Finite Group Databasebeta

← #14415459

Subgroups — #14415459

Order 1920.

1004 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
1#11756587C211yesmaximal, characteristic
2C2×#91837C211yesmaximal, characteristic, complemented
3C2×M16×D15C211yesmaximal, characteristic, complemented
4C3×C5×#408C211yesmaximal, characteristic, complemented
5#11751220C211yesmaximal, characteristic, complemented
6C2×#91104C211yesmaximal, characteristic, complemented
7#11751202C211yesmaximal, characteristic, complemented
8#11750831C212yesmaximal, complemented
9#11750830C212yesmaximal, complemented
10#11750829C212yesmaximal, complemented
11#11750826C212yesmaximal, complemented
12#10610390—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×C4×D15C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C22×C3×C5×C8C2211yescharacteristic
17#91837—24no—
18#91835C411yescharacteristic
19#91835C411yescharacteristic
20#91827C411yescharacteristic
21#91827C411yescharacteristic
22#91826C412yes—
23#91824C412yes—
24C4×#1220C2212yes—
25C4×#1220C2212yes—
26#91815C2211yescharacteristic
27C2×#6710C2211yescharacteristic
28C2×#6710C2212yescomplemented
29C22×C3⋊#168C2211yescharacteristic
30C22×C3⋊#168C2212yescomplemented
31M16×D15—24no—
32C2×#6705C2212yes—
33C2×C8×D15C2212yes—
34C3×C5×C22⋊C8C2212yescomplemented
35C3×C5×C22⋊C8C2212yescomplemented
36#91104C2211yescharacteristic
37#91104C2211yescharacteristic
38#91104C2211yescharacteristic
39#91104C2211yescharacteristic
40#91099C2212yescomplemented
41#91098C2212yescomplemented
42#91098C2212yescomplemented
43C8×Dic15C2212yescomplemented
44#78659—51nomaximal, Hall
45#66162—31no—
46C2×#4094—31no—
47C2×D5×M16—31no—
48C5×#408D311yescharacteristic, complemented
49#65408—31no—
50C2×#3952—31no—
51#65390—31no—
52#65019—32no—
53#65018—32no—
54#65017—32no—
55#65014—32no—
56C23×D15C2×C411yescharacteristic
57C3×C5×#41—22no—
58C3×C5×#41—22no—
59C2×C3×C5×Q8C2×C411yescharacteristic
60C2×C3×C5×D4C2×C412yes—
61C2×C3×C5×D4C2×C411yescharacteristic
62C22×C3×C4×C5C2312yes—
63C22×C3×C4×C5C2311yescharacteristic
64C2×#1220C2×C412yes—
65C2×#1220C2×C412yes—
66C22×Dic15C2×C411yescharacteristic
67C2×C4×D15—22no—
68C2×C4×D15—22no—
69C2×C4×D15C2×C411yescharacteristic
70C2×C4×D15C2×C411yescharacteristic
71C2×C4×D15C2311yescharacteristic
72C2×C4×D15C2311yescharacteristic
73C3×C5×M16—22no—
74C2×C3×C5×C8—22no—
75C2×C3×C5×C8C2312yes—
76C2×C3×C5×C8C2311yescharacteristic
77C2×C3×C5×C8C2311yescharacteristic
78#6719C2×C412yes—
79#6719C2×C412yes—
80#6719C2×C412yes—
81#6714C2×C412yes—
82#6714C2×C412yes—
83C4⋊Dic15C2×C411yescharacteristic
84C4⋊Dic15C2×C411yescharacteristic
85#6712C2×C412yes—
86#6712C2×C412yes—
87C4×Dic15C2312yes—
88#6710—24no—
89#6710D414yescomplemented
90C2×C3⋊#168—22no—
91C2×C3⋊#168—24no—
92C2×C3⋊#168C2311yescharacteristic
93C2×C3⋊#168C2311yescharacteristic
