Finite Group Databasebeta

← #34258476

Subgroups — #34258476

Order 20224.

666 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773406C211yesmaximal, characteristic
2#23772331C211yesmaximal, characteristic, complemented
3#436×D79C211yesmaximal, characteristic
4#23762292C211yesmaximal, characteristic, complemented
5#23762290C211yesmaximal, characteristic, complemented
6#23762288C211yesmaximal, characteristic, complemented
7#23762246C211yesmaximal, characteristic, complemented
8#23762244C211yesmaximal, characteristic, complemented
9#23762242C211yesmaximal, characteristic, complemented
10#23762240C211yesmaximal, characteristic, complemented
11#23762169C211yesmaximal, characteristic, complemented
12#23761823C211yesmaximal, characteristic
13#23761801C211yesmaximal, characteristic, complemented
14#23761682C211yesmaximal, characteristic, complemented
15C79×#1590C211yesmaximal, characteristic, complemented
16C2×M16×D79—21no—
17#19099117C2211yescharacteristic
18C4×D4×D79C2211yescharacteristic
19#19098895C411yescharacteristic
20#19098892C411yescharacteristic
21#19098852C411yescharacteristic
22#19098851C411yescharacteristic
23#19098767C411yescharacteristic
24#19098766C411yescharacteristic
25#19098715C411yescharacteristic
26#19098712C411yescharacteristic
27#19098693C2211yescharacteristic, complemented
28#19098691—21no—
29Q8×C79⋊C8C2211yescharacteristic, complemented
30#19098687—21no—
31#19098687C2211yescharacteristic, complemented
32C4×#11929613C2211yescharacteristic
33C4×#11929611C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098642C2211yescharacteristic
36#19098641C2211yescharacteristic
37#19098640C2211yescharacteristic
38#19098639—21no—
39#19098638—21no—
40#19098614C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098588—21no—
43#19098585C2211yescharacteristic
44C4⋊C8×D79C2211yescharacteristic
45C22⋊C8×D79—21no—
46#19098520—21no—
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic, complemented
49#19098313C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51#19098295C2211yescharacteristic, complemented
52Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
53#19098293C2211yescharacteristic, complemented
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic, complemented
58#19098285C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#446×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic, complemented
67C2×D4×D79C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929613C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76M16×D79—44no—
77C2×C8×D79—21no—
78C2×C8×D79D411yescharacteristic
79C2×C8×D79D411yescharacteristic
80#14579160C2×C411yescharacteristic
81#14579159C2×C411yescharacteristic
82#14579151—21no—
83#14579141C2×C411yescharacteristic
84#14579139—21no—
85#14579137C2×C411yescharacteristic
86C4×Dic158—21no—
87C4×Dic158C2311yescharacteristic
88C4×D316C2311yescharacteristic
89D4×Dic79C2×C411yescharacteristic
90#14579123C2×C411yescharacteristic
91C4×C22⋊D79—21no—
92(C22⋊D79)⋊C4—21no—
93#14579118—21no—
94C4⋊C4×D79C2×C411yescharacteristic
95C22⋊C4×D79—21no—
96C42×D79C2311yescharacteristic
97#14579093C2×C411yescharacteristic
98#14579092C2×C411yescharacteristic
99#14579092C2×C411yescharacteristic
100C4⋊(C79⋊C8)—21no—
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8—21no—
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8C2×C411yescharacteristic
109Dic79⋊C8C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579085C2×C411yescharacteristic
