Finite Group Databasebeta

← #12118765

Subgroups — #12118765

Order 1280.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10610743C211yesmaximal, characteristic, complemented
2#10610731C211yesmaximal, characteristic, complemented
3C4×#4050C211yesmaximal, characteristic
4C5×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5#10600586C212yesmaximal, complemented
6#10600584C211yesmaximal, characteristic, complemented
7#10600582C212yesmaximal, complemented
8#10600287C211yesmaximal, characteristic, complemented
9#10600287C211yesmaximal, characteristic, complemented
10#10599488C212yesmaximal
11#10599438C212yesmaximal, complemented
12C2×#4050C2211yescharacteristic
13C2×#4047C2212yes—
14C4×#709C2211yescharacteristic
15C4×#709C2211yescharacteristic
16C2×C5×C4⋊C8C2211yescharacteristic
17C2×C4×C5×C8C2212yes—
18C2×C4×C5×C8C2211yescharacteristic
19C4×C5×C4⋊C4C2211yescharacteristic
20#65402C2211yescharacteristic
21#65402C2211yescharacteristic
22#65399C2212yes—
23#65395C2211yescharacteristic
24#65391C2212yes—
25#65390C2212yes—
26C2×#3937C2212yes—
27C2×#3937C2212yes—
28#65213C2211yescharacteristic
29#65213C2211yescharacteristic
30#65129—22no—
31#65128—22no—
32#65126—21no—
33#65124—21no—
34#65122—21no—
35D5×C4⋊C8—21no—
36C4×Dic20C411yescharacteristic, complemented
37C4×Dic20C411yescharacteristic, complemented
38C4×D40C411yescharacteristic, complemented
39C4×D40C411yescharacteristic, complemented
40C4×#677C411yescharacteristic, complemented
41C4×#677C411yescharacteristic, complemented
42C4×#677C411yescharacteristic, complemented
43C4×#677C411yescharacteristic, complemented
44C5×#336C2212yes—
45#64772C2212yes—
46#64772C2212yes—
47#64743C2212yes—
48#64743C2212yes—
49#22783—51nomaximal, Sylow
50C2×#709C2311yescharacteristic
51C2×#709C2311yescharacteristic
52C22×C5×C8C2311yescharacteristic
53C22×C5×C8C2311yescharacteristic
54C22×C5×C8C2312yes—
55C2×C5×C4⋊C4—21no—
56C2×C42×C5C2312yes—
57C2×C42×C5C2311yescharacteristic
58C4×C22⋊D5—21no—
59C4×C22⋊D5—21no—
60C4×C22⋊D5—21no—
61C4×C22⋊D5—21no—
62#4072C2311yescharacteristic
63#4072C2311yescharacteristic
64C2×#681C2312yes—
65C2×#681C2312yes—
66C2×Dic20C2×C411yescharacteristic
67#4050—22no—
68#4050—22no—
69C2×D40C2×C411yescharacteristic
70C2×#677C2×C411yescharacteristic
71C2×#677C2×C411yescharacteristic
72#4047—24no—
73#4047—42no—
74C2×#676—22no—
75C2×C8×D5—22no—
76C4×D20C2×C411yescharacteristic
77C4×D20C2×C411yescharacteristic
78C4×D20C2×C411yescharacteristic
79C4×D20C2×C411yescharacteristic
80#4018—21no—
81#4018—21no—
82C42×D5—21no—
83C42×D5—21no—
84C4×Dic10C2×C411yescharacteristic
85C4×Dic10C2×C411yescharacteristic
86C4×Dic10C2×C411yescharacteristic
87C4×Dic10C2×C411yescharacteristic
88C5×C4⋊C8D411yescharacteristic
89C5×C4⋊C8D411yescharacteristic
90C5×C4⋊C8D412yes—
91C4×C5×C8—42no—
92C4×C5×C8C2×C411yescharacteristic
93C4×C5×C8C2×C411yescharacteristic
94C4×C5×C8C2×C411yescharacteristic
95C4×C5×C8C2×C411yescharacteristic
96C5×#95—21no—
97#3953C2×C411yescharacteristic
98#3953C2×C411yescharacteristic
99#3953C2×C411yescharacteristic
100#3953C2×C411yescharacteristic
101#3952—22no—
102#3952—22no—
103#3950C2×C411yescharacteristic
104#3950C2×C411yescharacteristic
105#3949C2×C411yescharacteristic
