Finite Group Databasebeta

← #14430984

Subgroups — #14430984

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5572C211yesmaximal, characteristic
2#11752661C211yesmaximal, characteristic
3#11751121C211yesmaximal, characteristic, complemented
4C3×#65311C211yesmaximal, characteristic, complemented
5#11749359C211yesmaximal, characteristic, complemented
6#11749334C211yesmaximal, characteristic, complemented
7#11749213C211yesmaximal, characteristic
8D3×#3967C211yesmaximal, characteristic
9#11747435C211yesmaximal, characteristic
10#11747417C211yesmaximal, characteristic, complemented
11#11747393C211yesmaximal, characteristic, complemented
12#11747191C211yesmaximal, characteristic
13#11746993C211yesmaximal, characteristic
14#11746961C211yesmaximal, characteristic, complemented
15#11746926C211yesmaximal, characteristic, complemented
16#10614281—31nomaximal, Hall
17C2×C5×D3×Q8C2211yescharacteristic
18C3×C4×C5×Q8C2211yescharacteristic
19#91758C2211yescharacteristic
20C5×Q8×Dic3C2211yescharacteristic
21C5×#892—21no—
22C5×#892C2211yescharacteristic
23C5×#887—21no—
24C5×#887C2211yescharacteristic
25C5×#872—21no—
26C5×#872C2211yescharacteristic
27C4×C5×Dic6—21no—
28C4×C5×Dic6C2211yescharacteristic
29C3×#4016C2211yescharacteristic
30#91469C2211yescharacteristic
31#91432—21no—
32D3×C4⋊Dic5C2211yescharacteristic
33#91417—21no—
34#91381C2211yescharacteristic
35#91343—21no—
36#91338C2211yescharacteristic
37#91328—21no—
38C2×#6650C2211yescharacteristic
39C2×D3×#168C2211yescharacteristic
40#91119C2211yescharacteristic
41#91119C2211yescharacteristic
42#91092C2211yescharacteristic
43#91092C2211yescharacteristic
44C4⋊(C3⋊#168)C2211yescharacteristic
45C3×#3967C2211yescharacteristic
46C3×#3967C2211yescharacteristic
47C3×#3940C2211yescharacteristic
48C3×#3939C2211yescharacteristic
49C3×C4⋊#168C2211yescharacteristic
50#90989C2211yescharacteristic
51#90988C2211yescharacteristic
52#90988C2211yescharacteristic
53#90986C2211yescharacteristic
54Dic3⋊#168C2211yescharacteristic
55Dic3⋊#168C2211yescharacteristic
56#90973C2211yescharacteristic
57#90973C2211yescharacteristic
58#90972C2211yescharacteristic
59#90972C2211yescharacteristic
60#81232—51nomaximal, Hall
61C5×#519—31no—
62#65853—31no—
63#65514—31no—
64C2×#3967—31no—
65#65311—31no—
66#65311—31no—
67#65311—31no—
68#65311D311yescharacteristic, complemented
69#65310—31no—
70#65310—31no—
71#65309—31no—
72#65309—31no—
73#65288—31no—
74#65287—31no—
75#65284—31no—
76C2×C3×C5×Q8C2311yescharacteristic
77C2×Dic30—21no—
78C5×D3×Q8—22no—
79C5×D3×Q8Q814yescomplemented
80C2×C4×C5×D3—21no—
81C2×C4×C5×D3C2311yescharacteristic
82C2×C5×Dic6—21no—
83C2×C5×Dic6C2311yescharacteristic
84C2×#1204—21no—
85C2×D3×Dic5—21no—
86C3×C5×C4⋊C4—21no—
87C3×C5×C4⋊C4C2311yescharacteristic
88C3×C42×C5—21no—
89C3×C42×C5C2311yescharacteristic
90C4⋊Dic15—21no—
91C4⋊Dic15C2311yescharacteristic
92C4⋊Dic15C2311yescharacteristic
93C2×C3⋊#168C2311yescharacteristic
