Finite Group Databasebeta

← #11991798

Subgroups — #11991798

Order 1280.

1040 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#537C211yesmaximal, characteristic
2#10602932C211yesmaximal, characteristic, complemented
3#10602930C211yesmaximal, characteristic, complemented
4#10602482C212yesmaximal, complemented
5#10602467C212yesmaximal, complemented
6C22×C5×#41C411yescharacteristic, complemented
7C22×C5×#41C2211yescharacteristic
8C5×#537—24no—
9C5×#537—24no—
10C2×C5×C22⋊Q8—21no—
11C2×C5×C22⋊Q8C411yescharacteristic, complemented
12C2×C5×C22⋊Q8C2211yescharacteristic
13C2×C5×#121—21no—
14C2×C5×#121—21no—
15C2×C5×#121—21no—
16C2×C5×#121C411yescharacteristic, complemented
17C2×C5×#121C2211yescharacteristic
18C22×C5×C4⋊C4C411yescharacteristic, complemented
19C2×#3968—21no—
20C2×#3966—21no—
21C2×#3966—21no—
22C2×#3965—21no—
23#65500C2212yes—
24#65496C2212yes—
25#64758—24no—
26#64757—24no—
27#8517—51nomaximal, Sylow
28C2×C5×#41—22no—
29C2×C5×#41—22no—
30C2×C5×#41—22no—
31C2×C5×#41—22no—
32C2×C5×#41—42no—
33C2×C5×#41—42no—
34C2×C5×#41—42no—
35C2×C5×#41—42no—
36C22×C5×Q8C2×C411yescharacteristic
37C22×C5×Q8D411yescharacteristic
38C22×C5×D4—21no—
39C22×C5×D4—21no—
40C22×C5×D4—21no—
41C22×C5×D4C2×C411yescharacteristic
42C22×C5×D4D411yescharacteristic
43C22×C5×D4D411yescharacteristic
44C22×C5×D4D411yescharacteristic
45C23×C4×C5—21no—
46C23×C4×C5—21no—
47C23×C4×C5C2×C411yescharacteristic
48C23×C4×C5C2×C411yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C5×C22⋊Q8—28no—
51C5×C22⋊Q8—22no—
52C5×C22⋊Q8—22no—
53C5×C22⋊Q8—22no—
54C5×C22⋊Q8D414yes—
55C5×#121—28no—
56C5×#121—28no—
57C5×#121—28no—
58C5×#121—22no—
59C5×#121—22no—
60C5×#121—22no—
61C5×#121D414yes—
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4C2×C411yescharacteristic
68C2×C5×C4⋊C4C2×C411yescharacteristic
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—21no—
72C2×C5×C22⋊C4—21no—
73C2×C22⋊Dic5—22no—
74C2×#681—22no—
75#3968—42no—
76#3966—42no—
77#3966—42no—
78#3965—42no—
79#3963—22no—
80#3963—22no—
81C22⋊#168—22no—
82C22⋊#168—22no—
83C2×#537D511yescharacteristic
84#1785—51no—
85#1782—51no—
86#1511—52no—
87#1510—52no—
88C24×C5—21no—
89C24×C5C22⋊C411yescharacteristic
90C24×C5C22⋊C411yescharacteristic
91C24×C5C2×D411yescharacteristic
92C5×#41—48no—
93C5×#41—48no—
94C5×#41—48no—
95C5×#41—48no—
96C2×C5×Q8—22no—
97C2×C5×Q8—21no—
98C2×C5×Q8—22no—
99C2×C5×Q8—41no—
100C2×C5×Q8—41no—
101C2×C5×Q8—41no—
102C2×C5×Q8C22⋊C412yes—
103C2×C5×D4—24no—
104C2×C5×D4—22no—
105C2×C5×D4—22no—
106C2×C5×D4—22no—
107C2×C5×D4—22no—
108C2×C5×D4—24no—
109C2×C5×D4—24no—
110C2×C5×D4—22no—
111C2×C5×D4—21no—
112C2×C5×D4—22no—
113C2×C5×D4—41no—
114C2×C5×D4—41no—
115C2×C5×D4—41no—
116C2×C5×D4—41no—
117C2×C5×D4—41no—
118C2×C5×D4—42no—
119C2×C5×D4—42no—
120C2×C5×D4—82no—
121C2×C5×D4—82no—
122C2×C5×D4C22⋊C412yes—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—21no—
126C22×C4×C5—21no—
127C22×C4×C5—24no—
128C22×C4×C5—21no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—24no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—41no—
135C22×C4×C5—41no—
136C22×C4×C5—41no—
137C22×C4×C5—41no—
138C22×C4×C5—42no—
139C22×C4×C5—42no—
140C22×C4×C5—82no—
141C22×C4×C5C22⋊C411yescharacteristic
