Finite Group Databasebeta

← #34260289

Subgroups — #34260289

Order 20224.

604 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773956C211yesmaximal, characteristic
2#23772498C211yesmaximal, characteristic, complemented
3#23772463C211yesmaximal, characteristic
4#23762300C211yesmaximal, characteristic, complemented
5#23762299C211yesmaximal, characteristic, complemented
6#23762245C211yesmaximal, characteristic, complemented
7#23762244C211yesmaximal, characteristic, complemented
8#23762235C211yesmaximal, characteristic, complemented
9#23762215C211yesmaximal, characteristic, complemented
10#23762212C211yesmaximal, characteristic, complemented
11#23762145C211yesmaximal, characteristic, complemented
12#23761819C211yesmaximal, characteristic, complemented
13#23761815C211yesmaximal, characteristic
14#23761688C211yesmaximal, characteristic, complemented
15C79×#1582C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17#19099110C2211yescharacteristic
18#19098960C411yescharacteristic
19#19098934C411yescharacteristic
20#19098912C411yescharacteristic
21#19098908C411yescharacteristic
22#19098795C411yescharacteristic
23#19098793C411yescharacteristic
24#19098791C411yescharacteristic
25#19098789C411yescharacteristic
26#19098691—21no—
27#19098690C2211yescharacteristic
28Q8×C79⋊C8C2211yescharacteristic
29C8×Dic158C2211yescharacteristic
30#19098687—21no—
31C4×#11929613C2211yescharacteristic
32C4×#11929611C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34#19098642C2211yescharacteristic
35#19098641C2211yescharacteristic
36#19098638—21no—
37#19098628C2211yescharacteristic
38C8×C22⋊D79—21no—
39#19098625—21no—
40#19098623—21no—
41#19098615C2211yescharacteristic
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098529—21no—
45#19098528—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098285C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929613C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79C2×C411yescharacteristic
76C2×C8×D79C2×C411yescharacteristic
77#14579160C2×C411yescharacteristic
78#14579160C2×C411yescharacteristic
79#14579159C2×C411yescharacteristic
80#14579157C2×C411yescharacteristic
81#14579151—21no—
82#14579147—21no—
83#14579141C2×C411yescharacteristic
84#14579139—21no—
85#14579137C2×C411yescharacteristic
86#14579136C2×C411yescharacteristic
87#14579134—21no—
88#14579132C2×C411yescharacteristic
89C4×Dic158—21no—
90C4×Dic158C2311yescharacteristic
91C4×D316C2311yescharacteristic
92C4×C22⋊D79—21no—
93#14579118—21no—
94C42×D79C2311yescharacteristic
95#14579093C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8D411yescharacteristic
104Dic79⋊C8D411yescharacteristic
105#14579086C2×C411yescharacteristic
106#14579085C2×C411yescharacteristic
107#14579084C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579082C2×C411yescharacteristic
110#14579081C2×C411yescharacteristic
111C22⋊(C79⋊C8)—21no—
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8D411yescharacteristic
117(C2×D79)⋊C8D411yescharacteristic
118#14579071—21no—
119#14579071—21no—
120C4×C79⋊C8—21no—
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79C2×C411yescharacteristic
128C8×Dic79C2×C411yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140#97×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—81no—
145#11929624—81no—
146C2×Dic158—21no—
147C2×Dic158—21no—
148C2×Dic158C22×C411yescharacteristic
149C2×D316—21no—
150C2×D316C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152C2×C22⋊D79—21no—
153C2×C22⋊D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79C22×C411yescharacteristic
156C2×C4×D79C2×D411yescharacteristic
157C2×C4×D79#4111yescharacteristic
158#11929613—41no—
159#11929612—22no—
160#11929611—22no—
161#11929608—81no—
162#11929607—42no—
163#11929607—81no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C8×D79—22no—
172C8×D79—41no—
173C8×D79—81no—
174C4⋊Dic79—21no—
175C4⋊Dic79C22×C411yescharacteristic
176C4⋊Dic79C22×C411yescharacteristic