94C2×C3⋊#168C2311yescharacteristic
95C5×(C2×D3)⋊C4—21no—
96C5×(C2×D3)⋊C4D411yescharacteristic
97C5×(C2×D3)⋊C4D411yescharacteristic
98C5×C4⋊Dic3—21no—
99C5×C4⋊Dic3C2311yescharacteristic
100C5×Dic3⋊C4—21no—
101C5×Dic3⋊C4—21no—
102C5×Dic3⋊C4D411yescharacteristic
103C5×Dic3⋊C4D411yescharacteristic
104C4×C5×Dic3—21no—
105C4×C5×Dic3—21no—
106C4×C5×Dic3—21no—
107C4×C5×Dic3—21no—
108C4×C5×Dic3C2311yescharacteristic
109C3×C4⋊Dic5C2311yescharacteristic
110C3×C4⋊Dic5C2311yescharacteristic
111C3×#683—21no—
112C3×#683—21no—
113C2×C3×#168C2311yescharacteristic
114C2×C3×#168C2311yescharacteristic
115Dic3⋊Dic5—21no—
116Dic3⋊Dic5—21no—
117#6669—21no—
118#6668—21no—
119(C2×D3)⋊Dic5—21no—
120Dic3×Dic5—21no—
121#6650—22no—
122D3×#168—22no—
123#5689—51no—
124#5572D511yescharacteristic
125C3×#479—51no—
126#5063—51no—
127#5020—51no—
128#4880—51no—
129#4877—51no—
130#4861—51no—
131#4854—51no—
132#4850—51no—
133#4839—51no—
134#4802—51no—
135D3×#103—51no—
136#4796—51no—
137#4793—51no—
138C22×C5×Q8—31no—
139C4×C5×Q8—31no—
140C4×C5×Q8—31no—
141C4×C5×Q8—31no—
142C4×C5×Q8—61no—
143C4×C5×Q8—61no—
144C4×C5×Q8C2×D311yescharacteristic
145C5×#117—31no—
146C5×#117—31no—
147C5×#117—61no—
148C5×#117—61no—
149#4072—31no—
150C2×C4⋊Dic5—31no—
151C2×#681—31no—
152C22×#168—31no—
153#4036—61no—
154#4034—61no—
155#4024—61no—
156#4024—61no—
157#4016—31no—
158#4016C2×D311yescharacteristic
159#4015—31no—
160#4015—31no—
161#3967—31no—
162#3967—31no—
163#3967—31no—
164#3967—31no—
165#3967—31no—
166#3967—31no—
167#3967C2×D311yescharacteristic
168#3967C2×D311yescharacteristic
169#3940—31no—
170#3940—31no—
171#3940—31no—
172#3940C2×D311yescharacteristic
173#3939—31no—
174#3939—31no—
175#3939—31no—
176#3939C2×D311yescharacteristic
177C4⋊#168—31no—
178C4⋊#168—31no—
179C4⋊#168—31no—
180C4⋊#168C2×D311yescharacteristic
181#3082—151noSylow
182C22×C5×D3C2×D411yescharacteristic
183C3×C5×Q8—21no—
184C3×C5×Q8C2×Q812yes—
185C2×C3×C4×C5—21no—
186C2×C3×C4×C5—21no—
187C2×C3×C4×C5C2×D411yescharacteristic
188C2×C3×C4×C5C2×D411yescharacteristic
189C2×C3×C4×C5C2411yescharacteristic
190C2×Dic15—21no—
191C2×Dic15—21no—
192Dic30—81no—
193C2×C5×Dic3—21no—
194C2×C5×Dic3—21no—
195C2×C5×Dic3C2×D411yescharacteristic
196C2×C5×Dic3C2×D411yescharacteristic
197C2×C5×Dic3C2×D411yescharacteristic
198C4×C5×D3—24no—
199C4×C5×D3C2×Q812yes—
200C4×C5×D3#4112yes—
201C5×Dic6—21no—
202C5×Dic6—24no—
203C5×Dic6C2×Q812yes—
204C2×C3×Dic5—21no—
205C2×C3×Dic5—21no—
206#1204—81no—
207D3×Dic5—81no—
208C3⋊#168—21no—
209C3⋊#168—21no—
210C3×#168—21no—
211C3×#168—21no—
212C2×D3×Q8C2×D511yescharacteristic
213C3×#125—51no—
214C3×C4×Q8C2×D511yescharacteristic
215Q8×Dic3C2×D511yescharacteristic
216C2×ΓL(1,𝔽25)—51no—
217C2×D3×C8—51no—
218#897—101no—
219#896—101no—
220#892—21no—
221#892—51no—
222#892C2×D511yescharacteristic
223D3×C4⋊C4—51no—
224#890—51no—
225#889—101no—
226#888—51no—
227#888—101no—
228#887—21no—
229#887C2×D511yescharacteristic
230#872—21no—
