Finite Group Databasebeta

← #12259625

Subgroups — #12259625

Order 1280.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#117C211yesmaximal, characteristic
2C2×#65850C211yesmaximal, characteristic, complemented
3C22×#4043C211yesmaximal, characteristic
4C5×#2316C211yesmaximal, characteristic, complemented
5#10613998C211yesmaximal, characteristic
6#10613961C211yesmaximal, characteristic, complemented
7#10613875C211yesmaximal, characteristic, complemented
8#10607936C214yesmaximal, complemented
9#10607928C214yesmaximal, complemented
10#10607715C214yesmaximal, complemented
11#10607696C214yesmaximal, complemented
12#10605740C214yesmaximal
13#10605609C214yesmaximal, complemented
14C23×C4×D5C2211yescharacteristic
15C23×Dic10C2211yescharacteristic
16C2×C5×#117C2211yescharacteristic
17C2×C5×#117C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×#685C2212yes—
20C2×#4072C2211yescharacteristic
21C2×#4072C2211yescharacteristic
22C22×C4⋊Dic5C2212yes—
23D5×#117—28no—
24#65850—28no—
25C2×#4043C4116yescomplemented
26C2×#4043C2212yes—
27C2×#4043C2212yes—
28C2×#4043C2212yes—
29C2×#4043C2212yes—
30C2×#4038C2212yes—
31C2×D5×C4⋊C4C2212yes—
32C2×#4033C2214yescomplemented
33C2×D5×C22⋊C4C2212yes—
34C2×#4023C2214yes—
35C2×#4023C2212yes—
36C2×#4018C2212yes—
37C2×C42×D5C2212yes—
38C2×C4×Dic10C2214yescomplemented
39C5×#389C2214yescomplemented
40C5×#389C2214yescomplemented
41C5×#380C2214yes—
42C4×C22⋊Dic5C2214yes—
43#65279C2218yes—
44#65278C2218yes—
45#65276C2212yes—
46#65276C2212yes—
47C4⋊C4×Dic5C2218yescomplemented
48C4⋊C4×Dic5C2214yescomplemented
49C4⋊C4×Dic5C2214yescomplemented
50#65263C2214yescomplemented
51#65262C2218yescomplemented
52#65244C2212yes—
53#65232C2212yes—
54#65232C2214yes—
55C4×#685C2212yes—
56C4×#685C2212yes—
57#65218C2214yescomplemented
58#64958C2218yes—
59#64955C2218yes—
60#64943C2218yescomplemented
61#64943C2218yescomplemented
62#34203—51nomaximal, Sylow
63C24×D5C2311yescharacteristic
64C23×C4×C5C2312yes—
65C23×C4×C5C2311yescharacteristic
66C23×Dic5C2311yescharacteristic
67C23×Dic5C2312yes—
68C22×C4×D5—24no—
69C22×C4×D5—24no—
70C22×C4×D5C2×C418yes—
71C22×C4×D5C2314yes—
72C22×C4×D5C2311yescharacteristic
73C22×C4×D5C2311yescharacteristic
74C22×C4×D5C2311yescharacteristic
75C22×C4×D5C2311yescharacteristic
76C22×Dic10—24no—
77C22×Dic10—24no—
78C22×Dic10C2×C418yescomplemented
79C22×Dic10C2312yes—
80C22×Dic10C2312yes—
81C5×#117—24no—
82C5×#117—24no—
83C2×C5×C4⋊C4C2×C418yescomplemented
84C2×C5×C4⋊C4C2312yes—
85C2×C5×C4⋊C4C2312yes—
86C2×C5×C4⋊C4C2318yes—
87C2×C5×C4⋊C4C2312yes—
88C2×C5×C4⋊C4C2312yes—
89C2×C5×C22⋊C4C2312yes—
90C2×C5×C22⋊C4C2312yes—
91C2×C42×C5C2312yes—
92C2×C42×C5C2312yes—
93C2×C22⋊Dic5C2312yes—
94C2×C22⋊Dic5C2312yes—
95C2×#685C2×C4116yes—
96C2×#685C2314yes—
97C2×#685C2314yes—
