Finite Group Databasebeta

← #34445294

Subgroups — #34445294

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D632C211yesmaximal, characteristic
2#23765643C211yesmaximal, characteristic, complemented
3#23765591C211yesmaximal, characteristic, complemented
4#23765580C211yesmaximal, characteristic, complemented
5#23765554C211yesmaximal, characteristic, complemented
6#23765515C211yesmaximal, characteristic, complemented
7#23765498C211yesmaximal, characteristic, complemented
8#23765440C211yesmaximal, characteristic, complemented
9#23765328C211yesmaximal, characteristic, complemented
10#23765328C211yesmaximal, characteristic, complemented
11#23765262C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765115C211yesmaximal, characteristic, complemented
14C2×#19098363C211yesmaximal, characteristic, complemented
15C79×#3336C211yesmaximal, characteristic, complemented
16C22×D632C2211yescharacteristic
17C22×D632C2211yescharacteristic
18D4×D316—21no—
19C2×#14579104C2211yescharacteristic
20C2×#14579104C2211yescharacteristic, complemented
21C2×#14579104C2211yescharacteristic, complemented
22C2×#14579101C2211yescharacteristic, complemented
23C22×C79⋊C16C2211yescharacteristic, complemented
24#19098940C2211yescharacteristic
25#19098799C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098759—21no—
28#19098699—21no—
29#19098699—21no—
30C4×D632C2211yescharacteristic
31#19098390C2211yescharacteristic, complemented
32#19098388C2211yescharacteristic, complemented
33#19098388C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098367C2211yescharacteristic, complemented
36#19098363C2211yescharacteristic
37#19098363C2211yescharacteristic
38#19098363C2211yescharacteristic
39#19098363C2211yescharacteristic, complemented
40#19098363C2211yescharacteristic, complemented
41#19098363C2211yescharacteristic, complemented
42#19098363C2211yescharacteristic, complemented
43#19098363C2211yescharacteristic, complemented
44#19098363C2211yescharacteristic, complemented
45C22⋊(C79⋊C16)C2211yescharacteristic, complemented
46C22⋊(C79⋊C16)C2211yescharacteristic, complemented
47#19098339C2211yescharacteristic, complemented
48#552×C79—21no—
49#499×C79C2211yescharacteristic, complemented
50#485×C79—21no—
51#483×C79—21no—
52#479×C79—21no—
53#467×C79C2211yescharacteristic, complemented
54#467×C79C2211yescharacteristic, complemented
55C4×Q16×C79C2211yescharacteristic, complemented
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic, complemented
58C2×#107×C79C2211yescharacteristic, complemented
59C22×D316—21no—
60C22×D316—21no—
61C2×D4×D79—41no—
62C2×D632—22no—
63C2×D632—22no—
64C2×D632—21no—
65C2×D632—21no—
66C2×D632—21no—
67C2×D632—41no—
68C2×D632—41no—
69C2×D632C2311yescharacteristic
70C2×D632C2311yescharacteristic
71C2×D632C2311yescharacteristic
72#14579158—21no—
73#14579136—41no—
74#14579135—21no—
75#14579135—21no—
76#14579131—41no—
77#14579131—41no—
78C4×D316—21no—
79#14579104—22no—
80#14579104—41no—
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#14579083—21no—
90#14579083—21no—
91#14579083—21no—
92C2×Q16×C79C2311yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C22×C8×C79C2311yescharacteristic
95#126×C79—41no—
96#125×C79—21no—
97#125×C79—41no—
98#123×C79—41no—
