Finite Group Databasebeta

← #34257922

Subgroups — #34257922

Order 20224.

698 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772510C211yesmaximal, characteristic, complemented
2#23772438C211yesmaximal, characteristic, complemented
3C4×#14579188C211yesmaximal, characteristic
4#23762279C211yesmaximal, characteristic, complemented
5#23762278C211yesmaximal, characteristic, complemented
6#23762263C211yesmaximal, characteristic, complemented
7#23762262C211yesmaximal, characteristic, complemented
8#23762240C211yesmaximal, characteristic, complemented
9#23762232C211yesmaximal, characteristic, complemented
10#23762231C211yesmaximal, characteristic, complemented
11#23762149C211yesmaximal, characteristic, complemented
12#23761781C211yesmaximal, characteristic, complemented
13D4⋊C8×D79C211yesmaximal, characteristic, complemented
14#23761688C211yesmaximal, characteristic, complemented
15C79×#1583C211yesmaximal, characteristic, complemented
16C2×#14579188C411yescharacteristic
17C4×#11929625C2211yescharacteristic
18C4×D4×D79C2211yescharacteristic
19Q8×C79⋊C8C2211yescharacteristic, complemented
20#19098674C411yescharacteristic
21#19098672C411yescharacteristic
22#19098666C411yescharacteristic
23#19098664C411yescharacteristic
24C4×#11929611C2211yescharacteristic
25C4×D632C2211yescharacteristic
26C4×#11929609C2211yescharacteristic
27#19098642C2211yescharacteristic
28#19098641C2211yescharacteristic
29#19098640C2211yescharacteristic
30#19098639—21no—
31#19098638—21no—
32C8×D316—21no—
33C8×D316C2211yescharacteristic, complemented
34#19098628—21no—
35#19098628C2211yescharacteristic, complemented
36#19098615C2211yescharacteristic
37#19098614—21no—
38#19098614C2211yescharacteristic
39#19098602C411yescharacteristic
40C4⋊C8×D79—21no—
41C4⋊C8×D79C2211yescharacteristic
42#19098566C411yescharacteristic
43#19098563C411yescharacteristic
44#19098526—21no—
45#19098525—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic, complemented
49#19098315C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic, complemented
54D4⋊(C79⋊C8)C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic, complemented
57#19098285C2211yescharacteristic, complemented
58#19098285C2211yescharacteristic, complemented
59#19098285C2211yescharacteristic, complemented
60C8×Q8×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic, complemented
67C2×#11929625C2×C411yescharacteristic
68C2×D4×D79C2×C411yescharacteristic
69C22×C4×D79—21no—
70#14579188—24no—
71C2×#11929611C2×C411yescharacteristic
72C2×D632C2×C411yescharacteristic
73C2×#11929609C2×C411yescharacteristic
74C2×#11929608—21no—
75C2×#11929607—21no—
76C2×#11929607—21no—
77C2×#11929607C2×C411yescharacteristic
78C2×#11929607C2×C411yescharacteristic
79C2×C8×D79—21no—
80C2×C8×D79—21no—
81C2×C8×D79—21no—
82C2×C8×D79—21no—
83Q8×Dic79C2×C411yescharacteristic
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86C4×D316C2311yescharacteristic
87D4×Dic79C2×C411yescharacteristic
