Finite Group Databasebeta

← #34445123

Subgroups — #34445123

Order 20224.

580 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775087C211yesmaximal, characteristic, complemented
2#23765634C211yesmaximal, characteristic, complemented
3#23765602C211yesmaximal, characteristic, complemented
4#23765591C211yesmaximal, characteristic, complemented
5#23765554C211yesmaximal, characteristic, complemented
6#23765494C211yesmaximal, characteristic, complemented
7#23765484C211yesmaximal, characteristic, complemented
8#23765455C211yesmaximal, characteristic, complemented
9#23765321C211yesmaximal, characteristic, complemented
10#23765311C211yesmaximal, characteristic, complemented
11C4×#14579102C211yesmaximal, characteristic, complemented
12D4×C79⋊C16C211yesmaximal, characteristic, complemented
13#23765105C211yesmaximal, characteristic, complemented
14C2×#19098363C211yesmaximal, characteristic, complemented
15C79×#3339C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×D632C2211yescharacteristic
18#19099209—21no—
19C2×#14579105C2211yescharacteristic, complemented
20C2×#14579102C2211yescharacteristic
21C2×#14579102C2211yescharacteristic
22C22×C79⋊C16C2211yescharacteristic
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#19098388C2211yescharacteristic, complemented
33#19098388C2211yescharacteristic, 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
47C4×C79⋊C16C2211yescharacteristic, complemented
48#553×C79—21no—
49#499×C79C2211yescharacteristic, complemented
50#482×C79—21no—
51#481×C79—21no—
52#478×C79—21no—
53#466×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic, complemented
55C4×D8×C79C2211yescharacteristic, complemented
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic, complemented
58C2×#107×C79C2211yescharacteristic
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—
82C2×C79⋊C16—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×D8×C79C2311yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95#126×C79—41no—
96#125×C79—21no—
97#123×C79—41no—
98#121×C79—21no—
99#121×C79—21no—
100#121×C79—41no—
101C4×D4×C79—21no—
102C4×D4×C79—41no—
103C4×D4×C79D411yescharacteristic
104#117×C79—21no—
105C2×C4⋊C4×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#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#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×D4×C79—21no—
153C2×D4×C79—41no—
154C2×D4×C79—41no—
155C2×D4×C79D811yescharacteristic
156C22×C4×C79—41no—
157C22×C4×C79—41no—
158C22×C4×C79C2×D411yescharacteristic
159C22×C4×C79C2×D411yescharacteristic
160D8×C79—21no—
161D8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79—21no—
164C2×C8×C79—21no—
165C2×C8×C79—21no—
166C2×C8×C79C2×D411yescharacteristic
167C2×C8×C79C2×D411yescharacteristic
168C2×C8×C79C2×D411yescharacteristic
169C2×C8×C79C2411yescharacteristic
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—21no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79D811yescharacteristic
177C22⋊C4×C79—41no—
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79D811yescharacteristic
182C22⋊C4×C79D811yescharacteristic
183C42×C79—41no—
184C42×C79C2×D411yescharacteristic
185C23×C79—41no—
186C23×C79C2×D811yescharacteristic
187C23×C79C2×D811yescharacteristic
188C22×D79—41no—
189C22×D79—41no—
190C22×D79—82no—
191C22×D79—82no—
192C22×D79—161no—
193D4×C79—22no—
194D4×C79—41no—
195D4×C79—41no—
196D4×C79—82no—
197D4×C79—161no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—41no—
203C2×C4×C79—41no—
204C2×C4×C79—41no—
205C2×C4×C79—161no—
206C2×C4×C79—161no—
207C2×C4×C79C2×D811yescharacteristic
208C2×C4×C79C2×D811yescharacteristic
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C22×D411yescharacteristic
