Finite Group Databasebeta

← #34445129

Subgroups — #34445129

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#23775803C211yesmaximal, characteristic, complemented
2#23765631C211yesmaximal, characteristic, complemented
3#23765600C211yesmaximal, characteristic, complemented
4#23765588C211yesmaximal, characteristic, complemented
5#23765557C211yesmaximal, characteristic, complemented
6#23765498C211yesmaximal, characteristic, complemented
7#23765484C211yesmaximal, characteristic, complemented
8#23765452C211yesmaximal, characteristic, complemented
9#23765323C211yesmaximal, characteristic, complemented
10#23765312C211yesmaximal, characteristic, complemented
11C4×#14579102C211yesmaximal, characteristic, complemented
12D4×C79⋊C16C211yesmaximal, characteristic, complemented
13#23765107C211yesmaximal, characteristic, complemented
14C2×#19098364C211yesmaximal, characteristic, complemented
15C79×#3280C211yesmaximal, characteristic, complemented
16#19099259—21no—
17C2×#14579104C2211yescharacteristic, complemented
18C2×#14579102C2211yescharacteristic
19C2×#14579102C2211yescharacteristic
20C22×C79⋊C16C2211yescharacteristic
21C22×C79⋊C16C2211yescharacteristic, complemented
22#19098965C2211yescharacteristic, complemented
23#19098888—21no—
24#19098882—21no—
25#19098864—21no—
26#19098799C2211yescharacteristic
27#19098799C2211yescharacteristic, complemented
28C4×D632C2211yescharacteristic, complemented
29#19098601C2211yescharacteristic, complemented
30C2×#14579095C2211yescharacteristic
31#19098390C2211yescharacteristic, complemented
32#19098388C2211yescharacteristic, complemented
33#19098388C2211yescharacteristic, complemented
34C4⋊(C79⋊C16)C2211yescharacteristic, complemented
35#19098367C2211yescharacteristic, complemented
36#19098367C2211yescharacteristic, complemented
37#19098367C2211yescharacteristic, complemented
38#19098364C2211yescharacteristic
39#19098364C2211yescharacteristic
40#19098364C2211yescharacteristic
41#19098364C2211yescharacteristic
42#19098364C2211yescharacteristic
43#19098364C2211yescharacteristic
44#19098363C2211yescharacteristic, complemented
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic, complemented
47C4×C79⋊C16C2211yescharacteristic, complemented
48C2×#135×C79C2211yescharacteristic
49C22×D8×C79C2211yescharacteristic
50#546×C79—21no—
51#495×C79C2211yescharacteristic
52#467×C79C2211yescharacteristic
53#466×C79C2211yescharacteristic
54#463×C79—21no—
55#450×C79—21no—
56#449×C79—21no—
57C4×D8×C79C2211yescharacteristic
58C8×D4×C79C2211yescharacteristic
59C2×D632C2311yescharacteristic
60#14579156—41no—
61#14579153—21no—
62#14579143—41no—
63#14579135—21no—
64#14579135—21no—
65#14579133—41no—
66C4×D316—21no—
67(C22⋊D79)⋊C4—41no—
68#14579120—21no—
69C2×C4⋊Dic79—21no—
70#14579104—41no—
71#14579102—22no—
72#14579102—41no—
73C2×C79⋊C16—41no—
74C2×C79⋊C16—41no—
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77C2×C79⋊C16C2311yescharacteristic
78C2×C79⋊C16C2311yescharacteristic
79#14579095—21no—
80#14579095C2311yescharacteristic
81#14579095C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579093—21no—
84#14579093—21no—
85#14579083—21no—
86#14579083—21no—
87#14579083—21no—
88C2×#41×C79—21no—
89C22×D4×C79—21no—
90#135×C79—22no—
91#135×C79—41no—
92C2×Q16×C79C2311yescharacteristic
93C2×SD16×C79—21no—
94C2×SD16×C79—21no—
95C2×D8×C79—22no—
96C2×D8×C79—21no—
97C2×D8×C79—41no—
98C2×D8×C79C2311yescharacteristic
99C2×D8×C79C2311yescharacteristic
100C22×C8×C79C2311yescharacteristic
101C22×C8×C79C2311yescharacteristic
102#124×C79—21no—
103#123×C79—41no—
104C22⋊Q8×C79—21no—
105#121×C79—21no—
106#121×C79—41no—
107#120×C79—41no—
108C4×D4×C79—21no—
109C4×D4×C79D411yescharacteristic
110C2×C22⋊C4×C79—41no—
111#107×C79C2311yescharacteristic
112C4⋊C8×C79D411yescharacteristic
113#103×C79—21no—
114#102×C79—21no—
115#102×C79—21no—
116C22⋊C8×C79D411yescharacteristic
117C22⋊C8×C79D411yescharacteristic
