Finite Group Databasebeta

← #34259285

Subgroups — #34259285

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773416C211yesmaximal, characteristic
2#23772419C211yesmaximal, characteristic, complemented
3#23772401C211yesmaximal, characteristic, complemented
4#23762311C211yesmaximal, characteristic, complemented
5#23762304C211yesmaximal, characteristic, complemented
6#23762279C211yesmaximal, characteristic, complemented
7#23762277C211yesmaximal, characteristic, complemented
8#23762271C211yesmaximal, characteristic, complemented
9#23762264C211yesmaximal, characteristic, complemented
10#23762216C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761857C211yesmaximal, characteristic, complemented
13#23761781C211yesmaximal, characteristic, complemented
14#23761688C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099095C2211yescharacteristic
17C4×#11929625C2211yescharacteristic
18#19098907C411yescharacteristic
19#19098886C411yescharacteristic
20#19098868C411yescharacteristic
21#19098862C411yescharacteristic
22#19098783C411yescharacteristic
23#19098777C411yescharacteristic
24#19098728C411yescharacteristic
25#19098707C411yescharacteristic
26#19098693C2211yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28#19098687C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098642—21no—
33#19098642C2211yescharacteristic
34#19098641C2211yescharacteristic
35#19098640—21no—
36#19098640C2211yescharacteristic
37#19098638—21no—
38#19098634—21no—
39#19098621—21no—
40#19098619—21no—
41#19098619C2211yescharacteristic
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098518—21no—
45C4×#11929607—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
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
59#19098285C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×#11929607D411yescharacteristic
75C2×#11929607D411yescharacteristic
76C2×C8×D79—21no—
77#14579154—21no—
78#14579152C2×C411yescharacteristic
79#14579148C2×C411yescharacteristic
80#14579143—21no—
81#14579141C2×C411yescharacteristic
82#14579132C2×C411yescharacteristic
83Q8×Dic79C2×C411yescharacteristic
84C4×Dic158C2311yescharacteristic
85C4×D316—21no—
86C4×D316C2311yescharacteristic
87#14579123—21no—
88#14579123C2×C411yescharacteristic
89C4×C22⋊D79—21no—
90#14579120—21no—
91#14579119—21no—
92#14579119C2×C411yescharacteristic
93C42×D79—21no—
94C42×D79C2311yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579094C2×C411yescharacteristic
97#14579093C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
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#14579071—21no—
119#14579071—21no—
120#14579071D411yescharacteristic
121#14579071D411yescharacteristic
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#11929625—22no—
145#11929625—41no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316—21no—
148C2×D316C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—22no—
157#11929613—41no—
158#11929609—41no—
159#11929608—42no—
160#11929607—41no—
161#11929607—41no—
162#11929607—42no—
163#11929607—42no—
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—42no—
172C4⋊Dic79—21no—
173C4⋊Dic79—21no—
174C4⋊Dic79—21no—
175C4⋊Dic79C22×C411yescharacteristic
176C4⋊Dic79C22×C411yescharacteristic
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—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—42no—
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—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79C4×D411yescharacteristic
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232D316—21no—
233D316—21no—
234D316—42no—
235C4×D79—21no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—41no—
240C4×D79—42no—
241C4×D79—41no—
242C4×D79—41no—
243C4×D79—81no—
244Dic158—21no—
245Dic158—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—21no—
250C8×C79—41no—
251C8×C79—41no—
252C8×C79—41no—
253C79⋊C8—22no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79—41no—
263C22×C79C2×C4×D411yescharacteristic
264C2×D79—21no—
265C2×D79—21no—
266C2×D79—21no—
267C2×D79—41no—
268C2×D79—41no—
269C2×D79—41no—
270C4×C79—22no—
271C4×C79—21no—
272C4×C79—41no—
273C4×C79—41no—
274C4×C79C2×#13111yescharacteristic
275C4×C79#56811yescharacteristic
276C4×C79#57511yescharacteristic
277C4×C79#57511yescharacteristic
278Dic79—21no—
279Dic79—21no—
280Dic79—21no—
281Dic79—41no—
282Dic79—41no—
283Dic79—41no—
284#18005—791nomaximal, Sylow
285C2×C79—81no—
286C2×C79#293311yescharacteristic
287C2×C79#294311yescharacteristic
288C2×C79#295611yescharacteristic
289D79—41no—
290D79—81no—
291#3102—791no—
292#2971—791no—
293#2969—791no—
294#1594—791no—
295#1592—791no—
296#1588—791no—
297#1588—791no—
298#1587—791no—
299#1586—791no—
300#1579—791no—
301SD16⋊C8D7911yescharacteristic, complemented
302#1572—791no—
303#1494—791no—
304#1476—791no—
305#1451—791no—
306C79#1800511yescharacteristic, complemented, Sylow
307#529—791no—
308C4×#41—791no—
309#485—791no—
310#482—791no—
311#479—791no—
312#478—791no—
313#464—791no—
314#463—791no—
315#452—791no—
316#449—791no—
317#446—791no—
318C8×Q8—791no—
319C8×Q8—791no—
320C8×Q8C2×D7911yescharacteristic
321C4×Q16—791no—
