Finite Group Databasebeta

← #12253784

Subgroups — #12253784

Order 1280.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#467C211yesmaximal, characteristic, complemented
2C2×#65468C211yesmaximal, characteristic, complemented
3#10615473C211yesmaximal, characteristic
4C22×#3944C211yesmaximal, characteristic, complemented
5#10602854C211yesmaximal, characteristic, complemented
6#10601877C212yesmaximal, complemented
7#10601876C212yesmaximal, complemented
8#10601875C212yesmaximal, complemented
9#10601874C212yesmaximal, complemented
10#10601391C211yesmaximal, characteristic, complemented
11#10601389C211yesmaximal, characteristic, complemented
12C22×C5×Q16C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C2×C5×C22⋊Q8—21no—
15#66125—21no—
16C2×#4050C2211yescharacteristic
17C5×#467—21no—
18C5×#467—21no—
19C5×#467C2212yes—
20C5×#467C2212yes—
21C2×C5×#107C2211yescharacteristic
22C2×C5×#103—21no—
23C2×#3961C2212yescomplemented
24#65468—24no—
25C2×#3960C2212yescomplemented
26#65405C2211yescharacteristic
27#65405C2211yescharacteristic
28#65404C2211yescharacteristic
29#65404C2211yescharacteristic
30#65403—21no—
31#65403—21no—
32#65402—21no—
33#65395C2211yescharacteristic
34C2×#3944—24no—
35C2×#3944C2211yescharacteristic
36C2×#3944C2212yes—
37C2×#3944C2211yescharacteristic
38C22×#672C2212yes—
39#64784C2212yescomplemented
40#64784C2212yescomplemented
41#64712C2212yescomplemented
42#64711C2212yescomplemented
43#64710C2212yescomplemented
44#64709C2212yescomplemented
45#33606—51nomaximal, Sylow
46C22×C5×Q8—21no—
47C23×C4×C5D411yescharacteristic
48C2×#709—21no—
49C2×C5×Q16—22no—
50C2×C5×Q16—21no—
51C2×C5×Q16—22no—
52C2×C5×Q16C2312yes—
53C22×C5×C8—22no—
54C22×C5×C8—22no—
55C22×C5×C8D411yescharacteristic
56C22×C5×C8D411yescharacteristic
57C22×C5×C8D411yescharacteristic
58C22×C5×C8C2311yescharacteristic
59C22×C5×C8C2311yescharacteristic
60C22×C5×C8C2311yescharacteristic
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8—22no—
63C5×C22⋊Q8—41no—
64C5×C22⋊Q8—41no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—41no—
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×#103—22no—
85C5×#103—22no—
86#3961—24no—
87#3960—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×#510—51no—
102C22×M32—51no—
103#3050—51no—
104C2×#467D511yescharacteristic, complemented
105#2220—52no—
106#2219—52no—
107#2217—52no—
108#2216—52no—
109#2159—51no—
110#2141—51no—
111#2139—51no—
112C24×C5D811yescharacteristic
113C2×C5×Q8—22no—
114C2×C5×Q8—41no—
115C2×C5×Q8—42no—
116C2×C5×Q8—42no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—41no—
120C22×C4×C5—41no—
121C22×C4×C5D811yescharacteristic
122C22×C4×C5D811yescharacteristic
123C22×C4×C5D811yescharacteristic
124C22×C4×C5C2×D411yescharacteristic
125C22×C4×C5C2×D411yescharacteristic
126C22×C4×C5C2×D411yescharacteristic
127C2×C22⋊D5—41no—
128C2×C22⋊D5—41no—
129C2×C22⋊D5—41no—
130#709—42no—
131#709—42no—
132C2×D20—21no—
133C2×C4×D5—41no—
134C2×Dic10—21no—
135C5×Q16—22no—
136C5×Q16—22no—
137C5×Q16—42no—
138C2×C5×C8—22no—
139C2×C5×C8—22no—
140C2×C5×C8—22no—
141C2×C5×C8—21no—
142C2×C5×C8—21no—
143C2×C5×C8—21no—
144C2×C5×C8—21no—
145C2×C5×C8C2×D412yes—
146C2×C5×C8C2×D411yescharacteristic
147C2×C5×C8C2×D411yescharacteristic
148C2×C5×C8C2×D411yescharacteristic
149C2×C5×C8#4112yes—
150C2×C5×C8C2411yescharacteristic
151C5×C4⋊C4—22no—
152C5×C4⋊C4—41no—
153C5×C4⋊C4—42no—
154C5×C4⋊C4—42no—
155C5×C22⋊C4—42no—
