66050D3semidirect
75. #6772D4semidirect
108. #5072D5semidirect
278. C3×C4×C5#135semidirect
452. C4×C5#950semidirect
505. C3×C5#3035semidirect
521. C3×C4#4058semidirect
574. C5#83296semidirect
593. C4#91490semidirect
596. C3#10612062semidirect

Maximal subgroups

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

subgroupindex
C3×#660502
C2×#914902
C4×#67722
#117511762
C5×#50722
#117490022
#117489782
#117489612
#117471042
#117470702
D5×C4⋊(C3⋊C8)2
#106120623
#832965