Finite Group Databasebeta

← #13045396

Subgroups — #13045396

Order 1280.

1862 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10618612C211yesmaximal, characteristic
2#10617551C211yesmaximal, characteristic, complemented
3#10617548C211yesmaximal, characteristic, complemented
4Q8×#704C211yesmaximal, characteristic, complemented
5C2×Q8×#170C211yesmaximal, characteristic, complemented
6#10617321C213yesmaximal, complemented
7#10617320C213yesmaximal, complemented
8#10617279C213yesmaximal, complemented
9#10617278C213yesmaximal, complemented
10#10617232C213yesmaximal, complemented
11#10617227C213yesmaximal, complemented
12#10617206C211yesmaximal, characteristic, complemented
13#10617203C211yesmaximal, characteristic, complemented
14#10617139C213yesmaximal, complemented
15#10617137C213yesmaximal, complemented
16C2×#4131C2216yes—
17C5×#548C411yescharacteristic
18#66168C411yescharacteristic
19#41×Dic5C2211yescharacteristic
20#66151C411yescharacteristic
21C2×Q8×Dic5C2211yescharacteristic
22#66015C413yes—
23#66014C2213yes—
24#65975C413yes—
25#65958C2213yes—
26#65952C413yes—
27#65930C2213yes—
28C4×#713C2211yescharacteristic
29#65903C411yescharacteristic
30#65899C413yes—
31C4×#711C2213yes—
32#65788C2213yes—
33#65788C2213yes—
34Q8×#170C2211yescharacteristic
35Q8×#170C2211yescharacteristic
36Q8×#170C2211yescharacteristic
37Q8×#170C2211yescharacteristic
38Q8×#170C2213yes—
39Q8×#170C2213yes—
40#65775C2213yes—
41#65775C2213yes—
42#65774C2213yes—
43#65774C2213yes—
44D4×#170C2213yes—
45D4×#170C2213yes—
46#65758C2213yes—
47#65758C2213yes—
48#65754C2213yes—
49C4×#704C2213yes—
50#65748C2213yes—
51C2×C4⋊#170C2213yes—
52C2×C4⋊#170C2213yes—
53C2×C4×#170C2213yes—
54#65708C2216yes—
55#65706C2213yes—
56#65706C2213yes—
57#65705C2213yes—
58#65705C2213yes—
59#65702C2213yes—
60#65702C2213yes—
61#65697C2213yes—
62#65697C2213yes—
63#65696C2213yes—
64#65696C2216yes—
65#65696C2211yescharacteristic
66#65695C2216yes—
67#65694C2216yes—
68#65693C2216yes—
69#65692C2216yes—
70#65681C2213yes—
71#65680C2213yes—
72D5⋊(C4⋊C8)C2213yes—
73#65678C2213yes—
74#60274—51nomaximal, Sylow
75C2×C5×#41C2×C411yescharacteristic
76C22×C5×Q8C2×C411yescharacteristic
77C2×#713C2×C411yescharacteristic
78C2×#711C2313yes—
79C2×#704C2313yes—
80C2×#704C2311yescharacteristic
81C2×#704C2316yes—
82C22×#170C2313yes—
83C22×#170C2311yescharacteristic
84C22×#170C2316yes—
85#4131—412no—
86C2×#700C2316yes—
87C2×D5⋊C8C2316yes—
88C5×#124C2×C413yes—
89C5×C22⋊Q8C2×C413yes—
90C5×#121C2×C413yes—
91C4×C5×Q8C2×C411yescharacteristic
92C4×C5×D4C2×C413yes—
93#4093C2×C413yes—
94#4092C2×C413yes—
95Q8×Dic5C2×C411yescharacteristic
96Q8×Dic5C2×C413yes—
97Q8×Dic5C2×C411yescharacteristic
