Finite Group Databasebeta

← #12102774

Subgroups — #12102774

Order 1280.

1110 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65170C212yesmaximal
2C2×#65126C211yesmaximal, characteristic
3D5×#384C211yesmaximal, characteristic
4C4×#4038C211yesmaximal, characteristic
5C5×#1947C211yesmaximal, characteristic, complemented
6#10601408C212yesmaximal, complemented
7#10601247C211yesmaximal, characteristic, complemented
8#10600361C211yesmaximal, characteristic, complemented
9#10600259C211yesmaximal, characteristic, complemented
10#10599660C212yesmaximal
11#10599589C212yesmaximal, complemented
12C22×#676C2212yes—
13C2×#4038C2211yescharacteristic
14C2×#4038C2211yescharacteristic
15C2×D5×C4⋊C4—21no—
16C2×D5×C4⋊C4—21no—
17C2×D5×C4⋊C4C2212yes—
18C2×C42×D5C2211yescharacteristic
19C2×C5×#107C2212yes—
20C2×C5×C4⋊C8C2211yescharacteristic
21C5×#384C2211yescharacteristic
22C4×C5×C4⋊C4C2211yescharacteristic
23C2×#3949C2212yes—
24C2×#3946C2212yes—
25#65265—21no—
26#65265—21no—
27#65265C2212yes—
28C4⋊C4×Dic5C2211yescharacteristic
29C4⋊C4×Dic5C2211yescharacteristic
30C2×#3940C2212yes—
31C2×#3939C2212yes—
32C4×#685—21no—
33#65224C2211yescharacteristic
34C4×C4⋊Dic5C2211yescharacteristic
35C42×Dic5—21no—
36C2×C4⋊#168C2211yescharacteristic
37#65170—28no—
38#65126C418yescomplemented
39#64955—21no—
40#64937—21no—
41C5×#340C2212yes—
42#64771C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#21171—51nomaximal, Sylow
46C22×C4×D5—22no—
47C22×C4×D5D411yescharacteristic
48C22×C4×D5D411yescharacteristic
49C22×C4×D5C2311yescharacteristic
50C22×C5×C8C2312yes—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C5×C4⋊C4C2312yes—
54C2×C5×C4⋊C4C2311yescharacteristic
55C2×C5×C4⋊C4C2311yescharacteristic
56C2×C42×C5—21no—
57C2×C42×C5C2311yescharacteristic
58C2×#685—21no—
59C2×#685—21no—
60C2×C4⋊Dic5—21no—
61C2×C4⋊Dic5—21no—
62C2×C4⋊Dic5C2312yes—
63C2×C4⋊Dic5C2311yescharacteristic
64C2×C4⋊Dic5C2311yescharacteristic
65C2×#683—22no—
66C2×#683—22no—
67C2×#683—22no—
68C2×C4×Dic5—21no—
69C2×C4×Dic5—21no—
70C2×C4×Dic5—21no—
71C2×C4×Dic5—21no—
72C2×C4×Dic5—21no—
73C2×C4×Dic5D411yescharacteristic
74C2×C4×Dic5D411yescharacteristic
75C2×C4×Dic5C2311yescharacteristic
76C22×#168C2312yes—
77C2×#676—24no—
78C2×#676—24no—
79C2×#676C2×C418yes—
80#4038—22no—
81#4038—22no—
82#4038D414yes—
83#4038Q814yes—
84D5×C4⋊C4—24no—
85D5×C4⋊C4—24no—
86D5×C4⋊C4—44no—
87D5×C4⋊C4—44no—
88C42×D5C2×C414yes—
89C42×D5Q814yes—
90C5×#107—24no—
91C5×C4⋊C8C2×C414yescomplemented
92C5×#95—21no—
93#3963—21no—
94#3963—21no—
95#3963—21no—
96#3949—24no—
97#3946C2×C418yescomplemented
98#3940—24no—
99#3939—24no—
100C4⋊#168C2×C414yescomplemented
101C2×#428—52no—
102C2×#423—51no—
103C2×#384—51no—
104C4×#117—51no—
105#1948—51no—
106#1948—51no—
107#1948—52no—
108#1947—52no—
109#1947—51no—
110#1947D511yescharacteristic, complemented
111#1736—52no—
112C23×D5C2×D411yescharacteristic
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—22no—
