Finite Group Databasebeta

← #34444728

Subgroups — #34444728

Order 20224.

760 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098998C211yesmaximal, characteristic, complemented
2C22×#14579102C211yesmaximal, characteristic, complemented
3D4×D632C211yesmaximal, characteristic, complemented
4#23765667C211yesmaximal, characteristic, complemented
5#23765501C211yesmaximal, characteristic, complemented
6#23765498C211yesmaximal, characteristic, complemented
7#23765452C211yesmaximal, characteristic, complemented
8#23765435C211yesmaximal, characteristic, complemented
9#23765328C211yesmaximal, characteristic, complemented
10#23765323C211yesmaximal, characteristic, complemented
11#23765312C211yesmaximal, characteristic, complemented
12#23765311C211yesmaximal, characteristic, complemented
13#23765274C211yesmaximal, characteristic, complemented
14#23765226C211yesmaximal, characteristic, complemented
15C79×#3280C211yesmaximal, characteristic, complemented
16C22×D632C2211yescharacteristic
17C22×D632C2211yescharacteristic, complemented
18D4×D316—21no—
19#19098998—24no—
20C2×#14579104C2211yescharacteristic, complemented
21C2×#14579104C2211yescharacteristic, complemented
22C2×#14579104C2211yescharacteristic, complemented
23C2×#14579102—24no—
24C2×#14579102C2211yescharacteristic
25C2×#14579102C2211yescharacteristic
26C2×#14579102C2211yescharacteristic, complemented
27C2×#14579102C2211yescharacteristic, complemented
28C2×#14579102C2211yescharacteristic, complemented
29C2×#14579102C2211yescharacteristic, complemented
30C2×#14579101C2211yescharacteristic, complemented
31C22×C79⋊C16C2211yescharacteristic, complemented
32#19098940C2211yescharacteristic, complemented
33#19098799C2211yescharacteristic
34#19098799C2211yescharacteristic, complemented
35#19098759—21no—
36#19098699—21no—
37#19098699—21no—
38C4×D632C2211yescharacteristic, complemented
39#19098390C2211yescharacteristic, complemented
40C4⋊(C79⋊C16)C2211yescharacteristic, complemented
41#19098367C2211yescharacteristic, complemented
42#19098364C2211yescharacteristic, complemented
43#19098364C2211yescharacteristic, complemented
44#19098363C2211yescharacteristic, complemented
45#19098363C2211yescharacteristic, complemented
46#19098363C2211yescharacteristic, complemented
47C22⋊(C79⋊C16)C2211yescharacteristic, complemented
48C22⋊(C79⋊C16)C2211yescharacteristic, complemented
49#19098339C2211yescharacteristic, complemented
50C2×#135×C79C2211yescharacteristic, complemented
51C22×D8×C79C2211yescharacteristic, complemented
52#546×C79—21no—
53#495×C79C2211yescharacteristic, complemented
54#467×C79C2211yescharacteristic, complemented
55#466×C79C2211yescharacteristic, complemented
56#463×C79—21no—
57#450×C79—21no—
58#449×C79—21no—
59C4×D8×C79C2211yescharacteristic, complemented
60C8×D4×C79C2211yescharacteristic
61C22×D316—21no—
62C22×D316—21no—
63C2×D4×D79—41no—
64C2×D632—22no—
65C2×D632—22no—
66C2×D632—21no—
67C2×D632—21no—
68C2×D632—21no—
69C2×D632—41no—
70C2×D632—41no—
71C2×D632C2311yescharacteristic
72C2×D632C2311yescharacteristic
73C2×D632C2311yescharacteristic
74#14579158—21no—
75#14579136—41no—
76#14579135—21no—
