Finite Group Databasebeta

← #12067009

Subgroups — #12067009

Order 1280.

1070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617887C211yesmaximal, characteristic
2#10610332C211yesmaximal, characteristic
3#10605100C211yesmaximal, characteristic
4C2×C5×#328C211yesmaximal, characteristic, complemented
5#10602483C211yesmaximal, characteristic, complemented
6#10600739C211yesmaximal, characteristic, complemented
7C2×#64672C211yesmaximal, characteristic, complemented
8#10599132C212yesmaximal, complemented
9D5×#328C212yesmaximal, complemented
10#10599107C212yesmaximal, complemented
11#10599102C212yesmaximal, complemented
12C2×C5×C22⋊Q8C2211yescharacteristic
13Q8×C22⋊D5C412yescomplemented
14C2×C4×C22⋊D5—21no—
15C2×C4×C22⋊D5C2211yescharacteristic
16C2×#4070C2211yescharacteristic
17#65970—22no—
18#65969C412yescomplemented
19#65968—21no—
20#65966C412yescomplemented
21D5×C22⋊Q8C2212yes—
22#65957—21no—
23#65956C2212yes—
24#65845—22no—
25#65845C412yescomplemented
26#65820—21no—
27#65817—21no—
28C2×C5×C22⋊C8C2211yescharacteristic
29C2×C5×#95C2211yescharacteristic
30#65501C2211yescharacteristic
31#65499—21no—
32C2×C22⋊#168C2211yescharacteristic
33C8×C22⋊D5—22no—
34#65014—22no—
35D5×C22⋊C8C2212yes—
36#65007C2212yes—
37#64957—21no—
38#64956C2211yescharacteristic
39D5×#95C2211yescharacteristic
40#64939C2211yescharacteristic
41#64938C2211yescharacteristic
42#64936—21no—
43C5×#328C2212yes—
44C5×#328C2212yes—
45#64758C2212yes—
46#64758C2212yes—
47#64687C2212yes—
48#64687C2212yes—
49#64672C2212yes—
50#64672C2212yes—
51#17614—51nomaximal, Sylow
52C22×C5×Q8C2×C411yescharacteristic, complemented
53C23×C4×C5—21no—
54C23×C4×C5C2311yescharacteristic
55C22×C22⋊D5D411yescharacteristic
56C2×Q8×D5C2×C412yescomplemented
57C22×C4×D5—21no—
58C22×C4×D5C2311yescharacteristic
59C22×Dic10C2×C411yescharacteristic, complemented
60C22×C5×C8—21no—
61C5×C22⋊Q8—21no—
62C5×C22⋊Q8—21no—
63C5×C22⋊Q8C2×C412yescomplemented
64C5×C22⋊Q8C2312yes—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4C2×C411yescharacteristic, complemented
67C2×C5×C22⋊C4—21no—
68C2×C22⋊Dic5—21no—
69C2×C22⋊Dic5—21no—
70C2×C22⋊Dic5D411yescharacteristic
71#4092—22no—
72#4092C2×C412yescomplemented
73Q8×Dic5C2×C412yescomplemented
74#4090—22no—
75#4090C2×C412yescomplemented
76C4×C22⋊D5—22no—
77C4×C22⋊D5—22no—
78C4×C22⋊D5—44no—
79C4×C22⋊D5C2×C412yes—
80C4×C22⋊D5C2×C412yes—
81C4×C22⋊D5D412yes—
82C4×C22⋊D5D412yes—
83C2×#685—21no—
84C2×#685D411yescharacteristic
85C2×C4⋊Dic5—21no—
86C2×C4⋊Dic5C2×C411yescharacteristic, complemented
87#4070—21no—
88#4070—21no—
89#4070—22no—
90#4070—22no—
91#4070C2×C412yescomplemented
92#4070C2312yes—
93C2×#683—21no—
94C2×#683—21no—
95C2×#683—21no—
96C2×#683—21no—
97C2×#683D411yescharacteristic
98C2×C4×Dic5—21no—
99C2×C4×Dic5C2311yescharacteristic
