Finite Group Databasebeta

← #34259003

Subgroups — #34259003

Order 20224.

658 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773956C211yesmaximal, characteristic
2#23772331C211yesmaximal, characteristic, complemented
3#23772107C211yesmaximal, characteristic
4#23762281C211yesmaximal, characteristic, complemented
5#23762279C211yesmaximal, characteristic, complemented
6#23762277C211yesmaximal, characteristic, complemented
7#23762275C211yesmaximal, characteristic, complemented
8#23762259C211yesmaximal, characteristic, complemented
9#23762257C211yesmaximal, characteristic, complemented
10#23762255C211yesmaximal, characteristic, complemented
11#23762157C211yesmaximal, characteristic, complemented
12#23761853C211yesmaximal, characteristic, complemented
13#23761849C211yesmaximal, characteristic
14#322×D79C211yesmaximal, characteristic, complemented
15C79×#1585C211yesmaximal, characteristic, complemented
16C2×#14579167—21no—
17#19099117C2211yescharacteristic
18#19099110C2211yescharacteristic
19#19098960C411yescharacteristic
20#19098934C411yescharacteristic
21#19098912C411yescharacteristic
22#19098908C411yescharacteristic
23#19098795C411yescharacteristic
24#19098793C411yescharacteristic
25#19098791C411yescharacteristic
26#19098789C411yescharacteristic
27#19098693C2211yescharacteristic
28#19098691—21no—
29Q8×C79⋊C8C2211yescharacteristic
30#19098687—21no—
31#19098687C2211yescharacteristic
32C4×#11929613C2211yescharacteristic
33C4×#11929611C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098641C2211yescharacteristic
36#19098640C2211yescharacteristic
37#19098638—21no—
38C8×D316C2211yescharacteristic
39C8×C22⋊D79—21no—
40#19098588—21no—
41#19098585C2211yescharacteristic
42#19098585C2211yescharacteristic
43#19098528—21no—
44#19098520—21no—
45C4×C8×D79C2211yescharacteristic
46C4×C8×D79C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098315C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929613C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71#14579167—44no—
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79C2×C411yescharacteristic
79C2×C8×D79C2×C411yescharacteristic
80C2×C8×D79D411yescharacteristic
81C2×C8×D79D411yescharacteristic
82#14579160C2×C411yescharacteristic
83#14579160C2×C411yescharacteristic
84#14579159C2×C411yescharacteristic
85#14579157C2×C411yescharacteristic
86#14579151—21no—
87#14579147—21no—
88#14579141C2×C411yescharacteristic
89#14579139—21no—
90#14579137C2×C411yescharacteristic
91#14579136C2×C411yescharacteristic
92#14579134—21no—
93#14579132C2×C411yescharacteristic
94C4×Dic158—21no—
95C4×Dic158C2311yescharacteristic
96C4×D316C2311yescharacteristic
97C4×C22⋊D79—21no—
98#14579118—21no—
99C42×D79C2311yescharacteristic
100#14579093C2×C411yescharacteristic
101#14579092C2×C411yescharacteristic
102#14579092C2×C411yescharacteristic
103C4⋊(C79⋊C8)—21no—
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109Dic79⋊C8—21no—
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579084C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579082C2×C411yescharacteristic
115#14579081C2×C411yescharacteristic
116C22⋊(C79⋊C8)—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119#14579071—21no—
120C4×C79⋊C8—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79C2×C411yescharacteristic
127C8×Dic79C2×C411yescharacteristic
128C8×Dic79D411yescharacteristic
129C8×Dic79D411yescharacteristic
130C2×SD16×C79C2×C411yescharacteristic
131C22×C8×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—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929624—42no—
146#11929624—42no—
147C2×Dic158—21no—
148C2×Dic158—21no—
149C2×Dic158C22×C411yescharacteristic
150C2×D316—21no—
151C2×D316C22×C411yescharacteristic
152C2×C22⋊D79—21no—
153C2×C22⋊D79—21no—
154C2×C22⋊D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79C22×C411yescharacteristic
157C2×C4×D79C2×D411yescharacteristic
158C2×C4×D79#4111yescharacteristic
159#11929613—41no—
160#11929612—41no—
161#11929611—41no—
162#11929608—42no—
163#11929607—42no—
164#11929607—42no—
165#11929607—42no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C8×D79—24no—
