Finite Group Databasebeta

← #34445240

Subgroups — #34445240

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775748C211yesmaximal, characteristic, complemented
2#23765636C211yesmaximal, characteristic, complemented
3#23765595C211yesmaximal, characteristic, complemented
4#23765583C211yesmaximal, characteristic, complemented
5#23765547C211yesmaximal, characteristic, complemented
6#23765501C211yesmaximal, characteristic, complemented
7#23765498C211yesmaximal, characteristic, complemented
8#23765452C211yesmaximal, characteristic, complemented
9#23765323C211yesmaximal, characteristic, complemented
10#23765312C211yesmaximal, characteristic, complemented
11#23765274C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765112C211yesmaximal, characteristic, complemented
14C2×#19098364C211yesmaximal, characteristic, complemented
15C79×#3280C211yesmaximal, characteristic, complemented
16#19099217—21no—
17C2×#14579104C2211yescharacteristic, complemented
18C2×#14579102C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20C2×#14579101C2211yescharacteristic, complemented
21C22×C79⋊C16C2211yescharacteristic
22#19098968C2211yescharacteristic, complemented
23#19098877—21no—
24#19098877—21no—
25#19098844—21no—
26#19098799C2211yescharacteristic
27#19098799C2211yescharacteristic, complemented
28C4×D632C2211yescharacteristic, complemented
29C2×#14579095C2211yescharacteristic
30C2×#14579095C2211yescharacteristic, complemented
31#19098390C2211yescharacteristic, complemented
32#19098390C2211yescharacteristic, complemented
33#19098390C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098367C2211yescharacteristic, complemented
36#19098367C2211yescharacteristic, complemented
37#19098367C2211yescharacteristic, complemented
38#19098364C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098363C2211yescharacteristic, complemented
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic, complemented
47#19098339C2211yescharacteristic, complemented
48C2×#135×C79C2211yescharacteristic
49C22×D8×C79C2211yescharacteristic
50#546×C79—21no—
51#495×C79C2211yescharacteristic
52#467×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#463×C79—21no—
55#450×C79—21no—
56#449×C79—21no—
57C4×D8×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×#11929623—41no—
60C2×D632C2311yescharacteristic
61#14579161—21no—
62#14579145—41no—
63#14579145—41no—
64#14579141—41no—
65#14579135—21no—
66#14579135—21no—
67C4×D316—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—21no—
70#14579104—41no—
71#14579102—22no—
72#14579102—41no—
73#14579101—41no—
74C2×C79⋊C16—41no—
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79#14579095—21no—
80#14579095—21no—
81#14579095—21no—
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579095C2311yescharacteristic
85#14579083—21no—
86#14579083—21no—
87#14579083—21no—
88C2×#41×C79—21no—
89C22×D4×C79—21no—
90#135×C79—22no—
91#135×C79—41no—
92C2×Q16×C79C2311yescharacteristic
93C2×SD16×C79—21no—
94C2×SD16×C79—21no—
95C2×D8×C79—22no—
96C2×D8×C79—21no—
97C2×D8×C79—41no—
98C2×D8×C79C2311yescharacteristic
99C2×D8×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102#124×C79—21no—
103#123×C79—41no—