98Q8×Dic5C2311yescharacteristic
99Q8×Dic5C2313yes—
100#4086C2×C413yes—
101#4085C2×C416yes—
102#4084C2×C413yes—
103D4×Dic5C2×C413yes—
104D4×Dic5C2×C413yes—
105D4×Dic5C2313yes—
106#4076C2×C413yes—
107#4076C2×C413yes—
108C4×C22⋊D5C2×C413yes—
109C4×C22⋊D5C2×C413yes—
110#4072C2×C413yes—
111C2×C4⋊Dic5C2×C413yes—
112C2×C4×Dic5C2313yes—
113C2×C4×Dic5C2313yes—
114#4044C2×C416yes—
115#4039C2313yes—
116#4038C2×C413yes—
117#4038C2313yes—
118#4036C2×C413yes—
119#4034C2×C413yes—
120#4033C2×C413yes—
121#4032C2×C416yes—
122#4031C2×C416yes—
123#4029C2×C416yes—
124#4027C2×C413yes—
125#4027C2×C413yes—
126#4027C2316yes—
127#4025C2×C416yes—
128#4023C2311yescharacteristic
129#4023C2313yes—
130C4×D20C2×C413yes—
131C42×D5C2313yes—
132#4016C2×C413yes—
133C4×Dic10C2×C413yes—
134C22⋊#170C2313yes—
135C22⋊#170C2313yes—
136C22⋊#170C2313yes—
137C22⋊#170C2313yes—
138#4004C2311yescharacteristic
139#4004C2313yes—
140#4004C2316yes—
141#4004C2311yescharacteristic
142#4004C2313yes—
143#4003C2313yes—
144#4003C2313yes—
145#4003C2313yes—
146#4003C2313yes—
147#4002C2313yes—
148#4002C2313yes—
149C4⋊#170C2313yes—
150C4⋊#170C2313yes—
151C4⋊#170C2311yescharacteristic
152C4⋊#170C2313yes—
153C4⋊#170C2313yes—
154C4⋊#170C2313yes—
155C4⋊#170C2316yes—
156C4⋊#170C2311yescharacteristic
157C4⋊#170C2313yes—
158C4×#170C2313yes—
159C4×#170C2313yes—
160C4×#170C2313yes—
161C4×#170C2313yes—
162C4×#170C2313yes—
163C4×#170C2313yes—
164#3480—51no—
165#2994—53no—
166#2994—53no—
167#2993—53no—
168#2993—53no—
169#2991—51no—
170#2991—51no—
171#2988—51no—
172#2988—51no—
173Q8×M16—51no—
174C2×C8×Q8—51no—
175#2935—53no—
176#2931—53no—
177#2922—53no—
178#2906—53no—
179C5×#41—24no—
180C2×C5×Q8—24no—
181C2×C5×Q8C22×C413yes—
182C2×C5×Q8C22×C411yescharacteristic
183C2×C5×D4C22×C413yes—
184C2×C5×D4C22×C413yes—
185C22×C4×C5C22×C413yes—
186C22×C4×C5C22×C413yes—
187C2×C22⋊D5C22×C413yes—
188C2×C22⋊D5C22×C413yes—
189C2×C22⋊D5C22×C416yes—
190C22×Dic5C22×C413yes—
191C22×Dic5C22×C413yes—
192C22×Dic5C2413yes—
193C22×Dic5C2411yescharacteristic
194#713—24no—
195#711—26no—
196#711—26no—
197C2×D20C22×C413yes—
198C2×C4×D5C22×C413yes—
199C2×C4×D5C22×C413yes—
200C2×C4×D5C2413yes—
201C2×Dic10C22×C413yes—
202#704—26no—
203#704—22no—
204#704—46no—
205C2×#170—26no—
206C2×#170—22no—
207C2×#170—46no—
208C2×#170C2413yes—
209C2×#170C2413yes—
210C2×#170C2413yes—
211C2×#170C2413yes—
212C2×#170C2411yescharacteristic
213C2×#170C2411yescharacteristic
214C2×#170C2411yescharacteristic
215C2×#170C2411yescharacteristic
216#700—46no—
217D5⋊C8—46no—
218C5×C4⋊C4C22×C413yes—
219C5×C4⋊C4C22×C413yes—
220C5×C22⋊C4C22×C411yescharacteristic
221C5×C22⋊C4C22×C413yes—
222C5×C22⋊C4C22×C416yes—
223C42×C5C22×C413yes—
