Finite Group Databasebeta

← #12098166

Subgroups — #12098166

Order 1280.

870 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
1C2×#65128C211yesmaximal, characteristic
2C2×#64998C211yesmaximal, characteristic
3#10605351C211yesmaximal, characteristic
4C5×#1771C211yesmaximal, characteristic, complemented
5#10601259C211yesmaximal, characteristic, complemented
6#10600625C211yesmaximal, characteristic, complemented
7#10600624C211yesmaximal, characteristic
8#10600603C211yesmaximal, characteristic, complemented
9#10600574C211yesmaximal, characteristic
10C4×#3946C211yesmaximal, characteristic, complemented
11C4⋊C4×#168C211yesmaximal, characteristic, complemented
12#10599442C211yesmaximal, characteristic
13#10599440C211yesmaximal, characteristic
14#10599397C211yesmaximal, characteristic, complemented
15#10599396C211yesmaximal, characteristic, complemented
16C22×#676C2211yescharacteristic
17C22×C8×D5C2211yescharacteristic
18C2×#4018C2211yescharacteristic
19C2×C5×C4⋊C8C2211yescharacteristic
20C2×C5×#97C2211yescharacteristic
21C2×C5×#97C2211yescharacteristic
22C2×C4×C5×C8C2211yescharacteristic
23C4×C5×C4⋊C4C2211yescharacteristic
24C2×#3952C2211yescharacteristic
25C2×#3952C2211yescharacteristic
26C2×#3947C2211yescharacteristic
27C2×#3947C2211yescharacteristic
28C2×#3947C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31C2×#3946C2211yescharacteristic
32C2×#3946C2211yescharacteristic
33C2×C8×Dic5C2211yescharacteristic
34#65279C412yes—
35#65267C412yes—
36C4×#685C2211yescharacteristic
37C4×#685C2211yescharacteristic
38C4×C4⋊Dic5C2211yescharacteristic
39#65221C2211yescharacteristic
40C42×Dic5C2211yescharacteristic
41C2×C4⋊#168C2211yescharacteristic
42C2×C4×#168C2211yescharacteristic
43C2×C4×#168C2211yescharacteristic
44C2×C4×#168C2211yescharacteristic
45#65128C418yescomplemented
46#64998C418yescomplemented
47#64959C412yes—
48#64944C412yes—
49C5×#336C2211yescharacteristic
50C5×#336C2211yescharacteristic
51#64770C2211yescharacteristic
52#64770C2211yescharacteristic
53#64770C2211yescharacteristic
54#64770C2211yescharacteristic
55#64744C2211yescharacteristic
56#64744C2211yescharacteristic
57#20553—51nomaximal, Sylow
58C22×C4×D5C2311yescharacteristic
59C22×C5×C8C2311yescharacteristic
60C22×C5×C8C2311yescharacteristic
61C22×C5×C8C2311yescharacteristic
62C22×C5×C8C2311yescharacteristic
63C2×C5×C4⋊C4C2×C412yes—
64C2×C42×C5C2311yescharacteristic
65C2×C42×C5C2311yescharacteristic
66C2×C42×C5C2311yescharacteristic
67C2×#685C2×C412yes—
68C2×#685C2×C412yes—
69C2×#685C2×C411yescharacteristic
70C2×#685C2×C411yescharacteristic
71C2×C4⋊Dic5C2×C412yes—
72C2×#683C2×C411yescharacteristic
73C2×#683C2×C411yescharacteristic
74C2×C4×Dic5C2×C412yes—
75C2×C4×Dic5C2×C412yes—
76C2×C4×Dic5C2311yescharacteristic
77C2×C4×Dic5C2311yescharacteristic
78C2×C4×Dic5C2311yescharacteristic
79C22×#168C2311yescharacteristic
80C22×#168C2311yescharacteristic
81C22×#168C2311yescharacteristic
82C22×#168C2311yescharacteristic
