Finite Group Databasebeta

← #34502072

Subgroups — #34502072

Order 20224.

2014 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099189C211yesmaximal, characteristic
2C22×#14579154C211yesmaximal, characteristic
3#23771737C212yesmaximal, complemented
4#23769980C214yesmaximal, complemented
5#23769978C214yesmaximal, complemented
6#23769853C214yesmaximal, complemented
7#23769851C214yesmaximal, complemented
8#23769219C212yesmaximal, complemented
9#23768777C212yesmaximal, complemented
10#23767883C212yesmaximal, complemented
11#23767681C211yesmaximal, characteristic
12#23767678C211yesmaximal, characteristic
13#23767678C211yesmaximal, characteristic
14#23765996C211yesmaximal, characteristic
15C79×#2483C211yesmaximal, characteristic
16#19099189—28no—
17C2×#14579154—28no—
18C2×#14579154C2214yes—
19C2×#14579154C2214yes—
20C2×#14579140C2212yes—
21C2×#14579140C2212yes—
22C2×#14579138C2212yes—
23C2×#14579138C2212yes—
24C2×#14579138C2212yes—
25C2×#14579138C2212yes—
26C22×C4⋊Dic79C2211yescharacteristic
27C22×Dic79⋊C4C2211yescharacteristic
28C22×Dic79⋊C4C2211yescharacteristic
29C22×C22⋊Dic79C2211yescharacteristic
30C22×C22⋊Dic79C2211yescharacteristic
31C22×C22⋊Dic79C2211yescharacteristic
32C22×C4×Dic79C2211yescharacteristic
33#19098505C2214yescomplemented
34#19098504C2214yescomplemented
35#19098504C2214yescomplemented
36#19098497C2214yescomplemented
37#19098496C2214yescomplemented
38#19098494C2214yescomplemented
39#19098491C2214yes—
40#19098491C2214yes—
41#19098488C2214yes—
42#19098484C2214yescomplemented
43#19098483C2214yescomplemented
44#19098467C2214yescomplemented
45#19098467C2214yescomplemented
46#19098462C2214yescomplemented
47#19098460C2214yescomplemented
48#19098460C2214yescomplemented
49#19098457C2214yes—
50#19098451C2214yes—
51#19098450C2214yescomplemented
52#19098446C2214yescomplemented
53#19098438C2212yes—
54#19098438C2212yes—
55#19098436C2212yes—
56#19098436C2212yes—
57#19098435C2212yes—
58#19098435C2212yes—
59#19098427C2214yescomplemented
60#19098426C2212yes—
61#19098426C2212yes—
62#19098424C2212yes—
63#19098424C2212yes—
64#19098424C2212yes—
65#19098424C2212yes—
66#19098421C2212yes—
67#19098419C2211yescharacteristic
68C22⋊C4×Dic79C2212yes—
69C22×C22⋊C4×C79C2211yescharacteristic
70C22×C22⋊C4×C79C2211yescharacteristic
71#397×C79C2214yes—
72#385×C79C2212yes—
73#385×C79C2212yes—
74#381×C79C2212yes—
75#381×C79C2212yes—
76#379×C79C2211yescharacteristic
77C22×Dic158C2312yes—
78C22×Dic158C2312yes—
79C23×Dic79C2311yescharacteristic
80C23×Dic79C2311yescharacteristic
81C23×Dic79C2311yescharacteristic
82C23×Dic79C2311yescharacteristic
83C23×Dic79C2311yescharacteristic
84C23×Dic79C2311yescharacteristic
85#14579154—216no—
86#14579154—216no—
87#14579140—28no—
88#14579140—28no—
89#14579138—28no—
90#14579138—28no—
91#14579138—28no—
92#14579138—28no—
93C2×C4⋊Dic79—24no—
