Finite Group Databasebeta

← #12149444

Subgroups — #12149444

Order 1280.

894 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10614231C211yesmaximal, characteristic
2C2×#65309C211yesmaximal, characteristic, complemented
3C2×C8×D20C211yesmaximal, characteristic
4#10605533C211yesmaximal, characteristic
5C4×C5×#103C211yesmaximal, characteristic, complemented
6#10601067C212yesmaximal, complemented
7#10601063C212yesmaximal, complemented
8#10600570C211yesmaximal, characteristic, complemented
9#10600260C211yesmaximal, characteristic, complemented
10#10599661C212yesmaximal, complemented
11#10599626C212yesmaximal, complemented
12C2×C4×C5×Q8C2211yescharacteristic
13C2×#4092C2212yes—
14C22×C8×D5C2212yes—
15C2×#4021—21no—
16C2×C4×D20C2211yescharacteristic
17C2×#4015C2211yescharacteristic
18C2×C5×#103C2211yescharacteristic
19C2×C5×#103C2211yescharacteristic
20C2×C4×C5×C8C2211yescharacteristic
21C4×C5×C4⋊C4C2211yescharacteristic
22C2×#3967C2212yes—
23C2×#3952C2212yes—
24#65309—24no—
25#65283—22no—
26#65276C2212yes—
27#65276C2212yes—
28#65276C2212yes—
29C2×#3940C2212yes—
30C2×#3940C2212yes—
31#65228—21no—
32#65224C2211yescharacteristic
33#65218—21no—
34#65217—21no—
35C2×C4⋊#168C2211yescharacteristic
36C8×D20—24no—
37#64963—22no—
38C5×#340C2211yescharacteristic
39C5×#340C2211yescharacteristic
40#64771C2212yes—
41#64771C2212yes—
42#64749C2212yes—
43#25341—51nomaximal, Sylow
44C22×C5×Q8C2311yescharacteristic
45C22×D20D411yescharacteristic
46C22×C4×D5C2312yes—
47C22×Dic10—21no—
48C22×C5×C8C2311yescharacteristic
49C22×C5×C8C2311yescharacteristic
50C4×C5×Q8—22no—
51C4×C5×Q8—22no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4C2311yescharacteristic
54C2×C5×C4⋊C4C2311yescharacteristic
55C2×C5×C4⋊C4C2311yescharacteristic
56C2×C42×C5—21no—
57C2×C42×C5—21no—
58C2×C42×C5C2311yescharacteristic
59#4092—24no—
60#4092—42no—
61C2×#685—22no—
62C2×#685—22no—
63C2×#685—22no—
64C2×#685—22no—
65C2×#685D412yes—
66C2×C4⋊Dic5—21no—
67C2×C4⋊Dic5—21no—
68C2×C4⋊Dic5—21no—
69C2×C4⋊Dic5D411yescharacteristic
70C2×C4⋊Dic5C2312yes—
71C2×C4⋊Dic5C2312yes—
72C2×#683—22no—
73C22×#168C2312yes—
74C2×C8×D5—24no—
75C2×C8×D5—24no—
76C2×C8×D5—42no—
77#4021—44no—
78C4×D20—22no—
79C4×D20—22no—
80#4015—22no—
81#4015—22no—
82C5×#103—22no—
83C5×#103—22no—
84C4×C5×C8—22no—
85C5×#95—21no—
86#3967—24no—
87#3963—22no—
88#3963—22no—
89#3963—22no—
90#3952—24no—
91#3940—24no—
92#3940—42no—
93C4⋊#168D414yes—
94C2×#475—51no—
95C2×C8×D4—51no—
96#2547—51no—
97#2538—51no—
98#2077—52no—
99#2035—52no—
100#1990—51no—
101#1948—51no—
102#1892—52no—
103#1807—52no—
104C4×#103D511yescharacteristic, complemented
105C23×D5C2×D412yes—
106C2×C5×Q8—21no—
107C2×C5×Q8—21no—
