Finite Group Databasebeta

← #14298560

Subgroups — #14298560

Order 1920.

612 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D15×#117C211yesmaximal, characteristic
2C4×#6667C211yesmaximal, characteristic
3#11748351C211yesmaximal, characteristic
4C5×#4532C211yesmaximal, characteristic, complemented
5C3×#64752C211yesmaximal, characteristic, complemented
6#11745978C211yesmaximal, characteristic, complemented
7#11745976C211yesmaximal, characteristic, complemented
8#10599547—31nomaximal, Hall
9C22×C4×D15C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11#91815C2211yescharacteristic
12#91781C411yescharacteristic, complemented
13#91781C411yescharacteristic, complemented
14C4⋊C4×D15C411yescharacteristic, complemented
15C4⋊C4×D15C411yescharacteristic, complemented
16C22⋊C4×D15—21no—
17#91766—21no—
18#91761—21no—
19C42×D15—21no—
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C5×#208C2211yescharacteristic
22C2×C3×C4×Dic5C2211yescharacteristic
23C2×C3×#681C2211yescharacteristic
24C2×#6667—21no—
25#90994—21no—
26#90957—21no—
27#90957—21no—
28#68754—51nomaximal, Hall
29D5×#117—31no—
30#65415—31no—
31C4×#685—31no—
32C5×#339—31no—
33#64775—31no—
34#64752D311yescharacteristic, complemented
35#64743—31no—
36C23×D15D411yescharacteristic
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15D411yescharacteristic
39C2×C4×D15—21no—
40C2×C4×D15—21no—
41C2×C4×D15—21no—
42C2×C4×D15—21no—
43C2×C4×D15—21no—
44C2×C4×D15—21no—
45C2×C4×D15C2×C411yescharacteristic
46C2×C4×D15C2×C411yescharacteristic
47C2×C4×D15D411yescharacteristic
48C2×C4×D15D411yescharacteristic
49C22×C5×Dic3—21no—
50C22×C3×Dic5—21no—
51C3×C5×C4⋊C4C2×C411yescharacteristic
52C3×C5×C4⋊C4C2×C411yescharacteristic
53C3×C5×C22⋊C4—21no—
54C3×C42×C5—21no—
55#6719—21no—
56#6714—21no—
57#6714—21no—
58C4⋊Dic15C2×C411yescharacteristic
59C4⋊Dic15C2×C411yescharacteristic
60#6712—21no—
61#6712—21no—
62C4×Dic15—21no—
63C4×Dic15—21no—
64C4×Dic15—21no—
65C4×C5×Dic3—21no—
66C4×C5×Dic3—21no—
67C5×#208—21no—
68C5×#208—21no—
69C2×C5×C3⋊C8—21no—
70C3×C4×Dic5—21no—
71C3×C4×Dic5—21no—
72C3×#681—21no—
73C3×#681—21no—
74C2×C3×#168—21no—
75#6667—24no—
76D3×#117—51no—
77#5126—51no—
78C4×(C2×D3)⋊C4—51no—
79C3×#339—51no—
80#4555—51no—
81#4532D511yescharacteristic, complemented
82#4523—51no—
83C22×C4×D5—31no—
84C2×C5×M16—31no—
85C5×#117C2×D311yescharacteristic
86C2×C42×C5—31no—
87C2×#685—61no—
88#4072—31no—
89C2×C4×Dic5C2×D311yescharacteristic
90C2×#681C2×D311yescharacteristic
91#4038—31no—
92#4038—31no—
93D5×C4⋊C4—31no—
94D5×C4⋊C4—31no—
95D5×C22⋊C4—61no—
96#4023—61no—
97#4018—61no—
98C42×D5—61no—
99#3963—61no—
100#3952—61no—
101#3952—61no—
102#1826—151noSylow
103C23×C3×C5C2×D411yescharacteristic
104C22×D15—21no—
105C22×D15—21no—
106C22×D15—22no—
107C22×D15—41no—
108C22×D15C22⋊C411yescharacteristic
109C22×D15C22⋊C411yescharacteristic
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5C22×C411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5#4111yescharacteristic
115C2×C3×C4×C5#4111yescharacteristic
116C2×C3×C4×C5#4111yescharacteristic
117C2×C3×C4×C5#4111yescharacteristic
118C2×Dic15—21no—
119C2×Dic15—21no—
120C2×Dic15—21no—
