Finite Group Databasebeta

← #11994866

Subgroups — #11994866

Order 1280.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2847C211yesmaximal, characteristic
2C22×C22⋊#168C211yesmaximal, characteristic, complemented
3#10602920C211yesmaximal, characteristic, complemented
4C2×#64757C212yesmaximal, complemented
5#10598115C212yesmaximal, complemented
6C24×C4×C5C2211yescharacteristic
7C2×C5×#121C411yescharacteristic, complemented
8C2×C5×#121C411yescharacteristic, complemented
9C2×C5×#121C2211yescharacteristic
10C2×C5×#120—21no—
11C22×C22⋊Dic5—21no—
12C23×#168—21no—
13C5×#386—21no—
14C5×#386—21no—
15C5×#386—21no—
16C5×#380C411yescharacteristic, complemented
17C5×#380C411yescharacteristic, complemented
18C5×#380C2211yescharacteristic
19C2×#3963—21no—
20C2×#3963C2212yes—
21C2×C22⋊#168—24no—
22C2×C22⋊#168—21no—
23C2×C22⋊#168C2212yes—
24#64757—24no—
25#9113—51nomaximal, Sylow
26C25×C5D411yescharacteristic
27C22×C5×D4—21no—
28C22×C5×D4—21no—
29C22×C5×D4—21no—
30C22×C5×D4C2×C411yescharacteristic
31C23×C4×C5—22no—
32C23×C4×C5—22no—
33C23×C4×C5—42no—
34C23×C4×C5—42no—
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5D411yescharacteristic
38C23×C4×C5D411yescharacteristic
39C23×C4×C5D411yescharacteristic
40C23×C4×C5C2311yescharacteristic
41C23×Dic5—22no—
42C5×#121—22no—
43C5×#121—22no—
44C5×#121—42no—
45C5×#121—42no—
46C5×#120—42no—
47C5×#120—42no—
48C2×C5×C4⋊C4C2×C411yescharacteristic
49C2×C5×C4⋊C4C2×C411yescharacteristic
50C2×C5×C4⋊C4C2×C411yescharacteristic
51C2×C5×C22⋊C4—21no—
52C2×C5×C22⋊C4—21no—
53C2×C5×C22⋊C4—21no—
54C2×C5×C22⋊C4—21no—
55C2×C5×C22⋊C4—21no—
56C2×C5×C22⋊C4—21no—
57C2×C22⋊Dic5—24no—
58C2×C22⋊Dic5—42no—
59C2×C22⋊Dic5—42no—
60C22×#168—24no—
61C22×#168—22no—
62C22×#168—42no—
63C22×#168—42no—
64C5×#95—21no—
65C5×#95—21no—
66C5×#95—21no—
67#3963—22no—
68#3963—22no—
69#3963—42no—
70C22⋊#168—24no—
71C22⋊#168—22no—
72C22⋊#168D414yescomplemented
73C22×C22⋊C8—51no—
74#2847D511yescharacteristic
75C2×#327—52no—
76C22⋊#95—51no—
77#1294—52no—
78C24×C5—22no—
79C24×C5—22no—
80C24×C5—22no—
81C24×C5—22no—
82C24×C5—21no—
83C24×C5—41no—
84C24×C5—41no—
85C24×C5C22⋊C411yescharacteristic
86C24×C5C22⋊C411yescharacteristic
87C24×C5D812yes—
88C24×C5SD1612yes—
89C24×C5C2×D411yescharacteristic
90C2×C5×D4—22no—
91C2×C5×D4—41no—
92C2×C5×D4—41no—
93C2×C5×D4—42no—
94C2×C5×D4—42no—
95C2×C5×D4—42no—
96C2×C5×D4—42no—
97C2×C5×D4—42no—
98C2×C5×D4—42no—
99C2×C5×D4—42no—
100C2×C5×D4—41no—
101C2×C5×D4—41no—
102C2×C5×D4—42no—
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—21no—
106C22×C4×C5—21no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—22no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—21no—
118C22×C4×C5—41no—
119C22×C4×C5—41no—
120C22×C4×C5—41no—
121C22×C4×C5—41no—
122C22×C4×C5—41no—
123C22×C4×C5—41no—
124C22×C4×C5—41no—
125C22×C4×C5—48no—
126C22×C4×C5—48no—
127C22×C4×C5C22⋊C411yescharacteristic
128C22×C4×C5C22⋊C411yescharacteristic
129C22×C4×C5C22×C411yescharacteristic
130C22×C4×C5C2×D412yes—
131C22×C4×C5C2×D411yescharacteristic
132C22×C4×C5#4112yes—
133C22×Dic5—24no—
134C22×Dic5—42no—
135C22×Dic5—42no—
136C22×Dic5—42no—
137C22×Dic5—42no—
138C22×Dic5—42no—
139C22×Dic5—42no—
140C5×C4⋊C4—22no—
141C5×C4⋊C4—41no—
142C5×C4⋊C4—41no—
143C5×C22⋊C4—42no—
144C5×C22⋊C4—42no—
145C5×C22⋊C4—42no—
146C5×C22⋊C4—41no—
147C5×C22⋊C4—41no—
148C5×C22⋊C4—42no—
149C5×C22⋊C4—41no—
150C5×C22⋊C4—41no—
151C22⋊Dic5—48no—
152C2×#168—24no—
153C2×#168—48no—
154C2×#168—42no—
155C2×#168—42no—
156C24×C4C2×D511yescharacteristic
157C23×C8—101no—
158C2×#121Dic511yescharacteristic, complemented