224C22⋊Dic5C22×C413yes—
225C22⋊Dic5C22×C411yescharacteristic
226C22⋊Dic5C22×C416yes—
227#685C22×C416yes—
228#685C22×C413yes—
229#685C22×C413yes—
230#685C22×C416yes—
231C4⋊Dic5C22×C413yes—
232C4⋊Dic5C22×C413yes—
233C4⋊Dic5C22×C413yes—
234C4⋊Dic5C22×C413yes—
235C4⋊Dic5C22×C416yes—
236#683C22×C411yescharacteristic
237#683C22×C413yes—
238#683C22×C416yes—
239#683C22×C413yes—
240#683C22×C411yescharacteristic
241C4×Dic5C22×C413yes—
242C4×Dic5C22×C413yes—
243C4×Dic5C22×C413yes—
244C4×Dic5C22×C413yes—
245C4×Dic5C22×C413yes—
246C4×Dic5C2413yes—
247C4×Dic5C2413yes—
248C4×Dic5C2411yescharacteristic
249C4×Dic5C2411yescharacteristic
250C2×#131—56no—
251#559—53no—
252#557—51no—
253#556—53no—
254#553—53no—
255#553—53no—
256#550—51no—
257#548—51no—
258#548AGL(1,𝔽5)11yescharacteristic
259#532—53no—
260#529—53no—
261#529—53no—
262C4×#41—51no—
263C4×#41—51no—
264C4×#41—53no—
265C2×C4×Q8—51no—
266#446—53no—
267#446—53no—
268C8×Q8—51no—
269C8×Q8—51no—
270C8×Q8—53no—
271C8×Q8—53no—
272C8×Q8—51no—
273C8×Q8—51no—
274#436—56no—
275#436—53no—
276#436—53no—
277#436—53no—
278#436—53no—
279#435—56no—
280#435—56no—
281C8×D4—56no—
282C8×D4—53no—
283C8×D4—53no—
284C8×D4—53no—
285C8×D4—53no—
286#433—53no—
287#433—53no—
288#433—53no—
289#433—53no—
290#431—53no—
291#431—53no—
292#431—53no—
293#431—53no—
294#424—53no—
295#424—56no—
296#424—56no—
297#424—51no—
298#423—53no—
299#423—53no—
300#423—53no—
301C2×C4⋊C8—53no—
302C2×C4⋊C8—53no—
303C2×C4⋊C8—53no—
304#408—53no—
305#408—53no—
306C4×M16—53no—
307C2×C4×C8—53no—
308C23×C5C23×C413yes—
309C23×C5C23×C411yescharacteristic
310C22×D5C23×C413yes—
311C5×Q8—26no—
312C5×Q8#13114yescomplemented
313C5×D4—26no—
314C5×D4—26no—
315C2×C4×C5—26no—
316C2×C4×C5—26no—
317C2×C4×C5C23×C411yescharacteristic
318C2×C4×C5C23×C413yes—
319C2×C4×C5C23×C413yes—
320C2×C4×C5C23×C411yescharacteristic
321C2×C4×C5C23×C413yes—
322C22⋊D5—26no—
323C22⋊D5—26no—
324C22⋊D5—46no—
325C2×Dic5—23no—
326C2×Dic5—23no—
327C2×Dic5—43no—
328C2×Dic5—43no—
329C2×Dic5C23×C411yescharacteristic
330C2×Dic5C23×C411yescharacteristic
331C2×Dic5C23×C413yes—
332C2×Dic5C23×C413yes—
333C2×Dic5C23×C413yes—
334C2×Dic5C23×C411yescharacteristic
335C2×Dic5H(5,𝔽2)12yes—
336C2×Dic52−1+412yes—
337C2×Dic5C2511yescharacteristic
338D20—26no—
339C4×D5—26no—
340C4×D5—23no—
341C4×D5—23no—
342C4×D5—43no—
343Dic10—26no—
344#170—23no—
345#170—23no—
346#170—23no—
347#170—23no—
348#170—21no—
349#170—21no—
350#170—21no—
351#170—21no—
352C2×#41—51no—
353C2×#41—53no—
354C2×#41C2×AGL(1,𝔽5)11yescharacteristic
355C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
356#131—2012no—
357C2×M16—53no—
358C2×M16—51no—
359C2×M16—56no—
360C2×M16—56no—
361C22×C8—53no—
