Finite Group Databasebeta

← #12102768

Subgroups — #12102768

Order 1280.

1082 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65170C211yesmaximal, characteristic
2C2×#65150C211yesmaximal, characteristic
3C2×#65126C211yesmaximal, characteristic
4#10605879C211yesmaximal, characteristic
5C4×D5×C4⋊C4C211yesmaximal, characteristic
6C5×#1949C211yesmaximal, characteristic, complemented
7#10601408C211yesmaximal, characteristic, complemented
8#10601356C211yesmaximal, characteristic, complemented
9#10601248C211yesmaximal, characteristic, complemented
10#10600333C211yesmaximal, characteristic, complemented
11#10600261C211yesmaximal, characteristic, complemented
12#10599660C211yesmaximal, characteristic
13#10599660C211yesmaximal, characteristic
14#10599589C211yesmaximal, characteristic, complemented
15#10599588C211yesmaximal, characteristic, complemented
16C22×#676C2211yescharacteristic
17C22×#676C2211yescharacteristic
18C2×#4038C2211yescharacteristic
19C2×#4038C2211yescharacteristic
20C2×D5×C4⋊C4C2211yescharacteristic
21C2×D5×C4⋊C4C2211yescharacteristic
22C2×C42×D5—21no—
23C2×C42×D5C2211yescharacteristic
24C2×C5×#107C2211yescharacteristic
25C2×C5×#106C2211yescharacteristic
26C2×C5×C4⋊C8C2211yescharacteristic
27C5×#382C2211yescharacteristic
28C4×C5×C4⋊C4C2211yescharacteristic
29C2×#3950C2211yescharacteristic
30C2×#3949C2211yescharacteristic
31C2×#3946C2211yescharacteristic
32C2×#3946C2211yescharacteristic
33#65265C2211yescharacteristic
34#65265C2211yescharacteristic
35C4⋊C4×Dic5C2211yescharacteristic
36C4⋊C4×Dic5C2211yescharacteristic
37C2×#3940C2211yescharacteristic
38C2×#3940C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40C2×#3939C2211yescharacteristic
41#65225C2211yescharacteristic
42C4×C4⋊Dic5C2211yescharacteristic
43C4×#683—21no—
44C2×C4⋊#168C2211yescharacteristic
45#65170—24no—
46#65150—24no—
47#65126C418yescomplemented
48#64956—21no—
49#64956—21no—
50D5×#95—21no—
51#64938—21no—
52#64938—21no—
53#64938—21no—
54C5×#340C2211yescharacteristic
55C5×#340C2211yescharacteristic
56#64771C2211yescharacteristic
57#64771C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#64749C2211yescharacteristic
61#64749C2211yescharacteristic
62#21170—51nomaximal, Sylow
63C22×C4×D5—21no—
64C22×C4×D5—21no—
65C22×C4×D5D411yescharacteristic
66C22×C4×D5D411yescharacteristic
67C22×C4×D5C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C42×C5—21no—
75C2×C42×C5C2311yescharacteristic
76C2×#685—21no—
77C2×#685—21no—
78C2×C4⋊Dic5C2311yescharacteristic
79C2×C4⋊Dic5C2311yescharacteristic
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×#683—21no—
83C2×#683—21no—
84C2×#683—21no—
85C2×#683—21no—
86C2×#683—21no—
87C2×#683—21no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5D411yescharacteristic
94C2×C4×Dic5D411yescharacteristic
95C2×C4×Dic5C2311yescharacteristic
96C22×#168C2311yescharacteristic
97C22×#168C2311yescharacteristic
98C2×#676—22no—
99C2×#676—22no—
100C2×#676—22no—
101C2×#676—22no—
102C2×#676C2×C414yes—
103C2×#676C2×C414yes—
104#4038—22no—
105#4038—22no—
106#4038—22no—
107#4038—22no—
108D5×C4⋊C4—22no—
109D5×C4⋊C4—22no—
110D5×C4⋊C4D414yes—
111D5×C4⋊C4Q814yes—
112C42×D5—44no—
113C42×D5C2×C414yes—
114C42×D5Q814yes—
115C5×#107—22no—
116C5×#106—22no—
117C5×C4⋊C8C2×C414yescomplemented
118C5×#95—21no—
119C5×#95—21no—
120C5×#95—21no—
121#3963—21no—
122#3963—21no—
123#3963—21no—
124#3963—21no—
125#3963—21no—
126#3950—22no—
127#3949—22no—
128#3946C2×C414yescomplemented
