Finite Group Databasebeta

← #34347209

Subgroups — #34347209

Order 20224.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773932C211yesmaximal, characteristic, complemented
2#23773262C211yesmaximal, characteristic
3#23773251C211yesmaximal, characteristic
4#23764913C211yesmaximal, characteristic, complemented
5#23764908C211yesmaximal, characteristic, complemented
6#23764748C211yesmaximal, characteristic, complemented
7#23764743C211yesmaximal, characteristic, complemented
8#23764183C211yesmaximal, characteristic, complemented
9#23764154C211yesmaximal, characteristic, complemented
10#23763633C211yesmaximal, characteristic
11#23763626C211yesmaximal, characteristic
12#23763567C211yesmaximal, characteristic, complemented
13#23763459C211yesmaximal, characteristic
14#23763242C211yesmaximal, characteristic
15C79×#2913C211yesmaximal, characteristic
16#19099191C2211yescharacteristic
17#19099191C2211yescharacteristic
18#19098971—21no—
19#19098971—21no—
20#19098963—21no—
21#19098960—21no—
22#19098870—21no—
23#19098870—21no—
24#19098863—21no—
25#19098863—21no—
26#19098858—21no—
27#19098853—21no—
28#19098848—21no—
29#19098843—21no—
30C2×#14579097C2211yescharacteristic
31#19098606C2211yescharacteristic
32#19098606C2211yescharacteristic
33#19098603C2211yescharacteristic
34#19098603C2211yescharacteristic
35C2×#14579094C2211yescharacteristic
36C2×#14579093C2211yescharacteristic
37C2×#14579092C2211yescharacteristic
38#19098590C2211yescharacteristic
39#19098590C2211yescharacteristic
40C2×#14579088C2211yescharacteristic
41#19098564C2211yescharacteristic
42#19098564C2211yescharacteristic
43#19098559C2211yescharacteristic
44#19098518C2211yescharacteristic
45C2×#14579071C2211yescharacteristic
46#19098432C2211yescharacteristic
47#19098428C2211yescharacteristic
48#19098332C2211yescharacteristic
49#19098332C2211yescharacteristic
50#19098331C2211yescharacteristic
51#19098331C2211yescharacteristic
52#19098327C2211yescharacteristic
53#19098324C2211yescharacteristic
54#19098322C2211yescharacteristic
55#19098322C2211yescharacteristic
56C2×#131×C79C2211yescharacteristic
57#520×C79C2211yescharacteristic
58#520×C79C2211yescharacteristic
59#428×C79—21no—
60#428×C79—21no—
61#428×C79C2211yescharacteristic
62#428×C79C2211yescharacteristic
63#427×C79—21no—
64#427×C79C2211yescharacteristic
65C2×#107×C79—21no—
66C2×#107×C79C2211yescharacteristic
67C2×#11929608C2311yescharacteristic
68C2×#11929608C2311yescharacteristic
69C2×#11929608C2311yescharacteristic
70C22×C79⋊C8C2311yescharacteristic
71#14579161—21no—
72#14579161—21no—
73#14579160—21no—
74#14579160—21no—
75#14579153—21no—
76#14579153—21no—
77#14579152—21no—
78#14579152—21no—
79#14579138—41no—
80#14579138—41no—
81#14579120C2311yescharacteristic
82C2×C4⋊Dic79C2311yescharacteristic
83C2×C4⋊Dic79C2311yescharacteristic
84C2×Dic79⋊C4—41no—
85C2×C4×Dic79C2311yescharacteristic
86#14579097—22no—
87#14579094—21no—
88#14579094—21no—
89#14579094—21no—
90#14579094—21no—
91#14579094—21no—
92#14579094—21no—
93#14579093—21no—
94#14579093—21no—
95#14579092—21no—
96#14579092—21no—
97#14579092—21no—
98#14579092—21no—
99#14579092—21no—
100#14579092—21no—
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)—21no—
104C4⋊(C79⋊C8)—21no—
105#14579088Q814yescomplemented
106#14579085—21no—
107#14579085—21no—
108#14579085D411yescharacteristic
109#14579085D411yescharacteristic
