Finite Group Databasebeta

← #12172013

Subgroups — #12172013

Order 1280.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2496C211yesmaximal, characteristic
2#10605196C211yesmaximal, characteristic
3#10600427C211yesmaximal, characteristic, complemented
4#10600425C211yesmaximal, characteristic, complemented
5#10600411C211yesmaximal, characteristic, complemented
6#10600394C211yesmaximal, characteristic, complemented
7#10600382C211yesmaximal, characteristic, complemented
8#10600361C211yesmaximal, characteristic, complemented
9#10600319C211yesmaximal, characteristic
10#10600319C211yesmaximal, characteristic
11#10600298C211yesmaximal, characteristic
12#10600295C211yesmaximal, characteristic
13C4×#3942C211yesmaximal, characteristic, complemented
14#10600253C211yesmaximal, characteristic, complemented
15#10600244C211yesmaximal, characteristic
16C2×C5×#126—21no—
17C22×C4⋊Dic5C2211yescharacteristic
18C2×#4070C2211yescharacteristic
19C2×C4×Dic10C2211yescharacteristic
20C5×#400—21no—
21C5×#397—21no—
22C5×#389C2211yescharacteristic
23C5×#389C2211yescharacteristic
24C5×#387—21no—
25C5×#384C2211yescharacteristic
26C5×#380C2211yescharacteristic
27C5×#380C2211yescharacteristic
28C4×C5×C4⋊C4C2211yescharacteristic
29C4×C5×C22⋊C4C2211yescharacteristic
30C2×C22⋊#168C2211yescharacteristic
31C2×C22⋊#168C2211yescharacteristic
32C2×#3942C2211yescharacteristic
33C2×#3942C2211yescharacteristic
34C2×#3942C2211yescharacteristic
35C2×#3942C2211yescharacteristic
36C2×#3940C2211yescharacteristic
37C2×#3940C2211yescharacteristic
38C2×#3940C2211yescharacteristic
39C2×#3940C2211yescharacteristic
40#65235—21no—
41C22⋊C4×Dic5—21no—
42C4×C4⋊Dic5C2211yescharacteristic
43#65218C2211yescharacteristic
44#65217C2211yescharacteristic
45C2×C4⋊#168C2211yescharacteristic
46C2×C4×#168C2211yescharacteristic
47#64944—21no—
48#64936—21no—
49#64749C2211yescharacteristic
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#27034—51nomaximal, Sylow
60C23×C4×C5C2311yescharacteristic
61C23×Dic5—21no—
62C22×Dic10C2311yescharacteristic
63C5×#126—44no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C5×C22⋊C4D411yescharacteristic
75C2×C5×C22⋊C4D411yescharacteristic
76C2×C42×C5—21no—
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C22⋊Dic5—21no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5—41no—
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86#4070—22no—
87#4070—22no—
88C2×#683—21no—
89C2×#683—21no—
90C2×#683—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C22×#168C2311yescharacteristic
97C4×Dic10—22no—
98C4×Dic10—41no—
99C5×#95—21no—
100C5×#95—21no—
101C5×#95—21no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95D411yescharacteristic
105C5×#95D411yescharacteristic
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110C22⋊#168—22no—
111C22⋊#168—41no—
112#3942—22no—
113#3942—22no—
114#3942—22no—
115#3942—41no—
116#3940—22no—
117#3940—22no—
118#3940—22no—
119#3940—41no—
120C4⋊#168—22no—
121C4×#168—41no—
122#2496D511yescharacteristic
123#2360—51no—
124#2072—51no—
125#2066—51no—
126#2035—51no—
127#2035—51no—
128#1948—51no—
129#1941—51no—
130#1923—51no—
131#1920—51no—
132#1919—51no—
133#1865—51no—
134#1853—51no—
135#1824—51no—
136C4×#103—51no—
137C24×C5C2×D411yescharacteristic
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2411yescharacteristic
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158C22×Dic5—42no—
159C22×Dic5—42no—
160C22×Dic5—81no—
161C2×Dic10—21no—
162C2×Dic10—21no—
163C2×Dic10—21no—
