Finite Group Databasebeta

← #34296451

Subgroups — #34296451

Order 20224.

730 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#23773743C211yesmaximal, characteristic
2#23772332C211yesmaximal, characteristic, complemented
3#23772110C211yesmaximal, characteristic
4#23764379C211yesmaximal, characteristic, complemented
5#23764230C211yesmaximal, characteristic
6#23764113C211yesmaximal, characteristic, complemented
7#23764112C211yesmaximal, characteristic, complemented
8#23764109C211yesmaximal, characteristic, complemented
9#23764108C211yesmaximal, characteristic, complemented
10#23763539C211yesmaximal, characteristic, complemented
11#23763239C211yesmaximal, characteristic, complemented
12#23763237C211yesmaximal, characteristic, complemented
13#23763235C211yesmaximal, characteristic, complemented
14#23763086C211yesmaximal, characteristic, complemented
15C79×#1764C211yesmaximal, characteristic, complemented
16C2×#14579167C2211yescharacteristic
17#19099116C2211yescharacteristic
18#19099109C2211yescharacteristic
19#19098930C412yescomplemented
20#19098926C412yescomplemented
21#19098924C412yescomplemented
22#19098921C412yescomplemented
23#19098691—22no—
24C8×Dic158—22no—
25#19098639—22no—
26C8×C22⋊D79—22no—
27#19098591C2211yescharacteristic
28#19098590C2211yescharacteristic
29#19098586C2211yescharacteristic
30#19098586C2211yescharacteristic
31C2×#14579089C2211yescharacteristic
32C2×#14579088C2211yescharacteristic
33C2×#14579087C2211yescharacteristic
34#19098560C2211yescharacteristic
35#19098559C2211yescharacteristic
36#19098557C2211yescharacteristic
37#19098531C2211yescharacteristic
38#19098530C2211yescharacteristic
39#19098518C2211yescharacteristic
40C2×#14579072C2211yescharacteristic
41C2×#14579071C2211yescharacteristic
42C2×#14579071C2211yescharacteristic
43C2×C8×Dic79C2211yescharacteristic
44C2×C8×Dic79C2211yescharacteristic
45#19098409C2211yescharacteristic
46#19098328C2211yescharacteristic
47#19098327C2211yescharacteristic
48#19098327C2211yescharacteristic
49#19098324C2211yescharacteristic
50#19098323C2211yescharacteristic
51#19098322C2211yescharacteristic
52C2×#131×C79C2211yescharacteristic
53C2×#104×C79C2211yescharacteristic
54#416×C79C2211yescharacteristic
55#405×C79C2211yescharacteristic
56C2×C4×C8×C79C2211yescharacteristic
57#341×C79C2211yescharacteristic
58#339×C79C2211yescharacteristic
59C2×#11929624C2311yescharacteristic
60#14579167—22no—
61#14579167—22no—
62C2×#11929608C2311yescharacteristic
63C2×#11929608C2311yescharacteristic
64C2×#11929607—21no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C2×C8×D79—21no—
68#14579160C2×C412yes—
69#14579159C2×C411yescharacteristic
70#14579157C2×C411yescharacteristic
71#14579152C2×C412yes—
72#14579150C2×C411yescharacteristic
73#14579148C2×C411yescharacteristic
74#14579138—22no—
75#14579133—22no—
76C4×Dic158—22no—
77C4×C22⋊D79—22no—
78#14579120C2311yescharacteristic
79C2×C4×Dic79C2311yescharacteristic
80C2×C4×Dic79C2311yescharacteristic
81C4⋊(C79⋊C8)—22no—
82Dic79⋊C8—22no—
83Dic79⋊C8—22no—
84Dic79⋊C8—22no—
85#14579089—22no—
86#14579088—22no—
87#14579087—22no—
88#14579086C2×C412yes—
89#14579085C2×C412yes—
90#14579084C2×C412yes—
91#14579083C2×C412yes—