110#14579082—21no—
111#14579082—21no—
112#14579082D411yescharacteristic
113#14579082D411yescharacteristic
114#14579071—21no—
115#14579071—22no—
116C8×Dic79—21no—
117C79⋊#95—41no—
118C2×#41×C79C2311yescharacteristic
119#131×C79—22no—
120#131×C79Q814yescomplemented
121C2×M16×C79—21no—
122C2×M16×C79C2311yescharacteristic
123C22×C8×C79—21no—
124C22×C8×C79C2311yescharacteristic
125C4×Q8×C79—21no—
126C4×Q8×C79—21no—
127C4×D4×C79—21no—
128C4×D4×C79—21no—
129C4×D4×C79—41no—
130C4×D4×C79—41no—
131#117×C79—21no—
132#117×C79—21no—
133#117×C79C2311yescharacteristic
134#117×C79C2311yescharacteristic
135C2×C4⋊C4×C79—21no—
136C2×C4⋊C4×C79—21no—
137C2×C4⋊C4×C79C2311yescharacteristic
138C2×C4⋊C4×C79C2311yescharacteristic
139#107×C79—21no—
140#107×C79—21no—
141#107×C79—21no—
142#107×C79—21no—
143#107×C79D411yescharacteristic
144#107×C79D411yescharacteristic
145#106×C79—21no—
146#106×C79—21no—
147#106×C79D411yescharacteristic
148#106×C79D411yescharacteristic
149C2×Dic158—41no—
150C2×Dic158—41no—
151C22×Dic79—21no—
152C22×Dic79—21no—
153C22×Dic79—21no—
154#11929608—21no—
155#11929608—21no—
156#11929608—21no—
157#11929608—21no—
158#11929608—21no—
159#11929608C2×Q812yes—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79—41no—
172C4⋊Dic79—41no—
173C4⋊Dic79C2×D411yescharacteristic
174C4⋊Dic79C2×D411yescharacteristic
175Dic79⋊C4—41no—
176Dic79⋊C4—41no—
177Dic79⋊C4—41no—
178Dic79⋊C4—41no—
179Dic79⋊C4—82no—
180C22⋊Dic79—21no—
181C4×Dic79—21no—
182C4×Dic79—21no—
183C4×Dic79C2×Q812yes—
184#41×C79—21no—
185#41×C79—21no—
186#41×C79C2×Q812yes—
187#41×C79C2×Q812yes—
188C2×Q8×C79C2×D411yescharacteristic
189C2×D4×C79—21no—
190C2×D4×C79C2×D411yescharacteristic
191C22×C4×C79—21no—
192C22×C4×C79—21no—
193C22×C4×C79—21no—
194C22×C4×C79—41no—
195C22×C4×C79—41no—
196C22×C4×C79C2411yescharacteristic
197M16×C79—24no—
198M16×C79—21no—
199M16×C79C2×Q812yes—
200C2×C8×C79—24no—
201C2×C8×C79—21no—
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C2×C8×C79C2×Q812yes—
205C2×C8×C79#4112yes—
206C4⋊C4×C79—21no—
207C4⋊C4×C79—21no—
208C4⋊C4×C79—21no—
209C4⋊C4×C79—21no—
210C4⋊C4×C79—41no—
211C4⋊C4×C79—41no—
212C4⋊C4×C79C2×D411yescharacteristic
213C4⋊C4×C79C2×D411yescharacteristic
214C4⋊C4×C79C2×D411yescharacteristic
215C4⋊C4×C79C2×D411yescharacteristic
216C22⋊C4×C79—21no—
217C22⋊C4×C79—21no—
218C22⋊C4×C79—41no—
219C22⋊C4×C79—41no—
220C42×C79—21no—
221C42×C79—21no—
222C42×C79—41no—
223C42×C79—41no—
224C23×C79—21no—
225C23×C79H(5,𝔽2)11yescharacteristic
226Q8×C79—21no—
227Q8×C79C4⋊Q812yes—
228D4×C79—22no—
229D4×C79—22no—
230D4×C79—21no—
231D4×C79C4⋊Q812yes—
232C2×C4×C79—21no—
233C2×C4×C79—22no—
234C2×C4×C79—22no—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—81no—
242C2×C4×C79—81no—
243C2×C4×C79—161no—
244C2×C4×C79—161no—
245C2×C4×C79C22×D411yescharacteristic
246C2×C4×C79C22×D411yescharacteristic
247C2×C4×C79C22×Q811yescharacteristic
248C2×C4×C79C22×Q811yescharacteristic
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C792−1+411yescharacteristic
251C2×Dic79—21no—
252C2×Dic79—21no—
253C2×Dic79—22no—
254C2×Dic79—41no—
255C2×Dic79—41no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—81no—
259C2×Dic79—81no—
260Dic158—161no—
261Dic158—161no—
262C8×C79—22no—
263C8×C79C22⋊Q812yes—
264C8×C79C22⋊Q812yes—
265C79⋊C8—21no—
266C79⋊C8—21no—
267C79⋊C8—22no—
268C79⋊C8—41no—
269C79⋊C8—41no—
270C22×C79—21no—
271C22×C79—21no—