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183C22⋊Dic79—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188C4×Dic79—21no—
189C4×Dic79C22×C411yescharacteristic
190C4×Dic79C2×D411yescharacteristic
191C4×Dic79#4111yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193C2×D4×C79C22×C411yescharacteristic
194C22×C4×C79—21no—
195SD16×C79—41no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—81no—
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C4⋊C4×C79—21no—
205C4⋊C4×C79C22×C411yescharacteristic
206C4⋊C4×C79C22×C411yescharacteristic
207C22⋊C4×C79—21no—
208C42×C79—21no—
209C42×C79C2411yescharacteristic
210C23×C79—21no—
211C22×D79—21no—
212C22×D79C4×D411yescharacteristic
213Q8×C79—21no—
214Q8×C79—21no—
215D4×C79—21no—
216D4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C22⋊D79—42no—
226C22⋊D79—81no—
227C22⋊D79—81no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79C4×D411yescharacteristic
232C2×Dic79C4×D411yescharacteristic
233C2×Dic79C4×D411yescharacteristic
234D316—21no—
235D316—21no—
236D316—81no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—21no—
241C4×D79—22no—
242C4×D79—41no—
243C4×D79—41no—
244Dic158—21no—
245Dic158—21no—
246Dic158—42no—
247Dic158—81no—
248C8×C79—21no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—41no—
253C8×C79—41no—
254C8×C79—41no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C79⋊C8—41no—
261C79⋊C8#13112yes—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79C2×C4×D411yescharacteristic
266C2×D79—21no—
267C2×D79—21no—
268C2×D79—21no—
269C2×D79—41no—
270C2×D79—41no—
271C2×D79—41no—
272C4×C79—21no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—41no—
276C4×C79C2×#13111yescharacteristic
277C4×C79#56811yescharacteristic
278C4×C79#57711yescharacteristic
279C4×C79#57711yescharacteristic
280Dic79—21no—
281Dic79—21no—
282Dic79—41no—
283Dic79—41no—
284Dic79—41no—
285Dic79#44412yes—
286#18089—791nomaximal, Sylow
287C2×C79—81no—
288C2×C79#293511yescharacteristic
289C2×C79#294911yescharacteristic
290C2×C79C2×#44411yescharacteristic
291D79—41no—
292D79—81no—
293#3167—791no—
294#2986—791no—
295#2979—791no—
296#1592—791no—
297#1592—791no—
298#1583—791no—
299#1583—791no—
300#1582—791no—
301#1582D7911yescharacteristic, complemented
302SD16⋊C8—791no—
303SD16⋊C8—791no—
304#1564—791no—
305#1485—791no—
306#1484—791no—
307#1451—791no—
308C79#1808911yescharacteristic, complemented, Sylow
309#533—791no—
310#532—791no—
311#498—791no—
312#492—791no—
313#487—791no—
314#486—791no—
315#465—791no—
316#465—791no—
317#465—791no—
318#465—791no—
319#446—791no—
320#446—1581no—
321#446C2×D7911yescharacteristic
322C8×Q8—791no—
323C8×Q8—791no—
324C8×Q8—1581no—
325C4×SD16—791no—
326C4×SD16—791no—
327C4×SD16—791no—
328C4×SD16C2×D7911yescharacteristic
329#436—791no—
330#436—791no—
331#436—1581no—
332C8×D4—791no—
333C8×D4—1581no—
334C8×D4C2×D7911yescharacteristic
335#433—1581no—
336#433—1581no—
337#431—791no—
338#431—791no—
339#431—791no—
340#408—1581no—
341#408—1581no—
342C4×M16—791no—
343C2×C4×C8—791no—
344#335—791no—
345#335—791no—
346#334—791no—
347#334C2×D7911yescharacteristic
348Q8⋊C8—791no—
349Q8⋊C8—791no—
350Q8⋊C8—791no—
351Q8⋊C8C2×D7911yescharacteristic
352D4⋊C8—791no—
353D4⋊C8—791no—
354D4⋊C8—791no—
355D4⋊C8C2×D7911yescharacteristic
356#322—791no—
357#322—791no—
358#322—791no—
359#322C2×D7911yescharacteristic
360C2×#41—1581no—
361C2×SD16—791no—
362C2×SD16—791no—
363C2×SD16—791no—
364C2×SD16C4×D7911yescharacteristic
365C2×M16—1581no—
366C2×M16—1581no—
367C2×M16—1581no—
368C22×C8—21no—
369C22×C8—791no—
370C22×C8—791no—
371C22×C8—1581no—
372C4⋊Q8—791no—
373C4⋊Q8—791no—
374C4⋊Q8—791no—
375#127—791no—
376#125—1581no—
377#124—1581no—
378C22⋊Q8—791no—
379C22⋊Q8—791no—
380C22⋊Q8—1581no—
381#121—791no—
382#121—791no—
383#121—1581no—
384C4×Q8—791no—
385C4×Q8—1581no—
386C4×Q8C22×D7911yescharacteristic
387C4×D4—791no—
388C4×D4—1581no—
389C4×D4C22×D7911yescharacteristic
390#117—1581no—
391C2×C42—791no—
392#106—791no—
393#106—791no—
394#106—791no—