104C22⋊Q8×C79—21no—
105#121×C79—21no—
106#121×C79—41no—
107#120×C79—41no—
108C4×D4×C79—21no—
109C4×D4×C79D411yescharacteristic
110C2×C22⋊C4×C79—41no—
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119#11929623—84no—
120C2×Dic158—41no—
121C2×D316—21no—
122C2×C22⋊D79—41no—
123C2×C22⋊D79—41no—
124C22×Dic79—41no—
125C22×Dic79—41no—
126C2×C4×D79—41no—
127D632—21no—
128D632—21no—
129C4⋊Dic79—21no—
130C4⋊Dic79—21no—
131C4⋊Dic79—21no—
132C4⋊Dic79—41no—
133C4⋊Dic79—41no—
134C4⋊Dic79—41no—
135(C2×D79)⋊C4—41no—
136(C2×D79)⋊C4—41no—
137(C2×D79)⋊C4—41no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C79⋊C16—21no—
142C24×C79—41no—
143#41×C79—42no—
144#41×C79—81no—
145C2×Q8×C79—21no—
146C2×D4×C79—21no—
147C2×D4×C79—21no—
148C2×D4×C79—42no—
149C2×D4×C79—41no—
150C2×D4×C79—41no—
151C2×D4×C79—41no—
152C2×D4×C79—81no—
153C2×D4×C79D811yescharacteristic
154C22×C4×C79—41no—
155C22×C4×C79—41no—
156C22×C4×C79C2×D411yescharacteristic
157C22×C4×C79C2×D411yescharacteristic
158Q16×C79—21no—
159Q16×C79—21no—
160SD16×C79—42no—
161SD16×C79—81no—
162D8×C79—21no—
163D8×C79—21no—
164D8×C79—21no—
165D8×C79—42no—
166D8×C79#4112yescomplemented
167C2×C8×C79—21no—
168C2×C8×C79—21no—
169C2×C8×C79—21no—
170C2×C8×C79—21no—
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79C2×D411yescharacteristic
173C2×C8×C79C2×D411yescharacteristic
174C2×C8×C79C2411yescharacteristic
175C4⋊C4×C79—21no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79D811yescharacteristic
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79—41no—
182C22⋊C4×C79—41no—
183C22⋊C4×C79D811yescharacteristic
184C22⋊C4×C79D811yescharacteristic
185C42×C79C2×D411yescharacteristic
186C23×C79—41no—
187C23×C79—41no—
188C23×C79—82no—
189C23×C79—82no—
190C23×C79—161no—
191C23×C79C2×D811yescharacteristic
192C23×C79C2×D811yescharacteristic
193C22×D79—41no—
194Q8×C79—41no—
195Q8×C79—41no—
196D4×C79—22no—
197D4×C79—22no—
198D4×C79—41no—
199D4×C79—41no—
200D4×C79—41no—
201D4×C79—82no—
202D4×C79—82no—
203D4×C79—161no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
210C2×C4×C79—82no—
211C2×C4×C79—161no—
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C2×D811yescharacteristic
214C2×C4×C79C2×D811yescharacteristic
215C2×C4×C79C2×D811yescharacteristic
216C2×C4×C79C22×D411yescharacteristic
217C22⋊D79—82no—
218C22⋊D79—82no—
219C2×Dic79—41no—
220C2×Dic79—41no—
221C2×Dic79—41no—
222C2×Dic79—82no—
223C2×Dic79—82no—
224D316—41no—
225D316—41no—
226C4×D79—82no—
227Dic158—82no—
228C8×C79—21no—
229C8×C79—21no—
230C8×C79—21no—
231C8×C79C2×#4111yescharacteristic
232C8×C79H(5,𝔽2)11yescharacteristic
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—21no—
236C22×C79—21no—
237C22×C79—21no—
238C22×C79—42no—
239C22×C79—81no—
240C22×C79—81no—
241C22×C79—81no—
242C22×C79—81no—
243C22×C79—81no—
244C22×C79—164no—
245C22×C79#50912yes—
246C22×C79C22×D811yescharacteristic
247C2×D79—81no—
248C2×D79—81no—
249C2×D79—81no—
250C4×C79—21no—
251C4×C79—21no—
252C4×C79—21no—
253C4×C79—21no—
254C4×C79—81no—
255C4×C79—81no—
256C4×C79#54611yescharacteristic
257C4×C79#57711yescharacteristic
258Dic79—81no—
259Dic79—81no—
260Dic79—81no—
261#33873—791nomaximal, Sylow
262C2×C79—22no—
263C2×C79—41no—
264C2×C79—82no—