129#3946C2×C414yescomplemented
130#3940—22no—
131#3940—22no—
132#3939—22no—
133#3939—22no—
134C4⋊#168C2×C414yescomplemented
135C2×#428—51no—
136C2×#428—51no—
137C2×#423—51no—
138#2285—51no—
139C2×C4×C4⋊C4—51no—
140#1949—51no—
141#1949—51no—
142#1949—51no—
143#1949D511yescharacteristic, complemented
144#1948—51no—
145#1948—51no—
146#1947—51no—
147#1947—51no—
148#1736—51no—
149#1736—51no—
150C23×D5C2×D411yescharacteristic
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2411yescharacteristic
158C22×Dic5—21no—
159C22×Dic5—21no—
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5C2×D411yescharacteristic
163C22×Dic5C2×D411yescharacteristic
164C22×Dic5C2×D411yescharacteristic
165C2×C4×D5—22no—
166C2×C4×D5—22no—
167C2×C4×D5—24no—
168C2×C4×D5—24no—
169C2×C4×D5—41no—
170C2×C4×D5—41no—
171C2×C4×D5—42no—
172C2×C4×D5—42no—
173C2×C4×D5—42no—
174C2×C4×D5—42no—
175C2×C4×D5C4⋊C414yes—
176C2×C4×D5C4⋊C414yes—
177C2×C4×D5C22×C412yes—
178C2×C4×D5C2×D412yes—
179C2×C4×D5C2×Q812yes—
180C2×C4×D5C2×Q812yes—
181C2×C4×D5#4112yes—
182C2×C4×D5#4112yes—
183C2×C5×C8—21no—
184C2×C5×C8—21no—
185C2×C5×C8—21no—
186C2×C5×C8—21no—
187C2×C5×C8C22×C412yes—
188C2×C5×C8C22×C412yes—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4C2×D412yes—
196C5×C4⋊C4C2×Q812yes—
197C42×C5—42no—
198C42×C5C22×C412yes—
199C42×C5C2×Q812yes—
200#685—42no—
201#685—42no—
202C4⋊Dic5—21no—
203C4⋊Dic5—21no—
204C4⋊Dic5—21no—
205C4⋊Dic5—21no—
206C4⋊Dic5—21no—
207C4⋊Dic5—21no—
208C4⋊Dic5C2×D412yes—
209C4⋊Dic5C2×Q812yes—
210#683—24no—
211#683—24no—
212#683—42no—
213#683—42no—
214#683—42no—
215#683—42no—
216C4×Dic5—42no—
217C4×Dic5—42no—
218C4×Dic5—42no—
219C4×Dic5—42no—
220C4×Dic5—42no—
221C4×Dic5C4⋊C414yes—
222C4×Dic5C4⋊C414yes—
223C4×Dic5C22×C412yes—
224C4×Dic5C2×Q812yes—
225C2×#168—21no—
226C2×#168—21no—
227C2×#168—21no—
228C2×#168—21no—
229C2×#168C22×C412yes—
230C2×#168C22×C412yes—
231#676—28no—
232#676—28no—
233C22×M16—51no—
234C22×M16—51no—
235C2×#117—51no—
236C2×#117—51no—
237C22×C4⋊C4—51no—
238C22×C4⋊C4—51no—
239C22×C42—51no—
240C22×C42—101no—
241#428—104no—
242#428—104no—
243C2×#107—51no—
244C2×#107—51no—
245C2×#107—51no—
246C2×#107C2×D511yescharacteristic
247C2×#106—51no—
248C2×#106—51no—
249C2×#106—51no—
250C2×#106C2×D511yescharacteristic
251#423—58no—
252C2×C4⋊C8—51no—
253C2×C4⋊C8—51no—
254C2×C4⋊C8—51no—
255C2×C4⋊C8C2×D511yescharacteristic
256#384—51no—
257#384—51no—
258#382—51no—
259#382C2×D511yescharacteristic
260#380—101no—
261#380—101no—
262C4×C4⋊C4—51no—
263C4×C4⋊C4—51no—
264C4×C4⋊C4—51no—
265C4×C4⋊C4—101no—
266C4×C4⋊C4—101no—
267C4×C4⋊C4—101no—
268C4×C4⋊C4—101no—
269C4×C4⋊C4C2×D511yescharacteristic
270C2×#95—101no—
271#340—51no—
272#340—51no—
273#340—51no—
274#340—51no—
275#340—51no—
276#340—51no—
277#340C2×D511yescharacteristic
278#340C2×D511yescharacteristic
279C23×C5C22×D411yescharacteristic
280C22×D5—21no—
281C22×D5—21no—
282C22×D5—21no—
283C22×D5—21no—
284C22×D5C2×C4⋊C412yes—
285C22×D5#12112yes—
286C22×D5C22⋊Q812yes—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—41no—
303C2×C4×C5C2×C4⋊C412yes—
304C2×C4×C5C2×C4⋊C412yes—
305C2×C4×C5C23×C411yescharacteristic
306C2×C4×C5C22×D411yescharacteristic
307C2×C4×C5C22×Q811yescharacteristic
308C2×C4×C5C22×Q811yescharacteristic
309C2×C4×C5C2×#4111yescharacteristic
310C2×C4×C5C2×#4111yescharacteristic
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—22no—
314C2×Dic5—22no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321C2×Dic5—41no—