164C2×Dic10—21no—
165C2×Dic10—21no—
166C2×Dic10#4112yes—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—42no—
176C5×C4⋊C4—42no—
177C5×C4⋊C4—42no—
178C5×C4⋊C4—42no—
179C5×C22⋊C4—41no—
180C5×C22⋊C4—42no—
181C5×C22⋊C4—42no—
182C5×C22⋊C4SD1614yes—
183C42×C5—21no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5—42no—
188C22⋊Dic5—24no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5—21no—
192C4⋊Dic5—21no—
193C4⋊Dic5—42no—
194C4⋊Dic5#4112yes—
195C4⋊Dic5#4112yes—
196#683—24no—
197#683—42no—
198#683—81no—
199C4×Dic5—42no—
200C4×Dic5—81no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168#4112yes—
213C2×#126—21no—
214C2×C22⋊Q8—51no—
215C2×C4×Q8—51no—
216C22×C4⋊C4—51no—
217C2×#106—51no—
218C2×#106—51no—
219C2×#106—51no—
220C2×#106—51no—
221C2×C4⋊C8—51no—
222C2×#103—51no—
223C2×#103—51no—
224C2×#103—51no—
225C2×#103—51no—
226C2×C22⋊C8—51no—
227C2×C22⋊C8—51no—
228C2×C4×C8—51no—
229#400—21no—
230#397—21no—
231#391—51no—
232#389—51no—
233#389—101no—
234#389C2×D511yescharacteristic
235#389C2×D511yescharacteristic
236#387—21no—
237#387—101no—
238#385—101no—
239#384C2×D511yescharacteristic
240#380C2×D511yescharacteristic
241#380C2×D511yescharacteristic
242C4×C4⋊C4—51no—
243C4×C4⋊C4C2×D511yescharacteristic
244C4×C22⋊C4—101no—
245C4×C22⋊C4C2×D511yescharacteristic
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#340—51no—
251#340—51no—
252#340—51no—
253#340—51no—
254#336—51no—
255#336—51no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—21no—
262C23×C5C2×SD1612yes—
263C23×C5C22×D411yescharacteristic
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5C2×SD1612yes—
294C2×C4×C5C2×SD1612yes—
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5C2×#4111yescharacteristic
297C2×C4×C5C2×#4111yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C2×C4×C5H(5,𝔽2)11yescharacteristic
300C2×C4×C52−1+411yescharacteristic
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—84no—
314Dic10—22no—
315Dic10—22no—
316Dic10—41no—
317Dic10—41no—
318#168—22no—
319#168—41no—
320#168—41no—
321#168—41no—
322C22×Q8—51no—
323C23×C4—101no—
324C23×C4C22×D511yescharacteristic
325C22×C8—51no—
326C22×C8—51no—
327C22×C8—51no—
328C22×C8—51no—
329#126—44no—
330C22⋊Q8—102no—
331C22⋊Q8—102no—
332C4×Q8—102no—
333C4×Q8—201no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4—101no—
343C2×C4⋊C4—101no—
344C2×C4⋊C4—101no—
345C2×C4⋊C4—101no—
346C2×C4⋊C4—201no—
347C2×C4⋊C4C22×D511yescharacteristic
348C2×C4⋊C4C22×D511yescharacteristic
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—101no—
354C2×C22⋊C4C22⋊D511yescharacteristic
355C2×C22⋊C4C22⋊D511yescharacteristic
356C2×C42—21no—
357C2×C42—101no—
358C2×C42—101no—
359C2×C42C22×D511yescharacteristic
360C2×C42C22×D511yescharacteristic
361#106—102no—
362#106—102no—
363#106—102no—
364#106—201no—
365C4⋊C8—102no—
366#103—102no—
367#103—102no—
368#103—102no—
369#103—201no—
370C22⋊C8—102no—
371C22⋊C8—201no—
372C4×C8—201no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—101no—
379#95—101no—
380#95—101no—
381#95—101no—
382#95C22⋊D511yescharacteristic
383#95C22⋊D511yescharacteristic
384C22×C5—22no—
385C22×C5—22no—
386C22×C5—22no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5—41no—
391C22×C5#52511yescharacteristic
392C22×C5#54611yescharacteristic
393C22×C5#54711yescharacteristic
394C22×C5C22×SD1611yescharacteristic
395C22×C5#57511yescharacteristic
396C22×C5#57711yescharacteristic
397C22×C5#57811yescharacteristic
398C4×C5—22no—
399C4×C5—22no—
400C4×C5—41no—
401C4×C5—41no—
