Finite Group Databasebeta

← #34258902

Subgroups — #34258902

Order 20224.

622 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#23773993C211yesmaximal, characteristic
2#23772544C211yesmaximal, characteristic, complemented
3#23772477C211yesmaximal, characteristic, complemented
4#23762304C211yesmaximal, characteristic, complemented
5#23762288C211yesmaximal, characteristic, complemented
6#23762283C211yesmaximal, characteristic, complemented
7#23762221C211yesmaximal, characteristic, complemented
8#23762209C211yesmaximal, characteristic, complemented
9#23762199C211yesmaximal, characteristic, complemented
10D8×C79⋊C8C211yesmaximal, characteristic, complemented
11#23762164C211yesmaximal, characteristic, complemented
12#23761857C211yesmaximal, characteristic, complemented
13#23761849C211yesmaximal, characteristic, complemented
14#23761679C211yesmaximal, characteristic, complemented
15C79×#1594C211yesmaximal, characteristic, complemented
16#19099110C2211yescharacteristic
17#19099095C2211yescharacteristic
18#19098971C411yescharacteristic
19#19098951C411yescharacteristic
20#19098926C411yescharacteristic
21#19098921C411yescharacteristic
22#19098815C411yescharacteristic
23#19098813C411yescharacteristic
24#19098810C411yescharacteristic
25#19098808C411yescharacteristic
26#19098693C2211yescharacteristic
27#19098687C2211yescharacteristic
28C4×#11929613C2211yescharacteristic
29C4×#11929612C2211yescharacteristic
30C4×#11929609C2211yescharacteristic
31#19098642C2211yescharacteristic
32#19098639—21no—
33#19098632—21no—
34#19098631—21no—
35#19098631—21no—
36D4×C79⋊C8C2211yescharacteristic
37D4×C79⋊C8C2211yescharacteristic
38#19098628C2211yescharacteristic
39#19098621—21no—
40#19098619C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098591—21no—
43#19098585C2211yescharacteristic
44#19098531—21no—
45#19098528—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098293C2211yescharacteristic
53#19098292C2211yescharacteristic
54D4⋊(C79⋊C8)C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C4×D8×C79C2211yescharacteristic
61#436×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64D4⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929613C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929607—21no—
72C2×#11929607—21no—
73C2×#11929607C2×C411yescharacteristic
74C2×#11929607C2×C411yescharacteristic
75C22×C79⋊C8—21no—
76C22×C79⋊C8—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79#14579160C2×C411yescharacteristic
80#14579157C2×C411yescharacteristic
81#14579154—21no—
82#14579152C2×C411yescharacteristic
83#14579148C2×C411yescharacteristic
84#14579147—21no—
85#14579143—21no—
86#14579141C2×C411yescharacteristic
87#14579136C2×C411yescharacteristic
88#14579134—21no—
89#14579132C2×C411yescharacteristic
90#14579132C2×C411yescharacteristic
91C4×Dic158C2311yescharacteristic
92C4×D316C2311yescharacteristic
93C4×C22⋊D79—21no—
94C4×C22⋊D79—21no—
95#14579120—21no—
96C42×D79C2311yescharacteristic
97#14579094C2×C411yescharacteristic
98#14579093C2×C411yescharacteristic
99#14579092C2×C411yescharacteristic
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579084C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111#14579082C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114C22⋊(C79⋊C8)—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8D411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120#14579071—21no—
121#14579071—21no—
122#14579071C2×C411yescharacteristic
123#14579071C2×C411yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×D8×C79C2×C411yescharacteristic
130C2×M16×C79—21no—
131C2×M16×C79—21no—
132C4×D4×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#107×C79C2×C411yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#102×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—81no—
145#11929624—81no—
146C2×Dic158—21no—
147C2×Dic158C22×C411yescharacteristic
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×C22⋊D79—21no—
151C2×C22⋊D79—21no—
152C2×C22⋊D79—21no—
153C2×C22⋊D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79C22×C411yescharacteristic
156C2×C4×D79C2×D411yescharacteristic
157C2×C4×D79#4111yescharacteristic
158#11929613—22no—
159#11929612—22no—
160#11929609—22no—
161#11929607—22no—