94C2×C3⋊#168C2312yes—
95#6705—24no—
96C8×D15—24no—
97#5828—51no—
98C2×#948—51no—
99C2×D3×M16—51no—
100C3×#408D511yescharacteristic, complemented
101#5119—51no—
102C2×(C2×D3)⋊C8—51no—
103#5101—51no—
104#4730—52no—
105#4729—52no—
106#4728—52no—
107#67⋊C8—52no—
108C2×C5×#41C2×D311yescharacteristic
109C22×C4×D5—31no—
110C2×C5×M16C2×D311yescharacteristic
111C22×C5×C8C2×D311yescharacteristic
112#4094—64no—
113#4092—31no—
114#4092—31no—
115#4084—31no—
116#4084—31no—
117#4083—32no—
118#4081—32no—
119C4×C22⋊D5—32no—
120C4×C22⋊D5—32no—
121#4072—31no—
122C2×#681—31no—
123C2×#681—32no—
124C22×#168—31no—
125C22×#168—32no—
126D5×M16—64no—
127C2×#676—32no—
128C2×C8×D5—32no—
129C5×C22⋊C8C2×D312yescomplemented
130C5×C22⋊C8C2×D312yescomplemented
131#3952—31no—
132#3952—31no—
133#3952—31no—
134#3952—31no—
135#3947—32no—
136#3946—32no—
137#3946—32no—
138C8×Dic5—32no—
139#2933—151noSylow
140C23×C3×C5C22×C412yes—
141C23×C3×C5C22×C411yescharacteristic
142C22×D15—22no—
143C22×D15—22no—
144C22×D15—41no—
145C22×D15C22×C411yescharacteristic
146C22×D15C22×C411yescharacteristic
147C3×C5×Q8—22no—
148C3×C5×D4—24no—
149C3×C5×D4—22no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5C22×C412yes—
153C2×C3×C4×C5C22×C411yescharacteristic
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5#4112yes—
156C2×C3×C4×C5C2411yescharacteristic
157#1220—24no—
158#1220—42no—
159C2×Dic15—22no—
160C2×Dic15C22×C412yes—
161C2×Dic15C22×C412yes—
162C2×Dic15C22×C411yescharacteristic
163C2×Dic15C22×C411yescharacteristic
164C4×D15—22no—
165C4×D15—22no—
166C4×D15—21no—
167C4×D15—21no—
168C4×D15—21no—
169C4×D15—21no—
170C3×C5×C8—22no—
171C3×C5×C8—21no—
172C3×C5×C8—21no—
173C3⋊#168—21no—
174C3⋊#168—21no—
175C3⋊#168C2×D414yes—
176C2×C3×#41C2×D511yescharacteristic
177C22×C4×D3—51no—
178C2×C3×M16C2×D511yescharacteristic
179C22×C3×C8C2×D511yescharacteristic
180#948—104no—
181#946—51no—
182#946—51no—
183#938—51no—
184#938—51no—
185#937—52no—
186#935—52no—
187C4×#67—52no—
188C4×#67—52no—
189#926—51no—
190C2×#208—51no—
191C2×#208—52no—
192C22×C3⋊C8—52no—
193C22×C3⋊C8—51no—
194D3×M16—104no—
195C2×ΓL(1,𝔽25)—52no—
196C2×D3×C8—52no—
197C3×C22⋊C8C2×D512yescomplemented
198C3×C22⋊C8C2×D512yescomplemented
199(C2×D3)⋊C8—51no—
200(C2×D3)⋊C8—51no—
201(C2×D3)⋊C8—51no—
202(C2×D3)⋊C8—51no—
203#815—52no—
204Dic3⋊C8—52no—
205Dic3⋊C8—52no—
206C8×Dic3—52no—
207C23×D5—31no—
208C5×#41—22no—
209C5×#41—22no—
210C2×C5×Q8C4×D311yescharacteristic
211C2×C5×D4C4×D312yes—
212C2×C5×D4C4×D311yescharacteristic
213C22×C4×C5C22×D312yes—
214C22×C4×C5C22×D311yescharacteristic
215C2×C22⋊D5—32no—
216C2×C22⋊D5—32no—
217C22×Dic5—31no—
218C2×C4×D5—31no—
219C2×C4×D5—31no—
220C2×C4×D5—31no—
221C2×C4×D5—31no—
222C2×C4×D5—62no—
223C2×C4×D5—62no—
224C5×M16—22no—
225C2×C5×C8—22no—
226C2×C5×C8C22×D311yescharacteristic
227C2×C5×C8C22×D311yescharacteristic
228C2×C5×C8C22×D312yes—