402C4×C5—41no—
403C4×C5—81no—
404C4×C5—81no—
405C4×C5—81no—
406C4×C5—81no—
407C4×C5#52911yescharacteristic
408C4×C5#53911yescharacteristic
409C4×C5#55311yescharacteristic
410C4×C5#55611yescharacteristic
411Dic5—42no—
412Dic5—42no—
413Dic5—42no—
414Dic5—81no—
415C24C2×C22⋊D511yescharacteristic
416C2×Q8—52no—
417C2×Q8—101no—
418C2×Q8—101no—
419C2×Q8—101no—
420C2×Q8—101no—
421C2×Q8—101no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—101no—
433C22×C4—101no—
434C22×C4—101no—
435C22×C4—101no—
436C22×C4—202no—
437C22×C4—202no—
438C22×C4—401no—
439C22×C4C2×C22⋊D511yescharacteristic
440C22×C4C2×C22⋊D511yescharacteristic
441C22×C4C2×C22⋊D511yescharacteristic
442C22×C4C2×C22⋊D511yescharacteristic
443C22×C4C2×C22⋊D511yescharacteristic
444C22×C4C23×D511yescharacteristic
445C2×C8—52no—
446C2×C8—101no—
447C2×C8—101no—
448C2×C8—101no—
449C2×C8—101no—
450C2×C8—101no—
451C2×C8—101no—
452C2×C8—101no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—52no—
470C4⋊C4—52no—
471C4⋊C4—104no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4—202no—
477C4⋊C4—202no—
478C4⋊C4—401no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—104no—
483C22⋊C4#68714yes—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—202no—
490C42—401no—
491C2×C5—42no—
492C2×C5#246611yescharacteristic
493C2×C5#310311yescharacteristic
494C2×C5#312011yescharacteristic
495C2×C5#329811yescharacteristic
496C2×C5#331111yescharacteristic
497C2×C5#333511yescharacteristic
498C2×C5#335211yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23C2×#68712yes—
506C23C22×C22⋊D511yescharacteristic
507Q8—102no—
508Q8—102no—
509Q8—201no—
510Q8—201no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—102no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4—201no—
552C2×C4—404no—
553C2×C4C2×#68712yes—
554C2×C4C2×#68712yes—
555C2×C4D5×#4111yescharacteristic
556C2×C4D5×#4111yescharacteristic
557C2×C4D5×#4111yescharacteristic
558C2×C4D5×#4111yescharacteristic
559C2×C4#414911yescharacteristic
560C2×C4#415011yescharacteristic
561C8—102no—
562C8—201no—
563C8—201no—
564C8—201no—
565C5#2703411yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C22×#68711yescharacteristic
574C22#6615611yescharacteristic
575C22#6615811yescharacteristic
576C22#6616211yescharacteristic
577C22#6616311yescharacteristic
578C22#6617111yescharacteristic
579C22#6617211yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—202no—
590C4—202no—
591C4—202no—
592C4—401no—
593C4#6603611yescharacteristic
594C4#6604011yescharacteristic
595C4#6604111yescharacteristic
596C4#6604311yescharacteristic
597C2—42no—
598C2#1061483611yescharacteristic
599C2#1061504611yescharacteristic
600C2#1061517711yescharacteristic
601C2#1061524811yescharacteristic
602C2#1061526211yescharacteristic
603C2#1061532811yescharacteristic
604C2#1061539611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2496C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2496D5
hypocenterC5#27034
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
3. #10600427C2semidirect
4. #10600425C2semidirect
5. #10600411C2semidirect
6. #10600394C2semidirect
7. #10600382C2semidirect
8. #10600361C2semidirect
13. C4×#3942C2semidirect
14. #10600253C2semidirect
565. C5#27034semidirect

Maximal subgroups

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

subgroupindex
C5×#24962
#106051962
#106004272
#106004252
#106004112
#106003942
#106003822
#106003612
#106003192
#106002982
#106002952
C4×#39422
#106002532
#106002442
#270345