121C2×Dic15—21no—
122C2×Dic15C22⋊C411yescharacteristic
123C2×Dic15C22⋊C411yescharacteristic
124C4×D15—21no—
125C4×D15—21no—
126C4×D15—21no—
127C4×D15—21no—
128C4×D15—21no—
129C4×D15—21no—
130C4×D15—21no—
131C4×D15—21no—
132C4×D15—42no—
133C4×D15—42no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—41no—
136C2×C5×Dic3—41no—
137C2×C3×Dic5—22no—
138C2×C3×Dic5—41no—
139C2×C3×Dic5—41no—
140C3×#168—41no—
141C5×C3⋊C8—41no—
142C22×C4×D3—51no—
143C2×C3×M16—51no—
144C3×#117C2×D511yescharacteristic
145C2×C3×C42—51no—
146C2×(C2×D3)⋊C4—101no—
147#926—51no—
148C2×C4×Dic3C2×D511yescharacteristic
149C2×#208C2×D511yescharacteristic
150#892—51no—
151#892—51no—
152D3×C4⋊C4—51no—
153D3×C4⋊C4—51no—
154D3×C22⋊C4—101no—
155#877—101no—
156#872—101no—
157C42×D3—101no—
158C3⋊#95—101no—
159(C2×D3)⋊C8—101no—
160(C2×D3)⋊C8—101no—
161C23×D5—31no—
162C22×C4×C5—61no—
163C22×C4×C5C22×D311yescharacteristic
164C22×Dic5—21no—
165C22×Dic5—31no—
166C2×C4×D5—31no—
167C2×C4×D5—31no—
168C2×C4×D5—31no—
169C2×C4×D5—31no—
170C2×C4×D5—61no—
171C2×C4×D5—61no—
172C2×C4×D5—61no—
173C2×C4×D5—61no—
174C2×C4×D5—61no—
175C2×C4×D5—61no—
176C5×M16—61no—
177C5×M16—61no—
178C2×C5×C8—61no—
179C5×C4⋊C4C4×D311yescharacteristic
180C5×C4⋊C4C4×D311yescharacteristic
181C5×C22⋊C4—21no—
182C42×C5—21no—
183C42×C5—61no—
184C42×C5—61no—
185C22⋊Dic5—61no—
186#685—64no—
187#685—61no—
188#685—61no—
189C4⋊Dic5—31no—
190C4⋊Dic5—31no—
191#683—61no—
192#683—61no—
193C4×Dic5—21no—
194C4×Dic5—21no—
195C4×Dic5—61no—
196C4×Dic5—61no—
197C4×Dic5—61no—
198#681—21no—
199#681—21no—
200C2×#168—21no—
201C2×#117—151no—
202#407—151no—
203C4×C22⋊C4—151no—
204#339—31no—
205#339—51no—
206#339—151no—
207#339—151no—
208C22×C3×C5—21no—
209C22×C3×C5—21no—
210C22×C3×C5C2×C22⋊C411yescharacteristic
211C22×C3×C5#12311yescharacteristic
212C22×C3×C5#12311yescharacteristic
213C2×D15—21no—
214C2×D15—21no—
215C2×D15—21no—
216C2×D15—21no—
217C2×D15—21no—
218C2×D15—42no—
219C2×D15—42no—
220C2×D15#10412yes—
221C3×C4×C5—41no—
222C3×C4×C5—41no—
223C3×C4×C5#11711yescharacteristic
224C3×C4×C5#11711yescharacteristic
225C3×C4×C5#12311yescharacteristic
226C3×C4×C5#12311yescharacteristic
227Dic15—21no—
228Dic15—21no—
229Dic15—41no—
230Dic15—41no—
231C3×Dic5—42no—
232C5×Dic3—42no—
233C23×D3—51no—
234C22×C3×C4—101no—
235C22×C3×C4C22×D511yescharacteristic
236C22×Dic3—21no—
237C22×Dic3—51no—
238C2×C4×D3—51no—
239C2×C4×D3—51no—
240C2×C4×D3—51no—
241C2×C4×D3—51no—
242C2×C4×D3—101no—
243C2×C4×D3—101no—
244C2×C4×D3—101no—
245C2×C4×D3—101no—
246C2×C4×D3—101no—
247C2×C4×D3—101no—
248C3×M16—101no—
249C3×M16—101no—
250C2×C3×C8—101no—
251C3×C4⋊C4C4×D511yescharacteristic
252C3×C4⋊C4C4×D511yescharacteristic
253C3×C22⋊C4—21no—
254C3×C42—21no—
255C3×C42—101no—
256C3×C42—101no—
257#217—101no—
258(C2×D3)⋊C4—104no—
259(C2×D3)⋊C4—101no—
260(C2×D3)⋊C4—101no—
261C4⋊Dic3—51no—
262C4⋊Dic3—51no—
263Dic3⋊C4—101no—
264Dic3⋊C4—101no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—101no—
268C4×Dic3—101no—
269C4×Dic3—101no—
270#208—21no—
271#208—21no—
272C2×C3⋊C8—21no—
273C23×C5D3×D411yescharacteristic
274C22×D5—31no—
275C22×D5—31no—
276C22×D5—62no—
277C22×D5—61no—
278C22×D5—61no—