77#14579135—21no—
78#14579131—41no—
79#14579131—41no—
80C4×D316—21no—
81#14579104—22no—
82#14579104—22no—
83#14579104—41no—
84#14579102—22no—
85#14579102—22no—
86#14579102—22no—
87#14579102—22no—
88#14579102—41no—
89#14579102D414yescomplemented
90#14579101—22no—
91C2×C79⋊C16—22no—
92C2×C79⋊C16C2311yescharacteristic
93C2×C79⋊C16C2311yescharacteristic
94C2×C79⋊C16C2311yescharacteristic
95C2×C79⋊C16C2311yescharacteristic
96#14579095C2311yescharacteristic
97#14579083—21no—
98#14579083—21no—
99#14579083—21no—
100C2×#41×C79—21no—
101C22×D4×C79—21no—
102#135×C79—22no—
103#135×C79—41no—
104C2×Q16×C79C2311yescharacteristic
105C2×SD16×C79—21no—
106C2×SD16×C79—21no—
107C2×D8×C79—22no—
108C2×D8×C79—21no—
109C2×D8×C79—41no—
110C2×D8×C79C2311yescharacteristic
111C2×D8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114#124×C79—21no—
115#123×C79—41no—
116C22⋊Q8×C79—21no—
117#121×C79—21no—
118#121×C79—41no—
119#120×C79—41no—
120C4×D4×C79—21no—
121C4×D4×C79D411yescharacteristic
122C2×C22⋊C4×C79—41no—
123#107×C79C2311yescharacteristic
124C4⋊C8×C79D411yescharacteristic
125#103×C79—21no—
126#102×C79—21no—
127#102×C79—21no—
128C22⋊C8×C79D411yescharacteristic
129C22⋊C8×C79D411yescharacteristic
130C4×C8×C79C2311yescharacteristic
131C23×D79—41no—
132C23×D79—41no—
133C2×D316—21no—
134C2×D316—21no—
135C2×D316—21no—
136C2×D316—42no—
137C2×D316—42no—
138C2×D316—41no—
139C2×D316—41no—
140C2×D316—41no—
141C2×D316—41no—
142C2×D316—81no—
143C2×D316—81no—
144C2×C22⋊D79—41no—
145C2×C22⋊D79—41no—
146D4×D79—84no—
147C2×C4×D79—41no—
148D632—21no—
149D632—21no—
150D632—21no—
151D632—21no—
152D632—21no—
153D632—42no—
154D632—42no—
155D632—81no—
156D632C2×D412yes—
157C4⋊Dic79—21no—
158(C2×D79)⋊C4—41no—
159(C2×D79)⋊C4—41no—
160(C2×D79)⋊C4—41no—
161C79⋊C16—21no—
162C79⋊C16—21no—
163C79⋊C16—21no—
164C79⋊C16C2×D412yescomplemented
165C24×C79—41no—
166#41×C79—42no—
167#41×C79—81no—
168C2×Q8×C79—21no—
169C2×D4×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79—42no—
172C2×D4×C79—41no—
173C2×D4×C79—41no—
174C2×D4×C79—41no—
175C2×D4×C79—81no—
176C2×D4×C79D811yescharacteristic
177C22×C4×C79—41no—
178C22×C4×C79—41no—
179C22×C4×C79C2×D411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181Q16×C79—21no—
182Q16×C79—21no—
183SD16×C79—42no—
184SD16×C79—81no—
185D8×C79—21no—
186D8×C79—21no—
187D8×C79—21no—
188D8×C79—42no—
189D8×C79C2×D412yescomplemented
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79C2×D411yescharacteristic
195C2×C8×C79C2×D411yescharacteristic
196C2×C8×C79C2×D411yescharacteristic
197C2×C8×C79C2411yescharacteristic
198C4⋊C4×C79—21no—
199C4⋊C4×C79—41no—
200C4⋊C4×C79D811yescharacteristic
201C22⋊C4×C79—41no—
202C22⋊C4×C79—41no—
203C22⋊C4×C79—41no—
204C22⋊C4×C79—41no—
205C22⋊C4×C79—41no—
206C22⋊C4×C79D811yescharacteristic
207C22⋊C4×C79D811yescharacteristic
208C42×C79C2×D411yescharacteristic
209C23×C79—41no—
210C23×C79—41no—
211C23×C79—82no—
212C23×C79—82no—
213C23×C79—161no—
214C23×C79C2×D811yescharacteristic
215C23×C79C2×D811yescharacteristic
216C22×D79—41no—
217C22×D79—41no—
218C22×D79—41no—
219C22×D79—82no—
220C22×D79—82no—
221C22×D79—82no—
222C22×D79—82no—