118C4×C8×C79C2311yescharacteristic
119C2×D316—21no—
120C2×C22⋊D79—41no—
121C2×C22⋊D79—41no—
122C22×Dic79—41no—
123C2×C4×D79—41no—
124D632—21no—
125D632—21no—
126C4⋊Dic79—21no—
127C4⋊Dic79—21no—
128C4⋊Dic79—21no—
129C4⋊Dic79—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132C22⋊Dic79—41no—
133(C2×D79)⋊C4—41no—
134(C2×D79)⋊C4—41no—
135(C2×D79)⋊C4—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C24×C79—41no—
142#41×C79—42no—
143#41×C79—81no—
144C2×Q8×C79—21no—
145C2×D4×C79—21no—
146C2×D4×C79—21no—
147C2×D4×C79—42no—
148C2×D4×C79—41no—
149C2×D4×C79—41no—
150C2×D4×C79—41no—
151C2×D4×C79—81no—
152C2×D4×C79D811yescharacteristic
153C22×C4×C79—41no—
154C22×C4×C79—41no—
155C22×C4×C79C2×D411yescharacteristic
156C22×C4×C79C2×D411yescharacteristic
157Q16×C79—21no—
158Q16×C79—21no—
159SD16×C79—42no—
160SD16×C79—81no—
161D8×C79—21no—
162D8×C79—21no—
163D8×C79—21no—
164D8×C79—42no—
165D8×C79#4112yes—
166C2×C8×C79—21no—
167C2×C8×C79—21no—
168C2×C8×C79—21no—
169C2×C8×C79—21no—
170C2×C8×C79C2×D411yescharacteristic
171C2×C8×C79C2×D411yescharacteristic
172C2×C8×C79C2×D411yescharacteristic
173C2×C8×C79C2411yescharacteristic
174C4⋊C4×C79—21no—
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×C79—41no—
182C22⋊C4×C79D811yescharacteristic
183C22⋊C4×C79D811yescharacteristic
184C42×C79C2×D411yescharacteristic
185C23×C79—41no—
186C23×C79—41no—
187C23×C79—82no—
188C23×C79—82no—
189C23×C79—161no—
190C23×C79C2×D811yescharacteristic
191C23×C79C2×D811yescharacteristic
192C22×D79—41no—
193Q8×C79—41no—
194Q8×C79—41no—
195D4×C79—22no—
196D4×C79—22no—
197D4×C79—41no—
198D4×C79—41no—
199D4×C79—41no—
200D4×C79—82no—
201D4×C79—82no—
202D4×C79—161no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—41no—
208C2×C4×C79—41no—
209C2×C4×C79—82no—
210C2×C4×C79—161no—
211C2×C4×C79C2×D811yescharacteristic
212C2×C4×C79C2×D811yescharacteristic
213C2×C4×C79C2×D811yescharacteristic
214C2×C4×C79C2×D811yescharacteristic
215C2×C4×C79C22×D411yescharacteristic
216C22⋊D79—82no—
217C22⋊D79—161no—
218C2×Dic79—41no—
219C2×Dic79—41no—
220C2×Dic79—41no—
221C2×Dic79—161no—
222D316—41no—
223D316—41no—
224C4×D79—161no—
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—42no—
236C22×C79—81no—
237C22×C79—81no—
238C22×C79—81no—
239C22×C79—81no—
240C22×C79—81no—
241C22×C79—164no—
242C22×C79#50812yes—
243C22×C79C22×D811yescharacteristic
244C2×D79—81no—
245C2×D79—81no—
246C2×D79—81no—
247C4×C79—21no—
248C4×C79—21no—
249C4×C79—21no—
250C4×C79—21no—
251C4×C79—81no—
252C4×C79—81no—
253C4×C79#54611yescharacteristic
254C4×C79#57711yescharacteristic
255Dic79—81no—
256Dic79—81no—
257Dic79—81no—
258#33864—791nomaximal, Sylow
259C2×C79—22no—
260C2×C79—41no—
261C2×C79—82no—
262C2×C79—161no—
263C2×C79#329411yescharacteristic
264C2×C79C2×#50811yescharacteristic
265C2×C79#341511yescharacteristic
266D79—161no—
267#3339—791no—
268#3280D7911yescharacteristic, complemented
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×#135C2×D7911yescharacteristic
284C22×D8C2×D7911yescharacteristic
285#553—1581no—
286#546—21no—
287C2×SD32—791no—
288C2×D16—791no—
289C2×D16—791no—
290C22×C16—791no—
291C22×C16—791no—
292#499—791no—
293#495C2×D7911yescharacteristic
294#482—1581no—
295#481—1581no—
296#478—1581no—
297#467C2×D7911yescharacteristic
298#466—791no—
299#466—791no—
300#466C2×D7911yescharacteristic
301#463—21no—
302#450—21no—
303#449—21no—
304C4×D8—791no—
305C4×D8C2×D7911yescharacteristic
306C8×D4C2×D7911yescharacteristic
307#427—791no—
308C2×#107—791no—
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—21no—
327C22×D4—21no—
328#135—22no—
329#135—41no—
330C2×Q16C22×D7911yescharacteristic
331C2×SD16—21no—
332C2×SD16—21no—
333C2×D8—22no—
334C2×D8—21no—
335C2×D8—41no—
336C2×D8—791no—
337C2×D8C22×D7911yescharacteristic
338C2×D8C22×D7911yescharacteristic
339C22×C8C22×D7911yescharacteristic
340C22×C8C22×D7911yescharacteristic
341#126—3161no—
342#125—1581no—
343#124—21no—
