Finite Group Databasebeta

← #34260032

Subgroups — #34260032

Order 20224.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773956C211yesmaximal, characteristic
2#23772348C211yesmaximal, characteristic, complemented
3#23772139C211yesmaximal, characteristic
4#23762293C211yesmaximal, characteristic, complemented
5#23762290C211yesmaximal, characteristic, complemented
6#23762245C211yesmaximal, characteristic, complemented
7#23762244C211yesmaximal, characteristic, complemented
8#23762235C211yesmaximal, characteristic, complemented
9C8×#11929612C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#334×D79C211yesmaximal, characteristic, complemented
12#23761801C211yesmaximal, characteristic, complemented
13#23761797C211yesmaximal, characteristic
14#23761679C211yesmaximal, characteristic, complemented
15C79×#1582C211yesmaximal, 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#19098691—21no—
28#19098690—21no—
29#19098690C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31C8×Dic158C2211yescharacteristic
32C4×#11929613C2211yescharacteristic
33C4×#11929611C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098642C2211yescharacteristic
36#19098641C2211yescharacteristic
37#19098637—21no—
38#19098634—21no—
39#19098628C2211yescharacteristic
40#19098614C2211yescharacteristic
41#19098588—21no—
42C4⋊C8×D79C2211yescharacteristic
43C4⋊C8×D79C2211yescharacteristic
44#19098528—21no—
45#19098520—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
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
71#14579167—44no—
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607D411yescharacteristic
75C2×#11929607D411yescharacteristic
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79C2×C8×D79C2×C411yescharacteristic
80C2×C8×D79C2×C411yescharacteristic
81#14579160C2×C411yescharacteristic
82#14579160C2×C411yescharacteristic
83#14579159C2×C411yescharacteristic
84#14579157C2×C411yescharacteristic
85#14579151—21no—
86#14579147—21no—
87#14579141C2×C411yescharacteristic
88#14579139—21no—
89#14579137C2×C411yescharacteristic
90#14579136C2×C411yescharacteristic
91#14579134—21no—
92#14579132C2×C411yescharacteristic
93C4×Dic158—21no—
94C4×Dic158C2311yescharacteristic
95C4×D316C2311yescharacteristic
96C4×C22⋊D79—21no—
97#14579118—21no—
98C42×D79C2311yescharacteristic
99#14579093C2×C411yescharacteristic
100#14579092C2×C411yescharacteristic
101#14579092C2×C411yescharacteristic
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)C2311yescharacteristic
104C4⋊(C79⋊C8)C2311yescharacteristic
105Dic79⋊C8—21no—
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(C2×D79)⋊C8—21no—
120#14579071—21no—
121#14579071D411yescharacteristic
122#14579071D411yescharacteristic
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79C2×C411yescharacteristic
129C8×Dic79C2×C411yescharacteristic
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—22no—
161#11929611—22no—
162#11929608—42no—
163#11929607—22no—
164#11929607—42no—
165#11929607SD1614yescomplemented
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—22no—
174C8×D79—41no—
175C8×D79—42no—
176C8×D79—42no—
177C8×D79—42no—
178C4⋊Dic79—21no—
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187C22⋊Dic79—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192C4×Dic79—21no—
193C4×Dic79C22×C411yescharacteristic
194C4×Dic79C2×D411yescharacteristic
195C4×Dic79#4111yescharacteristic
196C2×Q8×C79C22×C411yescharacteristic
197C2×D4×C79C22×C411yescharacteristic
198C22×C4×C79—21no—
199SD16×C79—41no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—42no—
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C4⋊C4×C79—21no—
209C4⋊C4×C79C22×C411yescharacteristic
210C4⋊C4×C79C22×C411yescharacteristic
211C22⋊C4×C79—21no—
212C42×C79—21no—
213C42×C79C2411yescharacteristic
214C23×C79—21no—
215C22×D79—21no—
