Finite Group Databasebeta

← #12157465

Subgroups — #12157465

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2357C211yesmaximal, characteristic
2#10605444C211yesmaximal, characteristic, complemented
3C4×#3964C211yesmaximal, characteristic
4#10602398C211yesmaximal, characteristic
5#10600446C211yesmaximal, characteristic
6#10600445C211yesmaximal, characteristic
7#10600432C211yesmaximal, characteristic
8#10600431C211yesmaximal, characteristic
9#10600340C211yesmaximal, characteristic, complemented
10#10600333C211yesmaximal, characteristic, complemented
11#10600284C211yesmaximal, characteristic, complemented
12#10600283C211yesmaximal, characteristic, complemented
13#10600277C211yesmaximal, characteristic, complemented
14#10600254C211yesmaximal, characteristic, complemented
15#10600253C211yesmaximal, characteristic, complemented
16C2×C4×C5×D4C2211yescharacteristic
17C2×#4016C2211yescharacteristic
18C2×#4015C2211yescharacteristic
19C5×#390C2211yescharacteristic
20C5×#390C2211yescharacteristic
21C5×#388—21no—
22C5×#388—21no—
23C5×#382C2211yescharacteristic
24C5×#381—21no—
25C4×C5×C4⋊C4C2211yescharacteristic
26C4×C5×C4⋊C4C2211yescharacteristic
27C4×C5×C4⋊C4C2211yescharacteristic
28C4×C5×C22⋊C4—21no—
29C2×#3964C2211yescharacteristic
30C2×#3964C2211yescharacteristic
31C2×#3964C2211yescharacteristic
32C2×#3964C2211yescharacteristic
33#65273—21no—
34#65268—21no—
35C2×#3940C2211yescharacteristic
36C2×#3939C2211yescharacteristic
37#65225C2211yescharacteristic
38C4×C4⋊Dic5C2211yescharacteristic
39#65217C2211yescharacteristic
40#65217C2211yescharacteristic
41C2×C4⋊#168C2211yescharacteristic
42C2×C4⋊#168C2211yescharacteristic
43C2×C4⋊#168C2211yescharacteristic
44C2×C4×#168C2211yescharacteristic
45#64936—21no—
46#64935—21no—
47#64749C2211yescharacteristic
48#64749C2211yescharacteristic
49#64749C2211yescharacteristic
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#26028—51nomaximal, Sylow
60C22×C5×D4C2311yescharacteristic
61C23×C4×C5—21no—
62C22×Dic10—21no—
63C4×C5×D4—22no—
64C4×C5×D4Q814yescomplemented
65C2×C5×C4⋊C4D411yescharacteristic
66C2×C5×C4⋊C4D411yescharacteristic
67C2×C5×C4⋊C4C2311yescharacteristic
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—21no—
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C5×C22⋊C4—21no—
75C2×C42×C5—21no—
76C2×C42×C5—21no—
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C42×C5C2311yescharacteristic
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×#683—21no—
89C2×C4×Dic5—21no—
90C22×#168C2311yescharacteristic
91C22×#168C2311yescharacteristic
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94#4016—22no—
95#4016—41no—
96#4015—22no—
97#4015—41no—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95—21no—
101C5×#95—21no—
102C5×#95D411yescharacteristic
103C5×#95D411yescharacteristic
104#3964—22no—
105#3964—22no—
106#3964—41no—
107#3964—41no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113#3940—22no—
114#3939—22no—
115C4⋊#168—22no—
116C4⋊#168—41no—
117C4⋊#168—41no—
118C4×#168—22no—
119#2550—51no—
120#2357D511yescharacteristic
121#2067—51no—
122#2065—51no—
123#2025—51no—
124#2023—51no—
125#1989—51no—
126#1988—51no—
127#1949—51no—
128#1939—51no—
129#1921—51no—
130#1920—51no—
131#1835—51no—
132#1804—51no—
133C4×#102—51no—
134C24×C5—21no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4C2×Q812yes—
141C22×C4×C5—21no—
142C22×C4×C5—24no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—81no—
146C22×C4×C5—81no—
147C22×C4×C5C2×D411yescharacteristic
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—
158C2×Dic10—42no—
159C2×Dic10—42no—
160C2×Dic10—81no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—21no—
167C5×C4⋊C4—41no—
168C5×C4⋊C4C2×Q812yes—