142C22×C4×C5C22⋊C411yescharacteristic
143C22×C4×C5C22×C411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D412yes—
146C22×C4×C5C2×D412yes—
147C22×Dic5—42no—
148C5×C4⋊C4—24no—
149C5×C4⋊C4—24no—
150C5×C4⋊C4—21no—
151C5×C4⋊C4—21no—
152C5×C4⋊C4—24no—
153C5×C4⋊C4C22⋊C412yes—
154C5×C4⋊C4C22⋊C412yes—
155C5×C22⋊C4—24no—
156C5×C22⋊C4—24no—
157C5×C22⋊C4—24no—
158C5×C22⋊C4—24no—
159C22⋊Dic5—42no—
160C22⋊Dic5—42no—
161#681—42no—
162#681—42no—
163C2×#168—42no—
164C22×#41Dic511yescharacteristic, complemented
165C22×#41C2×D511yescharacteristic
166#537—24no—
167#537—24no—
168C2×C22⋊Q8—21no—
169C2×C22⋊Q8Dic511yescharacteristic, complemented
170C2×C22⋊Q8C2×D511yescharacteristic
171C2×#121—21no—
172C2×#121—21no—
173C2×#121—21no—
174C2×#121Dic511yescharacteristic, complemented
175C2×#121C2×D511yescharacteristic
176C22×C4⋊C4Dic511yescharacteristic, complemented
177C2×#101—101no—
178C2×#100—101no—
179C2×#99—101no—
180C2×#99—101no—
181#407—52no—
182#381—52no—
183#328—104no—
184#327—104no—
185C23×C5—21no—
186C23×C5—21no—
187C23×C5—21no—
188C23×C5—21no—
189C23×C5—21no—
190C23×C5—22no—
191C23×C5—21no—
192C23×C5—21no—
193C23×C5—41no—
194C23×C5—41no—
195C23×C5—41no—
196C23×C5—41no—
197C23×C5—42no—
198C23×C5—42no—
199C23×C5—41no—
200C23×C5—41no—
201C23×C5—41no—
202C23×C5—41no—
203C23×C5#9912yes—
204C23×C5C2×C22⋊C411yescharacteristic
205C23×C5#12012yes—
206C5×Q8—44no—
207C5×Q8—42no—
208C5×Q8—42no—
209C5×D4—44no—
210C5×D4—44no—
211C5×D4—42no—
212C5×D4—42no—
213C5×D4—44no—
214C5×D4—48no—
215C5×D4—84no—
216C5×D4—84no—
217C2×C4×C5—24no—
218C2×C4×C5—24no—
219C2×C4×C5—21no—
220C2×C4×C5—21no—
221C2×C4×C5—21no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—22no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—21no—
229C2×C4×C5—21no—
230C2×C4×C5—21no—
231C2×C4×C5—42no—
232C2×C4×C5—42no—
233C2×C4×C5—48no—
234C2×C4×C5—42no—
235C2×C4×C5—42no—
236C2×C4×C5—42no—
237C2×C4×C5—42no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—84no—
241C2×C4×C5#9912yes—
242C2×C4×C5C2×C22⋊C411yescharacteristic
243C2×C4×C5C2×C22⋊C411yescharacteristic
244C2×C4×C5#12012yes—
245C2×Dic5—82no—
246C2×Dic5—82no—
247C2×Dic5—82no—
248#168—82no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—42no—
254C2×#41—42no—
255C2×#41—42no—
256C2×#41—42no—
257C22×Q8C2×Dic511yescharacteristic
258C22×Q8C22⋊D511yescharacteristic
259C22×D4—21no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C2×Dic511yescharacteristic
263C22×D4C22⋊D511yescharacteristic
264C22×D4C22⋊D511yescharacteristic
265C22×D4C22⋊D511yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×Dic511yescharacteristic
269C23×C4C2×Dic511yescharacteristic
270C23×C4C22×D511yescharacteristic
271C2×M16—102no—
272C22⋊Q8—28no—
273C22⋊Q8—22no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8C22⋊D514yes—
277#121—28no—
278#121—28no—
279#121—28no—
280#121—22no—
281#121—22no—
282#121—22no—
283#121C22⋊D514yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×Dic511yescharacteristic
290C2×C4⋊C4C2×Dic511yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—102no—
296#101—202no—
297#100—202no—
298#99—202no—
299#99—202no—
300C22⋊C8—102no—
301C22⋊C8—102no—
302#95—102no—
303#95—102no—
304C22×C5—21no—
305C22×C5—21no—
306C22×C5—22no—