174C8×D79—22no—
175C8×D79—22no—
176C8×D79—41no—
177C8×D79—41no—
178C8×D79—42no—
179C4⋊Dic79—21no—
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187Dic79⋊C4—21no—
188C22⋊Dic79—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192(C2×D79)⋊C4—21no—
193C4×Dic79—21no—
194C4×Dic79C22×C411yescharacteristic
195C4×Dic79C2×D411yescharacteristic
196C4×Dic79#4111yescharacteristic
197C2×Q8×C79C22×C411yescharacteristic
198C2×D4×C79C22×C411yescharacteristic
199C22×C4×C79—21no—
200SD16×C79—41no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—42no—
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C22×C411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C2×C8×C79C2×D411yescharacteristic
209C4⋊C4×C79—21no—
210C4⋊C4×C79C22×C411yescharacteristic
211C4⋊C4×C79C22×C411yescharacteristic
212C22⋊C4×C79—21no—
213C42×C79—21no—
214C42×C79C2411yescharacteristic
215C23×C79—21no—
216C22×D79—21no—
217C22×D79C4×D411yescharacteristic
218Q8×C79—21no—
219Q8×C79—21no—
220D4×C79—21no—
221D4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—42no—
231C22⋊D79—42no—
232C22⋊D79—42no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238C2×Dic79C4×D411yescharacteristic
239D316—21no—
240D316—21no—
241D316—42no—
242C4×D79—21no—
243C4×D79—21no—
244C4×D79—22no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79AGL(1,ℤ/8)12yes—
248C4×D79#13712yes—
249Dic158—21no—
250Dic158—21no—
251Dic158—42no—
252Dic158—42no—
253C8×C79—22no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—41no—
259C8×C79—41no—
260C79⋊C8—22no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79C2×C4×D411yescharacteristic
271C2×D79—21no—
272C2×D79—21no—
273C2×D79—41no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79#44012yes—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79#56811yescharacteristic
282C4×C79#56811yescharacteristic
283C4×C79C2×AGL(1,ℤ/8)11yescharacteristic
284C4×C79C2×#13711yescharacteristic
285Dic79—21no—
286Dic79—41no—
287Dic79—41no—
288Dic79—41no—
289Dic79#44012yes—
290Dic79#44312yes—
291#17979—791nomaximal, Sylow
292C2×C79—81no—
293C2×C79#293211yescharacteristic
294C2×C79C2×#44011yescharacteristic
295C2×C79C2×#44311yescharacteristic
296D79—22no—
297D79—81no—
298#3167—791no—
299#2960—791no—
300#2931—791no—
301#1588—791no—
302#1588—791no—
303#1588—791no—
304#1588—791no—
305#1585—791no—
306#1585—791no—
307#1585—791no—
308#1585D7911yescharacteristic, complemented
309#1567—791no—
310#1493—791no—
311#1492—791no—
312C2×#322—791no—
313C79#1797911yescharacteristic, complemented, Sylow
314C2×#131—1581no—
315#533—791no—
316#532—791no—
317#498—791no—
318#492—791no—
319#487—791no—
320#486—791no—
321#465—791no—
322#465—791no—
323#465—791no—
324#465—791no—
325#446—791no—
326#446—1581no—
327#446C2×D7911yescharacteristic
328C8×Q8—791no—
329C8×Q8—791no—
330C8×Q8—1581no—
331C4×SD16—791no—
332C4×SD16—791no—
333C4×SD16—791no—
334C4×SD16C2×D7911yescharacteristic
335#436—791no—
336#436—791no—
337#436—1581no—
338C8×D4—791no—
339C8×D4—1581no—
340C8×D4C2×D7911yescharacteristic
341#424—1581no—
342#423—791no—
343#423—791no—
344#408—1581no—
345#405—1581no—
346C2×C4×C8—791no—
347C2×C4×C8—791no—
348C2×C4×C8—791no—
349#335—791no—
350#335—791no—
351#335—791no—
352#335C2×D7911yescharacteristic
353Q8⋊C8—791no—
354Q8⋊C8—791no—
355Q8⋊C8—791no—
356Q8⋊C8C2×D7911yescharacteristic
357D4⋊C8—791no—
358D4⋊C8—791no—
359D4⋊C8—791no—
360D4⋊C8C2×D7911yescharacteristic
361#322—791no—
362#322—791no—
363#322—791no—
364#322C2×D7911yescharacteristic
365C2×#41—1581no—
366C2×SD16—791no—
367C2×SD16—791no—
368C2×SD16—791no—
369C2×SD16C4×D7911yescharacteristic
370#131—3164no—
371C2×M16—1581no—
372C2×M16—1581no—
373C2×M16—1581no—
374C2×M16—1581no—
375C22×C8—21no—
376C22×C8—791no—
377C22×C8—791no—
378C22×C8—791no—
379C22×C8—791no—
380C22×C8—1581no—
381C22×C8—1581no—
382C4⋊Q8—791no—
383C4⋊Q8—791no—
384C4⋊Q8—791no—
385#127—791no—
386#125—1581no—
387#124—1581no—
388C22⋊Q8—791no—
389C22⋊Q8—791no—
390C22⋊Q8—1581no—
391#121—791no—
392#121—791no—
393#121—1581no—
394C4×Q8—791no—
395C4×Q8—1581no—
396C4×Q8C22×D7911yescharacteristic
397C4×D4—791no—
398C4×D4—1581no—