106#3949C2×C411yescharacteristic
107#3948C2×C411yescharacteristic
108#3948C2×C411yescharacteristic
109#3948C2×C411yescharacteristic
110#3948C2×C411yescharacteristic
111#3947—22no—
112#3946—22no—
113#3946—22no—
114C8×Dic5—22no—
115#3937—24no—
116#3937—42no—
117C4⋊#168—21no—
118C4⋊#168—21no—
119C4⋊#168—21no—
120C4⋊#168—21no—
121C4×#135—51no—
122#2925—51no—
123#2906—51no—
124#1937—52no—
125#1922—52no—
126#1876—52no—
127#1837—51no—
128#1806—51no—
129#1806—51no—
130C8×C4⋊C4D511yescharacteristic, complemented
131#1764—52no—
132C22×C4×C5—22no—
133C22×C4×C5#4111yescharacteristic
134C22×C4×C5#4111yescharacteristic
135C22×C4×C5C2411yescharacteristic
136C2×C22⋊D5—21no—
137C2×C22⋊D5—21no—
138#709—21no—
139#709—21no—
140#709—21no—
141#709—21no—
142#709—21no—
143#709—21no—
144#709#4112yes—
145#709#4112yes—
146C2×D20C22×C411yescharacteristic
147C2×D20C22×C411yescharacteristic
148C2×C4×D5—21no—
149C2×C4×D5—21no—
150C2×C4×D5—21no—
151C2×C4×D5—21no—
152C2×C4×D5—21no—
153C2×C4×D5—21no—
154C2×Dic10C22×C411yescharacteristic
155C2×Dic10C22×C411yescharacteristic
156C2×C5×C8—22no—
157C2×C5×C8—21no—
158C2×C5×C8—21no—
159C2×C5×C8—21no—
160C2×C5×C8—22no—
161C2×C5×C8—21no—
162C2×C5×C8C22×C411yescharacteristic
163C2×C5×C8C22×C411yescharacteristic
164C2×C5×C8C22×C411yescharacteristic
165C2×C5×C8C22×C411yescharacteristic
166C2×C5×C8C2×D412yes—
167C2×C5×C8C2×D412yes—
168C5×C4⋊C4—24no—
169C42×C5—22no—
170C42×C5—22no—
171C42×C5C22×C411yescharacteristic
172C42×C5C22×C411yescharacteristic
173C42×C5C2×D411yescharacteristic
174C42×C5C2×D411yescharacteristic
175C22⋊Dic5—21no—
176C22⋊Dic5—21no—
177#685—21no—
178#685—21no—
179#685—21no—
180#685—21no—
181C4⋊Dic5C22×C411yescharacteristic
182C4⋊Dic5C22×C411yescharacteristic
183C4⋊Dic5C22×C411yescharacteristic
184C4⋊Dic5C22×C411yescharacteristic
185#683—21no—
186#683—21no—
187#683—21no—
188#683—21no—
189C4×Dic5—21no—
190C4×Dic5—21no—
191C4×Dic5—21no—
192C4×Dic5—21no—
193C4×Dic5—21no—
194C4×Dic5—21no—
195#681—22no—
196#681—22no—
197#681—22no—
198#681—22no—
199C2×#168—22no—
200C2×#168—22no—
201Dic20—22no—
202D40—22no—
203#677—22no—
204#677—22no—
205#676—44no—
206C8×D5—44no—
207C2×#135—51no—
208C2×#131—52no—
209C4×#41—51no—
210C4×#41—51no—
211C4×Q16—51no—
212C4×Q16—51no—
213C4×SD16—51no—
214C4×SD16—51no—
215C4×SD16—51no—
216C4×SD16—51no—
217C4×D8—51no—
218C4×D8—51no—
219#433—102no—
220#432—101no—
221#431—101no—
222#427—51no—
223#424—52no—
224#424—102no—
225#423—51no—
226#423—51no—
227#423—101no—
228C2×C4⋊C8—101no—
229C2×C4⋊C8C2×D511yescharacteristic
230C2×#104—52no—
231C2×#104—52no—
232#416—51no—
233#416—51no—
234#408—52no—
235#405—52no—
236C2×C4×C8C2×D512yes—
237C2×C4×C8C2×D511yescharacteristic
238C4×C4⋊C4C2×D511yescharacteristic
239#341—52no—
240#341—52no—
241#339—52no—
242#339—52no—
243#336C2×D512yes—
244C23×C5C2×#4111yescharacteristic
245C22×D5—21no—
246C22×D5—21no—
247C2×C4×C5—22no—
248C2×C4×C5—24no—
249C2×C4×C5—41no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5C4×D411yescharacteristic
253C2×C4×C5C4×D411yescharacteristic
254C2×C4×C5C4×D411yescharacteristic
255C2×C4×C5C4×D411yescharacteristic
256C2×C4×C5C23×C411yescharacteristic
257C2×C4×C5C22×D411yescharacteristic
258C2×C4×C5C2×#4111yescharacteristic
259C2×C4×C5C2×#4111yescharacteristic
