Finite Group Databasebeta

← #34445354

Subgroups — #34445354

Order 20224.

570 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775087C211yesmaximal, characteristic
2#23765639C211yesmaximal, characteristic, complemented
3#23765602C211yesmaximal, characteristic, complemented
4#23765580C211yesmaximal, characteristic, complemented
5#23765534C211yesmaximal, characteristic, complemented
6#23765515C211yesmaximal, characteristic, complemented
7#23765489C211yesmaximal, characteristic, complemented
8#23765462C211yesmaximal, characteristic, complemented
9#23765338C211yesmaximal, characteristic, complemented
10#23765328C211yesmaximal, characteristic, complemented
11#23765262C211yesmaximal, characteristic, complemented
12#23765226C211yesmaximal, characteristic, complemented
13#23765110C211yesmaximal, characteristic, complemented
14C2×#19098363C211yesmaximal, characteristic, complemented
15C79×#3338C211yesmaximal, characteristic, complemented
16C2×#14579183C2211yescharacteristic
17C22×D632C2211yescharacteristic
18#19099209—21no—
19C2×#14579107C2211yescharacteristic, complemented
20C2×#14579104C2211yescharacteristic
21C2×#14579104C2211yescharacteristic
22C2×#14579101C2211yescharacteristic, complemented
23C22×C79⋊C16C2211yescharacteristic
24#19098947C2211yescharacteristic
25#19098804C2211yescharacteristic
26#19098799C2211yescharacteristic
27#19098779—21no—
28#19098716—21no—
29#19098710—21no—
30C4×D632C2211yescharacteristic
31#19098388C2211yescharacteristic, complemented
32#19098388C2211yescharacteristic, 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#554×C79—21no—
49#500×C79C2211yescharacteristic, complemented
50#484×C79—21no—
51#484×C79—21no—
52#477×C79—21no—
53#467×C79C2211yescharacteristic
54#467×C79C2211yescharacteristic, complemented
55C4×Q16×C79C2211yescharacteristic, complemented
56C8×D4×C79C2211yescharacteristic
57C2×#107×C79C2211yescharacteristic
58C2×#107×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#125×C79—41no—
97C22⋊Q8×C79—21no—
98C22⋊Q8×C79—21no—
99C22⋊Q8×C79—41no—
100C22⋊Q8×C79—41no—
101C4×Q8×C79—21no—
102C4×D4×C79—41no—
103C4×D4×C79D411yescharacteristic
104C2×C4⋊C4×C79—21no—
105C2×C4⋊C4×C79—21no—
106#107×C79—21no—
107#107×C79—21no—
108#107×C79—21no—
109#107×C79C2311yescharacteristic
110#107×C79C2311yescharacteristic
111#107×C79C2311yescharacteristic
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×C79—41no—
156C22×C4×C79—41no—
157C22×C4×C79C2×D411yescharacteristic
158C22×C4×C79C2×D411yescharacteristic
159Q16×C79—21no—
160Q16×C79—21no—
161C2×C8×C79—21no—
162C2×C8×C79—21no—
163C2×C8×C79—21no—
164C2×C8×C79—21no—
165C2×C8×C79C2×D411yescharacteristic
166C2×C8×C79C2×D411yescharacteristic
167C2×C8×C79C2×D411yescharacteristic
168C2×C8×C79C2411yescharacteristic
169C4⋊C4×C79—21no—
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—41no—
173C4⋊C4×C79—41no—
174C4⋊C4×C79—41no—
175C4⋊C4×C79—41no—
176C4⋊C4×C79—41no—
177C4⋊C4×C79—41no—
178C4⋊C4×C79D811yescharacteristic
179C22⋊C4×C79—41no—
180C22⋊C4×C79—41no—
181C22⋊C4×C79D811yescharacteristic
182C22⋊C4×C79D811yescharacteristic
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—161no—
195D4×C79—22no—
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×C79—161no—
205C2×C4×C79—161no—
206C2×C4×C79C2×D811yescharacteristic
207C2×C4×C79C2×D811yescharacteristic
208C2×C4×C79C2×D811yescharacteristic
209C2×C4×C79C2×D811yescharacteristic
210C2×C4×C79C22×D411yescharacteristic
211C22⋊D79—82no—
212C22⋊D79—161no—
213C2×Dic79—41no—
