Finite Group Databasebeta

← #34445309

Subgroups — #34445309

Order 20224.

562 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775087C211yesmaximal, characteristic
2#23765639C211yesmaximal, characteristic, complemented
3#23765608C211yesmaximal, characteristic, complemented
4#23765591C211yesmaximal, characteristic, complemented
5#23765544C211yesmaximal, characteristic, complemented
6#23765506C211yesmaximal, characteristic, complemented
7#23765498C211yesmaximal, characteristic, complemented
8#23765462C211yesmaximal, characteristic, complemented
9#23765338C211yesmaximal, characteristic, complemented
10#23765328C211yesmaximal, characteristic, complemented
11#23765262C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765115C211yesmaximal, characteristic, complemented
14#23765105C211yesmaximal, characteristic, complemented
15C79×#3342C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×D632C2211yescharacteristic
18#19099209—21no—
19C2×#14579107C2211yescharacteristic, complemented
20C2×#14579104C2211yescharacteristic
21C2×#14579104C2211yescharacteristic
22C2×#14579101C2211yescharacteristic
23C22×C79⋊C16C2211yescharacteristic, complemented
24#19098947C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098779—21no—
28#19098716—21no—
29#19098710—21no—
30C4×D632C2211yescharacteristic
31#19098390C2211yescharacteristic, complemented
32#19098390C2211yescharacteristic, complemented
33#19098388C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098367C2211yescharacteristic, complemented
36#19098366C2211yescharacteristic, complemented
37#19098366C2211yescharacteristic, complemented
38#19098366C2211yescharacteristic, 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#556×C79—21no—
49#500×C79C2211yescharacteristic, complemented
50#485×C79—21no—
51#485×C79—21no—
52#475×C79—21no—
53#467×C79C2211yescharacteristic
54#467×C79C2211yescharacteristic, complemented
55C4×Q16×C79C2211yescharacteristic, complemented
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic
58#427×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#14579107—41no—
80#14579104—22no—
81#14579104—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×Q16×C79C2311yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95C4⋊Q8×C79—21no—
96#126×C79—41no—
97#126×C79—41no—
98#124×C79—41no—
99C22⋊Q8×C79—21no—
100C22⋊Q8×C79—21no—
101C4×Q8×C79—21no—
102C4×Q8×C79—41no—
103C4×D4×C79D411yescharacteristic
104#117×C79—21no—
105#117×C79—21no—
106#107×C79—21no—
107#107×C79C2311yescharacteristic
108#107×C79C2311yescharacteristic
109#107×C79C2311yescharacteristic
110#106×C79—21no—
111#106×C79—21no—
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#103×C79—21no—
115#103×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#4112yes—
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—
152C2×Q8×C79—21no—
153C2×Q8×C79—41no—
154C2×D4×C79D811yescharacteristic
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157Q16×C79—21no—
158Q16×C79—21no—
159C2×C8×C79—21no—
160C2×C8×C79—21no—
161C2×C8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79C2×D411yescharacteristic
164C2×C8×C79C2×D411yescharacteristic
165C2×C8×C79C2×D411yescharacteristic
166C2×C8×C79C2411yescharacteristic
167C4⋊C4×C79—21no—
168C4⋊C4×C79—21no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—41no—
171C4⋊C4×C79—41no—
172C4⋊C4×C79—41no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79D811yescharacteristic
175C22⋊C4×C79—41no—
176C22⋊C4×C79—41no—
177C22⋊C4×C79—41no—
178C22⋊C4×C79—41no—
179C22⋊C4×C79D811yescharacteristic
180C22⋊C4×C79D811yescharacteristic
181C42×C79—41no—
182C42×C79—41no—
183C42×C79—41no—
184C42×C79C2×D411yescharacteristic
185C23×C79C2×D811yescharacteristic
186C23×C79C2×D811yescharacteristic
187C22×D79—41no—
188C22×D79—41no—
189C22×D79—82no—
190C22×D79—82no—
191C22×D79—161no—
192Q8×C79—41no—
193Q8×C79—41no—
194Q8×C79—82no—
195D4×C79—41no—
196C2×C4×C79—21no—
197C2×C4×C79—21no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—41no—
201C2×C4×C79—41no—
202C2×C4×C79—41no—
203C2×C4×C79—41no—
204C2×C4×C79C2×D811yescharacteristic
205C2×C4×C79C2×D811yescharacteristic
206C2×C4×C79C2×D811yescharacteristic
207C2×C4×C79C2×D811yescharacteristic
208C2×C4×C79C22×D411yescharacteristic
209C22⋊D79—82no—
210C22⋊D79—161no—
211C2×Dic79—41no—
212C2×Dic79—41no—
213D316—22no—
214D316—41no—
215D316—41no—
216D316—41no—
217D316—82no—
218C4×D79—82no—
219C4×D79—161no—
220Dic158—41no—
221Dic158—41no—
222C8×C79—21no—
223C8×C79—21no—
224C8×C79—21no—
225C8×C79C2×#4111yescharacteristic
226C8×C79H(5,𝔽2)11yescharacteristic
227C22×C79—21no—
228C22×C79—21no—
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79C22×D811yescharacteristic
234C2×D79—42no—
235C2×D79—81no—
236C2×D79—81no—
237C2×D79—81no—
238C2×D79—81no—
239C2×D79—81no—
240C2×D79—164no—
241C4×C79—21no—
242C4×C79—21no—
243C4×C79—21no—
244C4×C79—21no—
245C4×C79—81no—
246C4×C79—81no—
247C4×C79—81no—
248C4×C79—81no—
249C4×C79#54611yescharacteristic
250C4×C79#57711yescharacteristic
251Dic79—81no—
252Dic79—81no—
253#33878—791nomaximal, Sylow
254C2×C79—41no—
