Finite Group Databasebeta

← #14302163

Subgroups — #14302163

Order 1920.

780 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D15C211yesmaximal, characteristic, complemented
2C2×#91465C211yesmaximal, characteristic
3C2×C3×#3955C211yesmaximal, characteristic, complemented
4C2×#90958C211yesmaximal, characteristic, complemented
5#11748351C211yesmaximal, characteristic, complemented
6C5×#4554C211yesmaximal, characteristic, complemented
7#11745986C211yesmaximal, characteristic, complemented
8#10599785—31nomaximal, Hall
9C22×C4×D15C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C22×#1202C412yes—
12C2×C3×C5×M16C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14M16×D15—22no—
15M16×D15—22no—
16C2×#6705—21no—
17C2×C8×D15—21no—
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21C2×#6665C412yes—
22#91465—21no—
23#91465—21no—
24#91465—21no—
25#91465—21no—
26C3×#3955—22no—
27#90958—22no—
28#90957—21no—
29#90957—21no—
30#70081—51nomaximal, Hall
31C2×D5×M16—31no—
32C2×#4041—31no—
33C2×#3955—31no—
34C2×#3955D311yescharacteristic, complemented
35#65415—31no—
36C5×#343—31no—
37#64750—31no—
38C23×D15C2×C411yescharacteristic
39C23×C3×D5C2×C412yes—
40C22×C3×C4×C5C2311yescharacteristic
41C22×Dic15C2×C411yescharacteristic
42C2×C4×D15—21no—
43C2×C4×D15—21no—
44C2×C4×D15—21no—
45C2×C4×D15—21no—
46C2×C4×D15D411yescharacteristic
47C2×C4×D15D411yescharacteristic
48C2×C4×D15D411yescharacteristic
49C2×C4×D15D411yescharacteristic
50C22×C5×Dic3C2×C412yes—
51C2×C3×D20—21no—
52C2×C3×D20—21no—
53C2×C3×D20—21no—
54C2×C3×D20—21no—
55C2×C3×D20—21no—
56C2×C3×D20—21no—
57C2×#1202—22no—
58C2×#1202—22no—
59C2×#1202—44no—
60C3×C5×M16—21no—
61C3×C5×M16D412yescomplemented
62C2×C3×C5×C8—21no—
63#6710—21no—
64#6710D412yescomplemented
65C2×C3⋊#168—21no—
66#6705—24no—
67C8×D15—24no—
68C5×C4⋊Dic3—21no—
69C5×C4⋊Dic3—21no—
70C5×#208—21no—
71C5×#208—21no—
72C2×C5×C3⋊C8—21no—
73C3×#681—21no—
74C3×#681—21no—
75C2×C3×#168—21no—
76#6665—22no—
77#6665—22no—
78C2×#938—51no—
79C2×D3×M16—51no—
80C2×C3×#100—51no—
81C2×#833—51no—
82#5126—51no—
83#4554—51no—
84#4554D511yescharacteristic, complemented
85C22×D20—32no—
86C22×D20C2×D311yescharacteristic
87C22×C4×D5—31no—
88C2×C5×M16—31no—
89C2×C5×M16C2×D311yescharacteristic
90C2×C5×C4⋊C4—31no—
91C2×#685—32no—
92C2×#681—31no—
93C2×#681C2×D311yescharacteristic
94D5×M16—62no—
95D5×M16—62no—
96C2×#676—61no—
97C2×C8×D5—61no—
98#4041—61no—
99#4041—61no—
100#4041—61no—
101#4041—61no—
102#3955—22no—
103#3955—62no—
104#3952—61no—
105#3952—61no—
106#1900—151noSylow
107C23×C3×C5C22×C411yescharacteristic
108C22×D15—21no—
109C22×D15—21no—
110C22×D15—22no—
111C22×D15—41no—
112C22×D15C22⋊C411yescharacteristic
113C22×D15C22⋊C411yescharacteristic
114C22×C3×D5—22no—
115C22×C3×D5—22no—
116C22×C3×D5—22no—
117C22×C3×D5—42no—
118C22×C3×D5—42no—
119C2×C3×C4×C5C2×D411yescharacteristic
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5#4111yescharacteristic
124C2×C3×C4×C5#4111yescharacteristic
125C2×Dic15—21no—
126C2×Dic15—21no—
127C2×Dic15C22⋊C411yescharacteristic
128C2×Dic15C22⋊C411yescharacteristic
129C4×D15—21no—
130C4×D15—21no—
131C4×D15—21no—
132C4×D15—21no—
133C4×D15—21no—
134C4×D15—21no—