231#872C2×D511yescharacteristic
232#869—51no—
233C4×Dic6—21no—
234C4×Dic6C2×D511yescharacteristic
235C3×#107—51no—
236C3×#106—51no—
237C3×C4⋊C8—51no—
238C3×#103—51no—
239C3×#103—51no—
240#835—51no—
241#835—51no—
242#818—51no—
243#818—51no—
244#816—51no—
245#816—51no—
246Dic3⋊C8—51no—
247Dic3⋊C8—51no—
248#810—51no—
249#810—51no—
250#808—51no—
251#808—51no—
252#808—51no—
253#807—51no—
254C4⋊(C3⋊C8)—51no—
255C2×C5×Q8—34no—
256C2×C5×Q8—31no—
257C2×C5×Q8—61no—
258C2×C5×Q8—62no—
259C2×C5×Q8C22×D311yescharacteristic
260C22×C4×C5—31no—
261C22×C4×C5—61no—
262C22×Dic5—61no—
263C2×Dic10—61no—
264C2×Dic10—61no—
265C5×C4⋊C4—21no—
266C5×C4⋊C4—31no—
267C5×C4⋊C4—31no—
268C5×C4⋊C4—31no—
269C5×C4⋊C4—61no—
270C5×C4⋊C4—61no—
271C5×C4⋊C4—61no—
272C5×C4⋊C4C22×D311yescharacteristic
273C5×C22⋊C4—31no—
274C5×C22⋊C4—31no—
275C5×C22⋊C4—61no—
276C42×C5—21no—
277C42×C5—31no—
278C42×C5—61no—
279C42×C5—61no—
280C42×C5—61no—
281C42×C5—61no—
282C42×C5C22×D311yescharacteristic
283C22⋊Dic5—61no—
284C4⋊Dic5—31no—
285C4⋊Dic5—31no—
286C4⋊Dic5—61no—
287C4⋊Dic5—61no—
288C4⋊Dic5—61no—
289C4⋊Dic5C22×D311yescharacteristic
290C4⋊Dic5C22×D311yescharacteristic
291#683—21no—
292#683—21no—
293#683—61no—
294#683—61no—
295C4×Dic5—61no—
296#681—62no—
297C2×#168—31no—
298C2×#168—31no—
299C2×#168—62no—
300C2×#168C22×D311yescharacteristic
301C2×#168C22×D311yescharacteristic
302#544—151no—
303#519—31no—
304#479—51no—
305#479—151no—
306#479—151no—
307#479—151no—
308#477—151no—
309#477—151no—
310#475—151no—
311#475—151no—
312#428—151no—
313#427—151no—
314#424—151no—
315#419—151no—
316C2×#103—151no—
317C22×C3×C5C22×D411yescharacteristic
318C2×C5×D3—21no—
319C2×C5×D3—21no—
320C2×C5×D3C22⋊Q812yes—
321C3×C4×C5—22no—
322C3×C4×C5—21no—
323C3×C4×C5—21no—
324C3×C4×C5—41no—
325C3×C4×C5C22×Q811yescharacteristic
326C3×C4×C5C2×#4111yescharacteristic
327Dic15—41no—
328Dic15—41no—
329C3×Dic5—41no—
330C3×Dic5—41no—
331C5×Dic3—22no—
332C5×Dic3—21no—
333C5×Dic3—21no—
334C5×Dic3—41no—
335C5×Dic3C22⋊Q812yes—
336C2×C3×Q8C22×D511yescharacteristic
337D3×Q8—22no—
338D3×Q8Dic1014yescomplemented
339C2×C4×D3—21no—
340C2×C4×D3—101no—
341C2×C4×D3C22×D511yescharacteristic
342C2×Dic6—21no—
343C2×Dic6—101no—
344C2×Dic6—101no—
345C2×Dic6C22×D511yescharacteristic
346C2×C3×C8—51no—
347C2×C3×C8—51no—
348C3×C4⋊C4—21no—
349C3×C4⋊C4—51no—
350C3×C4⋊C4—51no—
351C3×C4⋊C4—101no—
352C3×C4⋊C4—101no—
353C3×C4⋊C4C22×D511yescharacteristic
354C3×C42—21no—
355C3×C42C22×D511yescharacteristic
356(C2×D3)⋊C4—21no—
357(C2×D3)⋊C4—101no—
358(C2×D3)⋊C4C22⋊D511yescharacteristic
359(C2×D3)⋊C4C22⋊D511yescharacteristic
360C4⋊Dic3—21no—
361C4⋊Dic3—51no—
362C4⋊Dic3—51no—
363C4⋊Dic3—101no—
364C4⋊Dic3—101no—
365C4⋊Dic3C22×D511yescharacteristic
366Dic3⋊C4—21no—
367Dic3⋊C4—21no—
368Dic3⋊C4—101no—
369Dic3⋊C4—101no—
370Dic3⋊C4—101no—
371Dic3⋊C4C22⋊D511yescharacteristic
372Dic3⋊C4C22⋊D511yescharacteristic
373C4×Dic3—21no—
374C4×Dic3—21no—
375C4×Dic3—21no—
376C4×Dic3—21no—
