Finite Group Databasebeta

← #12121109

Subgroups — #12121109

Order 1280.

898 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65131C211yesmaximal, characteristic
2#10606091C211yesmaximal, characteristic
3#10605387C211yesmaximal, characteristic, complemented
4C5×#1948C211yesmaximal, characteristic, complemented
5#10601377C212yesmaximal
6#10601352C212yesmaximal, complemented
7#10601258C211yesmaximal, characteristic, complemented
8#10600375C211yesmaximal, characteristic, complemented
9C4×#3941C211yesmaximal, characteristic
10#10599609C212yesmaximal, complemented
11#10599573C212yesmaximal
12C2×#4039C2212yes—
13C2×D5×C4⋊C4—21no—
14C2×C4×D20C2211yescharacteristic
15C2×#4015C2211yescharacteristic
16C2×C5×#106C2212yes—
17C2×C5×C4⋊C8C2211yescharacteristic
18C5×#384C2211yescharacteristic
19C4×C5×C4⋊C4C2211yescharacteristic
20C2×#3953C2212yes—
21C2×#3949C2212yes—
22C2×C8×Dic5C2212yes—
23#65278—22no—
24#65276—21no—
25C4⋊C4×Dic5C2212yes—
26#65262C2212yes—
27#65262C2212yes—
28C2×#3941C2211yescharacteristic
29C2×#3941C2211yescharacteristic
30C4×C4⋊Dic5—21no—
31C4×C4⋊Dic5C2211yescharacteristic
32#65217—21no—
33C2×C4×#168C2211yescharacteristic
34#65131C418yescomplemented
35#64935—22no—
36C5×#340C2212yes—
37#64770C2212yes—
38#64749C2211yescharacteristic
39#64749C2211yescharacteristic
40#64749C2212yes—
41#64749C2212yes—
42#22963—51nomaximal, Sylow
43C22×D20C2311yescharacteristic
44C22×C4×D5—21no—
45C22×C4×D5—22no—
46C22×Dic10—21no—
47C22×C5×C8C2312yes—
48C2×C5×C4⋊C4—21no—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4C2311yescharacteristic
51C2×C5×C4⋊C4C2312yes—
52C2×C5×C4⋊C4C2311yescharacteristic
53C2×C42×C5—21no—
54C2×C42×C5C2311yescharacteristic
55C2×#685—21no—
56C2×#685—22no—
57C2×C4⋊Dic5—21no—
58C2×C4⋊Dic5D411yescharacteristic
59C2×C4⋊Dic5D411yescharacteristic
60C2×C4⋊Dic5C2312yes—
61C2×C4⋊Dic5C2311yescharacteristic
62C2×#683—22no—
63C2×#683—22no—
64C2×C4×Dic5—22no—
65C2×C4×Dic5C2312yes—
66C22×#168C2311yescharacteristic
67C22×#168C2311yescharacteristic
68#4039—24no—
69#4039—42no—
70D5×C4⋊C4—44no—
71C4×D20—22no—
72C4×D20C2×C414yes—
73#4015—22no—
74#4015C2×C414yescomplemented
75C5×#106—42no—
76C5×C4⋊C8C2×C414yescomplemented
77C5×#95—21no—
78#3963—22no—
79#3963—22no—
80#3963—22no—
81#3963D412yes—
82#3953C2×C418yes—
83#3949C2×C418yescomplemented
84C8×Dic5—42no—
85#3941—22no—
86#3941Q814yescomplemented
87C4×#168—22no—
88C2×#476—51no—
89#2369—51no—
90D4×C4⋊C4—51no—
91#1990—51no—
92#1948D511yescharacteristic, complemented
93#1941—52no—
94#1919—52no—
95#1864—52no—
96C8×C4⋊C4—51no—
97C4×#102—51no—
98C4×#102—52no—
99C23×D5—21no—
100C22×C4×C5—21no—