156C5×C22⋊C4—42no—
157C22⋊Dic5—41no—
158C22⋊Dic5—41no—
159C22⋊Dic5—41no—
160#685—41no—
161#685—41no—
162C4⋊Dic5—21no—
163C4⋊Dic5—21no—
164#683—41no—
165#683—41no—
166C4×Dic5—41no—
167Dic20—21no—
168Dic20—21no—
169D40—21no—
170D40—21no—
171#677—42no—
172#672—22no—
173#672C2×D414yes—
174C2×#135—51no—
175C22×Q16C2×D511yescharacteristic
176C23×C8C2×D511yescharacteristic
177#525—101no—
178C2×C22⋊Q8—21no—
179#510—104no—
180C2×Q32—52no—
181C2×SD32—52no—
182C2×M32—51no—
183C2×M32—51no—
184C2×M32—52no—
185C2×M32—104no—
186C22×C16—52no—
187#467—21no—
188#467—21no—
189#467—51no—
190#467—51no—
191#467C2×D512yes—
192#467C2×D512yes—
193#466—51no—
194#466—51no—
195#465—101no—
196#465—101no—
197#427—51no—
198C2×#107C2×D511yescharacteristic
199#416—101no—
200C2×#103—21no—
201#367—52no—
202#366—52no—
203#358—52no—
204#358—52no—
205#358—52no—
206#357—52no—
207C23×C5—22no—
208C23×C5—22no—
209C23×C5—41no—
210C23×C5C2×D811yescharacteristic
211C23×C5C2×D811yescharacteristic
212C23×C5C2×D811yescharacteristic
213C22×D5—41no—
214C5×Q8—42no—
215C5×Q8—42no—
216C5×Q8—82no—
217C2×C4×C5—22no—
218C2×C4×C5—21no—
219C2×C4×C5—21no—
220C2×C4×C5—21no—
221C2×C4×C5—21no—
222C2×C4×C5—22no—
223C2×C4×C5—22no—
224C2×C4×C5—42no—
225C2×C4×C5—42no—
226C2×C4×C5—82no—
227C2×C4×C5—82no—
228C2×C4×C5#12112yes—
229C2×C4×C5C2×D811yescharacteristic
230C2×C4×C5C2×D811yescharacteristic
231C2×C4×C5C2×D811yescharacteristic
232C2×C4×C5#13512yes—
233C2×C4×C5C22×D411yescharacteristic
234C22⋊D5—82no—
235C22⋊D5—82no—
236C22⋊D5—82no—
237C2×Dic5—41no—
238C2×Dic5—41no—
239C2×Dic5—41no—
240D20—41no—
241D20—41no—
242C4×D5—82no—
243Dic10—41no—
244Dic10—41no—
245C5×C8—21no—
246C5×C8—21no—
247C5×C8#12112yes—
248C5×C8C22×D411yescharacteristic
249C5×C8C2×#4111yescharacteristic
250C2×#41—101no—
251C22×Q8—21no—
252C23×C4C22⋊D511yescharacteristic
253#135—102no—
254#135—102no—
255C2×Q16—22no—
256C2×Q16—22no—
257C2×Q16—21no—
258C2×Q16—51no—
259C2×Q16C22×D512yes—
260C2×SD16—101no—
261C2×D8—51no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8C22⋊D511yescharacteristic
265C22×C8C22⋊D511yescharacteristic
266C22×C8C22⋊D511yescharacteristic
267C22×C8C22×D511yescharacteristic
268C22×C8C22×D511yescharacteristic
269C22×C8C22×D511yescharacteristic
270#123—201no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—41no—
274C22⋊Q8—41no—
275C22⋊Q8—101no—
276C22⋊Q8—101no—
277#121—101no—
278#121—101no—
279#120—201no—
280C4×D4—201no—
281C4×D4—201no—
282#117—101no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—41no—
285C2×C22⋊C4—41no—
286Q32—104no—
287SD32—104no—
288M32—54no—
289M32—104no—
290M32—102no—
291C2×C16—52no—
292C2×C16—52no—
293C2×C16—104no—
294#107—21no—
295#107—51no—
296#107—51no—
297#107C22×D512yes—
298#106—101no—
299#103—22no—
300#103—22no—
301#103—101no—
302#103—101no—
303#102—101no—
304#102—101no—
305C22×C5—21no—
306C22×C5—21no—
307C22×C5—21no—
308C22×C5—21no—
309C22×C5—21no—
310C22×C5—21no—
311C22×C5—21no—
312C22×C5—44no—
313C22×C5#46612yes—
314C22×C5#51012yescomplemented
315C22×C5C22×D811yescharacteristic
316C2×D5—81no—
317C2×D5—81no—
318C2×D5—81no—
319C4×C5—21no—
320C4×C5—21no—
321C4×C5—82no—
322C4×C5—82no—
323C4×C5#46612yes—
324C4×C5C2×#12111yescharacteristic
325C4×C5C2×#13511yescharacteristic
326Dic5—81no—
327Dic5—81no—
328Dic5—81no—
329C24#68611yescharacteristic
330#41—202no—
331#41—202no—
332C2×Q8—22no—
333C2×Q8—41no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—101no—