88#14579123—21no—
89#14579123C2×C411yescharacteristic
90#14579123C2×C411yescharacteristic
91C4×C22⋊D79—21no—
92(C22⋊D79)⋊C4—21no—
93#14579119—21no—
94#14579119C2×C411yescharacteristic
95C4⋊C4×D79C2×C411yescharacteristic
96C22⋊C4×D79—21no—
97C42×D79—21no—
98C42×D79C2311yescharacteristic
99#14579095C2×C411yescharacteristic
100#14579094C2×C411yescharacteristic
101#14579092C2×C411yescharacteristic
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)C2311yescharacteristic
104C4⋊(C79⋊C8)C2311yescharacteristic
105Dic79⋊C8—21no—
106Dic79⋊C8D411yescharacteristic
107Dic79⋊C8D411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579083C2×C411yescharacteristic
111#14579082C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113#14579081C2×C411yescharacteristic
114C22⋊(C79⋊C8)—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8D411yescharacteristic
121(C2×D79)⋊C8D411yescharacteristic
122#14579071C2×C411yescharacteristic
123#14579071C2×C411yescharacteristic
124C4×C79⋊C8—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79—21no—
129C8×Dic79—21no—
130C2×SD16×C79C2×C411yescharacteristic
131C2×M16×C79—21no—
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#106×C79C2×C411yescharacteristic
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145C23×D79—21no—
146#11929625—22no—
147#11929625M1614yescomplemented
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152D4×D79—22no—
153D4×D79—22no—
154C22×Dic79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79—21no—
160C2×C4×D79—21no—
161C2×C4×D79—81no—
162C2×C4×D79—81no—
163C2×C4×D79C22×C411yescharacteristic
164C2×C4×D79C2×D411yescharacteristic
165C2×C4×D79#4111yescharacteristic
166#11929611—22no—
167D632—22no—
168#11929609—22no—
169#11929608—81no—
170#11929607—22no—
171#11929607—22no—
172#11929607—81no—
173#11929607—81no—
174C2×C79⋊C8—21no—
175C2×C79⋊C8—21no—
176C2×C79⋊C8—21no—
177C2×C79⋊C8C22×C411yescharacteristic
178C2×C79⋊C8C22×C411yescharacteristic
179C2×C79⋊C8C2×D411yescharacteristic
180C2×C79⋊C8C2×D411yescharacteristic
181C8×D79—42no—
182C8×D79—81no—
183C8×D79—81no—
184C8×D79—81no—
185C4⋊Dic79—21no—
186C4⋊Dic79C22×C411yescharacteristic
187C4⋊Dic79C22×C411yescharacteristic
188Dic79⋊C4—21no—
189C22⋊Dic79—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192(C2×D79)⋊C4—21no—
193(C2×D79)⋊C4—21no—
194(C2×D79)⋊C4—21no—
195C4×Dic79—21no—
196C4×Dic79—21no—
197C4×Dic79C22×C411yescharacteristic
198C4×Dic79C2×D411yescharacteristic
199C4×Dic79#4111yescharacteristic
200C2×Q8×C79C22×C411yescharacteristic
201C2×D4×C79C22×C411yescharacteristic
202C22×C4×C79—21no—
203SD16×C79—22no—
204M16×C79—81no—
205C2×C8×C79—21no—
206C2×C8×C79—21no—
207C2×C8×C79—21no—
208C2×C8×C79C22×C411yescharacteristic
209C2×C8×C79C22×C411yescharacteristic
210C2×C8×C79C2×D411yescharacteristic
211C2×C8×C79C2×D411yescharacteristic
212C4⋊C4×C79—21no—
213C4⋊C4×C79C22×C411yescharacteristic
214C4⋊C4×C79C22×C411yescharacteristic
215C22⋊C4×C79—21no—
216C42×C79—21no—
217C42×C79C2411yescharacteristic