216C22×D79C4×D411yescharacteristic
217Q8×C79—21no—
218Q8×C79—21no—
219D4×C79—21no—
220D4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79C23×C411yescharacteristic
227C2×C4×C79C22×D411yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C22⋊D79—42no—
230C22⋊D79—42no—
231C22⋊D79—81no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79C4×D411yescharacteristic
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238D316—21no—
239D316—21no—
240D316—42no—
241C4×D79—21no—
242C4×D79—21no—
243C4×D79—41no—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79C2×SD1612yes—
247C4×D79#13512yes—
248Dic158—21no—
249Dic158—21no—
250Dic158—42no—
251Dic158—81no—
252C8×C79—21no—
253C8×C79—21no—
254C8×C79—21no—
255C8×C79—41no—
256C8×C79—41no—
257C8×C79—41no—
258C8×C79C2×SD1612yes—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C79⋊C8—41no—
264C79⋊C8#13112yescomplemented
265C79⋊C8C2×SD1612yes—
266C22×C79—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79C2×C4×D411yescharacteristic
270C2×D79—21no—
271C2×D79—21no—
272C2×D79—41no—
273C2×D79—41no—
274C2×D79—41no—
275C2×D79C4×SD1612yes—
276C4×C79—21no—
277C4×C79—41no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79C2×#13111yescharacteristic
281C4×C79#56811yescharacteristic
282C4×C79C22×SD1611yescharacteristic
283C4×C79C2×#13511yescharacteristic
284Dic79—21no—
285Dic79—21no—
286Dic79—41no—
287Dic79—41no—
288Dic79—41no—
289Dic79C4×SD1612yes—
290#18067—791nomaximal, Sylow
291C2×C79—81no—
292C2×C79#293511yescharacteristic
293C2×C79C2×C4×SD1611yescharacteristic
294C2×C79#295711yescharacteristic
295D79—22no—
296D79—81no—
297#3167—791no—
298#2962—791no—
299#2935—791no—
300#1590—791no—
301#1590—791no—
302#1583—791no—
303#1583—791no—
304#1582—791no—
305#1582D7911yescharacteristic, complemented
306C8×SD16—791no—
307C8×SD16—791no—
308C2×#334—791no—
309#1481—791no—
310#1480—791no—
311C8×M16—791no—
312C79#1806711yescharacteristic, complemented, Sylow
313C2×#131—1581no—
314#533—791no—
315#532—791no—
316#498—791no—
317#492—791no—
318#487—791no—
319#486—791no—
320#465—791no—
321#465—791no—
322#465—791no—
323#465—791no—
324#446—791no—
325#446—1581no—
326#446—1581no—
327#446C2×D7911yescharacteristic
328C8×Q8—791no—
329C8×Q8—791no—
330C4×SD16—791no—
331C4×SD16—791no—
332C4×SD16—791no—
333C4×SD16C2×D7911yescharacteristic
334#436—791no—
335#436—791no—
336#435—1581no—
337#435—1581no—
338C8×D4—791no—
339C8×D4C2×D7911yescharacteristic
340#431—791no—
341#424—1581no—
342C2×C4⋊C8—791no—
343C2×C4⋊C8—791no—
344#408—1581no—
345#405—1581no—
346C4×M16—791no—
347C2×C4×C8—791no—
348#334—791no—
349#334—791no—
350#334—791no—
351#334C2×D7911yescharacteristic
352Q8⋊C8—791no—
353Q8⋊C8—791no—
354Q8⋊C8—791no—
355Q8⋊C8C2×D7911yescharacteristic
356D4⋊C8—791no—
357D4⋊C8—791no—
358D4⋊C8—791no—
359D4⋊C8C2×D7911yescharacteristic
360#322—791no—
361#322C2×D7911yescharacteristic
362C82—791no—
363C82—791no—
364C2×#41—1581no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16—791no—
368C2×SD16C4×D7911yescharacteristic
369#131—3164no—
370C2×M16—791no—
371C2×M16—791no—
372C2×M16—1581no—
373C2×M16—1581no—
374C22×C8—21no—
375C22×C8—791no—
376C22×C8—791no—
377C22×C8—1581no—
378C22×C8—1581no—
379C22×C8—1581no—
380C4⋊Q8—791no—
381C4⋊Q8—791no—
382C4⋊Q8—791no—
383#127—791no—
384#125—1581no—
385#124—1581no—
386C22⋊Q8—791no—
387C22⋊Q8—791no—
388C22⋊Q8—1581no—
389#121—791no—
390#121—791no—
391#121—1581no—
392C4×Q8—791no—
393C4×Q8—1581no—
394C4×Q8C22×D7911yescharacteristic
395C4×D4—791no—
396C4×D4—1581no—
397C4×D4C22×D7911yescharacteristic
398#117—1581no—
399C2×C42—791no—
400#106—791no—
401#106—791no—
402#106—791no—
403#106C4×D7911yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—791no—
406C4⋊C8—791no—
407C4⋊C8—1581no—
408C4⋊C8—1581no—
409C4⋊C8—1581no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8C22×D7911yescharacteristic