307C22×C5—22no—
308C22×C5—22no—
309C22×C5—22no—
310C22×C5—42no—
311C22×C5—42no—
312C22×C5—41no—
313C22×C5—41no—
314C22×C5—41no—
315C22×C5—42no—
316C22×C5—41no—
317C22×C5—42no—
318C22×C5—41no—
319C22×C5—42no—
320C22×C5—81no—
321C22×C5—81no—
322C22×C5—81no—
323C22×C5—81no—
324C22×C5—82no—
325C22×C5—82no—
326C22×C5#37911yescharacteristic
327C22×C5C2×#9911yescharacteristic
328C22×C5C2×#9911yescharacteristic
329C4×C5—22no—
330C4×C5—22no—
331C4×C5—21no—
332C4×C5—21no—
333C4×C5—42no—
334C4×C5—42no—
335C4×C5—41no—
336C4×C5—42no—
337Dic5—162no—
338C24—21no—
339C24C22⋊Dic511yescharacteristic
340C24C22⋊Dic511yescharacteristic
341C24C2×C22⋊D511yescharacteristic
342#41—48no—
343#41—48no—
344#41—48no—
345#41—48no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8C22⋊Dic512yes—
353C2×D4—24no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4C22⋊Dic512yes—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—82no—
391C22×C4—202no—
392C22×C4C22⋊Dic511yescharacteristic
393C22×C4C22⋊Dic511yescharacteristic
394C22×C4C22×Dic511yescharacteristic
395C22×C4C2×C22⋊D511yescharacteristic
396C22×C4C2×C22⋊D512yes—
397C22×C4C2×C22⋊D512yes—
398M16—202no—
399M16—202no—
400C2×C8—202no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4C22⋊Dic512yes—
407C4⋊C4C22⋊Dic512yes—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—202no—
413C22⋊C4—202no—
414C2×C5—21no—
415C2×C5—21no—
416C2×C5—42no—
417C2×C5—42no—
418C2×C5—81no—
419C2×C5—82no—
420C2×C5#178111yescharacteristic
421C2×C5#179511yescharacteristic
422C2×C5#179611yescharacteristic
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23#396612yes—
442C23C2×C22⋊Dic511yescharacteristic
443C23#409912yes—
444Q8—44no—
445Q8—42no—
446Q8—42no—
447D4—44no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—44no—
452D4—48no—
453D4—84no—
454D4—84no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—84no—
479C2×C4—402no—
480C2×C4—402no—
481C2×C4—402no—
482C2×C4#396612yes—
483C2×C4C2×C22⋊Dic511yescharacteristic
484C2×C4C2×C22⋊Dic511yescharacteristic
485C2×C4#409912yes—
486C8—402no—
487C5#851711yescharacteristic, complemented, Sylow
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22C2×#396611yescharacteristic
511C22C2×#396611yescharacteristic
512C22#6553611yescharacteristic
513C4—22no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—802no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—42no—
526C2—82no—
527C2—81no—
528C2#1060292311yescharacteristic
529C2#1060293411yescharacteristic
530C2#1060293711yescharacteristic
subgroupG/H
centerC22#65536
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
Fitting, prime radicalC2×C5×#537C2
socleC22×C5#379
cosocleC23×C4×C5C23
hypercenterC2×#537D5
hypocenterC5#8517
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. #10602932C2semidirect
3. #10602930C2semidirect
4. #10602482C2semidirect
5. #10602467C2semidirect
6. C22×C5×#41C4semidirect
11. C2×C5×C22⋊Q8C4semidirect
16. C2×C5×#121C4semidirect
18. C22×C5×C4⋊C4C4semidirect
164. C22×#41Dic5semidirect
169. C2×C22⋊Q8Dic5semidirect
174. C2×#121Dic5semidirect
176. C22×C4⋊C4Dic5semidirect
487. C5#8517semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5372
#106029322
#106029302
#106024822
#106024672
#85175