Finite Group Databasebeta

← #34444773

Subgroups — #34444773

Order 20224.

692 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
1C2×#19098998C211yesmaximal, characteristic, complemented
2C2×#19098993C211yesmaximal, characteristic, complemented
3#23775087C211yesmaximal, characteristic, complemented
4#23765669C211yesmaximal, characteristic, complemented
5#23765506C211yesmaximal, characteristic, complemented
6#23765498C211yesmaximal, characteristic, complemented
7#23765455C211yesmaximal, characteristic, complemented
8#23765442C211yesmaximal, characteristic, complemented
9#23765340C211yesmaximal, characteristic, complemented
10#23765338C211yesmaximal, characteristic, complemented
11#23765328C211yesmaximal, characteristic, complemented
12#23765323C211yesmaximal, characteristic, complemented
13#23765274C211yesmaximal, characteristic, complemented
14#23765226C211yesmaximal, characteristic, complemented
15C79×#3283C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×D632C2211yescharacteristic
18#19099209—21no—
19#19098998—24no—
20#19098993—24no—
21C2×#14579107C2211yescharacteristic, complemented
22C2×#14579107C2211yescharacteristic, complemented
23C2×#14579105C2211yescharacteristic, complemented
24C2×#14579104C2211yescharacteristic
25C2×#14579104C2211yescharacteristic
26C2×#14579104C2211yescharacteristic
27C2×#14579104C2211yescharacteristic
28C2×#14579102C2211yescharacteristic
29C2×#14579102C2211yescharacteristic
30C2×#14579101C2211yescharacteristic
31C22×C79⋊C16C2211yescharacteristic, complemented
32#19098947C2211yescharacteristic
33#19098804C2211yescharacteristic
34#19098799C2211yescharacteristic
35#19098779—21no—
36#19098716—21no—
37#19098710—21no—
38C4×D632C2211yescharacteristic
39#19098390C2211yescharacteristic, complemented
40C4⋊(C79⋊C16)C2211yescharacteristic, complemented
41#19098367C2211yescharacteristic, complemented
42#19098367C2211yescharacteristic, complemented
43#19098366C2211yescharacteristic, complemented
44#19098364C2211yescharacteristic, complemented
45#19098363C2211yescharacteristic
46#19098363C2211yescharacteristic
47C22⋊(C79⋊C16)C2211yescharacteristic
48C22⋊(C79⋊C16)C2211yescharacteristic, complemented
49#19098339C2211yescharacteristic, complemented
50C2×#135×C79C2211yescharacteristic
51C2×#135×C79C2211yescharacteristic, complemented
52#547×C79—21no—
53#494×C79C2211yescharacteristic, complemented
54#467×C79C2211yescharacteristic
55#467×C79C2211yescharacteristic, complemented
56#461×C79—21no—
57#452×C79—21no—
58#452×C79—21no—
59C4×D8×C79C2211yescharacteristic, complemented
60C8×D4×C79C2211yescharacteristic
61C2×#11929624—21no—
62C22×D316—21no—
63#14579183—22no—
64#14579183—41no—
65C2×Dic316C2311yescharacteristic
66C2×#11929613—21no—
67C2×#11929613—21no—
68C2×D632—22no—
69C2×D632—21no—
70C2×D632—41no—
71C2×D632C2311yescharacteristic
72C2×D632C2311yescharacteristic
73#14579149—21no—
74#14579146—41no—
75#14579140—21no—
76#14579135—21no—
77#14579134—41no—
78#14579129—41no—
79C4×D316—21no—
80C22⋊C4×D79—41no—
81#14579107—22no—
82#14579107—41no—
83#14579105—22no—
84#14579104—22no—
85#14579104—22no—
86#14579104—22no—
87#14579104—41no—
88#14579102—22no—
89#14579102D414yescomplemented
90#14579101—22no—
91C2×C79⋊C16—22no—
92C2×C79⋊C16C2311yescharacteristic
93C2×C79⋊C16C2311yescharacteristic
94C2×C79⋊C16C2311yescharacteristic
95C2×C79⋊C16C2311yescharacteristic
96#14579095C2311yescharacteristic
97#14579086—21no—
98#14579083—21no—
99#14579083—21no—
100C2×#41×C79—21no—
101C2×#41×C79—21no—
102#135×C79—22no—
103#135×C79—22no—
104#135×C79—41no—