98C2×#685C2314yes—
99C2×#685C2314yes—
100C2×#685C2314yes—
101#4072—24no—
102#4072D418yes—
103C2×C4⋊Dic5C2×C4116yescomplemented
104C2×C4⋊Dic5C2312yes—
105C2×C4⋊Dic5C2314yes—
106C2×C4⋊Dic5C2314yes—
107C2×C4⋊Dic5C2312yes—
108C2×#683C2318yes—
109C2×#683C2314yes—
110C2×C4×Dic5C2318yes—
111C2×C4×Dic5C2314yes—
112C2×C4×Dic5C2312yes—
113C2×C4×Dic5C2318yes—
114C2×C4×Dic5C2312yes—
115#4043C2×C4116yes—
116#4043C2×C4116yes—
117#4043C2×C4116yescomplemented
118#4043C2×C4116yescomplemented
119#4038—28no—
120D5×C4⋊C4—28no—
121#4033—216no—
122D5×C22⋊C4—28no—
123#4023—216no—
124#4023—28no—
125#4018—28no—
126C42×D5—28no—
127C4×Dic10—216no—
128C5×#95C2318yes—
129#3963C2314yes—
130#3963C2314yes—
131#3963C2318yes—
132#3963C2318yes—
133#3963C2318yes—
134C22×C22⋊Q8—51no—
135C2×#519—51no—
136C22×#117—51no—
137#2362—54no—
138#2360—54no—
139#2344—54no—
140#2342—54no—
141#2323—54no—
142#2316D511yescharacteristic, complemented
143#2310—51no—
144C4×C22⋊Q8—54no—
145#2280—51no—
146#2274—51no—
147C24×C5C2411yescharacteristic
148C23×D5—22no—
149C23×D5—22no—
150C23×D5—24no—
151C23×D5—42no—
152C23×D5C22×C414yes—
153C23×D5C2411yescharacteristic
154C23×D5C2411yescharacteristic
155C22×C4×C5—44no—
156C22×C4×C5C22×C418yes—
157C22×C4×C5C22×C414yes—
158C22×C4×C5C2×D414yes—
159C22×C4×C5#4114yes—
160C22×C4×C5C2414yes—
161C22×C4×C5C2414yes—
162C22×C4×C5C2414yes—
163C22×C4×C5C2411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C22×Dic5—24no—
166C22×Dic5—22no—
167C22×Dic5—24no—
168C22×Dic5—22no—
169C22×Dic5C22×C418yes—
170C22×Dic5C22×C414yes—
171C22×Dic5C2414yes—
172C22×Dic5C2414yes—
173C22×Dic5C2411yescharacteristic
174C22×Dic5C2411yescharacteristic
175C22×Dic5C2414yes—
176C2×C4×D5—232no—
177C2×C4×D5—22no—
178C2×C4×D5—22no—
179C2×C4×D5—22no—
180C2×C4×D5—22no—
181C2×C4×D5—22no—
182C2×C4×D5—22no—
183C2×C4×D5—22no—
184C2×C4×D5—22no—
185C2×C4×D5—28no—
186C2×C4×D5—28no—
187C2×C4×D5—44no—
188C2×C4×D5C22×C414yes—
189C2×C4×D5C22×C414yes—
190C2×C4×D5C22×C414yes—
191C2×C4×D5C22×C414yes—
192C2×Dic10—232no—
193C2×Dic10—24no—
194C2×Dic10—24no—
195C2×Dic10—24no—
196C2×Dic10—24no—
197C2×Dic10C22×C418yes—
198C2×Dic10C22×C418yes—
199C5×C4⋊C4—24no—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—48no—
202C5×C4⋊C4C22×C418yes—
203C5×C4⋊C4C22×C418yes—
204C5×C22⋊C4—24no—
205C5×C22⋊C4—24no—
206C42×C5—24no—
207C42×C5—24no—
208C22⋊Dic5—24no—
209C22⋊Dic5C2×D418yes—
210#685—28no—
211#685—28no—
212#685—44no—
213#685C22×C4116yes—
214#685C22×C4116yes—
215C4⋊Dic5—24no—
216C4⋊Dic5C22×C4116yes—
217C4⋊Dic5C22×C4116yes—
218C4⋊Dic5C2×D418yes—
219#683—216no—
220#683—28no—
221C4×Dic5—24no—
222C4×Dic5—216no—
223C4×Dic5—28no—
224C4×Dic5—48no—
225C4×Dic5C2×D418yes—
226C23×Q8—51no—
227C24×C4—51no—
228C2×C22⋊Q8—516no—
229C2×C22⋊Q8—52no—
230C2×C22⋊Q8—52no—
231C2×C22⋊Q8—52no—