229C22⋊Dic5—32no—
230C22⋊Dic5—32no—
231C22⋊Dic5—32no—
232#685—32no—
233#685—32no—
234C4⋊Dic5—31no—
235C4⋊Dic5—31no—
236#683—32no—
237#683—32no—
238C4×Dic5—32no—
239#681—34no—
240#681—64no—
241C2×#168—32no—
242C2×#168—31no—
243C2×#168—31no—
244C2×#168—64no—
245C2×#168—62no—
246#676—64no—
247C8×D5—64no—
248C2×#131—151no—
249C22×M16—151no—
250#525—151no—
251#436—152no—
252#435—152no—
253#435—152no—
254C8×D4—152no—
255#424—151no—
256#408D1511yescharacteristic, complemented
257C2×C22⋊C8—151no—
258#405—151no—
259C22×C3×C5—21no—
260C22×C3×C5—21no—
261C22×C3×C5—22no—
262C22×C3×C5—22no—
263C22×C3×C5—22no—
264C22×C3×C5C4×D412yes—
265C22×C3×C5C23×C411yescharacteristic
266C2×D15—21no—
267C2×D15—21no—
268C2×D15—21no—
269C2×D15—21no—
270C2×D15—21no—
271C2×D15—44no—
272C2×D15#13112yescomplemented
273C3×C4×C5—22no—
274C3×C4×C5C4×D412yes—
275C3×C4×C5C22×D411yescharacteristic
276C3×C4×C5C2×#4111yescharacteristic
277Dic15—21no—
278Dic15—21no—
279Dic15—22no—
280Dic15—22no—
281C23×D3—51no—
282C3×#41—22no—
283C3×#41—22no—
284C2×C3×Q8C4×D511yescharacteristic
285C2×C3×D4C4×D512yes—
286C2×C3×D4C4×D511yescharacteristic
287C22×C3×C4C22×D512yes—
288C22×C3×C4C22×D511yescharacteristic
289C2×#67—52no—
290C2×#67—52no—
291C22×Dic3—51no—
292C2×C4×D3—51no—
293C2×C4×D3—51no—
294C2×C4×D3—51no—
295C2×C4×D3—51no—
296C2×C4×D3—102no—
297C2×C4×D3—102no—
298C3×M16—22no—
299C2×C3×C8—22no—
300C2×C3×C8C22×D511yescharacteristic
301C2×C3×C8C22×D512yes—
302C2×C3×C8C22×D511yescharacteristic
303#217—52no—
304#217—52no—
305#217—52no—
306(C2×D3)⋊C4—52no—
307(C2×D3)⋊C4—52no—
308C4⋊Dic3—51no—
309C4⋊Dic3—51no—
310Dic3⋊C4—52no—
311Dic3⋊C4—52no—
312C4×Dic3—52no—
313#208—54no—
314#208—104no—
315C2×C3⋊C8—52no—
316C2×C3⋊C8—51no—
317C2×C3⋊C8—51no—
318C2×C3⋊C8—104no—
319C2×C3⋊C8—102no—
320ΓL(1,𝔽25)—104no—
321D3×C8—104no—
322C23×C5C2×C4×D312yes—
323C23×C5C2×C4×D311yescharacteristic
324C22×D5—31no—
325C22×D5—31no—
326C22×D5—62no—
327C22×D5—62no—
328C22×D5—121no—
329C5×Q8—22no—
330C5×D4—24no—
331C5×D4—22no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5C2×C4×D312yes—
335C2×C4×C5C2×C4×D311yescharacteristic
336C2×C4×C5D3×D412yes—
337C2×C4×C5#23712yes—
338C2×C4×C5C23×D311yescharacteristic
339C22⋊D5—64no—
340C22⋊D5—122no—
341C2×Dic5—32no—
342C2×Dic5—32no—
343C2×Dic5—31no—
344C2×Dic5—31no—
345C2×Dic5—62no—
346C4×D5—62no—
347C4×D5—62no—
348C4×D5—61no—
349C4×D5—61no—
350C4×D5—61no—
351C4×D5—61no—
352C5×C8—22no—
353C5×C8—21no—
354C5×C8—21no—
355#168—34no—
356#168—61no—
357#168—61no—
358C2×#41C2×D1511yescharacteristic
359C23×C4—151no—
360#131—304no—
361C2×M16—151no—
362C2×M16—152no—
363C2×M16—152no—
364C2×M16—304no—
365C2×M16C2×D1511yescharacteristic
366C22×C8—151no—
367C22×C8—152no—
368C22×C8—152no—
369C22×C8C2×D1511yescharacteristic
370#123—152no—
371C22⋊Q8—151no—
372C22⋊Q8—151no—
373#121—151no—
374#121—151no—