116C22×C4×C5C2×D411yescharacteristic
117C22×C4×C5C2×D411yescharacteristic
118C22×C4×C5C2411yescharacteristic
119C22×Dic5—21no—
120C22×Dic5—21no—
121C22×Dic5—22no—
122C22×Dic5C2×D411yescharacteristic
123C22×Dic5C2×D411yescharacteristic
124C22×Dic5C2×D411yescharacteristic
125C2×C4×D5—22no—
126C2×C4×D5—22no—
127C2×C4×D5—22no—
128C2×C4×D5—22no—
129C2×C4×D5—44no—
130C2×C4×D5—44no—
131C2×C4×D5—44no—
132C2×C4×D5C4⋊C414yes—
133C2×C4×D5C4⋊C414yes—
134C2×C4×D5C22×C412yes—
135C2×C4×D5C2×D412yes—
136C2×C4×D5C2×Q812yes—
137C2×C4×D5C2×Q812yes—
138C2×C4×D5#4112yes—
139C2×C4×D5#4112yes—
140C2×C5×C8—22no—
141C2×C5×C8—22no—
142C2×C5×C8C22×C414yes—
143C5×C4⋊C4—21no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4—22no—
146C5×C4⋊C4—22no—
147C5×C4⋊C4—42no—
148C5×C4⋊C4—42no—
149C5×C4⋊C4C2×D412yes—
150C5×C4⋊C4C2×Q812yes—
151C42×C5—42no—
152C42×C5C22×C412yes—
153C42×C5C2×Q812yes—
154#685—24no—
155#685—24no—
156C4⋊Dic5—21no—
157C4⋊Dic5—21no—
158C4⋊Dic5—22no—
159C4⋊Dic5—22no—
160C4⋊Dic5—42no—
161C4⋊Dic5—42no—
162C4⋊Dic5C2×D412yes—
163C4⋊Dic5C2×Q812yes—
164#683—44no—
165#683—44no—
166#683—44no—
167C4×Dic5—24no—
168C4×Dic5—24no—
169C4×Dic5—42no—
170C4×Dic5—42no—
171C4×Dic5—42no—
172C4×Dic5C4⋊C414yes—
173C4×Dic5C4⋊C414yes—
174C4×Dic5C22×C412yes—
175C4×Dic5C2×Q812yes—
176C2×#168—22no—
177C2×#168—22no—
178C2×#168C22×C414yes—
179#676—216no—
180C22×M16—52no—
181C2×#117—51no—
182C2×#117—51no—
183C22×C4⋊C4—52no—
184C22×C4⋊C4—101no—
185C22×C4⋊C4—101no—
186C22×C42—51no—
187#428—108no—
188C2×#107—52no—
189C2×#107C2×D512yes—
190C2×#106—52no—
191C2×#106—52no—
192#423—58no—
193C2×C4⋊C8—52no—
194C2×C4⋊C8—51no—
195C2×C4⋊C8C2×D511yescharacteristic
196#384—52no—
197#384—51no—
198#384—101no—
199#384—101no—
200#384C2×D511yescharacteristic
201C4×C4⋊C4—51no—
202C4×C4⋊C4—51no—
203C4×C4⋊C4—51no—
204C4×C4⋊C4C2×D511yescharacteristic
205C4×C22⋊C4—101no—
206C4×C22⋊C4—101no—
207#376—101no—
208C43—101no—
209#340—52no—
210#340—52no—
211#340—52no—
212#340C2×D512yes—
213C23×C5C22×D411yescharacteristic
214C22×D5—21no—
215C22×D5—21no—
216C22×D5—21no—
217C22×D5—21no—
218C22×D5C2×C4⋊C412yes—
219C22×D5#12112yes—
220C22×D5C22⋊Q812yes—
221C2×C4×C5—21no—
222C2×C4×C5—21no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
225C2×C4×C5—41no—
226C2×C4×C5—41no—
227C2×C4×C5—41no—
228C2×C4×C5—41no—
229C2×C4×C5—42no—
230C2×C4×C5—42no—
231C2×C4×C5—42no—
232C2×C4×C5C2×C4⋊C412yes—
233C2×C4×C5C2×C4⋊C412yes—
234C2×C4×C5AGL(1,ℤ/8)12yes—
235C2×C4×C5#13712yes—
236C2×C4×C5C23×C411yescharacteristic
237C2×C4×C5C22×D411yescharacteristic
238C2×C4×C5C22×Q811yescharacteristic
239C2×C4×C5C22×Q811yescharacteristic
240C2×C4×C5C2×#4111yescharacteristic
241C2×C4×C5C2×#4111yescharacteristic
242C2×Dic5—21no—
243C2×Dic5—21no—
244C2×Dic5—22no—
245C2×Dic5—22no—
246C2×Dic5—41no—
247C2×Dic5—41no—
248C2×Dic5—41no—
249C2×Dic5—41no—
250C2×Dic5—42no—
251C2×Dic5—42no—
252C2×Dic5—42no—
253C2×Dic5C2×C4⋊C412yes—
254C2×Dic5C2×C4⋊C412yes—
255C2×Dic5C2×C4⋊C412yes—
256C2×Dic5#12112yes—