94C2×C4⋊Dic79C2312yes—
95C2×C4⋊Dic79C2314yes—
96C2×C4⋊Dic79C2312yes—
97C2×C4⋊Dic79C2314yes—
98C2×Dic79⋊C4—24no—
99C2×Dic79⋊C4—24no—
100C2×Dic79⋊C4C2312yes—
101C2×Dic79⋊C4C2314yes—
102C2×Dic79⋊C4C2312yes—
103C2×Dic79⋊C4C2312yes—
104C2×Dic79⋊C4C2312yes—
105C2×Dic79⋊C4C2314yes—
106C2×Dic79⋊C4C2314yes—
107C2×Dic79⋊C4C2314yes—
108C2×Dic79⋊C4C2314yes—
109C2×C22⋊Dic79—24no—
110C2×C22⋊Dic79—24no—
111C2×C22⋊Dic79D418yes—
112C2×C22⋊Dic79C2312yes—
113C2×C22⋊Dic79C2312yes—
114C2×C22⋊Dic79C2312yes—
115C2×C22⋊Dic79C2312yes—
116C2×C22⋊Dic79C2312yes—
117C2×C22⋊Dic79C2312yes—
118C2×C22⋊Dic79C2312yes—
119C2×C22⋊Dic79C2312yes—
120C2×C22⋊Dic79C2312yes—
121C2×C4×Dic79—24no—
122C2×C4×Dic79C2314yes—
123C2×C4×Dic79C2314yes—
124C79⋊#95C2314yes—
125C79⋊#95C2314yes—
126C79⋊#95C2314yes—
127C79⋊#95C2314yes—
128C79⋊#95C2314yes—
129C79⋊#95C2314yes—
130C79⋊#95C2314yes—
131C79⋊#95C2314yes—
132C79⋊#95C2314yes—
133C79⋊#95C2314yes—
134C79⋊#95C2314yes—
135C25×C79C2311yescharacteristic
136C23×C4×C79C2311yescharacteristic
137C23×C4×C79C2311yescharacteristic
138C23×C4×C79C2311yescharacteristic
139C23×C4×C79C2311yescharacteristic
140C2×C4⋊C4×C79C2314yes—
141C2×C22⋊C4×C79—24no—
142C2×C22⋊C4×C79—24no—
143C2×C22⋊C4×C79C2312yes—
144C2×C22⋊C4×C79C2312yes—
145C2×C22⋊C4×C79C2312yes—
146C2×C22⋊C4×C79C2312yes—
147C2×C22⋊C4×C79C2312yes—
148C2×C22⋊C4×C79C2312yes—
149C2×C22⋊C4×C79C2312yes—
150#95×C79C2314yes—
151#95×C79C2314yes—
152#95×C79C2314yes—
153C2×Dic158—24no—
154C2×Dic158—24no—
155C2×Dic158—44no—
156C2×Dic158—44no—
157C22×Dic79—22no—
158C22×Dic79—22no—
159C22×Dic79—22no—
160C22×Dic79—22no—
161C22×Dic79—24no—
162C22×Dic79—42no—
163C22×Dic79—42no—
164C22×Dic79—42no—
165C22×Dic79—42no—
166C22×Dic79—42no—
167C22×Dic79C2×D414yes—
168C22×Dic79C2×D414yes—
169C22×Dic79C2412yes—
170C22×Dic79C2412yes—
171C22×Dic79C2414yes—
172C22×Dic79C2412yes—
173C22×Dic79C2412yes—
174C22×Dic79C2412yes—
175C22×Dic79C2412yes—
176C4⋊Dic79—28no—
177C4⋊Dic79—24no—
178C4⋊Dic79—24no—
179C4⋊Dic79—44no—
180Dic79⋊C4—28no—
181Dic79⋊C4—28no—
182Dic79⋊C4—24no—
183Dic79⋊C4—24no—
184Dic79⋊C4—24no—
185Dic79⋊C4—24no—
186Dic79⋊C4—44no—
187Dic79⋊C4—44no—
188Dic79⋊C4—44no—
189C22⋊Dic79—24no—
190C22⋊Dic79—24no—
191C22⋊Dic79—24no—
192C22⋊Dic79—24no—
193C22⋊Dic79—24no—
194C22⋊Dic79—42no—
195C22⋊Dic79—42no—
196C22⋊Dic79C2×D418yes—
197C22⋊Dic79C2×D418yes—
198C4×Dic79—28no—
199C4×Dic79—44no—
200C24×C79—24no—
201C24×C79—42no—
202C24×C79—42no—
203C24×C79C2×D414yes—
204C24×C79C2411yescharacteristic
205C24×C79C2411yescharacteristic
206C24×C79C2411yescharacteristic
207C22×C4×C79—22no—
208C22×C4×C79—22no—
209C22×C4×C79—24no—
210C22×C4×C79—22no—
211C22×C4×C79—22no—
212C22×C4×C79—42no—
213C22×C4×C79—42no—
214C22×C4×C79—42no—
215C22×C4×C79C2412yes—
216C22×C4×C79C2412yes—
217C22×C4×C79C2412yes—
218C22×C4×C79C2414yes—
219C22×C4×C79C2412yes—
220C4⋊C4×C79—44no—
