Finite Group Databasebeta

← #34501944

Subgroups — #34501944

Order 20224.

1986 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099191C211yesmaximal, characteristic, complemented
2C22×#14579156C211yesmaximal, characteristic
3#23771737C212yesmaximal, complemented
4#23770300C214yesmaximal, complemented
5#23770296C214yesmaximal, complemented
6#23770231C214yesmaximal, complemented
7#23770227C214yesmaximal, complemented
8#23769211C212yesmaximal
9#23768557C212yesmaximal
10#23767876C212yesmaximal, complemented
11#23767687C211yesmaximal, characteristic, complemented
12#23767676C211yesmaximal, characteristic
13#23767670C211yesmaximal, characteristic
14#23766001C211yesmaximal, characteristic, complemented
15C79×#2484C211yesmaximal, characteristic, complemented
16#19099191—28no—
17C2×#14579161C2212yescomplemented
18C2×#14579160C2212yescomplemented
19C2×#14579156—28no—
20C2×#14579156C2214yes—
21C2×#14579156C2214yes—
22C2×#14579153C2212yescomplemented
23C2×#14579152C2212yescomplemented
24C2×#14579138C2212yes—
25C2×#14579138C2212yes—
26C2×#14579120C2211yescharacteristic
27C22×C4⋊Dic79C2211yescharacteristic
28C22×Dic79⋊C4C2211yescharacteristic
29C22×(C2×D79)⋊C4C2211yescharacteristic
30C22×(C2×D79)⋊C4C2211yescharacteristic
31C22×(C2×D79)⋊C4C2211yescharacteristic
32C22×C4×Dic79C2211yescharacteristic
33#19098502C2212yescomplemented
34#19098500C2212yes—
35#19098498C2212yes—
36#19098498C2214yes—
37#19098497C2214yescomplemented
38#19098496C2214yescomplemented
39#19098493C2212yescomplemented
40#19098492C2214yes—
41#19098490C2212yes—
42#19098489C2212yes—
43#19098487C2214yes—
44#19098485C2214yes—
45#19098476C2214yes—
46#19098476C2214yes—
47#19098467C2214yescomplemented
48#19098467C2214yescomplemented
49#19098460C2214yescomplemented
50#19098455C2214yes—
51#19098455C2214yes—
52#19098454C2214yes—
53#19098452C2214yes—
54#19098450C2214yescomplemented
55#19098448C2214yescomplemented
56#19098446C2214yescomplemented
57#19098446C2214yescomplemented
58#19098444C2214yescomplemented
59#19098438C2212yes—
60#19098436C2212yes—
61#19098435C2212yes—
62#19098432C2214yescomplemented
63#19098426C2212yes—
64#19098426C2211yescharacteristic
65#19098421C2212yes—
66#19098418C2212yes—
67#19098418C2212yes—
68C22⋊C4×Dic79C2212yes—
69C2×#117×C79C2211yescharacteristic
70C22×C4⋊C4×C79C2211yescharacteristic
71#400×C79C2212yescomplemented
72#398×C79C2212yescomplemented
73#387×C79C2214yescomplemented
74#385×C79C2212yes—
75#381×C79C2212yes—
76#381×C79C2211yescharacteristic
77C24×D79C2311yescharacteristic
78C22×Dic158C2312yes—
79C22×Dic158C2312yes—
80C23×Dic79C2311yescharacteristic
81C23×Dic79C2311yescharacteristic
82C23×Dic79C2311yescharacteristic
83#14579161—28no—
84#14579160—28no—
85#14579156—216no—
86#14579156—216no—
87#14579153—28no—
88#14579152—28no—
89#14579138—28no—
90#14579138—28no—
91#14579120—24no—