232C2×C22⋊Q8—52no—
233#519—108no—
234C2×C4×Q8—54no—
235C2×C4×Q8—54no—
236C2×#117—51no—
237C2×#117—51no—
238C2×#117—52no—
239C2×#117—54no—
240C2×#117—52no—
241C2×#117—52no—
242C2×#117—108no—
243C2×#117C2×D511yescharacteristic
244C2×#117C2×D511yescharacteristic
245C22×C4⋊C4—52no—
246C22×C4⋊C4—52no—
247C22×C4⋊C4C2×D511yescharacteristic
248C22×C22⋊C4—52no—
249C22×C22⋊C4—52no—
250C22×C42—52no—
251#391—54no—
252#391—54no—
253#389—58no—
254#389C2×D514yescomplemented
255#389C2×D514yescomplemented
256#388—58no—
257#388—58no—
258#387—58no—
259#387—58no—
260#385—58no—
261#381—54no—
262#381—52no—
263#380—52no—
264#380—52no—
265#380C2×D514yes—
266#379—52no—
267C4×C4⋊C4—54no—
268C4×C4⋊C4—58no—
269C4×C4⋊C4—54no—
270C4×C22⋊C4—54no—
271C4×C22⋊C4—52no—
272C4×C22⋊C4—52no—
273C4×C22⋊C4—58no—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5C23×C412yes—
277C23×C5C22×D412yes—
278C23×C5C2×#4112yes—
279C23×C5C2511yescharacteristic
280C22×D5—21no—
281C22×D5—21no—
282C22×D5—22no—
283C22×D5—28no—
284C22×D5—21no—
285C22×D5—21no—
286C22×D5—22no—
287C22×D5—22no—
288C22×D5—28no—
289C22×D5—44no—
290C22×D5—48no—
291C22×D5—48no—
292C22×D5C23×C412yes—
293C22×D5C23×C412yes—
294C22×D52−1+414yes—
295C2×C4×C5—28no—
296C2×C4×C5—24no—
297C2×C4×C5—24no—
298C2×C4×C5—24no—
299C2×C4×C5—24no—
300C2×C4×C5—24no—
301C2×C4×C5—24no—
302C2×C4×C5—48no—
303C2×C4×C5C4×D4116yes—
304C2×C4×C5C23×C418yes—
305C2×C4×C5C23×C412yes—
306C2×C4×C5C23×C412yes—
307C2×C4×C5C22×D412yes—
308C2×C4×C5C22×D412yes—
309C2×C4×C5C2×#4112yes—
310C2×C4×C5C2×#4112yes—
311C2×Dic5—216no—
312C2×Dic5—28no—
313C2×Dic5—21no—
314C2×Dic5—24no—
315C2×Dic5—21no—
316C2×Dic5—24no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—24no—
320C2×Dic5—24no—
321C2×Dic5—24no—
322C2×Dic5—24no—
323C2×Dic5—24no—
324C2×Dic5C23×C412yes—
325C2×Dic5C23×C412yes—
326C2×Dic5C23×C418yes—
327C2×Dic5C22×D418yes—
328C4×D5—28no—
329C4×D5—28no—
330C4×D5—28no—
331C4×D5—28no—
332C4×D5—416no—
333Dic10—216no—
334Dic10—216no—
335C25—51no—
336C22×Q8—58no—
337C22×Q8—52no—
338C22×Q8—52no—
339C22×Q8—104no—
340C22×Q8—104no—
341C23×C4—52no—
342C23×C4—51no—
343C23×C4—58no—
344C23×C4—51no—
345C23×C4—51no—
346C23×C4—51no—
347C23×C4—51no—
348C23×C4—54no—
349C23×C4—104no—
350C23×C4—104no—
351C23×C4C22×D512yes—
352C23×C4C22×D511yescharacteristic
353C22⋊Q8—516no—
354C22⋊Q8—516no—
355C22⋊Q8—516no—
356C22⋊Q8—516no—
357C4×Q8—1016no—
358C4×Q8—1016no—
359#117—24no—
360#117—24no—
361#117—58no—
362#117—108no—
363#117—108no—
364#117—1016no—
365#117—108no—
366#117—104no—
367C2×C4⋊C4—516no—
368C2×C4⋊C4—52no—
369C2×C4⋊C4—54no—
370C2×C4⋊C4—54no—
371C2×C4⋊C4—54no—
372C2×C4⋊C4—58no—
373C2×C4⋊C4—52no—
374C2×C4⋊C4—108no—
375C2×C4⋊C4C4×D518yescomplemented
376C2×C4⋊C4C22×D512yes—
377C2×C4⋊C4C22×D512yes—
378C2×C4⋊C4C22×D518yes—