101C22×C4×C5—21no—
102C22×C4×C5—22no—
103C22×C4×C5C2×D411yescharacteristic
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5C2411yescharacteristic
106C22×Dic5—22no—
107C22×Dic5—21no—
108C22×Dic5C2×D412yes—
109C22×Dic5C2×D412yes—
110C2×D20—21no—
111C2×D20—21no—
112C2×D20—21no—
113C2×D20C22×C412yes—
114C2×D20C2×Q812yes—
115C2×D20#4112yes—
116C2×C4×D5—24no—
117C2×C4×D5—42no—
118C2×C4×D5—42no—
119C2×C4×D5—44no—
120C2×C4×D5—44no—
121C2×C4×D5—82no—
122C2×Dic10—24no—
123C2×Dic10—81no—
124C2×Dic10—81no—
125C2×C5×C8—22no—
126C2×C5×C8—22no—
127C2×C5×C8C22×C414yes—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4—21no—
132C5×C4⋊C4—22no—
133C5×C4⋊C4—42no—
134C5×C4⋊C4—81no—
135C5×C4⋊C4C2×Q812yes—
136C42×C5—42no—
137C42×C5C22×C412yes—
138C42×C5#4112yes—
139#685—24no—
140#685—44no—
141C4⋊Dic5—24no—
142C4⋊Dic5—22no—
143C4⋊Dic5—22no—
144C4⋊Dic5—21no—
145C4⋊Dic5SD1614yes—
146C4⋊Dic5C22×C412yes—
147C4⋊Dic5C22×C414yes—
148#683—44no—
149#683—82no—
150C4×Dic5—22no—
151C4×Dic5—22no—
152C4×Dic5—82no—
153C2×#168—21no—
154C2×#168—21no—
155C2×#168—21no—
156C2×#168—21no—
157C2×#168—21no—
158C2×#168C2×Q812yes—
159C2×C4⋊Q8—51no—
160C2×C4×D4—52no—
161C2×C4×D4—51no—
162C22×C4⋊C4—101no—
163#476—58no—
164C2×#106—52no—
165C2×#106C2×D512yes—
166C2×C4⋊C8C2×D511yescharacteristic
167C2×#102—52no—
168C2×#102—51no—
169C2×#102—51no—
170C2×C4×C8—51no—
171C2×C4×C8—52no—
172#391—102no—
173#389—52no—
174#389—52no—
175#389—101no—
176#385—102no—
177#384C2×D511yescharacteristic
178#380—101no—
179C4×C4⋊C4—51no—
180C4×C4⋊C4—52no—
181C4×C4⋊C4—101no—
182C4×C4⋊C4C2×D511yescharacteristic
183#340—52no—
184#340—51no—
185#340—51no—
186#340—52no—
187#340C2×D512yes—
188#336—52no—
189C23×C5C22×D411yescharacteristic
190C22×D5—22no—
191C22×D5—22no—
192C22×D5—22no—
193C22×D5—41no—
194C22×D5—41no—
195C22×D5—41no—
196C22×D5—41no—
197C2×C4×C5—21no—
198C2×C4×C5—21no—
199C2×C4×C5—22no—
200C2×C4×C5—41no—
201C2×C4×C5—41no—
202C2×C4×C5—42no—
203C2×C4×C5—41no—
204C2×C4×C5—42no—
205C2×C4×C5—41no—
206C2×C4×C5—42no—
207C2×C4×C5—41no—
208C2×C4×C5C4×D412yes—
209C2×C4×C5C4×D412yes—
210C2×C4×C5C2×SD1612yes—
211C2×C4×C5#13512yes—
212C2×C4×C5C23×C411yescharacteristic
213C2×C4×C5C22×Q811yescharacteristic
214C2×C4×C5C2×#4111yescharacteristic
215C2×C4×C5C2×#4111yescharacteristic
216C2×C4×C5H(5,𝔽2)11yescharacteristic
217C2×C4×C52−1+411yescharacteristic
218C2×Dic5—22no—
219C2×Dic5—22no—
220C2×Dic5—24no—
221C2×Dic5—22no—
222C2×Dic5—22no—
223C2×Dic5—22no—
224C2×Dic5—22no—
225C2×Dic5—41no—
226C2×Dic5—41no—
227C2×Dic5—42no—
228C2×Dic5—42no—
229C2×Dic5C2×SD1614yes—
230D20—22no—
231D20—22no—
232D20—22no—
