Finite Group Databasebeta

← #34258015

Subgroups — #34258015

Order 20224.

682 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#23772365C211yesmaximal, characteristic, complemented
2#23772363C211yesmaximal, characteristic, complemented
3C4×#14579182C211yesmaximal, characteristic
4#23762295C211yesmaximal, characteristic, complemented
5#23762294C211yesmaximal, characteristic, complemented
6#23762292C211yesmaximal, characteristic, complemented
7#23762291C211yesmaximal, characteristic, complemented
8#23762289C211yesmaximal, characteristic, complemented
9#23762283C211yesmaximal, characteristic, complemented
10#23762282C211yesmaximal, characteristic, complemented
11#23762145C211yesmaximal, characteristic, complemented
12#23761781C211yesmaximal, characteristic, complemented
13#23761779C211yesmaximal, characteristic, complemented
14C82×D79C211yesmaximal, characteristic, complemented
15C79×#1590C211yesmaximal, characteristic, complemented
16C2×#14579182C411yescharacteristic
17C4×#11929625C2211yescharacteristic
18C4×#11929623C2211yescharacteristic
19#19098694C2211yescharacteristic
20#19098693C2211yescharacteristic
21#19098690C2211yescharacteristic
22C4×#11929614C2211yescharacteristic
23#19098655C411yescharacteristic
24C4×#11929613C2211yescharacteristic
25SD16×Dic79C411yescharacteristic
26#19098650C411yescharacteristic
27C4×#11929609C2211yescharacteristic
28#19098643C411yescharacteristic
29#19098642C2211yescharacteristic
30#19098641C2211yescharacteristic
31#19098640C2211yescharacteristic
32#19098637—21no—
33#19098636—21no—
34#19098633—21no—
35#19098632—21no—
36#19098615—21no—
37#19098615C2211yescharacteristic
38#19098615C2211yescharacteristic
39#19098613—21no—
40#19098598C411yescharacteristic
41#19098558C411yescharacteristic
42#19098556C411yescharacteristic
43#19098515—21no—
44M16×Dic79—21no—
45C4×C8×D79C2211yescharacteristic
46C4×C8×D79C2211yescharacteristic
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
57C8×C79⋊C8C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929623C2×C411yescharacteristic
69#14579182—24no—
70C2×#11929614C2×C411yescharacteristic
71C2×#11929613C2×C411yescharacteristic
72C2×#11929609C2×C411yescharacteristic
73C2×#11929608—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C2×C8×D79—21no—
77C2×C8×D79C2×C411yescharacteristic
78C2×C8×D79C2×C411yescharacteristic
79C2×C8×D79D411yescharacteristic
80C2×C8×D79D411yescharacteristic
81Q8×Dic79C2×C411yescharacteristic
82C4×Dic158C2311yescharacteristic
83#14579126C2×C411yescharacteristic
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86D4×Dic79C2×C411yescharacteristic
87#14579123—21no—
88#14579123C2×C411yescharacteristic
89C4×C22⋊D79—21no—
90(C22⋊D79)⋊C4—21no—
91#14579119—21no—
92#14579119C2×C411yescharacteristic
93#14579119C2×C411yescharacteristic
94#14579117—21no—
95C2×C4×Dic79—21no—
96C42×D79—21no—
97C42×D79C2311yescharacteristic
98#14579094C2×C411yescharacteristic
99#14579094C2×C411yescharacteristic
100#14579093C2×C411yescharacteristic
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)C2311yescharacteristic
103C4⋊(C79⋊C8)C2311yescharacteristic
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106#14579086C2×C411yescharacteristic
107#14579085C2×C411yescharacteristic
108#14579084C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111#14579081C2×C411yescharacteristic
112C22⋊(C79⋊C8)—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117#14579072—21no—
118#14579071—21no—
119#14579071—21no—
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79C2×C411yescharacteristic
127C8×Dic79C2×C411yescharacteristic
128C8×Dic79D411yescharacteristic
129C8×Dic79D411yescharacteristic