108C2×C5×Q8—21no—
109C2×C5×Q8—21no—
110C2×C5×Q8—21no—
111C2×C5×Q8#4112yes—
112C22×C4×C5—21no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5C2×D411yescharacteristic
117C22×C4×C5C2×D411yescharacteristic
118C22×C4×C5C2411yescharacteristic
119C22×Dic5—22no—
120C22×Dic5—22no—
121C22×Dic5C2×D412yes—
122C2×D20—22no—
123C2×D20—22no—
124C2×D20SD1614yes—
125C2×C4×D5—22no—
126C2×C4×D5—22no—
127C2×C4×D5—22no—
128C2×C4×D5—22no—
129C2×C4×D5—22no—
130C2×C4×D5—22no—
131C2×Dic10—42no—
132C2×Dic10—42no—
133C2×Dic10—42no—
134C2×C5×C8—21no—
135C2×C5×C8—21no—
136C2×C5×C8—21no—
137C2×C5×C8—21no—
138C2×C5×C8—21no—
139C2×C5×C8#4112yes—
140C5×C4⋊C4—21no—
141C5×C4⋊C4—21no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4—21no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—42no—
147C42×C5—42no—
148C42×C5—42no—
149C42×C5C2×D412yes—
150C42×C5#4112yes—
151#685—24no—
152#685—44no—
153#685—44no—
154#685—44no—
155#685—82no—
156C4⋊Dic5—22no—
157C4⋊Dic5—22no—
158C4⋊Dic5—22no—
159C4⋊Dic5—22no—
160C4⋊Dic5—22no—
161C4⋊Dic5—42no—
162C4⋊Dic5—42no—
163C4⋊Dic5—81no—
164#683—44no—
165C2×#168—22no—
166C2×#168—22no—
167C2×#168C2×D414yes—
168C8×D5—48no—
169C23×C8—52no—
170C2×C4⋊Q8—51no—
171C2×#124—101no—
172C2×C22⋊Q8—52no—
173C2×C4×Q8C2×D511yescharacteristic
174C2×C4×D4—51no—
175#475—104no—
176C8×D4—104no—
177C2×#106—52no—
178C2×#106—52no—
179C2×C4⋊C8—51no—
180C2×#103—52no—
181C2×#103C2×D511yescharacteristic
182C2×#103C2×D511yescharacteristic
183C2×C22⋊C8—52no—
184C2×C4×C8C2×D511yescharacteristic
185#397—102no—
186#396—102no—
187#391—101no—
188#390—101no—
189#389—101no—
190#384—51no—
191#380—52no—
192#380—52no—
193#380—52no—
194C4×C4⋊C4C2×D511yescharacteristic
195#340—52no—
196#340—52no—
197#340—52no—
198#340C2×D511yescharacteristic
199#340C2×D511yescharacteristic
200C23×C5C22×D411yescharacteristic
201C22×D5—22no—
202C22×D5—22no—
203C22×D5—22no—
204C22×D5—22no—
205C22×D5—22no—
206C22×D5—22no—
207C22×D5C2×SD1614yes—
208C5×Q8—22no—
209C5×Q8—22no—
210C5×Q8—41no—
211C5×Q8—41no—
212C2×C4×C5—21no—
213C2×C4×C5—21no—
214C2×C4×C5—22no—
215C2×C4×C5—41no—
216C2×C4×C5—41no—
217C2×C4×C5—41no—
218C2×C4×C5—41no—
219C2×C4×C5—41no—
220C2×C4×C5—41no—
221C2×C4×C5—41no—
222C2×C4×C5—41no—
223C2×C4×C5—41no—
224C2×C4×C5—41no—
225C2×C4×C5—41no—
226C2×C4×C5#12112yes—
227C2×C4×C5#12112yes—
228C2×C4×C5C2×SD1612yes—
229C2×C4×C5#13512yes—
230C2×C4×C5C22×D411yescharacteristic
231C2×C4×C5C2×#4111yescharacteristic
232C2×C4×C5C2×#4111yescharacteristic
233C2×C4×C5C2×#4111yescharacteristic
234C2×C4×C5H(5,𝔽2)11yescharacteristic
235C2×C4×C52−1+411yescharacteristic
236C2×Dic5—22no—
237C2×Dic5—22no—
238C2×Dic5—22no—
239C2×Dic5—42no—
240C2×Dic5—42no—
241C2×Dic5—42no—