362C22×C8—51no—
363C22×C8—56no—
364C22×C8—56no—
365C4⋊Q8—53no—
366#127—53no—
367#126—56no—
368#126—56no—
369#125—53no—
370#125—53no—
371#124—53no—
372#124—56no—
373#124C2×AGL(1,𝔽5)13yes—
374#123—56no—
375#123—56no—
376C22⋊Q8—53no—
377C22⋊Q8C2×AGL(1,𝔽5)13yes—
378#121—53no—
379#121—56no—
380#121—53no—
381#121—53no—
382#121C2×AGL(1,𝔽5)13yes—
383C4×Q8—51no—
384C4×Q8—53no—
385C4×Q8—51no—
386C4×Q8—53no—
387C4×Q8—51no—
388C4×Q8—53no—
389C4×Q8—53no—
390C4×Q8C2×AGL(1,𝔽5)11yescharacteristic
391C4×D4—53no—
392C4×D4—53no—
393C4×D4—53no—
394C4×D4—53no—
395C4×D4—53no—
396C4×D4—53no—
397C4×D4—53no—
398C4×D4—53no—
399C4×D4—56no—
400C4×D4—53no—
401C4×D4C2×AGL(1,𝔽5)13yes—
402#117—53no—
403#117—53no—
404#117—53no—
405#117—51no—
406#117—53no—
407C2×C4⋊C4—53no—
408C2×C42—53no—
409C2×C42—53no—
410C2×C42—53no—
411C4⋊C8—53no—
412C4⋊C8—53no—
413C4⋊C8—53no—
414C4⋊C8—53no—
415C4⋊C8—53no—
416C4⋊C8—56no—
417C4⋊C8—53no—
418C4⋊C8—51no—
419C4⋊C8—51no—
420C4⋊C8—53no—
421C4⋊C8—53no—
422C4⋊C8—56no—
423C4⋊C8—51no—
424C4⋊C8—51no—
425C22⋊C8—53no—
426C22⋊C8—53no—
427C22⋊C8—53no—
428C22⋊C8—53no—
429C22⋊C8—53no—
430C22⋊C8—53no—
431C22⋊C8—53no—
432C22⋊C8—53no—
433#97—53no—
434#97—53no—
435C4×C8—53no—
436C4×C8—53no—
437C4×C8—53no—
438C4×C8—53no—
439C4×C8—53no—
440C4×C8—53no—
441C22×C5—21no—
442C22×C5—21no—
443C22×C5—23no—
444C22×C5—23no—
445C22×C5—23no—
446C22×C5#52012yes—
447C22×C5C24×C411yescharacteristic
448C2×D5—23no—
449C2×D5—23no—
450C2×D5—23no—
451C4×C5—23no—
452C4×C5—21no—
453C4×C5—23no—
454C4×C5—21no—
455C4×C5C2×#13116yes—
456Dic5—23no—
457Dic5—21no—
458Dic5—23no—
459Dic5—21no—
460Dic5—23no—
461Dic5#52012yes—
462Dic5C2×H(5,𝔽2)11yescharacteristic
463Dic5C2×2−1+411yescharacteristic
464#41—24no—
465#41—106no—
466#41—106no—
467#41—104no—
468C2×Q8—24no—
469C2×Q8—53no—
470C2×Q8C22×AGL(1,𝔽5)13yes—
471C2×Q8C22×AGL(1,𝔽5)11yescharacteristic
472C2×D4—53no—
473C2×D4—53no—
474C2×D4—56no—
475C2×D4—53no—
476C2×D4C22×AGL(1,𝔽5)13yes—
477C2×D4C22×AGL(1,𝔽5)13yes—
478C22×C4—51no—
479C22×C4—53no—
480C22×C4—53no—
481C22×C4—53no—
482C22×C4—53no—
483C22×C4—53no—
484C22×C4—53no—
485C22×C4C22×AGL(1,𝔽5)13yes—
486C22×C4C22×AGL(1,𝔽5)13yes—
487M16—106no—
488M16—102no—
489M16—206no—
490M16—206no—
491C2×C8—51no—
492C2×C8—51no—
493C2×C8—51no—
494C2×C8—51no—
495C2×C8—53no—
496C2×C8—53no—
497C2×C8—53no—
498C2×C8—53no—
499C2×C8—106no—
500C2×C8—102no—
501C2×C8—206no—
502C2×C8—206no—
503C4⋊C4—53no—
504C4⋊C4—53no—
505C4⋊C4—56no—
506C4⋊C4—51no—
507C4⋊C4—51no—
508C4⋊C4—53no—
509C4⋊C4—53no—
510C4⋊C4—53no—
511C4⋊C4—53no—
512C4⋊C4—56no—
513C4⋊C4C22×AGL(1,𝔽5)13yes—
514C4⋊C4C22×AGL(1,𝔽5)13yes—
515C22⋊C4—51no—
516C22⋊C4—53no—
517C22⋊C4—56no—
518C22⋊C4—53no—
519C22⋊C4—56no—