279C22×D5—121no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—62no—
283C2×C4×C5—121no—
284C2×C4×C5—121no—
285C2×C4×C5C2×C4×D311yescharacteristic
286C2×C4×C5#23511yescharacteristic
287C2×C4×C5#23511yescharacteristic
288C2×C4×C5D3×D411yescharacteristic
289C2×C4×C5#23711yescharacteristic
290C2×C4×C5#23911yescharacteristic
291C2×Dic5—22no—
292C2×Dic5—31no—
293C2×Dic5—31no—
294C2×Dic5—41no—
295C2×Dic5—41no—
296C2×Dic5—61no—
297C2×Dic5—61no—
298C2×Dic5—61no—
299C2×Dic5—61no—
300C4×D5—61no—
301C4×D5—61no—
302C4×D5—61no—
303C4×D5—61no—
304C4×D5—61no—
305C4×D5—61no—
306C4×D5—61no—
307C4×D5—61no—
308C4×D5—122no—
309C4×D5—122no—
310C5×C8—121no—
311#168—41no—
312C23×C4—151no—
313C2×M16—31no—
314C2×M16—51no—
315#117—151no—
316#117—151no—
317#117—151no—
318#117—301no—
319#117—301no—
320#117D3×D511yescharacteristic
321C2×C4⋊C4—151no—
322C2×C4⋊C4—151no—
323C2×C22⋊C4—301no—
324C2×C22⋊C4—301no—
325C2×C42—31no—
326C2×C42—51no—
327C2×C42—301no—
328C22⋊C8—301no—
329C22⋊C8—301no—
330#95—301no—
331C2×C3×C5—21no—
332C2×C3×C5—21no—
333C2×C3×C5#38111yescharacteristic
334C2×C3×C5C2×#10411yescharacteristic
335C2×C3×C5#42111yescharacteristic
336D15—22no—
337D15—41no—
338C23×C3D4×D511yescharacteristic
339C22×D3—51no—
340C22×D3—51no—
341C22×D3—102no—
342C22×D3—101no—
343C22×D3—101no—
344C22×D3—201no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—102no—
348C2×C3×C4—201no—
349C2×C3×C4—201no—
350C2×C3×C4C2×C4×D511yescharacteristic
351C2×C3×C4#70911yescharacteristic
352C2×C3×C4#70911yescharacteristic
353C2×C3×C4D4×D511yescharacteristic
354C2×C3×C4#71111yescharacteristic
355C2×C3×C4#71311yescharacteristic
356C2×Dic3—22no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—51no—
360C2×Dic3—51no—
361C2×Dic3—101no—
362C2×Dic3—101no—
363C2×Dic3—101no—
364C2×Dic3—101no—
365C4×D3—101no—
366C4×D3—101no—
367C4×D3—101no—
368C4×D3—101no—
369C4×D3—101no—
370C4×D3—101no—
371C4×D3—101no—
372C4×D3—101no—
373C4×D3—202no—
374C4×D3—202no—
375C3×C8—201no—
376C3⋊C8—41no—
377C22×C5—21no—
378C22×C5—21no—
379C22×C5D3×C22⋊C411yescharacteristic
380C22×C5#88311yescharacteristic
381C22×C5#89411yescharacteristic
382C2×D5—32no—
383C2×D5—61no—
384C2×D5—61no—
385C2×D5—61no—
386C2×D5—61no—
387C2×D5—61no—
388C2×D5—122no—
389C2×D5—122no—
390C4×C5—41no—
391C4×C5—41no—
392C4×C5—122no—
393C4×C5#87211yescharacteristic
394C4×C5#88311yescharacteristic
395C4×C5#89211yescharacteristic
396C4×C5#89411yescharacteristic
397Dic5—42no—
398Dic5—61no—
399Dic5—61no—
400Dic5—121no—
401Dic5—121no—
402C24—151no—
403C22×C4—61no—
404C22×C4—101no—
405C22×C4—151no—
406C22×C4—151no—
407C22×C4—151no—
408C22×C4—151no—
409C22×C4—151no—
410C22×C4—301no—
411C22×C4—301no—
412C22×C4—301no—
413C22×C4—301no—
414C22×C4—301no—
415C22×C4—301no—
416C22×C4C2×D3×D511yescharacteristic
417M16—61no—
418M16—61no—
419M16—101no—
420M16—101no—
421C2×C8—61no—
422C2×C8—101no—
423C4⋊C4—151no—
424C4⋊C4—151no—
425C4⋊C4—301no—
426C4⋊C4—301no—
427C4⋊C4#120011yescharacteristic
428C4⋊C4#120011yescharacteristic
429C22⋊C4—21no—
430C22⋊C4—301no—
431C22⋊C4—301no—
432C22⋊C4—304no—
433C22⋊C4—301no—
434C42—21no—
435C42—61no—
436C42—61no—
437C42—101no—
438C42—101no—
439C42—301no—