99C22⋊Q8×C79—21no—
100C22⋊Q8×C79—21no—
101C4×Q8×C79—21no—
102C4×D4×C79D411yescharacteristic
103#117×C79—21no—
104#117×C79—41no—
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#103×C79—21no—
114#103×C79—21no—
115#103×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C23×D79—41no—
120C23×D79—41no—
121C2×D316—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—42no—
125C2×D316—42no—
126C2×D316—41no—
127C2×D316—41no—
128C2×D316—41no—
129C2×D316—41no—
130C2×D316—81no—
131C2×D316—81no—
132C2×C22⋊D79—41no—
133C2×C22⋊D79—41no—
134D4×D79—84no—
135C2×C4×D79—41no—
136D632—21no—
137D632—21no—
138D632—21no—
139D632—21no—
140D632—21no—
141D632—42no—
142D632—42no—
143D632—81no—
144D632#4112yes—
145C4⋊Dic79—21no—
146(C2×D79)⋊C4—41no—
147(C2×D79)⋊C4—41no—
148(C2×D79)⋊C4—41no—
149C79⋊C16—21no—
150C79⋊C16—21no—
151C79⋊C16—21no—
152C79⋊C16—21no—
153C2×Q8×C79—21no—
154C2×D4×C79D811yescharacteristic
155C22×C4×C79—41no—
156C22×C4×C79C2×D411yescharacteristic
157C22×C4×C79C2×D411yescharacteristic
158Q16×C79—21no—
159Q16×C79—21no—
160C2×C8×C79—21no—
161C2×C8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79—21no—
164C2×C8×C79C2×D411yescharacteristic
165C2×C8×C79C2×D411yescharacteristic
166C2×C8×C79C2×D411yescharacteristic
167C2×C8×C79C2411yescharacteristic
168C4⋊C4×C79—21no—
169C4⋊C4×C79—21no—
170C4⋊C4×C79—21no—
171C4⋊C4×C79—41no—
172C4⋊C4×C79—41no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79D811yescharacteristic
178C22⋊C4×C79—41no—
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79D811yescharacteristic
182C22⋊C4×C79D811yescharacteristic
183C42×C79—41no—
184C42×C79—41no—
185C42×C79C2×D411yescharacteristic
186C23×C79C2×D811yescharacteristic
187C23×C79C2×D811yescharacteristic
188C22×D79—41no—
189C22×D79—41no—
190C22×D79—41no—
191C22×D79—82no—
192C22×D79—82no—
193C22×D79—82no—
194C22×D79—82no—
195C22×D79—161no—
196C22×D79—161no—
197Q8×C79—41no—
198Q8×C79—41no—
199D4×C79—22no—
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—41no—
208C2×C4×C79—82no—
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C2×D811yescharacteristic
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C22×D411yescharacteristic
214C22⋊D79—82no—
215C22⋊D79—82no—
216C2×Dic79—41no—
217D316—22no—
218D316—41no—
219D316—41no—
220D316—41no—
221D316—41no—
222D316—41no—
223D316—82no—
224D316—82no—
225D316—82no—
226D316—161no—
227C4×D79—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#50912yes—
239C22×C79C22×D811yescharacteristic
240C2×D79—42no—
241C2×D79—81no—
242C2×D79—81no—
243C2×D79—81no—
244C2×D79—81no—
245C2×D79—81no—
246C2×D79—81no—
247C2×D79—81no—
248C2×D79—81no—
249C2×D79—164no—
250C2×D79—164no—
251C4×C79—21no—
252C4×C79—21no—
253C4×C79—21no—
254C4×C79—21no—
255C4×C79—81no—
256C4×C79—81no—
257C4×C79—81no—
258C4×C79—81no—
259C4×C79#54611yescharacteristic
260C4×C79#57711yescharacteristic
261Dic79—81no—
262#33877—791nomaximal, Sylow
263C2×C79—22no—
264C2×C79—41no—
265C2×C79#329411yescharacteristic
266C2×C79C2×#50911yescharacteristic
267C2×C79#341611yescharacteristic
268D79—82no—
269D79—161no—
270D79—161no—
271#3336D7911yescharacteristic, complemented
272D4×D8—791no—
273#2244—791no—
274#2234—791no—
275#2232—791no—
276#2228—791no—
277#2220—791no—
