Finite Group Databasebeta

← #34445234

Subgroups — #34445234

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#23775087C211yesmaximal, characteristic, complemented
2#23765639C211yesmaximal, characteristic, complemented
3#23765597C211yesmaximal, characteristic, complemented
4#23765585C211yesmaximal, characteristic, complemented
5#23765544C211yesmaximal, characteristic, complemented
6#23765501C211yesmaximal, characteristic, complemented
7#23765494C211yesmaximal, characteristic, complemented
8#23765455C211yesmaximal, characteristic, complemented
9#23765321C211yesmaximal, characteristic, complemented
10#23765311C211yesmaximal, characteristic, complemented
11#23765274C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765110C211yesmaximal, characteristic, complemented
14C2×#19098363C211yesmaximal, characteristic, complemented
15C79×#3334C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×D632C2211yescharacteristic
18#19099209—21no—
19C2×#14579105C2211yescharacteristic, complemented
20C2×#14579102C2211yescharacteristic
21C2×#14579102C2211yescharacteristic
22C2×#14579101C2211yescharacteristic, complemented
23C22×C79⋊C16C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098779—21no—
28#19098716—21no—
29#19098710—21no—
30C4×D632C2211yescharacteristic
31#19098390C2211yescharacteristic, complemented
32#19098390C2211yescharacteristic, complemented
33#19098390C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098366C2211yescharacteristic, complemented
36#19098366C2211yescharacteristic, complemented
37#19098366C2211yescharacteristic, complemented
38#19098364C2211yescharacteristic, complemented
39#19098363C2211yescharacteristic
40#19098363C2211yescharacteristic
41#19098363C2211yescharacteristic
42#19098363C2211yescharacteristic
43#19098363C2211yescharacteristic
44#19098363C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic, complemented
47#19098339C2211yescharacteristic, complemented
48#547×C79—21no—
49#500×C79C2211yescharacteristic, complemented
50#480×C79—21no—
51#480×C79—21no—
52#474×C79—21no—
53#466×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic, complemented
55C4×D8×C79C2211yescharacteristic, complemented
56C8×D4×C79C2211yescharacteristic
57C2×#107×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic, complemented
59C2×#11929624—21no—
60C22×D316—21no—
61#14579183—22no—
62#14579183—41no—
63C2×Dic316C2311yescharacteristic
64C2×#11929613—21no—
65C2×#11929613—21no—
66C2×D632—22no—
67C2×D632—21no—
68C2×D632—41no—
69C2×D632C2311yescharacteristic
70C2×D632C2311yescharacteristic
71#14579149—21no—
72#14579146—41no—
73#14579140—21no—
74#14579135—21no—
75#14579134—41no—
76#14579129—41no—
77C4×D316—21no—
78C22⋊C4×D79—41no—
79#14579105—41no—
80#14579102—22no—
81#14579102—41no—
82#14579101—41no—
83C2×C79⋊C16—41no—
84C2×C79⋊C16C2311yescharacteristic
85C2×C79⋊C16C2311yescharacteristic
86C2×C79⋊C16C2311yescharacteristic
87C2×C79⋊C16C2311yescharacteristic
88#14579095C2311yescharacteristic
89#14579086—21no—
90#14579083—21no—
91#14579083—21no—
92C2×#41×C79—41no—
93C2×D8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95C22×C8×C79C2311yescharacteristic
96C4⋊Q8×C79—21no—
97#123×C79—41no—
98#123×C79—41no—
99C22⋊Q8×C79—41no—
100#121×C79—21no—
101#121×C79—21no—
102C4×D4×C79—21no—
103C4×D4×C79D411yescharacteristic
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—21no—
106#107×C79—21no—
107#107×C79—21no—
108#107×C79—21no—
109#107×C79C2311yescharacteristic
110#107×C79C2311yescharacteristic
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#102×C79—21no—
114#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C23×D79—41no—
120#11929624—42no—
121#11929624—81no—
122C2×Dic158—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316—42no—
126C2×D316—41no—
127C2×D316—81no—
128C2×C22⋊D79—41no—
129C2×C22⋊D79—41no—
130C2×C4×D79—41no—
131C2×C4×D79—41no—
132Dic316—21no—
133Dic316—21no—
134#11929613—42no—