395#106C4×D7911yescharacteristic
396C4⋊C8—21no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8C22×D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406#103—791no—
407#103—791no—
408#103—791no—
409#103C4×D7911yescharacteristic
410#102—791no—
411#102—791no—
412#102—791no—
413#102C4×D7911yescharacteristic
414C22⋊C8—21no—
415C22⋊C8—791no—
416C22⋊C8—791no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422#97—21no—
423#97—1581no—
424#97—1581no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—791no—
429C4×C8—791no—
430C4×C8—1581no—
431C4×C8—1581no—
432C4×C8—1581no—
433C4×C8—1581no—
434C4×C8C22×D7911yescharacteristic
435C4×C8C22×D7911yescharacteristic
436C4×C8C22×D7911yescharacteristic
437#41—6321no—
438#41—6321no—
439C2×Q8—791no—
440C2×Q8—1581no—
441C2×Q8—1581no—
442C2×Q8C2×C4×D7911yescharacteristic
443C2×D4—791no—
444C2×D4—1581no—
445C2×D4—1581no—
446C2×D4—1581no—
447C2×D4—1581no—
448C2×D4C2×C4×D7911yescharacteristic
449C22×C4—21no—
450C22×C4—791no—
451C22×C4—791no—
452C22×C4—791no—
453C22×C4—1581no—
454SD16—41no—
455SD16—3161no—
456SD16—1582no—
457SD16—1582no—
458M16—3162no—
459M16—6321no—
460M16—6321no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—81no—
465C2×C8—3161no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—6321no—
471C2×C8—1582no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8C2×C4×D7911yescharacteristic
476C2×C8C2×C4×D7911yescharacteristic
477C2×C8D4×D7911yescharacteristic
478C2×C8D4×D7911yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—791no—
481C4⋊C4—791no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4C2×C4×D7911yescharacteristic
490C4⋊C4C2×C4×D7911yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—1581no—
493C22⋊C4—1581no—
494C22⋊C4—1581no—
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C42—21no—
498C42—791no—
499C42—791no—
500C42—791no—
501C42—1581no—
502C42C23×D7911yescharacteristic
503C23—21no—
504C23—791no—
505C23—1581no—
506Q8—21no—
507Q8—21no—
508Q8—3162no—
509Q8—6321no—
510Q8—1581no—
511Q8—1581no—
512D4—21no—
513D4—21no—
514D4—3162no—
515D4—6321no—
516D4—6321no—
517D4—6321no—
518D4—1581no—
519D4—1581no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—791no—
528C2×C4—791no—
529C2×C4—791no—
530C2×C4—1582no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4C22×C4×D7911yescharacteristic
539C2×C4C2×D4×D7911yescharacteristic
540C2×C4#41×D7911yescharacteristic
541C8—21no—
542C8—21no—
543C8—21no—
544C8—21no—
545C8—41no—
546C8—41no—
547C8—41no—
548C8—3161no—
549C8—3161no—
550C8—3161no—
551C8—792no—
552C8—1581no—
553C8—1581no—
554C8—1581no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—3161no—
559C22—3161no—
560C22—3161no—
561C22—1581no—
562C22—1581no—
563C22—1581no—
564C22C4×D4×D7911yescharacteristic
565C4—21no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—3161no—
570C4—3161no—
571C4—3161no—
572C4—792no—
573C4—1581no—
574C4—1581no—
575C4#131×D7911yescharacteristic
576C4#1909932911yescharacteristic
577C4#1909936911yescharacteristic
578C4#1909937011yescharacteristic
579C2—81no—
580C2—3161no—
581C2—6321no—
582C2#2377214111yescharacteristic
583C2#2377225411yescharacteristic
584C2#444×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1582C2
socleC22×C79C2×C4×D4
prime radical#19099110C22
cosocleC42×C79C24
hypercenter#1582D79
hypocenterC79#18089
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #23772498C2semidirect
4. #23762300C2semidirect
5. #23762299C2semidirect
6. #23762245C2semidirect
7. #23762244C2semidirect
8. #23762235C2semidirect
9. #23762215C2semidirect
10. #23762212C2semidirect
11. #23762145C2semidirect
12. #23761819C2semidirect
14. #23761688C2semidirect
15. C79×#1582C2semidirect
301. #1582D79semidirect
308. C79#18089semidirect

Maximal subgroups

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

subgroupindex
#237739562
#237724982
#237724632
#237623002
#237622992
#237622452
#237622442
#237622352
#237622152
#237622122
#237621452
#237618192
#237618152
#237616882
C79×#15822
#1808979