Finite Group Databasebeta

← #34258367

Subgroups — #34258367

Order 20224.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773467C211yesmaximal, characteristic
2#23772507C211yesmaximal, characteristic
3#23772491C211yesmaximal, characteristic, complemented
4#23762270C211yesmaximal, characteristic, complemented
5#23762258C211yesmaximal, characteristic, complemented
6#23762253C211yesmaximal, characteristic, complemented
7#23762233C211yesmaximal, characteristic, complemented
8#23762217C211yesmaximal, characteristic, complemented
9#23762212C211yesmaximal, characteristic, complemented
10C8×Dic316C211yesmaximal, characteristic, complemented
11#23762182C211yesmaximal, characteristic, complemented
12#23761831C211yesmaximal, characteristic
13#23761801C211yesmaximal, characteristic, complemented
14#23761699C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×#11929623C2211yescharacteristic
18#19098907C411yescharacteristic
19#19098903C411yescharacteristic
20#19098847C411yescharacteristic
21#19098841C411yescharacteristic
22#19098772C411yescharacteristic
23#19098766C411yescharacteristic
24#19098726C411yescharacteristic
25#19098722C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098693—21no—
28#19098692—21no—
29#19098690C2211yescharacteristic
30C8×Dic158C2211yescharacteristic
31C8×Dic158C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×Dic316C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36D4×C79⋊C8C2211yescharacteristic
37C8×C22⋊D79—21no—
38(C22⋊D79)⋊C8—21no—
39#19098623—21no—
40#19098622—21no—
41#19098619C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098613—21no—
46C2×Dic79⋊C8—21no—
47C4⋊C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098293C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098285C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C22×C79⋊C8—21no—
72C2×C8×D79—21no—
73#14579160C2×C411yescharacteristic
74#14579159C2×C411yescharacteristic
75#14579151—21no—
76#14579141C2×C411yescharacteristic
77#14579139—21no—
78#14579137C2×C411yescharacteristic
79C4×Dic158—21no—
80C4×Dic158C2311yescharacteristic
81C4×Dic158C2311yescharacteristic
82#14579126C2×C411yescharacteristic
83D4×Dic79C2×C411yescharacteristic
84C4×C22⋊D79—21no—
85(C22⋊D79)⋊C4—21no—
86#14579119C2×C411yescharacteristic
87#14579118—21no—
88#14579117—21no—
89C2×C4×Dic79—21no—
90C42×D79C2311yescharacteristic
91#14579095C2×C411yescharacteristic
92#14579094C2×C411yescharacteristic
93#14579092C2×C411yescharacteristic
94C4⋊(C79⋊C8)—21no—
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579082C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8D411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120#14579072—21no—
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×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—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929623—22no—
145#11929623—22no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150C22×Dic79—21no—
151C22×Dic79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—22no—
156#11929614—22no—
157#11929612—22no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—81no—
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C8×D79—81no—
167C4⋊Dic79—21no—
168C4⋊Dic79C22×C411yescharacteristic
169C4⋊Dic79C22×C411yescharacteristic
170Dic79⋊C4—21no—
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177C22⋊Dic79—21no—
178(C2×D79)⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181C4×Dic79—21no—
182C4×Dic79—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79C22×C411yescharacteristic
186C4×Dic79C2×D411yescharacteristic
187C4×Dic79#4111yescharacteristic
188C2×Q8×C79C22×C411yescharacteristic
189C2×D4×C79C22×C411yescharacteristic
190C22×C4×C79—21no—
191SD16×C79—22no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—81no—
196C2×C8×C79C22×C411yescharacteristic
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C4⋊C4×C79—21no—
201C4⋊C4×C79C22×C411yescharacteristic
202C4⋊C4×C79C22×C411yescharacteristic
203C22⋊C4×C79—21no—
204C42×C79—21no—
205C42×C79C2411yescharacteristic
206C23×C79—21no—
207C22×D79C4×D411yescharacteristic
208Q8×C79—21no—
209Q8×C79—21no—
210D4×C79—21no—
211D4×C79#13112yes—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C22⋊D79—24no—
221C2×Dic79—24no—
222C2×Dic79—21no—
223C2×Dic79—21no—
224C2×Dic79—21no—
225C2×Dic79—21no—
226C2×Dic79—81no—
227C2×Dic79C4×D411yescharacteristic
228C2×Dic79C4×D411yescharacteristic
229C2×Dic79C4×D411yescharacteristic
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—21no—
234C4×D79—41no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—81no—
239Dic158#13112yes—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—21no—
244C8×C79—41no—
245C8×C79—41no—
246C8×C79—41no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C22×C79—22no—
255C22×C79—41no—
256C22×C79—41no—
257C22×C79C2×C4×D411yescharacteristic
258C2×D79—21no—
259C2×D79—21no—
260C2×D79—21no—
261C4×C79—21no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79C2×#13111yescharacteristic
266C4×C79C2×#13111yescharacteristic
267C4×C79#57711yescharacteristic
268C4×C79#57811yescharacteristic
269Dic79—22no—
270Dic79—21no—
271Dic79—21no—
272Dic79—21no—
273Dic79—41no—
274Dic79—41no—
275Dic79—41no—
276#17921—791nomaximal, Sylow
277C2×C79—42no—
278C2×C79#292911yescharacteristic
279C2×C79#294711yescharacteristic