375#120—152no—
376C4×D4—152no—
377C4×D4—152no—
378#117—151no—
379C4⋊C8—152no—
380C4⋊C8—152no—
381C22⋊C8—151no—
382C22⋊C8—151no—
383C22⋊C8—151no—
384C22⋊C8—151no—
385C22⋊C8C2×D1512yescomplemented
386C22⋊C8C2×D1512yescomplemented
387#97—152no—
388C4×C8—152no—
389C2×C3×C5—22no—
390C2×C3×C5—42no—
391C2×C3×C5C2×C4×D411yescharacteristic
392C2×C3×C5C2×#13111yescharacteristic
393C2×C3×C5#56811yescharacteristic
394D15—22no—
395D15—41no—
396C23×C3C2×C4×D512yes—
397C23×C3C2×C4×D511yescharacteristic
398C22×D3—51no—
399C22×D3—51no—
400C22×D3—102no—
401C22×D3—102no—
402C22×D3—201no—
403C3×Q8—22no—
404C3×D4—24no—
405C3×D4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4C2×C4×D512yes—
409C2×C3×C4C2×C4×D511yescharacteristic
410C2×C3×C4D4×D512yes—
411C2×C3×C4#71112yes—
412C2×C3×C4C23×D511yescharacteristic
413#67—104no—
414#67—202no—
415C2×Dic3—52no—
416C2×Dic3—52no—
417C2×Dic3—51no—
418C2×Dic3—51no—
419C2×Dic3—102no—
420C4×D3—101no—
421C4×D3—101no—
422C4×D3—101no—
423C4×D3—101no—
424C4×D3—102no—
425C4×D3—102no—
426C3×C8—22no—
427C3×C8—21no—
428C3×C8—21no—
429C3⋊C8—54no—
430C3⋊C8—101no—
431C3⋊C8—101no—
432C22×C5—21no—
433C22×C5—21no—
434C22×C5—22no—
435C22×C5—22no—
436C22×C5—22no—
437C22×C5#67⋊C412yes—
438C22×C5C22×C4×D311yescharacteristic
439C2×D5—32no—
440C2×D5—61no—
441C2×D5—61no—
442C2×D5—61no—
443C2×D5—61no—
444C2×D5—61no—
445C2×D5—124no—
446C4×C5—22no—
447C4×C5#67⋊C412yes—
448C4×C5C2×D3×D411yescharacteristic
449C4×C5C2×#23711yescharacteristic
450Dic5—61no—
451Dic5—61no—
452Dic5—62no—
453Dic5—62no—
454C24—151no—
455#41—22no—
456#41—22no—
457C2×Q8C4×D1511yescharacteristic
458C2×D4—152no—
459C2×D4—152no—
460C2×D4C4×D1512yes—
461C2×D4C4×D1511yescharacteristic
462C22×C4—151no—
463C22×C4—151no—
464C22×C4—151no—
465C22×C4—151no—
466C22×C4—151no—
467C22×C4—302no—
468C22×C4—302no—
469C22×C4C22×D1512yes—
470C22×C4C22×D1511yescharacteristic
471M16—22no—
472M16—154no—
473M16—304no—
474M16—304no—
475C2×C8—22no—
476C2×C8—152no—
477C2×C8—151no—
478C2×C8—151no—
479C2×C8—304no—
480C2×C8—304no—
481C2×C8—302no—
482C2×C8C22×D1511yescharacteristic
483C2×C8C22×D1511yescharacteristic
484C2×C8C22×D1512yes—
485C4⋊C4—151no—
486C4⋊C4—151no—
487C4⋊C4—152no—
488C4⋊C4—152no—
489C22⋊C4—152no—
490C22⋊C4—152no—
491C22⋊C4—152no—
492C22⋊C4—152no—
493C22⋊C4—152no—
494C42—152no—
495C3×C5#293311yescharacteristic, complemented, Hall
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—22no—
501C22×C3#402712yes—
502C22×C3C22×C4×D511yescharacteristic
503C2×D3—52no—
504C2×D3—101no—
505C2×D3—101no—
506C2×D3—101no—
507C2×D3—101no—
508C2×D3—101no—
509C2×D3—204no—
510C3×C4—22no—
511C3×C4#402712yes—
512C3×C4C2×D4×D511yescharacteristic
513C3×C4C2×#71111yescharacteristic
514Dic3—101no—
515Dic3—101no—
516Dic3—102no—
517Dic3—102no—
518C2×C5—22no—
519C2×C5—42no—
520C2×C5C2×#67⋊C411yescharacteristic