218C23×C79—21no—
219C22×D79—21no—
220C22×D79—21no—
221C22×D79—42no—
222C22×D79—81no—
223C22×D79—81no—
224C22×D79C4×D411yescharacteristic
225Q8×C79—21no—
226Q8×C79C2×M1612yes—
227D4×C79—21no—
228D4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79C23×C411yescharacteristic
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C22⋊D79—42no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—81no—
241C2×Dic79C4×D411yescharacteristic
242C2×Dic79C4×D411yescharacteristic
243C2×Dic79C4×D411yescharacteristic
244D316—24no—
245D316—21no—
246D316—21no—
247D316—21no—
248D316C2×M1612yes—
249C4×D79—21no—
250C4×D79—21no—
251C4×D79—24no—
252C4×D79—41no—
253C4×D79—41no—
254C4×D79—41no—
255C4×D79—41no—
256C4×D79—41no—
257C4×D79—81no—
258C4×D79—81no—
259C4×D79—81no—
260C4×D79C2×M1612yes—
261C4×D79#13112yes—
262C8×C79—21no—
263C8×C79—21no—
264C8×C79—21no—
265C8×C79—21no—
266C8×C79—41no—
267C8×C79—41no—
268C8×C79—41no—
269C79⋊C8—22no—
270C79⋊C8—21no—
271C79⋊C8—21no—
272C79⋊C8—21no—
273C79⋊C8—21no—
274C79⋊C8—41no—
275C79⋊C8—41no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79—41no—
279C22×C79C2×C4×D411yescharacteristic
280C2×D79—21no—
281C2×D79—21no—
282C2×D79—22no—
283C2×D79—41no—
284C2×D79—41no—
285C2×D79—41no—
286C2×D79—41no—
287C2×D79—41no—
288C2×D79—84no—
289C2×D79#43512yes—
290C4×C79—22no—
291C4×C79—21no—
292C4×C79—41no—
293C4×C79—41no—
294C4×C79C22×M1611yescharacteristic
295C4×C79C2×#13111yescharacteristic
296C4×C79#57611yescharacteristic
297C4×C79#57711yescharacteristic
298Dic79—21no—
299Dic79—21no—
300Dic79—41no—
301Dic79—41no—
302Dic79#43512yes—
303#17875—791nomaximal, Sylow
304C2×C79—81no—
305C2×C79C2×#43511yescharacteristic
306C2×C79#294611yescharacteristic
307C2×C79#295611yescharacteristic
308D79—22no—
309D79—42no—
310D79—81no—
311#2988—791no—
312#2975—791no—
313C4×AGL(1,ℤ/8)—791no—
314#1588—791no—
315#1588—791no—
316#1586—791no—
317#1586—791no—
318#1583—791no—
319#1583D7911yescharacteristic, complemented
320#1581—791no—
321#1581—791no—
322#1565—791no—
323#1476—791no—
324C2×D4⋊C8—791no—
325#1451—791no—
326C79#1787511yescharacteristic, complemented, Sylow
327C2×AGL(1,ℤ/8)—791no—
328C4×#41—791no—
329C2×C4×D4—791no—
330C8×Q8—791no—
331C8×Q8C2×D7911yescharacteristic, complemented
332#442—791no—
333#442—791no—
334#440—791no—
335#440—791no—
336C4×SD16—791no—
337C4×SD16C2×D7911yescharacteristic
338C4×D8—791no—
339C4×D8—791no—
340#436—791no—
341#436—791no—
342#436—791no—
343#436—1581no—
344#436—1581no—
345#436C2×D7911yescharacteristic
346C8×D4—791no—
347C8×D4—791no—
348C8×D4—1581no—
349C8×D4—1581no—
350#431—791no—
351#431—791no—
352#431—1581no—
353#428—791no—
354C2×C4⋊C8—791no—
355C2×C4⋊C8—1581no—
356#418—791no—
357#417—791no—
358#407—1581no—
359#407—1581no—
360C4×M16—791no—
361C2×C4×C8—791no—
362#335—791no—
363#335—791no—
364#334—791no—
365#334C2×D7911yescharacteristic, complemented
366Q8⋊C8—791no—
367Q8⋊C8C2×D7911yescharacteristic, complemented
368D4⋊C8—791no—
369D4⋊C8—791no—
