Finite Group Databasebeta

← #34257902

Subgroups — #34257902

Order 20224.

648 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#23772515C211yesmaximal, characteristic
2#23772479C211yesmaximal, characteristic, complemented
3C4×#14579190C211yesmaximal, characteristic
4#23762279C211yesmaximal, characteristic, complemented
5#23762276C211yesmaximal, characteristic, complemented
6#23762271C211yesmaximal, characteristic, complemented
7#23762270C211yesmaximal, characteristic, complemented
8#23762252C211yesmaximal, characteristic, complemented
9#23762247C211yesmaximal, characteristic, complemented
10#23762246C211yesmaximal, characteristic, complemented
11#23762149C211yesmaximal, characteristic, complemented
12#23761779C211yesmaximal, characteristic
13Q8⋊C8×D79C211yesmaximal, characteristic, complemented
14#23761688C211yesmaximal, characteristic, complemented
15C79×#1583C211yesmaximal, characteristic, complemented
16C2×#14579190C411yescharacteristic
17C4×#11929623C2211yescharacteristic
18C4×Q8×D79C2211yescharacteristic
19#19098693C2211yescharacteristic
20#19098690C2211yescharacteristic
21Q8×C79⋊C8C2211yescharacteristic
22C8×Dic158—21no—
23C8×Dic158C2211yescharacteristic
24#19098687—21no—
25#19098687C2211yescharacteristic
26#19098669C411yescharacteristic
27#19098667C411yescharacteristic
28#19098664C411yescharacteristic
29#19098660C411yescharacteristic
30C4×Dic316C2211yescharacteristic
31C4×#11929614C2211yescharacteristic
32C4×#11929612C2211yescharacteristic
33#19098641C2211yescharacteristic
34#19098637—21no—
35#19098632—21no—
36#19098617—21no—
37#19098615—21no—
38#19098615C2211yescharacteristic
39#19098614—21no—
40#19098614C2211yescharacteristic
41#19098602C411yescharacteristic
42#19098586—21no—
43C4⋊C8×D79C2211yescharacteristic
44#19098570C411yescharacteristic
45#19098561C411yescharacteristic
46C4×#11929607C2211yescharacteristic
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
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×Q8×D79C2×C411yescharacteristic
69#14579190—24no—
70C2×Dic316C2×C411yescharacteristic
71C2×#11929614C2×C411yescharacteristic
72C2×#11929612C2×C411yescharacteristic
73C2×#11929608—21no—
74C2×#11929607C2×C411yescharacteristic
75C2×#11929607C2×C411yescharacteristic
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78Q8×Dic79C2×C411yescharacteristic
79C4×Dic158—21no—
80C4×Dic158C2311yescharacteristic
81C4×Dic158C2311yescharacteristic
82#14579126—21no—
83#14579126C2×C411yescharacteristic
84#14579126C2×C411yescharacteristic
85D4×Dic79C2×C411yescharacteristic
86C4×C22⋊D79—21no—
87(C22⋊D79)⋊C4—21no—
88#14579119C2×C411yescharacteristic
89#14579117—21no—
90C4⋊C4×D79—21no—
91C4⋊C4×D79C2×C411yescharacteristic
92C2×C4×Dic79—21no—
93C42×D79—21no—
94C42×D79C2311yescharacteristic
95#14579095C2×C411yescharacteristic
96#14579094C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)C2311yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8D411yescharacteristic
107Dic79⋊C8D411yescharacteristic
108#14579086C2×C411yescharacteristic
109#14579086C2×C411yescharacteristic
110#14579085C2×C411yescharacteristic
111#14579084C2×C411yescharacteristic
112#14579084C2×C411yescharacteristic
113#14579082C2×C411yescharacteristic
114C22⋊(C79⋊C8)—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8D411yescharacteristic
117(C2×D79)⋊C8D411yescharacteristic
118#14579071—21no—
119#14579071—21no—
120#14579071C2×C411yescharacteristic
121#14579071C2×C411yescharacteristic
122C4×C79⋊C8—21no—
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
129C8×Dic79—21no—
130C2×SD16×C79C2×C411yescharacteristic
131C2×M16×C79—21no—
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#106×C79C2×C411yescharacteristic
135C4⋊C8×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79C2×C411yescharacteristic
139#102×C79C2×C411yescharacteristic
140C22⋊C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145#11929623—22no—
146#11929623—22no—
147C2×Dic158—21no—
148C2×Dic158C22×C411yescharacteristic
149C2×Dic158C22×C411yescharacteristic
150Q8×D79—22no—
151Q8×D79M1614yescomplemented
152C2×C22⋊D79—21no—
153C22×Dic79—21no—
