Finite Group Databasebeta

← #12253773

Subgroups — #12253773

Order 1280.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#466C211yesmaximal, characteristic, complemented
2C2×#65436C211yesmaximal, characteristic, complemented
3#10615473C211yesmaximal, characteristic, complemented
4C22×#3944C211yesmaximal, characteristic, complemented
5#10602826C211yesmaximal, characteristic, complemented
6#10601867C212yesmaximal, complemented
7#10601866C212yesmaximal, complemented
8#10601865C212yesmaximal, complemented
9#10601864C212yesmaximal, complemented
10#10601391C211yesmaximal, characteristic, complemented
11#10601389C211yesmaximal, characteristic, complemented
12C22×C5×D8C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C2×C5×#121—21no—
15#66125—21no—
16C2×#4050C2211yescharacteristic
17C5×#466—21no—
18C5×#466—21no—
19C5×#466C2212yes—
20C5×#466C2212yes—
21C2×C5×#107C2211yescharacteristic
22C2×C5×#102—21no—
23C2×#3959C2212yescomplemented
24#65436—24no—
25C2×#3958C2212yescomplemented
26#65405C2211yescharacteristic, complemented
27#65405C2211yescharacteristic, complemented
28#65404C2211yescharacteristic, complemented
29#65404C2211yescharacteristic, complemented
30#65403—21no—
31#65403—21no—
32#65402—21no—
33#65395C2211yescharacteristic
34C2×#3944—24no—
35C2×#3944C2211yescharacteristic
36C2×#3944C2211yescharacteristic
37C2×#3944C2212yescomplemented
38C22×#672C2212yescomplemented
39#64782C2212yescomplemented
40#64782C2212yescomplemented
41#64712C2212yescomplemented
42#64711C2212yescomplemented
43#64710C2212yescomplemented
44#64709C2212yescomplemented
45#33605—51nomaximal, Sylow
46C22×C5×D4—21no—
47C22×C5×D4—41no—
48C23×C4×C5D411yescharacteristic
49C2×#709—21no—
50C2×C5×D8—22no—
51C2×C5×D8—21no—
52C2×C5×D8—22no—
53C2×C5×D8C2312yes—
54C22×C5×C8—22no—
55C22×C5×C8—22no—
56C22×C5×C8D411yescharacteristic
57C22×C5×C8D411yescharacteristic
58C22×C5×C8D411yescharacteristic
59C22×C5×C8C2311yescharacteristic
60C22×C5×C8C2311yescharacteristic
61C22×C5×C8C2311yescharacteristic
62C5×#121—22no—
63C5×#121—22no—
64C5×#121—41no—
65C5×#121—41no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C22⋊C4—41no—
68#4099—41no—
69#4076—21no—
70#4076—21no—
71#4075—41no—
72C4×C22⋊D5—41no—
73C4×C22⋊D5—41no—
74#4072—21no—
75#4070—21no—
76#4070—21no—
77C2×Dic20C2311yescharacteristic
78#4050—22no—
79#4050—22no—
80C2×D40C2311yescharacteristic
81C2×#677—21no—
82C5×#107—21no—
83C5×#107C2312yes—
84C5×#102—22no—
85C5×#102—22no—
86#3959—24no—
87#3958—24no—
88#3953—21no—
89#3953—21no—
90#3950C2311yescharacteristic
91#3950C2311yescharacteristic
92#3949—21no—
93#3948—21no—
94#3948—21no—
95#3944—22no—
96#3944—24no—
97#3944D414yescomplemented
98C2×#672—24no—
99C2×#672C2312yes—
100C2×#672C2312yes—
101C2×#509—51no—
102C22×M32—51no—
103#3050—51no—
104C2×#466D511yescharacteristic, complemented
105#2220—52no—
106#2218—52no—
107#2217—52no—
108#2215—52no—
109#2159—51no—
110#2140—51no—
111#2139—51no—
112C24×C5—41no—
113C24×C5D811yescharacteristic
114C2×C5×D4—22no—
115C2×C5×D4—42no—
116C2×C5×D4—42no—
117C2×C5×D4—42no—
118C2×C5×D4—41no—
119C2×C5×D4—42no—
120C2×C5×D4—84no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—41no—
124C22×C4×C5D811yescharacteristic
125C22×C4×C5D811yescharacteristic
126C22×C4×C5D811yescharacteristic
127C22×C4×C5C2×D411yescharacteristic
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×D411yescharacteristic
130C2×C22⋊D5—41no—
131C2×C22⋊D5—41no—
132C2×C22⋊D5—41no—
133#709—42no—
134#709—42no—
135C2×D20—21no—
136C2×C4×D5—41no—
137C2×Dic10—21no—
138C5×D8—22no—
139C5×D8—22no—
140C5×D8—42no—
141C2×C5×C8—22no—
142C2×C5×C8—22no—
143C2×C5×C8—22no—
144C2×C5×C8—21no—