257C2×Dic5C22⋊Q812yes—
258C2×Dic5AGL(1,ℤ/8)12yes—
259C2×Dic5#13712yes—
260C4×D5—28no—
261C4×D5—28no—
262C4×D5—88no—
263C4×D5C4×D414yes—
264C4×D5C4×Q814yes—
265C4×D5#12514yes—
266C4×D5C4⋊Q814yes—
267C5×C8—24no—
268#168—24no—
269C23×C4—51no—
270C23×C4—51no—
271C23×C4—51no—
272C23×C4—102no—
273C2×M16—58no—
274C2×M16—104no—
275C2×M16—104no—
276C22×C8—52no—
277C22×C8C22×D512yes—
278#117—54no—
279#117—54no—
280#117—102no—
281#117—102no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—51no—
285C2×C4⋊C4—51no—
286C2×C4⋊C4—52no—
287C2×C4⋊C4—104no—
288C2×C4⋊C4—104no—
289C2×C4⋊C4—102no—
290C2×C4⋊C4—102no—
291C2×C4⋊C4—102no—
292C2×C4⋊C4—101no—
293C2×C4⋊C4—101no—
294C2×C4⋊C4—204no—
295C2×C4⋊C4—204no—
296C2×C4⋊C4C22×D512yes—
297C2×C4⋊C4C22×D511yescharacteristic
298C2×C4⋊C4C22×D511yescharacteristic
299C2×C22⋊C4—101no—
300C2×C22⋊C4—101no—
301C2×C42—21no—
302C2×C42—51no—
303C2×C42—51no—
304C2×C42—51no—
305C2×C42—54no—
306C2×C42—54no—
307C2×C42—101no—
308C2×C42—101no—
309C2×C42—101no—
310C2×C42—101no—
311C2×C42—101no—
312C2×C42C22×D511yescharacteristic
313#107—24no—
314#107—104no—
315#106—104no—
316#106—104no—
317C4⋊C8—58no—
318C4⋊C8—54no—
319C4⋊C8C4×D514yescomplemented
320#95—21no—
321#95—101no—
322#95—101no—
323#95—101no—
324C22×C5C22×C4⋊C411yescharacteristic
325C22×C5C2×#12111yescharacteristic
326C22×C5C2×C22⋊Q811yescharacteristic
327C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C5C2×#13711yescharacteristic
330C22×C5C2×#13711yescharacteristic
331C2×D5—22no—
332C2×D5—22no—
333C2×D5—22no—
334C2×D5—22no—
335C2×D5—22no—
336C2×D5—22no—
337C2×D5#38914yes—
338C4×C5—22no—
339C4×C5—42no—
340C4×C5—42no—
341C4×C5—82no—
342C4×C5#42814yescomplemented
343C4×C5C2×C4×D411yescharacteristic
344C4×C5C2×C4×Q811yescharacteristic
345C4×C5C2×#12511yescharacteristic
346C4×C5C2×C4⋊Q811yescharacteristic
347Dic5—22no—
348Dic5—42no—
349Dic5—42no—
350Dic5—82no—
351Dic5#38914yes—
352Dic5#42814yescomplemented
353C24—51no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—51no—
358C22×C4—51no—
359C22×C4—52no—
360C22×C4—52no—
361C22×C4—52no—
362C22×C4—52no—
363C22×C4—52no—
364C22×C4—52no—
365C22×C4—51no—
366C22×C4—54no—
367C22×C4—54no—
368C22×C4—102no—
369C22×C4—102no—
370C22×C4—102no—
371C22×C4—102no—
372C22×C4—101no—
373C22×C4—101no—
374C22×C4—102no—
375C22×C4—204no—
376C22×C4—204no—
377C22×C4—204no—
378C22×C4D4×D511yescharacteristic
379C22×C4D4×D511yescharacteristic
380C22×C4C23×D511yescharacteristic
381M16—1016no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—54no—
385C2×C8—102no—
386C2×C8—102no—
387C2×C8C2×C4×D514yes—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—52no—
395C4⋊C4—52no—
396C4⋊C4—101no—
397C4⋊C4—101no—
398C4⋊C4—102no—
399C4⋊C4—102no—
400C4⋊C4—202no—
401C4⋊C4—204no—
402C4⋊C4—204no—
403C4⋊C4—204no—
404C4⋊C4—202no—
405C4⋊C4D4×D512yes—
406C4⋊C4Q8×D512yes—
407C22⋊C4—104no—
408C22⋊C4—104no—
409C42—42no—
410C42—52no—
411C42—52no—
412C42—54no—
413C42—54no—
414C42—104no—
415C42—104no—