278#2217—791no—
279#2207—791no—
280#2187—791no—
281#2187—791no—
282#2175—791no—
283#2168—791no—
284#2140—791no—
285C2×#357—791no—
286C79#3387711yescharacteristic, complemented, Sylow
287C22×D8—791no—
288C22×D8—791no—
289#552—21no—
290D42—1581no—
291C2×SD32—791no—
292C2×SD32—791no—
293C2×SD32—791no—
294C2×M32—791no—
295C22×C16—791no—
296#499C2×D7911yescharacteristic, complemented
297#493—791no—
298#485—21no—
299#483—21no—
300#479—21no—
301#467C2×D7911yescharacteristic, complemented
302#467C2×D7911yescharacteristic, complemented
303#466—791no—
304#466—791no—
305#459—1581no—
306#447—1581no—
307#447—1581no—
308C4×Q16C2×D7911yescharacteristic, complemented
309C4×D8—791no—
310C8×D4C2×D7911yescharacteristic
311#427C2×D7911yescharacteristic, complemented
312C2×#107C2×D7911yescharacteristic, complemented
313#367—791no—
314#366—791no—
315#366—791no—
316C4⋊C16—791no—
317#358—791no—
318#357—791no—
319#357—791no—
320#357—791no—
321#357—791no—
322#357—791no—
323#357—791no—
324#357—791no—
325#357—791no—
326#357—791no—
327C22⋊C16—791no—
328C22⋊C16—791no—
329#346—791no—
330C22×D4—3161no—
331C22×D4—1581no—
332C22×D4—1581no—
333C2×Q16C22×D7911yescharacteristic
334C2×D8—3161no—
335C2×D8—3161no—
336C2×D8—791no—
337C2×D8—791no—
338C2×D8—791no—
339C2×D8—1582no—
340C2×D8—1582no—
341C2×D8—1581no—
342C2×D8—1581no—
343C2×D8—1581no—
344C22×C8C22×D7911yescharacteristic
345C22×C8C22×D7911yescharacteristic
346#127—1581no—
347#126—41no—
348#125—21no—
349#125—41no—
350#123—41no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353#121—3161no—
354#121—1581no—
355#121—1581no—
356#120—3161no—
357#120—3161no—
358C4×Q8—21no—
359C4×D4—1581no—
360C4×D4C22⋊D7911yescharacteristic
361#117—21no—
362#117—41no—
363C2×C4⋊C4—21no—
364SD32—3161no—
365SD32—3161no—
366SD32—1582no—
367M32—3161no—
368C2×C16—3161no—
369C2×C16—791no—
370C2×C16—791no—
371C2×C16—791no—
372C2×C16—791no—
373#107—21no—
374#107—791no—
375#107C22×D7911yescharacteristic
376#107C22×D7911yescharacteristic
377#107C22×D7911yescharacteristic
378#106—21no—
379#106—21no—
380C4⋊C8C22⋊D7911yescharacteristic
381#103—21no—
382#103—21no—
383#103—21no—
384#102—1581no—
385#102—1581no—
386#102—1581no—
387C22⋊C8C22⋊D7911yescharacteristic
388C22⋊C8C22⋊D7911yescharacteristic
389C4×C8C22×D7911yescharacteristic
390C24—3161no—
391C24—3161no—
392C2×Q8—21no—
393C2×D4—3162no—
394C2×D4—3162no—
395C2×D4—3161no—
396C2×D4—3161no—
397C2×D4—3161no—
398C2×D4—3161no—
399C2×D4—3161no—
400C2×D4—3161no—
401C2×D4—6321no—
402C2×D4—6321no—
403C2×D4—6324no—
404C2×D4—1581no—
405C2×D4—1581no—
406C2×D4—1581no—
407C2×D4#1192960911yescharacteristic
408C22×C4—41no—
409C22×C4—3161no—
410C22×C4C2×C22⋊D7911yescharacteristic
411C22×C4C2×C22⋊D7911yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—3162no—
415D8—3162no—
416D8—792no—
417D8—6321no—
418D8—1581no—
419D8—1581no—
420D8—1581no—
421D8—1581no—
422D8—1581no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D7911yescharacteristic
428C2×C8C2×C22⋊D7911yescharacteristic
429C2×C8C2×C22⋊D7911yescharacteristic
430C2×C8C23×D7911yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—1581no—
441C4⋊C4#1192960911yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—3161no—
446C22⋊C4—3161no—
447C22⋊C4—3161no—