92#14579082C2×C412yes—
93#14579081C2×C412yes—
94C22⋊(C79⋊C8)—22no—
95(C2×D79)⋊C8—22no—
96#14579072—22no—
97#14579071—21no—
98#14579071—22no—
99#14579071C2×C412yes—
100#14579071C2×C412yes—
101C8×Dic79—21no—
102C8×Dic79—22no—
103C8×Dic79D412yes—
104C8×Dic79D412yes—
105C79⋊#95—22no—
106C2×#41×C79C2311yescharacteristic
107#131×C79—22no—
108#131×C79—22no—
109C2×M16×C79—21no—
110C2×M16×C79C2311yescharacteristic
111C2×M16×C79C2311yescharacteristic
112C22×C8×C79—21no—
113C22×C8×C79C2311yescharacteristic
114C22×C8×C79C2311yescharacteristic
115#117×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117#104×C79—22no—
118#103×C79C2×C412yes—
119#102×C79C2×C412yes—
120#97×C79—21no—
121C4×C8×C79—21no—
122C4×C8×C79—22no—
123#11929624—21no—
124#11929624—21no—
125#11929624—21no—
126#11929624—21no—
127C2×Dic158—22no—
128C2×Dic158C22×C411yescharacteristic
129C2×D316C22×C411yescharacteristic
130C2×C22⋊D79—22no—
131C2×C22⋊D79—21no—
132C22×Dic79—21no—
133C22×Dic79#4112yes—
134C2×C4×D79—21no—
135#11929608—21no—
136#11929608—21no—
137#11929608—21no—
138#11929608—21no—
139#11929607—42no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8C22×C412yes—
147C2×C79⋊C8C2×D412yes—
148C8×D79—42no—
149C4⋊Dic79—22no—
150C4⋊Dic79C22×C412yes—
151Dic79⋊C4—22no—
152Dic79⋊C4—22no—
153Dic79⋊C4—22no—
154C22⋊Dic79—22no—
155C22⋊Dic79—21no—
156(C2×D79)⋊C4—22no—
157C4×Dic79—21no—
158C4×Dic79—21no—
159C4×Dic79C22×C411yescharacteristic
160C4×Dic79C22×C411yescharacteristic
161C4×Dic79C2×D412yes—
162C4×Dic79#4112yes—
163#41×C79—21no—
164#41×C79—21no—
165#41×C79—21no—
166#41×C79—21no—
167C2×Q8×C79C22×C411yescharacteristic
168C2×D4×C79—21no—
169C2×D4×C79C22×C411yescharacteristic
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79C2411yescharacteristic
173M16×C79—21no—
174M16×C79—21no—
175M16×C79—21no—
176M16×C79—21no—
177M16×C79—42no—
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—42no—
183C2×C8×C79C22×C412yes—
184C2×C8×C79C2×D411yescharacteristic
185C2×C8×C79C2×D411yescharacteristic
186C2×C8×C79#4112yes—
187C2×C8×C79#4112yes—
188C4⋊C4×C79C22×C412yes—
189C22⋊C4×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79#4112yes—
193C23×C79—21no—
194C23×C79C2×#4111yescharacteristic
195C22×D79—21no—
196Q8×C79—21no—
197Q8×C79—21no—
198D4×C79—21no—
199D4×C79—21no—
200D4×C79—41no—
201D4×C79—41no—
202C2×C4×C79—21no—
203C2×C4×C79—22no—
204C2×C4×C79—22no—
205C2×C4×C79—42no—
206C2×C4×C79—41no—
207C2×C4×C79—41no—
208C2×C4×C79C23×C411yescharacteristic
209C2×C4×C79C22×D411yescharacteristic
210C2×C4×C79C2×#4111yescharacteristic
211C2×C4×C79C2×#4111yescharacteristic
212C2×C4×C79C2×#4111yescharacteristic
213C2×C4×C79C2×#4111yescharacteristic
214C22⋊D79—44no—
215C22⋊D79—41no—
216C22⋊D79—41no—
217C2×Dic79—22no—
218C2×Dic79—21no—
219C2×Dic79—22no—
220C2×Dic79—42no—
221C2×Dic79—42no—
222C2×Dic79C4×D412yes—
223C2×Dic79C4×D412yes—
224D316—21no—
225D316—21no—
226C4×D79—41no—
227C4×D79—41no—
228Dic158—21no—
229Dic158—21no—
230Dic158—44no—
231C8×C79—22no—
232C8×C79—41no—
233C8×C79—41no—
234C8×C79#12412yes—
235C8×C79#12412yes—
236C79⋊C8—22no—
237C79⋊C8—22no—
238C79⋊C8—41no—
239C79⋊C8—41no—
240C22×C79—22no—
241C22×C79—42no—
242C22×C79—41no—
243C22×C79C2×C4×D411yescharacteristic