169C5×C22⋊C4—24no—
170C5×C22⋊C4—42no—
171C5×C22⋊C4—42no—
172C5×C22⋊C4—81no—
173C5×C22⋊C4—81no—
174C42×C5—21no—
175C42×C5—21no—
176C42×C5—21no—
177C42×C5—21no—
178C42×C5—81no—
179C42×C5—81no—
180C42×C5C2×Q812yes—
181C42×C5#4112yes—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188C4⋊Dic5—42no—
189C4⋊Dic5#4112yes—
190C4⋊Dic5#4112yes—
191#683—42no—
192#683—42no—
193#683—42no—
194#683—81no—
195C4×Dic5—81no—
196C2×#168—21no—
197C2×#168—21no—
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×C4⋊Q8—51no—
209C2×#125—51no—
210C2×C4×D4C2×D511yescharacteristic
211C2×#107—51no—
212C2×#106—51no—
213C2×C4⋊C8—51no—
214C2×C4⋊C8—51no—
215C2×C4⋊C8—51no—
216C2×#102—51no—
217C2×#102—51no—
218C2×#102—51no—
219C2×#102—51no—
220C2×C4×C8—51no—
221#398—101no—
222#395—101no—
223#391—101no—
224#390C2×D511yescharacteristic
225#390C2×D511yescharacteristic
226#389—51no—
227#389—51no—
228#389—101no—
229#388—21no—
230#388—21no—
231#382—51no—
232#382C2×D511yescharacteristic
233#381—21no—
234C4×C4⋊C4—51no—
235C4×C4⋊C4C2×D511yescharacteristic
236C4×C4⋊C4C2×D511yescharacteristic
237C4×C4⋊C4C2×D511yescharacteristic
238C4×C22⋊C4—21no—
239#340—51no—
240#340—51no—
241#340—51no—
242#340—51no—
243#340—51no—
244#340—51no—
245#340—51no—
246#340—51no—
247#340—51no—
248#340—51no—
249#336—51no—
250#336—51no—
251C23×C5—22no—
252C23×C5—41no—
253C23×C5—41no—
254C23×C5—41no—
255C23×C5—41no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5C22×D411yescharacteristic
259C5×D4—22no—
260C5×D4—22no—
261C5×D4—41no—
262C5×D4—41no—
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—22no—
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—21no—
278C2×C4×C5—21no—
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—84no—
288C2×C4×C5C22⋊Q812yes—
289C2×C4×C5C22⋊Q812yes—
290C2×C4×C5#13512yes—
291C2×C4×C5C22×Q811yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C2×C4×C5C2×#4111yescharacteristic
294C2×C4×C5C2×#4111yescharacteristic
295C2×C4×C5H(5,𝔽2)11yescharacteristic
296C2×C4×C52−1+411yescharacteristic
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309Dic10—84no—
310#168—41no—
311#168—41no—
312#168—41no—
313#168—41no—
314C22×Q8—101no—
315C22×D4C22×D511yescharacteristic
316C23×C4—21no—
317C22×C8—51no—
318C22×C8—51no—
319C22×C8—51no—
320C22×C8—51no—
321C4⋊Q8—102no—
322C4⋊Q8—201no—
323#125—102no—
324#125—201no—
325C4×D4—22no—
326C4×D4Dic1014yescomplemented
327C2×C4⋊C4—51no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—101no—
332C2×C4⋊C4—101no—
333C2×C4⋊C4—101no—
334C2×C4⋊C4—101no—
335C2×C4⋊C4—101no—
336C2×C4⋊C4C22⋊D511yescharacteristic
337C2×C4⋊C4C22⋊D511yescharacteristic
338C2×C4⋊C4C22×D511yescharacteristic
339C2×C4⋊C4C22×D511yescharacteristic
340C2×C4⋊C4C22×D511yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—101no—
349C2×C42C22×D511yescharacteristic
350C2×C42C22×D511yescharacteristic
351C2×C42C22×D511yescharacteristic
352#107—102no—
353#106—102no—
354C4⋊C8—102no—
355C4⋊C8—201no—
356C4⋊C8—201no—
357#102—102no—
358#102—102no—
359#102—201no—
360#102—201no—
361C4×C8—102no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—101no—
367#95—101no—
368#95—101no—
369#95—101no—
370#95—101no—
371#95C22⋊D511yescharacteristic
372#95C22⋊D511yescharacteristic
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—81no—
379C22×C5—81no—
380C22×C5C2×C22⋊Q811yescharacteristic
381C22×C5#54611yescharacteristic
382C22×C5#54811yescharacteristic
383C22×C5C2×#13511yescharacteristic
384C22×C5#57511yescharacteristic
385C22×C5#57611yescharacteristic
386C22×C5#57711yescharacteristic
387C4×C5—22no—
388C4×C5—22no—
389C4×C5—22no—
390C4×C5—42no—
391C4×C5—41no—
392C4×C5—41no—
393C4×C5—41no—
394C4×C5—81no—
395C4×C5—81no—
396C4×C5#53311yescharacteristic
397C4×C5#54411yescharacteristic