162#11929607—22no—
163#11929607—81no—
164#11929607—81no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—81no—
169C2×C79⋊C8—81no—
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C22×C411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C2×C79⋊C8C2×D411yescharacteristic
174C8×D79—42no—
175C8×D79—81no—
176C4⋊Dic79—21no—
177C4⋊Dic79—21no—
178C4⋊Dic79C22×C411yescharacteristic
179C4⋊Dic79C22×C411yescharacteristic
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183C22⋊Dic79—21no—
184C22⋊Dic79—21no—
185C22⋊Dic79—21no—
186C22⋊Dic79—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190C4×Dic79—21no—
191C4×Dic79C22×C411yescharacteristic
192C4×Dic79C2×D411yescharacteristic
193C4×Dic79#4111yescharacteristic
194C2×D4×C79C22×C411yescharacteristic
195C2×D4×C79C22×C411yescharacteristic
196C22×C4×C79—21no—
197C22×C4×C79—21no—
198D8×C79—22no—
199M16×C79—81no—
200M16×C79—81no—
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×C79C22×C411yescharacteristic
209C4⋊C4×C79C22×C411yescharacteristic
210C22⋊C4×C79—21no—
211C22⋊C4×C79—21no—
212C42×C79C2411yescharacteristic
213C23×C79—21no—
214C23×C79—21no—
215C22×D79—21no—
216C22×D79C4×D411yescharacteristic
217D4×C79—21no—
218D4×C79—21no—
219D4×C79—21no—
220D4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C22⋊D79—81no—
231C22⋊D79—81no—
232C22⋊D79—81no—
233C22⋊D79—81no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79C4×D411yescharacteristic
238C2×Dic79C4×D411yescharacteristic
239C2×Dic79C4×D411yescharacteristic
240D316—21no—
241D316—21no—
242D316—81no—
243C4×D79—21no—
244C4×D79—21no—
245C4×D79—21no—
246C4×D79—21no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250Dic158—21no—
251Dic158—21no—
252Dic158—81no—
253C8×C79—21no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—41no—
257C8×C79—41no—
258C8×C79—41no—
259C8×C79#13112yes—
260C79⋊C8—22no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—21no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C79⋊C8#13112yes—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79—41no—
273C22×C79C2×C4×D411yescharacteristic
274C2×D79—21no—
275C2×D79—21no—
276C2×D79—21no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C4×C79—21no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79C2×#13111yescharacteristic
284C4×C79C2×#13111yescharacteristic
285C4×C79#57611yescharacteristic
286C4×C79#57711yescharacteristic
287Dic79—21no—
288Dic79—21no—
289Dic79—21no—
290Dic79—41no—
291Dic79—41no—
292Dic79—41no—
293#17968—791nomaximal, Sylow
294C2×C79—81no—
295C2×C79—81no—
296C2×C79#293111yescharacteristic
297C2×C79#294611yescharacteristic
298C2×C79#295611yescharacteristic
299D79—41no—
300D79—81no—
301#3171—791no—
302#2993—791no—
303#2981—791no—
304#1594D7911yescharacteristic, complemented
305#1592—791no—
306#1590—791no—
307#1589—791no—
308#1580—791no—
309SD16⋊C8—791no—
310C8×SD16—791no—
311C8×D8—791no—
312#1569—791no—
313#1494—791no—
314#1492—791no—
315C8×M16—791no—
316C79#1796811yescharacteristic, complemented, Sylow
317#532—791no—
318#529—791no—
319#501—791no—
320#496—791no—
321#490—791no—
322#489—791no—
323#469—791no—
324#469—791no—
325#468—791no—
326#468—791no—
327#446—791no—
328C8×Q8—791no—
329C4×SD16—791no—
330C4×SD16—791no—
331C4×D8—791no—
332C4×D8C2×D7911yescharacteristic
333#436—791no—
334#436—1581no—
335#436C2×D7911yescharacteristic
336#436C2×D7911yescharacteristic
337#435—1581no—
338#435—1581no—
339#435—1581no—
340C8×D4—791no—
341C8×D4—791no—
342C8×D4—791no—
343#433—1581no—
344#432—791no—
345#431—791no—
346#424—1581no—
347#423—791no—
348#408—1581no—
349#408—1581no—
350C4×M16—791no—
351C2×C4×C8—791no—
352#335—791no—
353#335—791no—
354#334—791no—
355#334C2×D7911yescharacteristic
356Q8⋊C8—791no—
357Q8⋊C8—791no—
358D4⋊C8—791no—
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8C2×D7911yescharacteristic
363D4⋊C8C2×D7911yescharacteristic
364#322—791no—
365#322C2×D7911yescharacteristic
366C82—791no—
367C82—791no—
368C2×#41—1581no—
369C2×SD16—791no—
370C2×SD16—791no—
371C2×D8—791no—
372C2×D8C4×D7911yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—791no—
376C2×M16—791no—
377C2×M16—1581no—
378C2×M16—1581no—
379C22×C8—1581no—
380C22×C8—1581no—
381C22×C8—1581no—
382C22×C8—1581no—
383C4⋊Q8—791no—
384#127—791no—
385#126—1581no—