92C2×C4⋊Dic79D418yes—
93C2×C4⋊Dic79C2312yes—
94C2×C4⋊Dic79C2312yes—
95C2×C4⋊Dic79C2311yescharacteristic
96C2×C4⋊Dic79C2311yescharacteristic
97C2×C4⋊Dic79C2314yes—
98C2×C4⋊Dic79C2312yes—
99C2×C4⋊Dic79C2312yes—
100C2×Dic79⋊C4—24no—
101C2×Dic79⋊C4C2312yes—
102C2×Dic79⋊C4C2314yes—
103C2×Dic79⋊C4C2312yes—
104C2×Dic79⋊C4C2312yes—
105C2×Dic79⋊C4C2312yes—
106C2×Dic79⋊C4C2314yes—
107C2×Dic79⋊C4C2314yes—
108C2×Dic79⋊C4C2314yes—
109C2×C22⋊Dic79C2312yes—
110C2×C22⋊Dic79C2312yes—
111C2×(C2×D79)⋊C4—24no—
112C2×(C2×D79)⋊C4—24no—
113C2×(C2×D79)⋊C4—24no—
114C2×(C2×D79)⋊C4C2314yes—
115C2×(C2×D79)⋊C4C2312yes—
116C2×(C2×D79)⋊C4C2312yes—
117C2×(C2×D79)⋊C4C2314yes—
118C2×(C2×D79)⋊C4C2314yes—
119C2×(C2×D79)⋊C4C2314yes—
120C2×(C2×D79)⋊C4C2314yes—
121C2×(C2×D79)⋊C4C2314yes—
122C2×C4×Dic79—24no—
123C2×C4×Dic79C2312yes—
124C2×C4×Dic79C2314yes—
125C2×C4×Dic79C2314yes—
126C79⋊#95C2314yes—
127C79⋊#95C2314yes—
128C79⋊#95C2312yes—
129C79⋊#95C2314yes—
130C79⋊#95C2314yes—
131C79⋊#95C2312yes—
132C79⋊#95C2314yes—
133C79⋊#95C2314yes—
134C79⋊#95C2314yes—
135C23×C4×C79C2311yescharacteristic
136C23×C4×C79C2311yescharacteristic
137C23×C4×C79C2311yescharacteristic
138#117×C79—24no—
139C2×C4⋊C4×C79—24no—
140C2×C4⋊C4×C79C2312yes—
141C2×C4⋊C4×C79C2312yes—
142C2×C4⋊C4×C79C2311yescharacteristic
143C2×C4⋊C4×C79C2311yescharacteristic
144C2×C4⋊C4×C79C2314yes—
145C2×C22⋊C4×C79C2312yes—
146C2×C22⋊C4×C79C2312yes—
147C2×C42×C79C2312yes—
148#95×C79C2312yes—
149#95×C79C2314yes—
150#95×C79C2312yes—
151#95×C79C2314yes—
152#95×C79C2314yes—
153C23×D79—22no—
154C23×D79—24no—
155C23×D79—42no—
156C23×D79—42no—
157C23×D79C2412yes—
158C2×Dic158—24no—
159C2×Dic158—24no—
160C2×Dic158—44no—
161C2×Dic158—44no—
162C22×Dic79—22no—
163C22×Dic79—42no—
164C22×Dic79—42no—
165C22×Dic79—42no—
166C22×Dic79C2×D414yes—
167C22×Dic79C2×D414yes—
168C22×Dic79C2412yes—
169C22×Dic79C2414yes—
170C22×Dic79C2412yes—
171C22×Dic79C2412yes—
172C22×Dic79C2412yes—
173C22×Dic79C2412yes—
174C4⋊Dic79—24no—
175C4⋊Dic79—24no—
176C4⋊Dic79—22no—
177C4⋊Dic79—22no—
178C4⋊Dic79—44no—
179C4⋊Dic79C2×D418yes—
180C4⋊Dic79C2×D418yes—
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—
191(C2×D79)⋊C4—28no—
192(C2×D79)⋊C4—28no—
193(C2×D79)⋊C4—24no—
194(C2×D79)⋊C4—24no—
195(C2×D79)⋊C4—44no—
196(C2×D79)⋊C4—44no—
197(C2×D79)⋊C4—44no—
198(C2×D79)⋊C4—44no—
199C4×Dic79—28no—
200C4×Dic79—24no—
201C4×Dic79—44no—
202C24×C79C2411yescharacteristic
203C22×C4×C79—24no—
204C22×C4×C79—22no—
205C22×C4×C79—22no—
206C22×C4×C79—24no—
207C22×C4×C79—42no—
208C22×C4×C79C2×D414yes—
209C22×C4×C79C2412yes—
210C22×C4×C79C2414yes—
211C22×C4×C79C2412yes—
212C22×C4×C79C2412yes—
213C22×C4×C79C2412yes—
214C22×C4×C79C2411yescharacteristic
215C22×C4×C79C2411yescharacteristic
216C4⋊C4×C79—24no—
217C4⋊C4×C79—24no—
218C4⋊C4×C79—22no—
219C4⋊C4×C79—22no—
220C4⋊C4×C79—44no—
221C22⋊C4×C79—24no—
222C22⋊C4×C79—24no—
