Finite Group Databasebeta

← #12172028

Subgroups — #12172028

Order 1280.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2495C211yesmaximal, characteristic
2#10605196C211yesmaximal, characteristic
3#10600428C211yesmaximal, characteristic, complemented
4#10600424C211yesmaximal, characteristic, complemented
5#10600412C211yesmaximal, characteristic, complemented
6#10600393C211yesmaximal, characteristic, complemented
7#10600353C211yesmaximal, characteristic, complemented
8#10600333C211yesmaximal, characteristic, complemented
9#10600320C211yesmaximal, characteristic
10#10600319C211yesmaximal, characteristic
11#10600297C211yesmaximal, characteristic
12#10600294C211yesmaximal, characteristic
13C4×#3942C211yesmaximal, characteristic, complemented
14#10600253C211yesmaximal, characteristic, complemented
15#10600244C211yesmaximal, characteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C2×#4070C2211yescharacteristic
18C2×C4×Dic10C2211yescharacteristic
19C5×#400—21no—
20C5×#397—21no—
21C5×#389C2211yescharacteristic
22C5×#389C2211yescharacteristic
23C5×#388—21no—
24C5×#383—21no—
25C5×#382C2211yescharacteristic
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×#3939C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39C2×#3939C2211yescharacteristic
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#27035—51nomaximal, Sylow
60C23×C4×C5C2311yescharacteristic
61C23×Dic5—21no—
62C22×Dic10C2311yescharacteristic
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4C2311yescharacteristic
66C2×C5×C4⋊C4C2311yescharacteristic
67C2×C5×C4⋊C4C2311yescharacteristic
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4D411yescharacteristic
72C2×C5×C22⋊C4D411yescharacteristic
73C2×C42×C5—21no—
74C2×C42×C5C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×C22⋊Dic5—21no—
77C2×C4⋊Dic5—22no—
78C2×C4⋊Dic5—21no—
79C2×C4⋊Dic5—41no—
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83#4070—22no—
84#4070—22no—
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5—21no—
90C22×#168C2311yescharacteristic
91C22×#168C2311yescharacteristic
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C4×Dic10—22no—
95C4×Dic10—41no—
96C5×#95—21no—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95—21no—
101C5×#95—21no—
102C5×#95—21no—
103C5×#95D411yescharacteristic
104C5×#95D411yescharacteristic
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109C22⋊#168—22no—
110C22⋊#168—41no—
111#3942—22no—
112#3942—22no—
113#3942—22no—
114#3942—41no—
115#3940—22no—
116#3939—22no—
117#3939—22no—
118#3939—41no—
119C4⋊#168—22no—
120C4×#168—41no—
121#2495D511yescharacteristic
122#2360—51no—
123#2074—51no—
124#2068—51no—
125#2036—51no—
126#2035—51no—
127#1949—51no—
128#1942—51no—
129#1923—51no—
130#1920—51no—
131#1918—51no—
132#1863—51no—
133#1852—51no—
134#1823—51no—
135C4×#103—51no—
136C24×C5C2×D411yescharacteristic
137C22×C4×C5—21no—
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×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2411yescharacteristic
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—42no—
158C22×Dic5—42no—
159C22×Dic5—81no—
160C2×Dic10—21no—
161C2×Dic10—21no—
162C2×Dic10—21no—
163C2×Dic10—21no—
164C2×Dic10—21no—
165C2×Dic10#4112yes—
166C5×C4⋊C4—21no—
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—81no—
175C5×C4⋊C4—81no—
176C5×C22⋊C4—22no—
177C5×C22⋊C4—22no—
178C5×C22⋊C4—42no—
179C5×C22⋊C4—81no—
180C42×C5—21no—
181C42×C5—21no—
182C42×C5—21no—
183C42×C5—21no—
184C42×C5—42no—
185C22⋊Dic5—24no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—42no—
191C4⋊Dic5#4112yes—