398C4×C5#55511yescharacteristic
399C4×C5#55611yescharacteristic
400Dic5—42no—
401Dic5—42no—
402Dic5—81no—
403Dic5—81no—
404C24—21no—
405C2×Q8—202no—
406C2×Q8—202no—
407C2×Q8—401no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×Dic1012yes—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—101no—
421C22×C4—101no—
422C22×C4—101no—
423C22×C4—101no—
424C22×C4C2×C22⋊D511yescharacteristic
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C2×C22⋊D511yescharacteristic
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C2×C22⋊D511yescharacteristic
429C22×C4C2×C22⋊D511yescharacteristic
430C22×C4C23×D511yescharacteristic
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—101no—
434C2×C8—101no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—101no—
440C2×C8—101no—
441C2×C8—101no—
442C2×C8—101no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—52no—
451C4⋊C4—52no—
452C4⋊C4—101no—
453C4⋊C4—101no—
454C4⋊C4—101no—
455C4⋊C4—101no—
456C4⋊C4—101no—
457C4⋊C4—101no—
458C4⋊C4—202no—
459C4⋊C4—202no—
460C4⋊C4—202no—
461C4⋊C4—202no—
462C4⋊C4—401no—
463C4⋊C4C2×Dic1012yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—81no—
474C42—81no—
475C42—401no—
476C42C2×Dic1012yes—
477C42#70912yes—
478C2×C5—82no—
479C2×C5#243911yescharacteristic
480C2×C5#307611yescharacteristic
481C2×C5#308111yescharacteristic
482C2×C5#329711yescharacteristic
483C2×C5#332611yescharacteristic
484C2×C5#336511yescharacteristic
485C2×C5#337011yescharacteristic
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C22×C22⋊D511yescharacteristic
494Q8—404no—
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
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—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—84no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—201no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4#407012yes—
537C2×C4#407012yes—
538C2×C4#409612yes—
539C2×C4C22×Dic1011yescharacteristic
540C2×C4C2×#70911yescharacteristic
541C2×C4D5×#4111yescharacteristic
542C2×C4D5×#4111yescharacteristic
543C2×C4#414911yescharacteristic
544C2×C4#415011yescharacteristic
545C8—201no—
546C8—201no—
547C8—201no—
548C8—201no—
549C5#2602811yescharacteristic, complemented, Sylow
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#407011yescharacteristic
558C22C2×#409611yescharacteristic
559C22#6615611yescharacteristic
560C22#6616311yescharacteristic
561C22#6616511yescharacteristic
562C22#6616911yescharacteristic
563C22#6617011yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—42no—
571C4—81no—
572C4—81no—
573C4—202no—
574C4—202no—
575C4—401no—
576C4—401no—
577C4#6585111yescharacteristic
578C4#6585311yescharacteristic
579C4#6586611yescharacteristic
580C4#6586711yescharacteristic
581C2—82no—
582C2#1061397911yescharacteristic
583C2#1061423811yescharacteristic
584C2#1061427011yescharacteristic
585C2#1061534511yescharacteristic
586C2#1061535211yescharacteristic
587C2#1061544511yescharacteristic
588C2#1061544811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2357C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
hypercenter#2357D5
hypocenterC5#26028
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #10605444C2semidirect
9. #10600340C2semidirect
10. #10600333C2semidirect
11. #10600284C2semidirect
12. #10600283C2semidirect
13. #10600277C2semidirect
14. #10600254C2semidirect
15. #10600253C2semidirect
64. C4×C5×D4Q8semidirect
326. C4×D4Dic10semidirect
549. C5#26028semidirect

Maximal subgroups

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

subgroupindex
C5×#23572
#106054442
C4×#39642
#106023982
#106004462
#106004452
#106004322
#106004312
#106003402
#106003332
#106002842
#106002832
#106002772
#106002542
#106002532
#260285