370D4⋊C8—791no—
371D4⋊C8—791no—
372D4⋊C8—791no—
373D4⋊C8C2×D7911yescharacteristic
374#322—791no—
375#322—791no—
376#322—791no—
377#322C2×D7911yescharacteristic, complemented
378C2×#41—791no—
379C22×D4—791no—
380C23×C4—1581no—
381AGL(1,ℤ/8)—1584no—
382C2×SD16—791no—
383C2×SD16C4×D7911yescharacteristic
384C2×D8—791no—
385C2×D8—791no—
386C2×M16—21no—
387C2×M16—791no—
388C2×M16—791no—
389C2×M16—1581no—
390C2×M16—1581no—
391C2×M16—1581no—
392C22×C8—1581no—
393C22×C8—1581no—
394C22×C8—1581no—
395C22×C8—1581no—
396C4×Q8—791no—
397C4×Q8C22×D7911yescharacteristic
398C4×D4—791no—
399C4×D4—791no—
400C4×D4—791no—
401C4×D4—791no—
402C4×D4—791no—
403C4×D4—1581no—
404C4×D4—1581no—
405C4×D4—1581no—
406C4×D4—1581no—
407C4×D4C22×D7911yescharacteristic
408#117—791no—
409#117—1581no—
410C2×C4⋊C4—791no—
411C2×C22⋊C4—1581no—
412C2×C42—791no—
413C2×C42—1581no—
414#107—791no—
415#107—791no—
416#106—791no—
417#106C4×D7911yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—791no—
420C4⋊C8—791no—
421C4⋊C8—791no—
422C4⋊C8—791no—
423C4⋊C8—1581no—
424C4⋊C8—1581no—
425C4⋊C8C22×D7911yescharacteristic
426C4⋊C8C22×D7911yescharacteristic
427#103—791no—
428#103C4×D7911yescharacteristic
429#102—791no—
430#102—791no—
431#102—791no—
432#102—791no—
433#102—791no—
434#102C4×D7911yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—791no—
437C22⋊C8—791no—
438C22⋊C8—1581no—
439C22⋊C8—1581no—
440C22⋊C8—1581no—
441C22⋊C8—1581no—
442C22⋊C8—1581no—
443C22⋊C8—1581no—
444#97—791no—
445#97—791no—
446C4×C8—21no—
447C4×C8—791no—
448C4×C8—791no—
449C4×C8—791no—
450C4×C8—1581no—
451C4×C8—1581no—
452C4×C8—1581no—
453C4×C8C22×D7911yescharacteristic
454C4×C8C22×D7911yescharacteristic
455C4×C8C22×D7911yescharacteristic
456C24—1581no—
457#41—794no—
458#41—1582no—
459C2×Q8C2×C4×D7911yescharacteristic
460C2×D4—791no—
461C2×D4—791no—
462C2×D4—1581no—
463C2×D4—1582no—
464C2×D4—1582no—
465C2×D4—1581no—
466C2×D4C2×C4×D7911yescharacteristic
467C22×C4—21no—
468C22×C4—791no—
469C22×C4—791no—
470C22×C4—791no—
471C22×C4—6321no—
472C22×C4—6321no—
473C22×C4—1581no—
474C22×C4—1581no—
475C22×C4—1581no—
476C22×C4—1581no—
477C22×C4—1581no—
478C22×C4—1581no—
479C22×C4—1581no—
480SD16—22no—
481SD16—1582no—
482D8—1582no—
483D8—1582no—
484M16—81no—
485M16—6321no—
486M16—6321no—
487M16—6321no—
488M16—1582no—
489M16—1582no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—3162no—
494C2×C8—791no—
495C2×C8—791no—
496C2×C8—791no—
497C2×C8—791no—
498C2×C8—6321no—
499C2×C8—6321no—
500C2×C8—6321no—
501C2×C8—1581no—
502C2×C8—1581no—
503C2×C8—1581no—
504C2×C8C2×C4×D7911yescharacteristic
505C2×C8C2×C4×D7911yescharacteristic
506C2×C8D4×D7911yescharacteristic
507C2×C8D4×D7911yescharacteristic
508C4⋊C4—21no—
509C4⋊C4—791no—
510C4⋊C4—791no—
511C4⋊C4—1581no—
512C4⋊C4—1581no—
513C4⋊C4C2×C4×D7911yescharacteristic
514C4⋊C4C2×C4×D7911yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—1581no—
517C22⋊C4—1581no—
518C22⋊C4—1581no—
519C22⋊C4—1581no—
520C22⋊C4—1581no—