416C42—202no—
417C42—202no—
418C42—202no—
419C42C2×C4×D512yes—
420C42Q8×D512yes—
421C2×C5C2×#38911yescharacteristic
422C2×C5C2×#42811yescharacteristic
423C2×C5C2×#42811yescharacteristic
424C2×C5#303811yescharacteristic
425C2×C5#303911yescharacteristic
426C2×C5#307911yescharacteristic
427C2×C5#307911yescharacteristic
428D5—24no—
429C23—52no—
430C23—52no—
431C23—52no—
432C23—101no—
433C23—101no—
434C23—101no—
435C23—101no—
436C23C2×D4×D511yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—52no—
449C2×C4—52no—
450C2×C4—52no—
451C2×C4—52no—
452C2×C4—52no—
453C2×C4—52no—
454C2×C4—54no—
455C2×C4—54no—
456C2×C4—54no—
457C2×C4—54no—
458C2×C4—52no—
459C2×C4—108no—
460C2×C4—101no—
461C2×C4—101no—
462C2×C4—108no—
463C2×C4—102no—
464C2×C4—102no—
465C2×C4—201no—
466C2×C4—201no—
467C2×C4—201no—
468C2×C4—201no—
469C2×C4—202no—
470C2×C4—202no—
471C2×C4—202no—
472C2×C4—408no—
473C2×C4D5×C4⋊C412yes—
474C2×C4D5×C4⋊C412yes—
475C2×C4#405712yes—
476C2×C4#406412yes—
477C2×C4C22×C4×D511yescharacteristic
478C2×C4C2×D4×D511yescharacteristic
479C2×C4C2×Q8×D511yescharacteristic
480C2×C4C2×Q8×D511yescharacteristic
481C2×C4D5×#4111yescharacteristic
482C2×C4D5×#4111yescharacteristic
483C8—24no—
484C8—104no—
485C5#2117111yescharacteristic, complemented, Sylow
486C22—54no—
487C22—102no—
488C22—102no—
489C22—102no—
490C22—102no—
491C22—102no—
492C22—102no—
493C22C2×D5×C4⋊C411yescharacteristic
494C22D5×#12111yescharacteristic
495C22D5×C22⋊Q811yescharacteristic
496C22C2×#405711yescharacteristic
497C22C2×#406411yescharacteristic
498C22D5×AGL(1,ℤ/8)11yescharacteristic
499C22D5×#13711yescharacteristic
500C4—22no—
501C4—42no—
502C4—42no—
503C4—54no—
504C4—54no—
505C4—82no—
506C4—102no—
507C4—202no—
508C4—202no—
509C4—402no—
510C4#6517014yescomplemented
511C4C4×D4×D511yescharacteristic
512C4C4×Q8×D511yescharacteristic
513C4D5×#12511yescharacteristic
514C4D5×C4⋊Q811yescharacteristic
515C2—104no—
516C2D5×#38911yescharacteristic
517C2C2×#6517011yescharacteristic
518C2D5×#42811yescharacteristic
519C2#1061206411yescharacteristic
520C2#1061220811yescharacteristic
521C2#1061269711yescharacteristic
522C2#1061284011yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1947C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1947D5
hypocenterC5#21171
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
5. C5×#1947C2semidirect
6. #10601408C2semidirect
7. #10601247C2semidirect
8. #10600361C2semidirect
9. #10600259C2semidirect
11. #10599589C2semidirect
38. #65126C4semidirect
91. C5×C4⋊C8C2×C4semidirect
97. #3946C2×C4semidirect
100. C4⋊#168C2×C4semidirect
110. #1947D5semidirect
319. C4⋊C8C4×D5semidirect
342. C4×C5#428semidirect
352. Dic5#428semidirect
485. C5#21171semidirect
510. C4#65170semidirect

Maximal subgroups

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

subgroupindex
C2×#651702
C2×#651262
D5×#3842
C4×#40382
C5×#19472
#106014082
#106012472
#106003612
#106002592
#105996602
#105995892
#211715