223C22×D79—161no—
224C22×D79—161no—
225Q8×C79—41no—
226Q8×C79—41no—
227D4×C79—22no—
228D4×C79—22no—
229D4×C79—41no—
230D4×C79—41no—
231D4×C79—41no—
232D4×C79—82no—
233D4×C79—82no—
234D4×C79—161no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—82no—
242C2×C4×C79—161no—
243C2×C4×C79C2×D811yescharacteristic
244C2×C4×C79C2×D811yescharacteristic
245C2×C4×C79C2×D811yescharacteristic
246C2×C4×C79C2×D811yescharacteristic
247C2×C4×C79C22×D411yescharacteristic
248C22⋊D79—82no—
249C22⋊D79—82no—
250C2×Dic79—41no—
251D316—22no—
252D316—41no—
253D316—41no—
254D316—41no—
255D316—41no—
256D316—41no—
257D316—82no—
258D316—82no—
259D316—82no—
260D316—161no—
261C4×D79—82no—
262C8×C79—21no—
263C8×C79—21no—
264C8×C79—21no—
265C8×C79C22×D411yescharacteristic
266C8×C79H(5,𝔽2)11yescharacteristic
267C22×C79—21no—
268C22×C79—21no—
269C22×C79—21no—
270C22×C79—21no—
271C22×C79—21no—
272C22×C79—42no—
273C22×C79—81no—
274C22×C79—81no—
275C22×C79—81no—
276C22×C79—81no—
277C22×C79—81no—
278C22×C79—164no—
279C22×C79#50912yescomplemented
280C22×C79C22×D811yescharacteristic
281C2×D79—42no—
282C2×D79—81no—
283C2×D79—81no—
284C2×D79—81no—
285C2×D79—81no—
286C2×D79—81no—
287C2×D79—81no—
288C2×D79—81no—
289C2×D79—81no—
290C2×D79—164no—
291C2×D79—164no—
292C4×C79—21no—
293C4×C79—21no—
294C4×C79—21no—
295C4×C79—21no—
296C4×C79—81no—
297C4×C79—81no—
298C4×C79D4211yescharacteristic
299C4×C79#57611yescharacteristic
300Dic79—81no—
301#33830—791nomaximal, Sylow
302C2×C79—22no—
303C2×C79—41no—
304C2×C79—82no—
305C2×C79—161no—
306C2×C79D4×D811yescharacteristic
307C2×C79C2×#50911yescharacteristic
308C2×C79#341511yescharacteristic
309D79—82no—
310D79—161no—
311D79—161no—
312C2×#509—791no—
313C22×D16—791no—
314#3280D7911yescharacteristic, complemented
315D4×D8—791no—
316#2250—791no—
317#2218—791no—
318#2217—791no—
319#2210—791no—
320#2206—791no—
321#2187—791no—
322#2186—791no—
323#2184—791no—
324#2184—791no—
325#2177—791no—
326#2168—791no—
327C79#3383011yescharacteristic, complemented, Sylow
328C2×#135C2×D7911yescharacteristic, complemented
329C22×D8—791no—
330C22×D8—791no—
331C22×D8C2×D7911yescharacteristic, complemented
332#546—21no—
333D42—1581no—
334#509—1584no—
335C2×SD32—791no—
336C2×SD32—791no—
337C2×SD32—791no—
338C2×D16—791no—
339C2×D16—791no—
340C2×D16—791no—
341C2×D16—791no—
342C2×D16—791no—
343C2×D16—791no—
344C2×D16—1584no—
345C2×M32—791no—
346C22×C16—791no—
347#495C2×D7911yescharacteristic, complemented
348#493—791no—
349#467C2×D7911yescharacteristic, complemented
350#466—791no—
351#466—791no—
352#466C2×D7911yescharacteristic, complemented
353#463—21no—
354#459—1581no—
355#450—21no—
356#449—21no—
357#447—1581no—
358#447—1581no—
359C4×D8—791no—
360C4×D8C2×D7911yescharacteristic, complemented
361C8×D4C2×D7911yescharacteristic
362#367—791no—
363C4⋊C16—791no—
364#358—791no—
365#357—791no—
366#357—791no—
367#357—791no—
368#357—791no—
369#357—791no—
370C22⋊C16—791no—
371C22⋊C16—791no—
372#346—791no—
373C2×#41—21no—
374C22×D4—21no—
375C22×D4—3161no—
376C22×D4—1581no—
377C22×D4—1581no—
378#135—22no—
379#135—41no—
380C2×Q16C22×D7911yescharacteristic
381C2×SD16—21no—