83C2×#676—41no—
84C2×#676—41no—
85C2×#676C2×C414yes—
86C2×C8×D5—41no—
87C2×C8×D5—41no—
88C2×C8×D5C2×C414yes—
89#4018C2×C414yes—
90#4018C2×C414yes—
91C5×C4⋊C8C2×C414yes—
92C5×#97—41no—
93C5×#97C2×C414yes—
94C4×C5×C8—41no—
95C5×#95C2×C412yes—
96#3963C2×C412yes—
97#3963C2×C412yes—
98#3963C2×C412yes—
99#3963C2×C412yes—
100#3963C2×C412yes—
101#3952C2×C414yes—
102#3952C2×C414yes—
103#3947—41no—
104#3947—41no—
105#3947—41no—
106#3946C2×C414yes—
107#3946C2×C414yes—
108#3946C2×C414yes—
109#3946C2×C414yes—
110C8×Dic5—41no—
111C4⋊#168C2×C414yes—
112C4×#168—41no—
113C4×#168—41no—
114C4×#168C2×C414yes—
115C2×#433—51no—
116C2×#424—51no—
117#2298—51no—
118#1771—51no—
119#1771D511yescharacteristic, complemented
120#1770—51no—
121C8×C4⋊C4—51no—
122#1768—51no—
123#1768—51no—
124#1767—51no—
125C4×C4⋊C8—51no—
126#1753—51no—
127#1753—51no—
128#1752—51no—
129C8×C22⋊C4—51no—
130C23×D5C22×C411yescharacteristic
131C22×C4×C5C22×C412yes—
132C22×C4×C5C22×C411yescharacteristic
133C22×C4×C5C22×C411yescharacteristic
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2411yescharacteristic
136C22×Dic5C22×C412yes—
137C22×Dic5C22×C412yes—
138C22×Dic5C22×C411yescharacteristic
139C22×Dic5C22×C411yescharacteristic
140C22×Dic5C22×C411yescharacteristic
141C2×C4×D5—21no—
142C2×C4×D5—21no—
143C2×C4×D5—21no—
144C2×C4×D5—21no—
145C2×C4×D5C22×C412yes—
146C2×C4×D5C22×C412yes—
147C2×C5×C8—21no—
148C2×C5×C8—21no—
149C2×C5×C8—21no—
150C2×C5×C8—21no—
151C2×C5×C8—21no—
152C2×C5×C8—21no—
153C2×C5×C8—21no—
154C2×C5×C8—21no—
155C2×C5×C8C22×C412yes—
156C2×C5×C8C22×C412yes—
157C2×C5×C8C22×C412yes—
158C2×C5×C8C22×C412yes—
159C5×C4⋊C4—24no—
160C42×C5—21no—
161C42×C5—21no—
162C42×C5—21no—
163C42×C5—21no—
164C42×C5C22×C412yes—
165C42×C5C22×C412yes—
166#685—24no—
167#685—42no—
168#685C4214yes—
169#685C4214yes—
170C4⋊Dic5—42no—
171#683C4214yes—
172#683C4214yes—
173C4×Dic5—24no—
174C4×Dic5—21no—
175C4×Dic5—21no—
176C4×Dic5—21no—
177C4×Dic5—21no—
178C4×Dic5—42no—
179C4×Dic5C22×C412yes—
180C4×Dic5C22×C412yes—
181C2×#168—21no—
182C2×#168—21no—
183C2×#168—21no—
184C2×#168—21no—
185C2×#168—21no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168C22×C412yes—
190C2×#168C22×C412yes—
191C2×#168C22×C412yes—
192C2×#168C22×C412yes—
193#676—44no—
194C8×D5—44no—
195C22×M16—51no—
196C23×C8—51no—
197C2×#117—51no—
198#433—58no—
199#424—58no—
200C2×C4⋊C8—51no—
201C2×C4⋊C8—51no—
202C2×C4⋊C8—51no—
203C2×C4⋊C8—51no—
204C2×C4⋊C8—51no—
205C2×C4⋊C8C2×D511yescharacteristic
206C2×C22⋊C8—51no—
207C2×C22⋊C8—51no—
208C2×#97—51no—
209C2×#97—51no—
210C2×#97—51no—
211C2×#97C2×D511yescharacteristic
212C2×#97C2×D511yescharacteristic
213C2×C4×C8—51no—
214C2×C4×C8—51no—
215C2×C4×C8—51no—
216C2×C4×C8—51no—
217C2×C4×C8C2×D511yescharacteristic
218#388—52no—
219#388—52no—
220#387—52no—
221#387—52no—
222C4×C4⋊C4—51no—
223C4×C4⋊C4C2×D511yescharacteristic
224C4×C22⋊C4—51no—
225C4×C22⋊C4—51no—
226#376—51no—
227C43—51no—
228#336—51no—
229#336—51no—