344#123—41no—
345#123—3161no—
346C22⋊Q8—21no—
347#121—21no—
348#121—41no—
349#121—3161no—
350#121—1581no—
351#121—1581no—
352#120—41no—
353C4×D4—21no—
354C4×D4—3161no—
355C4×D4—1581no—
356C4×D4C22⋊D7911yescharacteristic
357#117—1581no—
358C2×C4⋊C4—1581no—
359C2×C22⋊C4—41no—
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—791no—
370#107—791no—
371#107—791no—
372#107—1581no—
373#107C22×D7911yescharacteristic
374#106—1581no—
375#106—1581no—
376C4⋊C8C22⋊D7911yescharacteristic
377#103—21no—
378#102—21no—
379#102—21no—
380#102—1581no—
381#102—1581no—
382#102—1581no—
383C22⋊C8C22⋊D7911yescharacteristic
384C22⋊C8C22⋊D7911yescharacteristic
385C4×C8C22×D7911yescharacteristic
386C24—41no—
387#41—42no—
388#41—81no—
389C2×Q8—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—81no—
397C2×D4—3161no—
398C2×D4—3161no—
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—21no—
408Q16—21no—
409SD16—42no—
410SD16—81no—
411D8—21no—
412D8—21no—
413D8—21no—
414D8—42no—
415D8—1581no—
416D8—1581no—
417D8#1192962312yes—
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—41no—
428C4⋊C4—3161no—
429C4⋊C4—3161no—
430C4⋊C4—3161no—
431C4⋊C4—1581no—
432C4⋊C4—1581no—
433C4⋊C4—1581no—
434C4⋊C4#1192960911yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—3161no—
441C22⋊C4—3161no—
442C22⋊C4—3161no—
443C22⋊C4—3161no—
444C22⋊C4#1192960911yescharacteristic
445C22⋊C4#1192960911yescharacteristic
446C42—3161no—
447C42C2×C22⋊D7911yescharacteristic
448C16—1581no—
449C16—1581no—
450C16—1581no—
451C16—1581no—
452C23—41no—
453C23—41no—
454C23—82no—
455C23—82no—
456C23—161no—
457C23—3161no—
458C23C2×#1192960911yescharacteristic
459C23C2×#1192960911yescharacteristic
460Q8—41no—
461Q8—41no—
462D4—22no—
463D4—22no—
464D4—41no—
465D4—41no—
466D4—41no—
467D4—82no—
468D4—82no—
469D4—161no—
470D4—3161no—
471D4—3161no—
472D4—6322no—
473D4—12641no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—82no—
481C2×C4—161no—
482C2×C4—3161no—
483C2×C4—3161no—
484C2×C4—3161no—
485C2×C4—12641no—
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—
495C8C2×#1192962311yescharacteristic
496C8#1457920611yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—164no—
509C22—6321no—
510C22—6321no—
511C22—6321no—
512C22#1909899312yes—
513C22C22×#1192960911yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—6321no—
521C4—6321no—
522C4—6321no—
523C4#1909920711yescharacteristic
524C4#1909937411yescharacteristic
525C2—22no—
526C2—41no—
527C2—82no—
528C2—161no—
529C2—12641no—
530C2#2377521311yescharacteristic
531C2C2×#1909899311yescharacteristic
532C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3280C2
socleC22×C79C22×D8
hypercenter#3280D79
hypocenterC79#33864
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #23775803C2semidirect
2. #23765631C2semidirect
3. #23765600C2semidirect
4. #23765588C2semidirect
5. #23765557C2semidirect
6. #23765498C2semidirect
7. #23765484C2semidirect
8. #23765452C2semidirect
9. #23765323C2semidirect
10. #23765312C2semidirect
11. C4×#14579102C2semidirect
12. D4×C79⋊C16C2semidirect
13. #23765107C2semidirect
14. C2×#19098364C2semidirect
15. C79×#3280C2semidirect
17. C2×#14579104C22semidirect
21. C22×C79⋊C16C22semidirect
22. #19098965C22semidirect
27. #19098799C22semidirect
28. C4×D632C22semidirect
29. #19098601C22semidirect
31. #19098390C22semidirect
32. #19098388C22semidirect
33. #19098388C22semidirect
34. C4⋊(C79⋊C16)C22semidirect
35. #19098367C22semidirect
36. #19098367C22semidirect
37. #19098367C22semidirect
44. #19098363C22semidirect
46. C22⋊(C79⋊C16)C22semidirect
47. C4×C79⋊C16C22semidirect
268. #3280D79semidirect
282. C79#33864semidirect

Maximal subgroups

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

subgroupindex
#237758032
#237656312
#237656002
#237655882
#237655572
#237654982
#237654842
#237654522
#237653232
#237653122
C4×#145791022
D4×C79⋊C162
#237651072
C2×#190983642
C79×#32802
#3386479