382C2×SD16—21no—
383C2×D8—22no—
384C2×D8—21no—
385C2×D8—41no—
386C2×D8—3161no—
387C2×D8—3161no—
388C2×D8—791no—
389C2×D8—791no—
390C2×D8—791no—
391C2×D8—1582no—
392C2×D8—1582no—
393C2×D8—1581no—
394C2×D8—1581no—
395C2×D8—1581no—
396C2×D8C22×D7911yescharacteristic
397C2×D8C22×D7911yescharacteristic
398C22×C8C22×D7911yescharacteristic
399C22×C8C22×D7911yescharacteristic
400#127—1581no—
401#124—21no—
402#123—41no—
403C22⋊Q8—21no—
404#121—21no—
405#121—41no—
406#121—3161no—
407#121—1581no—
408#121—1581no—
409#120—41no—
410#120—3161no—
411#120—3161no—
412C4×D4—21no—
413C4×D4—1581no—
414C4×D4C22⋊D7911yescharacteristic
415C2×C22⋊C4—41no—
416SD32—3161no—
417SD32—1582no—
418SD32—1582no—
419D16—3161no—
420D16—794no—
421D16—1582no—
422D16—1582no—
423D16—1582no—
424D16—1582no—
425M32—1582no—
426C2×C16—791no—
427C2×C16—791no—
428C2×C16—791no—
429C2×C16—791no—
430C2×C16—1582no—
431#107—791no—
432#107C22×D7911yescharacteristic
433C4⋊C8C22⋊D7911yescharacteristic
434#103—21no—
435#102—21no—
436#102—21no—
437#102—1581no—
438#102—1581no—
439#102—1581no—
440C22⋊C8C22⋊D7911yescharacteristic
441C22⋊C8C22⋊D7911yescharacteristic
442C4×C8C22×D7911yescharacteristic
443C24—41no—
444C24—3161no—
445C24—3161no—
446#41—42no—
447#41—81no—
448C2×Q8—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—41no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—81no—
456C2×D4—3162no—
457C2×D4—3162no—
458C2×D4—3161no—
459C2×D4—3161no—
460C2×D4—3161no—
461C2×D4—3161no—
462C2×D4—3161no—
463C2×D4—3161no—
464C2×D4—6321no—
465C2×D4—6321no—
466C2×D4—6324no—
467C2×D4—1581no—
468C2×D4—1581no—
469C2×D4—1581no—
470C2×D4#1192960911yescharacteristic
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—3161no—
474C22×C4C2×C22⋊D7911yescharacteristic
475C22×C4C2×C22⋊D7911yescharacteristic
476Q16—21no—
477Q16—21no—
478SD16—42no—
479SD16—81no—
480D8—21no—
481D8—21no—
482D8—21no—
483D8—42no—
484D8—3162no—
485D8—3162no—
486D8—792no—
487D8—6321no—
488D8—1581no—
489D8—1581no—
490D8—1581no—
491D8—1581no—
492D8—1581no—
493D8D4×D7912yescomplemented
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C2×C22⋊D7911yescharacteristic
499C2×C8C2×C22⋊D7911yescharacteristic
500C2×C8C2×C22⋊D7911yescharacteristic
501C2×C8C23×D7911yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—41no—
504C4⋊C4—1581no—
505C4⋊C4#1192960911yescharacteristic
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—3161no—
512C22⋊C4—3161no—
513C22⋊C4—3161no—
514C22⋊C4#1192960911yescharacteristic
515C22⋊C4#1192960911yescharacteristic
516C42C2×C22⋊D7911yescharacteristic
517C16—792no—
518C16—1581no—
519C16—1581no—
520C16—1581no—
521C23—41no—
522C23—41no—
523C23—82no—
524C23—82no—
525C23—161no—
526C23—3161no—
527C23—3161no—
528C23—3161no—
529C23—6322no—
530C23—6322no—
531C23—6322no—
532C23—6322no—
533C23—12641no—
534C23—12641no—
535C23C2×#1192960911yescharacteristic
536C23C2×#1192960911yescharacteristic
537Q8—41no—
538Q8—41no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—82no—
545D4—82no—
546D4—161no—
547D4—3161no—
548D4—3161no—
549D4—3161no—
550D4—3161no—
551D4—3161no—
552D4—6322no—
553D4—6322no—
554D4—6322no—
555D4—6322no—