233D20C4×Q814yes—
234C4×D5—48no—
235C4×D5—88no—
236Dic10—84no—
237C5×C8—24no—
238#168—22no—
239#168—41no—
240C22×Q8—101no—
241C22×D4—51no—
242C23×C4—102no—
243C23×C4—101no—
244C22×C8—51no—
245C22×C8—51no—
246C22×C8C22×D512yes—
247C4⋊Q8—54no—
248C4⋊Q8—102no—
249C4×D4—54no—
250C4×D4—104no—
251C4×D4—102no—
252C4×D4—202no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—51no—
256C2×C4⋊C4—51no—
257C2×C4⋊C4—52no—
258C2×C4⋊C4—51no—
259C2×C4⋊C4—101no—
260C2×C4⋊C4—102no—
261C2×C4⋊C4—102no—
262C2×C4⋊C4—204no—
263C2×C4⋊C4C22×D512yes—
264C2×C4⋊C4C22×D511yescharacteristic
265C2×C4⋊C4C22×D511yescharacteristic
266C2×C22⋊C4—101no—
267C2×C22⋊C4—102no—
268C2×C42—21no—
269C2×C42—52no—
270C2×C42—102no—
271C2×C42C22×D511yescharacteristic
272#106—42no—
273#106—58no—
274C4⋊C8C4×D514yescomplemented
275#102—58no—
276#102—54no—
277#102—102no—
278C4×C8—102no—
279C4×C8—202no—
280#95—21no—
281#95—52no—
282#95—102no—
283#95—102no—
284#95—102no—
285C22×C5C2×C4×D411yescharacteristic
286C22×C5#54611yescharacteristic
287C22×C5D4×Q811yescharacteristic
288C22×C5C22×SD1611yescharacteristic
289C22×C5C2×#13511yescharacteristic
290C22×C5#57611yescharacteristic
291C22×C5#57711yescharacteristic
292C2×D5—24no—
293C2×D5—42no—
294C2×D5—42no—
295C2×D5—42no—
296C2×D5—42no—
297C2×D5—42no—
298C2×D5—42no—
299C4×C5—22no—
300C4×C5—42no—
301C4×C5—82no—
302C4×C5—81no—
303C4×C5C4×SD1614yes—
304C4×C5C2×C4×Q811yescharacteristic
305C4×C5#52011yescharacteristic
306C4×C5#55211yescharacteristic
307C4×C5#55411yescharacteristic
308Dic5—24no—
309Dic5—42no—
310Dic5—44no—
311Dic5—42no—
312C24—101no—
313C2×Q8—104no—
314C2×Q8—401no—
315C2×Q8—401no—
316C2×D4—52no—
317C2×D4—52no—
318C2×D4—52no—
319C2×D4—101no—
320C2×D4—101no—
321C2×D4—101no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—52no—
326C22×C4—52no—
327C22×C4—102no—
328C22×C4—101no—
329C22×C4—104no—
330C22×C4—204no—
331C22×C4—204no—
332C22×C4—202no—
333C22×C4—202no—
334C22×C4—402no—
335C22×C4D4×D511yescharacteristic
336C22×C4D4×D511yescharacteristic
337C22×C4C23×D511yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—52no—
341C2×C8—101no—
342C2×C8—101no—
343C2×C8—101no—
344C2×C8—101no—
345C2×C8—101no—
346C2×C8C2×C4×D514yes—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—54no—
354C4⋊C4—54no—
355C4⋊C4—52no—
356C4⋊C4—81no—
357C4⋊C4—101no—
358C4⋊C4—102no—
359C4⋊C4—104no—
360C4⋊C4—102no—
361C4⋊C4—204no—
362C4⋊C4—402no—
363C4⋊C4Q8×D512yes—
364C22⋊C4—104no—
365C22⋊C4—204no—
366C42—42no—
367C42—102no—
368C42—102no—
369C42—402no—
370C42C2×C4×D512yes—
371C42#71312yes—
372C2×C5#236011yescharacteristic
373C2×C5C2×C4×SD1611yescharacteristic