221C22⋊C4×C79—24no—
222C22⋊C4×C79—24no—
223C22⋊C4×C79—24no—
224C22⋊C4×C79—24no—
225C22⋊C4×C79—24no—
226C22⋊C4×C79—42no—
227C22⋊C4×C79—42no—
228C23×C79—22no—
229C23×C79—22no—
230C23×C79—22no—
231C23×C79—22no—
232C23×C79—22no—
233C23×C79—22no—
234C23×C79—22no—
235C23×C79—22no—
236C23×C79—28no—
237C23×C79—44no—
238C23×C79—44no—
239C23×C79—88no—
240C23×C79C22×D412yes—
241C23×C79C22×D412yes—
242C23×C79C22×D412yes—
243C23×C792−1+414yes—
244C23×C79C2511yescharacteristic
245C2×C4×C79—22no—
246C2×C4×C79—22no—
247C2×C4×C79—22no—
248C2×C4×C79—22no—
249C2×C4×C79—24no—
250C2×C4×C79—24no—
251C2×C4×C79—24no—
252C2×C4×C79—24no—
253C2×C4×C79—24no—
254C2×C4×C79—24no—
255C2×C4×C79—24no—
256C2×C4×C79—44no—
257C2×C4×C79—44no—
258C2×C4×C79—82no—
259C2×C4×C79—82no—
260C2×Dic79—24no—
261C2×Dic79—24no—
262C2×Dic79—22no—
263C2×Dic79—22no—
264C2×Dic79—22no—
265C2×Dic79—22no—
266C2×Dic79—24no—
267C2×Dic79—24no—
268C2×Dic79—24no—
269C2×Dic79—24no—
270C2×Dic79—24no—
271C2×Dic79—24no—
272C2×Dic79—24no—
273C2×Dic79—44no—
274C2×Dic79—44no—
275C2×Dic79—44no—
276C2×Dic79—44no—
277C2×Dic79—82no—
278C2×Dic79—82no—
279C2×Dic79C22×D414yes—
280C2×Dic79C22×D414yes—
281Dic158—84no—
282Dic158—84no—
283C22×C79—22no—
284C22×C79—22no—
285C22×C79—28no—
286C22×C79—22no—
287C22×C79—22no—
288C22×C79—22no—
289C22×C79—22no—
290C22×C79—22no—
291C22×C79—22no—
292C22×C79—44no—
293C22×C79—44no—
294C22×C79—88no—
295C22×C79#53614yes—
296C22×C79C23×D411yescharacteristic
297C22×C79C2×2−1+412yes—
298C22×C79#58512yes—
299C22×C79#58512yes—
300C4×C79—42no—
301C4×C79—42no—
302C4×C79—42no—
303C4×C79—42no—
304C4×C79—44no—
305Dic79—24no—
306Dic79—24no—
307Dic79—42no—
308Dic79—42no—
309Dic79—42no—
310Dic79—42no—
311Dic79—44no—
312#36970—791nomaximal, Sylow
313C2×C79—24no—
314C2×C79—42no—
315C2×C79—42no—
316C2×C79C2×#53611yescharacteristic
317C2×C79#348511yescharacteristic
318C2×C79#348511yescharacteristic
319C2×C79#356414yes—
320C2×#543—791no—
321C22×#126—791no—
322#2855—792no—
323#2698—794no—
324#2698—794no—
325#2689—794no—
326#2689—794no—
327#2627—792no—
328#2583—792no—
329#2503—792no—
330#2483—791no—
331#2483—791no—
332#2483—791no—
333#2483D7911yescharacteristic
334#2312—791no—
335C79#3697011yescharacteristic, complemented, Sylow
336#543—1588no—
337C2×#126—794no—
338C2×#126—794no—
339C2×#126—1588no—
340C2×C22⋊Q8—792no—
341C2×C22⋊Q8—792no—
342C2×C22⋊Q8—792no—
343C2×C22⋊Q8—792no—
344C2×C22⋊Q8—792no—
345C2×C22⋊Q8—792no—
346C22×C4⋊C4—791no—
347C22×C4⋊C4—791no—
348C22×C4⋊C4—791no—
349C22×C22⋊C4—791no—
350C22×C22⋊C4—791no—
351C22×C22⋊C4—791no—
352C22×C22⋊C4C2×D7911yescharacteristic
353C22×C22⋊C4C2×D7911yescharacteristic
354C22×C42—791no—
355#400—794no—
356#400—794no—
357#400—794no—
358#398—794no—
359#398—794no—
360#398—794no—
361#397—794no—
362#397—794no—
363#397—794no—
364#397C2×D7914yes—
365#395—794no—
366#395—794no—
367#391—794no—