154C22×Dic79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79C22×C411yescharacteristic
158C2×C4×D79C2×D411yescharacteristic
159C2×C4×D79#4111yescharacteristic
160Dic316—22no—
161#11929614—22no—
162#11929612—22no—
163#11929608—81no—
164#11929607—22no—
165#11929607—22no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C8×D79—42no—
174C8×D79—81no—
175C4⋊Dic79—21no—
176C4⋊Dic79C22×C411yescharacteristic
177C4⋊Dic79C22×C411yescharacteristic
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183C22⋊Dic79—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79—21no—
189C4×Dic79—21no—
190C4×Dic79—21no—
191C4×Dic79C22×C411yescharacteristic
192C4×Dic79C2×D411yescharacteristic
193C4×Dic79#4111yescharacteristic
194C2×Q8×C79C22×C411yescharacteristic
195C2×D4×C79C22×C411yescharacteristic
196C22×C4×C79—21no—
197SD16×C79—22no—
198M16×C79—81no—
199C2×C8×C79—21no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C4⋊C4×C79—21no—
207C4⋊C4×C79C22×C411yescharacteristic
208C4⋊C4×C79C22×C411yescharacteristic
209C22⋊C4×C79—21no—
210C42×C79—21no—
211C42×C79C2411yescharacteristic
212C23×C79—21no—
213C22×D79C4×D411yescharacteristic
214Q8×C79—21no—
215Q8×C79C2×M1612yes—
216D4×C79—21no—
217D4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79C23×C411yescharacteristic
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C22⋊D79—42no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—42no—
232C2×Dic79—81no—
233C2×Dic79C4×D411yescharacteristic
234C2×Dic79C4×D411yescharacteristic
235C2×Dic79C4×D411yescharacteristic
236C4×D79—24no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—41no—
240C4×D79—81no—
241C4×D79C2×M1612yes—
242C4×D79#13112yes—
243Dic158—21no—
244Dic158—24no—
245Dic158—21no—
246Dic158—21no—
247Dic158C2×M1612yes—
248C8×C79—21no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—41no—
253C8×C79—41no—
254C8×C79—41no—
255C79⋊C8—22no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79C2×C4×D411yescharacteristic
266C2×D79—21no—
267C2×D79—21no—
268C2×D79#43512yes—
269C4×C79—22no—
270C4×C79—21no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79C22×M1611yescharacteristic
274C4×C79C2×#13111yescharacteristic
275C4×C79#57711yescharacteristic
276C4×C79#57811yescharacteristic
277Dic79—22no—
278Dic79—21no—
279Dic79—21no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283Dic79#43512yes—
284#17874—791nomaximal, Sylow
285C2×C79—81no—
286C2×C79C2×#43511yescharacteristic
287C2×C79#294711yescharacteristic
288C2×C79#295611yescharacteristic
289D79—22no—
290#2989—791no—
291#2982—791no—
292C4×#137—791no—
293#1588—791no—
294#1588—791no—
295#1587—791no—
296#1587—791no—
297#1584—791no—
298#1584—791no—
299#1583—791no—
300#1583D7911yescharacteristic, complemented
301#1565—791no—
302#1476—791no—
303C2×Q8⋊C8—791no—
304#1451—791no—
305C79#1787411yescharacteristic, complemented, Sylow
306C2×#137—791no—
307C4×#41—791no—
308C2×C4×Q8—791no—
309#446—791no—
310#446—791no—
311C8×Q8—791no—
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—1581no—
315C8×Q8—1581no—
316C8×Q8C2×D7911yescharacteristic
317#441—791no—
318#441—791no—
319#440—791no—
320#440—791no—
321C4×Q16—791no—
322C4×Q16—791no—
323C4×SD16—791no—
324C4×SD16C2×D7911yescharacteristic
325#436—791no—
326#436C2×D7911yescharacteristic
327#435—1581no—
328#435—1581no—
329#432—1581no—
330#431—791no—
331#431—791no—
332#431—1581no—
333#431—1581no—
334#428—791no—
335#423—1581no—
336C2×C4⋊C8—791no—
337#419—791no—
338#417—791no—
339C4×M16—791no—
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⋊C8—791no—
349Q8⋊C8—791no—
350Q8⋊C8C2×D7911yescharacteristic
351D4⋊C8—791no—
352D4⋊C8C2×D7911yescharacteristic
353#322—791no—
354#322—791no—
355#322—791no—
356#322C2×D7911yescharacteristic
357C2×#41—791no—
358C22×Q8—791no—
359#137—1584no—
360C2×Q16—791no—
361C2×Q16—791no—
362C2×SD16—791no—
363C2×SD16C4×D7911yescharacteristic
364C2×M16—21no—
365C2×M16—791no—
366C2×M16—791no—
367C2×M16—1581no—
368C22×C8—1581no—
369C22×C8—1581no—
370C4×Q8—791no—
371C4×Q8—791no—
372C4×Q8—791no—
373C4×Q8—791no—
374C4×Q8—791no—
375C4×Q8—1581no—
376C4×Q8—1581no—
377C4×Q8C22×D7911yescharacteristic
378C4×D4—791no—
379C4×D4—1581no—
380C4×D4—1581no—
381C4×D4C22×D7911yescharacteristic