280C2×C79#295711yescharacteristic
281D79—41no—
282#3107—791no—
283#2988—791no—
284#2985—791no—
285#1587—791no—
286#1585—791no—
287#1584—791no—
288#1582—791no—
289#1579—791no—
290SD16⋊C8—791no—
291C8×Q16—791no—
292C8×SD16D7911yescharacteristic, complemented
293#1573—791no—
294#1488—791no—
295#1481—791no—
296#1454—791no—
297C79#1792111yescharacteristic, complemented, Sylow
298#533—791no—
299C4×#41—791no—
300#485—791no—
301#485—791no—
302#475—791no—
303#474—791no—
304#462—791no—
305#461—791no—
306#452—791no—
307#452—791no—
308#446—791no—
309#446—791no—
310#446—1581no—
311#446—1581no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C8×Q8C2×D7911yescharacteristic
316C4×Q16—791no—
317C4×Q16—791no—
318C4×SD16—791no—
319C4×SD16C2×D7911yescharacteristic
320C8×D4—791no—
321C8×D4—1581no—
322C8×D4—1581no—
323C8×D4C2×D7911yescharacteristic
324#433—1581no—
325#433—1581no—
326#432—791no—
327#431—791no—
328#431—791no—
329#431—791no—
330#431—1581no—
331C2×C4⋊C8—791no—
332C2×C4⋊C8—1581no—
333#335—791no—
334#335—791no—
335#334—791no—
336#334C2×D7911yescharacteristic
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8C2×D7911yescharacteristic
343D4⋊C8—791no—
344D4⋊C8C2×D7911yescharacteristic
345#322—791no—
346#322—791no—
347#322—791no—
348C82C2×D7911yescharacteristic
349C2×#41—791no—
350C2×Q16—791no—
351C2×Q16—791no—
352C2×SD16—791no—
353C2×SD16C4×D7911yescharacteristic
354C22×C8—21no—
355C22×C8—1581no—
356C22×C8—1581no—
357C4⋊Q8—791no—
358C4⋊Q8—791no—
359#125—1581no—
360C22⋊Q8—791no—
361C22⋊Q8—791no—
362C22⋊Q8—1581no—
363C4×Q8—791no—
364C4×Q8—791no—
365C4×Q8—791no—
366C4×Q8—1581no—
367C4×Q8C22×D7911yescharacteristic
368C4×D4—791no—
369C4×D4—1581no—
370C4×D4—1581no—
371C4×D4C22×D7911yescharacteristic
372#117—791no—
373#117—1581no—
374#117—1581no—
375C2×C42—791no—
376C2×C42—1581no—
377#107—791no—
378#107—791no—
379#106—791no—
380#106C4×D7911yescharacteristic
381C4⋊C8—21no—
382C4⋊C8—791no—
383C4⋊C8—791no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8C22×D7911yescharacteristic
396C4⋊C8C22×D7911yescharacteristic
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103C4×D7911yescharacteristic
403#102—791no—
404#102C4×D7911yescharacteristic
405C22⋊C8—21no—
406C22⋊C8—791no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—1581no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—1581no—
414#97—1581no—
415#97—1581no—
416C4×C8—21no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C4×C8C22×D7911yescharacteristic
426#41—1582no—
427#41—1582no—
428C2×Q8—791no—
429C2×Q8—791no—
430C2×Q8—1581no—
431C2×Q8C2×C4×D7911yescharacteristic
432C2×D4—1581no—
433C2×D4C2×C4×D7911yescharacteristic
434C22×C4—21no—
435C22×C4—791no—
436C22×C4—791no—
437C22×C4—791no—
438C22×C4—1581no—
439C22×C4—1581no—
440Q16—1582no—
441Q16—1582no—
442SD16—22no—
443SD16—1582no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—81no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—791no—
452C2×C8—6321no—
453C2×C8—6321no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8C2×C4×D7911yescharacteristic
458C2×C8C2×C4×D7911yescharacteristic
459C2×C8D4×D7911yescharacteristic
460C2×C8D4×D7911yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—791no—
463C4⋊C4—791no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C42—21no—
480C42—791no—
481C42—791no—
482C42—791no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C42—1581no—
487C42C23×D7911yescharacteristic
488C23—21no—
489C23—791no—
490Q8—21no—
491Q8—21no—
492Q8—792no—
493Q8—6321no—
494Q8—1581no—
495Q8—1581no—
496Q8—1581no—
497D4—21no—
498D4—1584no—
499D4#1457916712yes—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—3161no—
506C2×C4—791no—
507C2×C4—791no—
508C2×C4—791no—
509C2×C4—6321no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1584no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4—1581no—
519C2×C4C22×C4×D7911yescharacteristic
520C2×C4C2×D4×D7911yescharacteristic
521C2×C4#41×D7911yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—3161no—
530C8—3161no—
531C8—3161no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—1581no—
540C22—1581no—
541C22—1581no—
542C22C4×D4×D7911yescharacteristic
543C4—21no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—3161no—
548C4—3161no—
549C4—3161no—
550C4—1581no—
551C4—1581no—
552C4—1581no—
553C4—1582no—
554C4#131×D7911yescharacteristic
555C4C2×#1457916711yescharacteristic
556C4#1909936911yescharacteristic
557C4#1909937711yescharacteristic
558C2—42no—
559C2—3161no—
560C2#2377208911yescharacteristic
561C2#2377223611yescharacteristic
562C2#2377231211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929623C2×C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#17921
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
3. #23772491C2semidirect
4. #23762270C2semidirect
5. #23762258C2semidirect
6. #23762253C2semidirect
7. #23762233C2semidirect
8. #23762217C2semidirect
9. #23762212C2semidirect
10. C8×Dic316C2semidirect
11. #23762182C2semidirect
13. #23761801C2semidirect
14. #23761699C2semidirect
15. C8×SD16×C79C2semidirect
292. C8×SD16D79semidirect
297. C79#17921semidirect

Maximal subgroups

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

subgroupindex
#237734672
#237725072
#237724912
#237622702
#237622582
#237622532
#237622332
#237622172
#237622122
C8×Dic3162
#237621822
#237618312
#237618012
#237616992
C8×SD16×C792
#1792179