Finite Group Databasebeta

← #12158343

Subgroups — #12158343

Order 1280.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2357C211yesmaximal, characteristic
2#10605454C211yesmaximal, characteristic, complemented
3C4×#3964C211yesmaximal, characteristic
4#10602398C211yesmaximal, characteristic
5#10600448C211yesmaximal, characteristic
6#10600447C211yesmaximal, characteristic
7#10600445C211yesmaximal, characteristic
8#10600445C211yesmaximal, characteristic
9#10600342C211yesmaximal, characteristic, complemented
10#10600333C211yesmaximal, characteristic, complemented
11#10600277C211yesmaximal, characteristic, complemented
12#10600261C211yesmaximal, characteristic, complemented
13#10600260C211yesmaximal, characteristic, complemented
14#10600254C211yesmaximal, characteristic, complemented
15#10600251C211yesmaximal, characteristic, complemented
16C2×C4×C5×D4C2211yescharacteristic
17C5×#390C2211yescharacteristic
18C5×#390C2211yescharacteristic
19C5×#388—21no—
20C5×#388—21no—
21C5×#382C2211yescharacteristic
22C5×#381—21no—
23C4×C5×C4⋊C4C2211yescharacteristic
24C4×C5×C4⋊C4C2211yescharacteristic
25C4×C5×C4⋊C4C2211yescharacteristic
26C4×C5×C22⋊C4—21no—
27C2×#3964C2211yescharacteristic
28C2×#3964C2211yescharacteristic
29C2×#3964C2211yescharacteristic
30C2×#3964C2211yescharacteristic
31#65275—21no—
32#65273—21no—
33C2×#3940C2211yescharacteristic
34C2×#3940C2211yescharacteristic
35C2×#3940C2211yescharacteristic
36C2×#3939C2211yescharacteristic
37C2×#3939C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39#65225C2211yescharacteristic
40#65225C2211yescharacteristic
41#65224C2211yescharacteristic
42C4×C4⋊Dic5C2211yescharacteristic
43#65217C2211yescharacteristic
44#65217C2211yescharacteristic
45C2×C4⋊#168C2211yescharacteristic
46C2×C4⋊#168C2211yescharacteristic
47C2×C4⋊#168C2211yescharacteristic
48C2×C4×#168C2211yescharacteristic
49#64944—21no—
50#64943—21no—
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64744C2211yescharacteristic
58#64744C2211yescharacteristic
59#26094—51nomaximal, Sylow
60C22×C5×D4C2311yescharacteristic
61C23×C4×C5—21no—
62C4×C5×D4—22no—
63C4×C5×D4Q814yes—
64C2×C5×C4⋊C4D411yescharacteristic
65C2×C5×C4⋊C4D411yescharacteristic
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⋊C4—21no—
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C42×C5—21no—
75C2×C42×C5—21no—
76C2×C42×C5C2311yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C4⋊Dic5—21no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×#683—21no—
86C2×#683—21no—
87C2×C4×Dic5—21no—
88C22×#168C2311yescharacteristic
89C22×#168C2311yescharacteristic
90C22×#168C2311yescharacteristic
91C22×#168C2311yescharacteristic
92C5×#95—21no—
93C5×#95—21no—
94C5×#95—21no—
95C5×#95—21no—
96C5×#95D411yescharacteristic
97C5×#95D411yescharacteristic
98#3964—22no—
99#3964—22no—
100#3964—41no—
101#3964—41no—
102#3963—21no—
103#3963—21no—
104#3963—21no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3940—41no—
110#3940—41no—
111#3940—41no—
112#3939—22no—
113#3939—22no—
114#3939—41no—
115C4⋊#168—22no—
116C4⋊#168—22no—
117C4⋊#168—22no—
118C4×#168—22no—
119#2570—51no—
120#2357D511yescharacteristic
121#2093—51no—
122#2092—51no—
123#2065—51no—
124#2065—51no—
125#1949—51no—
126#1949—51no—
127#1948—51no—
128#1939—51no—
129#1921—51no—
130#1918—51no—
131#1846—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—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—41no—
165C5×C4⋊C4C2×Q812yes—
166C5×C22⋊C4—24no—
167C5×C22⋊C4—42no—
168C5×C22⋊C4—42no—
169C5×C22⋊C4—81no—