100C22×#168—21no—
101C2×C8×D5—22no—
102#4043—22no—
103#4043C2×C412yescomplemented
104#4042—22no—
105#4042—22no—
106#4042—22no—
107D5×C4⋊C4—22no—
108D5×C4⋊C4C2×C412yescomplemented
109#4035—22no—
110#4035—22no—
111#4035—22no—
112#4034—22no—
113#4034C2×C412yescomplemented
114#4033—22no—
115#4033C2×C412yescomplemented
116#4027—22no—
117#4027—22no—
118D5×C22⋊C4—22no—
119#4024—22no—
120#4024—22no—
121#4024—22no—
122#4024—22no—
123#4023—22no—
124C5×C22⋊C8—21no—
125C5×C22⋊C8C2312yes—
126C5×#95—21no—
127C5×#95C2311yescharacteristic
128C5×#95C2311yescharacteristic
129#3963—21no—
130#3963—21no—
131#3963C2311yescharacteristic
132#3963C2311yescharacteristic
133C22⋊#168—21no—
134C22⋊#168C2312yes—
135#3952—22no—
136#3946—22no—
137C8×Dic5—22no—
138#3501—51no—
139#2972—51no—
140#2349—51no—
141#1520—51no—
142#1520—52no—
143#1520—51no—
144#1518—52no—
145#1503—52no—
146C2×#328—51no—
147C2×#328—52no—
148C2×#328D511yescharacteristic, complemented
149C24×C5C2×D411yescharacteristic
150C23×D5C2×D411yescharacteristic
151C2×C5×Q8—21no—
152C2×C5×Q8—81no—
153C2×C5×Q8C22×C412yes—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—42no—
159C22×C4×C5—42no—
160C22×C4×C5—81no—
161C22×C4×C5—81no—
162C22×C4×C5C22×C411yescharacteristic
163C22×C4×C5C22×C411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2411yescharacteristic
168C2×C22⋊D5C22⋊C412yes—
169C2×C22⋊D5C22⋊C412yes—
170C2×C22⋊D5SD1614yes—
171C2×C22⋊D5Q1614yes—
172C22×Dic5—21no—
173C22×Dic5—21no—
174C22×Dic5—21no—
175C22×Dic5—21no—
176C22×Dic5C2×D411yescharacteristic
177C22×Dic5C2×D411yescharacteristic
178C22×Dic5C2×D411yescharacteristic
179Q8×D5—82no—
180C2×C4×D5—22no—
181C2×C4×D5—22no—
182C2×C4×D5—42no—
183C2×C4×D5—42no—
184C2×C4×D5—81no—
185C2×C4×D5—81no—
186C2×C4×D5C22×C412yes—
187C2×C4×D5C2×D412yes—
188C2×Dic10—21no—
189C2×Dic10—22no—
190C2×Dic10—22no—
191C2×Dic10—81no—
192C2×Dic10C22×C412yes—
193C2×C5×C8—22no—
194C2×C5×C8—81no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—22no—
198C5×C4⋊C4C22×C412yes—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C22⋊Dic5—22no—
202C22⋊Dic5—22no—
203C22⋊Dic5—42no—
204C22⋊Dic5C22⋊C412yes—
205C22⋊Dic5C22⋊C412yes—
206#685—22no—
207#685—22no—
208#685—42no—
209#685C22⋊C412yes—
210#685C22⋊C412yes—
211C4⋊Dic5—21no—
212C4⋊Dic5—22no—
213C4⋊Dic5—22no—
214C4⋊Dic5—81no—
215C4⋊Dic5C22×C412yes—
216#683—22no—
217#683—22no—
218#683—22no—
219#683—22no—
220#683—22no—
221#683—22no—
222#683—22no—
223#683—22no—
224#683—42no—
225#683—42no—
226#683—81no—
227#683C22⋊C412yes—
228#683C22⋊C412yes—
229C4×Dic5—22no—
230C4×Dic5—22no—
231C4×Dic5—42no—
232C4×Dic5C22×C412yes—
233C4×Dic5C2×D412yes—
234C2×#168—22no—
235C2×#168—81no—
236C8×D5—82no—
237#554—52no—
238#554—52no—
239#554—102no—
240#554—102no—
241D4×Q8—52no—
242D4×Q8—52no—
243#544—101no—
244#543—101no—