159C2×#121Dic511yescharacteristic, complemented
160C2×#121C2×D511yescharacteristic
161C2×#120—21no—
162C22×C22⋊C4—101no—
163C2×C22⋊C8—52no—
164C2×C22⋊C8—104no—
165C2×C22⋊C8—101no—
166#386—21no—
167#386—21no—
168#386—21no—
169#380Dic511yescharacteristic, complemented
170#380Dic511yescharacteristic, complemented
171#380C2×D511yescharacteristic
172C2×#95—52no—
173C2×#95—101no—
174#327—104no—
175C23×C5—21no—
176C23×C5—21no—
177C23×C5—21no—
178C23×C5—21no—
179C23×C5—21no—
180C23×C5—21no—
181C23×C5—21no—
182C23×C5—21no—
183C23×C5—24no—
184C23×C5—24no—
185C23×C5—21no—
186C23×C5—21no—
187C23×C5—21no—
188C23×C5—21no—
189C23×C5—21no—
190C23×C5—21no—
191C23×C5—24no—
192C23×C5—24no—
193C23×C5—42no—
194C23×C5—42no—
195C23×C5—44no—
196C23×C5—44no—
197C23×C5—44no—
198C23×C5—44no—
199C23×C5—41no—
200C23×C5—41no—
201C23×C5—41no—
202C23×C5—41no—
203C23×C5—41no—
204C23×C5—41no—
205C23×C5—41no—
206C23×C5#9912yes—
207C23×C5#10212yes—
208C23×C5#10212yes—
209C23×C5#10412yes—
210C23×C5C2×C22⋊C411yescharacteristic
211C23×C5#12112yes—
212C23×C5C2×D811yescharacteristic
213C23×C5C2×SD1611yescharacteristic
214C23×C5#13512yes—
215C5×D4—82no—
216C5×D4—84no—
217C5×D4—84no—
218C5×D4—82no—
219C5×D4—82no—
220C2×C4×C5—24no—
221C2×C4×C5—24no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5—21no—
227C2×C4×C5—21no—
228C2×C4×C5—21no—
229C2×C4×C5—21no—
230C2×C4×C5—21no—
231C2×C4×C5—42no—
232C2×C4×C5—42no—
233C2×C4×C5—44no—
234C2×C4×C5—44no—
235C2×C4×C5—44no—
236C2×C4×C5—42no—
237C2×C4×C5—42no—
238C2×C4×C5—41no—
239C2×C4×C5—41no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5#9912yes—
248C2×C4×C5C4×D412yes—
249C2×C4×C5#12011yescharacteristic
250C2×C4×C5#12311yescharacteristic
251C2×Dic5—44no—
252C2×Dic5—44no—
253C2×Dic5—44no—
254C2×Dic5—82no—
255C2×Dic5—82no—
256C2×Dic5—82no—
257C2×Dic5—82no—
258#168—44no—
259C25C22⋊D511yescharacteristic
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4C2×Dic511yescharacteristic
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—42no—
267C23×C4—42no—
268C23×C4—102no—
269C23×C4C2×Dic511yescharacteristic
270C23×C4C2×Dic511yescharacteristic
271C23×C4C22⋊D511yescharacteristic
272C23×C4C22⋊D511yescharacteristic
273C23×C4C22⋊D511yescharacteristic
274C23×C4C22×D511yescharacteristic
275C22×C8—104no—
276C22×C8—102no—
277C22×C8—202no—
278C22×C8—202no—
279#121—22no—
280#121—22no—
281#121—42no—
282#121—42no—
283#120—42no—
284#120—42no—
285C2×C4⋊C4C2×Dic511yescharacteristic
286C2×C4⋊C4C2×Dic511yescharacteristic
287C2×C4⋊C4C2×Dic511yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—104no—
295C2×C22⋊C4—202no—
296C2×C22⋊C4—202no—
297C22⋊C8—54no—
298C22⋊C8—104no—
299C22⋊C8—102no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—102no—
304#95—102no—
305#95—202no—
306C22×C5—22no—
307C22×C5—22no—
308C22×C5—22no—
309C22×C5—22no—
310C22×C5—22no—
311C22×C5—21no—
312C22×C5—21no—
313C22×C5—22no—
314C22×C5—22no—
315C22×C5—22no—
316C22×C5—22no—
317C22×C5—21no—
318C22×C5—22no—
319C22×C5—21no—
320C22×C5—22no—
321C22×C5—22no—
322C22×C5—22no—
323C22×C5—48no—
324C22×C5—48no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5—41no—
328C22×C5—41no—
329C22×C5—81no—
330C22×C5—81no—
331C22×C5—81no—
332C22×C5—81no—
333C22×C5—81no—
334C22×C5—81no—
335C22×C5—81no—
336C22×C5#32714yescomplemented
337C22×C5#38611yescharacteristic
338C22×C5C2×#9911yescharacteristic
339C22×C5C2×#9911yescharacteristic
340C22×C5C2×#10211yescharacteristic
341C22×C5#41611yescharacteristic
342C22×C5C2×#10411yescharacteristic
343C22×C5#42111yescharacteristic
344C22×C5#46512yes—
345C22×C5#46612yes—
346C4×C5—21no—
347C4×C5—22no—
348C4×C5—21no—