377C4×Dic3—101no—
378C4×Dic3C22×D511yescharacteristic
379C2×C3⋊C8—51no—
380C2×C3⋊C8—51no—
381ΓL(1,𝔽25)—102no—
382D3×C8—102no—
383C23×C5—31no—
384C5×Q8—21no—
385C5×Q8—32no—
386C5×Q8—64no—
387C5×Q8—61no—
388C5×Q8D3×Q812yes—
389C2×C4×C5—21no—
390C2×C4×C5—21no—
391C2×C4×C5—31no—
392C2×C4×C5—32no—
393C2×C4×C5—32no—
394C2×C4×C5—31no—
395C2×C4×C5—31no—
396C2×C4×C5—64no—
397C2×C4×C5—61no—
398C2×C4×C5—61no—
399C2×C4×C5D3×D411yescharacteristic
400C2×C4×C5D3×D411yescharacteristic
401C2×C4×C5C23×D311yescharacteristic
402C2×Dic5—21no—
403C2×Dic5—21no—
404C2×Dic5—61no—
405C2×Dic5—61no—
406C2×Dic5—241no—
407Dic10—241no—
408Dic10—241no—
409#168—21no—
410#168—21no—
411#168—61no—
412#168—61no—
413C22×Q8—31no—
414C2×M16—151no—
415C22×C8—151no—
416C4⋊Q8—151no—
417C4⋊Q8—151no—
418C4⋊Q8—301no—
419#125—51no—
420#125—151no—
421#125—301no—
422C22⋊Q8—301no—
423C22⋊Q8—301no—
424C4×Q8—31no—
425C4×Q8—31no—
426C4×Q8—31no—
427C4×Q8—61no—
428C4×Q8—61no—
429C4×Q8D3×D511yescharacteristic
430#117—31no—
431#117—31no—
432#117—61no—
433#117—61no—
434#117—151no—
435C2×C4⋊C4—151no—
436#107—51no—
437#107—151no—
438#107—151no—
439#107—151no—
440#106—51no—
441#106—151no—
442#106—151no—
443#106—151no—
444C4⋊C8—51no—
445C4⋊C8—151no—
446C4⋊C8—151no—
447C4⋊C8—151no—
448#103—51no—
449#103—51no—
450#103—151no—
451#103—151no—
452#103—151no—
453#103—151no—
454#103—151no—
455#103—151no—
456C2×C3×C5C2×C22⋊Q811yescharacteristic
457C2×C3×C5#57711yescharacteristic
458C2×C3×C5#57811yescharacteristic
459C5×D3—22no—
460C22×D3C2×C22⋊D511yescharacteristic
461C3×Q8—21no—
462C3×Q8C2×Dic1012yes—
463C2×C3×C4—21no—
464C2×C3×C4—21no—
465C2×C3×C4—101no—
466C2×C3×C4—101no—
467C2×C3×C4C2×C22⋊D511yescharacteristic
468C2×C3×C4C2×C22⋊D511yescharacteristic
469C2×C3×C4C23×D511yescharacteristic
470C2×Dic3—21no—
471C2×Dic3—21no—
472C2×Dic3—101no—
473C2×Dic3—101no—
474C2×Dic3C2×C22⋊D511yescharacteristic
475C2×Dic3C2×C22⋊D511yescharacteristic
476C2×Dic3C2×C22⋊D511yescharacteristic
477C4×D3—24no—
478C4×D3—401no—
479C4×D3C2×Dic1012yes—
480C4×D3#70912yes—
481Dic6—21no—
482Dic6—24no—
483Dic6—401no—
484Dic6—401no—
485Dic6C2×Dic1012yes—
486C3×C8—101no—
487C3×C8—101no—
488C3⋊C8—101no—
489C3⋊C8—101no—
490C22×C5—32no—
491C22×C5—61no—
492C22×C5—61no—
493C22×C5C2×D3×D411yescharacteristic
494C4×C5—21no—
495C4×C5—21no—
496C4×C5—22no—
497C4×C5—32no—
498C4×C5—41no—
499C4×C5—61no—
500C4×C5—61no—
501C4×C5—62no—
502C4×C5—121no—
503C4×C5C2×D3×Q811yescharacteristic
504C4×C5D3×#4111yescharacteristic
505Dic5—41no—
506Dic5—41no—
507Dic5—121no—
508Dic5—121no—
509C2×Q8—34no—
510C2×Q8—31no—
511C2×Q8—61no—
512C2×Q8—62no—
513C2×Q8—301no—
514C2×Q8—301no—
515C2×Q8C2×D3×D511yescharacteristic
516C22×C4—31no—
517C22×C4—61no—
518C22×C4—301no—
519M16—302no—
520C2×C8—51no—
521C2×C8—51no—
522C2×C8—151no—
523C2×C8—151no—
524C2×C8—302no—
525C4⋊C4—21no—
526C4⋊C4—31no—
527C4⋊C4—31no—
528C4⋊C4—31no—
529C4⋊C4—51no—
530C4⋊C4—51no—
531C4⋊C4—61no—
532C4⋊C4—61no—