192C4⋊Dic5#4112yes—
193#683—24no—
194#683—42no—
195#683—81no—
196C4×Dic5—42no—
197C4×Dic5—81no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
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#4112yes—
210C2×C22⋊Q8—51no—
211C2×C4×Q8—51no—
212C22×C4⋊C4—51no—
213C2×#107—51no—
214C2×#107—51no—
215C2×#107—51no—
216C2×#106—51no—
217C2×C4⋊C8—51no—
218C2×#103—51no—
219C2×#103—51no—
220C2×#103—51no—
221C2×#103—51no—
222C2×C22⋊C8—51no—
223C2×C22⋊C8—51no—
224C2×C4×C8—51no—
225#400—21no—
226#397—21no—
227#391—51no—
228#389—51no—
229#389—101no—
230#389C2×D511yescharacteristic
231#389C2×D511yescharacteristic
232#388—21no—
233#387—101no—
234#385—101no—
235#383—21no—
236#382C2×D511yescharacteristic
237#380C2×D511yescharacteristic
238#380C2×D511yescharacteristic
239C4×C4⋊C4—51no—
240C4×C4⋊C4C2×D511yescharacteristic
241C4×C22⋊C4—101no—
242C4×C22⋊C4C2×D511yescharacteristic
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#340—51no—
250#340—51no—
251#336—51no—
252#336—51no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5#13512yes—
260C23×C5C22×D411yescharacteristic
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
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—41no—
276C2×C4×C5—41no—
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#13712yes—
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5C2×#4111yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5H(5,𝔽2)11yescharacteristic
297C2×C4×C52−1+411yescharacteristic
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—84no—
311Dic10—22no—
312Dic10—22no—
313Dic10—41no—
314Dic10—41no—
315#168—22no—
316#168—41no—
317#168—41no—
318#168—41no—
319C22×Q8—51no—
320C23×C4—101no—
321C23×C4C22×D511yescharacteristic
322C22×C8—51no—
323C22×C8—51no—
324C22×C8—51no—
325C22×C8—51no—
326C22⋊Q8—102no—
327C22⋊Q8—102no—
328C4×Q8—102no—
329C4×Q8—201no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—102no—
336C2×C4⋊C4—101no—
337C2×C4⋊C4—101no—
338C2×C4⋊C4—101no—
339C2×C4⋊C4—101no—
340C2×C4⋊C4—201no—
341C2×C4⋊C4C22×D511yescharacteristic
342C2×C4⋊C4C22×D511yescharacteristic
343C2×C4⋊C4C22×D511yescharacteristic
344C2×C4⋊C4C22×D511yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—101no—
348C2×C22⋊C4C22⋊D511yescharacteristic
349C2×C22⋊C4C22⋊D511yescharacteristic
350C2×C42—21no—
351C2×C42—101no—
352C2×C42—101no—
353C2×C42C22×D511yescharacteristic
354C2×C42C22×D511yescharacteristic
355#107—102no—
356#107—102no—
357#107—201no—
358#106—102no—
359C4⋊C8—102no—
360#103—102no—
361#103—102no—
362#103—102no—
363#103—201no—
364C22⋊C8—102no—
365C22⋊C8—201no—
366C4×C8—201no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—101no—
375#95—101no—
376#95—101no—
377#95—101no—
378#95C22⋊D511yescharacteristic
379#95C22⋊D511yescharacteristic
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—41no—
384C22×C5—41no—
385C22×C5—41no—
386C22×C5—41no—
387C22×C5#52511yescharacteristic
388C22×C5#54611yescharacteristic
389C22×C5#54711yescharacteristic
390C22×C5C2×#13511yescharacteristic
391C22×C5C2×#13711yescharacteristic
392C22×C5#57611yescharacteristic
393C22×C5#57811yescharacteristic
394C4×C5—22no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—41no—
398C4×C5—41no—
399C4×C5—81no—
400C4×C5—81no—
401C4×C5—81no—
402C4×C5—81no—
403C4×C5#52911yescharacteristic
404C4×C5#53911yescharacteristic
405C4×C5#55311yescharacteristic
406C4×C5#55611yescharacteristic
407Dic5—42no—
408Dic5—42no—
409Dic5—42no—
410Dic5—81no—
411C24C2×C22⋊D511yescharacteristic
412C2×Q8—52no—
413C2×Q8—101no—
414C2×Q8—101no—
415C2×Q8—101no—
416C2×Q8—101no—
