Finite Group Databasebeta

← #14302153

Subgroups — #14302153

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×#91449C211yesmaximal, characteristic
2C2×D5×#208C211yesmaximal, characteristic, complemented
3C2×C5×#823C211yesmaximal, characteristic, complemented
4C3×#65415C211yesmaximal, characteristic, complemented
5C2×#90959C211yesmaximal, characteristic, complemented
6#11746375C211yesmaximal, characteristic, complemented
7#11745987C211yesmaximal, characteristic, complemented
8#10600874—31nomaximal, Hall
9C22×C5×D12C2211yescharacteristic
10C22×C3×C4×D5C2211yescharacteristic
11C22×#1201C412yes—
12C2×C3×C5×M16C2211yescharacteristic
13C2×C4⋊Dic15C2211yescharacteristic
14C2×#6710C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C2×C3×#681C2211yescharacteristic
17C2×(C2×D3)⋊Dic5C412yes—
18#91449—21no—
19#91449—21no—
20#91449—21no—
21#91449—21no—
22C2×#6649—21no—
23D5×#208—22no—
24D5×#208—22no—
25C2×D5×C3⋊C8—21no—
26C5×#823—22no—
27C3×#3952—21no—
28C3×#3952—21no—
29#90959—22no—
30#69985—51nomaximal, Hall
31C2×#4084—31no—
32C2×D5×M16—31no—
33C2×C5×#100—31no—
34C2×#3965—31no—
35#65415D311yescharacteristic, complemented
36#64774—31no—
37#64774—31no—
38C23×C5×D3C2×C412yes—
39C23×C3×D5C2×C411yescharacteristic
40C22×C3×C4×C5C2311yescharacteristic
41C22×Dic15C2×C412yes—
42C2×C5×D12—21no—
43C2×C5×D12—21no—
44C2×C5×D12—21no—
45C2×C5×D12—21no—
46C2×C5×D12—21no—
47C2×C5×D12—21no—
48C22×C3×Dic5C2×C411yescharacteristic
49C2×C3×C4×D5—21no—
50C2×C3×C4×D5—21no—
51C2×C3×C4×D5—21no—
52C2×C3×C4×D5—21no—
53C2×C3×C4×D5D411yescharacteristic
54C2×C3×C4×D5D411yescharacteristic
55C2×C3×C4×D5D411yescharacteristic
56C2×C3×C4×D5D411yescharacteristic
57C2×#1201—22no—
58C2×#1201—22no—
59C2×#1201—44no—
60C3×C5×M16—21no—
61C3×C5×M16—21no—
62C2×C3×C5×C8—21no—
63C4⋊Dic15—21no—
64C4⋊Dic15—21no—
65#6710—21no—
66#6710D412yescomplemented
67C2×C3⋊#168—21no—
68C5×#208—21no—
69C5×#208D412yescomplemented
70C2×C5×C3⋊C8—21no—
71C3×#681—21no—
72C3×#681—21no—
73C2×C3×#168—21no—
74(C2×D3)⋊Dic5—22no—
75(C2×D3)⋊Dic5—22no—
76#6649—24no—
77D5×C3⋊C8—24no—
78C2×#930—51no—
79C22×#208—51no—
80C3×#407—51no—
81C2×#823—51no—
82C2×#823D511yescharacteristic, complemented
83#4554—51no—
84#4554—51no—
85C22×C5×D4—31no—
86C22×C22⋊D5—32no—
87C22×C4×D5C2×D311yescharacteristic
88C2×C5×M16—31no—
89C2×C5×M16C2×D311yescharacteristic
90C2×C22⋊Dic5—32no—
91#4084—61no—
92#4084—61no—
93#4084—61no—
94#4084—61no—
95C2×C4⋊Dic5—31no—
96C2×#681—31no—
97C2×#681C2×D311yescharacteristic
98D5×M16—62no—
99D5×M16—62no—
100C2×#676—61no—
101C2×C8×D5—61no—
102C5×#100—62no—
103#3965—62no—
104#3952—21no—
105#3952—21no—
106#1900—151noSylow
107C23×C3×C5C22×C411yescharacteristic
108C22×C5×D3—22no—
109C22×C5×D3—22no—
110C22×C5×D3—22no—
111C22×C5×D3—42no—
112C22×C5×D3—42no—
113C22×C3×D5—21no—
114C22×C3×D5—21no—
115C22×C3×D5—22no—
116C22×C3×D5—41no—
117C22×C3×D5C22⋊C411yescharacteristic
118C22×C3×D5C22⋊C411yescharacteristic
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—22no—
126C2×Dic15—42no—
127C5×D12—41no—
128C5×D12—41no—
129C5×D12—41no—
130C5×D12—41no—
131C2×C3×Dic5—21no—