272C22×C79—22no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79#54411yescharacteristic
276C22×C79D4211yescharacteristic
277C22×C79#55411yescharacteristic
278C4×C79—22no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79—81no—
284C4×C79—81no—
285C4×C79C2×C22⋊Q811yescharacteristic
286C4×C79C2×C4⋊Q811yescharacteristic
287C4×C79#54711yescharacteristic
288C4×C79D4×Q811yescharacteristic
289Dic79—42no—
290Dic79—41no—
291Dic79—41no—
292Dic79—81no—
293Dic79—81no—
294#26085—791nomaximal, Sylow
295C2×C79—21no—
296C2×C79—21no—
297C2×C79—42no—
298C2×C79#244011yescharacteristic
299C2×C79#328811yescharacteristic
300C2×C79#328911yescharacteristic
301#3163—791no—
302#3084—791no—
303#3083—791no—
304#2913D7911yescharacteristic
305#2090—791no—
306#2089—791no—
307#2061—791no—
308#2060—791no—
309#1952—791no—
310#1946—791no—
311#1860—791no—
312#1859—791no—
313#1847—791no—
314#1815—791no—
315#1765—791no—
316C79#2608511yescharacteristic, complemented, Sylow
317C2×#131C2×D7911yescharacteristic
318#544—791no—
319#544—791no—
320#520C2×D7911yescharacteristic
321#520C2×D7911yescharacteristic
322#501—1581no—
323#501—1581no—
324#499—1581no—
325#498—1581no—
326#479—1581no—
327#479—1581no—
328#478—1581no—
329#478—1581no—
330#477—1581no—
331#476—1581no—
332#475—1581no—
333#474—1581no—
334C2×#108—791no—
335#428—21no—
336#428—21no—
337#428—791no—
338#428—791no—
339#428—791no—
340#428—791no—
341#428C2×D7911yescharacteristic
342#428C2×D7911yescharacteristic
343#427—21no—
344#427C2×D7911yescharacteristic
345C2×#107—21no—
346C2×#107—791no—
347C2×#107C2×D7911yescharacteristic
348C2×#106—791no—
349C2×#106—791no—
350#424—791no—
351#424—791no—
352C2×#104—791no—
353#417—791no—
354#417—791no—
355#416—791no—
356#405—791no—
357C2×#97—791no—
358#384—791no—
359#382—791no—
360#343—791no—
361#343—791no—
362#343—791no—
363#343—791no—
364#341—791no—
365#339—791no—
366#339—791no—
367#339—791no—
368C2×#41C22×D7911yescharacteristic
369#131—22no—
370#131Dic15814yescomplemented
371C2×M16—21no—
372C2×M16—791no—
373C2×M16—791no—
374C2×M16—791no—
375C2×M16C22×D7911yescharacteristic
376C22×C8—21no—
377C22×C8—791no—
378C22×C8C22×D7911yescharacteristic
379C4⋊Q8—1581no—
380C4⋊Q8—1581no—
381C4⋊Q8—1581no—
382C4⋊Q8—1581no—
383#125—1581no—
384#125—1581no—
385#125—1581no—
386#125—1581no—
387C22⋊Q8—3161no—
388C22⋊Q8—3161no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—41no—
394C4×D4—41no—
395#117—21no—
396#117—21no—
397#117—791no—
398#117C22×D7911yescharacteristic
399#117C22×D7911yescharacteristic
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—3161no—
403C2×C4⋊C4—791no—
404C2×C4⋊C4—791no—
405C2×C4⋊C4C22×D7911yescharacteristic
406C2×C4⋊C4C22×D7911yescharacteristic
407C2×C42—791no—
408#108—1582no—
409#107—21no—
410#107—21no—
411#107—21no—
412#107—21no—
413#107—1581no—
414#107—1581no—
415#107—1581no—
416#107—1581no—
417#107—1581no—
418#107—1581no—
419#107C22⋊D7911yescharacteristic
420#107C22⋊D7911yescharacteristic
421#106—21no—
422#106—21no—
423#106—1581no—
424#106—1581no—
425#106—1581no—
426#106—1581no—
427#106—1581no—
428#106—1581no—
429#106—1581no—
430#106—1581no—
431#106C22⋊D7911yescharacteristic
432#106C22⋊D7911yescharacteristic
433C4⋊C8—1581no—
434C4⋊C8—1581no—
435C4⋊C8—1581no—
436C4⋊C8—1581no—
437#104—794no—
438#103—791no—
439#103—791no—
440#103—1581no—
441#103—1581no—
442#102—791no—
443#102—791no—
444#102—1581no—