349C4×C5—22no—
350C4×C5—41no—
351C4×C5—81no—
352C4×C5—81no—
353C4×C5—81no—
354Dic5—84no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—41no—
361C24—41no—
362C24#68612yes—
363C24#68712yes—
364C24C22⋊Dic511yescharacteristic
365C24C22⋊Dic511yescharacteristic
366C24C2×C22⋊D511yescharacteristic
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—42no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—104no—
405C22×C4—202no—
406C22×C4—202no—
407C22×C4—202no—
408C22×C4—202no—
409C22×C4—202no—
410C22×C4—202no—
411C22×C4C22⋊Dic511yescharacteristic
412C22×C4C22⋊Dic511yescharacteristic
413C22×C4D4×D512yes—
414C22×C4#71112yes—
415C22×C4C22×Dic511yescharacteristic
416C22×C4C2×C22⋊D511yescharacteristic
417C2×C8—104no—
418C2×C8—208no—
419C2×C8—202no—
420C2×C8—202no—
421C4⋊C4—22no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—42no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—208no—
433C2×C5—22no—
434C2×C5—24no—
435C2×C5—22no—
436C2×C5—42no—
437C2×C5—161no—
438C2×C5C2×#32711yescharacteristic
439C2×C5#150111yescharacteristic
440C2×C5#151611yescharacteristic
441C2×C5#151911yescharacteristic
442C2×C5#187111yescharacteristic
443C2×C5#189311yescharacteristic
444C2×C5#189811yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—24no—
454C23—24no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—24no—
462C23—21no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23#396412yes—
477C23#396412yes—
478C23#396612yes—
479C23#396912yes—
480C23C2×#68611yescharacteristic
481C23C2×#68711yescharacteristic
482C23#408512yes—
483C23#409612yes—
484C23C2×C22⋊Dic511yescharacteristic
485D4—82no—
486D4—84no—
487D4—82no—
488D4—82no—
489D4—84no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—204no—
518C2×C4—204no—
519C2×C4—204no—
520C2×C4—402no—
521C2×C4—402no—
522C2×C4—402no—
523C2×C4—402no—
524C2×C4#396612yes—
525C2×C4D4×Dic512yes—
526C2×C4#408111yescharacteristic
527C2×C4#408311yescharacteristic
528C8—204no—
529C5#911311yescharacteristic, complemented, Sylow
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—22no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#6475714yescomplemented
561C22#6532712yes—
562C22#6533012yes—
563C22C2×#396411yescharacteristic
564C22C2×#396611yescharacteristic
565C22C2×#396611yescharacteristic
566C22#6550911yescharacteristic
567C22#6552211yescharacteristic
568C22C2×#396911yescharacteristic
569C22#6552511yescharacteristic
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—404no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—161no—
584C2#1060246311yescharacteristic
585C2C2×#6475711yescharacteristic
586C2#1060246811yescharacteristic
587C2#1060247411yescharacteristic
588C2#1060250011yescharacteristic
589C2#1060250311yescharacteristic
590C2#1060250611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic5
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
Fitting, prime radicalC5×#2847C2
socleC23×C5C2×C22⋊C4
cosocleC23×C4×C5C23
hypercenter#2847D5
hypocenterC5#9113
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊#168C2semidirect
3. #10602920C2semidirect
4. C2×#64757C2semidirect
5. #10598115C2semidirect
7. C2×C5×#121C4semidirect
8. C2×C5×#121C4semidirect
16. C5×#380C4semidirect
17. C5×#380C4semidirect
72. C22⋊#168D4semidirect
158. C2×#121Dic5semidirect
159. C2×#121Dic5semidirect
169. #380Dic5semidirect
170. #380Dic5semidirect
336. C22×C5#327semidirect
529. C5#9113semidirect
560. C22#64757semidirect

Maximal subgroups

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

subgroupindex
C5×#28472
C22×C22⋊#1682
#106029202
C2×#647572
#105981152
#91135