132C2×C3×Dic5—21no—
133C2×C3×Dic5C22⋊C411yescharacteristic
134C2×C3×Dic5C22⋊C411yescharacteristic
135C3×C4×D5—21no—
136C3×C4×D5—21no—
137C3×C4×D5—21no—
138C3×C4×D5—21no—
139C3×C4×D5—21no—
140C3×C4×D5—21no—
141C3×C4×D5—21no—
142C3×C4×D5—21no—
143#1201—44no—
144#1201—44no—
145C3×C5×C8—41no—
146C3⋊#168—22no—
147C3×#168—41no—
148C5×C3⋊C8—22no—
149C23×C3×C4—51no—
150C22×#67—52no—
151C22×D12C2×D511yescharacteristic
152C2×C3×M16—51no—
153C2×C3×M16C2×D511yescharacteristic
154#930—101no—
155#930—101no—
156#930—101no—
157#930—101no—
158C2×(C2×D3)⋊C4—52no—
159C2×C4⋊Dic3—51no—
160C2×#208—51no—
161C2×#208—101no—
162C2×#208—102no—
163C2×#208—102no—
164C2×#208C2×D511yescharacteristic
165C22×C3⋊C8—101no—
166C3×C22⋊C8—101no—
167C3×C22⋊C8—101no—
168#823—22no—
169#823—102no—
170C24×C5—32no—
171C23×D5C4×D311yescharacteristic
172C2×C5×D4—61no—
173C2×C5×D4—61no—
174C2×C5×D4—61no—
175C2×C5×D4—61no—
176C2×C5×D4—61no—
177C2×C5×D4—61no—
178C22×C4×C5C22×D311yescharacteristic
179C2×C22⋊D5—62no—
180C2×C22⋊D5—62no—
181C2×C22⋊D5—124no—
182C22×Dic5—32no—
183C22×Dic5C4×D311yescharacteristic
184C2×C4×D5—21no—
185C2×C4×D5—21no—
186C2×C4×D5—21no—
187C2×C4×D5—21no—
188C2×C4×D5D1211yescharacteristic
189C2×C4×D5D1211yescharacteristic
190C2×C4×D5#6711yescharacteristic
191C2×C4×D5#6711yescharacteristic
192C5×M16—21no—
193C5×M16—21no—
194C5×M16—32no—
195C5×M16—61no—
196C2×C5×C8—21no—
197C2×C5×C8—61no—
198C22⋊Dic5—62no—
199C22⋊Dic5—62no—
200C4⋊Dic5—61no—
201C4⋊Dic5—61no—
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—31no—
213C2×#100—151no—
214#407—51no—
215#343—151no—
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×C5×D3—42no—
223C2×C5×D3—42no—
224C2×C5×D3—42no—
225C2×C5×D3—82no—
226C2×C5×D3—82no—
227C2×C3×D5—21no—
228C2×C3×D5—21no—
229C2×C3×D5—21no—
230C2×C3×D5—21no—
231C2×C3×D5—21no—
232C2×C3×D5—42no—
233C2×C3×D5—42no—
234C2×C3×D5#10012yescomplemented
235C3×C4×C5#12011yescharacteristic
236C3×C4×C5#12111yescharacteristic
237C3×C4×C5#12111yescharacteristic
238C3×C4×C5#12311yescharacteristic
239Dic15—42no—
240C3×Dic5—21no—
241C3×Dic5—21no—
242C24×C3—51no—
243C23×D3C4×D512yes—
244C22×C3×C4—51no—
245C22×C3×C4—51no—
246C22×C3×C4—51no—
247C22×C3×C4—51no—
248C22×C3×C4—51no—
249C22×C3×C4—101no—
250C22×C3×C4—101no—
251C22×C3×C4—101no—
252C22×C3×C4—101no—
253C22×C3×C4C22×D511yescharacteristic
254C2×#67—102no—
255C2×#67—102no—
256C2×#67—204no—
257C22×Dic3—52no—
258C2×D12—21no—
259C2×D12—21no—
260C2×D12—21no—
261C2×D12—21no—
262C2×D12—21no—
263C2×D12—21no—
264C3×M16—21no—
265C3×M16—21no—
266C3×M16—101no—
267C3×M16—101no—
268C2×C3×C8—21no—
269C2×C3×C8—101no—
270(C2×D3)⋊C4—102no—
271(C2×D3)⋊C4—102no—
272C4⋊Dic3—101no—
273C4⋊Dic3—101no—
274#208—21no—
275#208—52no—
276#208—104no—
277#208—101no—
278#208C22⋊D512yescomplemented
279C2×C3⋊C8—21no—
280C2×C3⋊C8—104no—
281C2×C3⋊C8—101no—
282C23×C5—62no—
283C23×C5—62no—
284C23×C5—62no—
285C23×C5—122no—
286C23×C5—122no—
287C23×C5C2×C4×D311yescharacteristic
288C22×D5—21no—
289C22×D5—21no—
290C22×D5—22no—
291C22×D5—41no—
292C22×D5(C2×D3)⋊C411yescharacteristic
293C22×D5(C2×D3)⋊C411yescharacteristic