212C22⋊D79—82no—
213C22⋊D79—161no—
214C2×Dic79—41no—
215C2×Dic79—41no—
216D316—22no—
217D316—41no—
218D316—41no—
219D316—41no—
220D316—82no—
221C4×D79—82no—
222C4×D79—161no—
223Dic158—41no—
224Dic158—41no—
225C8×C79—21no—
226C8×C79—21no—
227C8×C79—21no—
228C8×C79C2×#4111yescharacteristic
229C8×C79H(5,𝔽2)11yescharacteristic
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79—21no—
235C22×C79—81no—
236C22×C79—81no—
237C22×C79—81no—
238C22×C79#50812yes—
239C22×C79C22×D811yescharacteristic
240C2×D79—42no—
241C2×D79—81no—
242C2×D79—81no—
243C2×D79—81no—
244C2×D79—81no—
245C2×D79—81no—
246C2×D79—164no—
247C4×C79—21no—
248C4×C79—21no—
249C4×C79—21no—
250C4×C79—21no—
251C4×C79—81no—
252C4×C79—81no—
253C4×C79—81no—
254C4×C79#54611yescharacteristic
255C4×C79#57711yescharacteristic
256Dic79—81no—
257Dic79—81no—
258#33864—791nomaximal, Sylow
259C2×C79—22no—
260C2×C79—41no—
261C2×C79—161no—
262C2×C79#329411yescharacteristic
263C2×C79C2×#50811yescharacteristic
264C2×C79#341511yescharacteristic
265D79—82no—
266D79—161no—
267#3339D7911yescharacteristic, complemented
268#3280—791no—
269#2242—791no—
270#2236—791no—
271#2234—791no—
272#2228—791no—
273#2217—791no—
274#2215—791no—
275#2210—791no—
276#2186—791no—
277#2184—791no—
278C4×D16—791no—
279D4×C16—791no—
280#2138—791no—
281C2×#357—791no—
282C79#3386411yescharacteristic, complemented, Sylow
283C2×#135—791no—
284C22×D8—791no—
285#553—21no—
286#546—1581no—
287C2×SD32—791no—
288C2×D16—791no—
289C2×D16—791no—
290C22×C16—791no—
291C22×C16—791no—
292#499C2×D7911yescharacteristic, complemented
293#495—791no—
294#482—21no—
295#481—21no—
296#478—21no—
297#467—791no—
298#466—791no—
299#466C2×D7911yescharacteristic
300#466C2×D7911yescharacteristic, complemented
301#463—1581no—
302#450—1581no—
303#449—1581no—
304C4×D8—791no—
305C4×D8C2×D7911yescharacteristic, complemented
306C8×D4C2×D7911yescharacteristic
307#427C2×D7911yescharacteristic, complemented
308C2×#107C2×D7911yescharacteristic
309#367—791no—
310#366—791no—
311#366—791no—
312C4⋊C16—791no—
313#358—791no—
314#358—791no—
315#358—791no—
316#357—791no—
317#357—791no—
318#357—791no—
319#357—791no—
320#357—791no—
321#357—791no—
322#357—791no—
323C22⋊C16—791no—
324C22⋊C16—791no—
325C4×C16—791no—
326C2×#41—1581no—
327C22×D4—1581no—
328#135—3161no—
329#135—1582no—
330C2×Q16—791no—
331C2×SD16—1581no—
332C2×SD16—1581no—
333C2×D8—3161no—
334C2×D8—791no—
335C2×D8—791no—
336C2×D8—1582no—
337C2×D8—1581no—
338C2×D8C22×D7911yescharacteristic
339C22×C8C22×D7911yescharacteristic
340C22×C8C22×D7911yescharacteristic
341#126—41no—
342#125—21no—
343#124—1581no—
344#123—41no—
345#123—3161no—
346C22⋊Q8—1581no—
347#121—21no—
348#121—21no—
349#121—41no—
350#121—3161no—
351#121—1581no—
352#120—3161no—
353C4×D4—21no—
354C4×D4—41no—
355C4×D4—1581no—
356C4×D4C22⋊D7911yescharacteristic
357#117—21no—
358C2×C4⋊C4—21no—
359C2×C22⋊C4—3161no—
360SD32—3161no—
361D16—3161no—
362D16—1582no—
363C2×C16—3161no—
364C2×C16—3161no—
365C2×C16—791no—
366C2×C16—791no—
367C2×C16—791no—
368C2×C16—791no—
369#107—21no—
370#107—791no—
371#107C22×D7911yescharacteristic
372#107C22×D7911yescharacteristic
373#107C22×D7911yescharacteristic
374#106—21no—
375#106—21no—
376C4⋊C8C22⋊D7911yescharacteristic
377#103—1581no—
378#102—21no—
379#102—21no—
380#102—21no—
381#102—1581no—
382#102—1581no—
383C22⋊C8C22⋊D7911yescharacteristic
384C22⋊C8C22⋊D7911yescharacteristic
385C4×C8C22×D7911yescharacteristic
386C24—3161no—
387#41—3162no—
388#41—6321no—
389C2×Q8—1581no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—3162no—
394C2×D4—3161no—
395C2×D4—3161no—
396C2×D4—3161no—
397C2×D4—6321no—