265C2×C79—161no—
266C2×C79#329411yescharacteristic
267C2×C79C2×#50911yescharacteristic
268C2×C79#341611yescharacteristic
269D79—161no—
270#3334—791no—
271#3280D7911yescharacteristic, complemented
272#2243—791no—
273#2235—791no—
274#2233—791no—
275#2226—791no—
276#2218—791no—
277#2217—791no—
278#2210—791no—
279#2186—791no—
280#2184—791no—
281#2177—791no—
282#2168—791no—
283#2139—791no—
284C2×#357—791no—
285C79#3387311yescharacteristic, complemented, Sylow
286C2×#135C2×D7911yescharacteristic
287C22×D8C2×D7911yescharacteristic
288#547—1581no—
289#546—21no—
290C2×SD32—791no—
291C2×D16—791no—
292C2×D16—791no—
293C2×M32—791no—
294C22×C16—791no—
295#500—791no—
296#495C2×D7911yescharacteristic
297#480—1581no—
298#480—1581no—
299#474—1581no—
300#467C2×D7911yescharacteristic
301#466—791no—
302#466—791no—
303#466C2×D7911yescharacteristic
304#463—21no—
305#450—21no—
306#449—21no—
307C4×D8—791no—
308C4×D8C2×D7911yescharacteristic
309C8×D4C2×D7911yescharacteristic
310C2×#107—791no—
311C2×#107—791no—
312#367—791no—
313#367—791no—
314#367—791no—
315C4⋊C16—791no—
316#358—791no—
317#358—791no—
318#358—791no—
319#357—791no—
320#357—791no—
321#357—791no—
322#357—791no—
323#357—791no—
324#357—791no—
325#357—791no—
326C22⋊C16—791no—
327C22⋊C16—791no—
328#346—791no—
329C2×#41—21no—
330C2×#41—3161no—
331C22×D4—21no—
332#135—22no—
333#135—41no—
334C2×Q16C22×D7911yescharacteristic
335C2×SD16—21no—
336C2×SD16—21no—
337C2×D8—22no—
338C2×D8—21no—
339C2×D8—41no—
340C2×D8—791no—
341C2×D8C22×D7911yescharacteristic
342C2×D8C22×D7911yescharacteristic
343C22×C8C22×D7911yescharacteristic
344C22×C8C22×D7911yescharacteristic
345C4⋊Q8—1581no—
346#124—21no—
347#123—41no—
348#123—3161no—
349#123—3161no—
350C22⋊Q8—21no—
351C22⋊Q8—3161no—
352#121—21no—
353#121—41no—
354#121—1581no—
355#121—1581no—
356#120—41no—
357C4×D4—21no—
358C4×D4—1581no—
359C4×D4C22⋊D7911yescharacteristic
360C2×C4⋊C4—1581no—
361C2×C4⋊C4—1581no—
362C2×C22⋊C4—41no—
363SD32—3161no—
364D16—3161no—
365D16—1582no—
366M32—3161no—
367C2×C16—3161no—
368C2×C16—791no—
369C2×C16—791no—
370C2×C16—791no—
371C2×C16—791no—
372#107—791no—
373#107—791no—
374#107—791no—
375#107—1581no—
376#107—1581no—
377#107—1581no—
378#107C22×D7911yescharacteristic
379C4⋊C8C22⋊D7911yescharacteristic
380#103—21no—
381#102—21no—
382#102—21no—
383#102—1581no—
384#102—1581no—
385#102—1581no—
386C22⋊C8C22⋊D7911yescharacteristic
387C22⋊C8C22⋊D7911yescharacteristic
388C4×C8C22×D7911yescharacteristic
389C24—41no—
390#41—42no—
391#41—81no—
392#41—6324no—
393C2×Q8—21no—
394C2×Q8—3161no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—42no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—81no—
402C2×D4—3161no—
403C2×D4—3161no—
404C2×D4—1581no—
405C2×D4#1192960911yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—3161no—
409C22×C4—3161no—
410C22×C4—3161no—
411C22×C4C2×C22⋊D7911yescharacteristic
412C22×C4C2×C22⋊D7911yescharacteristic
413Q16—21no—
414Q16—21no—
415SD16—42no—
416SD16—81no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—42no—
421D8—1581no—
422D8—1581no—
423D8#1192962312yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×C22⋊D7911yescharacteristic
429C2×C8C2×C22⋊D7911yescharacteristic