417C2×Q8—101no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—101no—
429C22×C4—101no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—202no—
433C22×C4—202no—
434C22×C4—401no—
435C22×C4C2×C22⋊D511yescharacteristic
436C22×C4C2×C22⋊D511yescharacteristic
437C22×C4C2×C22⋊D511yescharacteristic
438C22×C4C2×C22⋊D511yescharacteristic
439C22×C4C2×C22⋊D511yescharacteristic
440C22×C4C23×D511yescharacteristic
441C2×C8—52no—
442C2×C8—101no—
443C2×C8—101no—
444C2×C8—101no—
445C2×C8—101no—
446C2×C8—101no—
447C2×C8—101no—
448C2×C8—101no—
449C2×C8—101no—
450C2×C8—101no—
451C2×C8—101no—
452C2×C8—101no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—52no—
462C4⋊C4—52no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—104no—
466C4⋊C4—101no—
467C4⋊C4—101no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—202no—
471C4⋊C4—202no—
472C4⋊C4—401no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—104no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—202no—
484C42—401no—
485C2×C5—42no—
486C2×C5#246611yescharacteristic
487C2×C5#310811yescharacteristic
488C2×C5#312411yescharacteristic
489C2×C5#329711yescharacteristic
490C2×C5#331211yescharacteristic
491C2×C5#333611yescharacteristic
492C2×C5#335711yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#409612yes—
500C23C22×C22⋊D511yescharacteristic
501Q8—102no—
502Q8—102no—
503Q8—201no—
504Q8—201no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
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—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
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—102no—
536C2×C4—102no—
537C2×C4—102no—
538C2×C4—201no—
539C2×C4—201no—
540C2×C4—201no—
541C2×C4—201no—
542C2×C4—201no—
543C2×C4—201no—
544C2×C4—201no—
545C2×C4—201no—
546C2×C4—201no—
547C2×C4—404no—
548C2×C4#409712yes—
549C2×C4D5×#4111yescharacteristic
550C2×C4D5×#4111yescharacteristic
551C2×C4D5×#4111yescharacteristic
552C2×C4D5×#4111yescharacteristic
553C2×C4#414911yescharacteristic
554C2×C4#415011yescharacteristic
555C8—102no—
556C8—201no—
557C8—201no—
558C8—201no—
559C5#2703511yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22C2×#409611yescharacteristic
568C22C2×#409711yescharacteristic
569C22#6615811yescharacteristic
570C22#6616211yescharacteristic
571C22#6616311yescharacteristic
572C22#6616911yescharacteristic
573C22#6617211yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—202no—
584C4—202no—
585C4—202no—
586C4—401no—
587C4#6603611yescharacteristic
588C4#6604011yescharacteristic
589C4#6604111yescharacteristic
590C4#6604311yescharacteristic
591C2—42no—
592C2#1061483611yescharacteristic
593C2#1061488311yescharacteristic
594C2#1061519211yescharacteristic
595C2#1061532111yescharacteristic
596C2#1061533411yescharacteristic
597C2#1061534511yescharacteristic
598C2#1061538211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2495C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2495D5
hypocenterC5#27035
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2495

Complements

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

NH ≅ G/Ntype
3. #10600428C2semidirect
4. #10600424C2semidirect
5. #10600412C2semidirect
6. #10600393C2semidirect
7. #10600353C2semidirect
8. #10600333C2semidirect
13. C4×#3942C2semidirect
14. #10600253C2semidirect
559. C5#27035semidirect

Maximal subgroups

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

subgroupindex
C5×#24952
#106051962
#106004282
#106004242
#106004122
#106003932
#106003532
#106003332
#106003202
#106003192
#106002972
#106002942
C4×#39422
#106002532
#106002442
#270355