379C2×C4⋊C4C22×D512yes—
380C2×C4⋊C4C22×D512yes—
381C2×C22⋊C4—52no—
382C2×C22⋊C4—516no—
383C2×C22⋊C4—54no—
384C2×C22⋊C4—54no—
385C2×C22⋊C4—52no—
386C2×C22⋊C4—54no—
387C2×C22⋊C4—54no—
388C2×C22⋊C4—54no—
389C2×C22⋊C4—108no—
390C2×C22⋊C4C22×D512yes—
391C2×C22⋊C4C22×D512yes—
392C2×C42—52no—
393C2×C42—54no—
394C2×C42—58no—
395C2×C42—52no—
396C2×C42—58no—
397C2×C42—108no—
398C2×C42C22×D512yes—
399C2×C42C22×D512yes—
400#95—54no—
401#95—58no—
402#95—54no—
403#95—58no—
404#95—58no—
405#95C22×D518yes—
406C22×C5—22no—
407C22×C5—22no—
408C22×C5—24no—
409C22×C5—24no—
410C22×C5C2×C4×D414yes—
411C22×C5C24×C411yescharacteristic
412C22×C5C23×D411yescharacteristic
413C22×C5C22×#4111yescharacteristic
414C22×C5C2×2−1+412yes—
415C22×C5#58512yes—
416C2×D5—22no—
417C2×D5—22no—
418C2×D5—24no—
419C2×D5—24no—
420C2×D5—22no—
421C2×D5—22no—
422C2×D5—22no—
423C2×D5—416no—
424C2×D5—42no—
425C2×D5—42no—
426C2×D5#51914yescomplemented
427C4×C5—28no—
428C4×C5—44no—
429C4×C5—44no—
430C4×C5C2×C4×D414yes—
431C4×C5C2×C4×D414yes—
432Dic5—28no—
433Dic5—22no—
434Dic5—22no—
435Dic5—28no—
436Dic5—44no—
437C24—51no—
438C24—51no—
439C24—54no—
440C24—104no—
441C24—102no—
442C24—102no—
443C24—202no—
444C24C23×D511yescharacteristic
445C2×Q8—58no—
446C2×Q8—58no—
447C2×Q8—104no—
448C2×Q8—104no—
449C2×Q8—104no—
450C2×Q8—104no—
451C2×Q8—1032no—
452C22×C4—44no—
453C22×C4—58no—
454C22×C4—54no—
455C22×C4—51no—
456C22×C4—54no—
457C22×C4—51no—
458C22×C4—54no—
459C22×C4—54no—
460C22×C4—54no—
461C22×C4—54no—
462C22×C4—54no—
463C22×C4—54no—
464C22×C4—108no—
465C22×C4—108no—
466C22×C4—102no—
467C22×C4—104no—
468C22×C4—102no—
469C22×C4—104no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—1032no—
479C22×C4—204no—
480C22×C4C2×C4×D518yes—
481C22×C4C2×C4×D514yes—
482C22×C4D4×D514yes—
483C22×C4#71314yes—
484C22×C4C23×D514yes—
485C22×C4C23×D514yes—
486C22×C4C23×D514yes—
487C22×C4C23×D511yescharacteristic
488C22×C4C23×D511yescharacteristic
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—48no—
492C4⋊C4—516no—
493C4⋊C4—516no—
494C4⋊C4—58no—
495C4⋊C4—108no—
496C4⋊C4—1016no—
497C4⋊C4—104no—
498C4⋊C4C2×C4×D518yes—
499C4⋊C4C2×C4×D518yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—58no—
503C22⋊C4—516no—
504C22⋊C4—516no—
505C22⋊C4—108no—
506C22⋊C4—108no—
507C22⋊C4—104no—
508C22⋊C4—204no—
509C42—24no—
510C42—24no—
511C42—58no—
512C42—108no—
513C42—1016no—
514C42—104no—
515C42—208no—
516C2×C5—24no—
517C2×C5C22×C4×D411yescharacteristic
518C2×C5C2×#51911yescharacteristic
519C2×C5#342811yescharacteristic
520C2×C5#348912yes—
521C2×C5#349512yes—
522D5—24no—
523D5—42no—
524C23—22no—
525C23—22no—
526C23—52no—
527C23—54no—
528C23—52no—
529C23—108no—
530C23—101no—
531C23—101no—
532C23—102no—
533C23—108no—
534C23—101no—
535C23—101no—
536C23—102no—
537C23—102no—