105#135×C79—41no—
106C2×Q16×C79—21no—
107C2×Q16×C79C2311yescharacteristic
108C2×Q16×C79C2311yescharacteristic
109C2×SD16×C79—21no—
110C2×SD16×C79—21no—
111C2×D8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C22×C8×C79C2311yescharacteristic
114C4⋊Q8×C79—21no—
115#123×C79—41no—
116#123×C79—41no—
117C22⋊Q8×C79—21no—
118C22⋊Q8×C79—21no—
119#121×C79—41no—
120C4×D4×C79—21no—
121C4×D4×C79D411yescharacteristic
122C2×C4⋊C4×C79—41no—
123#107×C79C2311yescharacteristic
124C4⋊C8×C79D411yescharacteristic
125#103×C79—21no—
126#103×C79—21no—
127#102×C79—21no—
128C22⋊C8×C79D411yescharacteristic
129C22⋊C8×C79D411yescharacteristic
130C4×C8×C79C2311yescharacteristic
131C23×D79—41no—
132#11929624—42no—
133#11929624—81no—
134C2×Dic158—21no—
135C2×D316—21no—
136C2×D316—21no—
137C2×D316—42no—
138C2×D316—41no—
139C2×D316—81no—
140C2×C22⋊D79—41no—
141C2×C22⋊D79—41no—
142C2×C4×D79—41no—
143C2×C4×D79—41no—
144Dic316—21no—
145Dic316—21no—
146#11929613—42no—
147#11929613—81no—
148D632—21no—
149D632—21no—
150D632—21no—
151D632—42no—
152D632C2×D412yes—
153C4⋊Dic79—21no—
154Dic79⋊C4—41no—
155C22⋊Dic79—41no—
156(C2×D79)⋊C4—41no—
157(C2×D79)⋊C4—41no—
158(C2×D79)⋊C4—41no—
159(C2×D79)⋊C4—41no—
160C79⋊C16—21no—
161C79⋊C16—21no—
162C79⋊C16—21no—
163C79⋊C16C2×D412yes—
164#41×C79—42no—
165#41×C79—42no—
166#41×C79—81no—
167#41×C79—81no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×D4×C79—21no—
171C2×D4×C79—41no—
172C2×D4×C79—41no—
173C2×D4×C79D811yescharacteristic
174C22×C4×C79—41no—
175C22×C4×C79—41no—
176C22×C4×C79—41no—
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179Q16×C79—21no—
180Q16×C79—21no—
181Q16×C79—21no—
182Q16×C79—21no—
183Q16×C79—81no—
184SD16×C79—42no—
185SD16×C79—42no—
186D8×C79—21no—
187D8×C79C2×D412yes—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79C2×D411yescharacteristic
193C2×C8×C79C2×D411yescharacteristic
194C2×C8×C79C2×D411yescharacteristic
195C2×C8×C79C2411yescharacteristic
196C4⋊C4×C79—21no—
197C4⋊C4×C79—41no—
198C4⋊C4×C79—41no—
199C4⋊C4×C79—41no—
200C4⋊C4×C79—41no—
201C4⋊C4×C79D811yescharacteristic
202C22⋊C4×C79—41no—
203C22⋊C4×C79—41no—
204C22⋊C4×C79—41no—
205C22⋊C4×C79D811yescharacteristic
206C22⋊C4×C79D811yescharacteristic
207C42×C79C2×D411yescharacteristic
208C23×C79—41no—
209C23×C79C2×D811yescharacteristic
210C23×C79C2×D811yescharacteristic
211C22×D79—41no—
212C22×D79—41no—
213C22×D79—82no—
214C22×D79—82no—
215C22×D79—161no—
216Q8×C79—41no—
217Q8×C79—41no—
218Q8×C79—41no—
219Q8×C79—41no—
220D4×C79—22no—
221D4×C79—41no—
222D4×C79—41no—
223D4×C79—82no—
224D4×C79—82no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—82no—
233C2×C4×C79—82no—
234C2×C4×C79—82no—
235C2×C4×C79C2×D811yescharacteristic
236C2×C4×C79C2×D811yescharacteristic
237C2×C4×C79C2×D811yescharacteristic
238C2×C4×C79C2×D811yescharacteristic
239C2×C4×C79C22×D411yescharacteristic
240C22⋊D79—82no—
241C22⋊D79—161no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244D316—22no—
245D316—41no—
246D316—41no—
247D316—41no—
248D316—82no—
249C4×D79—82no—
250C4×D79—161no—
251Dic158—41no—
252Dic158—41no—
253C8×C79—21no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79C22×D411yescharacteristic
257C8×C79H(5,𝔽2)11yescharacteristic
258C22×C79—21no—
259C22×C79—21no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79—21no—
263C22×C79—21no—
264C22×C79—42no—
265C22×C79—81no—
266C22×C79—81no—
267C22×C79C22×D811yescharacteristic
268C2×D79—42no—
269C2×D79—81no—
270C2×D79—81no—
271C2×D79—81no—
272C2×D79—81no—
273C2×D79—81no—
274C2×D79—164no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79—21no—