337C2×D4—101no—
338C2×D4—201no—
339C2×D4—201no—
340C2×D4—201no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—201no—
346C22×C4#68611yescharacteristic
347C22×C4#68611yescharacteristic
348C22×C4#68611yescharacteristic
349C22×C4C2×C22⋊D511yescharacteristic
350C22×C4C2×C22⋊D511yescharacteristic
351C22×C4C2×C22⋊D511yescharacteristic
352Q16—22no—
353Q16—22no—
354Q16—42no—
355Q16—101no—
356Q16—101no—
357SD16—202no—
358D8—101no—
359D8—101no—
360C2×C8—22no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8D4×D512yes—
368C2×C8#71112yes—
369C2×C8C2×C22⋊D511yescharacteristic
370C2×C8C2×C22⋊D511yescharacteristic
371C2×C8C2×C22⋊D511yescharacteristic
372C2×C8C23×D511yescharacteristic
373C4⋊C4—22no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—101no—
378C4⋊C4—101no—
379C4⋊C4—201no—
380C4⋊C4—201no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—201no—
384C22⋊C4—201no—
385C22⋊C4—201no—
386C22⋊C4—201no—
387C22⋊C4—201no—
388C42—201no—
389C16—54no—
390C16—102no—
391C2×C5—22no—
392C2×C5—22no—
393C2×C5—41no—
394C2×C5C2×#46611yescharacteristic
395C2×C5C2×#51011yescharacteristic
396C2×C5#341411yescharacteristic
397D5—161no—
398C23—22no—
399C23—22no—
400C23—41no—
401C23—201no—
402C23C2×#68611yescharacteristic
403C23C2×#68611yescharacteristic
404C23C2×#68611yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—82no—
408Q8—201no—
409Q8—201no—
410D4—201no—
411D4—201no—
412D4—402no—
413D4—402no—
414D4—402no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—201no—
427C2×C4—201no—
428C2×C4—201no—
429C2×C4—402no—
430C2×C4C2×#68611yescharacteristic
431C2×C4C2×#68611yescharacteristic
432C2×C4C2×#68611yescharacteristic
433C2×C4#408512yes—
434C2×C4#409612yes—
435C2×C4C22×C22⋊D511yescharacteristic
436C8—21no—
437C8—21no—
438C8#408512yes—
439C8C2×D4×D511yescharacteristic
440C8C2×#71111yescharacteristic
441C5#3360611yescharacteristic, complemented, Sylow
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—401no—
451C22—401no—
452C22—401no—
453C22#6532712yes—
454C22#6546812yescomplemented
455C22C22×#68611yescharacteristic
456C4—21no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—401no—
461C4—401no—
462C4—401no—
463C4#6532712yes—
464C4C2×#408511yescharacteristic
465C4C2×#409611yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—801no—
470C2C2×#6532711yescharacteristic
471C2C2×#6546811yescharacteristic
472C2#1061562511yescharacteristic
subgroupG/H
centerC22C22×#686
commutator, cosocleC2×C5×C8C24
FrattiniC2×C8C23×D5
FittingC2×C5×#467C2
socleC22×C5C22×D8
prime radical#10615473C2
hypercenterC2×#467D5
hypocenterC5#33606
LCSG ▷ C2×C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×C5×#467C2semidirect
2. C2×#65468C2semidirect
4. C22×#3944C2semidirect
5. #10602854C2semidirect
6. #10601877C2semidirect
7. #10601876C2semidirect
8. #10601875C2semidirect
9. #10601874C2semidirect
10. #10601391C2semidirect
11. #10601389C2semidirect
23. C2×#3961C22semidirect
25. C2×#3960C22semidirect
39. #64784C22semidirect
40. #64784C22semidirect
41. #64712C22semidirect
42. #64711C22semidirect
43. #64710C22semidirect
44. #64709C22semidirect
97. #3944D4semidirect
104. C2×#467D5semidirect
314. C22×C5#510semidirect
441. C5#33606semidirect
454. C22#65468semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4672
C2×#654682
#106154732
C22×#39442
#106028542
#106018772
#106018762
#106018752
#106018742
#106013912
#106013892
#336065