260C2×C4×C5H(5,𝔽2)11yescharacteristic
261C2×C4×C52−1+411yescharacteristic
262C22⋊D5—22no—
263C22⋊D5—22no—
264C22⋊D5—41no—
265C22⋊D5—41no—
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5—21no—
269C2×Dic5—21no—
270C2×Dic5—21no—
271C2×Dic5—21no—
272D20—21no—
273D20—21no—
274D20—21no—
275D20#11712yes—
276C4×D5—22no—
277C4×D5—22no—
278C4×D5—41no—
279C4×D5—41no—
280C4×D5—42no—
281C4×D5—42no—
282C4×D5—81no—
283C4×D5—81no—
284Dic10—21no—
285Dic10—21no—
286Dic10—21no—
287Dic10#11712yes—
288C5×C8—21no—
289C5×C8—21no—
290C5×C8—21no—
291C5×C8—21no—
292C5×C8—22no—
293C5×C8—22no—
294#168—42no—
295#168—42no—
296C2×#41—51no—
297C2×#41—51no—
298#135—102no—
299#135—102no—
300C2×Q16—51no—
301C2×SD16—51no—
302C2×SD16—51no—
303C2×D8—51no—
304#131—104no—
305#131—202no—
306C2×M16—52no—
307C2×M16—52no—
308C2×M16—102no—
309C22×C8—102no—
310C22×C8C22×D511yescharacteristic
311C22×C8C22×D511yescharacteristic
312C22×C8C22×D512yes—
313C4×Q8—51no—
314C4×Q8—51no—
315C4×Q8—51no—
316C4×Q8—51no—
317C4×D4—51no—
318C4×D4—51no—
319C4×D4—51no—
320C4×D4—51no—
321C4×D4—101no—
322C4×D4—101no—
323C4×D4—101no—
324C4×D4—101no—
325#117—51no—
326#117—51no—
327#117—101no—
328#117—101no—
329C2×C4⋊C4—21no—
330C2×C42—101no—
331C2×C42—101no—
332C2×C42C22×D512yes—
333C2×C42C22×D511yescharacteristic
334#107—51no—
335#107—51no—
336#106—51no—
337#106—51no—
338C4⋊C8—102no—
339C4⋊C8—102no—
340C4⋊C8—101no—
341C4⋊C8—101no—
342C4⋊C8—101no—
343C4⋊C8—101no—
344C4⋊C8D2011yescharacteristic
345C4⋊C8D2011yescharacteristic
346C4⋊C8D2012yes—
347#104—104no—
348#104—202no—
349#103—51no—
350#103—51no—
351#103—51no—
352#103—51no—
353#102—51no—
354#102—51no—
355#102—51no—
356#102—51no—
357C22⋊C8—102no—
358C22⋊C8—102no—
359#97—102no—
360C4×C8—42no—
361C4×C8—102no—
362C4×C8C4×D511yescharacteristic
363C4×C8C4×D511yescharacteristic
364C4×C8C4×D511yescharacteristic
365C4×C8C4×D511yescharacteristic
366#95—21no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5C2×C4×D411yescharacteristic
370C22×C5#55211yescharacteristic
371C22×C5#55311yescharacteristic
372C2×D5—22no—
373C2×D5—41no—
374C2×D5—41no—
375C2×D5—41no—
376C2×D5—41no—
377C2×D5—41no—
378C4×C5—21no—
379C4×C5—21no—
380C4×C5—21no—
381C4×C5—44no—
382C4×C5#44312yes—
383C4×C5C2×#11711yescharacteristic
384C4×C5#52011yescharacteristic
385C4×C5#54611yescharacteristic
386C4×C5#54811yescharacteristic
387Dic5—22no—
388Dic5—41no—
389Dic5—41no—
390Dic5—41no—
391Dic5—41no—
392Dic5—41no—
393#41—52no—
394#41—52no—
395#41—101no—
396#41—101no—
397#41—101no—
398#41—101no—
399#41—101no—
400#41—101no—
401C2×Q8—51no—
402C2×Q8—51no—
403C2×D4—51no—
404C2×D4—51no—
405C2×D4—101no—
406C2×D4—101no—
407C22×C4—22no—
408C22×C4—101no—
409C22×C4—101no—
410C22×C4—101no—
411C22×C4—101no—
412C22×C4—101no—
413C22×C4—101no—
414C22×C4#70911yescharacteristic
415C22×C4#70911yescharacteristic
416C22×C4C23×D511yescharacteristic
417Q16—102no—
418SD16—102no—
419SD16—102no—
420D8—102no—
421M16—102no—
422M16—102no—
423M16—102no—
424M16—102no—
425M16—204no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—102no—
433C2×C8—102no—
434C2×C8—204no—
435C2×C8C2×C4×D511yescharacteristic
436C2×C8C2×C4×D511yescharacteristic
437C2×C8C2×C4×D511yescharacteristic
438C2×C8C2×C4×D511yescharacteristic