230#336—51no—
231#336—51no—
232#336—51no—
233#336—51no—
234#336C2×D511yescharacteristic
235#336C2×D511yescharacteristic
236C23×C5C23×C411yescharacteristic
237C22×D5—21no—
238C22×D5—21no—
239C22×D5—22no—
240C22×D5—22no—
241C22×D5C2×C4212yes—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5C2×C4212yes—
250C2×C4×C5C2×C4212yes—
251C2×C4×C5#13114yes—
252C2×C4×C5C23×C411yescharacteristic
253C2×C4×C5C23×C411yescharacteristic
254C2×C4×C5H(5,𝔽2)11yescharacteristic
255C2×C4×C5H(5,𝔽2)11yescharacteristic
256C2×C4×C52−1+411yescharacteristic
257C2×C4×C52−1+411yescharacteristic
258C2×Dic5—21no—
259C2×Dic5—21no—
260C2×Dic5—22no—
261C2×Dic5—22no—
262C2×Dic5—22no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5C2×C4212yes—
269C2×Dic5C2×C4212yes—
270C2×Dic5C2×C4212yes—
271C4×D5—22no—
272C4×D5—22no—
273C4×D5—22no—
274C4×D5—22no—
275C5×C8—22no—
276C5×C8—22no—
277C5×C8—22no—
278C5×C8—22no—
279#168—22no—
280#168—22no—
281#168—22no—
282#168—22no—
283C23×C4—51no—
284C2×M16—54no—
285C2×M16—201no—
286C2×M16—201no—
287C22×C8—51no—
288C22×C8—51no—
289C22×C8—51no—
290C22×C8—51no—
291C22×C8—54no—
292C22×C8—201no—
293C22×C8—201no—
294C22×C8C22×D511yescharacteristic
295C22×C8C22×D511yescharacteristic
296C22×C8C22×D511yescharacteristic
297C22×C8C22×D511yescharacteristic
298#117—54no—
299#117—54no—
300C2×C4⋊C4—51no—
301C2×C4⋊C4—51no—
302C2×C4⋊C4—52no—
303C2×C4⋊C4C4×D512yes—
304C2×C22⋊C4—51no—
305C2×C22⋊C4—51no—
306C2×C22⋊C4—52no—
307C2×C22⋊C4—52no—
308C2×C42—51no—
309C2×C42—52no—
310C2×C42—51no—
311C2×C42—51no—
312C2×C42—52no—
313C2×C42C22×D511yescharacteristic
314C2×C42C22×D511yescharacteristic
315C2×C42C22×D511yescharacteristic
316C4⋊C8—54no—
317C4⋊C8—54no—
318C4⋊C8—54no—
319C4⋊C8—54no—
320C4⋊C8—54no—
321C4⋊C8C4×D514yes—
322C22⋊C8—54no—
323C22⋊C8—54no—
324#97—41no—
325#97—201no—
326#97—201no—
327#97—201no—
328#97C4×D514yes—
329C4×C8—41no—
330C4×C8—54no—
331C4×C8—201no—
332C4×C8—201no—
333C4×C8—201no—
334#95—52no—
335#95—52no—
336#95—52no—
337#95—52no—
338#95—52no—
339#95C4×D512yes—
340C22×C5C22×C4211yescharacteristic
341C22×C5#52011yescharacteristic
342C22×C5#52011yescharacteristic
343C22×C5C2×#13112yes—
344C22×C5#56812yes—
345C2×D5—22no—
346C2×D5—22no—
347C2×D5—22no—
348C2×D5—42no—
349C2×D5—42no—
350C4×C5—22no—
351C4×C5—22no—
352C4×C5—24no—
353C4×C5—42no—
354C4×C5#51811yescharacteristic
355C4×C5#51911yescharacteristic
356C4×C5#52011yescharacteristic
357C4×C5#52011yescharacteristic
358Dic5—22no—
359Dic5—22no—
360Dic5—22no—
361Dic5—42no—
362Dic5—42no—
363C24—51no—
364C22×C4—51no—
365C22×C4—51no—
366C22×C4—51no—
367C22×C4—52no—
368C22×C4—52no—
369C22×C4—52no—
370C22×C4—52no—
371C22×C4—101no—
372C22×C4—101no—
373C22×C4—101no—
374C22×C4—101no—
375C22×C4C2×C4×D512yes—
376C22×C4C2×C4×D511yescharacteristic
377C22×C4C2×C4×D511yescharacteristic
378C22×C4C2×C4×D512yes—
379C22×C4C23×D511yescharacteristic
380M16—204no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—52no—
390C2×C8—52no—
391C2×C8—52no—