374C2×C5#294611yescharacteristic
375C2×C5#329511yescharacteristic
376C2×C5#330311yescharacteristic
377C2×C5Q8×SD1611yescharacteristic
378C2×C5#338011yescharacteristic
379D5—44no—
380C23—102no—
381C23—102no—
382C23—102no—
383C23—201no—
384C23—201no—
385C23—201no—
386C23—201no—
387C23C2×D4×D511yescharacteristic
388Q8—404no—
389D4—54no—
390D4—102no—
391D4—102no—
392D4—102no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—54no—
405C2×C4—102no—
406C2×C4—102no—
407C2×C4—102no—
408C2×C4—104no—
409C2×C4—102no—
410C2×C4—102no—
411C2×C4—102no—
412C2×C4—201no—
413C2×C4—201no—
414C2×C4—202no—
415C2×C4—202no—
416C2×C4—208no—
417C2×C4—408no—
418C2×C4#403912yes—
419C2×C4#403912yes—
420C2×C4D5×SD1612yes—
421C2×C4#406212yes—
422C2×C4C22×C4×D511yescharacteristic
423C2×C4#414411yescharacteristic
424C2×C4C2×Q8×D511yescharacteristic
425C2×C4C2×#71311yescharacteristic
426C2×C4D5×#4111yescharacteristic
427C2×C4#415011yescharacteristic
428C8—24no—
429C8—102no—
430C8—201no—
431C5#2296311yescharacteristic, complemented, Sylow
432C22—104no—
433C22—202no—
434C22—202no—
435C22—202no—
436C22—202no—
437C22—202no—
438C22—202no—
439C22C2×#403911yescharacteristic
440C22#6594611yescharacteristic
441C22#6596611yescharacteristic
442C22C2×D5×SD1611yescharacteristic
443C22C2×#406211yescharacteristic
444C22#6610811yescharacteristic
445C22#6611211yescharacteristic
446C4—22no—
447C4—42no—
448C4—82no—
449C4—81no—
450C4—104no—
451C4—202no—
452C4—202no—
453C4—204no—
454C4#6514114yes—
455C4#6588011yescharacteristic
456C4C4×Q8×D511yescharacteristic
457C4#6602811yescharacteristic
458C4#6606411yescharacteristic
459C2—204no—
460C2#1060793611yescharacteristic
461C2C2×#6514111yescharacteristic
462C2#1061093211yescharacteristic
463C2#1061212611yescharacteristic
464C2#1061218211yescharacteristic
465C2#1061277011yescharacteristic
466C2#1061278511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1948C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#1948D5
hypocenterC5#22963
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
3. #10605387C2semidirect
4. C5×#1948C2semidirect
6. #10601352C2semidirect
7. #10601258C2semidirect
8. #10600375C2semidirect
10. #10599609C2semidirect
34. #65131C4semidirect
74. #4015C2×C4semidirect
76. C5×C4⋊C8C2×C4semidirect
83. #3949C2×C4semidirect
86. #3941Q8semidirect
92. #1948D5semidirect
274. C4⋊C8C4×D5semidirect
431. C5#22963semidirect

Maximal subgroups

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

subgroupindex
C2×#651312
#106060912
#106053872
C5×#19482
#106013772
#106013522
#106012582
#106003752
C4×#39412
#105996092
#105995732
#229635