430C2×C8C2×C22⋊D7911yescharacteristic
431C2×C8C23×D7911yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—41no—
434C4⋊C4—3161no—
435C4⋊C4—3161no—
436C4⋊C4—3161no—
437C4⋊C4—1581no—
438C4⋊C4—1581no—
439C4⋊C4—1581no—
440C4⋊C4#1192960911yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—3161no—
447C22⋊C4—3161no—
448C22⋊C4—3161no—
449C22⋊C4#1192960911yescharacteristic
450C22⋊C4#1192960911yescharacteristic
451C42C2×C22⋊D7911yescharacteristic
452C16—1581no—
453C16—1581no—
454C16—1581no—
455C16—1581no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23—82no—
460C23—161no—
461C23—3161no—
462C23C2×#1192960911yescharacteristic
463C23C2×#1192960911yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—6322no—
467D4—22no—
468D4—22no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—82no—
473D4—82no—
474D4—161no—
475D4—3161no—
476D4—3161no—
477D4—6322no—
478D4—6322no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—161no—
487C2×C4—3161no—
488C2×C4—3161no—
489C2×C4—3161no—
490C2×C4—6322no—
491C2×C4—6322no—
492C2×C4—6322no—
493C2×C4C2×#1192960911yescharacteristic
494C2×C4C2×#1192960911yescharacteristic
495C2×C4C2×#1192960911yescharacteristic
496C2×C4C2×#1192960911yescharacteristic
497C2×C4C22×C22⋊D7911yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8C2×#1192962311yescharacteristic
502C8#1457920611yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—164no—
515C22—6321no—
516C22—6321no—
517C22—6321no—
518C22#1909899812yes—
519C22C22×#1192960911yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—6321no—
527C4—6321no—
528C4—6321no—
529C4#1909920711yescharacteristic
530C4#1909937411yescharacteristic
531C2—22no—
532C2—41no—
533C2—82no—
534C2—161no—
535C2—12641no—
536C2#2377521311yescharacteristic
537C2C2×#1909899811yescharacteristic
538C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3280C2
socleC22×C79C22×D8
hypercenter#3280D79
hypocenterC79#33873
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. #23775748C2semidirect
2. #23765636C2semidirect
3. #23765595C2semidirect
4. #23765583C2semidirect
5. #23765547C2semidirect
6. #23765501C2semidirect
7. #23765498C2semidirect
8. #23765452C2semidirect
9. #23765323C2semidirect
10. #23765312C2semidirect
11. #23765274C2semidirect
12. #23765226C2semidirect
13. #23765112C2semidirect
14. C2×#19098364C2semidirect
15. C79×#3280C2semidirect
17. C2×#14579104C22semidirect
20. C2×#14579101C22semidirect
22. #19098968C22semidirect
27. #19098799C22semidirect
28. C4×D632C22semidirect
30. C2×#14579095C22semidirect
31. #19098390C22semidirect
32. #19098390C22semidirect
33. #19098390C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098367C22semidirect
36. #19098367C22semidirect
37. #19098367C22semidirect
44. #19098363C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. #19098339C22semidirect
166. D8×C79#41semidirect
271. #3280D79semidirect
285. C79#33873semidirect
423. D8#11929623semidirect

Maximal subgroups

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

subgroupindex
#237757482
#237656362
#237655952
#237655832
#237655472
#237655012
#237654982
#237654522
#237653232
#237653122
#237652742
#237652262
#237651122
C2×#190983642
C79×#32802
#3387379