255C2×C79—41no—
256C2×C79#329411yescharacteristic
257C2×C79#341411yescharacteristic
258C2×C79#341611yescharacteristic
259D79—82no—
260D79—161no—
261#3342D7911yescharacteristic, complemented
262#3280—791no—
263#2243—791no—
264#2237—791no—
265#2234—791no—
266#2226—791no—
267#2219—791no—
268#2217—791no—
269#2211—791no—
270#2189—791no—
271#2187—791no—
272#2175—791no—
273#2168—791no—
274#2140—791no—
275#2138—791no—
276C79#3387811yescharacteristic, complemented, Sylow
277C2×#135—791no—
278C22×D8—791no—
279#556—21no—
280#546—1581no—
281C2×Q32—791no—
282C2×SD32—791no—
283C2×SD32—791no—
284C2×M32—791no—
285C22×C16—791no—
286#500C2×D7911yescharacteristic, complemented
287#495—791no—
288#485—21no—
289#485—21no—
290#475—21no—
291#467—791no—
292#467C2×D7911yescharacteristic
293#467C2×D7911yescharacteristic, complemented
294#466—791no—
295#463—1581no—
296#450—1581no—
297#449—1581no—
298C4×Q16C2×D7911yescharacteristic, complemented
299C4×D8—791no—
300C8×D4C2×D7911yescharacteristic
301#427C2×D7911yescharacteristic
302#427C2×D7911yescharacteristic, complemented
303#367—791no—
304#367—791no—
305#366—791no—
306C4⋊C16—791no—
307#358—791no—
308#358—791no—
309#358—791no—
310#358—791no—
311#357—791no—
312#357—791no—
313#357—791no—
314#357—791no—
315#357—791no—
316#357—791no—
317C22⋊C16—791no—
318C22⋊C16—791no—
319#346—791no—
320C2×#41—1581no—
321C22×D4—1581no—
322#135—3161no—
323#135—1582no—
324C2×Q16—791no—
325C2×Q16C22×D7911yescharacteristic
326C2×SD16—1581no—
327C2×SD16—1581no—
328C2×D8—3161no—
329C2×D8—791no—
330C2×D8—791no—
331C2×D8—1582no—
332C2×D8—1581no—
333C22×C8C22×D7911yescharacteristic
334C22×C8C22×D7911yescharacteristic
335C4⋊Q8—21no—
336#126—41no—
337#126—41no—
338#124—41no—
339#124—1581no—
340#123—3161no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1581no—
344#121—3161no—
345#121—1581no—
346#120—3161no—
347C4×Q8—21no—
348C4×Q8—41no—
349C4×D4—1581no—
350C4×D4C22⋊D7911yescharacteristic
351#117—21no—
352#117—21no—
353C2×C22⋊C4—3161no—
354Q32—3161no—
355SD32—3161no—
356SD32—1582no—
357M32—3161no—
358C2×C16—3161no—
359C2×C16—791no—
360C2×C16—791no—
361C2×C16—791no—
362C2×C16—791no—
363#107—21no—
364#107—791no—
365#107C22×D7911yescharacteristic
366#107C22×D7911yescharacteristic
367#107C22×D7911yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D7911yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—1581no—
375#102—1581no—
376#102—1581no—
377C22⋊C8C22⋊D7911yescharacteristic
378C22⋊C8C22⋊D7911yescharacteristic
379C4×C8C22×D7911yescharacteristic
380C24—3161no—
381#41—3162no—
382#41—6321no—
383C2×Q8—21no—
384C2×Q8—41no—
385C2×Q8—1581no—
386C2×D4—3162no—
387C2×D4—3161no—
388C2×D4—3161no—
389C2×D4—3161no—
390C2×D4—6321no—
391C2×D4—1581no—
392C2×D4—1581no—
393C2×D4#1192960911yescharacteristic
394C22×C4—3161no—
395C22×C4—3161no—
396C22×C4C2×C22⋊D7911yescharacteristic
397C22×C4C2×C22⋊D7911yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1581no—
401Q16—1581no—
402SD16—3162no—
403SD16—6321no—
404D8—3162no—
405D8—792no—
406D8—1581no—
407D8—1581no—
408D8—1581no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D7911yescharacteristic
414C2×C8C2×C22⋊D7911yescharacteristic
415C2×C8C2×C22⋊D7911yescharacteristic
416C2×C8C23×D7911yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—3161no—
425C4⋊C4—1581no—
426C4⋊C4#1192960911yescharacteristic
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—3161no—
432C22⋊C4—3161no—
433C22⋊C4—3161no—
434C22⋊C4—3161no—
435C22⋊C4—3161no—
436C22⋊C4#1192960911yescharacteristic
437C22⋊C4#1192960911yescharacteristic
438C42—41no—
439C42—41no—
440C42—41no—
441C42C2×C22⋊D7911yescharacteristic
442C16—1581no—
443C16—1581no—
444C16—1581no—
445C16—1581no—
446C23—3161no—
447C23—3161no—
448C23—6322no—
449C23—6322no—
450C23—12641no—
451C23C2×#1192960911yescharacteristic
452C23C2×#1192960911yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—82no—
456Q8—3161no—
457Q8—3161no—
458D4—41no—
459D4—3161no—
460D4—3161no—
461D4—3161no—
462D4—6322no—
463D4—6322no—
464D4—1582no—
465D4—12641no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—3161no—
475C2×C4—3161no—
476C2×C4—6322no—
477C2×C4—12641no—
478C2×C4C2×#1192960911yescharacteristic
479C2×C4C2×#1192960911yescharacteristic
480C2×C4C2×#1192960911yescharacteristic
481C2×C4C2×#1192960911yescharacteristic
482C2×C4C22×C22⋊D7911yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D7911yescharacteristic
487C8#1457920711yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—3162no—
495C22—6321no—
496C22—6321no—
497C22—6321no—
498C22—6321no—
499C22—6321no—
500C22—12644no—
501C22C22×#1192960911yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—6321no—
511C4—6321no—
512C4#1909920111yescharacteristic
513C4#1909937111yescharacteristic
514C2—41no—
515C2—41no—
516C2—6322no—
517C2—12641no—
518C2#2377520711yescharacteristic
519C2#2377653411yescharacteristic
520C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3342C2
socleC22×C79C22×D8
prime radical#23775087C2
hypercenter#3342D79
hypocenterC79#33878
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3342