445#102—1581no—
446#97—1581no—
447#97—1582no—
448C4×C8—1581no—
449#95—3161no—
450#41—21no—
451#41—21no—
452#41Q8×D7912yes—
453#41C2×Dic15812yes—
454C2×Q8—3161no—
455C2×Q8—3161no—
456C2×Q8D4×D7911yescharacteristic
457C2×D4—21no—
458C2×D4D4×D7911yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4C23×D7911yescharacteristic
468M16—21no—
469M16—24no—
470M16—792no—
471M16—1581no—
472M16—1581no—
473M16—1581no—
474M16—1581no—
475M16—1581no—
476M16C2×Dic15812yes—
477C2×C8—21no—
478C2×C8—24no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8C2×C22⋊D7911yescharacteristic
487C2×C8C2×C22⋊D7911yescharacteristic
488C2×C8C2×Dic15812yes—
489C2×C8#1192962412yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—3161no—
497C4⋊C4—3161no—
498C4⋊C4—3161no—
499C4⋊C4—3161no—
500C4⋊C4—3161no—
501C4⋊C4—3161no—
502C4⋊C4—791no—
503C4⋊C4—791no—
504C4⋊C4—6322no—
505C4⋊C4—1581no—
506C4⋊C4—1581no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4C2×C22⋊D7911yescharacteristic
510C4⋊C4C2×C22⋊D7911yescharacteristic
511C4⋊C4C2×C22⋊D7911yescharacteristic
512C4⋊C4C2×C22⋊D7911yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—1581no—
518C42—21no—
519C42—21no—
520C42—41no—
521C42—41no—
522C42—792no—
523C42—1581no—
524C42—1581no—
525C23—21no—
526C23#1457920611yescharacteristic
527Q8—21no—
528Q8—12641no—
529Q8—12641no—
530Q8#1457916012yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4#1457916012yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—161no—
547C2×C4—161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—6321no—
553C2×C4—6321no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1582no—
557C2×C4C2×D4×D7911yescharacteristic
558C2×C4C22×C22⋊D7911yescharacteristic
559C2×C4C2×Q8×D7911yescharacteristic
560C2×C4C22×Dic15811yescharacteristic
561C2×C4C2×#1192962411yescharacteristic
562C2×C4#1457920911yescharacteristic
563C8—22no—
564C8—3161no—
565C8—3161no—
566C8—1581no—
567C8—1581no—
568C8—1582no—
569C8#1457914012yes—
570C8#1457914012yes—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#1909919411yescharacteristic
577C22D4×C22⋊D7911yescharacteristic
578C22#1909926811yescharacteristic
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—3162no—
587C4—3161no—
588C4—3161no—
589C4—6321no—
590C4—6321no—
591C4C2×#1457914011yescharacteristic
592C4C2×#1457916011yescharacteristic
593C4#1909921411yescharacteristic
594C4Q8×C22⋊D7911yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2#2376709611yescharacteristic
599C2#2377516411yescharacteristic
600C2#2377517111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2913C2
socleC22×C79D42
prime radicalC2×#41×C79C23
hypercenter#2913D79
hypocenterC79#26085
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2913

Complements

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

NH ≅ G/Ntype
1. #23773932C2semidirect
4. #23764913C2semidirect
5. #23764908C2semidirect
6. #23764748C2semidirect
7. #23764743C2semidirect
8. #23764183C2semidirect
9. #23764154C2semidirect
12. #23763567C2semidirect
105. #14579088Q8semidirect
120. #131×C79Q8semidirect
316. C79#26085semidirect
370. #131Dic158semidirect

Maximal subgroups

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

subgroupindex
#237739322
#237732622
#237732512
#237649132
#237649082
#237647482
#237647432
#237641832
#237641542
#237636332
#237636262
#237635672
#237634592
#237632422
C79×#29132
#2608579