112#14579084C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579082C2×C411yescharacteristic
115#14579081C2×C411yescharacteristic
116C22⋊(C79⋊C8)—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8C2×C411yescharacteristic
121(C2×D79)⋊C8C2×C411yescharacteristic
122#14579071—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79D411yescharacteristic
129C8×Dic79D411yescharacteristic
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—
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145C23×D79—21no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×D316C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150D4×D79—41no—
151D4×D79—41no—
152C22×Dic79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—42no—
157C2×C4×D79—42no—
158C2×C4×D79C22×C411yescharacteristic
159C2×C4×D79C2×D411yescharacteristic
160C2×C4×D79#4111yescharacteristic
161#11929613—41no—
162#11929612—41no—
163#11929611—41no—
164#11929608—42no—
165#11929607—42no—
166#11929607—42no—
167#11929607—42no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C22×C411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C2×C79⋊C8C2×D411yescharacteristic
175C8×D79—22no—
176C8×D79—42no—
177C8×D79SD1614yescomplemented
178C4⋊Dic79—21no—
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184Dic79⋊C4—21no—
185Dic79⋊C4—21no—
186Dic79⋊C4—21no—
187C22⋊Dic79—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192C4×Dic79—21no—
193C4×Dic79C22×C411yescharacteristic
194C4×Dic79C2×D411yescharacteristic
195C4×Dic79#4111yescharacteristic
196C2×Q8×C79C22×C411yescharacteristic
197C2×D4×C79C22×C411yescharacteristic
198C22×C4×C79—21no—
199SD16×C79—41no—
200M16×C79—42no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79—21no—
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C22×C411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C4⋊C4×C79—21no—
209C4⋊C4×C79C22×C411yescharacteristic
210C4⋊C4×C79C22×C411yescharacteristic
211C22⋊C4×C79—21no—
212C42×C79—21no—
213C42×C79C2411yescharacteristic
214C23×C79—21no—
215C22×D79—21no—
216C22×D79—42no—
217C22×D79—42no—
218C22×D79—42no—
219C22×D79C4×D411yescharacteristic
220Q8×C79—21no—
221Q8×C79—21no—
222D4×C79—21no—
223D4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79C23×C411yescharacteristic
230C2×C4×C79C22×D411yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C22⋊D79—42no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—42no—
237C2×Dic79C4×D411yescharacteristic
238C2×Dic79C4×D411yescharacteristic
239C2×Dic79C4×D411yescharacteristic
240D316—21no—
241D316—21no—
242C4×D79—21no—
243C4×D79—21no—
244C4×D79—41no—
245C4×D79—42no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79—42no—
249C4×D79C2×SD1612yes—
250C4×D79#13512yes—
251Dic158—21no—
252Dic158—21no—
253Dic158—42no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—41no—
258C8×C79—41no—
259C8×C79—41no—
260C8×C79C2×SD1612yes—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C79⋊C8C2×SD1612yes—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79C2×C4×D411yescharacteristic
272C2×D79—21no—
273C2×D79—21no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—84no—
278C2×D79C4×SD1612yes—
279C4×C79—21no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79#56811yescharacteristic
284C4×C79#56811yescharacteristic
285C4×C79C22×SD1611yescharacteristic
286C4×C79C2×#13511yescharacteristic
287Dic79—21no—
288Dic79—21no—
289Dic79—41no—
290Dic79—41no—
291Dic79—41no—
292Dic79C4×SD1612yes—
