Finite Group Databasebeta

← #12129299

Subgroups — #12129299

Order 1280.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2285C211yesmaximal, characteristic, complemented
2C4×#4076C211yesmaximal, characteristic
3C4×C22⋊#168C211yesmaximal, characteristic, complemented
4#10600380C211yesmaximal, characteristic, complemented
5#10600378C211yesmaximal, characteristic, complemented
6#10600365C211yesmaximal, characteristic, complemented
7#10600363C211yesmaximal, characteristic, complemented
8#10600317C212yesmaximal
9#10600300C212yesmaximal, complemented
10C4×#3940C212yesmaximal, complemented
11C4×#3941C212yesmaximal, complemented
12C2×C5×#117—21no—
13C22×C42×C5C2211yescharacteristic
14C2×#4076C2211yescharacteristic
15C2×C4×C22⋊D5—21no—
16C2×C4×D20C2211yescharacteristic
17C5×#384C2211yescharacteristic
18C5×#384C2211yescharacteristic
19C5×#382—21no—
20C5×#380—21no—
21C5×#380C2212yes—
22C4×C5×C4⋊C4—21no—
23C4×C5×C4⋊C4C2212yes—
24#65502—21no—
25#65501C2211yescharacteristic
26C2×C22⋊#168C2212yes—
27C2×#3941C2212yes—
28C2×#3941C2212yes—
29C2×#3940C2212yes—
30C2×#3940C2212yes—
31#65229—21no—
32#65228C2211yescharacteristic
33C4×#685—21no—
34C4×C4⋊Dic5C2211yescharacteristic
35#65217C2211yescharacteristic
36C2×C4×#168C2212yes—
37#64749C2212yes—
38#64749C2212yes—
39#64749C2212yes—
40#64749C2212yes—
41#64744C2211yescharacteristic
42#64744C2211yescharacteristic
43#23755—51nomaximal, Sylow
44C23×C4×C5D411yescharacteristic
45C23×C4×C5D411yescharacteristic
46C23×C4×C5C2311yescharacteristic
47C22×C22⋊D5—21no—
48C22×D20C2311yescharacteristic
49C22×C4×D5—21no—
50C5×#117—44no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4—21no—
54C2×C5×C4⋊C4—21no—
55C2×C5×C4⋊C4C2312yes—
56C2×C5×C4⋊C4C2312yes—
57C2×C5×C22⋊C4—22no—
58C2×C42×C5—24no—
59C2×C42×C5—22no—
60C2×C42×C5D411yescharacteristic
61C2×C42×C5D411yescharacteristic
62C2×C42×C5C2311yescharacteristic
63C2×C42×C5C2311yescharacteristic
64C2×C22⋊Dic5—21no—
65#4076—22no—
66#4076—22no—
67C4×C22⋊D5—44no—
68C2×#685—21no—
69C2×#685—21no—
70C2×#685—21no—
71C2×#685—21no—
72C2×C4⋊Dic5C2311yescharacteristic
73C2×C4⋊Dic5C2311yescharacteristic
74C2×C4⋊Dic5C2311yescharacteristic
75C2×#683—21no—
76C2×C4×Dic5—21no—
77C22×#168C2312yes—
78C22×#168C2312yes—
79C4×D20—24no—
80C5×#95—22no—
81C5×#95—22no—
82#3963—21no—
83#3963—21no—
84#3963—21no—
85C22⋊#168—24no—
86#3941—24no—
87#3941—24no—
88#3940—24no—
89#3940—24no—
90C4×#168—24no—
91C4×#121—51no—
92#2285D511yescharacteristic, complemented
93#2100—51no—
94#2094—51no—
95#2084—51no—
96#2077—52no—
97#2071—51no—
98#2034—52no—
99C4×#106—52no—
100C4×#102—52no—
101C4×C22⋊C8—51no—
102C24×C5C2×D411yescharacteristic
103C23×D5—21no—
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5—22no—
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—41no—
115C22×C4×C5SD1614yes—
116C22×C4×C5C2×D411yescharacteristic
117C22×C4×C5C2×D411yescharacteristic
118C22×C4×C5C2×D411yescharacteristic
119C22×C4×C5C2×D411yescharacteristic
120C22×C4×C5C2×D411yescharacteristic
121C22×C4×C5C2411yescharacteristic
122C2×C22⋊D5—24no—
123C2×C22⋊D5—42no—
124C2×C22⋊D5—42no—
125C22×Dic5—21no—
126C22×Dic5—21no—
127C22×Dic5—21no—
128C2×D20—22no—
129C2×D20—22no—
130C2×D20—21no—
131C2×D20#4112yes—
132C2×C4×D5—42no—
133C2×C4×D5—42no—
134C2×C4×D5—42no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—22no—
139C5×C4⋊C4—42no—
140C5×C4⋊C4—42no—
141C5×C4⋊C4—42no—
142C5×C4⋊C4—42no—
143C5×C22⋊C4—44no—
144C42×C5—22no—
145C42×C5—22no—
146C42×C5—22no—
147C42×C5—44no—
148C42×C5#4114yes—
149C22⋊Dic5—42no—
150#685—24no—
151#685—42no—