135#11929613—81no—
136D632—21no—
137D632—21no—
138D632—21no—
139D632—42no—
140D632#4112yescomplemented
141C4⋊Dic79—21no—
142Dic79⋊C4—41no—
143C22⋊Dic79—41no—
144(C2×D79)⋊C4—41no—
145(C2×D79)⋊C4—41no—
146(C2×D79)⋊C4—41no—
147(C2×D79)⋊C4—41no—
148C79⋊C16—21no—
149C79⋊C16—21no—
150C79⋊C16—21no—
151C79⋊C16—21no—
152#41×C79—84no—
153C2×Q8×C79—41no—
154C2×D4×C79—21no—
155C2×D4×C79—41no—
156C2×D4×C79—41no—
157C2×D4×C79D811yescharacteristic
158C22×C4×C79—41no—
159C22×C4×C79—41no—
160C22×C4×C79—41no—
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163D8×C79—21no—
164D8×C79—21no—
165C2×C8×C79—21no—
166C2×C8×C79—21no—
167C2×C8×C79—21no—
168C2×C8×C79—21no—
169C2×C8×C79C2×D411yescharacteristic
170C2×C8×C79C2×D411yescharacteristic
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79C2411yescharacteristic
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79—41no—
178C4⋊C4×C79—41no—
179C4⋊C4×C79D811yescharacteristic
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×C79C2×D811yescharacteristic
188C23×C79C2×D811yescharacteristic
189C22×D79—41no—
190C22×D79—41no—
191C22×D79—82no—
192C22×D79—82no—
193C22×D79—161no—
194Q8×C79—82no—
195D4×C79—22no—
196D4×C79—41no—
197D4×C79—41no—
198D4×C79—82no—
199D4×C79—82no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—41no—
205C2×C4×C79—41no—
206C2×C4×C79—41no—
207C2×C4×C79—82no—
208C2×C4×C79—82no—
209C2×C4×C79—82no—
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C2×D811yescharacteristic
214C2×C4×C79C22×D411yescharacteristic
215C22⋊D79—82no—
216C22⋊D79—161no—
217C2×Dic79—41no—
218C2×Dic79—41no—
219D316—22no—
220D316—41no—
221D316—41no—
222D316—41no—
223D316—82no—
224C4×D79—82no—
225C4×D79—161no—
226Dic158—41no—
227Dic158—41no—
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—81no—
239C22×C79—81no—
240C22×C79—81no—
241C22×C79#50912yes—
242C22×C79C22×D811yescharacteristic
243C2×D79—42no—
244C2×D79—81no—
245C2×D79—81no—
246C2×D79—81no—
247C2×D79—81no—
248C2×D79—81no—
249C2×D79—164no—
250C4×C79—21no—
251C4×C79—21no—
252C4×C79—21no—
253C4×C79—21no—
254C4×C79—81no—
255C4×C79—81no—
256C4×C79—81no—
257C4×C79#54611yescharacteristic
258C4×C79#57711yescharacteristic
259Dic79—81no—
260Dic79—81no—
261#33873—791nomaximal, Sylow
262C2×C79—22no—
263C2×C79—41no—
264C2×C79—161no—
265C2×C79#329411yescharacteristic
266C2×C79C2×#50911yescharacteristic
267C2×C79#341611yescharacteristic
268D79—82no—
269D79—161no—
270#3334D7911yescharacteristic, complemented
271#3280—791no—
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×#135—791no—
287C22×D8—791no—
288#547—21no—
289#546—1581no—
290C2×SD32—791no—
291C2×D16—791no—
292C2×D16—791no—
293C2×M32—791no—
294C22×C16—791no—
295#500C2×D7911yescharacteristic, complemented
296#495—791no—
297#480—21no—
298#480—21no—
299#474—21no—
300#467—791no—
301#466—791no—
302#466C2×D7911yescharacteristic
303#466C2×D7911yescharacteristic, complemented
304#463—1581no—
305#450—1581no—
306#449—1581no—
307C4×D8—791no—
308C4×D8C2×D7911yescharacteristic, complemented
309C8×D4C2×D7911yescharacteristic
310C2×#107C2×D7911yescharacteristic
311C2×#107C2×D7911yescharacteristic, complemented
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—41no—
330C2×#41—1581no—
331C22×D4—1581no—
332#135—3161no—
333#135—1582no—
334C2×Q16—791no—
335C2×SD16—1581no—
336C2×SD16—1581no—
337C2×D8—3161no—
338C2×D8—791no—
339C2×D8—791no—
340C2×D8—1582no—
341C2×D8—1581no—
342C2×D8C22×D7911yescharacteristic
343C22×C8C22×D7911yescharacteristic
344C22×C8C22×D7911yescharacteristic
345C4⋊Q8—21no—
346#124—1581no—
347#123—41no—
348#123—41no—
349#123—3161no—
350C22⋊Q8—41no—
351C22⋊Q8—1581no—
352#121—21no—
353#121—21no—
354#121—3161no—
355#121—1581no—
356#120—3161no—
357C4×D4—21no—
358C4×D4—1581no—
359C4×D4C22⋊D7911yescharacteristic
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—3161no—
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—21no—
373#107—21no—
374#107—21no—
375#107—791no—