382#117—791no—
383#117—1581no—
384C2×C4⋊C4—791no—
385C2×C4⋊C4—1581no—
386C2×C42—791no—
387C2×C42—1581no—
388C2×C42—1581no—
389#107—791no—
390#107—791no—
391#106—791no—
392#106C4×D7911yescharacteristic
393C4⋊C8—21no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8C22×D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406#103—791no—
407#103—791no—
408#103—791no—
409#103—791no—
410#103—791no—
411#103C4×D7911yescharacteristic
412#102—791no—
413#102C4×D7911yescharacteristic
414C22⋊C8—21no—
415C22⋊C8—791no—
416C22⋊C8—791no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419#97—791no—
420#97—791no—
421#97—1581no—
422#97—1581no—
423C4×C8—21no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—1581no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8—1581no—
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434C4×C8C22×D7911yescharacteristic
435#41—1582no—
436#41—1582no—
437C2×Q8—794no—
438C2×Q8—791no—
439C2×Q8—791no—
440C2×Q8—1581no—
441C2×Q8—1582no—
442C2×Q8C2×C4×D7911yescharacteristic
443C2×D4—1581no—
444C2×D4C2×C4×D7911yescharacteristic
445C22×C4—21no—
446C22×C4—791no—
447C22×C4—791no—
448C22×C4—791no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453Q16—1582no—
454Q16—1582no—
455SD16—22no—
456SD16—1582no—
457M16—81no—
458M16—6321no—
459M16—1582no—
460M16—1582no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—3162no—
465C2×C8—791no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—6321no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8C2×C4×D7911yescharacteristic
474C2×C8C2×C4×D7911yescharacteristic
475C2×C8D4×D7911yescharacteristic
476C2×C8D4×D7911yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—791no—
479C4⋊C4—791no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4C2×C4×D7911yescharacteristic
487C4⋊C4C2×C4×D7911yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—21no—
492C42—791no—
493C42—791no—
494C42—791no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42—1581no—
499C42—1581no—
500C42—1581no—
501C42C23×D7911yescharacteristic
502C23—21no—
503C23—791no—
504Q8—21no—
505Q8—792no—
506Q8—1584no—
507Q8—1581no—
508Q8—1581no—
509Q8—1581no—
510Q8M16×D7912yes—
511D4—21no—
512D4—21no—
513D4—3162no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—3161no—
520C2×C4—3162no—
521C2×C4—791no—
522C2×C4—792no—
523C2×C4—792no—
524C2×C4—791no—
525C2×C4—791no—
526C2×C4—6321no—
527C2×C4—6321no—
528C2×C4—1584no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1581no—
535C2×C4C22×C4×D7911yescharacteristic
536C2×C4C2×D4×D7911yescharacteristic
537C2×C4#41×D7911yescharacteristic
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—3161no—
546C8—3161no—
547C8—1582no—
548C8—1581no—
549C8—1581no—
550C8—1581no—
551C8—1581no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—792no—
556C22—1581no—
557C22—1581no—
558C22C4×D4×D7911yescharacteristic
559C4—21no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—3161no—
564C4—3161no—
565C4—3161no—
566C4—792no—
567C4—1581no—
568C4—1581no—
569C4—1582no—
570C4C2×M16×D7911yescharacteristic
571C4#131×D7911yescharacteristic
572C4#1909936911yescharacteristic
573C4#1909937711yescharacteristic
574C2—81no—
575C2—1582no—
576C2#435×D7911yescharacteristic
577C2#2377223611yescharacteristic
578C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1583C2
socleC22×C79C2×C4×D4
prime radicalC2×#11929623C2×C4
cosocleC42×C79C24
hypercenter#1583D79
hypocenterC79#17874
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
2. #23772479C2semidirect
4. #23762279C2semidirect
5. #23762276C2semidirect
6. #23762271C2semidirect
7. #23762270C2semidirect
8. #23762252C2semidirect
9. #23762247C2semidirect
10. #23762246C2semidirect
11. #23762149C2semidirect
13. Q8⋊C8×D79C2semidirect
14. #23761688C2semidirect
15. C79×#1583C2semidirect
151. Q8×D79M16semidirect
300. #1583D79semidirect
305. C79#17874semidirect

Maximal subgroups

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

subgroupindex
#237725152
#237724792
C4×#145791902
#237622792
#237622762
#237622712
#237622702
#237622522
#237622472
#237622462
#237621492
#237617792
Q8⋊C8×D792
#237616882
C79×#15832
#1787479