135C4×D15—21no—
136C4×D15—21no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—42no—
139C3×D20—41no—
140C3×D20—41no—
141C3×D20—41no—
142C3×D20—41no—
143#1202—44no—
144#1202—44no—
145C3×C5×C8—22no—
146C3⋊#168—22no—
147C3×#168—41no—
148C5×C3⋊C8—41no—
149C22×C3×D4—51no—
150C22×#67—52no—
151C22×C4×D3—51no—
152C2×C3×M16—51no—
153C2×C3×M16C2×D511yescharacteristic
154C2×#217—52no—
155#938—101no—
156#938—101no—
157#938—101no—
158#938—101no—
159C2×C4⋊Dic3C2×D511yescharacteristic
160C2×#208—51no—
161C2×#208C2×D511yescharacteristic
162D3×M16—102no—
163D3×M16—102no—
164C2×ΓL(1,𝔽25)—101no—
165C2×D3×C8—101no—
166C3×#100—102no—
167#833—102no—
168(C2×D3)⋊C8—101no—
169(C2×D3)⋊C8—101no—
170C23×D5—31no—
171C23×D5C4×D312yes—
172C22×C4×C5—32no—
173C22×C4×C5C22×D311yescharacteristic
174C22×Dic5—31no—
175C2×D20—21no—
176C2×D20—21no—
177C2×D20—21no—
178C2×D20—21no—
179C2×D20—21no—
180C2×D20—21no—
181C2×D20—62no—
182C2×D20—62no—
183C2×D20—124no—
184C2×C4×D5—31no—
185C2×C4×D5—31no—
186C2×C4×D5—31no—
187C2×C4×D5—31no—
188C2×C4×D5—61no—
189C2×C4×D5—61no—
190C2×C4×D5—61no—
191C2×C4×D5—61no—
192C5×M16—21no—
193C5×M16—61no—
194C5×M16—61no—
195C5×M16#6712yescomplemented
196C2×C5×C8—21no—
197C2×C5×C8—61no—
198C5×C4⋊C4—61no—
199C5×C4⋊C4—61no—
200#685—62no—
201#685—62no—
202#681—21no—
203#681—21no—
204#681—32no—
205#681—61no—
206C2×#168—21no—
207C2×#168—61no—
208#676—64no—
209C8×D5—64no—
210C22×M16—151no—
211C2×#121—151no—
212C2×#100—51no—
213C2×#100—151no—
214#407—151no—
215#343—31no—
216#343—151no—
217C22×C3×C5—21no—
218C22×C3×C5—21no—
219C22×C3×C5C2×C22⋊C411yescharacteristic
220C22×C3×C5C4×D411yescharacteristic
221C22×C3×C5C4×D411yescharacteristic
222C2×D15—21no—
223C2×D15—21no—
224C2×D15—21no—
225C2×D15—21no—
226C2×D15—21no—
227C2×D15—42no—
228C2×D15—42no—
229C2×D15#10012yescomplemented
230C2×C3×D5—42no—
231C2×C3×D5—42no—
232C2×C3×D5—42no—
233C2×C3×D5—82no—
234C2×C3×D5—82no—
235C3×C4×C5#12011yescharacteristic
236C3×C4×C5#12111yescharacteristic
237C3×C4×C5#12111yescharacteristic
238C3×C4×C5#12311yescharacteristic
239Dic15—21no—
240Dic15—21no—
241C5×Dic3—42no—
242C24×C3—52no—
243C23×D3—51no—
244C2×C3×D4—101no—
245C2×C3×D4—101no—
246C2×C3×D4—101no—
247C2×C3×D4—101no—
248C2×C3×D4—101no—
249C2×C3×D4—101no—
250C22×C3×C4C22×D511yescharacteristic
251C2×#67—102no—
252C2×#67—102no—
253C2×#67—204no—
254C22×Dic3—51no—
255C22×Dic3C4×D512yes—
256C2×C4×D3—51no—
257C2×C4×D3—51no—
258C2×C4×D3—51no—
259C2×C4×D3—51no—
260C2×C4×D3—101no—
261C2×C4×D3—101no—
262C2×C4×D3—101no—
263C2×C4×D3—101no—
264C3×M16—21no—
265C3×M16—101no—
266C3×M16—101no—
267C3×M16D2012yescomplemented
268C2×C3×C8—21no—
269C2×C3×C8—101no—
270#217—102no—
271#217—102no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274#208—21no—
275#208—21no—
276#208—52no—
277#208—101no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—101no—
280ΓL(1,𝔽25)—104no—
281D3×C8—104no—
282C23×C5C2×C4×D311yescharacteristic
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—22no—
286C22×D5—31no—
287C22×D5—31no—
288C22×D5—42no—
289C22×D5—42no—
290C22×D5—62no—
291C22×D5—61no—
292C22×D5—61no—
293C22×D5—121no—
294C2×C4×C5—62no—
295C2×C4×C5—122no—
296C2×C4×C5#23511yescharacteristic
