Finite Group Databasebeta

← #12069012

Subgroups — #12069012

Order 1280.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614252C211yesmaximal, characteristic, complemented
2C4×#4096C211yesmaximal, characteristic
3#10610744C211yesmaximal, characteristic, complemented
4C5×#1477C211yesmaximal, characteristic, complemented
5#10601548C211yesmaximal, characteristic, complemented
6#10601535C211yesmaximal, characteristic, complemented
7C2×#64699C211yesmaximal, characteristic, complemented
8#10599013C212yesmaximal, complemented
9#10599012C212yesmaximal, complemented
10#10598952C212yesmaximal, complemented
11#10598946C212yesmaximal, complemented
12C4×C5×#41C2211yescharacteristic
13C2×#4096C411yescharacteristic
14C4×#709C2211yescharacteristic
15C5×#431C2211yescharacteristic
16C2×C5×C4⋊C8C2211yescharacteristic
17#65522C411yescharacteristic
18C8×C22⋊D5—22no—
19C4×#689C411yescharacteristic
20C4×#689C2211yescharacteristic
21C4×#688C411yescharacteristic
22C4×#688C2211yescharacteristic
23#65313C2212yes—
24C4×#687C411yescharacteristic
25C4×#687C2211yescharacteristic
26C4×#686C411yescharacteristic
27C4×#686C2211yescharacteristic
28#65301C2212yes—
29#65300—22no—
30#65291C411yescharacteristic
31#65287C411yescharacteristic
32#65215—21no—
33#65214C2211yescharacteristic
34#65213—21no—
35C2×C4⋊#168C2211yescharacteristic
36C2×C4×#168C2211yescharacteristic
37#65123C2212yes—
38#65122—21no—
39D5×C4⋊C8—21no—
40#65119C2212yes—
41C5×Q8⋊C8C2212yes—
42C5×D4⋊C8C2212yes—
43#64699C2212yes—
44#64699C2212yes—
45#64679C2212yes—
46#64676C2212yes—
47#64675C2212yes—
48#64674C2212yes—
49#17881—51nomaximal, Sylow
50C2×C5×#41C2×C411yescharacteristic
51C2×#709C2×C411yescharacteristic
52C22×C5×C8—21no—
53C4×C5×Q8C2×C411yescharacteristic
54C4×C5×Q8C2311yescharacteristic
55C4×C5×D4—21no—
56C4×C5×D4—21no—
57C4×C5×D4C2×C411yescharacteristic
58C4×C5×D4C2311yescharacteristic
59C5×#117—21no—
60C5×#117C2×C411yescharacteristic
61C2×C42×C5—21no—
62C2×C42×C5C2311yescharacteristic
63#4096—24no—
64C2×#689C2×C411yescharacteristic
65C2×#688C2×C411yescharacteristic
66C2×#687C2×C411yescharacteristic
67C2×#686C2×C411yescharacteristic
68C4×C22⋊D5—21no—
69C4×C22⋊D5—21no—
70#4072C2×C411yescharacteristic
71C2×#681—22no—
72C22×#168—21no—
73C22×#168C2×C411yescharacteristic
74C22×#168C2×C411yescharacteristic
75C2×C8×D5—22no—
76C4×D20C2×C411yescharacteristic
77C4×D20C2311yescharacteristic
78#4018—21no—
79C42×D5—21no—
80C4×Dic10C2×C411yescharacteristic
81C4×Dic10C2311yescharacteristic
82C5×C4⋊C8—21no—
83C5×C4⋊C8—21no—
84C5×C4⋊C8C2312yes—
85C5×C22⋊C8—21no—
86C4×C5×C8C2312yes—
87#3967C2×C411yescharacteristic
88#3967C2×C411yescharacteristic
89#3964C2×C411yescharacteristic
90#3964C2×C411yescharacteristic
91C22⋊#168—22no—
92C22⋊#168D412yes—
93#3952—22no—
94#3946—22no—
95C8×Dic5—22no—
96#3942C2×C411yescharacteristic
97#3942C2×C411yescharacteristic