368#391—794no—
369#389—794no—
370#389—794no—
371#389—794no—
372#388—794no—
373#388—794no—
374#387—794no—
375#387—794no—
376#385—792no—
377#385—792no—
378#385—792no—
379#385—792no—
380#385—792no—
381#385—792no—
382#385C2×D7912yes—
383#385C2×D7912yes—
384#382—794no—
385#381—792no—
386#381—792no—
387#381—792no—
388#381—792no—
389#381—792no—
390#381—792no—
391#381C2×D7912yes—
392#381C2×D7912yes—
393#380—792no—
394#379—791no—
395#379C2×D7911yescharacteristic
396C4×C22⋊C4—792no—
397C25C22×D7911yescharacteristic
398C22×Q8—792no—
399C22×Q8—792no—
400C23×C4—791no—
401C23×C4—791no—
402C23×C4—791no—
403C23×C4—791no—
404C23×C4—791no—
405C23×C4—791no—
406C23×C4C22×D7911yescharacteristic
407C23×C4C22×D7911yescharacteristic
408C23×C4C22×D7911yescharacteristic
409C23×C4C22×D7911yescharacteristic
410#126—15816no—
411#126—15816no—
412C22⋊Q8—1588no—
413C22⋊Q8—1588no—
414C22⋊Q8—1588no—
415C22⋊Q8—1588no—
416C22⋊Q8—1588no—
417C22⋊Q8—1588no—
418C2×C4⋊C4—794no—
419C2×C4⋊C4—794no—
420C2×C4⋊C4—794no—
421C2×C4⋊C4—794no—
422C2×C4⋊C4—794no—
423C2×C4⋊C4—794no—
424C2×C4⋊C4—794no—
425C2×C4⋊C4—792no—
426C2×C4⋊C4—792no—
427C2×C4⋊C4—792no—
428C2×C4⋊C4—792no—
429C2×C4⋊C4—792no—
430C2×C4⋊C4—792no—
431C2×C4⋊C4—1584no—
432C2×C4⋊C4—1584no—
433C2×C4⋊C4—1584no—
434C2×C4⋊C4C22×D7914yes—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—792no—
438C2×C22⋊C4—792no—
439C2×C22⋊C4—798no—
440C2×C22⋊C4—792no—
441C2×C22⋊C4—792no—
442C2×C22⋊C4—792no—
443C2×C22⋊C4—792no—
444C2×C22⋊C4—792no—
445C2×C22⋊C4—792no—
446C2×C22⋊C4—792no—
447C2×C22⋊C4—1584no—
448C2×C22⋊C4—1584no—
449C2×C22⋊C4C22×D7912yes—
450C2×C22⋊C4C22×D7912yes—
451C2×C22⋊C4C22×D7912yes—
452C2×C22⋊C4C22×D7912yes—
453C2×C22⋊C4C22×D7912yes—
454C2×C22⋊C4C22×D7912yes—
455C2×C22⋊C4C22×D7912yes—
456C2×C42—794no—
457C2×C42—794no—
458C2×C42—1584no—
459#95—794no—
460#95—794no—
461#95—794no—
462#95—794no—
463#95—794no—
464#95—794no—
465#95—794no—
466#95—794no—
467#95—794no—
468#95—794no—
469#95—794no—
470#95C22×D7914yes—
471#95C22×D7914yes—
472#95C22×D7914yes—
473C24—24no—
474C24—42no—
475C24—42no—
476C24D4×D7914yes—
477C24C23×D7911yescharacteristic
478C24C23×D7911yescharacteristic
479C24C23×D7911yescharacteristic
480C2×Q8—3164no—
481C2×Q8—3164no—
482C2×Q8—1584no—
483C2×Q8—1584no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—3162no—
493C22×C4—3162no—
494C22×C4—3162no—
495C22×C4—3162no—
496C22×C4—3162no—
497C22×C4—794no—
498C22×C4—794no—
499C22×C4—792no—
500C22×C4—792no—
501C22×C4—794no—
502C22×C4—792no—
503C22×C4—792no—
504C22×C4—792no—
505C22×C4—792no—
506C22×C4—1584no—
507C22×C4—1582no—
508C22×C4—1582no—
509C22×C4—1582no—
510C22×C4—1582no—
511C22×C4C23×D7914yes—
512C22×C4C23×D7912yes—
513C22×C4C23×D7912yes—
514C22×C4C23×D7912yes—
515C22×C4C23×D7912yes—
516C4⋊C4—44no—
517C4⋊C4—3164no—
518C4⋊C4—3164no—
519C4⋊C4—3164no—
520C4⋊C4—3164no—
521C4⋊C4—1588no—