297C2×C4×C5D3×D411yescharacteristic
298C2×C4×C5D3×D411yescharacteristic
299C2×C4×C5D3×D411yescharacteristic
300C2×C4×C5#23711yescharacteristic
301C2×C4×C5C2×#6711yescharacteristic
302C2×Dic5—31no—
303C2×Dic5—31no—
304C2×Dic5—61no—
305C2×Dic5—61no—
306D20—41no—
307D20—41no—
308D20—41no—
309D20—41no—
310D20—124no—
311D20—124no—
312C4×D5—61no—
313C4×D5—61no—
314C4×D5—61no—
315C4×D5—61no—
316C4×D5—61no—
317C4×D5—61no—
318C4×D5—61no—
319C4×D5—61no—
320C5×C8—22no—
321C5×C8—121no—
322#168—41no—
323#168—62no—
324C22×D4—51no—
325C22×D4—152no—
326C23×C4—151no—
327C2×M16—31no—
328C2×M16—51no—
329C2×M16—151no—
330C2×M16—301no—
331C2×M16—302no—
332C2×M16—302no—
333C2×M16D3×D511yescharacteristic
334C22×C8—301no—
335#121—301no—
336#121—301no—
337#121—301no—
338#121—301no—
339C2×C4⋊C4—31no—
340C2×C22⋊C4—152no—
341#100—102no—
342#100—302no—
343C22⋊C8—301no—
344C22⋊C8—301no—
345C2×C3×C5—21no—
346C2×C3×C5—21no—
347C2×C3×C5#38611yescharacteristic
348C2×C3×C5C2×#10011yescharacteristic
349C2×C3×C5#41311yescharacteristic
350D15—22no—
351D15—41no—
352C3×D5—82no—
353C23×C3—102no—
354C23×C3—102no—
355C23×C3—102no—
356C23×C3—202no—
357C23×C3—202no—
358C23×C3C2×C4×D511yescharacteristic
359C22×D3—51no—
360C22×D3—51no—
361C22×D3—102no—
362C22×D3—101no—
363C22×D3—101no—
364C22×D3—201no—
365C3×D4—201no—
366C3×D4—201no—
367C3×D4—201no—
368C3×D4—201no—
369C2×C3×C4C2×D2011yescharacteristic
370C2×C3×C4#70911yescharacteristic
371C2×C3×C4D4×D511yescharacteristic
372C2×C3×C4D4×D511yescharacteristic
373C2×C3×C4D4×D511yescharacteristic
374C2×C3×C4#71311yescharacteristic
375#67—204no—
376#67—204no—
377C2×Dic3—22no—
378C2×Dic3—42no—
379C2×Dic3—51no—
380C2×Dic3—51no—
381C2×Dic3—101no—
382C2×Dic3—101no—
383C4×D3—101no—
384C4×D3—101no—
385C4×D3—101no—
386C4×D3—101no—
387C4×D3—101no—
388C4×D3—101no—
389C4×D3—101no—
390C4×D3—101no—
391C3×C8—22no—
392C3×C8—201no—
393C3⋊C8—41no—
394C3⋊C8—102no—
395C22×C5—21no—
396C22×C5—21no—
397C22×C5D3×C22⋊C411yescharacteristic
398C22×C5#67⋊C411yescharacteristic
399C22×C5C4×#6711yescharacteristic
400C2×D5—32no—
401C2×D5—42no—
402C2×D5—42no—
403C2×D5—42no—
404C2×D5—61no—
405C2×D5—61no—
406C2×D5—61no—
407C2×D5—61no—
408C2×D5—61no—
409C2×D5—82no—
410C2×D5—82no—
411C2×D5—122no—
412C2×D5—122no—
413C4×C5—122no—
414C4×C5#88311yescharacteristic
415C4×C5#88411yescharacteristic
416C4×C5#93711yescharacteristic
417C4×C5#93811yescharacteristic
418Dic5—61no—
419Dic5—61no—
420C24—52no—
421C24—151no—
422C2×D4—101no—
423C2×D4—101no—
424C2×D4—101no—
425C2×D4—101no—
426C2×D4—101no—
427C2×D4—101no—
428C2×D4—302no—
429C2×D4—302no—
430C2×D4—604no—
431C22×C4—32no—
432C22×C4—151no—
433C22×C4—151no—
434C22×C4—151no—
435C22×C4—151no—
436C22×C4—151no—
437C22×C4—301no—
438C22×C4—301no—
439C22×C4—301no—
440C22×C4—301no—
441C22×C4C2×D3×D511yescharacteristic
442M16—21no—
443M16—61no—
444M16—61no—
445M16—101no—
446M16—101no—
447M16—152no—
448M16—304no—
449M16—301no—
450M16#120212yescomplemented
451C2×C8—21no—
452C2×C8—61no—
453C2×C8—101no—
454C2×C8—304no—
455C2×C8—301no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C22⋊C4—302no—
459C22⋊C4—302no—
460C3×C5#190011yescharacteristic, complemented, Hall