170C5×C22⋊C4—81no—
171C42×C5—21no—
172C42×C5—21no—
173C42×C5—21no—
174C42×C5—21no—
175C42×C5—81no—
176C42×C5—81no—
177C42×C5C2×Q812yes—
178C42×C5#4112yes—
179C4⋊Dic5—21no—
180C4⋊Dic5—21no—
181C4⋊Dic5—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—21no—
187C4⋊Dic5—42no—
188C4⋊Dic5—81no—
189#683—42no—
190#683—81no—
191C4×Dic5—81no—
192C2×#168—21no—
193C2×#168—21no—
194C2×#168—21no—
195C2×#168—21no—
196C2×#168—21no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168#4112yes—
203C2×#168#4112yes—
204C2×C4×D4C2×D511yescharacteristic
205C2×#107—51no—
206C2×#107—51no—
207C2×#107—51no—
208C2×#106—51no—
209C2×#106—51no—
210C2×#106—51no—
211C2×C4⋊C8—51no—
212C2×C4⋊C8—51no—
213C2×C4⋊C8—51no—
214C2×#102—51no—
215C2×#102—51no—
216C2×#102—51no—
217C2×#102—51no—
218C2×C4×C8—51no—
219#400—101no—
220#398—101no—
221#390C2×D511yescharacteristic
222#390C2×D511yescharacteristic
223#389—51no—
224#389—51no—
225#388—21no—
226#388—21no—
227#387—101no—
228#387—101no—
229#384—51no—
230#382—51no—
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#336—51no—
246#336—51no—
247C23×C5—22no—
248C23×C5—41no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—41no—
252C23×C5—41no—
253C23×C5—41no—
254C23×C5C22×D411yescharacteristic
255C5×D4—22no—
256C5×D4—22no—
257C5×D4—41no—
258C5×D4—41no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—22no—
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—84no—
284C2×C4×C5C22⋊Q812yes—
285C2×C4×C5C22⋊Q812yes—
286C2×C4×C5#13512yes—
287C2×C4×C5C22×Q811yescharacteristic
288C2×C4×C5C2×#4111yescharacteristic
289C2×C4×C5C2×#4111yescharacteristic
290C2×C4×C5C2×#4111yescharacteristic
291C2×C4×C5H(5,𝔽2)11yescharacteristic
292C2×C4×C52−1+411yescharacteristic
293C2×Dic5—41no—
294C2×Dic5—41no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305#168—22no—
306#168—22no—
307#168—41no—
308#168—41no—
309C22×D4C22×D511yescharacteristic
310C23×C4—21no—
311C22×C8—51no—
312C22×C8—51no—
313C22×C8—51no—
314C22×C8—51no—
315C4×D4—22no—
316C4×D4Dic1014yes—
317C2×C4⋊C4—51no—
318C2×C4⋊C4—51no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—51no—
321C2×C4⋊C4—101no—
322C2×C4⋊C4—101no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—101no—
325C2×C4⋊C4C22⋊D511yescharacteristic
326C2×C4⋊C4C22⋊D511yescharacteristic
327C2×C4⋊C4C22×D511yescharacteristic
328C2×C4⋊C4C22×D511yescharacteristic
329C2×C4⋊C4C22×D511yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—101no—
338C2×C42C22×D511yescharacteristic
339C2×C42C22×D511yescharacteristic
340C2×C42C22×D511yescharacteristic
341#107—102no—
342#107—102no—
343#107—201no—
344#106—201no—
345#106—201no—
346#106—201no—
347C4⋊C8—102no—
348C4⋊C8—102no—
349C4⋊C8—102no—
350#102—102no—
351#102—102no—
352#102—201no—
353#102—201no—
354C4×C8—102no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—101no—
360#95—101no—
361#95—101no—
362#95—101no—
363#95—101no—
364#95—101no—
365#95—101no—
366#95C22⋊D511yescharacteristic
367#95C22⋊D511yescharacteristic
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—42no—
371C22×C5—81no—
372C22×C5—81no—
373C22×C5—81no—
374C22×C5—81no—
375C22×C5C2×C22⋊Q811yescharacteristic
376C22×C5#54711yescharacteristic
377C22×C5#55011yescharacteristic
378C22×C5C2×#13511yescharacteristic
379C22×C5#57511yescharacteristic
380C22×C5#57611yescharacteristic
381C22×C5#57711yescharacteristic
382C4×C5—22no—
383C4×C5—22no—
384C4×C5—22no—
385C4×C5—42no—
386C4×C5—41no—
387C4×C5—41no—
388C4×C5—41no—
389C4×C5—81no—
390C4×C5—81no—
391C4×C5#53311yescharacteristic
392C4×C5#54411yescharacteristic