293#17930—791nomaximal, Sylow
294C2×C79—81no—
295C2×C79#293011yescharacteristic
296C2×C79C2×C4×SD1611yescharacteristic
297C2×C79#295511yescharacteristic
298D79—22no—
299D79—81no—
300#3101—791no—
301#2960—791no—
302C2×#436—791no—
303#1590—791no—
304#1590—791no—
305#1590—791no—
306#1590D7911yescharacteristic, complemented
307#1583—791no—
308#1583—791no—
309#1583—791no—
310#1583—791no—
311#1570—791no—
312#1486—791no—
313#1481—791no—
314#1450—791no—
315C79#1793011yescharacteristic, complemented, Sylow
316C22×M16—1581no—
317#533—791no—
318C2×C4×D4—791no—
319#483—791no—
320#483—791no—
321#476—791no—
322#476—791no—
323#461—791no—
324#461—791no—
325#450—791no—
326#450—791no—
327#446—791no—
328#446—1581no—
329#446C2×D7911yescharacteristic, complemented
330C8×Q8—791no—
331C8×Q8—791no—
332C8×Q8—1581no—
333C4×SD16—791no—
334C4×SD16—791no—
335C4×SD16—791no—
336C4×SD16C2×D7911yescharacteristic
337#436—791no—
338#436—791no—
339#436—791no—
340#436—1581no—
341#436—1581no—
342#436C2×D7911yescharacteristic
343#431—791no—
344#431—791no—
345#424—1581no—
346#423—791no—
347C2×C4⋊C8—791no—
348C2×C22⋊C8—1581no—
349#405—1581no—
350C2×C4×C8—791no—
351#334—791no—
352#334—791no—
353#334—791no—
354#334C2×D7911yescharacteristic, complemented
355Q8⋊C8—791no—
356Q8⋊C8—791no—
357Q8⋊C8—791no—
358Q8⋊C8C2×D7911yescharacteristic, complemented
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8C2×D7911yescharacteristic
363#322—791no—
364#322—791no—
365C82—791no—
366C82C2×D7911yescharacteristic, complemented
367C22×D4—791no—
368C23×C4—1581no—
369C2×SD16—791no—
370C2×SD16—791no—
371C2×SD16—791no—
372C2×SD16C4×D7911yescharacteristic
373C2×M16—21no—
374C2×M16—3164no—
375C2×M16—1581no—
376C2×M16—1581no—
377C2×M16—1581no—
378C2×M16—1581no—
379C22×C8—791no—
380C22×C8—791no—
381C22×C8—1581no—
382C4⋊Q8—791no—
383C4⋊Q8—791no—
384#125—1581no—
385C22⋊Q8—791no—
386C22⋊Q8—791no—
387C22⋊Q8—1581no—
388C4×Q8—791no—
389C4×Q8—1581no—
390C4×Q8C22×D7911yescharacteristic
391C4×D4—791no—
392C4×D4—791no—
393C4×D4—791no—
394C4×D4—1581no—
395C4×D4—1581no—
396C4×D4C22×D7911yescharacteristic
397#117—1581no—
398C2×C4⋊C4—791no—
399C2×C22⋊C4—1581no—
400C2×C42—791no—
401#106—791no—
402#106—791no—
403#106—791no—
404#106C4×D7911yescharacteristic
405C4⋊C8—21no—
406C4⋊C8—791no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—1581no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8—1581no—
415C4⋊C8—1581no—
416C4⋊C8C22×D7911yescharacteristic
417C4⋊C8C22×D7911yescharacteristic
418#103—791no—
419#103—791no—
420#103—791no—
421#103C4×D7911yescharacteristic
422#102—791no—
423#102—791no—
424#102—791no—
425#102C4×D7911yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—791no—
428C22⋊C8—791no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433#97—21no—
434#97—1581no—
435C4×C8—791no—
436C4×C8—791no—
437C4×C8—791no—
438C4×C8—791no—
439C4×C8—791no—
440C4×C8—1581no—
441C4×C8—1581no—
442C4×C8C22×D7911yescharacteristic
443C4×C8C22×D7911yescharacteristic
444C4×C8C22×D7911yescharacteristic
445C24—1581no—
446C2×Q8—791no—
447C2×Q8—1581no—
448C2×Q8C2×C4×D7911yescharacteristic
449C2×D4—3161no—
450C2×D4—3161no—
451C2×D4—791no—
452C2×D4—1581no—
453C2×D4C2×C4×D7911yescharacteristic
454C22×C4—21no—
455C22×C4—3162no—
456C22×C4—3162no—
457C22×C4—791no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—1581no—
461C22×C4—1581no—
462C22×C4—1581no—
463C22×C4—1581no—
464SD16—41no—
465SD16—3161no—
466SD16—3161no—
467SD16—3161no—
468M16—42no—