214C2×Dic79—41no—
215D316—22no—
216D316—41no—
217D316—41no—
218D316—41no—
219D316—82no—
220C4×D79—82no—
221C4×D79—161no—
222Dic158—41no—
223Dic158—41no—
224C8×C79—21no—
225C8×C79—21no—
226C8×C79—21no—
227C8×C79C2×#4111yescharacteristic
228C8×C79H(5,𝔽2)11yescharacteristic
229C22×C79—21no—
230C22×C79—21no—
231C22×C79—21no—
232C22×C79—21no—
233C22×C79—21no—
234C22×C79#51012yes—
235C22×C79C22×D811yescharacteristic
236C2×D79—42no—
237C2×D79—81no—
238C2×D79—81no—
239C2×D79—81no—
240C2×D79—81no—
241C2×D79—81no—
242C2×D79—164no—
243C4×C79—21no—
244C4×C79—21no—
245C4×C79—21no—
246C4×C79—21no—
247C4×C79—81no—
248C4×C79—81no—
249C4×C79—81no—
250C4×C79—81no—
251C4×C79#54611yescharacteristic
252C4×C79#57711yescharacteristic
253Dic79—81no—
254Dic79—81no—
255#33881—791nomaximal, Sylow
256C2×C79—22no—
257C2×C79—41no—
258C2×C79#329411yescharacteristic
259C2×C79C2×#51011yescharacteristic
260C2×C79#341511yescharacteristic
261D79—82no—
262D79—161no—
263#3338D7911yescharacteristic, complemented
264#3280—791no—
265#2243—791no—
266#2236—791no—
267#2232—791no—
268#2224—791no—
269#2220—791no—
270#2216—791no—
271#2211—791no—
272#2189—791no—
273#2187—791no—
274#2175—791no—
275#2168—791no—
276#2139—791no—
277C2×#357—791no—
278C79#3388111yescharacteristic, complemented, Sylow
279C2×#135—791no—
280C22×D8—791no—
281#554—21no—
282#546—1581no—
283C2×Q32—791no—
284C2×SD32—791no—
285C2×SD32—791no—
286C2×M32—791no—
287C22×C16—791no—
288#500C2×D7911yescharacteristic, complemented
289#495—791no—
290#484—21no—
291#484—21no—
292#477—21no—
293#467—791no—
294#467C2×D7911yescharacteristic
295#467C2×D7911yescharacteristic, complemented
296#466—791no—
297#463—1581no—
298#450—1581no—
299#449—1581no—
300C4×Q16C2×D7911yescharacteristic, complemented
301C4×D8—791no—
302C8×D4C2×D7911yescharacteristic
303C2×#107C2×D7911yescharacteristic
304C2×#107C2×D7911yescharacteristic, complemented
305#366—791no—
306#366—791no—
307#366—791no—
308C4⋊C16—791no—
309#358—791no—
310#358—791no—
311#358—791no—
312#358—791no—
313#357—791no—
314#357—791no—
315#357—791no—
316#357—791no—
317#357—791no—
318#357—791no—
319C22⋊C16—791no—
320C22⋊C16—791no—
321#346—791no—
322C2×#41—1581no—
323C22×D4—1581no—
324#135—3161no—
325#135—1582no—
326C2×Q16—791no—
327C2×Q16C22×D7911yescharacteristic
328C2×SD16—1581no—
329C2×SD16—1581no—
330C2×D8—3161no—
331C2×D8—791no—
332C2×D8—791no—
333C2×D8—1582no—
334C2×D8—1581no—
335C22×C8C22×D7911yescharacteristic
336C22×C8C22×D7911yescharacteristic
337C4⋊Q8—21no—
338#125—41no—
339#124—1581no—
340#123—3161no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—41no—
344C22⋊Q8—41no—
345C22⋊Q8—1581no—
346#121—3161no—
347#121—1581no—
348#120—3161no—
349C4×Q8—21no—
350C4×D4—41no—
351C4×D4—1581no—
352C4×D4C22⋊D7911yescharacteristic
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C22⋊C4—3161no—
356Q32—3161no—
357SD32—3161no—
358SD32—1582no—
359M32—3161no—
360C2×C16—3161no—
361C2×C16—791no—
362C2×C16—791no—
363C2×C16—791no—
364C2×C16—791no—
365#107—21no—
366#107—21no—
367#107—21no—
368#107—791no—
369#107C22×D7911yescharacteristic
370#107C22×D7911yescharacteristic
371#107C22×D7911yescharacteristic
372C4⋊C8C22⋊D7911yescharacteristic
373#103—21no—
374#103—21no—
375#103—21no—
376#103—1581no—
377#102—1581no—
378#102—1581no—
379C22⋊C8C22⋊D7911yescharacteristic
380C22⋊C8C22⋊D7911yescharacteristic
381C4×C8C22×D7911yescharacteristic
382C24—3161no—
383#41—3162no—
384#41—6321no—
385C2×Q8—21no—
386C2×Q8—41no—
387C2×Q8—1581no—
388C2×D4—3162no—