521C2×C5D3×#13111yescharacteristic
522C2×C5#575011yescharacteristic
523D5—62no—
524D5—121no—
525C23—151no—
526C23—151no—
527C23—302no—
528C23—302no—
529C23—601no—
530C23C2×C4×D1512yes—
531C23C2×C4×D1511yescharacteristic
532Q8—22no—
533D4—24no—
534D4—22no—
535D4—304no—
536D4—602no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—152no—
540C2×C4—152no—
541C2×C4—151no—
542C2×C4—151no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4—302no—
546C2×C4—301no—
547C2×C4—301no—
548C2×C4—301no—
549C2×C4—301no—
550C2×C4C2×C4×D1512yes—
551C2×C4C2×C4×D1511yescharacteristic
552C2×C4D4×D1512yes—
553C2×C4#681912yes—
554C2×C4C23×D1511yescharacteristic
555C8—22no—
556C8—21no—
557C8—21no—
558C8—154no—
559C8—301no—
560C8—301no—
561C2×C3—22no—
562C2×C3—42no—
563C2×C3C2×#402711yescharacteristic
564C2×C3D5×#13111yescharacteristic
565C2×C3#6608411yescharacteristic
566D3—102no—
567D3—201no—
568C5#7865911yescharacteristic, complemented, Sylow
569C22—21no—
570C22—21no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—152no—
575C22—301no—
576C22—301no—
577C22—301no—
578C22—301no—
579C22—301no—
580C22—604no—
581C22#9177012yes—
582C22C22×C4×D1511yescharacteristic
583C4—22no—
584C4—301no—
585C4—301no—
586C4—302no—
587C4—302no—
588C4#9177012yes—
589C4C2×D4×D1511yescharacteristic
590C4C2×#681911yescharacteristic
591C3#1061039011yescharacteristic, complemented, Sylow
592C2—22no—
593C2—42no—
594C2—302no—
595C2—601no—
596C2C2×#9177011yescharacteristic
597C2D15×#13111yescharacteristic
598C2#1175650911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D15
commutator, socleC22×C3×C5C23×C4
FittingC3×C5×#408C2
prime radical#11756587C2
cosocleC2×C3×C4×C5C24
hypercenter#408D15
hypocenterC3×C5#2933
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#91837C2semidirect
3. C2×M16×D15C2semidirect
4. C3×C5×#408C2semidirect
5. #11751220C2semidirect
6. C2×#91104C2semidirect
7. #11751202C2semidirect
8. #11750831C2semidirect
9. #11750830C2semidirect
10. #11750829C2semidirect
11. #11750826C2semidirect
28. C2×#6710C22semidirect
30. C22×C3⋊#168C22semidirect
34. C3×C5×C22⋊C8C22semidirect
35. C3×C5×C22⋊C8C22semidirect
40. #91099C22semidirect
41. #91098C22semidirect
42. #91098C22semidirect
43. C8×Dic15C22semidirect
48. C5×#408D3semidirect
89. #6710D4semidirect
100. C3×#408D5semidirect
129. C5×C22⋊C8C2×D3semidirect
130. C5×C22⋊C8C2×D3semidirect
197. C3×C22⋊C8C2×D5semidirect
198. C3×C22⋊C8C2×D5semidirect
256. #408D15semidirect
272. C2×D15#131semidirect
385. C22⋊C8C2×D15semidirect
386. C22⋊C8C2×D15semidirect
495. C3×C5#2933semidirect
568. C5#78659semidirect
591. C3#10610390semidirect

Maximal subgroups

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

subgroupindex
#117565872
C2×#918372
C2×M16×D152
C3×C5×#4082
#117512202
C2×#911042
#117512022
#117508312
#117508302
#117508292
#117508262
#106103903
#786595