Finite Group Databasebeta

← #34258397

Subgroups — #34258397

Order 20224.

626 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773429C211yesmaximal, characteristic
2#23772479C211yesmaximal, characteristic, complemented
3#23772474C211yesmaximal, characteristic
4#23762279C211yesmaximal, characteristic, complemented
5#23762277C211yesmaximal, characteristic, complemented
6#23762246C211yesmaximal, characteristic, complemented
7#23762240C211yesmaximal, characteristic, complemented
8#23762214C211yesmaximal, characteristic, complemented
9C8×#11929611C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761845C211yesmaximal, characteristic, complemented
13#23761815C211yesmaximal, characteristic
14#23761696C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099110C2211yescharacteristic
17C4×Q8×D79C2211yescharacteristic
18#19098883C411yescharacteristic
19#19098880C411yescharacteristic
20#19098847C411yescharacteristic
21#19098846C411yescharacteristic
22#19098762C411yescharacteristic
23#19098761C411yescharacteristic
24#19098703C411yescharacteristic
25#19098700C411yescharacteristic
26Q8×C79⋊C8C2211yescharacteristic
27C8×Dic158—21no—
28C8×Dic158C2211yescharacteristic
29#19098687—21no—
30#19098687C2211yescharacteristic
31C4×#11929613C2211yescharacteristic
32C4×#11929611C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34#19098641C2211yescharacteristic
35#19098640C2211yescharacteristic
36#19098637—21no—
37#19098634—21no—
38C8×D316C2211yescharacteristic
39#19098619C2211yescharacteristic
40#19098615—21no—
41#19098615C2211yescharacteristic
42#19098614—21no—
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098529—21no—
46#19098528—21no—
47C4×#11929607C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×Q8×D79C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×C8×D79—21no—
76#14579160C2×C411yescharacteristic
77#14579157C2×C411yescharacteristic
78#14579147—21no—
79#14579136C2×C411yescharacteristic
80#14579134—21no—
81#14579132C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×Dic158—21no—
84C4×Dic158C2311yescharacteristic
85#14579126—21no—
86#14579126C2×C411yescharacteristic
87C4×D316C2311yescharacteristic
88C4×C22⋊D79—21no—
89C4⋊C4×D79—21no—
90C4⋊C4×D79C2×C411yescharacteristic
91C42×D79—21no—
92C42×D79C2311yescharacteristic
93#14579093C2×C411yescharacteristic
94#14579092C2×C411yescharacteristic
95#14579092C2×C411yescharacteristic
96C4⋊(C79⋊C8)—21no—
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8C2×C411yescharacteristic
102Dic79⋊C8C2×C411yescharacteristic
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)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8D411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120#14579071—21no—
121#14579071—21no—
122C4×C79⋊C8—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
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—
140C4×C8×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
149Q8×D79—22no—
150Q8×D79—22no—
151C2×D316—21no—
152C2×D316C22×C411yescharacteristic
153C2×C22⋊D79—21no—
154C2×C22⋊D79—21no—
155C2×C22⋊D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79C22×C411yescharacteristic
160C2×C4×D79C2×D411yescharacteristic
161C2×C4×D79#4111yescharacteristic
162#11929613—22no—
163#11929612—22no—
164#11929611—22no—
165#11929608—81no—
166#11929607—42no—
167#11929607—81no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C22×C411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C8×D79—81no—
176C4⋊Dic79—21no—
177C4⋊Dic79C22×C411yescharacteristic
178C4⋊Dic79C22×C411yescharacteristic
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—