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—202no—
464C22×C3—202no—
465C22×C3—202no—
466C22×C3—402no—
467C22×C3—402no—
468C22×C3C4×D2011yescharacteristic
469C22×C3D5×C22⋊C411yescharacteristic
470C22×C3#403911yescharacteristic
471C2×D3—52no—
472C2×D3—101no—
473C2×D3—101no—
474C2×D3—101no—
475C2×D3—101no—
476C2×D3—101no—
477C2×D3—202no—
478C2×D3—202no—
479C3×C4#402811yescharacteristic
480C3×C4#403011yescharacteristic
481C3×C4#404011yescharacteristic
482C3×C4#404111yescharacteristic
483Dic3—42no—
484Dic3—101no—
485Dic3—101no—
486C2×C5—21no—
487C2×C5—21no—
488C2×C5D3×#10011yescharacteristic
489C2×C5#474511yescharacteristic
490C2×C5#495611yescharacteristic
491D5—62no—
492D5—82no—
493D5—121no—
494C23—102no—
495C23—102no—
496C23—102no—
497C23—151no—
498C23—151no—
499C23—202no—
500C23—202no—
501C23—302no—
502C23—301no—
503C23—301no—
504C23—601no—
505C23C4×D3×D511yescharacteristic
506D4—201no—
507D4—201no—
508D4—201no—
509D4—201no—
510D4—604no—
511D4—604no—
512C2×C4—62no—
513C2×C4—122no—
514C2×C4—151no—
515C2×C4—151no—
516C2×C4—301no—
517C2×C4—301no—
518C2×C4—301no—
519C2×C4—301no—
520C2×C4—301no—
521C2×C4—301no—
522C2×C4—301no—
523C2×C4—301no—
524C2×C4—301no—
525C2×C4—301no—
526C2×C4#676611yescharacteristic
527C2×C4#677111yescharacteristic
528C2×C4#677711yescharacteristic
529C2×C4C2×#120211yescharacteristic
530C2×C4#679011yescharacteristic
531C2×C4#679011yescharacteristic
532C8—22no—
533C8—121no—
534C8—201no—
535C8—302no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—402no—
539C2×C3#6495411yescharacteristic
540C2×C3D5×#10011yescharacteristic
541C2×C3#6504011yescharacteristic
542D3—102no—
543D3—201no—
544C5#7008111yescharacteristic, complemented, Sylow
545C22—21no—
546C22—21no—
547C22—152no—
548C22—202no—
549C22—202no—
550C22—202no—
551C22—301no—
552C22—301no—
553C22—301no—
554C22—301no—
555C22—301no—
556C22—402no—
557C22—402no—
558C22—602no—
559C22—602no—
560C22#9139611yescharacteristic
561C22C4×#120211yescharacteristic
562C22#9147311yescharacteristic
563C4—122no—
564C4—301no—
565C4—301no—
566C4#9135411yescharacteristic
567C4#9146111yescharacteristic
568C4#9146511yescharacteristic
569C4#9157211yescharacteristic
570C3#1059978511yescharacteristic, complemented, Sylow
571C2—21no—
572C2—21no—
573C2—302no—
574C2—402no—
575C2—601no—
576C2#1174822411yescharacteristic
577C2#1174823811yescharacteristic
578C2#1174891711yescharacteristic
subgroupG/H
centerC22#91473
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#91465C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1900
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×M16×D15C2semidirect
3. C2×C3×#3955C2semidirect
4. C2×#90958C2semidirect
5. #11748351C2semidirect
6. C5×#4554C2semidirect
7. #11745986C2semidirect
34. C2×#3955D3semidirect
61. C3×C5×M16D4semidirect
64. #6710D4semidirect
84. #4554D5semidirect
195. C5×M16#67semidirect
229. C2×D15#100semidirect
267. C3×M16D20semidirect
450. M16#1202semidirect
460. C3×C5#1900semidirect
544. C5#70081semidirect
570. C3#10599785semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D152
C2×#914652
C2×C3×#39552
C2×#909582
#117483512
C5×#45542
#117459862
#105997853
#700815