439C2×C8C2×D2012yes—
440C2×C8C2×D2012yes—
441C4⋊C4—24no—
442C4⋊C4—51no—
443C4⋊C4—51no—
444C4⋊C4—51no—
445C4⋊C4—51no—
446C4⋊C4—101no—
447C4⋊C4—101no—
448C4⋊C4—101no—
449C4⋊C4—101no—
450C22⋊C4—101no—
451C22⋊C4—101no—
452C22⋊C4—101no—
453C22⋊C4—101no—
454C22⋊C4—101no—
455C22⋊C4—101no—
456C42—22no—
457C42—22no—
458C42—101no—
459C42—101no—
460C42—101no—
461C42—101no—
462C42—101no—
463C42—101no—
464C42C2×C4×D511yescharacteristic
465C42C2×C4×D511yescharacteristic
466C42C2×D2011yescharacteristic
467C42C2×D2011yescharacteristic
468C2×C5—21no—
469C2×C5—21no—
470C2×C5#235211yescharacteristic
471C2×C5C2×#44311yescharacteristic
472C2×C5#295711yescharacteristic
473D5—42no—
474D5—81no—
475C23—101no—
476C23—101no—
477C23C2×#70911yescharacteristic
478Q8—52no—
479Q8—101no—
480Q8—101no—
481Q8—101no—
482D4—52no—
483D4—102no—
484D4—102no—
485D4—101no—
486D4—101no—
487D4—101no—
488D4—201no—
489D4—201no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—101no—
496C2×C4—102no—
497C2×C4—102no—
498C2×C4—101no—
499C2×C4—101no—
500C2×C4—101no—
501C2×C4—101no—
502C2×C4—101no—
503C2×C4—201no—
504C2×C4—201no—
505C2×C4—202no—
506C2×C4—202no—
507C2×C4—401no—
508C2×C4—401no—
509C2×C4C4×D2011yescharacteristic
510C2×C4C4×D2011yescharacteristic
511C2×C4C4×D2011yescharacteristic
512C2×C4C4×D2011yescharacteristic
513C2×C4C22×C4×D511yescharacteristic
514C2×C4C22×D2011yescharacteristic
515C2×C4#414411yescharacteristic
516C2×C4#414711yescharacteristic
517C2×C4D5×#4111yescharacteristic
518C2×C4D5×#4111yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—21no—
523C8—22no—
524C8—22no—
525C8—202no—
526C8—202no—
527C5#2278311yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—102no—
531C22—201no—
532C22—201no—
533C22—201no—
534C22—201no—
535C22—201no—
536C22C2×C4×D2011yescharacteristic
537C22#6589211yescharacteristic
538C22#6591511yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—44no—
543C4—102no—
544C4—201no—
545C4—201no—
546C4—201no—
547C4—201no—
548C4—201no—
549C4#6498912yes—
550C4#6583811yescharacteristic
551C4D5×#11711yescharacteristic
552C4#6588811yescharacteristic
553C4#6591011yescharacteristic
554C2—21no—
555C2—21no—
556C2—202no—
557C2—401no—
558C2#1060550211yescharacteristic
559C2C2×#6498911yescharacteristic
560C2#1061031111yescharacteristic
subgroupG/H
centerC2×C4C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×C8×C4⋊C4C2
socleC22×C5C2×C4×D4
prime radicalC2×#4050C22
cosocleC22×C4×C5C24
hypercenterC8×C4⋊C4D5
hypocenterC5#22783
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10610743C2semidirect
2. #10610731C2semidirect
4. C5×C8×C4⋊C4C2semidirect
5. #10600586C2semidirect
6. #10600584C2semidirect
7. #10600582C2semidirect
8. #10600287C2semidirect
9. #10600287C2semidirect
11. #10599438C2semidirect
36. C4×Dic20C4semidirect
37. C4×Dic20C4semidirect
38. C4×D40C4semidirect
39. C4×D40C4semidirect
40. C4×#677C4semidirect
41. C4×#677C4semidirect
42. C4×#677C4semidirect
43. C4×#677C4semidirect
130. C8×C4⋊C4D5semidirect
527. C5#22783semidirect

Maximal subgroups

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

subgroupindex
#106107432
#106107312
C4×#40502
C5×C8×C4⋊C42
#106005862
#106005842
#106005822
#106002872
#105994882
#105994382
#227835