279C4×C79—81no—
280C4×C79—81no—
281C4×C79—81no—
282C4×C79D4211yescharacteristic
283C4×C79#57611yescharacteristic
284Dic79—81no—
285Dic79—81no—
286#33833—791nomaximal, Sylow
287C2×C79—41no—
288C2×C79—41no—
289C2×C79—82no—
290C2×C79D4×D811yescharacteristic
291C2×C79#341411yescharacteristic
292C2×C79#341611yescharacteristic
293D79—82no—
294D79—161no—
295C2×#509—791no—
296C2×#508—791no—
297#3283D7911yescharacteristic, complemented
298#3280—791no—
299#2251—791no—
300#2219—791no—
301#2217—791no—
302#2210—791no—
303#2208—791no—
304#2189—791no—
305#2189—791no—
306#2187—791no—
307#2186—791no—
308#2177—791no—
309#2168—791no—
310C79#3383311yescharacteristic, complemented, Sylow
311C2×#135—791no—
312C2×#135C2×D7911yescharacteristic
313C2×#135C2×D7911yescharacteristic, complemented
314C22×D8—791no—
315#547—21no—
316#546—1581no—
317#509—1584no—
318#508—1584no—
319C2×Q32—791no—
320C2×Q32—791no—
321C2×SD32—791no—
322C2×SD32—791no—
323C2×SD32—791no—
324C2×SD32—791no—
325C2×SD32—791no—
326C2×D16—791no—
327C2×D16—791no—
328C2×M32—791no—
329C22×C16—791no—
330#495—791no—
331#494C2×D7911yescharacteristic, complemented
332#467—791no—
333#467C2×D7911yescharacteristic
334#467C2×D7911yescharacteristic, complemented
335#466—791no—
336#463—1581no—
337#461—21no—
338#452—21no—
339#452—21no—
340#450—1581no—
341#449—1581no—
342C4×D8—791no—
343C4×D8C2×D7911yescharacteristic, complemented
344C8×D4C2×D7911yescharacteristic
345#367—791no—
346C4⋊C16—791no—
347#358—791no—
348#358—791no—
349#358—791no—
350#357—791no—
351#357—791no—
352#357—791no—
353C22⋊C16—791no—
354C22⋊C16—791no—
355#346—791no—
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—1581no—
359C22×D4—1581no—
360#135—22no—
361#135—22no—
362#135—41no—
363#135—41no—
364#135—3161no—
365#135—1582no—
366C2×Q16—21no—
367C2×Q16—791no—
368C2×Q16C22×D7911yescharacteristic
369C2×Q16C22×D7911yescharacteristic
370C2×SD16—21no—
371C2×SD16—21no—
372C2×SD16—1581no—
373C2×SD16—1581no—
374C2×D8—3161no—
375C2×D8—791no—
376C2×D8—791no—
377C2×D8—1582no—
378C2×D8—1581no—
379C2×D8C22×D7911yescharacteristic
380C22×C8C22×D7911yescharacteristic
381C22×C8C22×D7911yescharacteristic
382C4⋊Q8—21no—
383#124—1581no—
384#123—41no—
385#123—41no—
386#123—3161no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—1581no—
390#121—41no—
391#121—3161no—
392#121—1581no—
393#120—3161no—
394C4×D4—21no—
395C4×D4—1581no—
396C4×D4C22⋊D7911yescharacteristic
397C2×C4⋊C4—41no—
398C2×C22⋊C4—3161no—
399Q32—3161no—
400Q32—1582no—
401SD32—3161no—
402SD32—1582no—
403SD32—1582no—
404SD32—1582no—
405SD32—1582no—
406D16—794no—
407D16—1582no—
408M32—1582no—
409C2×C16—791no—
410C2×C16—791no—
411C2×C16—791no—
412C2×C16—791no—
413C2×C16—1582no—
414#107—791no—
415#107C22×D7911yescharacteristic
416C4⋊C8C22⋊D7911yescharacteristic
417#103—21no—
418#103—21no—
419#103—1581no—
420#102—21no—
421#102—1581no—
422#102—1581no—
423C22⋊C8C22⋊D7911yescharacteristic
424C22⋊C8C22⋊D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426C24—3161no—
427#41—42no—
428#41—42no—
429#41—81no—
430#41—81no—
431#41—3162no—
432#41—6321no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—1581no—
436C2×D4—21no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—3162no—
440C2×D4—3161no—
441C2×D4—3161no—
442C2×D4—3161no—
443C2×D4—6321no—
444C2×D4—1581no—
445C2×D4—1581no—
446C2×D4#1192960911yescharacteristic
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—3161no—
451C22×C4—3161no—
452C22×C4C2×C22⋊D7911yescharacteristic
453C22×C4C2×C22⋊D7911yescharacteristic