522C4⋊C4—1588no—
523C4⋊C4—1584no—
524C4⋊C4—1584no—
525C4⋊C4—1584no—
526C4⋊C4—1584no—
527C4⋊C4—1588no—
528C4⋊C4—1584no—
529C4⋊C4—1584no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—3162no—
538C22⋊C4—3162no—
539C22⋊C4—798no—
540C22⋊C4—798no—
541C22⋊C4—1584no—
542C22⋊C4—1584no—
543C22⋊C4—1584no—
544C22⋊C4—1584no—
545C22⋊C4—1584no—
546C42—3164no—
547C42—1588no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—28no—
557C23—44no—
558C23—44no—
559C23—88no—
560C23C2×D4×D7912yes—
561C23C2×D4×D7912yes—
562C23C2×D4×D7912yes—
563C23#1457920814yes—
564C23C24×D7911yescharacteristic
565Q8—6324no—
566Q8—6324no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—3164no—
583C2×C4—3164no—
584C2×C4—3164no—
585C2×C4—3164no—
586C2×C4—794no—
587C2×C4—794no—
588C2×C4—6322no—
589C2×C4—6322no—
590C2×C4—1584no—
591C2×C4—1584no—
592C2×C4—1584no—
593C2×C4—1584no—
594C2×C4—1584no—
595C2×C4—1584no—
596C2×C4—1584no—
597C2×C4—1582no—
598C2×C4—1582no—
599C2×C4—1582no—
600C2×C4—1582no—
601C2×C4—1584no—
602C2×C4—1584no—
603C22—22no—
604C22—22no—
605C22—28no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—44no—
613C22—44no—
614C22—88no—
615C22#1909913514yes—
616C22C22×D4×D7911yescharacteristic
617C22C2×#1457920812yes—
618C22#1909939812yes—
619C22#1909939912yes—
620C4—42no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—44no—
625C4—3162no—
626C4—3162no—
627C4—3162no—
628C4—3162no—
629C4—3164no—
630C4—1584no—
631C4—1584no—
632C2—24no—
633C2—42no—
634C2—42no—
635C2C2×#1909913511yescharacteristic
636C2#2377703811yescharacteristic
637C2#2377704111yescharacteristic
638C2#2377801214yes—
subgroupG/H
center, FrattiniC23C24×D79
commutator, socle, cosocleC23×C79C25
FittingC79×#2483C2
prime radicalC25×C79C23
hypercenter#2483D79
hypocenterC79#36970
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2483

Complements

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

NH ≅ G/Ntype
3. #23771737C2semidirect
4. #23769980C2semidirect
5. #23769978C2semidirect
6. #23769853C2semidirect
7. #23769851C2semidirect
8. #23769219C2semidirect
9. #23768777C2semidirect
10. #23767883C2semidirect
33. #19098505C22semidirect
34. #19098504C22semidirect
35. #19098504C22semidirect
36. #19098497C22semidirect
37. #19098496C22semidirect
38. #19098494C22semidirect
42. #19098484C22semidirect
43. #19098483C22semidirect
44. #19098467C22semidirect
45. #19098467C22semidirect
46. #19098462C22semidirect
47. #19098460C22semidirect
48. #19098460C22semidirect
51. #19098450C22semidirect
52. #19098446C22semidirect
59. #19098427C22semidirect
335. C79#36970semidirect

Maximal subgroups

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

subgroupindex
C2×#190991892
C22×#145791542
#237717372
#237699802
#237699782
#237698532
#237698512
#237692192
#237687772
#237678832
#237676812
#237676782
#237659962
C79×#24832
#3697079