Complements

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

NH ≅ G/Ntype
2. #23765639C2semidirect
3. #23765608C2semidirect
4. #23765591C2semidirect
5. #23765544C2semidirect
6. #23765506C2semidirect
7. #23765498C2semidirect
8. #23765462C2semidirect
9. #23765338C2semidirect
10. #23765328C2semidirect
11. #23765262C2semidirect
12. #23765226C2semidirect
13. #23765115C2semidirect
14. #23765105C2semidirect
15. C79×#3342C2semidirect
19. C2×#14579107C22semidirect
23. C22×C79⋊C16C22semidirect
31. #19098390C22semidirect
32. #19098390C22semidirect
33. #19098388C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098367C22semidirect
36. #19098366C22semidirect
37. #19098366C22semidirect
38. #19098366C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. #19098339C22semidirect
49. #500×C79C22semidirect
54. #467×C79C22semidirect
55. C4×Q16×C79C22semidirect
58. #427×C79C22semidirect
261. #3342D79semidirect
276. C79#33878semidirect
286. #500C2×D79semidirect
293. #467C2×D79semidirect
298. C4×Q16C2×D79semidirect
302. #427C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237750872
#237656392
#237656082
#237655912
#237655442
#237655062
#237654982
#237654622
#237653382
#237653282
#237652622
#237652262
#237651152
#237651052
C79×#33422
#3387879