386#125—791no—
387#124—791no—
388#124—1581no—
389#123—1581no—
390C22⋊Q8—791no—
391#121—791no—
392#121—791no—
393#121—791no—
394#121—1581no—
395C4×Q8—791no—
396C4×D4—791no—
397C4×D4—1581no—
398C4×D4—1581no—
399C4×D4C22×D7911yescharacteristic
400C4×D4C22×D7911yescharacteristic
401#117—1581no—
402C2×C42—791no—
403#107—791no—
404#107C4×D7911yescharacteristic
405#106—791no—
406#106—791no—
407C4⋊C8—791no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—1581no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8C22×D7911yescharacteristic
415C4⋊C8C22×D7911yescharacteristic
416#103—791no—
417#103—791no—
418#102—791no—
419#102—791no—
420#102—791no—
421#102—791no—
422#102C4×D7911yescharacteristic
423#102C4×D7911yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—791no—
427C22⋊C8—791no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8—1581no—
432C22⋊C8—1581no—
433#97—791no—
434#97—791no—
435#97—1581no—
436#97—1581no—
437C4×C8—791no—
438C4×C8—791no—
439C4×C8—791no—
440C4×C8—1581no—
441C4×C8—1581no—
442C4×C8C22×D7911yescharacteristic
443C4×C8C22×D7911yescharacteristic
444C4×C8C22×D7911yescharacteristic
445#41—6321no—
446#41—6321no—
447C2×Q8—791no—
448C2×Q8—1581no—
449C2×D4—791no—
450C2×D4—1581no—
451C2×D4—1581no—
452C2×D4—1581no—
453C2×D4—1581no—
454C2×D4—1581no—
455C2×D4C2×C4×D7911yescharacteristic
456C2×D4C2×C4×D7911yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—791no—
462C22×C4—1581no—
463SD16—1582no—
464SD16—1582no—
465D8—22no—
466D8—1582no—
467M16—81no—
468M16—81no—
469M16—6321no—
470M16—6321no—
471M16—1582no—
472M16—1582no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—3162no—
477C2×C8—791no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—6321no—
482C2×C8—6321no—
483C2×C8—6321no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8C2×C4×D7911yescharacteristic
488C2×C8C2×C4×D7911yescharacteristic
489C2×C8D4×D7911yescharacteristic
490C2×C8D4×D7911yescharacteristic
491C4⋊C4—791no—
492C4⋊C4—791no—
493C4⋊C4—1581no—
494C4⋊C4—1581no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4C2×C4×D7911yescharacteristic
499C4⋊C4C2×C4×D7911yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C42—791no—
510C42—791no—
511C42—791no—
512C42—1581no—
513C42C23×D7911yescharacteristic
514C23—21no—
515C23—21no—
516C23—791no—
517C23—1581no—
518Q8—6321no—
519Q8—1581no—
520Q8—1581no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—6321no—
526D4—6321no—
527D4—6321no—
528D4—6321no—
529D4—6321no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—791no—
542C2×C4—791no—
543C2×C4—791no—
544C2×C4—1581no—
545C2×C4—1581no—
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—792no—
563C8—1582no—
564C8—1581no—
565C8—1581no—
566C8—1581no—
567C8#1457916912yes—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—1581no—
578C22—1581no—
579C22—1581no—
580C22C4×D4×D7911yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—3161no—
585C4—3161no—
586C4—3161no—
587C4—1581no—
588C4—1581no—
589C4—1581no—
590C4#131×D7911yescharacteristic
591C4C2×#1457916911yescharacteristic
592C4#1909936711yescharacteristic
593C4#1909937311yescharacteristic
594C2—81no—
595C2—81no—
596C2—3161no—
597C2—6321no—
598C2#2377210911yescharacteristic
599C2#2377222911yescharacteristic
600C2#2377230811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1594C2
socleC22×C79C2×C4×D4
prime radical#23773993C2
cosocleC42×C79C24
hypercenter#1594D79
hypocenterC79#17968
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1594

Complements

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

NH ≅ G/Ntype
2. #23772544C2semidirect
3. #23772477C2semidirect
4. #23762304C2semidirect
5. #23762288C2semidirect
6. #23762283C2semidirect
7. #23762221C2semidirect
8. #23762209C2semidirect
9. #23762199C2semidirect
10. D8×C79⋊C8C2semidirect
11. #23762164C2semidirect
12. #23761857C2semidirect
13. #23761849C2semidirect
14. #23761679C2semidirect
15. C79×#1594C2semidirect
304. #1594D79semidirect
316. C79#17968semidirect

Maximal subgroups

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

subgroupindex
#237739932
#237725442
#237724772
#237623042
#237622882
#237622832
#237622212
#237622092
#237621992
D8×C79⋊C82
#237621642
#237618572
#237618492
#237616792
C79×#15942
#1796879