242C2×Dic5—42no—
243C2×Dic5—42no—
244C2×Dic5—42no—
245D20—28no—
246C4×D5—42no—
247C4×D5—42no—
248C4×D5—42no—
249C4×D5—42no—
250Dic10—84no—
251C5×C8—22no—
252C5×C8—41no—
253#168—24no—
254C22×Q8—101no—
255C22×Q8C22×D511yescharacteristic
256C22×D4—51no—
257C23×C4—52no—
258C22×C8—52no—
259C22×C8—104no—
260C22×C8—104no—
261C22×C8—202no—
262C22×C8C22×D511yescharacteristic
263C22×C8C22×D511yescharacteristic
264C4⋊Q8—102no—
265C4⋊Q8—102no—
266#124—204no—
267C22⋊Q8—104no—
268C22⋊Q8—202no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×D4—102no—
272C4×D4—102no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—51no—
275C2×C4⋊C4—52no—
276C2×C4⋊C4—52no—
277C2×C4⋊C4—102no—
278C2×C4⋊C4—101no—
279C2×C4⋊C4—101no—
280C2×C4⋊C4—101no—
281C2×C4⋊C4C22×D511yescharacteristic
282C2×C4⋊C4C22×D511yescharacteristic
283C2×C4⋊C4C22×D511yescharacteristic
284C2×C22⋊C4—52no—
285C2×C22⋊C4—102no—
286C2×C22⋊C4—102no—
287C2×C22⋊C4—102no—
288C2×C22⋊C4—102no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C22×D511yescharacteristic
292#106—104no—
293#106—202no—
294C4⋊C8—54no—
295#103—22no—
296#103—22no—
297#103—104no—
298C22⋊C8—104no—
299C4×C8—22no—
300#95—21no—
301#95—102no—
302#95—102no—
303#95—102no—
304C22×C5C2×#12111yescharacteristic
305C22×C5#54611yescharacteristic
306C22×C5#54711yescharacteristic
307C22×C5C22×SD1611yescharacteristic
308C22×C5C2×#13511yescharacteristic
309C22×C5#57711yescharacteristic
310C22×C5#57811yescharacteristic
311C2×D5—24no—
312C2×D5—24no—
313C2×D5—24no—
314C2×D5—42no—
315C2×D5—42no—
316C2×D5—42no—
317C2×D5—42no—
318C4×C5—22no—
319C4×C5—42no—
320C4×C5—81no—
321C4×C5—81no—
322C4×C5—81no—
323C4×C5#46514yes—
324C4×C5#52911yescharacteristic
325C4×C5#54211yescharacteristic
326C4×C5#54611yescharacteristic
327C4×C5#54811yescharacteristic
328Dic5—42no—
329Dic5—82no—
330Dic5—82no—
331C24—52no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—202no—
338C2×Q8—202no—
339C2×Q8—202no—
340C2×Q8#70912yes—
341C2×D4—54no—
342C2×D4—102no—
343C2×D4—102no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—52no—
349C22×C4—102no—
350C22×C4—102no—
351C22×C4—102no—
352C22×C4—102no—
353C22×C4—102no—
354C22×C4—102no—
355C22×C4—102no—
356C22×C4—102no—
357C22×C4D4×D511yescharacteristic
358C22×C4D4×D511yescharacteristic
359C22×C4C23×D511yescharacteristic
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—54no—
366C2×C8—102no—
367C2×C8—102no—
368C2×C8—208no—
369C2×C8#70912yes—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—42no—
377C4⋊C4—102no—
378C4⋊C4—102no—
379C4⋊C4—102no—
380C4⋊C4—102no—
381C4⋊C4—102no—
382C4⋊C4—204no—
383C4⋊C4—202no—
384C4⋊C4—202no—
385C4⋊C4—401no—
386C22⋊C4—104no—
387C22⋊C4—204no—
388C22⋊C4—204no—