186C4×Dic79—21no—
187C4×Dic79—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—22no—
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#13112yes—
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—81no—
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—24no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—21no—
241C4×D79—21no—
242C4×D79—41no—
243C4×D79—41no—
244C4×D79—41no—
245C4×D79—81no—
246Dic158—24no—
247Dic158—21no—
248Dic158—81no—
249Dic158#13112yes—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—21no—
254C8×C79—41no—
255C8×C79—41no—
256C8×C79—41no—
257C79⋊C8—22no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—21no—
261C79⋊C8—21no—
262C79⋊C8—41no—
263C79⋊C8—41no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79C2×C4×D411yescharacteristic
268C2×D79—21no—
269C2×D79—21no—
270C2×D79—21no—
271C2×D79—41no—
272C2×D79—41no—
273C2×D79—41no—
274C4×C79—22no—
275C4×C79—21no—
276C4×C79—41no—
277C4×C79—41no—
278C4×C79C2×#13111yescharacteristic
279C4×C79C2×#13111yescharacteristic
280C4×C79#57711yescharacteristic
281C4×C79#57711yescharacteristic
282Dic79—21no—
283Dic79—22no—
284Dic79—21no—
285Dic79—21no—
286Dic79—41no—
287Dic79—41no—
288#17923—791nomaximal, Sylow
289C2×C79—81no—
290C2×C79#292911yescharacteristic
291C2×C79#294911yescharacteristic
292C2×C79#295611yescharacteristic
293D79—41no—
294D79—81no—
295#3103—791no—
296#2982—791no—
297#2981—791no—
298#1588—791no—
299#1588—791no—
300#1583—791no—
301#1583—791no—
302SD16⋊C8—791no—
303SD16⋊C8D7911yescharacteristic, complemented
304C8×SD16—791no—
305C8×SD16—791no—
306#1572—791no—
307#1491—791no—
308#1484—791no—
309#1453—791no—
310C79#1792311yescharacteristic, complemented, Sylow
311#532—791no—
312C2×C4×Q8—791no—
313#481—791no—
314#481—791no—
315#475—791no—
316#475—791no—
317#460—791no—
318#460—791no—
319#448—791no—
320#448—791no—
321C8×Q8—791no—
322C8×Q8—791no—
323C8×Q8—791no—
324C8×Q8—1581no—
325C8×Q8—1581no—
326C8×Q8C2×D7911yescharacteristic
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330C4×SD16C2×D7911yescharacteristic
331#436—791no—
332#436—791no—
333#435—1581no—
334#435—1581no—
335C8×D4—791no—
336C8×D4C2×D7911yescharacteristic
337#432—791no—
338#431—791no—
339#431—791no—
340#431—791no—
341#431—1581no—
342#431—1581no—
343#408—1581no—
344#408—1581no—
345C4×M16—791no—
346#335—791no—
347#335C2×D7911yescharacteristic
348#334—791no—
349#334—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353Q8⋊C8C2×D7911yescharacteristic
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8C2×D7911yescharacteristic
358#322—791no—
359#322—791no—
360#322C2×D7911yescharacteristic
361C82—791no—
362C2×#41—1581no—
363C22×Q8—791no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16C4×D7911yescharacteristic
368C2×M16—1581no—
369C2×M16—1581no—
370C2×M16—1581no—
371C22×C8—21no—
372C22×C8—1581no—
373C4⋊Q8—791no—
374#127—791no—
375#124—1581no—
376#121—791no—
377#121—791no—
378#121—1581no—
379C4×Q8—791no—
380C4×Q8—791no—
381C4×Q8—791no—
382C4×Q8—1581no—
383C4×Q8—1581no—
384C4×Q8C22×D7911yescharacteristic
385C4×D4—791no—
386C4×D4—1581no—
387C4×D4C22×D7911yescharacteristic
388C2×C4⋊C4—791no—
389C2×C4⋊C4—1581no—
390C2×C42—791no—
391C2×C42—1581no—
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—791no—
402C4⋊C8—791no—
403C4⋊C8—1581no—
404C4⋊C8—1581no—
405C4⋊C8—1581no—
406C4⋊C8C22×D7911yescharacteristic
407C4⋊C8C22×D7911yescharacteristic
408#103—791no—
409#103—791no—
410#103—791no—
411#103C4×D7911yescharacteristic
412#102—791no—
413#102—791no—
414#102—791no—