520C22⋊C4—53no—
521C22⋊C4—56no—
522C22⋊C4C22×AGL(1,𝔽5)11yescharacteristic
523C22⋊C4C22×AGL(1,𝔽5)13yes—
524C22⋊C4C22×AGL(1,𝔽5)16yes—
525C42—53no—
526C42—53no—
527C42—51no—
528C42—51no—
529C42—53no—
530C42—53no—
531C42—53no—
532C42—53no—
533C42—53no—
534C42C22×AGL(1,𝔽5)13yes—
535C2×C5—22no—
536C2×C5—43no—
537C2×C5C2×#52011yescharacteristic
538C2×C5C22×#13111yescharacteristic
539C2×C5#357311yescharacteristic
540D5—43no—
541C23—53no—
542C23C23×AGL(1,𝔽5)13yes—
543C23C23×AGL(1,𝔽5)11yescharacteristic
544Q8—26no—
545Q8—106no—
546Q8#413114yescomplemented
547D4—26no—
548D4—26no—
549D4—106no—
550D4—106no—
551D4—106no—
552D4—206no—
553C2×C4—26no—
554C2×C4—26no—
555C2×C4—52no—
556C2×C4—52no—
557C2×C4—51no—
558C2×C4—51no—
559C2×C4—51no—
560C2×C4—53no—
561C2×C4—51no—
562C2×C4—53no—
563C2×C4—53no—
564C2×C4—103no—
565C2×C4—103no—
566C2×C4—103no—
567C2×C4—103no—
568C2×C4—106no—
569C2×C4—203no—
570C2×C4—203no—
571C2×C4—203no—
572C2×C4C23×AGL(1,𝔽5)11yescharacteristic
573C2×C4C23×AGL(1,𝔽5)13yes—
574C2×C4C23×AGL(1,𝔽5)11yescharacteristic
575C2×C4C23×AGL(1,𝔽5)13yes—
576C2×C4C23×AGL(1,𝔽5)13yes—
577C8—103no—
578C8—103no—
579C8—101no—
580C8—101no—
581C8—103no—
582C8—103no—
583C8—101no—
584C8—101no—
585C5#6027411yescharacteristic, complemented, Sylow
586C22—21no—
587C22—21no—
588C22—23no—
589C22—23no—
590C22—23no—
591C22—103no—
592C22—103no—
593C22—103no—
594C22#6626612yes—
595C22C24×AGL(1,𝔽5)11yescharacteristic
596C4—21no—
597C4—23no—
598C4—23no—
599C4—21no—
600C4—51no—
601C4—51no—
602C4—52no—
603C4—101no—
604C4—103no—
605C4—103no—
606C4—101no—
607C4—103no—
608C4C2×#413116yes—
609C2—22no—
610C2—43no—
611C2—203no—
612C2C22×#413111yescharacteristic
613C2C2×#6626611yescharacteristic
614C2#1061941511yescharacteristic
subgroupG/H
center, FrattiniC22C24×AGL(1,𝔽5)
commutator, socleC22×C5C24×C4
FittingC5×#548C4
prime radical#10618612C2
cosocleC2×Dic5C25
hypercenter#548AGL(1,𝔽5)
hypocenterC5#60274
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
2. #10617551C2semidirect
3. #10617548C2semidirect
4. Q8×#704C2semidirect
5. C2×Q8×#170C2semidirect
6. #10617321C2semidirect
7. #10617320C2semidirect
8. #10617279C2semidirect
9. #10617278C2semidirect
10. #10617232C2semidirect
11. #10617227C2semidirect
12. #10617206C2semidirect
13. #10617203C2semidirect
14. #10617139C2semidirect
15. #10617137C2semidirect
312. C5×Q8#131semidirect
546. Q8#4131semidirect
585. C5#60274semidirect

Maximal subgroups

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

subgroupindex
#106186122
#106175512
#106175482
Q8×#7042
C2×Q8×#1702
#106173212
#106173202
#106172792
#106172782
#106172322
#106172272
#106172062
#106172032
#106171392
#106171372
#602745