414C4⋊C8C22×D7911yescharacteristic
415#103—791no—
416#103—791no—
417#103—791no—
418#103C4×D7911yescharacteristic
419#102—791no—
420#102—791no—
421#102—791no—
422#102C4×D7911yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427C22⋊C8—1581no—
428#97—21no—
429#97—791no—
430#97—791no—
431#97—1581no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—791no—
435C4×C8—791no—
436C4×C8—791no—
437C4×C8—1581no—
438C4×C8—1581no—
439C4×C8C22×D7911yescharacteristic
440C4×C8C22×D7911yescharacteristic
441C4×C8C22×D7911yescharacteristic
442#41—3162no—
443#41—3162no—
444C2×Q8—791no—
445C2×Q8—1581no—
446C2×Q8—1581no—
447C2×Q8C2×C4×D7911yescharacteristic
448C2×D4—791no—
449C2×D4—1581no—
450C2×D4—1581no—
451C2×D4—1581no—
452C2×D4—1581no—
453C2×D4C2×C4×D7911yescharacteristic
454C22×C4—21no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—791no—
458C22×C4—1581no—
459SD16—41no—
460SD16—3161no—
461SD16—1582no—
462SD16—1582no—
463M16—3162no—
464M16—3162no—
465M16—794no—
466M16—1582no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—42no—
471C2×C8—3161no—
472C2×C8—3162no—
473C2×C8—3162no—
474C2×C8—3162no—
475C2×C8—791no—
476C2×C8—791no—
477C2×C8—791no—
478C2×C8—791no—
479C2×C8—1582no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8C2×C4×D7911yescharacteristic
484C2×C8C2×C4×D7911yescharacteristic
485C2×C8D4×D7911yescharacteristic
486C2×C8D4×D7911yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—791no—
489C4⋊C4—791no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4C2×C4×D7911yescharacteristic
498C4⋊C4C2×C4×D7911yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—1581no—
501C22⋊C4—1581no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C42—21no—
506C42—791no—
507C42—791no—
508C42—791no—
509C42—1581no—
510C42C23×D7911yescharacteristic
511C23—21no—
512C23—791no—
513C23—1581no—
514Q8—21no—
515Q8—21no—
516Q8—3162no—
517Q8—6321no—
518Q8—1581no—
519Q8—1581no—
520D4—21no—
521D4—21no—
522D4—3162no—
523D4—3162no—
524D4—3162no—
525D4—6321no—
526D4—1581no—
527D4—1581no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—791no—
537C2×C4—792no—
538C2×C4—792no—
539C2×C4—791no—
540C2×C4—791no—
541C2×C4—1581no—
542C2×C4—1581no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4C22×C4×D7911yescharacteristic
547C2×C4C2×D4×D7911yescharacteristic
548C2×C4#41×D7911yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—3161no—
556C8—3161no—
557C8—3161no—
558C8—792no—
559C8—792no—
560C8—1581no—
561C8—1581no—
562C8SD16×D7912yes—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—3161no—
567C22—3161no—
568C22—3161no—
569C22—792no—
570C22—1581no—
571C22—1581no—
572C22C4×D4×D7911yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—3161no—
578C4—3161no—
579C4—3161no—
580C4—792no—
581C4—1581no—
582C4—1581no—
583C4#131×D7911yescharacteristic
584C4#1909932911yescharacteristic
585C4C2×SD16×D7911yescharacteristic
586C4#135×D7911yescharacteristic
587C2—81no—
588C2—6321no—
589C2—1582no—
590C2#2377214111yescharacteristic
591C2C4×SD16×D7911yescharacteristic
592C2#2377231411yescharacteristic
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#18067
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. #23772348C2semidirect
4. #23762293C2semidirect
5. #23762290C2semidirect
6. #23762245C2semidirect
7. #23762244C2semidirect
8. #23762235C2semidirect
9. C8×#11929612C2semidirect
10. #23762197C2semidirect
11. #334×D79C2semidirect
12. #23761801C2semidirect
14. #23761679C2semidirect
15. C79×#1582C2semidirect
165. #11929607SD16semidirect
264. C79⋊C8#131semidirect
305. #1582D79semidirect
312. C79#18067semidirect

Maximal subgroups

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

subgroupindex
#237739562
#237723482
#237721392
#237622932
#237622902
#237622452
#237622442
#237622352
C8×#119296122
#237621972
#334×D792
#237618012
#237617972
#237616792
C79×#15822
#1806779