322C4×SD16—791no—
323C4×SD16C2×D7911yescharacteristic
324C4×D8—791no—
325#436—791no—
326#436—791no—
327#436—791no—
328#436—1581no—
329#436—1581no—
330#436—1581no—
331#435—1581no—
332C8×D4C2×D7911yescharacteristic
333#433—1581no—
334#432—791no—
335#432—1581no—
336#431—791no—
337#431—791no—
338#405—1581no—
339C4×M16—791no—
340C4×M16—1581no—
341C2×C4×C8—791no—
342#335—791no—
343#335C2×D7911yescharacteristic
344#334—791no—
345#334—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8—791no—
349Q8⋊C8C2×D7911yescharacteristic
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8—791no—
353D4⋊C8C2×D7911yescharacteristic
354#322—791no—
355#322—791no—
356#322—791no—
357#322C2×D7911yescharacteristic
358C2×#41—791no—
359C2×Q16—791no—
360C2×SD16—791no—
361C2×SD16C4×D7911yescharacteristic
362C2×D8—791no—
363C2×M16—791no—
364C2×M16—791no—
365C2×M16—1581no—
366C2×M16—1581no—
367C2×M16—1581no—
368C22×C8—21no—
369C22×C8—1581no—
370#126—1581no—
371#125—791no—
372#124—791no—
373#123—1581no—
374C22⋊Q8—791no—
375#121—791no—
376C4×Q8—791no—
377C4×Q8—791no—
378C4×Q8C22×D7911yescharacteristic
379C4×D4—791no—
380C4×D4—791no—
381C4×D4—1581no—
382C4×D4—1581no—
383C4×D4—1581no—
384C4×D4C22×D7911yescharacteristic
385#117—791no—
386#117—1581no—
387#117—1581no—
388C2×C42—791no—
389C2×C42—1581no—
390#107—791no—
391#107—791no—
392#106—791no—
393#106C4×D7911yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8C22×D7911yescharacteristic
403C4⋊C8C22×D7911yescharacteristic
404#103—791no—
405#103—791no—
406#103—791no—
407#103C4×D7911yescharacteristic
408#102—791no—
409#102—791no—
410#102—791no—
411#102C4×D7911yescharacteristic
412C22⋊C8—21no—
413C22⋊C8—791no—
414C22⋊C8—791no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—791no—
421#97—791no—
422#97—1581no—
423#97—1581no—
424C4×C8—21no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8—1581no—
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434C4×C8C22×D7911yescharacteristic
435#41—3161no—
436#41—1582no—
437C2×Q8—791no—
438C2×Q8C2×C4×D7911yescharacteristic
439C2×D4—791no—
440C2×D4—1581no—
441C2×D4—1581no—
442C2×D4C2×C4×D7911yescharacteristic
443C22×C4—21no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—791no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450Q16—1582no—
451SD16—22no—
452SD16—3161no—
453D8—3161no—
454M16—3162no—
455M16—3161no—
456M16—3161no—
457M16—3162no—
458M16—3162no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—42no—
463C2×C8—3162no—
464C2×C8—791no—
465C2×C8—791no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8C2×C4×D7911yescharacteristic
472C2×C8C2×C4×D7911yescharacteristic
473C2×C8D4×D7911yescharacteristic
474C2×C8D4×D7911yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—791no—
477C4⋊C4—791no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4C2×C4×D7911yescharacteristic
484C4⋊C4C2×C4×D7911yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C42—21no—
494C42—791no—
495C42—791no—
496C42—791no—
497C42—1581no—
498C42—1581no—
499C42—1581no—
500C42C23×D7911yescharacteristic
501C23—21no—
502C23—791no—
503C23—1581no—
504Q8—21no—
505Q8—1581no—
506Q8—1581no—
507Q8#1457916712yes—
508D4—21no—
509D4—21no—
510D4—3162no—
511D4—6321no—
512D4—1581no—
513D4—1581no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—3162no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—791no—
524C2×C4—791no—
525C2×C4—791no—
526C2×C4—6321no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4C22×C4×D7911yescharacteristic
535C2×C4C2×D4×D7911yescharacteristic
536C2×C4#41×D7911yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—3161no—
545C8—3161no—
546C8—1582no—
547C8—1581no—
548C8—1581no—
549C8—1581no—
550C8—1581no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—3161no—
555C22—3161no—
556C22—3161no—
557C22—1581no—
558C22—1581no—
559C22—1581no—
560C22C4×D4×D7911yescharacteristic
561C4—21no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—3161no—
566C4—3161no—
567C4—3161no—
568C4—1581no—
569C4—1581no—
570C4—1581no—
571C4C2×#1457916711yescharacteristic
572C4#1909932811yescharacteristic
573C4#1909935911yescharacteristic
574C4#1909936211yescharacteristic
575C2—81no—
576C2—3161no—
577C2—6321no—
578C2#2377212011yescharacteristic
579C2#2377219511yescharacteristic
580C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radical#19098707C4
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#18005
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. #23772419C2semidirect
3. #23772401C2semidirect
4. #23762311C2semidirect
5. #23762304C2semidirect
6. #23762279C2semidirect
7. #23762277C2semidirect
8. #23762271C2semidirect
9. #23762264C2semidirect
10. #23762216C2semidirect
11. #23762179C2semidirect
12. #23761857C2semidirect
13. #23761781C2semidirect
14. #23761688C2semidirect
15. C79×SD16⋊C8C2semidirect
301. SD16⋊C8D79semidirect
306. C79#18005semidirect

Maximal subgroups

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

subgroupindex
#237734162
#237724192
#237724012
#237623112
#237623042
#237622792
#237622772
#237622712
#237622642
#237622162
#237621792
#237618572
#237617812
#237616882
C79×SD16⋊C82
#1800579