244C22×C79#52911yescharacteristic
245C22×C79#52911yescharacteristic
246C2×D79—42no—
247C2×D79—41no—
248C4×C79—42no—
249C4×C79—42no—
250C4×C79—41no—
251C4×C79C4×#4111yescharacteristic
252C4×C79C4×#4111yescharacteristic
253C4×C79C2×#12411yescharacteristic
254C4×C79#53211yescharacteristic
255Dic79—22no—
256Dic79—22no—
257Dic79—42no—
258Dic79—42no—
259Dic79—41no—
260#21661—791nomaximal, Sylow
261C2×C79—21no—
262C2×C79—21no—
263C2×C79—81no—
264C2×C79C4×#12411yescharacteristic
265C2×C79#295512yes—
266D79—81no—
267#3135—791no—
268#2960—791no—
269#2931—791no—
270#1995—791no—
271#1963—791no—
272#1938—791no—
273#1938—791no—
274#1938—791no—
275#1938—791no—
276#1838—791no—
277#1764—791no—
278#1764—791no—
279#1764—791no—
280#1764D7911yescharacteristic, complemented
281#1726—791no—
282C79#2166111yescharacteristic, complemented, Sylow
283C2×#131—791no—
284C2×#131C2×D7911yescharacteristic
285#533—791no—
286#532—791no—
287#491—792no—
288#490—792no—
289#489—792no—
290#489—792no—
291#446—1582no—
292C8×Q8—1582no—
293#436—1582no—
294C8×D4—1582no—
295#424—791no—
296#424—791no—
297#423—791no—
298#423—791no—
299C2×#104—791no—
300C2×#104—791no—
301C2×#104—791no—
302C2×#104C2×D7911yescharacteristic
303#416—791no—
304#416—791no—
305#416—791no—
306#416C2×D7911yescharacteristic
307#408—791no—
308#408—791no—
309#405—791no—
310#405C2×D7911yescharacteristic
311C2×#97—791no—
312C2×#97—791no—
313C2×#97—791no—
314C2×C4×C8—791no—
315C2×C4×C8—791no—
316C2×C4×C8C2×D7911yescharacteristic
317#376—791no—
318#341—791no—
319#341—791no—
320#341—791no—
321#341C2×D7911yescharacteristic
322#339—791no—
323#339—791no—
324#339—791no—
325#339C2×D7911yescharacteristic
326C2×#41—791no—
327C2×#41C22×D7911yescharacteristic
328#131—22no—
329#131—22no—
330#131—1582no—
331#131—1582no—
332C2×M16—21no—
333C2×M16—791no—
334C2×M16—791no—
335C2×M16—1581no—
336C2×M16C22×D7911yescharacteristic
337C2×M16C22×D7911yescharacteristic
338C22×C8—21no—
339C22×C8—791no—
340C22×C8—791no—
341C22×C8—1581no—
342C22×C8C22×D7911yescharacteristic
343C22×C8C22×D7911yescharacteristic
344C4⋊Q8—792no—
345C4⋊Q8—791no—
346#127—791no—
347#125—792no—
348#124—791no—
349#124—791no—
350C22⋊Q8—1582no—
351#121—1582no—
352C4×Q8—1582no—
353C4×D4—1582no—
354#117—791no—
355#117C22×D7911yescharacteristic
356C2×C42—791no—
357C2×C42—791no—
358C2×C42C22×D7911yescharacteristic
359C4⋊C8—1582no—
360C4⋊C8—1582no—
361C4⋊C8—1582no—
362C4⋊C8—1582no—
363#104—22no—
364#104—1582no—
365#104—1582no—
366#104—1582no—
367#103—792no—
368#103—792no—
369#103—792no—
370#103C4×D7912yes—
371#102—792no—
372#102—792no—
373#102—792no—
374#102C4×D7912yes—
375C22⋊C8—1582no—
376C22⋊C8—1582no—
377#97—21no—
378#97—792no—
379#97—792no—
380#97—1582no—
381#97—1581no—
382#97—1582no—
383C4×C8—21no—
384C4×C8—22no—
385C4×C8—792no—
386C4×C8—792no—
387C4×C8—1581no—
388C4×C8—1582no—
389#95—1582no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394#41—1581no—
395#41—1581no—
396#41—1581no—
397#41—1581no—
398C2×Q8—791no—
399C2×Q8—1582no—
400C2×Q8C2×C4×D7911yescharacteristic
401C2×D4—21no—
402C2×D4—791no—
403C2×D4—1582no—
404C2×D4—1581no—
405C2×D4C2×C4×D7911yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—792no—
409C22×C4—1581no—