294C5×D4—121no—
295C5×D4—121no—
296C5×D4—121no—
297C5×D4—121no—
298C2×C4×C5C2×D1211yescharacteristic
299C2×C4×C5D3×D411yescharacteristic
300C2×C4×C5D3×D411yescharacteristic
301C2×C4×C5#23711yescharacteristic
302C2×C4×C5#23711yescharacteristic
303C2×C4×C5C2×#6711yescharacteristic
304C22⋊D5—124no—
305C22⋊D5—124no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—62no—
309C2×Dic5—122no—
310C2×Dic5(C2×D3)⋊C411yescharacteristic
311C2×Dic5(C2×D3)⋊C411yescharacteristic
312C4×D5—21no—
313C4×D5—21no—
314C4×D5—21no—
315C4×D5—21no—
316C4×D5—21no—
317C4×D5—21no—
318C4×D5—21no—
319C4×D5—21no—
320C5×C8—41no—
321C5×C8—62no—
322#168—41no—
323#168—62no—
324C22×D4—31no—
325C22×D4—152no—
326C23×C4—51no—
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—151no—
340C2×C22⋊C4—152no—
341#100—62no—
342#100—302no—
343C22⋊C8—101no—
344C22⋊C8—101no—
345C2×C3×C5—21no—
346C2×C3×C5—21no—
347C2×C3×C5#38611yescharacteristic
348C2×C3×C5C2×#10011yescharacteristic
349C2×C3×C5#41311yescharacteristic
350C3×D5—22no—
351C3×D5—41no—
352C5×D3—82no—
353C23×C3—51no—
354C23×C3—51no—
355C23×C3—102no—
356C23×C3—101no—
357C23×C3—101no—
358C23×C3—201no—
359C23×C3C2×C4×D511yescharacteristic
360C22×D3—22no—
361C22×D3—22no—
362C22×D3—22no—
363C22×D3—42no—
364C22×D3—42no—
365C2×C3×C4—51no—
366C2×C3×C4—51no—
367C2×C3×C4—101no—
368C2×C3×C4—101no—
369C2×C3×C4—101no—
370C2×C3×C4—101no—
371C2×C3×C4—101no—
372C2×C3×C4—101no—
373C2×C3×C4—101no—
374C2×C3×C4—101no—
375C2×C3×C4—101no—
376C2×C3×C4—101no—
377C2×C3×C4#70911yescharacteristic
378C2×C3×C4D4×D511yescharacteristic
379C2×C3×C4D4×D511yescharacteristic
380C2×C3×C4D4×D511yescharacteristic
381C2×C3×C4#71111yescharacteristic
382C2×C3×C4C2×C22⋊D511yescharacteristic
383#67—204no—
384#67—204no—
385C2×Dic3—102no—
386C2×Dic3—202no—
387D12—41no—
388D12—41no—
389D12—41no—
390D12—41no—
391C3×C8—41no—
392C3×C8—201no—
393C3⋊C8—22no—
394C3⋊C8—102no—
395C22×C5—21no—
396C22×C5—21no—
397C22×C5—122no—
398C22×C5—122no—
399C22×C5—122no—
400C22×C5—242no—
401C22×C5—242no—
402C22×C5#67⋊C411yescharacteristic
403C22×C5#67⋊C411yescharacteristic
404C22×C5C2×(C2×D3)⋊C411yescharacteristic
405C2×D5—21no—
406C2×D5—21no—
407C2×D5—21no—
408C2×D5—21no—
409C2×D5—21no—
410C2×D5—42no—
411C2×D5—42no—
412C2×D5#82312yescomplemented
413C4×C5#88211yescharacteristic
414C4×C5#88611yescharacteristic
415C4×C5#93911yescharacteristic
416C4×C5#93911yescharacteristic
417Dic5—21no—
418Dic5—21no—
419Dic5—122no—
420C24—32no—
421C24—51no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—302no—
429C2×D4—302no—
430C2×D4—604no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—51no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—101no—
437C22×C4—101no—
438C22×C4—101no—
439C22×C4—101no—
440C22×C4—152no—
441C22×C4C2×D3×D511yescharacteristic
442M16—21no—
443M16—21no—
444M16—32no—
445M16—61no—
446M16—101no—
447M16—101no—
448M16—152no—
449M16—304no—
450M16—301no—
451C2×C8—21no—
452C2×C8—61no—
453C2×C8—101no—
454C2×C8—304no—
455C2×C8—301no—
456C4⋊C4—301no—
457C4⋊C4—301no—
458C22⋊C4—302no—
459C22⋊C4—302no—
460C3×C5#190011yescharacteristic, complemented, Hall
