Finite Group Databasebeta

← #34258494

Subgroups — #34258494

Order 20224.

604 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773422C211yesmaximal, characteristic
2#23772374C211yesmaximal, characteristic, complemented
3#23772363C211yesmaximal, characteristic, complemented
4#23762295C211yesmaximal, characteristic, complemented
5#23762292C211yesmaximal, characteristic, complemented
6#23762283C211yesmaximal, characteristic, complemented
7#23762247C211yesmaximal, characteristic, complemented
8#23762244C211yesmaximal, characteristic, complemented
9#23762235C211yesmaximal, characteristic, complemented
10#23762232C211yesmaximal, characteristic, complemented
11#23762177C211yesmaximal, characteristic, complemented
12#23761831C211yesmaximal, characteristic, complemented
13#23761807C211yesmaximal, characteristic, complemented
14#23761682C211yesmaximal, characteristic, complemented
15C79×#1590C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×#11929623C2211yescharacteristic
18#19098902C411yescharacteristic
19#19098889C411yescharacteristic
20#19098873C411yescharacteristic
21#19098865C411yescharacteristic
22#19098788C411yescharacteristic
23#19098780C411yescharacteristic
24#19098722C411yescharacteristic
25#19098709C411yescharacteristic
26#19098693C2211yescharacteristic
27#19098690C2211yescharacteristic
28Q8×C79⋊C8C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098642C2211yescharacteristic
33#19098641C2211yescharacteristic
34#19098637—21no—
35#19098636—21no—
36#19098632—21no—
37#19098628—21no—
38#19098628C2211yescharacteristic
39#19098623—21no—
40#19098619C2211yescharacteristic
41#19098615C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098613—21no—
45#19098520—21no—
46M16×Dic79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×C8×D79—21no—
74C2×C8×D79D411yescharacteristic
75C2×C8×D79D411yescharacteristic
76#14579156—21no—
77#14579152C2×C411yescharacteristic
78#14579150C2×C411yescharacteristic
79#14579146—21no—
80#14579137C2×C411yescharacteristic
81#14579136C2×C411yescharacteristic
82C4×Dic158C2311yescharacteristic
83#14579126C2×C411yescharacteristic
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86D4×Dic79C2×C411yescharacteristic
87C4×C22⋊D79—21no—
88(C22⋊D79)⋊C4—21no—
89#14579119C2×C411yescharacteristic
90#14579118—21no—
91#14579117—21no—
92C2×C4×Dic79—21no—
93C42×D79C2311yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)C2311yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105#14579086C2×C411yescharacteristic
106#14579085C2×C411yescharacteristic
107#14579084C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579082C2×C411yescharacteristic
110#14579081C2×C411yescharacteristic
111C22⋊(C79⋊C8)—21no—
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8C2×C411yescharacteristic
119#14579071—21no—
120#14579071—21no—
121#14579071—21no—
122C4×C79⋊C8—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C2×M16×C79—21no—
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79C2311yescharacteristic
133#106×C79C2×C411yescharacteristic
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140#97×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929623—41no—
145#11929623—41no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316—21no—
148C2×D316C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150C22×Dic79—21no—
151C22×Dic79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—41no—
157#11929613—41no—
158#11929609—41no—
159#11929608—42no—
160#11929607—42no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C8×D79—22no—
169C8×D79—22no—
170C8×D79—42no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×D4×C79C22×C411yescharacteristic
193C22×C4×C79—21no—
194SD16×C79—41no—
195M16×C79—42no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C4⋊C4×C79—21no—
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C22⋊C4×C79—21no—
207C42×C79—21no—
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C22×D79—21no—
211C22×D79C4×D411yescharacteristic
212Q8×C79—21no—
213Q8×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×D411yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C22⋊D79—42no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—42no—
229C2×Dic79—42no—
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232C2×Dic79C4×D411yescharacteristic
233D316—21no—
234D316—21no—
235D316—42no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—41no—
241C4×D79—41no—
242C4×D79—41no—
243Dic158—21no—
244Dic158—21no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—41no—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79#13512yes—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8#13512yes—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×C4×D411yescharacteristic
263C2×D79—21no—
264C2×D79—21no—
265C2×D79—21no—
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—21no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79#56811yescharacteristic
274C4×C79#56811yescharacteristic
275C4×C79C2×#13511yescharacteristic
276C4×C79C2×#13511yescharacteristic
277Dic79—21no—
278Dic79—21no—
279Dic79—21no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283#17932—791nomaximal, Sylow
284C2×C79—81no—
285C2×C79#293011yescharacteristic
286C2×C79C4×#13511yescharacteristic
287C2×C79#295511yescharacteristic
288D79—41no—
289D79—81no—
290#3102—791no—
291#2966—791no—
292#2965—791no—
293#1591—791no—