223C42×C79—24no—
224C23×C79—22no—
225C23×C79—22no—
226C23×C79C22×D412yes—
227C23×C792−1+414yes—
228C23×C79C2511yescharacteristic
229C22×D79—22no—
230C22×D79—24no—
231C22×D79—24no—
232C22×D79—22no—
233C22×D79—22no—
234C22×D79—44no—
235C22×D79—44no—
236C22×D79—44no—
237C22×D79—88no—
238C2×C4×C79—28no—
239C2×C4×C79—22no—
240C2×C4×C79—22no—
241C2×C4×C79—22no—
242C2×C4×C79—22no—
243C2×C4×C79—22no—
244C2×C4×C79—22no—
245C2×C4×C79—24no—
246C2×C4×C79—24no—
247C2×C4×C79—24no—
248C2×C4×C79—24no—
249C2×C4×C79—24no—
250C2×C4×C79—24no—
251C2×C4×C79—22no—
252C2×C4×C79—22no—
253C2×C4×C79—44no—
254C2×C4×C79—44no—
255C2×C4×C79C22×D412yes—
256C2×C4×C79C22×D412yes—
257C2×Dic79—24no—
258C2×Dic79—24no—
259C2×Dic79—22no—
260C2×Dic79—22no—
261C2×Dic79—22no—
262C2×Dic79—24no—
263C2×Dic79—24no—
264C2×Dic79—24no—
265C2×Dic79—24no—
266C2×Dic79—24no—
267C2×Dic79—24no—
268C2×Dic79—44no—
269C2×Dic79—44no—
270C2×Dic79—44no—
271C2×Dic79C22×D414yes—
272C2×Dic79C22×D414yes—
273Dic158—84no—
274Dic158—84no—
275C22×C79—22no—
276C22×C79—22no—
277C22×C79—28no—
278C22×C79#53614yes—
279C22×C79C23×D411yescharacteristic
280C22×C79C2×2−1+412yes—
281C22×C79#58512yes—
282C22×C79#58512yes—
283C2×D79—42no—
284C2×D79—42no—
285C2×D79—42no—
286C2×D79—44no—
287C2×D79—44no—
288C2×D79—88no—
289C4×C79—22no—
290C4×C79—22no—
291C4×C79—42no—
292C4×C79—42no—
293C4×C79—42no—
294C4×C79—42no—
295C4×C79—44no—
296Dic79—24no—
297Dic79—24no—
298Dic79—42no—
299Dic79—42no—
300Dic79—42no—
301Dic79—44no—
302#36962—791nomaximal, Sylow
303C2×C79—24no—
304C2×C79C2×#53611yescharacteristic
305C2×C79#348511yescharacteristic
306C2×C79#348711yescharacteristic
307C2×C79#356414yes—
308D79—82no—
309C2×#544—791no—
310C22×#126—791no—
311#2855—792no—
312#2721—794no—
313#2721—794no—
314#2716—794no—
315#2716—794no—
316#2626—792no—
317#2561—792no—
318#2502—792no—
319#2484—791no—
320#2484D7911yescharacteristic, complemented
321#2482—791no—
322#2482—791no—
323#2313—791no—
324C79#3696211yescharacteristic, complemented, Sylow
325#544—1588no—
326C2×C4⋊Q8—792no—
327C2×C4⋊Q8—792no—
328C2×#126—794no—
329C2×#126—794no—
330C2×#126—1588no—
331C2×#125—792no—
332C2×#125—792no—
333C2×C22⋊Q8—792no—
334C2×C22⋊Q8—792no—
335C2×#117—791no—
336C2×#117C2×D7911yescharacteristic
337C22×C4⋊C4—791no—
338C22×C4⋊C4—791no—
339C22×C4⋊C4C2×D7911yescharacteristic
340C22×C22⋊C4—791no—
341C22×C22⋊C4—791no—
342C22×C22⋊C4—791no—
343C22×C42—791no—
344#400—792no—
345#400C2×D7912yescomplemented
346#399—794no—
347#399—792no—
348#399—792no—
349#398—792no—
350#398—794no—
351#398—794no—
352#398C2×D7912yescomplemented
353#397—792no—
354#397—794no—
355#397—792no—
356#396—794no—
357#396—794no—
358#393—794no—
359#393—794no—
360#391—794no—
361#391—794no—
362#389—794no—
363#388—794no—
364#388—794no—
365#388—794no—
366#388—794no—
367#387—794no—
368#387—794no—
369#387—794no—
370#387—794no—