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—52no—
464C22×C3—101no—
465C22×C3—101no—
466C22×C3—101no—
467C22×C3—101no—
468C22×C3—101no—
469C22×C3—202no—
470C22×C3—202no—
471C22×C3D5×C22⋊C411yescharacteristic
472C22×C3#402711yescharacteristic
473C22×C3C4×C22⋊D511yescharacteristic
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—82no—
478C2×D3—82no—
479C3×C4—101no—
480C3×C4—101no—
481C3×C4#402911yescharacteristic
482C3×C4#403011yescharacteristic
483C3×C4#408311yescharacteristic
484C3×C4#408411yescharacteristic
485Dic3—202no—
486C2×C5—21no—
487C2×C5—21no—
488C2×C5—242no—
489C2×C5#495811yescharacteristic
490C2×C5C2×#82311yescharacteristic
491C2×C5#513211yescharacteristic
492D5—22no—
493D5—41no—
494C23—51no—
495C23—51no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23—102no—
500C23—101no—
501C23—101no—
502C23—122no—
503C23—122no—
504C23—201no—
505C23C4×D3×D511yescharacteristic
506D4—121no—
507D4—121no—
508D4—121no—
509D4—121no—
510D4—604no—
511D4—604no—
512C2×C4—51no—
513C2×C4—51no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—302no—
525C2×C4—602no—
526C2×C4#676511yescharacteristic
527C2×C4D5×D1211yescharacteristic
528C2×C4#677711yescharacteristic
529C2×C4#678011yescharacteristic
530C2×C4D5×#6711yescharacteristic
531C2×C4D3×C22⋊D511yescharacteristic
532C8—41no—
533C8—62no—
534C8—201no—
535C8—302no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—102no—
539C2×C3—201no—
540C2×C3D5×#10011yescharacteristic
541C2×C3#6503411yescharacteristic
542C2×C3#6524511yescharacteristic
543D3—82no—
544C5#6998511yescharacteristic, complemented, Sylow
545C22—21no—
546C22—21no—
547C22—52no—
548C22—101no—
549C22—101no—
550C22—101no—
551C22—101no—
552C22—101no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—202no—
557C22—202no—
558C22—242no—
559C22—242no—
560C22#9144111yescharacteristic
561C22#9144211yescharacteristic
562C22D5×(C2×D3)⋊C411yescharacteristic
563C4—101no—
564C4—101no—
565C4—602no—
566C4#9140811yescharacteristic
567C4#9141411yescharacteristic
568C4#9147511yescharacteristic
569C4#9154911yescharacteristic
570C3#1060087411yescharacteristic, complemented, Sylow
571C2—21no—
572C2—21no—
573C2—102no—
574C2—201no—
575C2—242no—
576C2D5×#82311yescharacteristic
577C2#1174823011yescharacteristic
578C2#1174890611yescharacteristic
subgroupG/H
centerC22D5×(C2×D3)⋊C4
commutatorC23×C3×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×#91449C2
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
2. C2×D5×#208C2semidirect
3. C2×C5×#823C2semidirect
4. C3×#65415C2semidirect
5. C2×#90959C2semidirect
6. #11746375C2semidirect
7. #11745987C2semidirect
35. #65415D3semidirect
66. #6710D4semidirect
69. C5×#208D4semidirect
82. C2×#823D5semidirect
234. C2×C3×D5#100semidirect
278. #208C22⋊D5semidirect
412. C2×D5#823semidirect
460. C3×C5#1900semidirect
544. C5#69985semidirect
570. C3#10600874semidirect

Maximal subgroups

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

subgroupindex
C2×#914492
C2×D5×#2082
C2×C5×#8232
C3×#654152
C2×#909592
#117463752
#117459872
#106008743
#699855