152#685—42no—
153#685—42no—
154C4⋊Dic5—21no—
155C4⋊Dic5—22no—
156C4⋊Dic5—22no—
157C4⋊Dic5#4112yes—
158#683—42no—
159C4×Dic5—42no—
160C2×#168—22no—
161C2×#168—22no—
162C2×#168—22no—
163C2×#168—22no—
164C2×#168—22no—
165C2×#168#4114yes—
166C2×#121—51no—
167C2×C4×D4—51no—
168C2×C4×D4—101no—
169C2×#117—21no—
170C22×C42C2×D511yescharacteristic
171C2×#106—52no—
172C2×#106—52no—
173C2×#102—52no—
174C2×#102—52no—
175C2×C22⋊C8—52no—
176C2×C4×C8—52no—
177#390—51no—
178#389—51no—
179#388—101no—
180#386—101no—
181#384C2×D511yescharacteristic
182#384C2×D511yescharacteristic
183#382—21no—
184#380—21no—
185#380—51no—
186#380C2×D512yes—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—51no—
189C4×C4⋊C4C2×D512yes—
190C4×C22⋊C4—101no—
191#340—52no—
192#340—52no—
193#340—52no—
194#340—52no—
195#336—51no—
196#336—51no—
197C23×C5—21no—
198C23×C5—21no—
199C23×C5—21no—
200C23×C5—21no—
201C23×C5—21no—
202C23×C5—21no—
203C23×C5C2×SD1612yes—
204C23×C5C22×D411yescharacteristic
205C22×D5—22no—
206C22×D5—41no—
207C22×D5—41no—
208C22×D5—41no—
209C22×D5—41no—
210C22×D5—41no—
211C22×D5—41no—
212C2×C4×C5—21no—
213C2×C4×C5—21no—
214C2×C4×C5—21no—
215C2×C4×C5—21no—
216C2×C4×C5—21no—
217C2×C4×C5—21no—
218C2×C4×C5—28no—
219C2×C4×C5—21no—
220C2×C4×C5—21no—
221C2×C4×C5—21no—
222C2×C4×C5—44no—
223C2×C4×C5—42no—
224C2×C4×C5—42no—
225C2×C4×C5—42no—
226C2×C4×C5—42no—
227C2×C4×C5—42no—
228C2×C4×C5—42no—
229C2×C4×C5—42no—
230C2×C4×C5—42no—
231C2×C4×C5#12312yes—
232C2×C4×C5#12312yes—
233C2×C4×C5C2×SD1612yes—
234C2×C4×C5C2×SD1612yes—
235C2×C4×C5#13512yes—
236C2×C4×C5AGL(1,ℤ/8)12yes—
237C2×C4×C5C2×#4112yes—
238C2×C4×C5C2×#4111yescharacteristic
239C2×C4×C5C2×#4111yescharacteristic
240C2×C4×C5C2×#4112yes—
241C22⋊D5—48no—
242C2×Dic5—22no—
243C2×Dic5—41no—
244C2×Dic5—41no—
245C2×Dic5—41no—
246C2×Dic5—41no—
247C2×Dic5—41no—
248C2×Dic5—41no—
249C2×Dic5—41no—
250C2×Dic5—41no—
251D20—24no—
252D20—42no—
253C4×D5—84no—
254#168—24no—
255#168—42no—
256C22×D4—51no—
257C22×D4—101no—
258C23×C4—101no—
259C23×C4C22⋊D511yescharacteristic
260C23×C4C22⋊D511yescharacteristic
261C23×C4C22×D511yescharacteristic
262C22×C8—52no—
263C22×C8—52no—
264#121—102no—
265#121—102no—
266C4×D4—104no—
267C4×D4—204no—
268#117—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—51no—
274C2×C4⋊C4—51no—
275C2×C4⋊C4—51no—
276C2×C4⋊C4—101no—
277C2×C4⋊C4C22×D512yes—
278C2×C4⋊C4C22×D512yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—101no—
281C2×C22⋊C4—101no—
282C2×C22⋊C4—101no—
283C2×C22⋊C4—101no—
284C2×C22⋊C4—101no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—101no—
288C2×C42C22⋊D511yescharacteristic
289C2×C42C22⋊D511yescharacteristic
290C2×C42C22×D511yescharacteristic
291C2×C42C22×D511yescharacteristic
292#106—104no—
293#106—104no—
294#102—104no—
295#102—104no—
296C22⋊C8—104no—
297C4×C8—104no—
298#95—22no—
299#95—22no—
300#95—101no—
301#95—101no—
302#95—101no—
303C22×C5—22no—
304C22×C5—22no—
305C22×C5—22no—
306C22×C5—41no—
307C22×C5—41no—
308C22×C5—41no—
309C22×C5—41no—
310C22×C5C2×#12311yescharacteristic
311C22×C5#52511yescharacteristic
312C22×C5#53211yescharacteristic
313C22×C5C22×SD1611yescharacteristic
314C22×C5C2×#13511yescharacteristic
315C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C5#57511yescharacteristic
317C2×D5—42no—
318C2×D5—42no—
319C2×D5—42no—
320C2×D5—81no—
321C2×D5—81no—
322C2×D5—81no—
323C2×D5—81no—
324C4×C5—22no—
325C4×C5—22no—
326C4×C5—22no—
327C4×C5—41no—
328C4×C5—82no—
329C4×C5—82no—
330C4×C5#48114yes—
331C4×C5#52912yes—
332C4×C5#52912yes—
333Dic5—42no—
334Dic5—81no—