294#1590—791no—
295#1590D7911yescharacteristic, complemented
296#1589—791no—
297#1584—791no—
298#1583—791no—
299#1582—791no—
300#1581—791no—
301#1572—791no—
302#1488—791no—
303#1482—791no—
304#1450—791no—
305C79#1793211yescharacteristic, complemented, Sylow
306#529—791no—
307C4×#41—791no—
308#485—791no—
309#482—791no—
310#479—791no—
311#478—791no—
312#464—791no—
313#463—791no—
314#452—791no—
315#449—791no—
316#446—791no—
317#446—791no—
318#446C2×D7911yescharacteristic
319C8×Q8—791no—
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16C2×D7911yescharacteristic
323C4×D8—791no—
324#436—791no—
325#436—791no—
326#436C2×D7911yescharacteristic
327#435—1581no—
328#435—1581no—
329#435—1581no—
330C8×D4—791no—
331C8×D4—1581no—
332#433—1581no—
333#432—791no—
334#431—791no—
335#431—791no—
336#431—791no—
337#431—1581no—
338#405—1581no—
339C4×M16—1581no—
340C2×C4×C8—791no—
341#335—791no—
342#335—791no—
343#334—791no—
344#334C2×D7911yescharacteristic
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8C2×D7911yescharacteristic
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8C2×D7911yescharacteristic
353#322—791no—
354#322—791no—
355C82—791no—
356C82C2×D7911yescharacteristic
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16C4×D7911yescharacteristic
361C2×D8—791no—
362C2×M16—21no—
363C2×M16—1581no—
364C2×M16—1581no—
365C22×C8—791no—
366C22×C8—791no—
367C22×C8—1581no—
368#126—1581no—
369#125—791no—
370#124—791no—
371#123—1581no—
372C22⋊Q8—791no—
373#121—791no—
374C4×Q8—791no—
375C4×Q8—791no—
376C4×Q8C22×D7911yescharacteristic
377C4×D4—791no—
378C4×D4—791no—
379C4×D4—1581no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4C22×D7911yescharacteristic
383#117—791no—
384#117—1581no—
385#117—1581no—
386C2×C42—791no—
387C2×C42—1581no—
388#107—791no—
389#107—791no—
390#106—791no—
391#106C4×D7911yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8C22×D7911yescharacteristic
402C4⋊C8C22×D7911yescharacteristic
403#103—791no—
404#103—791no—
405#103—791no—
406#103C4×D7911yescharacteristic
407#102—791no—
408#102—791no—
409#102—791no—
410#102C4×D7911yescharacteristic
411C22⋊C8—21no—
412C22⋊C8—791no—
413C22⋊C8—791no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—21no—
421#97—1581no—
422#97—1581no—
423#97—1581no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—791no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434#41—3161no—
435#41—3161no—
436C2×Q8—791no—
437C2×Q8C2×C4×D7911yescharacteristic
438C2×D4—791no—
439C2×D4—1581no—
440C2×D4—1581no—
441C2×D4C2×C4×D7911yescharacteristic
442C22×C4—21no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449Q16—3161no—
450SD16—41no—
451SD16—3161no—
452D8—3161no—
453M16—42no—
454M16—3162no—
455M16—3162no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—3162no—
460C2×C8—791no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—1582no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1582no—
469C2×C8C2×C4×D7911yescharacteristic
470C2×C8C2×C4×D7911yescharacteristic
471C2×C8D4×D7911yescharacteristic
472C2×C8D4×D7911yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—791no—
475C4⋊C4—791no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—21no—
492C42—791no—
493C42—791no—
494C42—791no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—791no—
501C23—1581no—
502Q8—21no—
503Q8—21no—
504Q8—1581no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—3162no—
509D4—3162no—
510D4—1581no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—3161no—
518C2×C4—3162no—
519C2×C4—3162no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—791no—
523C2×C4—791no—
524C2×C4—791no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4C22×C4×D7911yescharacteristic
533C2×C4C2×D4×D7911yescharacteristic
534C2×C4#41×D7911yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—3161no—
542C8—3161no—
543C8—3161no—
544C8—792no—
545C8—1581no—
546C8—1581no—
547C8—1581no—
548C8#1457918212yes—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—3161no—
553C22—3161no—
554C22—3161no—
555C22—1581no—
556C22—1581no—
557C22—1581no—
558C22C4×D4×D7911yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—3161no—
564C4—3161no—
565C4—3161no—
566C4—1581no—
567C4—1581no—
568C4—1581no—
569C4#1909932811yescharacteristic
570C4#1909932911yescharacteristic
571C4#135×D7911yescharacteristic
572C4C2×#1457918211yescharacteristic
573C2—81no—
574C2—3161no—
575C2—6321no—
576C2#2377210011yescharacteristic
577C2C4×#1457918211yescharacteristic
578C2#2377229811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1590C2
socleC22×C79C2×C4×D4
prime radical#19098709C4
cosocleC42×C79C24
hypercenter#1590D79
hypocenterC79#17932
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #23772374C2semidirect
3. #23772363C2semidirect
4. #23762295C2semidirect
5. #23762292C2semidirect
6. #23762283C2semidirect
7. #23762247C2semidirect
8. #23762244C2semidirect
9. #23762235C2semidirect
10. #23762232C2semidirect
11. #23762177C2semidirect
12. #23761831C2semidirect
13. #23761807C2semidirect
14. #23761682C2semidirect
15. C79×#1590C2semidirect
295. #1590D79semidirect
305. C79#17932semidirect

Maximal subgroups

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

subgroupindex
#237734222
#237723742
#237723632
#237622952
#237622922
#237622832
#237622472
#237622442
#237622352
#237622322
#237621772
#237618312
#237618072
#237616822
C79×#15902
#1793279