538C23—204no—
539C23—208no—
540C23—208no—
541C23C22×C4×D512yes—
542C23C2×D4×D512yes—
543C23C2×#71312yes—
544C23C24×D511yescharacteristic
545Q8—1016no—
546Q8—1016no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—48no—
555C2×C4—58no—
556C2×C4—52no—
557C2×C4—52no—
558C2×C4—58no—
559C2×C4—108no—
560C2×C4—1016no—
561C2×C4—101no—
562C2×C4—104no—
563C2×C4—101no—
564C2×C4—101no—
565C2×C4—101no—
566C2×C4—104no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—108no—
570C2×C4—108no—
571C2×C4—108no—
572C2×C4—108no—
573C2×C4—104no—
574C2×C4—104no—
575C2×C4—104no—
576C2×C4—2016no—
577C2×C4#4039116yes—
578C2×C4C22×C4×D518yes—
579C2×C4C22×C4×D512yes—
580C2×C4C22×C4×D512yes—
581C2×C4C2×D4×D512yes—
582C2×C4C2×D4×D512yes—
583C2×C4C2×#71312yes—
584C2×C4C2×#71312yes—
585C5#3420311yescharacteristic, complemented, Sylow
586C22—24no—
587C22—22no—
588C22—24no—
589C22—22no—
590C22—54no—
591C22—102no—
592C22—104no—
593C22—102no—
594C22—104no—
595C22—102no—
596C22—102no—
597C22—102no—
598C22—2016no—
599C22—202no—
600C22—202no—
601C22C2×#403914yes—
602C22C23×C4×D511yescharacteristic
603C22C22×D4×D511yescharacteristic
604C22C22×#71311yescharacteristic
605C22#6628512yes—
606C22D5×2−1+412yes—
607C4—28no—
608C4—44no—
609C4—44no—
610C4—102no—
611C4—102no—
612C4—108no—
613C4—108no—
614C4—204no—
615C4C2×#403914yes—
616C4C2×#403914yes—
617C2—24no—
618C2—104no—
619C2—202no—
620C2C22×#403911yescharacteristic
621C2#1061775711yescharacteristic
622C2D5×#51911yescharacteristic
623C2#1061828112yes—
624C2#1061849912yes—
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2316C2
socle, cosocleC23×C5C25
prime radicalC24×D5C23
hypercenter#2316D5
hypocenterC5#34203
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#65850C2semidirect
4. C5×#2316C2semidirect
6. #10613961C2semidirect
7. #10613875C2semidirect
8. #10607936C2semidirect
9. #10607928C2semidirect
10. #10607715C2semidirect
11. #10607696C2semidirect
13. #10605609C2semidirect
25. C2×#4043C4semidirect
32. C2×#4033C22semidirect
38. C2×C4×Dic10C22semidirect
39. C5×#389C22semidirect
40. C5×#389C22semidirect
47. C4⋊C4×Dic5C22semidirect
48. C4⋊C4×Dic5C22semidirect
49. C4⋊C4×Dic5C22semidirect
50. #65263C22semidirect
51. #65262C22semidirect
57. #65218C22semidirect
60. #64943C22semidirect
61. #64943C22semidirect
78. C22×Dic10C2×C4semidirect
83. C2×C5×C4⋊C4C2×C4semidirect
103. C2×C4⋊Dic5C2×C4semidirect
117. #4043C2×C4semidirect
118. #4043C2×C4semidirect
142. #2316D5semidirect
254. #389C2×D5semidirect
255. #389C2×D5semidirect
375. C2×C4⋊C4C4×D5semidirect
426. C2×D5#519semidirect
585. C5#34203semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#1172
C2×#658502
C22×#40432
C5×#23162
#106139982
#106139612
#106138752
#106079362
#106079282
#106077152
#106076962
#106057402
#106056092
#342035