322C2×Dic5—41no—
323C2×Dic5—41no—
324C2×Dic5—41no—
325C2×Dic5C2×C4⋊C412yes—
326C2×Dic5C2×C4⋊C412yes—
327C2×Dic5C2×C4⋊C412yes—
328C2×Dic5#12112yes—
329C2×Dic5C22⋊Q812yes—
330C2×Dic5AGL(1,ℤ/8)12yes—
331C2×Dic5#13712yes—
332C4×D5—28no—
333C4×D5—44no—
334C4×D5—48no—
335C4×D5—48no—
336C4×D5C4×D414yes—
337C4×D5C4×Q814yes—
338C4×D5#12514yes—
339C4×D5C4⋊Q814yes—
340C5×C8—22no—
341C5×C8—22no—
342#168—22no—
343#168—22no—
344C23×C4—51no—
345C23×C4—51no—
346C23×C4—51no—
347C23×C4—101no—
348C23×C4—101no—
349C2×M16—54no—
350C2×M16—54no—
351C2×M16—102no—
352C2×M16—102no—
353C2×M16—102no—
354C2×M16—102no—
355C22×C8—51no—
356C22×C8—51no—
357C22×C8C22×D511yescharacteristic
358C22×C8C22×D511yescharacteristic
359#117—102no—
360#117—102no—
361#117—102no—
362#117—102no—
363C2×C4⋊C4—51no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—54no—
366C2×C4⋊C4—54no—
367C2×C4⋊C4—51no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—102no—
370C2×C4⋊C4—102no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—101no—
373C2×C4⋊C4—101no—
374C2×C4⋊C4—101no—
375C2×C4⋊C4—101no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4C22×D511yescharacteristic
378C2×C4⋊C4C22×D511yescharacteristic
379C2×C4⋊C4C22×D511yescharacteristic
380C2×C4⋊C4C22×D511yescharacteristic
381C2×C22⋊C4—101no—
382C2×C22⋊C4—101no—
383C2×C42—21no—
384C2×C42—51no—
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—54no—
388C2×C42—54no—
389C2×C42—101no—
390C2×C42—101no—
391C2×C42—101no—
392C2×C42—101no—
393C2×C42—101no—
394C2×C42—204no—
395C2×C42C22×D511yescharacteristic
396#107—22no—
397#107—102no—
398#107—102no—
399#107—102no—
400#106—22no—
401#106—102no—
402#106—102no—
403#106—102no—
404C4⋊C8—54no—
405C4⋊C8—54no—
406C4⋊C8—54no—
407C4⋊C8C4×D514yescomplemented
408#95—21no—
409#95—21no—
410#95—21no—
411#95—101no—
412#95—101no—
413#95—101no—
414#95—101no—
415#95—101no—
416C22×C5C22×C4⋊C411yescharacteristic
417C22×C5C2×#12111yescharacteristic
418C22×C5C2×C22⋊Q811yescharacteristic
419C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
420C22×C5C2×#13711yescharacteristic
421C22×C5#57511yescharacteristic
422C22×C5#57511yescharacteristic
423C2×D5—22no—
424C2×D5—22no—
425C2×D5—22no—
426C2×D5—22no—
427C2×D5—22no—
428C2×D5—22no—
429C2×D5#38914yes—
430C4×C5—22no—
431C4×C5—22no—
432C4×C5—42no—
433C4×C5—42no—
434C4×C5—81no—
435C4×C5—81no—
436C4×C5C2×C4×D411yescharacteristic
437C4×C5C2×C4×Q811yescharacteristic
438C4×C5C2×#12511yescharacteristic
439C4×C5C2×C4⋊Q811yescharacteristic
440Dic5—22no—
441Dic5—42no—
442Dic5—42no—
443Dic5—81no—
444Dic5—81no—
445Dic5#38914yes—
446Dic5#42814yescomplemented
447C24—51no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—51no—
453C22×C4—51no—
454C22×C4—52no—
455C22×C4—52no—
456C22×C4—52no—
457C22×C4—52no—
458C22×C4—52no—
459C22×C4—52no—
460C22×C4—51no—
461C22×C4—54no—
462C22×C4—54no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—104no—
466C22×C4—104no—
467C22×C4—101no—
468C22×C4—101no—
469C22×C4—101no—
470C22×C4—101no—
471C22×C4—201no—
472C22×C4—201no—
473C22×C4—202no—
474C22×C4—202no—
475C22×C4—202no—
476C22×C4—202no—
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4C23×D511yescharacteristic
480M16—108no—
481M16—108no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—52no—
487C2×C8—52no—
488C2×C8—101no—
489C2×C8—101no—
490C2×C8—101no—
491C2×C8—101no—
492C2×C8C2×C4×D512yes—
493C2×C8C2×C4×D512yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—52no—