98#3941C2×C411yescharacteristic
99#3941C2×C411yescharacteristic
100#3940C2×C411yescharacteristic
101#3940C2×C411yescharacteristic
102#3939C2×C411yescharacteristic
103#3939C2×C411yescharacteristic
104C4⋊#168—21no—
105C4⋊#168—22no—
106C4⋊#168D411yescharacteristic
107C4⋊#168D411yescharacteristic
108C4⋊#168C2312yes—
109#3935—22no—
110C4×#168C2×C411yescharacteristic
111C4×#168C2×C411yescharacteristic
112C4×#168C2311yescharacteristic
113C4×#168C2311yescharacteristic
114C4×#168C2312yes—
115#2989—51no—
116#2988—51no—
117C4×#135—51no—
118#1588—52no—
119#1585—52no—
120#1584—52no—
121#1581—52no—
122C2×#335—51no—
123#1477—51no—
124#1477D511yescharacteristic, complemented
125#1452—51no—
126C5×#41—22no—
127C5×#41—22no—
128C2×C5×Q8C22×C411yescharacteristic
129C2×C5×D4—21no—
130C2×C5×D4C22×C411yescharacteristic
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5C22×C411yescharacteristic
135C22×C4×C5C2×D411yescharacteristic
136C22×C4×C5#4111yescharacteristic
137C2×C22⋊D5—21no—
138#709—22no—
139#709M1614yescomplemented
140C2×D20C22×C411yescharacteristic
141C2×C4×D5—21no—
142C2×C4×D5—21no—
143C2×C4×D5—21no—
144C2×Dic10C22×C411yescharacteristic
145C2×C5×C8—22no—
146C2×C5×C8—22no—
147C2×C5×C8—81no—
148C5×C4⋊C4—21no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4C22×C411yescharacteristic
151C5×C4⋊C4C22×C411yescharacteristic
152C5×C22⋊C4—21no—
153C5×C22⋊C4—21no—
154C5×C22⋊C4—21no—
155C42×C5—21no—
156C42×C5—21no—
157C42×C5—21no—
158C42×C5C22×C411yescharacteristic
159C42×C5C2×D411yescharacteristic
160C42×C5#4111yescharacteristic
161C42×C5C2411yescharacteristic
162C22⋊Dic5—21no—
163#689—22no—
164#688—22no—
165#687—22no—
166#686—22no—
167#685—21no—
168#685—21no—
169C4⋊Dic5C22×C411yescharacteristic
170C4⋊Dic5C22×C411yescharacteristic
171#683—21no—
172#683—21no—
173C4×Dic5—21no—
174C4×Dic5—21no—
175C4×Dic5—21no—
176#681—82no—
177C2×#168—22no—
178C2×#168—22no—
179C2×#168—22no—
180C2×#168—81no—
181C2×#168C22×C411yescharacteristic
182C2×#168C22×C411yescharacteristic
183C2×#168C22×C411yescharacteristic
184C2×#168C22×C411yescharacteristic
185C2×#168C2×D412yes—
186C2×#168C2×D412yes—
187C8×D5—82no—
188C2×#135—51no—
189C4×#41—51no—
190C4×#41C2×D511yescharacteristic
191#446—52no—
192C8×Q8—52no—
193C4×Q16—51no—
194C4×Q16—51no—
195C4×SD16—51no—
196C4×SD16—51no—
197C4×SD16—51no—
198C4×SD16—51no—
199C4×D8—51no—
200C4×D8—51no—
201#436—52no—
202#435—102no—
203C8×D4—52no—
204C8×D4—102no—
205#432—101no—
206#431—51no—
207#431—101no—
208#431C2×D511yescharacteristic
209#427—51no—
210#423—101no—
211C2×C4⋊C8—51no—
212C2×C4⋊C8—101no—
213C2×C4⋊C8C2×D511yescharacteristic
214#416—51no—
215#416—51no—
216C2×C4×C8—51no—
217#335—52no—
218#335—52no—
219Q8⋊C8—52no—
220Q8⋊C8C2×D512yes—
221D4⋊C8—52no—
222D4⋊C8C2×D512yes—
223#322—52no—
224#322—52no—
225C23×C5—21no—
226C23×C5C4×D411yescharacteristic
227C22×D5—21no—
228C5×Q8—21no—
229C5×Q8—21no—