556D4—6322no—
557D4—1582no—
558D4—12641no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—82no—
566C2×C4—161no—
567C2×C4—3161no—
568C2×C4—6322no—
569C2×C4C2×#1192960911yescharacteristic
570C2×C4C2×#1192960911yescharacteristic
571C2×C4C2×#1192960911yescharacteristic
572C2×C4C2×#1192960911yescharacteristic
573C2×C4C22×C22⋊D7911yescharacteristic
574C8—21no—
575C8—21no—
576C8—21no—
577C8C2×D4×D7911yescharacteristic
578C8#1457920611yescharacteristic
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—164no—
591C22—3162no—
592C22—6321no—
593C22—6321no—
594C22—6321no—
595C22—6321no—
596C22—6321no—
597C22—6321no—
598C22—6321no—
599C22—6321no—
600C22—12644no—
601C22—12644no—
602C22#1909899812yescomplemented
603C22C22×#1192960911yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—81no—
609C4—81no—
610C4—6321no—
611C4D4×C22⋊D7911yescharacteristic
612C4#1909936611yescharacteristic
613C2—22no—
614C2—41no—
615C2—82no—
616C2—161no—
617C2—6322no—
618C2—12641no—
619C2—12641no—
620C2D4×#1192960911yescharacteristic
621C2C2×#1909899811yescharacteristic
622C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3280C2
socleC22×C79C22×D8
hypercenter#3280D79
hypocenterC79#33830
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. C2×#19098998C2semidirect
2. C22×#14579102C2semidirect
3. D4×D632C2semidirect
4. #23765667C2semidirect
5. #23765501C2semidirect
6. #23765498C2semidirect
7. #23765452C2semidirect
8. #23765435C2semidirect
9. #23765328C2semidirect
10. #23765323C2semidirect
11. #23765312C2semidirect
12. #23765311C2semidirect
13. #23765274C2semidirect
14. #23765226C2semidirect
15. C79×#3280C2semidirect
17. C22×D632C22semidirect
20. C2×#14579104C22semidirect
21. C2×#14579104C22semidirect
22. C2×#14579104C22semidirect
26. C2×#14579102C22semidirect
27. C2×#14579102C22semidirect
28. C2×#14579102C22semidirect
29. C2×#14579102C22semidirect
30. C2×#14579101C22semidirect
31. C22×C79⋊C16C22semidirect
32. #19098940C22semidirect
34. #19098799C22semidirect
38. C4×D632C22semidirect
39. #19098390C22semidirect
40. C4⋊(C79⋊C16)C22semidirect
41. #19098367C22semidirect
42. #19098364C22semidirect
43. #19098364C22semidirect
44. #19098363C22semidirect
45. #19098363C22semidirect
46. #19098363C22semidirect
47. C22⋊(C79⋊C16)C22semidirect
48. C22⋊(C79⋊C16)C22semidirect
49. #19098339C22semidirect
50. C2×#135×C79C22semidirect
51. C22×D8×C79C22semidirect
53. #495×C79C22semidirect
54. #467×C79C22semidirect
55. #466×C79C22semidirect
59. C4×D8×C79C22semidirect
89. #14579102D4semidirect
164. C79⋊C16C2×D4semidirect
189. D8×C79C2×D4semidirect
279. C22×C79#509semidirect
314. #3280D79semidirect
327. C79#33830semidirect
328. C2×#135C2×D79semidirect
331. C22×D8C2×D79semidirect
347. #495C2×D79semidirect
349. #467C2×D79semidirect
352. #466C2×D79semidirect
360. C4×D8C2×D79semidirect
493. D8D4×D79semidirect
602. C22#19098998semidirect

Maximal subgroups

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

subgroupindex
C2×#190989982
C22×#145791022
D4×D6322
#237656672
#237655012
#237654982
#237654522
#237654352
#237653282
#237653232
#237653122
#237653112
#237652742
#237652262
C79×#32802
#3383079