145C2×C5×C8—21no—
146C2×C5×C8—21no—
147C2×C5×C8—21no—
148C2×C5×C8C2×D412yes—
149C2×C5×C8C2×D411yescharacteristic
150C2×C5×C8C2×D411yescharacteristic
151C2×C5×C8C2×D411yescharacteristic
152C2×C5×C8#4112yes—
153C2×C5×C8C2411yescharacteristic
154C5×C4⋊C4—22no—
155C5×C4⋊C4—41no—
156C5×C22⋊C4—42no—
157C5×C22⋊C4—42no—
158C22⋊Dic5—41no—
159C22⋊Dic5—41no—
160C22⋊Dic5—41no—
161#685—41no—
162#685—41no—
163C4⋊Dic5—21no—
164C4⋊Dic5—21no—
165#683—41no—
166#683—41no—
167C4×Dic5—41no—
168Dic20—21no—
169Dic20—21no—
170D40—21no—
171D40—21no—
172#677—42no—
173#672—22no—
174#672C2×D414yescomplemented
175C2×#135—51no—
176C22×D8C2×D511yescharacteristic
177C23×C8C2×D511yescharacteristic
178#525—101no—
179C2×#121—21no—
180#509—104no—
181C2×SD32—52no—
182C2×D16—52no—
183C2×M32—51no—
184C2×M32—51no—
185C2×M32—52no—
186C2×M32—104no—
187C22×C16—52no—
188#467—51no—
189#467—51no—
190#466—21no—
191#466—21no—
192#466—51no—
193#466—51no—
194#466C2×D512yes—
195#466C2×D512yes—
196#465—101no—
197#465—101no—
198#427—51no—
199C2×#107C2×D511yescharacteristic
200#416—101no—
201C2×#102—21no—
202#367—52no—
203#366—52no—
204#358—52no—
205#357—52no—
206#357—52no—
207#357—52no—
208C23×C5—22no—
209C23×C5—22no—
210C23×C5—41no—
211C23×C5—42no—
212C23×C5—82no—
213C23×C5—82no—
214C23×C5—82no—
215C23×C5C2×D811yescharacteristic
216C23×C5C2×D811yescharacteristic
217C23×C5C2×D811yescharacteristic
218C22×D5—41no—
219C5×D4—42no—
220C5×D4—42no—
221C5×D4—84no—
222C5×D4—84no—
223C5×D4—82no—
224C2×C4×C5—22no—
225C2×C4×C5—21no—
226C2×C4×C5—21no—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5—22no—
230C2×C4×C5—22no—
231C2×C4×C5—42no—
232C2×C4×C5—82no—
233C2×C4×C5#12112yes—
234C2×C4×C5C2×D811yescharacteristic
235C2×C4×C5C2×D811yescharacteristic
236C2×C4×C5C2×D811yescharacteristic
237C2×C4×C5#13512yes—
238C2×C4×C5C22×D411yescharacteristic
239C22⋊D5—82no—
240C22⋊D5—82no—
241C22⋊D5—82no—
242C2×Dic5—41no—
243C2×Dic5—41no—
244C2×Dic5—41no—
245D20—41no—
246D20—41no—
247C4×D5—82no—
248Dic10—41no—
249Dic10—41no—
250C5×C8—21no—
251C5×C8—21no—
252C5×C8#12112yes—
253C5×C8C22×D411yescharacteristic
254C5×C8C2×#4111yescharacteristic
255C2×#41—101no—
256C22×D4—21no—
257C22×D4—41no—
258C23×C4C22⋊D511yescharacteristic
259#135—102no—
260#135—102no—
261C2×Q16—51no—
262C2×SD16—101no—
263C2×D8—22no—
264C2×D8—22no—
265C2×D8—21no—
266C2×D8—51no—
267C2×D8C22×D512yes—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8C22⋊D511yescharacteristic
271C22×C8C22⋊D511yescharacteristic
272C22×C8C22⋊D511yescharacteristic
273C22×C8C22×D511yescharacteristic
274C22×C8C22×D511yescharacteristic
275C22×C8C22×D511yescharacteristic
276#123—201no—
277C22⋊Q8—101no—
278C22⋊Q8—101no—
279#121—22no—
280#121—22no—
281#121—41no—
282#121—41no—
283#121—101no—
284#121—101no—
285#120—201no—
286C4×D4—201no—
287C4×D4—201no—
288#117—101no—
289C2×C4⋊C4—21no—
290C2×C22⋊C4—41no—
291SD32—104no—
292D16—104no—
293M32—54no—
294M32—104no—
295M32—102no—
296C2×C16—52no—
297C2×C16—52no—
298C2×C16—104no—
299#107—21no—
300#107—51no—
301#107—51no—
302#107C22×D512yes—
303#106—101no—
304#103—101no—
305#103—101no—
306#102—22no—
307#102—22no—
308#102—101no—
309#102—101no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—44no—
318C22×C5—82no—
319C22×C5—82no—
320C22×C5—82no—
321C22×C5—164no—
322C22×C5#46612yes—
323C22×C5#50912yescomplemented
324C22×C5C22×D811yescharacteristic
325C2×D5—81no—
326C2×D5—81no—
327C2×D5—81no—
328C4×C5—21no—
329C4×C5—21no—
330C4×C5—82no—
331C4×C5#46612yes—
332C4×C5C2×#12111yescharacteristic
333C4×C5C2×#13511yescharacteristic
334Dic5—81no—
335Dic5—81no—
336Dic5—81no—
337C24—41no—
338C24#68611yescharacteristic
339#41—202no—