501C4⋊C4—52no—
502C4⋊C4—104no—
503C4⋊C4—104no—
504C4⋊C4—101no—
505C4⋊C4—101no—
506C4⋊C4—101no—
507C4⋊C4—101no—
508C4⋊C4—101no—
509C4⋊C4—101no—
510C4⋊C4—202no—
511C4⋊C4—202no—
512C4⋊C4—202no—
513C4⋊C4—202no—
514C4⋊C4D4×D512yes—
515C4⋊C4Q8×D512yes—
516C22⋊C4—202no—
517C22⋊C4—202no—
518C42—42no—
519C42—52no—
520C42—52no—
521C42—54no—
522C42—54no—
523C42—202no—
524C42—202no—
525C42—202no—
526C42—202no—
527C42—202no—
528C42C2×C4×D512yes—
529C42Q8×D512yes—
530C2×C5C2×#38911yescharacteristic
531C2×C5C2×#42811yescharacteristic
532C2×C5#291111yescharacteristic
533C2×C5#303611yescharacteristic
534C2×C5#303711yescharacteristic
535C2×C5#307711yescharacteristic
536C2×C5#307811yescharacteristic
537D5—24no—
538C23—52no—
539C23—52no—
540C23—52no—
541C23—101no—
542C23—101no—
543C23—101no—
544C23—101no—
545C23C2×D4×D511yescharacteristic
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—52no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—52no—
568C2×C4—54no—
569C2×C4—54no—
570C2×C4—54no—
571C2×C4—54no—
572C2×C4—52no—
573C2×C4—108no—
574C2×C4—101no—
575C2×C4—101no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—204no—
579C2×C4—208no—
580C2×C4—208no—
581C2×C4—201no—
582C2×C4—201no—
583C2×C4—201no—
584C2×C4—201no—
585C2×C4—201no—
586C2×C4—201no—
587C2×C4—201no—
588C2×C4—201no—
589C2×C4—201no—
590C2×C4—201no—
591C2×C4D5×C4⋊C412yes—
592C2×C4D5×C4⋊C412yes—
593C2×C4C22×C4×D511yescharacteristic
594C2×C4C2×D4×D511yescharacteristic
595C2×C4C2×Q8×D511yescharacteristic
596C2×C4C2×Q8×D511yescharacteristic
597C2×C4D5×#4111yescharacteristic
598C2×C4D5×#4111yescharacteristic
599C8—22no—
600C8—22no—
601C8—102no—
602C8—102no—
603C5#2117011yescharacteristic, complemented, Sylow
604C22—54no—
605C22—102no—
606C22—102no—
607C22—102no—
608C22—102no—
609C22—102no—
610C22—102no—
611C22C2×D5×C4⋊C411yescharacteristic
612C22D5×#12111yescharacteristic
613C22D5×C22⋊Q811yescharacteristic
614C22#6610211yescharacteristic
615C22#6610211yescharacteristic
616C22D5×AGL(1,ℤ/8)11yescharacteristic
617C22D5×#13711yescharacteristic
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—54no—
623C4—54no—
624C4—81no—
625C4—81no—
626C4—102no—
627C4—202no—
628C4—202no—
629C4—401no—
630C4—401no—
631C4C4×D4×D511yescharacteristic
632C4C4×Q8×D511yescharacteristic
633C4D5×#12511yescharacteristic
634C4D5×C4⋊Q811yescharacteristic
635C2—104no—
636C2D5×#38911yescharacteristic
637C2#1061090811yescharacteristic
638C2D5×#42811yescharacteristic
639C2#1061226511yescharacteristic
640C2#1061232911yescharacteristic
641C2#1061289711yescharacteristic
642C2#1061296111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1949C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1949D5
hypocenterC5#21170
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
6. C5×#1949C2semidirect
7. #10601408C2semidirect
8. #10601356C2semidirect
9. #10601248C2semidirect
10. #10600333C2semidirect
11. #10600261C2semidirect
14. #10599589C2semidirect
15. #10599588C2semidirect
47. #65126C4semidirect
117. C5×C4⋊C8C2×C4semidirect
128. #3946C2×C4semidirect
129. #3946C2×C4semidirect
134. C4⋊#168C2×C4semidirect
143. #1949D5semidirect
407. C4⋊C8C4×D5semidirect
446. Dic5#428semidirect
603. C5#21170semidirect

Maximal subgroups

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

subgroupindex
C2×#651702
C2×#651502
C2×#651262
#106058792
C4×D5×C4⋊C42
C5×#19492
#106014082
#106013562
#106012482
#106003332
#106002612
#105996602
#105995892
#105995882
#211705