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—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929625—22no—
146#11929625—22no—
147#11929623—22no—
148#11929623—22no—
149C2×Dic158C22×C411yescharacteristic
150C2×D316—21no—
151C2×D316C22×C411yescharacteristic
152C2×C22⋊D79—21no—
153C22×Dic79—21no—
154C22×Dic79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79C22×C411yescharacteristic
159C2×C4×D79C2×D411yescharacteristic
160C2×C4×D79#4111yescharacteristic
161#11929614—22no—
162#11929613—22no—
163#11929609—22no—
164#11929608—42no—
165#11929607—42no—
166#11929607—42no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C8×D79—24no—
175C8×D79—22no—
176C8×D79—22no—
177C8×D79—22no—
178C8×D79M1614yescomplemented
179C4⋊Dic79—21no—
180C4⋊Dic79C22×C411yescharacteristic
181C4⋊Dic79C22×C411yescharacteristic
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184C22⋊Dic79—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189C4×Dic79—21no—
190C4×Dic79—21no—
191C4×Dic79—21no—
192C4×Dic79—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—22no—
201M16×C79—42no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79—21no—
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—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—42no—
235C2×Dic79—42no—
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×D79—42no—
248C4×D79—42no—
249C4×D79C2×M1612yes—
250C4×D79#13112yes—
251Dic158—21no—
252Dic158—21no—
253C8×C79—22no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—41no—
257C8×C79—41no—
258C8×C79C2×M1612yes—
259C8×C79#13512yescomplemented
260C79⋊C8—22no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—41no—
264C79⋊C8—41no—
265C79⋊C8C2×M1612yes—
266C79⋊C8#13512yescomplemented
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#43612yes—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79C22×M1611yescharacteristic
282C4×C79C2×#13111yescharacteristic
283C4×C79C2×#13511yescharacteristic
284C4×C79C2×#13511yescharacteristic
285Dic79—21no—
286Dic79—41no—
287Dic79—41no—
288Dic79—41no—
289Dic79#43612yes—
290Dic79#44312yes—
291#17886—791nomaximal, Sylow
292C2×C79—81no—
293C2×C79C2×#43611yescharacteristic
294C2×C79C4×#13511yescharacteristic
295C2×C79C2×#44311yescharacteristic
296D79—22no—
297D79—81no—
298#2965—791no—
299#2965—791no—
300C4×#135—791no—
301#1591—791no—
302#1591—791no—
303#1590—791no—
304#1590—791no—
305#1590—791no—
306#1590D7911yescharacteristic, complemented
307#1589—791no—
308#1589—791no—
309#1564—791no—
310#1476—791no—
311#1476—791no—
312C2×C82—791no—
313C79#1788611yescharacteristic, complemented, Sylow
314C2×#135—791no—
315C4×#41—791no—
316C4×#41—791no—
317#446—791no—
318#446—791no—
319#446—791no—
320#446C2×D7911yescharacteristic
321C4×Q16—791no—
322C4×Q16—791no—
323C4×SD16—791no—
324C4×SD16—791no—
325C4×SD16—791no—
326C4×SD16C2×D7911yescharacteristic
327C4×D8—791no—
328C4×D8—791no—
329#436—791no—
330#436—791no—
331#436—791no—
332#436C2×D7911yescharacteristic
333#435—1581no—
334#435—1581no—
335#435—1581no—
336#435—1581no—
337#431—791no—
338#431—791no—
339#431—1581no—
340#431—1581no—
341#427—791no—
342#416—791no—
343#416—791no—
344C4×M16—1581no—
345C4×M16—1581no—
346C2×C4×C8—791no—
347C2×C4×C8—791no—
348C2×C4×C8—791no—
349#335—791no—
350#335—791no—
351#334—791no—
352#334C2×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
361C82—791no—
362C82—791no—
363C82—791no—
364C82C2×D7911yescharacteristic
365C2×#41—791no—
366C2×#41—791no—
367#135—1584no—
368C2×Q16—791no—
369C2×SD16—791no—
370C2×SD16C4×D7911yescharacteristic
371C2×D8—791no—
372C2×M16—21no—
373C2×M16—1581no—
374C2×M16—1581no—
375C2×M16—1581no—
376C22×C8—791no—
377C22×C8—791no—
378C22×C8—791no—
379C22×C8—791no—
380C22×C8—1581no—
381C4×Q8—791no—