454Q16—21no—
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—81no—
459Q16—1581no—
460Q16—1581no—
461SD16—42no—
462SD16—42no—
463SD16—3162no—
464SD16—6321no—
465D8—21no—
466D8—3162no—
467D8—792no—
468D8—1581no—
469D8—1581no—
470D8—1581no—
471D8D4×D7912yes—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8C2×C22⋊D7911yescharacteristic
477C2×C8C2×C22⋊D7911yescharacteristic
478C2×C8C2×C22⋊D7911yescharacteristic
479C2×C8C23×D7911yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—3161no—
486C4⋊C4—1581no—
487C4⋊C4#1192960911yescharacteristic
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—3161no—
492C22⋊C4—3161no—
493C22⋊C4—3161no—
494C22⋊C4—3161no—
495C22⋊C4—3161no—
496C22⋊C4#1192960911yescharacteristic
497C22⋊C4#1192960911yescharacteristic
498C42C2×C22⋊D7911yescharacteristic
499C16—792no—
500C16—1581no—
501C16—1581no—
502C16—1581no—
503C23—41no—
504C23—3161no—
505C23—3161no—
506C23—6322no—
507C23—6322no—
508C23—12641no—
509C23C2×#1192960911yescharacteristic
510C23C2×#1192960911yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—3161no—
516Q8—3161no—
517D4—22no—
518D4—41no—
519D4—41no—
520D4—82no—
521D4—82no—
522D4—3161no—
523D4—3161no—
524D4—3161no—
525D4—6322no—
526D4—6322no—
527D4—1582no—
528D4—12641no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—6322no—
542C2×C4—12641no—
543C2×C4C2×#1192960911yescharacteristic
544C2×C4C2×#1192960911yescharacteristic
545C2×C4C2×#1192960911yescharacteristic
546C2×C4C2×#1192960911yescharacteristic
547C2×C4C22×C22⋊D7911yescharacteristic
548C8—21no—
549C8—21no—
550C8—21no—
551C8C2×D4×D7911yescharacteristic
552C8#1457920611yescharacteristic
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—81no—
561C22—81no—
562C22—3162no—
563C22—6321no—
564C22—6321no—
565C22—6321no—
566C22—6321no—
567C22—6321no—
568C22—12644no—
569C22C22×#1192960911yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—6321no—
578C4—6321no—
579C4D4×C22⋊D7911yescharacteristic
580C4#1909936611yescharacteristic
581C2—41no—
582C2—41no—
583C2—82no—
584C2—6322no—
585C2—12641no—
586C2D4×#1192960911yescharacteristic
587C2#2377653411yescharacteristic
588C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3283C2
socleC22×C79C22×D8
hypercenter#3283D79
hypocenterC79#33833
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. C2×#19098998C2semidirect
2. C2×#19098993C2semidirect
3. #23775087C2semidirect
4. #23765669C2semidirect
5. #23765506C2semidirect
6. #23765498C2semidirect
7. #23765455C2semidirect
8. #23765442C2semidirect
9. #23765340C2semidirect
10. #23765338C2semidirect
11. #23765328C2semidirect
12. #23765323C2semidirect
13. #23765274C2semidirect
14. #23765226C2semidirect
15. C79×#3283C2semidirect
21. C2×#14579107C22semidirect
22. C2×#14579107C22semidirect
23. C2×#14579105C22semidirect
31. C22×C79⋊C16C22semidirect
39. #19098390C22semidirect
40. C4⋊(C79⋊C16)C22semidirect
41. #19098367C22semidirect
42. #19098367C22semidirect
43. #19098366C22semidirect
44. #19098364C22semidirect
48. C22⋊(C79⋊C16)C22semidirect
49. #19098339C22semidirect
51. C2×#135×C79C22semidirect
53. #494×C79C22semidirect
55. #467×C79C22semidirect
59. C4×D8×C79C22semidirect
89. #14579102D4semidirect
297. #3283D79semidirect
310. C79#33833semidirect
313. C2×#135C2×D79semidirect
331. #494C2×D79semidirect
334. #467C2×D79semidirect
343. C4×D8C2×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989982
C2×#190989932
#237750872
#237656692
#237655062
#237654982
#237654552
#237654422
#237653402
#237653382
#237653282
#237653232
#237652742
#237652262
C79×#32832
#3383379