245#543—101no—
246#533—52no—
247C2×C22⋊Q8—51no—
248C2×C22⋊Q8—52no—
249C2×C22⋊Q8C2×D511yescharacteristic
250#518—101no—
251C2×C4×D4—51no—
252C2×C4×D4—101no—
253C8×D4—102no—
254C8×D4—102no—
255#431—52no—
256C2×C22⋊C8—51no—
257C2×C22⋊C8—52no—
258C2×C22⋊C8C2×D511yescharacteristic
259#389—101no—
260#385—101no—
261#385—101no—
262#382—51no—
263#380—51no—
264#380—51no—
265C4×C4⋊C4—51no—
266C2×#95—51no—
267C2×#95C2×D511yescharacteristic
268#328—52no—
269#328—52no—
270#328—52no—
271#328—52no—
272#328—52no—
273#328—52no—
274#328C2×D512yes—
275#328C2×D512yes—
276C23×C5—41no—
277C23×C5—41no—
278C23×C5C2×C22⋊C411yescharacteristic
279C23×C5C2×C22⋊C411yescharacteristic
280C23×C5C2×SD1612yes—
281C23×C5C2×Q1612yes—
282C23×C5C22×D411yescharacteristic
283C22×D5—41no—
284C22×D5—41no—
285C22×D5C2×C22⋊C412yes—
286C22×D5C2×SD1612yes—
287C22×D5C2×Q1612yes—
288C5×Q8—42no—
289C5×Q8—81no—
290C2×C4×C5—22no—
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×C5—41no—
304C2×C4×C5—81no—
305C2×C4×C5—81no—
306C2×C4×C5—84no—
307C2×C4×C5C2×C22⋊C411yescharacteristic
308C2×C4×C5C2×C22⋊C411yescharacteristic
309C2×C4×C5C23×C411yescharacteristic
310C2×C4×C5C22×D411yescharacteristic
311C22⋊D5#10318yes—
312C22⋊D5#10318yes—
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—81no—
324C2×Dic5—81no—
325C2×Dic5C2×C22⋊C412yes—
326C2×Dic5C2×C22⋊C412yes—
327C2×Dic5C2×C22⋊C412yes—
328C2×Dic5C2×SD1612yes—
329C2×Dic5C2×Q1612yes—
330C4×D5—42no—
331C4×D5—42no—
332C4×D5—82no—
333C4×D5—82no—
334C4×D5—84no—
335Dic10—42no—
336Dic10—82no—
337Dic10—82no—
338Dic10—81no—
339C5×C8—42no—
340#168—42no—
341C22×Q8—52no—
342C22×Q8—51no—
343C22×Q8C4×D511yescharacteristic, complemented
344C22×D4—51no—
345C23×C4—21no—
346C23×C4—51no—
347C23×C4—101no—
348C23×C4C22×D511yescharacteristic
349C22×C8—21no—
350C22×C8—102no—
351C22×C8—101no—
352C4⋊Q8—52no—
353C4⋊Q8—52no—
354C4⋊Q8—102no—
355C4⋊Q8—102no—
356#125—102no—
357#125—102no—
358#125—102no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8—52no—
362C22⋊Q8—52no—
363C22⋊Q8—52no—
364C22⋊Q8—52no—
365C22⋊Q8—102no—
366C22⋊Q8—102no—
367C22⋊Q8—102no—
368C22⋊Q8—102no—
369C22⋊Q8—102no—
370C22⋊Q8—102no—
371C22⋊Q8—102no—
372C22⋊Q8—102no—
373C22⋊Q8—102no—
374C22⋊Q8—102no—
375C22⋊Q8—102no—
376C22⋊Q8—101no—
377C22⋊Q8—101no—
378C22⋊Q8C4×D512yescomplemented
379C22⋊Q8C22×D512yes—
380C4×Q8—52no—
381C4×Q8—52no—
382C4×Q8—102no—
383C4×D4—52no—
384C4×D4—52no—
385C4×D4—52no—
386C4×D4—52no—
387C4×D4—102no—
388C4×D4—102no—
389C4×D4—102no—
390C4×D4—102no—
391C4×D4—204no—
392#117—102no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—51no—
395C2×C4⋊C4—51no—
396C2×C4⋊C4—52no—
397C2×C4⋊C4—102no—
398C2×C4⋊C4—101no—
399C2×C4⋊C4—101no—
400C2×C4⋊C4—101no—
401C2×C4⋊C4—101no—
402C2×C4⋊C4—101no—
403C2×C4⋊C4C4×D511yescharacteristic, complemented
404C2×C22⋊C4—21no—