382C4×Q8—791no—
383C4×Q8—791no—
384C4×Q8C22×D7911yescharacteristic
385C4×D4—791no—
386C4×D4—791no—
387C4×D4—791no—
388C4×D4—1581no—
389C4×D4—1581no—
390C4×D4—1581no—
391C4×D4—1581no—
392C4×D4C22×D7911yescharacteristic
393#117—791no—
394#117—791no—
395#117—1581no—
396#117—1581no—
397C2×C42—791no—
398C2×C42—1581no—
399C2×C42—1581no—
400#107—791no—
401#107—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⋊C8C22×D7911yescharacteristic
411C4⋊C8C22×D7911yescharacteristic
412#103—791no—
413#103—791no—
414#103—791no—
415#103C4×D7911yescharacteristic
416#102—791no—
417#102—791no—
418#102—791no—
419#102C4×D7911yescharacteristic
420C22⋊C8—21no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426#97—21no—
427#97—1581no—
428#97—1581no—
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—1582no—
444#41—1582no—
445#41—1582no—
446#41—1582no—
447C2×Q8—791no—
448C2×Q8C2×C4×D7911yescharacteristic
449C2×D4—791no—
450C2×D4—1581no—
451C2×D4—1581no—
452C2×D4C2×C4×D7911yescharacteristic
453C22×C4—21no—
454C22×C4—791no—
455C22×C4—791no—
456C22×C4—791no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4—1581no—
461C22×C4—1581no—
462Q16—1582no—
463SD16—22no—
464SD16—1582no—
465D8—1582no—
466M16—42no—
467M16—3162no—
468M16—3162no—
469M16—3162no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—791no—
474C2×C8—791no—
475C2×C8—794no—
476C2×C8—791no—
477C2×C8—791no—
478C2×C8—1582no—
479C2×C8—1582no—
480C2×C8—1584no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1582no—
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⋊C4C2×C4×D7911yescharacteristic
496C4⋊C4C2×C4×D7911yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—1581no—
499C22⋊C4—1581no—
500C22⋊C4—1581no—
501C22⋊C4—1581no—
502C22⋊C4—1581no—
503C42—21no—
504C42—791no—
505C42—791no—
506C42—791no—
507C42—1581no—
508C42—1581no—
509C42—1581no—
510C42—1581no—
511C42—1581no—
512C42C23×D7911yescharacteristic
513C23—21no—
514C23—791no—
515C23—1581no—
516Q8—21no—
517Q8—21no—
518Q8—1581no—
519Q8—1581no—
520D4—21no—
521D4—21no—
522D4—3162no—
523D4—3162no—
524D4—1581no—
525D4—1581no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—3162no—
534C2×C4—3162no—
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—41no—
555C8—41no—
556C8—3161no—
557C8—3161no—
558C8—792no—
559C8—792no—
560C8—1582no—
561C8—1581no—
562C8—1581no—
563C8M16×D7912yes—
564C8#1457918212yescomplemented
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—
585C4C2×M16×D7911yescharacteristic
586C4#131×D7911yescharacteristic
587C4#135×D7911yescharacteristic
588C4C2×#1457918211yescharacteristic
589C2—81no—
590C2—6321no—
591C2—1582no—
592C2#436×D7911yescharacteristic
593C2C4×#1457918211yescharacteristic
594C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1590C2
socleC22×C79C2×C4×D4
prime radicalC2×#14579182C4
cosocleC42×C79C24
hypercenter#1590D79
hypocenterC79#17886
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
1. #23772365C2semidirect
2. #23772363C2semidirect
4. #23762295C2semidirect
5. #23762294C2semidirect
6. #23762292C2semidirect
7. #23762291C2semidirect
8. #23762289C2semidirect
9. #23762283C2semidirect
10. #23762282C2semidirect
11. #23762145C2semidirect
12. #23761781C2semidirect
13. #23761779C2semidirect
14. C82×D79C2semidirect
15. C79×#1590C2semidirect
178. C8×D79M16semidirect
259. C8×C79#135semidirect
266. C79⋊C8#135semidirect
306. #1590D79semidirect
313. C79#17886semidirect
564. C8#14579182semidirect

Maximal subgroups

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

subgroupindex
#237723652
#237723632
C4×#145791822
#237622952
#237622942
#237622922
#237622912
#237622892
#237622832
#237622822
#237621452
#237617812
#237617792
C82×D792
C79×#15902
#1788679