521C22⋊C4—1581no—
522C42—21no—
523C42—791no—
524C42—791no—
525C42—791no—
526C42—1581no—
527C42—1581no—
528C42C23×D7911yescharacteristic
529C23—21no—
530C23—3162no—
531C23—791no—
532C23—6321no—
533C23—6321no—
534C23—1581no—
535C23—1581no—
536Q8—21no—
537Q8M16×D7912yes—
538D4—21no—
539D4—21no—
540D4—3162no—
541D4—792no—
542D4—1584no—
543D4—1581no—
544D4—1581no—
545D4—1581no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—791no—
557C2×C4—792no—
558C2×C4—792no—
559C2×C4—791no—
560C2×C4—791no—
561C2×C4—6321no—
562C2×C4—6321no—
563C2×C4—6321no—
564C2×C4—6321no—
565C2×C4—1584no—
566C2×C4—1581no—
567C2×C4—1581no—
568C2×C4—1581no—
569C2×C4—1581no—
570C2×C4C22×C4×D7911yescharacteristic
571C2×C4C2×D4×D7911yescharacteristic
572C2×C4#41×D7911yescharacteristic
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—3161no—
581C8—3161no—
582C8—1582no—
583C8—1581no—
584C8—1581no—
585C8—1581no—
586C8—1581no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—3161no—
591C22—3161no—
592C22—3161no—
593C22—3161no—
594C22—3161no—
595C22—792no—
596C22—6324no—
597C22—1582no—
598C22—1581no—
599C22—1581no—
600C22C4×D4×D7911yescharacteristic
601C4—21no—
602C4—22no—
603C4—41no—
604C4—41no—
605C4—3161no—
606C4—3161no—
607C4—792no—
608C4—1581no—
609C4—1581no—
610C4C2×M16×D7911yescharacteristic
611C4#131×D7911yescharacteristic
612C4#1909936511yescharacteristic
613C4#1909936911yescharacteristic
614C2—81no—
615C2—3162no—
616C2—6321no—
617C2—1582no—
618C2#435×D7911yescharacteristic
619C2#2377222711yescharacteristic
620C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1583C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579188C4
cosocleC42×C79C24
hypercenter#1583D79
hypocenterC79#17875
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
1. #23772510C2semidirect
2. #23772438C2semidirect
4. #23762279C2semidirect
5. #23762278C2semidirect
6. #23762263C2semidirect
7. #23762262C2semidirect
8. #23762240C2semidirect
9. #23762232C2semidirect
10. #23762231C2semidirect
11. #23762149C2semidirect
12. #23761781C2semidirect
13. D4⋊C8×D79C2semidirect
14. #23761688C2semidirect
15. C79×#1583C2semidirect
19. Q8×C79⋊C8C22semidirect
33. C8×D316C22semidirect
35. #19098628C22semidirect
48. #19098316C22semidirect
49. #19098315C22semidirect
50. #19098313C22semidirect
51. Q8⋊(C79⋊C8)C22semidirect
53. #19098292C22semidirect
56. #19098290C22semidirect
57. #19098285C22semidirect
58. #19098285C22semidirect
59. #19098285C22semidirect
60. C8×Q8×C79C22semidirect
63. #334×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. #322×C79C22semidirect
147. #11929625M16semidirect
319. #1583D79semidirect
326. C79#17875semidirect
331. C8×Q8C2×D79semidirect
365. #334C2×D79semidirect
367. Q8⋊C8C2×D79semidirect
377. #322C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237725102
#237724382
C4×#145791882
#237622792
#237622782
#237622632
#237622622
#237622402
#237622322
#237622312
#237621492
#237617812
D4⋊C8×D792
#237616882
C79×#15832
#1787579