405C2×C22⋊C4—51no—
406C2×C22⋊C4—51no—
407C2×C22⋊C4—102no—
408C2×C22⋊C4—101no—
409C2×C22⋊C4—101no—
410C2×C22⋊C4—101no—
411C2×C42—51no—
412C2×C42—101no—
413C4⋊C8—102no—
414C22⋊C8—21no—
415C22⋊C8—52no—
416C22⋊C8—102no—
417C22⋊C8—101no—
418C22⋊C8C22×D512yes—
419C4×C8—102no—
420#95—21no—
421#95—51no—
422#95—51no—
423#95—101no—
424#95—101no—
425#95C22×D511yescharacteristic
426#95C22×D511yescharacteristic
427C22×C5—21no—
428C22×C5—21no—
429C22×C5—21no—
430C22×C5—21no—
431C22×C5—21no—
432C22×C5—21no—
433C22×C5—42no—
434C22×C5—42no—
435C22×C5C2×#10312yes—
436C22×C5C2×#10312yes—
437C22×C5C22×C22⋊C411yescharacteristic
438C22×C5C22×SD1611yescharacteristic
439C22×C5C22×Q1611yescharacteristic
440C2×D5—22no—
441C2×D5—22no—
442C2×D5—42no—
443C2×D5—42no—
444C2×D5C2×#10314yes—
445C4×C5—21no—
446C4×C5—21no—
447C4×C5—21no—
448C4×C5—21no—
449C4×C5—42no—
450C4×C5—42no—
451C4×C5—42no—
452C4×C5—81no—
453Dic5—22no—
454Dic5—22no—
455Dic5—42no—
456Dic5—42no—
457Dic5—81no—
458Dic5—81no—
459Dic5C2×#10314yes—
460C24—51no—
461C24D4×D511yescharacteristic
462C2×Q8—21no—
463C2×Q8—52no—
464C2×Q8—81no—
465C2×Q8—102no—
466C2×Q8—102no—
467C2×Q8—101no—
468C2×Q8—401no—
469C2×Q8—402no—
470C2×Q8C2×C4×D512yes—
471C2×D4—52no—
472C2×D4—52no—
473C2×D4—54no—
474C2×D4—54no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—51no—
482C22×C4—51no—
483C22×C4—51no—
484C22×C4—52no—
485C22×C4—52no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—101no—
491C22×C4—101no—
492C22×C4—101no—
493C22×C4—101no—
494C22×C4—202no—
495C22×C4—202no—
496C22×C4—401no—
497C22×C4—401no—
498C22×C4C2×C4×D511yescharacteristic
499C22×C4C2×C4×D511yescharacteristic
500C22×C4D4×D511yescharacteristic
501C22×C4D4×D511yescharacteristic
502C22×C4D4×D511yescharacteristic
503C22×C4C23×D511yescharacteristic
504C2×C8—22no—
505C2×C8—81no—
506C2×C8—102no—
507C2×C8—402no—
508C2×C8—401no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—52no—
513C4⋊C4—52no—
514C4⋊C4—52no—
515C4⋊C4—102no—
516C4⋊C4—102no—
517C4⋊C4—102no—
518C4⋊C4—102no—
519C4⋊C4—102no—
520C4⋊C4—102no—
521C4⋊C4—102no—
522C4⋊C4—102no—
523C4⋊C4—102no—
524C4⋊C4—101no—
525C4⋊C4—102no—
526C4⋊C4—202no—
527C4⋊C4—202no—
528C4⋊C4—401no—
529C4⋊C4—401no—
530C4⋊C4C2×C4×D512yes—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—52no—
534C22⋊C4—52no—
535C22⋊C4—52no—
536C22⋊C4—52no—
537C22⋊C4—102no—
538C22⋊C4—102no—
539C22⋊C4—102no—
540C22⋊C4—102no—
541C22⋊C4—202no—
542C22⋊C4—202no—
543C42—52no—
544C42—52no—
545C42—102no—
546C42—102no—
547C42—202no—
548C2×C5—21no—
549C2×C5—21no—
550C2×C5—22no—
551C2×C5—22no—
552C2×C5#288311yescharacteristic
553C2×C5C22×#10311yescharacteristic
554C2×C5#290111yescharacteristic
555D5—24no—
556C23—41no—
557C23—41no—
558C23—52no—
559C23—52no—
560C23—52no—
561C23—201no—
562C23—201no—
563C23D5×C22⋊C411yescharacteristic