335Dic5—81no—
336C24—101no—
337C24C2×C22⋊D511yescharacteristic
338C2×D4—52no—
339C2×D4—104no—
340C2×D4—102no—
341C2×D4—102no—
342C2×D4—101no—
343C2×D4—202no—
344C2×D4—202no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—101no—
357C22×C4—101no—
358C22×C4—101no—
359C22×C4—202no—
360C22×C4—202no—
361C22×C4—202no—
362C22×C4#68814yes—
363C22×C4C2×C22⋊D511yescharacteristic
364C22×C4C2×C22⋊D511yescharacteristic
365C22×C4C2×C22⋊D511yescharacteristic
366C22×C4C2×C22⋊D511yescharacteristic
367C22×C4C2×C22⋊D511yescharacteristic
368C22×C4C23×D511yescharacteristic
369C2×C8—54no—
370C2×C8—102no—
371C2×C8—102no—
372C2×C8—102no—
373C2×C8—102no—
374C2×C8—102no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—52no—
384C4⋊C4—101no—
385C4⋊C4—102no—
386C4⋊C4—102no—
387C4⋊C4—202no—
388C22⋊C4—44no—
389C22⋊C4—104no—
390C22⋊C4—202no—
391C22⋊C4—202no—
392C22⋊C4—202no—
393C22⋊C4—202no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—44no—
398C42—202no—
399C42#70914yes—
400C2×C5—42no—
401C2×C5#239511yescharacteristic
402C2×C5C2×#48111yescharacteristic
403C2×C5#309111yescharacteristic
404C2×C5#310111yescharacteristic
405C2×C5#310611yescharacteristic
406C2×C5#316711yescharacteristic
407C2×C5#317111yescharacteristic
408D5—82no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—102no—
416C23—201no—
417C23—201no—
418C23—201no—
419C23—201no—
420C23—201no—
421C23—201no—
422C23C2×#68812yes—
423C23C22×C22⋊D511yescharacteristic
424D4—104no—
425D4—208no—
426D4—202no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—28no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—102no—
447C2×C4—201no—
448C2×C4—201no—
449C2×C4—201no—
450C2×C4—201no—
451C2×C4—201no—
452C2×C4—201no—
453C2×C4—201no—
454C2×C4—201no—
455C2×C4—404no—
456C2×C4#407512yes—
457C2×C4#407512yes—
458C2×C4#407812yes—
459C2×C4C2×#68812yes—
460C2×C4C2×#68812yes—
461C2×C4#409612yes—
462C2×C4C2×#70912yes—
463C2×C4D5×#4111yescharacteristic
464C2×C4D5×#4111yescharacteristic
465C2×C4D5×#4112yes—
466C8—104no—
467C8—202no—
468C5#2375511yescharacteristic, complemented, Sylow
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—202no—
477C22—202no—
478C22—202no—
479C22—401no—
480C22—401no—
481C22—401no—
482C22—401no—
483C22C2×#407511yescharacteristic
484C22C2×#407811yescharacteristic
485C22C22×#68811yescharacteristic
486C22C2×#409611yescharacteristic
487C22#6615611yescharacteristic
488C22#6616211yescharacteristic
489C22#6616611yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—41no—
494C4—82no—
495C4—82no—
496C4—202no—
497C4—401no—
498C4—401no—
499C4#6525714yes—
500C4#6586212yes—
501C4#6586212yes—
502C2—42no—
503C2—402no—
504C2C2×#6525711yescharacteristic
505C2#1061394611yescharacteristic
506C2#1061400511yescharacteristic
507C2#1061502311yescharacteristic
508C2#1061503011yescharacteristic
509C2#1061507411yescharacteristic
510C2#1061508211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2285C2
socleC23×C5C22×D4
prime radicalC2×#4076C22
hypercenter#2285D5
hypocenterC5#23755
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C5×#2285C2semidirect
3. C4×C22⋊#168C2semidirect
4. #10600380C2semidirect
5. #10600378C2semidirect
6. #10600365C2semidirect
7. #10600363C2semidirect
9. #10600300C2semidirect
10. C4×#3940C2semidirect
11. C4×#3941C2semidirect
92. #2285D5semidirect
468. C5#23755semidirect

Maximal subgroups

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

subgroupindex
C5×#22852
C4×#40762
C4×C22⋊#1682
#106003802
#106003782
#106003652
#106003632
#106003172
#106003002
C4×#39402
C4×#39412
#237555