469M16—3162no—
470M16—3162no—
471M16—3162no—
472M16—3162no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—3162no—
477C2×C8—791no—
478C2×C8—794no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—1582no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8C2×C4×D7911yescharacteristic
487C2×C8C2×C4×D7911yescharacteristic
488C2×C8D4×D7911yescharacteristic
489C2×C8D4×D7911yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—791no—
492C4⋊C4—791no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4C2×C4×D7911yescharacteristic
501C4⋊C4C2×C4×D7911yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C42—21no—
509C42—791no—
510C42—791no—
511C42—791no—
512C42—1581no—
513C42C23×D7911yescharacteristic
514C23—21no—
515C23—3162no—
516C23—3162no—
517C23—3162no—
518C23—791no—
519C23—1581no—
520Q8—21no—
521Q8—21no—
522Q8—3162no—
523Q8—1581no—
524Q8—1581no—
525D4—21no—
526D4—21no—
527D4—3162no—
528D4—1581no—
529D4—1581no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—3161no—
536C2×C4—3162no—
537C2×C4—3162no—
538C2×C4—3162no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—791no—
542C2×C4—792no—
543C2×C4—792no—
544C2×C4—791no—
545C2×C4—791no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1581no—
549C2×C4—1581no—
550C2×C4—1581no—
551C2×C4C22×C4×D7911yescharacteristic
552C2×C4C2×D4×D7911yescharacteristic
553C2×C4#41×D7911yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—3161no—
561C8—3161no—
562C8—3161no—
563C8—792no—
564C8—1581no—
565C8—1581no—
566C8—1581no—
567C8SD16×D7912yes—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—3161no—
572C22—3161no—
573C22—3161no—
574C22—792no—
575C22—6324no—
576C22—1581no—
577C22—1581no—
578C22C4×D4×D7911yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—3161no—
584C4—3161no—
585C4—3161no—
586C4—792no—
587C4—1581no—
588C4—1581no—
589C4#1909932811yescharacteristic
590C4#1909932911yescharacteristic
591C4C2×SD16×D7911yescharacteristic
592C4#135×D7911yescharacteristic
593C2—81no—
594C2—6321no—
595C2—1582no—
596C2#2377210011yescharacteristic
597C2C4×SD16×D7911yescharacteristic
598C2#2377229811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1590C2
socleC22×C79C2×C4×D4
prime radicalC2×D4×D79C2×C4
cosocleC42×C79C24
hypercenter#1590D79
hypocenterC79#17930
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. #23772331C2semidirect
4. #23762292C2semidirect
5. #23762290C2semidirect
6. #23762288C2semidirect
7. #23762246C2semidirect
8. #23762244C2semidirect
9. #23762242C2semidirect
10. #23762240C2semidirect
11. #23762169C2semidirect
13. #23761801C2semidirect
14. #23761682C2semidirect
15. C79×#1590C2semidirect
27. #19098693C22semidirect
29. Q8×C79⋊C8C22semidirect
31. #19098687C22semidirect
48. #19098314C22semidirect
49. #19098313C22semidirect
50. #19098313C22semidirect
51. #19098295C22semidirect
52. Q8⋊(C79⋊C8)C22semidirect
53. #19098293C22semidirect
57. #19098285C22semidirect
58. #19098285C22semidirect
59. C8×C79⋊C8C22semidirect
60. #446×C79C22semidirect
63. #334×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. C82×C79C22semidirect
177. C8×D79SD16semidirect
306. #1590D79semidirect
315. C79#17930semidirect
329. #446C2×D79semidirect
354. #334C2×D79semidirect
358. Q8⋊C8C2×D79semidirect
366. C82C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734062
#237723312
#436×D792
#237622922
#237622902
#237622882
#237622462
#237622442
#237622422
#237622402
#237621692
#237618232
#237618012
#237616822
C79×#15902
#1793079