392C2×C8—52no—
393C2×C8—101no—
394C2×C8—101no—
395C2×C8—101no—
396C2×C8—101no—
397C2×C8—101no—
398C2×C8—101no—
399C2×C8—101no—
400C2×C8—101no—
401C2×C8—204no—
402C2×C8C2×C4×D512yes—
403C2×C8C2×C4×D512yes—
404C2×C8C2×C4×D512yes—
405C2×C8C2×C4×D512yes—
406C4⋊C4—24no—
407C4⋊C4—54no—
408C4⋊C4—54no—
409C4⋊C4—202no—
410C22⋊C4—54no—
411C22⋊C4—54no—
412C22⋊C4—104no—
413C22⋊C4—202no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—52no—
419C42—52no—
420C42—101no—
421C42—101no—
422C42—101no—
423C42—101no—
424C42—104no—
425C42—202no—
426C42C2×C4×D512yes—
427C42C2×C4×D512yes—
428C2×C5#227411yescharacteristic
429C2×C5C4×#13111yescharacteristic
430C2×C5#287511yescharacteristic
431C2×C5#299212yes—
432C2×C5#299312yes—
433D5—42no—
434C23—52no—
435C23—102no—
436C23—102no—
437C23—101no—
438C23—101no—
439C23C22×C4×D511yescharacteristic
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—52no—
448C2×C4—52no—
449C2×C4—52no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—102no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—102no—
458C2×C4—102no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—102no—
464C2×C4C42×D512yes—
465C2×C4C42×D512yes—
466C2×C4#405314yes—
467C2×C4C22×C4×D511yescharacteristic
468C2×C4C22×C4×D511yescharacteristic
469C2×C4#414411yescharacteristic
470C2×C4#414711yescharacteristic
471C2×C4#414911yescharacteristic
472C2×C4#415011yescharacteristic
473C8—22no—
474C8—22no—
475C8—22no—
476C8—22no—
477C8—102no—
478C8—102no—
479C8—102no—
480C8—102no—
481C5#2055311yescharacteristic, complemented, Sylow
482C22—102no—
483C22—102no—
484C22—102no—
485C22—202no—
486C22—202no—
487C22C2×C42×D511yescharacteristic
488C22#6588011yescharacteristic
489C22#6590811yescharacteristic
490C22C2×#405312yes—
491C22#6608412yes—
492C4—22no—
493C4—22no—
494C4—24no—
495C4—42no—
496C4—102no—
497C4—102no—
498C4—102no—
499C4—202no—
500C4—202no—
501C4#6583811yescharacteristic
502C4#6585711yescharacteristic
503C4#6587911yescharacteristic
504C4#6590611yescharacteristic
505C2—202no—
506C2#1060548111yescharacteristic
507C2C4×#405311yescharacteristic
508C2#1061028111yescharacteristic
509C2#1061119412yes—
510C2#1061167812yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC5×#1771C2
socleC23×C5C23×C4
prime radicalC23×D5C22×C4
cosocleC22×C4×C5C24
hypercenter#1771D5
hypocenterC5#20553
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
4. C5×#1771C2semidirect
5. #10601259C2semidirect
6. #10600625C2semidirect
8. #10600603C2semidirect
10. C4×#3946C2semidirect
11. C4⋊C4×#168C2semidirect
14. #10599397C2semidirect
15. #10599396C2semidirect
45. #65128C4semidirect
46. #64998C4semidirect
119. #1771D5semidirect
481. C5#20553semidirect

Maximal subgroups

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

subgroupindex
C2×#651282
C2×#649982
#106053512
C5×#17712
#106012592
#106006252
#106006242
#106006032
#106005742
C4×#39462
C4⋊C4×#1682
#105994422
#105994402
#105993972
#105993962
#205535