399C4×D4C22×D7911yescharacteristic
400#117—1581no—
401C2×C42—791no—
402#106—791no—
403#106—791no—
404#106—791no—
405#106C4×D7911yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—1581no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8C22×D7911yescharacteristic
415C4⋊C8C22×D7911yescharacteristic
416#103—791no—
417#103—791no—
418#103—791no—
419#103C4×D7911yescharacteristic
420#102—791no—
421#102—791no—
422#102—791no—
423#102C4×D7911yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427C22⋊C8—1581no—
428#97—21no—
429#97—1581no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—791no—
436C4×C8—791no—
437C4×C8—1581no—
438C4×C8—1581no—
439C4×C8—1581no—
440C4×C8C22×D7911yescharacteristic
441C4×C8C22×D7911yescharacteristic
442C4×C8C22×D7911yescharacteristic
443#41—3162no—
444#41—3162no—
445C2×Q8—791no—
446C2×Q8—1581no—
447C2×Q8—1581no—
448C2×Q8C2×C4×D7911yescharacteristic
449C2×D4—791no—
450C2×D4—1581no—
451C2×D4—1581no—
452C2×D4—1581no—
453C2×D4—1581no—
454C2×D4C2×C4×D7911yescharacteristic
455C22×C4—21no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—1581no—
460SD16—41no—
461SD16—3161no—
462SD16—3161no—
463SD16—3161no—
464M16—3162no—
465M16—3162no—
466M16—3162no—
467M16—3162no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—42no—
472C2×C8—3162no—
473C2×C8—3161no—
474C2×C8—3161no—
475C2×C8—791no—
476C2×C8—791no—
477C2×C8—791no—
478C2×C8—791no—
479C2×C8—1582no—
480C2×C8—1582no—
481C2×C8—1584no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8C2×C4×D7911yescharacteristic
486C2×C8C2×C4×D7911yescharacteristic
487C2×C8D4×D7911yescharacteristic
488C2×C8D4×D7911yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—791no—
491C4⋊C4—791no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4C2×C4×D7911yescharacteristic
500C4⋊C4C2×C4×D7911yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C42—21no—
508C42—791no—
509C42—791no—
510C42—791no—
511C42—1581no—
512C42C23×D7911yescharacteristic
513C23—21no—
514C23—791no—
515C23—1581no—
516Q8—21no—
517Q8—21no—
518Q8—3162no—
519Q8—3162no—
520Q8—1581no—
521Q8—1581no—
522D4—21no—
523D4—21no—
524D4—3162no—
525D4—3162no—
526D4—3162no—
527D4—3162no—
528D4—1581no—
529D4—1581no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—791no—
538C2×C4—791no—
539C2×C4—792no—
540C2×C4—792no—
541C2×C4—791no—
542C2×C4—1582no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4C22×C4×D7911yescharacteristic
549C2×C4C2×D4×D7911yescharacteristic
550C2×C4#41×D7911yescharacteristic
551C8—22no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—3161no—
559C8—3161no—
560C8—1582no—
561C8—1581no—
562C8—1581no—
563C8—1581no—
564C8—1581no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—3161no—
569C22—3161no—
570C22—3161no—
571C22—792no—
572C22—1581no—
573C22—1581no—
574C22C4×D4×D7911yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—3161no—
580C4—3161no—
581C4—3161no—
582C4—792no—
583C4—792no—
584C4—1581no—
585C4#1909932811yescharacteristic
586C4#1909932911yescharacteristic
587C4AGL(1,ℤ/8)×D7911yescharacteristic
588C4#137×D7911yescharacteristic
589C2—81no—
590C2—6321no—
591C2—1582no—
592C2#2377211411yescharacteristic
593C2#440×D7911yescharacteristic
594C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1585C2
socleC22×C79C2×C4×D4
prime radical#19099110C22
cosocleC42×C79C24
hypercenter#1585D79
hypocenterC79#17979
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1585

Complements

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

NH ≅ G/Ntype
2. #23772331C2semidirect
4. #23762281C2semidirect
5. #23762279C2semidirect
6. #23762277C2semidirect
7. #23762275C2semidirect
8. #23762259C2semidirect
9. #23762257C2semidirect
10. #23762255C2semidirect
11. #23762157C2semidirect
12. #23761853C2semidirect
14. #322×D79C2semidirect
15. C79×#1585C2semidirect
308. #1585D79semidirect
313. C79#17979semidirect

Maximal subgroups

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

subgroupindex
#237739562
#237723312
#237721072
#237622812
#237622792
#237622772
#237622752
#237622592
#237622572
#237622552
#237621572
#237618532
#237618492
#322×D792
C79×#15852
#1797979