376#107C22×D7911yescharacteristic
377#107C22×D7911yescharacteristic
378#107C22×D7911yescharacteristic
379C4⋊C8C22⋊D7911yescharacteristic
380#103—1581no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—1581no—
385#102—1581no—
386C22⋊C8C22⋊D7911yescharacteristic
387C22⋊C8C22⋊D7911yescharacteristic
388C4×C8C22×D7911yescharacteristic
389C24—3161no—
390#41—84no—
391#41—3162no—
392#41—6321no—
393C2×Q8—41no—
394C2×Q8—1581no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—3162no—
399C2×D4—3161no—
400C2×D4—3161no—
401C2×D4—3161no—
402C2×D4—6321no—
403C2×D4—1581no—
404C2×D4—1581no—
405C2×D4#1192960911yescharacteristic
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—3161no—
410C22×C4—3161no—
411C22×C4C2×C22⋊D7911yescharacteristic
412C22×C4C2×C22⋊D7911yescharacteristic
413Q16—1581no—
414Q16—1581no—
415SD16—3162no—
416SD16—6321no—
417D8—21no—
418D8—21no—
419D8—3162no—
420D8—792no—
421D8—1581no—
422D8—1581no—
423D8—1581no—
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—21no—
434C4⋊C4—21no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—3161no—
439C4⋊C4—1581no—
440C4⋊C4#1192960911yescharacteristic
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—3161no—
445C22⋊C4—3161no—
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—3161no—
458C23—3161no—
459C23—6322no—
460C23—6322no—
461C23—12641no—
462C23C2×#1192960911yescharacteristic
463C23C2×#1192960911yescharacteristic
464Q8—82no—
465Q8—3161no—
466Q8—3161no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—82no—
471D4—82no—
472D4—3161no—
473D4—3161no—
474D4—3161no—
475D4—6322no—
476D4—6322no—
477D4—1582no—
478D4—12641no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—3161no—
490C2×C4—3161no—
491C2×C4—6322no—
492C2×C4—12641no—
493C2×C4C2×#1192960911yescharacteristic
494C2×C4C2×#1192960911yescharacteristic
495C2×C4C2×#1192960911yescharacteristic
496C2×C4C2×#1192960911yescharacteristic
497C2×C4C22×C22⋊D7911yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D7911yescharacteristic
502C8#1457920711yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—3162no—
512C22—6321no—
513C22—6321no—
514C22—6321no—
515C22—6321no—
516C22—6321no—
517C22—12644no—
518C22#1909899912yes—
519C22C22×#1192960911yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—6321no—
528C4—6321no—
529C4#1909920111yescharacteristic
530C4#1909937111yescharacteristic
531C2—22no—
532C2—41no—
533C2—161no—
534C2—6322no—
535C2—12641no—
536C2#2377520711yescharacteristic
537C2C2×#1909899911yescharacteristic
538C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3334C2
socleC22×C79C22×D8
hypercenter#3334D79
hypocenterC79#33873
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #23775087C2semidirect
2. #23765639C2semidirect
3. #23765597C2semidirect
4. #23765585C2semidirect
5. #23765544C2semidirect
6. #23765501C2semidirect
7. #23765494C2semidirect
8. #23765455C2semidirect
9. #23765321C2semidirect
10. #23765311C2semidirect
11. #23765274C2semidirect
12. #23765226C2semidirect
13. #23765110C2semidirect
14. C2×#19098363C2semidirect
15. C79×#3334C2semidirect
19. C2×#14579105C22semidirect
22. C2×#14579101C22semidirect
31. #19098390C22semidirect
32. #19098390C22semidirect
33. #19098390C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098366C22semidirect
36. #19098366C22semidirect
37. #19098366C22semidirect
38. #19098364C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. #19098339C22semidirect
49. #500×C79C22semidirect
54. #466×C79C22semidirect
55. C4×D8×C79C22semidirect
58. C2×#107×C79C22semidirect
140. D632#41semidirect
270. #3334D79semidirect
285. C79#33873semidirect
295. #500C2×D79semidirect
303. #466C2×D79semidirect
308. C4×D8C2×D79semidirect
311. C2×#107C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237750872
#237656392
#237655972
#237655852
#237655442
#237655012
#237654942
#237654552
#237653212
#237653112
#237652742
#237652262
#237651102
C2×#190983632
C79×#33342
#3387379