415#102C4×D7911yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—791no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—791no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426#97—1581no—
427#97—1581no—
428C4×C8—21no—
429C4×C8—791no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—1581no—
433C4×C8—1581no—
434C4×C8—1581no—
435C4×C8—1581no—
436C4×C8C22×D7911yescharacteristic
437C4×C8C22×D7911yescharacteristic
438C4×C8C22×D7911yescharacteristic
439#41—6321no—
440#41—6321no—
441C2×Q8—791no—
442C2×Q8—1581no—
443C2×Q8—1581no—
444C2×Q8—1582no—
445C2×Q8—1582no—
446C2×Q8C2×C4×D7911yescharacteristic
447C2×D4—791no—
448C2×D4—1581no—
449C2×D4—1581no—
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—
460SD16—22no—
461SD16—1582no—
462SD16—1582no—
463SD16—1582no—
464M16—3162no—
465M16—6321no—
466M16—6321no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—81no—
471C2×C8—791no—
472C2×C8—791no—
473C2×C8—791no—
474C2×C8—791no—
475C2×C8—6321no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8C2×C4×D7911yescharacteristic
480C2×C8C2×C4×D7911yescharacteristic
481C2×C8D4×D7911yescharacteristic
482C2×C8D4×D7911yescharacteristic
483C4⋊C4—21no—
484C4⋊C4—791no—
485C4⋊C4—791no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4—1581no—
491C4⋊C4C2×C4×D7911yescharacteristic
492C4⋊C4C2×C4×D7911yescharacteristic
493C22⋊C4—21no—
494C22⋊C4—1581no—
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C42—21no—
498C42—791no—
499C42—791no—
500C42—791no—
501C42—1581no—
502C42—1581no—
503C42—1581no—
504C42C23×D7911yescharacteristic
505C23—21no—
506C23—791no—
507C23—1581no—
508Q8—21no—
509Q8—792no—
510Q8—6321no—
511Q8—1584no—
512Q8—1581no—
513Q8#1457916712yes—
514D4—21no—
515D4—21no—
516D4—6321no—
517D4—6321no—
518D4—6321no—
519D4—6321no—
520D4—1581no—
521D4—1581no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—791no—
531C2×C4—791no—
532C2×C4—791no—
533C2×C4—6321no—
534C2×C4—1584no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1581no—
540C2×C4—1581no—
541C2×C4—1581no—
542C2×C4C22×C4×D7911yescharacteristic
543C2×C4C2×D4×D7911yescharacteristic
544C2×C4#41×D7911yescharacteristic
545C8—21no—
546C8—21no—
547C8—21no—
548C8—21no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—3161no—
553C8—3161no—
554C8—1582no—
555C8—1581no—
556C8—1581no—
557C8—1581no—
558C8—1581no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—3161no—
563C22—3161no—
564C22—3161no—
565C22—1581no—
566C22—1581no—
567C22—1581no—
568C22C4×D4×D7911yescharacteristic
569C4—21no—
570C4—22no—
571C4—41no—
572C4—41no—
573C4—3161no—
574C4—3161no—
575C4—1581no—
576C4—1581no—
577C4—1581no—
578C4—1582no—
579C4#131×D7911yescharacteristic
580C4C2×#1457916711yescharacteristic
581C4#1909937211yescharacteristic
582C4#1909937311yescharacteristic
583C2—81no—
584C2—3161no—
585C2—6321no—
586C2#2377208911yescharacteristic
587C2#2377225611yescharacteristic
588C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radical#19099110C22
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#17923
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #23772479C2semidirect
4. #23762279C2semidirect
5. #23762277C2semidirect
6. #23762246C2semidirect
7. #23762240C2semidirect
8. #23762214C2semidirect
9. C8×#11929611C2semidirect
10. C8×#11929612C2semidirect
11. #23762179C2semidirect
12. #23761845C2semidirect
14. #23761696C2semidirect
15. C79×SD16⋊C8C2semidirect
303. SD16⋊C8D79semidirect
310. C79#17923semidirect

Maximal subgroups

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

subgroupindex
#237734292
#237724792
#237724742
#237622792
#237622772
#237622462
#237622402
#237622142
C8×#119296112
C8×#119296122
#237621792
#237618452
#237618152
#237616962
C79×SD16⋊C82
#1792379