440C42—301no—
441C42—301no—
442C3×C5#182611yescharacteristic, complemented, Hall
443C22×C3—21no—
444C22×C3—21no—
445C22×C3D5×C22⋊C411yescharacteristic
446C22×C3#402911yescharacteristic
447C22×C3#404011yescharacteristic
448C2×D3—52no—
449C2×D3—101no—
450C2×D3—101no—
451C2×D3—101no—
452C2×D3—101no—
453C2×D3—101no—
454C2×D3—202no—
455C2×D3—202no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—202no—
459C3×C4#401811yescharacteristic
460C3×C4#402911yescharacteristic
461C3×C4#403811yescharacteristic
462C3×C4#404011yescharacteristic
463Dic3—42no—
464Dic3—101no—
465Dic3—101no—
466Dic3—201no—
467Dic3—201no—
468C2×C5—21no—
469C2×C5—21no—
470C2×C5#466411yescharacteristic
471C2×C5D3×#10411yescharacteristic
472C2×C5#482111yescharacteristic
473D5—62no—
474D5—121no—
475C23—151no—
476C23—151no—
477C23—302no—
478C23—301no—
479C23—301no—
480C23—601no—
481C23#679011yescharacteristic
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—62no—
485C2×C4—102no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—151no—
489C2×C4—151no—
490C2×C4—201no—
491C2×C4—201no—
492C2×C4—301no—
493C2×C4—301no—
494C2×C4—301no—
495C2×C4—301no—
496C2×C4—301no—
497C2×C4—301no—
498C2×C4—301no—
499C2×C4—301no—
500C2×C4—301no—
501C2×C4—301no—
502C2×C4—301no—
503C2×C4—301no—
504C2×C4—602no—
505C2×C4—602no—
506C2×C4#676611yescharacteristic
507C2×C4#676811yescharacteristic
508C2×C4#677111yescharacteristic
509C2×C4#677111yescharacteristic
510C2×C4C2×#120011yescharacteristic
511C2×C4#679011yescharacteristic
512C8—121no—
513C8—201no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3#6495311yescharacteristic
517C2×C3D5×#10411yescharacteristic
518C2×C3#6511011yescharacteristic
519D3—102no—
520D3—201no—
521C5#6875411yescharacteristic, complemented, Sylow
522C22—21no—
523C22—21no—
524C22—152no—
525C22—301no—
526C22—301no—
527C22—301no—
528C22—301no—
529C22—301no—
530C22—602no—
531C22—602no—
532C22#9135411yescharacteristic
533C22#9135511yescharacteristic
534C22#9156111yescharacteristic
535C4—41no—
536C4—41no—
537C4—122no—
538C4—202no—
539C4—301no—
540C4—301no—
541C4—601no—
542C4—601no—
543C4#9134711yescharacteristic
544C4#9135411yescharacteristic
545C4#9135511yescharacteristic
546C4#9138611yescharacteristic
547C3#1059954711yescharacteristic, complemented, Sylow
548C2—21no—
549C2—21no—
550C2—302no—
551C2—601no—
552C2#1174860611yescharacteristic
553C2#1174943011yescharacteristic
554C2#1174943111yescharacteristic
subgroupG/H
centerC2×C4#6790
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×D15D4
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1826
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. C5×#4532C2semidirect
5. C3×#64752C2semidirect
6. #11745978C2semidirect
7. #11745976C2semidirect
12. #91781C4semidirect
13. #91781C4semidirect
14. C4⋊C4×D15C4semidirect
15. C4⋊C4×D15C4semidirect
34. #64752D3semidirect
81. #4532D5semidirect
442. C3×C5#1826semidirect
521. C5#68754semidirect
547. C3#10599547semidirect

Maximal subgroups

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

subgroupindex
D15×#1172
C4×#66672
#117483512
C5×#45322
C3×#647522
#117459782
#117459762
#105995473
#687545