230C5×D4—21no—
231C5×D4—21no—
232C5×D4—42no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—41no—
241C2×C4×C5—42no—
242C2×C4×C5—81no—
243C2×C4×C5C4×D411yescharacteristic
244C2×C4×C5C4×D411yescharacteristic
245C2×C4×C5C4×D411yescharacteristic
246C2×C4×C5C2×M1612yes—
247C2×C4×C5#13112yes—
248C2×C4×C5C23×C411yescharacteristic
249C2×C4×C5C22×D411yescharacteristic
250C2×C4×C5C2×#4111yescharacteristic
251C22⋊D5—24no—
252C2×Dic5—21no—
253C2×Dic5—21no—
254C2×Dic5—21no—
255D20—21no—
256D20C2×M1612yes—
257C4×D5—24no—
258C4×D5—41no—
259C4×D5—41no—
260C4×D5—81no—
261Dic10—21no—
262Dic10C2×M1612yes—
263C5×C8—42no—
264C5×C8—42no—
265#168—21no—
266#168—21no—
267#168—21no—
268#168—21no—
269#168—22no—
270#168—22no—
271#168—42no—
272C2×#41—51no—
273C2×#41C4×D511yescharacteristic
274#135—104no—
275C2×Q16—51no—
276C2×SD16—51no—
277C2×SD16—51no—
278C2×D8—51no—
279C2×M16—102no—
280C22×C8—21no—
281C22×C8—51no—
282C22×C8—51no—
283C22×C8—102no—
284C22×C8—101no—
285C4×Q8—51no—
286C4×Q8—51no—
287C4×Q8C4×D511yescharacteristic
288C4×Q8C22×D511yescharacteristic
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—51no—
292C4×D4—51no—
293C4×D4—101no—
294C4×D4—101no—
295C4×D4C4×D511yescharacteristic
296C4×D4C22×D511yescharacteristic
297#117—21no—
298#117—51no—
299#117—101no—
300#117C4×D511yescharacteristic
301C2×C42—21no—
302C2×C42—101no—
303C2×C42C22×D511yescharacteristic
304#107—51no—
305#107—51no—
306#106—51no—
307#106—51no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—51no—
311C4⋊C8—51no—
312C4⋊C8—52no—
313C4⋊C8—102no—
314C4⋊C8—102no—
315C4⋊C8—101no—
316C4⋊C8C22×D512yes—
317#103—51no—
318#103—51no—
319#103—51no—
320#103—51no—
321#102—51no—
322#102—51no—
323#102—51no—
324#102—51no—
325C22⋊C8—21no—
326C22⋊C8—52no—
327C22⋊C8—102no—
328C22⋊C8—102no—
329#97—102no—
330C4×C8—51no—
331C4×C8—51no—
332C4×C8—51no—
333C4×C8—51no—
334C4×C8—52no—
335C4×C8—102no—
336C4×C8C22×D512yes—
337C22×C5—21no—
338C22×C5—21no—
339C22×C5—41no—
340C22×C5—41no—
341C22×C5—41no—
342C22×C5#43512yes—
343C22×C5C2×C4×D411yescharacteristic
344C2×D5—22no—
345C2×D5—41no—
346C2×D5—41no—
347C4×C5—21no—
348C4×C5—21no—
349C4×C5—21no—
350C4×C5—41no—
351C4×C5—41no—
352C4×C5—41no—
353C4×C5#43512yes—
354C4×C5C22×M1611yescharacteristic
355C4×C5C2×#13111yescharacteristic
356C4×C5#57611yescharacteristic
357C4×C5#57811yescharacteristic
358Dic5—22no—
359Dic5—41no—
360Dic5—41no—
361#41—22no—
362#41—22no—
363#41—54no—
364#41—102no—
365C2×Q8—51no—
366C2×Q8C2×C4×D511yescharacteristic
367C2×D4—21no—
368C2×D4—51no—
369C2×D4—101no—
370C2×D4C2×C4×D511yescharacteristic
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—101no—
375C22×C4—101no—
376C22×C4—101no—
377C22×C4C2×C4×D511yescharacteristic
378C22×C4D4×D511yescharacteristic
379C22×C4#71111yescharacteristic
380Q16—102no—
381SD16—102no—
382SD16—102no—
383D8—102no—
384M16—402no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—51no—