371#387—794no—
372#387C2×D7914yescomplemented
373#385—792no—
374#385—792no—
375#385—792no—
376#385C2×D7912yes—
377#384—794no—
378#381—792no—
379#381—791no—
380#381C2×D7912yes—
381#381C2×D7911yescharacteristic
382#380—792no—
383#379—792no—
384#379—792no—
385C4×C22⋊C4—792no—
386C25—791no—
387C22×Q8—792no—
388C22×Q8—792no—
389C23×C4—791no—
390C23×C4—791no—
391C23×C4—791no—
392C23×C4C22×D7911yescharacteristic
393C23×C4C22×D7911yescharacteristic
394C23×C4C22×D7911yescharacteristic
395C4⋊Q8—1588no—
396C4⋊Q8—1588no—
397#126—15816no—
398#126—15816no—
399#125—1588no—
400#125—1588no—
401C22⋊Q8—1588no—
402C22⋊Q8—1588no—
403#117—24no—
404#117—1584no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—794no—
407C2×C4⋊C4—794no—
408C2×C4⋊C4—794no—
409C2×C4⋊C4—794no—
410C2×C4⋊C4—798no—
411C2×C4⋊C4—792no—
412C2×C4⋊C4—792no—
413C2×C4⋊C4—794no—
414C2×C4⋊C4—791no—
415C2×C4⋊C4—791no—
416C2×C4⋊C4—792no—
417C2×C4⋊C4—792no—
418C2×C4⋊C4—792no—
419C2×C4⋊C4—792no—
420C2×C4⋊C4—792no—
421C2×C4⋊C4—792no—
422C2×C4⋊C4—1584no—
423C2×C4⋊C4C22×D7912yes—
424C2×C4⋊C4C22×D7912yes—
425C2×C4⋊C4C22×D7911yescharacteristic
426C2×C4⋊C4C22×D7911yescharacteristic
427C2×C4⋊C4C22×D7914yes—
428C2×C22⋊C4—794no—
429C2×C22⋊C4—794no—
430C2×C22⋊C4—794no—
431C2×C22⋊C4—794no—
432C2×C22⋊C4—792no—
433C2×C22⋊C4—792no—
434C2×C22⋊C4—794no—
435C2×C22⋊C4—792no—
436C2×C22⋊C4—792no—
437C2×C22⋊C4—794no—
438C2×C22⋊C4—1584no—
439C2×C22⋊C4—1584no—
440C2×C22⋊C4—1584no—
441C2×C22⋊C4C22×D7912yes—
442C2×C22⋊C4C22×D7912yes—
443C2×C42—794no—
444C2×C42—792no—
445C2×C42—794no—
446C2×C42—1584no—
447C2×C42C22×D7912yes—
448#95—794no—
449#95—794no—
450#95—792no—
451#95—792no—
452#95—794no—
453#95—794no—
454#95—794no—
455#95—794no—
456#95—794no—
457#95C22×D7912yes—
458#95C22×D7912yes—
459#95C22×D7914yes—
460#95C22×D7914yes—
461#95C22×D7914yes—
462C24—3162no—
463C24—3162no—
464C24—792no—
465C24—1582no—
466C24—1584no—
467C24C23×D7911yescharacteristic
468C2×Q8—3164no—
469C2×Q8—3164no—
470C2×Q8—1584no—
471C2×Q8—1584no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—3162no—
478C22×C4—3162no—
479C22×C4—3162no—
480C22×C4—794no—
481C22×C4—794no—
482C22×C4—792no—
483C22×C4—792no—
484C22×C4—794no—
485C22×C4—792no—
486C22×C4—792no—
487C22×C4—792no—
488C22×C4—1582no—
489C22×C4D4×D7914yes—
490C22×C4C23×D7912yes—
491C22×C4C23×D7914yes—
492C22×C4C23×D7912yes—
493C22×C4C23×D7912yes—
494C22×C4C23×D7912yes—
495C22×C4C23×D7911yescharacteristic
496C22×C4C23×D7911yescharacteristic
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—3164no—
503C4⋊C4—3164no—
504C4⋊C4—3164no—
505C4⋊C4—3164no—
506C4⋊C4—798no—
507C4⋊C4—798no—
508C4⋊C4—1588no—
509C4⋊C4—1584no—
510C4⋊C4—1584no—
511C4⋊C4—1584no—
512C4⋊C4—1584no—
513C4⋊C4—1584no—
514C4⋊C4—1584no—
515C4⋊C4—1582no—
516C4⋊C4—1582no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—3164no—
520C22⋊C4—3164no—
521C22⋊C4—3164no—