389C22⋊C4—204no—
390C22⋊C4—402no—
391C42—42no—
392C42—42no—
393C42#70912yes—
394C42D4×D512yes—
395C2×C5#242711yescharacteristic
396C2×C5C2×#46511yescharacteristic
397C2×C5#306311yescharacteristic
398C2×C5#329811yescharacteristic
399C2×C5#330411yescharacteristic
400C2×C5#333511yescharacteristic
401C2×C5#334211yescharacteristic
402D5—44no—
403C23—54no—
404C23—102no—
405C23—102no—
406C23—102no—
407C23—102no—
408C23—102no—
409C23—102no—
410C23C2×D4×D511yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415Q8—404no—
416D4—108no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—102no—
432C2×C4—102no—
433C2×C4—102no—
434C2×C4—202no—
435C2×C4—202no—
436C2×C4—202no—
437C2×C4—202no—
438C2×C4—202no—
439C2×C4—202no—
440C2×C4—202no—
441C2×C4—202no—
442C2×C4—202no—
443C2×C4—202no—
444C2×C4#403012yes—
445C2×C4#403012yes—
446C2×C4D5×SD1612yes—
447C2×C4#406512yes—
448C2×C4C2×#70911yescharacteristic
449C2×C4C2×#70911yescharacteristic
450C2×C4C2×#70911yescharacteristic
451C2×C4C2×D4×D511yescharacteristic
452C2×C4#414911yescharacteristic
453C2×C4#415011yescharacteristic
454C8—22no—
455C8—41no—
456C8—104no—
457C5#2534111yescharacteristic, complemented, Sylow
458C22—104no—
459C22—104no—
460C22—104no—
461C22—202no—
462C22—202no—
463C22—202no—
464C22—202no—
465C22C2×#403011yescharacteristic
466C22#6588411yescharacteristic
467C22#6588511yescharacteristic
468C22C2×D5×SD1611yescharacteristic
469C22C2×#406511yescharacteristic
470C22#6610411yescharacteristic
471C22#6610511yescharacteristic
472C4—22no—
473C4—42no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—202no—
478C4—402no—
479C4—402no—
480C4#6509614yes—
481C4#6581311yescharacteristic
482C4#6586811yescharacteristic
483C4#6588411yescharacteristic
484C4#6588611yescharacteristic
485C2—204no—
486C2#1060521711yescharacteristic
487C2C2×#6509611yescharacteristic
488C2#1061057211yescharacteristic
489C2#1061140911yescharacteristic
490C2#1061141911yescharacteristic
491C2#1061146811yescharacteristic
492C2#1061147411yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC4×C5×#103C2
socleC23×C5C22×D4
prime radicalC22×D20D4
hypercenterC4×#103D5
hypocenterC5#25341
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#65309C2semidirect
5. C4×C5×#103C2semidirect
6. #10601067C2semidirect
7. #10601063C2semidirect
8. #10600570C2semidirect
9. #10600260C2semidirect
10. #10599661C2semidirect
11. #10599626C2semidirect
104. C4×#103D5semidirect
457. C5#25341semidirect

Maximal subgroups

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

subgroupindex
#106142312
C2×#653092
C2×C8×D202
#106055332
C4×C5×#1032
#106010672
#106010632
#106005702
#106002602
#105996612
#105996262
#253415