389C2×D4—3161no—
390C2×D4—3161no—
391C2×D4—3161no—
392C2×D4—6321no—
393C2×D4—1581no—
394C2×D4—1581no—
395C2×D4#1192960911yescharacteristic
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—3161no—
399C22×C4—3161no—
400C22×C4C2×C22⋊D7911yescharacteristic
401C22×C4C2×C22⋊D7911yescharacteristic
402Q16—21no—
403Q16—21no—
404Q16—1581no—
405Q16—1581no—
406SD16—3162no—
407SD16—6321no—
408D8—3162no—
409D8—792no—
410D8—1581no—
411D8—1581no—
412D8—1581no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8C2×C22⋊D7911yescharacteristic
418C2×C8C2×C22⋊D7911yescharacteristic
419C2×C8C2×C22⋊D7911yescharacteristic
420C2×C8C23×D7911yescharacteristic
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—3161no—
431C4⋊C4—1581no—
432C4⋊C4#1192960911yescharacteristic
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—3161no—
436C22⋊C4—3161no—
437C22⋊C4—3161no—
438C22⋊C4—3161no—
439C22⋊C4—3161no—
440C22⋊C4#1192960911yescharacteristic
441C22⋊C4#1192960911yescharacteristic
442C42—41no—
443C42C2×C22⋊D7911yescharacteristic
444C16—1581no—
445C16—1581no—
446C16—1581no—
447C16—1581no—
448C23—3161no—
449C23—3161no—
450C23—6322no—
451C23—6322no—
452C23—12641no—
453C23C2×#1192960911yescharacteristic
454C23C2×#1192960911yescharacteristic
455Q8—41no—
456Q8—41no—
457Q8—161no—
458Q8—3161no—
459Q8—3161no—
460D4—22no—
461D4—3161no—
462D4—3161no—
463D4—3161no—
464D4—6322no—
465D4—6322no—
466D4—1582no—
467D4—12641no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—161no—
477C2×C4—161no—
478C2×C4—3161no—
479C2×C4—3161no—
480C2×C4—6322no—
481C2×C4—12641no—
482C2×C4C2×#1192960911yescharacteristic
483C2×C4C2×#1192960911yescharacteristic
484C2×C4C2×#1192960911yescharacteristic
485C2×C4C2×#1192960911yescharacteristic
486C2×C4C22×C22⋊D7911yescharacteristic
487C8—21no—
488C8—21no—
489C8—21no—
490C8#41×D7911yescharacteristic
491C8#1457920711yescharacteristic
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—3162no—
498C22—6321no—
499C22—6321no—
500C22—6321no—
501C22—6321no—
502C22—6321no—
503C22—12644no—
504C22#1909900412yes—
505C22C22×#1192960911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—6321no—
515C4—6321no—
516C4#1909920111yescharacteristic
517C4#1909937111yescharacteristic
518C2—22no—
519C2—41no—
520C2—6322no—
521C2—12641no—
522C2#2377520711yescharacteristic
523C2C2×#1909900411yescharacteristic
524C2#2377653911yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3338C2
socleC22×C79C22×D8
prime radical#23775087C2
hypercenter#3338D79
hypocenterC79#33881
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #23765639C2semidirect
3. #23765602C2semidirect
4. #23765580C2semidirect
5. #23765534C2semidirect
6. #23765515C2semidirect
7. #23765489C2semidirect
8. #23765462C2semidirect
9. #23765338C2semidirect
10. #23765328C2semidirect
11. #23765262C2semidirect
12. #23765226C2semidirect
13. #23765110C2semidirect
14. C2×#19098363C2semidirect
15. C79×#3338C2semidirect
19. C2×#14579107C22semidirect
22. C2×#14579101C22semidirect
31. #19098388C22semidirect
32. #19098388C22semidirect
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. C2×#107×C79C22semidirect
263. #3338D79semidirect
278. C79#33881semidirect
288. #500C2×D79semidirect
295. #467C2×D79semidirect
300. C4×Q16C2×D79semidirect
304. C2×#107C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237750872
#237656392
#237656022
#237655802
#237655342
#237655152
#237654892
#237654622
#237653382
#237653282
#237652622
#237652262
#237651102
C2×#190983632
C79×#33382
#3388179