393C4×C5#55211yescharacteristic
394C4×C5#55311yescharacteristic
395Dic5—81no—
396Dic5—81no—
397Dic5—81no—
398Dic5—81no—
399C24—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4C2×Dic1012yes—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—101no—
413C22×C4—101no—
414C22×C4—101no—
415C22×C4—101no—
416C22×C4C2×C22⋊D511yescharacteristic
417C22×C4C2×C22⋊D511yescharacteristic
418C22×C4C2×C22⋊D511yescharacteristic
419C22×C4C2×C22⋊D511yescharacteristic
420C22×C4C2×C22⋊D511yescharacteristic
421C22×C4C2×C22⋊D511yescharacteristic
422C22×C4C23×D511yescharacteristic
423C2×C8—52no—
424C2×C8—52no—
425C2×C8—101no—
426C2×C8—101no—
427C2×C8—101no—
428C2×C8—101no—
429C2×C8—101no—
430C2×C8—101no—
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—101no—
434C2×C8—101no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—101no—
443C4⋊C4—101no—
444C4⋊C4—101no—
445C4⋊C4—101no—
446C4⋊C4—101no—
447C4⋊C4—101no—
448C4⋊C4—101no—
449C4⋊C4—101no—
450C4⋊C4—202no—
451C4⋊C4—202no—
452C4⋊C4—401no—
453C4⋊C4—401no—
454C4⋊C4C2×Dic1012yes—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—81no—
465C42—81no—
466C42—401no—
467C42C2×Dic1012yes—
468C42#70912yes—
469C2×C5—82no—
470C2×C5#244111yescharacteristic
471C2×C5#307611yescharacteristic
472C2×C5#308111yescharacteristic
473C2×C5#333711yescharacteristic
474C2×C5#335511yescharacteristic
475C2×C5#339611yescharacteristic
476C2×C5#340111yescharacteristic
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C22×C22⋊D511yescharacteristic
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—201no—
515C2×C4—201no—
516C2×C4—201no—
517C2×C4—201no—
518C2×C4—201no—
519C2×C4—201no—
520C2×C4—201no—
521C2×C4—201no—
522C2×C4—201no—
523C2×C4—201no—
524C2×C4—201no—
525C2×C4—201no—
526C2×C4#407012yes—
527C2×C4#407012yes—
528C2×C4#409612yes—
529C2×C4C22×Dic1011yescharacteristic
530C2×C4C2×#70911yescharacteristic
531C2×C4D5×#4111yescharacteristic
532C2×C4D5×#4111yescharacteristic
533C2×C4#414911yescharacteristic
534C2×C4#415011yescharacteristic
535C8—102no—
536C8—102no—
537C8—201no—
538C8—201no—
539C5#2609411yescharacteristic, complemented, Sylow
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#407011yescharacteristic
548C22C2×#409611yescharacteristic
549C22#6615611yescharacteristic
550C22#6615811yescharacteristic
551C22#6616811yescharacteristic
552C22#6616911yescharacteristic
553C22#6617011yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—401no—
564C4—401no—
565C4—401no—
566C4—401no—
567C4#6585111yescharacteristic
568C4#6585311yescharacteristic
569C4#6586311yescharacteristic
570C4#6586411yescharacteristic
571C2—82no—
572C2#1061398011yescharacteristic
573C2#1061423811yescharacteristic
574C2#1061427011yescharacteristic
575C2#1061533711yescharacteristic
576C2#1061534411yescharacteristic
577C2#1061544311yescharacteristic
578C2#1061545111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2357C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
hypercenter#2357D5
hypocenterC5#26094
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. #10605454C2semidirect
9. #10600342C2semidirect
10. #10600333C2semidirect
11. #10600277C2semidirect
12. #10600261C2semidirect
13. #10600260C2semidirect
14. #10600254C2semidirect
15. #10600251C2semidirect
539. C5#26094semidirect

Maximal subgroups

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

subgroupindex
C5×#23572
#106054542
C4×#39642
#106023982
#106004482
#106004472
#106004452
#106003422
#106003332
#106002772
#106002612
#106002602
#106002542
#106002512
#260945