533C4⋊C4—61no—
534C4⋊C4—101no—
535C4⋊C4—101no—
536C4⋊C4—151no—
537C4⋊C4—151no—
538C4⋊C4—301no—
539C4⋊C4—301no—
540C4⋊C4—301no—
541C4⋊C4—301no—
542C4⋊C4—301no—
543C4⋊C4C2×D3×D511yescharacteristic
544C22⋊C4—31no—
545C22⋊C4—31no—
546C22⋊C4—61no—
547C22⋊C4—301no—
548C42—21no—
549C42—31no—
550C42—61no—
551C42—61no—
552C42—61no—
553C42—61no—
554C42—301no—
555C42C2×D3×D511yescharacteristic
556C3×C5#308211yescharacteristic, complemented, Hall
557C22×C3C22×C22⋊D511yescharacteristic
558C2×D3—21no—
559C2×D3—21no—
560C2×D3#407012yes—
561C3×C4—22no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—41no—
565C3×C4—201no—
566C3×C4—201no—
567C3×C4C22×Dic1011yescharacteristic
568C3×C4C2×#70911yescharacteristic
569Dic3—22no—
570Dic3—21no—
571Dic3—21no—
572Dic3—41no—
573Dic3—201no—
574Dic3—201no—
575Dic3#407012yes—
576C2×C5—62no—
577C2×C5D3×C22⋊Q811yescharacteristic
578C2×C5#577111yescharacteristic
579C2×C5#577811yescharacteristic
580C23—31no—
581Q8—21no—
582Q8—32no—
583Q8—64no—
584Q8—61no—
585Q8—1201no—
586Q8—1201no—
587Q8D3×Dic1012yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—31no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—31no—
594C2×C4—31no—
595C2×C4—64no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—101no—
599C2×C4—101no—
600C2×C4—301no—
601C2×C4—301no—
602C2×C4—1201no—
603C2×C4D3×C22⋊D511yescharacteristic
604C2×C4D3×C22⋊D511yescharacteristic
605C2×C4C22×D3×D511yescharacteristic
606C8—101no—
607C8—101no—
608C8—301no—
609C8—301no—
610C2×C3C2×#407011yescharacteristic
611C2×C3#6617111yescharacteristic
612C2×C3#6617211yescharacteristic
613D3—22no—
614C5#8123211yescharacteristic, complemented, Sylow
615C22—32no—
616C22—61no—
617C22—61no—
618C22C2×D3×C22⋊D511yescharacteristic
619C4—21no—
620C4—21no—
621C4—22no—
622C4—32no—
623C4—41no—
624C4—61no—
625C4—61no—
626C4—62no—
627C4—121no—
628C4—201no—
629C4—201no—
630C4—601no—
631C4—601no—
632C4C2×D3×Dic1011yescharacteristic
633C4D3×#70911yescharacteristic
634C3#1061428111yescharacteristic, complemented, Sylow
635C2—62no—
636C2D3×#407011yescharacteristic
637C2#1175509211yescharacteristic
638C2#1175534611yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radicalC22×C5×D3C2×D4
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3082
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #11751121C2semidirect
4. C3×#65311C2semidirect
5. #11749359C2semidirect
6. #11749334C2semidirect
10. #11747417C2semidirect
11. #11747393C2semidirect
14. #11746961C2semidirect
15. #11746926C2semidirect
68. #65311D3semidirect
79. C5×D3×Q8Q8semidirect
338. D3×Q8Dic10semidirect
556. C3×C5#3082semidirect
614. C5#81232semidirect
634. C3#10614281semidirect

Maximal subgroups

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

subgroupindex
C5×#55722
#117526612
#117511212
C3×#653112
#117493592
#117493342
#117492132
D3×#39672
#117474352
#117474172
#117473932
#117471912
#117469932
#117469612
#117469262
#106142813
#812325