340#41—202no—
341C2×Q8—101no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—84no—
349C2×D4—101no—
350C2×D4—201no—
351C2×D4—201no—
352C2×D4—201no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—201no—
357C22×C4#68611yescharacteristic
358C22×C4#68611yescharacteristic
359C22×C4#68611yescharacteristic
360C22×C4C2×C22⋊D511yescharacteristic
361C22×C4C2×C22⋊D511yescharacteristic
362C22×C4C2×C22⋊D511yescharacteristic
363Q16—101no—
364Q16—101no—
365SD16—202no—
366D8—22no—
367D8—22no—
368D8—42no—
369D8—101no—
370D8—101no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8D4×D512yes—
379C2×C8#71112yes—
380C2×C8C2×C22⋊D511yescharacteristic
381C2×C8C2×C22⋊D511yescharacteristic
382C2×C8C2×C22⋊D511yescharacteristic
383C2×C8C23×D511yescharacteristic
384C4⋊C4—22no—
385C4⋊C4—41no—
386C4⋊C4—101no—
387C4⋊C4—101no—
388C4⋊C4—201no—
389C4⋊C4—201no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—201no—
393C22⋊C4—201no—
394C22⋊C4—201no—
395C22⋊C4—201no—
396C22⋊C4—201no—
397C42—201no—
398C16—54no—
399C16—102no—
400C2×C5—22no—
401C2×C5—22no—
402C2×C5—41no—
403C2×C5—162no—
404C2×C5C2×#46611yescharacteristic
405C2×C5C2×#50911yescharacteristic
406C2×C5#341411yescharacteristic
407D5—161no—
408C23—22no—
409C23—22no—
410C23—41no—
411C23—42no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—201no—
416C23C2×#68611yescharacteristic
417C23C2×#68611yescharacteristic
418C23C2×#68611yescharacteristic
419Q8—201no—
420Q8—201no—
421D4—42no—
422D4—42no—
423D4—84no—
424D4—84no—
425D4—82no—
426D4—201no—
427D4—201no—
428D4—402no—
429D4—402no—
430D4—402no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—201no—
441C2×C4—201no—
442C2×C4—201no—
443C2×C4—402no—
444C2×C4C2×#68611yescharacteristic
445C2×C4C2×#68611yescharacteristic
446C2×C4C2×#68611yescharacteristic
447C2×C4#408512yes—
448C2×C4#409612yes—
449C2×C4C22×C22⋊D511yescharacteristic
450C8—21no—
451C8—21no—
452C8#408512yes—
453C8C2×D4×D511yescharacteristic
454C8C2×#71111yescharacteristic
455C5#3360511yescharacteristic, complemented, Sylow
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22—401no—
469C22—401no—
470C22—401no—
471C22#6532712yes—
472C22#6543612yescomplemented
473C22C22×#68611yescharacteristic
474C4—21no—
475C4—21no—
476C4—82no—
477C4—401no—
478C4—401no—
479C4—401no—
480C4#6532712yes—
481C4C2×#408511yescharacteristic
482C4C2×#409611yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—162no—
487C2—801no—
488C2C2×#6532711yescharacteristic
489C2C2×#6543611yescharacteristic
490C2#1061562511yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
FittingC2×C5×#466C2
socleC22×C5C22×D8
hypercenterC2×#466D5
hypocenterC5#33605
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×C5×#466C2semidirect
2. C2×#65436C2semidirect
3. #10615473C2semidirect
4. C22×#3944C2semidirect
5. #10602826C2semidirect
6. #10601867C2semidirect
7. #10601866C2semidirect
8. #10601865C2semidirect
9. #10601864C2semidirect
10. #10601391C2semidirect
11. #10601389C2semidirect
23. C2×#3959C22semidirect
25. C2×#3958C22semidirect
26. #65405C22semidirect
27. #65405C22semidirect
28. #65404C22semidirect
29. #65404C22semidirect
37. C2×#3944C22semidirect
38. C22×#672C22semidirect
39. #64782C22semidirect
40. #64782C22semidirect
41. #64712C22semidirect
42. #64711C22semidirect
43. #64710C22semidirect
44. #64709C22semidirect
97. #3944D4semidirect
104. C2×#466D5semidirect
174. #672C2×D4semidirect
323. C22×C5#509semidirect
455. C5#33605semidirect
472. C22#65436semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4662
C2×#654362
#106154732
C22×#39442
#106028262
#106018672
#106018662
#106018652
#106018642
#106013912
#106013892
#336055