388C2×C8—51no—
389C2×C8—51no—
390C2×C8—51no—
391C2×C8—52no—
392C2×C8—52no—
393C2×C8—81no—
394C2×C8—102no—
395C2×C8—102no—
396C2×C8—102no—
397C2×C8—402no—
398C2×C8—401no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—51no—
402C4⋊C4—51no—
403C4⋊C4—101no—
404C4⋊C4—101no—
405C4⋊C4C2×C4×D511yescharacteristic
406C4⋊C4C2×C4×D511yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—101no—
411C22⋊C4—101no—
412C22⋊C4—101no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—101no—
417C42—101no—
418C42—101no—
419C42C2×C4×D511yescharacteristic
420C42D4×D511yescharacteristic
421C42#71111yescharacteristic
422C42C23×D511yescharacteristic
423C2×C5—22no—
424C2×C5—81no—
425C2×C5C2×#43511yescharacteristic
426C2×C5#295111yescharacteristic
427C2×C5#295711yescharacteristic
428D5—42no—
429C23—21no—
430C23—101no—
431C23#402711yescharacteristic
432Q8—21no—
433Q8—21no—
434Q8—52no—
435Q8—101no—
436D4—21no—
437D4—21no—
438D4—42no—
439D4—52no—
440D4—104no—
441D4—101no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—101no—
453C2×C4—104no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—201no—
457C2×C4—201no—
458C2×C4—401no—
459C2×C4#402711yescharacteristic
460C2×C4#402711yescharacteristic
461C2×C4#402711yescharacteristic
462C2×C4#404712yes—
463C2×C4D5×M1612yes—
464C2×C4C22×C4×D511yescharacteristic
465C2×C4C2×D4×D511yescharacteristic
466C2×C4C2×#71111yescharacteristic
467C8—42no—
468C8—42no—
469C8—101no—
470C8—101no—
471C8—101no—
472C8—101no—
473C8—102no—
474C8—102no—
475C8—202no—
476C5#1788111yescharacteristic, complemented, Sylow
477C22—21no—
478C22—21no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—102no—
483C22—201no—
484C22—201no—
485C22#6501712yes—
486C22C2×#402711yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—102no—
494C4—201no—
495C4—201no—
496C4#6501712yes—
497C4C2×#404711yescharacteristic
498C4C2×D5×M1611yescharacteristic
499C4#6610811yescharacteristic
500C4#6611311yescharacteristic
501C2—22no—
502C2—81no—
503C2—202no—
504C2C2×#6501711yescharacteristic
505C2#1061059911yescharacteristic
506C2#1061072111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#1477C2
socleC22×C5C2×C4×D4
prime radicalC2×#4096C4
cosocleC42×C5C24
hypercenter#1477D5
hypocenterC5#17881
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
1. #10614252C2semidirect
3. #10610744C2semidirect
4. C5×#1477C2semidirect
5. #10601548C2semidirect
6. #10601535C2semidirect
7. C2×#64699C2semidirect
8. #10599013C2semidirect
9. #10599012C2semidirect
10. #10598952C2semidirect
11. #10598946C2semidirect
124. #1477D5semidirect
139. #709M16semidirect
476. C5#17881semidirect

Maximal subgroups

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

subgroupindex
#106142522
C4×#40962
#106107442
C5×#14772
#106015482
#106015352
C2×#646992
#105990132
#105990122
#105989522
#105989462
#178815