522C22⋊C4—3164no—
523C22⋊C4—1588no—
524C22⋊C4—1588no—
525C22⋊C4—1584no—
526C22⋊C4—1584no—
527C22⋊C4—1584no—
528C22⋊C4—1584no—
529C42—24no—
530C42—3164no—
531C42—1588no—
532C42—1584no—
533C23—22no—
534C23—22no—
535C23—3164no—
536C23—3164no—
537C23—3164no—
538C23—6328no—
539C23—1584no—
540C23—1584no—
541C23—1582no—
542C23—1582no—
543C23—1582no—
544C23C2×D4×D7912yes—
545C23#1457920814yes—
546C23C24×D7911yescharacteristic
547Q8—6324no—
548Q8—6324no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—3164no—
567C2×C4—3164no—
568C2×C4—3164no—
569C2×C4—794no—
570C2×C4—794no—
571C2×C4—1584no—
572C2×C4—1584no—
573C2×C4—1584no—
574C2×C4—1584no—
575C2×C4—1584no—
576C2×C4—1584no—
577C2×C4—1582no—
578C2×C4—1582no—
579C2×C4—1582no—
580C2×C4—1584no—
581C2×C4—1584no—
582C2×C4C2×D4×D7912yes—
583C2×C4C2×D4×D7912yes—
584C22—22no—
585C22—22no—
586C22—28no—
587C22—3162no—
588C22—3162no—
589C22—3162no—
590C22—3164no—
591C22—3164no—
592C22—6328no—
593C22#1909913514yes—
594C22C22×D4×D7911yescharacteristic
595C22C2×#1457920812yes—
596C22#1909939912yes—
597C22#1909940012yes—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—44no—
605C4—3162no—
606C4—3162no—
607C4—3162no—
608C4—3164no—
609C4—1584no—
610C4—1584no—
611C2—24no—
612C2—6322no—
613C2C2×#1909913511yescharacteristic
614C2#2377704311yescharacteristic
615C2#2377705411yescharacteristic
616C2#2377802014yes—
subgroupG/H
center, FrattiniC23C24×D79
commutator, socle, cosocleC23×C79C25
FittingC79×#2484C2
prime radicalC24×D79C23
hypercenter#2484D79
hypocenterC79#36962
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2484

Complements

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

NH ≅ G/Ntype
1. C2×#19099191C2semidirect
3. #23771737C2semidirect
4. #23770300C2semidirect
5. #23770296C2semidirect
6. #23770231C2semidirect
7. #23770227C2semidirect
10. #23767876C2semidirect
11. #23767687C2semidirect
14. #23766001C2semidirect
15. C79×#2484C2semidirect
17. C2×#14579161C22semidirect
18. C2×#14579160C22semidirect
22. C2×#14579153C22semidirect
23. C2×#14579152C22semidirect
33. #19098502C22semidirect
37. #19098497C22semidirect
38. #19098496C22semidirect
39. #19098493C22semidirect
47. #19098467C22semidirect
48. #19098467C22semidirect
49. #19098460C22semidirect
54. #19098450C22semidirect
55. #19098448C22semidirect
56. #19098446C22semidirect
57. #19098446C22semidirect
58. #19098444C22semidirect
62. #19098432C22semidirect
71. #400×C79C22semidirect
72. #398×C79C22semidirect
73. #387×C79C22semidirect
320. #2484D79semidirect
324. C79#36962semidirect
345. #400C2×D79semidirect
352. #398C2×D79semidirect
372. #387C2×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190991912
C22×#145791562
#237717372
#237703002
#237702962
#237702312
#237702272
#237692112
#237685572
#237678762
#237676872
#237676762
#237676702
#237660012
C79×#24842
#3696279