410C22×C4—1581no—
411C22×C4C23×D7911yescharacteristic
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—42no—
417M16—3162no—
418M16—1581no—
419M16—1581no—
420M16—1581no—
421M16—1581no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—42no—
427C2×C8—3162no—
428C2×C8—792no—
429C2×C8—792no—
430C2×C8—1581no—
431C2×C8—1581no—
432C2×C8—1581no—
433C2×C8—1581no—
434C2×C8—1581no—
435C2×C8—1581no—
436C2×C8C2×C4×D7912yes—
437C2×C8D4×D7911yescharacteristic
438C2×C8D4×D7911yescharacteristic
439C2×C8#1192962312yes—
440C2×C8#1192962412yes—
441C4⋊C4—792no—
442C4⋊C4—1582no—
443C4⋊C4—1582no—
444C4⋊C4—1582no—
445C4⋊C4—1582no—
446C4⋊C4C2×C4×D7912yes—
447C22⋊C4—21no—
448C22⋊C4—1582no—
449C22⋊C4—1582no—
450C22⋊C4—1581no—
451C42—21no—
452C42—21no—
453C42—791no—
454C42—791no—
455C42—792no—
456C42—792no—
457C42—1581no—
458C42—1581no—
459C42#1192962412yes—
460C23—21no—
461C23—1581no—
462C23#41×D7911yescharacteristic
463Q8—21no—
464Q8—21no—
465Q8—3164no—
466Q8—1581no—
467Q8—1581no—
468D4—21no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—3161no—
473D4—3161no—
474D4—3164no—
475D4—1581no—
476D4—1581no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—3162no—
484C2×C4—3161no—
485C2×C4—3161no—
486C2×C4—3162no—
487C2×C4—792no—
488C2×C4—792no—
489C2×C4—1582no—
490C2×C4—1581no—
491C2×C4—1582no—
492C2×C4C22×C4×D7911yescharacteristic
493C2×C4C2×D4×D7911yescharacteristic
494C2×C4#41×D7911yescharacteristic
495C2×C4C2×#1192962311yescharacteristic
496C2×C4C2×#1192962411yescharacteristic
497C2×C4C2×#1192962411yescharacteristic
498C8—22no—
499C8—41no—
500C8—41no—
501C8—3161no—
502C8—3161no—
503C8—1582no—
504C8—1582no—
505C8#1457914712yes—
506C8#1457914712yes—
507C22—22no—
508C22—42no—
509C22—41no—
510C22—3162no—
511C22—3161no—
512C22C4×D4×D7911yescharacteristic
513C22#1909909111yescharacteristic
514C22#1909909311yescharacteristic
515C4—42no—
516C4—42no—
517C4—41no—
518C4—3162no—
519C4—3162no—
520C4—3161no—
521C4—1582no—
522C4—1582no—
523C4C4×#1192962311yescharacteristic
524C4C4×#1192962411yescharacteristic
525C4C2×#1457914711yescharacteristic
526C4#1909911011yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—6321no—
531C2C4×#1457914711yescharacteristic
532C2#2377229812yes—
subgroupG/H
centerC2×C8D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1764C2
socleC22×C79C2×C4×D4
prime radical#19099109C22
cosocleC22×C4×C79C24
hypercenter#1764D79
hypocenterC79#21661
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. #23772332C2semidirect
4. #23764379C2semidirect
6. #23764113C2semidirect
7. #23764112C2semidirect
8. #23764109C2semidirect
9. #23764108C2semidirect
10. #23763539C2semidirect
11. #23763239C2semidirect
12. #23763237C2semidirect
13. #23763235C2semidirect
14. #23763086C2semidirect
15. C79×#1764C2semidirect
19. #19098930C4semidirect
20. #19098926C4semidirect
21. #19098924C4semidirect
22. #19098921C4semidirect
280. #1764D79semidirect
282. C79#21661semidirect

Maximal subgroups

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

subgroupindex
#237737432
#237723322
#237721102
#237643792
#237642302
#237641132
#237641122
#237641092
#237641082
#237635392
#237632392
#237632372
#237632352
#237630862
C79×#17642
#2166179