564C23D5×C22⋊C411yescharacteristic
565C23D5×SD1612yes—
566C23D5×Q1612yes—
567C23C2×D4×D511yescharacteristic
568Q8—42no—
569Q8—81no—
570Q8—202no—
571Q8—401no—
572Q8—402no—
573Q8—402no—
574D4—58no—
575D4—58no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—52no—
591C2×C4—52no—
592C2×C4—52no—
593C2×C4—52no—
594C2×C4—52no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—84no—
598C2×C4—102no—
599C2×C4—102no—
600C2×C4—202no—
601C2×C4—202no—
602C2×C4—201no—
603C2×C4—201no—
604C2×C4—201no—
605C2×C4—201no—
606C2×C4—201no—
607C2×C4—201no—
608C2×C4—201no—
609C2×C4—201no—
610C2×C4—402no—
611C2×C4—402no—
612C2×C4—404no—
613C2×C4—401no—
614C2×C4—401no—
615C2×C4D5×C22⋊C411yescharacteristic
616C2×C4D5×C22⋊C411yescharacteristic
617C2×C4C22×C4×D511yescharacteristic
618C2×C4C2×D4×D511yescharacteristic
619C8—42no—
620C8—202no—
621C5#1761411yescharacteristic, complemented, Sylow
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—42no—
630C22—54no—
631C22—102no—
632C22—102no—
633C22—202no—
634C22—202no—
635C22D5×#10312yes—
636C22D5×#10312yes—
637C22C2×D5×C22⋊C411yescharacteristic
638C22C2×D5×SD1611yescharacteristic
639C22C2×D5×Q1611yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—54no—
648C4—81no—
649C4—102no—
650C4—102no—
651C4—202no—
652C4—202no—
653C4—401no—
654C4—401no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—104no—
660C2#1061041911yescharacteristic
661C2C2×D5×#10311yescharacteristic
662C2#1061071411yescharacteristic
subgroupG/H
centerC22C2×D5×C22⋊C4
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#328C2
socleC22×C5C22×C22⋊C4
prime radicalC22×C22⋊D5D4
cosocleC22×C4×C5C24
hypercenterC2×#328D5
hypocenterC5#17614
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
4. C2×C5×#328C2semidirect
5. #10602483C2semidirect
6. #10600739C2semidirect
7. C2×#64672C2semidirect
8. #10599132C2semidirect
9. D5×#328C2semidirect
10. #10599107C2semidirect
11. #10599102C2semidirect
13. Q8×C22⋊D5C4semidirect
18. #65969C4semidirect
20. #65966C4semidirect
25. #65845C4semidirect
52. C22×C5×Q8C2×C4semidirect
56. C2×Q8×D5C2×C4semidirect
59. C22×Dic10C2×C4semidirect
63. C5×C22⋊Q8C2×C4semidirect
66. C2×C5×C4⋊C4C2×C4semidirect
72. #4092C2×C4semidirect
73. Q8×Dic5C2×C4semidirect
75. #4090C2×C4semidirect
86. C2×C4⋊Dic5C2×C4semidirect
91. #4070C2×C4semidirect
103. #4043C2×C4semidirect
108. D5×C4⋊C4C2×C4semidirect
113. #4034C2×C4semidirect
115. #4033C2×C4semidirect
148. C2×#328D5semidirect
343. C22×Q8C4×D5semidirect
378. C22⋊Q8C4×D5semidirect
403. C2×C4⋊C4C4×D5semidirect
621. C5#17614semidirect

Maximal subgroups

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

subgroupindex
#106178872
#106103322
#106051002
C2×C5×#3282
#106024832
#106007392
C2×#646722
#105991322
D5×#3282
#105991072
#105991022
#176145