448C22⋊C4#1192960911yescharacteristic
449C22⋊C4#1192960911yescharacteristic
450C42—41no—
451C42—41no—
452C42C2×C22⋊D7911yescharacteristic
453C16—1581no—
454C16—1581no—
455C16—1581no—
456C16—1581no—
457C23—3161no—
458C23—3161no—
459C23—3161no—
460C23—6322no—
461C23—6322no—
462C23—6322no—
463C23—6322no—
464C23—12641no—
465C23—12641no—
466C23C2×#1192960911yescharacteristic
467C23C2×#1192960911yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—3161no—
472D4—3161no—
473D4—3161no—
474D4—3161no—
475D4—3161no—
476D4—6322no—
477D4—6322no—
478D4—6322no—
479D4—6322no—
480D4—6322no—
481D4—1582no—
482D4—12641no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—3161no—
493C2×C4—6322no—
494C2×C4C2×#1192960911yescharacteristic
495C2×C4C2×#1192960911yescharacteristic
496C2×C4C2×#1192960911yescharacteristic
497C2×C4C2×#1192960911yescharacteristic
498C2×C4C22×C22⋊D7911yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D7911yescharacteristic
503C8#1457920711yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—3162no—
510C22—6321no—
511C22—6321no—
512C22—6321no—
513C22—6321no—
514C22—6321no—
515C22—6321no—
516C22—6321no—
517C22—6321no—
518C22—12644no—
519C22—12644no—
520C22#1909899812yes—
521C22C22×#1192960911yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—6321no—
531C4#1909920111yescharacteristic
532C4#1909937111yescharacteristic
533C2—22no—
534C2—41no—
535C2—6322no—
536C2—12641no—
537C2—12641no—
538C2#2377520711yescharacteristic
539C2C2×#1909899811yescharacteristic
540C2#2377654411yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3336C2
socleC22×C79C22×D8
prime radicalD4×D632C2
hypercenter#3336D79
hypocenterC79#33877
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #23765643C2semidirect
3. #23765591C2semidirect
4. #23765580C2semidirect
5. #23765554C2semidirect
6. #23765515C2semidirect
7. #23765498C2semidirect
8. #23765440C2semidirect
9. #23765328C2semidirect
10. #23765328C2semidirect
11. #23765262C2semidirect
12. #23765226C2semidirect
13. #23765115C2semidirect
14. C2×#19098363C2semidirect
15. C79×#3336C2semidirect
20. C2×#14579104C22semidirect
21. C2×#14579104C22semidirect
22. C2×#14579101C22semidirect
23. C22×C79⋊C16C22semidirect
31. #19098390C22semidirect
32. #19098388C22semidirect
33. #19098388C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098367C22semidirect
39. #19098363C22semidirect
40. #19098363C22semidirect
41. #19098363C22semidirect
42. #19098363C22semidirect
43. #19098363C22semidirect
44. #19098363C22semidirect
45. C22⋊(C79⋊C16)C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. #19098339C22semidirect
49. #499×C79C22semidirect
53. #467×C79C22semidirect
54. #467×C79C22semidirect
55. C4×Q16×C79C22semidirect
57. #427×C79C22semidirect
58. C2×#107×C79C22semidirect
271. #3336D79semidirect
286. C79#33877semidirect
296. #499C2×D79semidirect
301. #467C2×D79semidirect
302. #467C2×D79semidirect
308. C4×Q16C2×D79semidirect
311. #427C2×D79semidirect
312. C2×#107C2×D79semidirect

Maximal subgroups

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

subgroupindex
D4×D6322
#237656432
#237655912
#237655802
#237655542
#237655152
#237654982
#237654402
#237653282
#237652622
#237652262
#237651152
C2×#190983632
C79×#33362
#3387779