398C2×D4—1581no—
399C2×D4—1581no—
400C2×D4#1192960911yescharacteristic
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—3161no—
404C22×C4—3161no—
405C22×C4C2×C22⋊D7911yescharacteristic
406C22×C4C2×C22⋊D7911yescharacteristic
407Q16—1581no—
408Q16—1581no—
409SD16—3162no—
410SD16—6321no—
411D8—21no—
412D8—21no—
413D8—3162no—
414D8—792no—
415D8—1581no—
416D8—1581no—
417D8—1581no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C22⋊D7911yescharacteristic
423C2×C8C2×C22⋊D7911yescharacteristic
424C2×C8C2×C22⋊D7911yescharacteristic
425C2×C8C23×D7911yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—3161no—
433C4⋊C4—1581no—
434C4⋊C4#1192960911yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—3161no—
440C22⋊C4—3161no—
441C22⋊C4—3161no—
442C22⋊C4—3161no—
443C22⋊C4—3161no—
444C22⋊C4#1192960911yescharacteristic
445C22⋊C4#1192960911yescharacteristic
446C42—41no—
447C42C2×C22⋊D7911yescharacteristic
448C16—1581no—
449C16—1581no—
450C16—1581no—
451C16—1581no—
452C23—41no—
453C23—3161no—
454C23—3161no—
455C23—6322no—
456C23—6322no—
457C23—12641no—
458C23C2×#1192960911yescharacteristic
459C23C2×#1192960911yescharacteristic
460Q8—3161no—
461Q8—3161no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—82no—
466D4—161no—
467D4—3161no—
468D4—3161no—
469D4—3161no—
470D4—6322no—
471D4—6322no—
472D4—1582no—
473D4—12641no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—161no—
482C2×C4—161no—
483C2×C4—3161no—
484C2×C4—3161no—
485C2×C4—6322no—
486C2×C4—12641no—
487C2×C4C2×#1192960911yescharacteristic
488C2×C4C2×#1192960911yescharacteristic
489C2×C4C2×#1192960911yescharacteristic
490C2×C4C2×#1192960911yescharacteristic
491C2×C4C22×C22⋊D7911yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8#41×D7911yescharacteristic
496C8#1457920711yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—3162no—
506C22—6321no—
507C22—6321no—
508C22—6321no—
509C22—6321no—
510C22—6321no—
511C22—12644no—
512C22#1909899312yes—
513C22C22×#1192960911yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—6321no—
522C4—6321no—
523C4#1909920111yescharacteristic
524C4#1909937111yescharacteristic
525C2—22no—
526C2—41no—
527C2—161no—
528C2—6322no—
529C2—12641no—
530C2#2377520711yescharacteristic
531C2C2×#1909899311yescharacteristic
532C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3339C2
socleC22×C79C22×D8
hypercenter#3339D79
hypocenterC79#33864
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. #23775087C2semidirect
2. #23765634C2semidirect
3. #23765602C2semidirect
4. #23765591C2semidirect
5. #23765554C2semidirect
6. #23765494C2semidirect
7. #23765484C2semidirect
8. #23765455C2semidirect
9. #23765321C2semidirect
10. #23765311C2semidirect
11. C4×#14579102C2semidirect
12. D4×C79⋊C16C2semidirect
13. #23765105C2semidirect
14. C2×#19098363C2semidirect
15. C79×#3339C2semidirect
19. C2×#14579105C22semidirect
23. C22×C79⋊C16C22semidirect
31. #19098390C22semidirect
32. #19098388C22semidirect
33. #19098388C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098366C22semidirect
36. #19098366C22semidirect
37. #19098366C22semidirect
38. #19098364C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. C4×C79⋊C16C22semidirect
49. #499×C79C22semidirect
54. #466×C79C22semidirect
55. C4×D8×C79C22semidirect
57. #427×C79C22semidirect
267. #3339D79semidirect
282. C79#33864semidirect
292. #499C2×D79semidirect
300. #466C2×D79semidirect
305. C4×D8C2×D79semidirect
307. #427C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237750872
#237656342
#237656022
#237655912
#237655542
#237654942
#237654842
#237654552
#237653212
#237653112
C4×#145791022
D4×C79⋊C162
#237651052
C2×#190983632
C79×#33392
#3386479