Finite Group Databasebeta

← #34258904

Subgroups — #34258904

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#23773991C211yesmaximal, characteristic
2#23772548C211yesmaximal, characteristic, complemented
3#23772474C211yesmaximal, characteristic, complemented
4#23762313C211yesmaximal, characteristic, complemented
5#23762303C211yesmaximal, characteristic, complemented
6#23762287C211yesmaximal, characteristic, complemented
7#23762282C211yesmaximal, characteristic, complemented
8#23762222C211yesmaximal, characteristic, complemented
9#23762210C211yesmaximal, characteristic, complemented
10C8×D632C211yesmaximal, characteristic, complemented
11#23762164C211yesmaximal, characteristic, complemented
12#23761859C211yesmaximal, characteristic, complemented
13#23761849C211yesmaximal, characteristic, complemented
14C8×#11929607C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099110C2211yescharacteristic
17#19099100C2211yescharacteristic
18#19098970C411yescharacteristic
19#19098949C411yescharacteristic
20#19098926C411yescharacteristic
21#19098924C411yescharacteristic
22#19098817C411yescharacteristic
23#19098815C411yescharacteristic
24#19098812C411yescharacteristic
25#19098806C411yescharacteristic
26#19098694C2211yescharacteristic
27C4×#11929611C2211yescharacteristic
28C4×D632C2211yescharacteristic
29C4×#11929609C2211yescharacteristic
30#19098642—21no—
31#19098641C2211yescharacteristic
32#19098640C2211yescharacteristic
33#19098640C2211yescharacteristic
34#19098637—21no—
35#19098634—21no—
36#19098633—21no—
37C8×D316C2211yescharacteristic
38C8×D316C2211yescharacteristic
39#19098622—21no—
40#19098619C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098588—21no—
43#19098585C2211yescharacteristic
44#19098529—21no—
45#19098528—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51Q8⋊(C79⋊C8)C2211yescharacteristic
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic
54D4⋊(C79⋊C8)C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929611C2×C411yescharacteristic
69C2×D632C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607C2×C411yescharacteristic
76C2×#11929607C2×C411yescharacteristic
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79C2×C8×D79—21no—
80#14579160C2×C411yescharacteristic
81#14579157C2×C411yescharacteristic
82#14579156—21no—
83#14579152C2×C411yescharacteristic
84#14579150C2×C411yescharacteristic
85#14579147—21no—
86#14579146—21no—
87#14579137C2×C411yescharacteristic
88#14579136C2×C411yescharacteristic
89#14579136C2×C411yescharacteristic
90#14579134—21no—
91#14579132C2×C411yescharacteristic
92C4×D316—21no—
93C4×D316C2311yescharacteristic
94C4×D316C2311yescharacteristic
95C4×C22⋊D79—21no—
96#14579118—21no—
97C42×D79C2311yescharacteristic
98#14579095C2×C411yescharacteristic
99#14579094C2×C411yescharacteristic
100#14579092C2×C411yescharacteristic
101C4⋊(C79⋊C8)—21no—
102C4⋊(C79⋊C8)C2311yescharacteristic
103C4⋊(C79⋊C8)C2311yescharacteristic
104Dic79⋊C8—21no—
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579085C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111#14579081C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8D411yescharacteristic
120(C2×D79)⋊C8D411yescharacteristic
121#14579072—21no—
122#14579071—21no—
123#14579071C2×C411yescharacteristic
124#14579071C2×C411yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
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#11929624—81no—
145#11929624—81no—
146C2×Dic158—21no—
147C2×D316—21no—
148C2×D316—21no—
149C2×D316C22×C411yescharacteristic
150C2×D316C22×C411yescharacteristic
151C2×C22⋊D79—21no—
152C2×C22⋊D79—21no—
153C2×C22⋊D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79C22×C411yescharacteristic
157C2×C4×D79C2×D411yescharacteristic
158C2×C4×D79#4111yescharacteristic
159#11929611—22no—
160D632—22no—
161#11929609—22no—
162#11929608—81no—
163#11929607—22no—
164#11929607—22no—
165#11929607—81no—
166#11929607—81no—
167#11929607—81no—
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—42no—
176C8×D79—81no—
177C8×D79—81no—
178C4⋊Dic79—21no—
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184C22⋊Dic79—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192C4×Dic79C22×C411yescharacteristic
193C4×Dic79C2×D411yescharacteristic
194C4×Dic79#4111yescharacteristic
195C2×Q8×C79C22×C411yescharacteristic
196C2×D4×C79C22×C411yescharacteristic
197C22×C4×C79—21no—
198SD16×C79—22no—
199C2×C8×C79—21no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—81no—
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C4⋊C4×C79—21no—
208C4⋊C4×C79C22×C411yescharacteristic
209C4⋊C4×C79C22×C411yescharacteristic
210C22⋊C4×C79—21no—
211C42×C79—21no—
212C42×C79C2411yescharacteristic
213C23×C79—21no—
214C22×D79—21no—
215C22×D79—21no—
216C22×D79C4×D411yescharacteristic
217Q8×C79—21no—
218Q8×C79—21no—
219D4×C79—21no—
220D4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79C23×C411yescharacteristic
227C2×C4×C79C22×D411yescharacteristic
228C2×C4×C79C2×#4111yescharacteristic
229C22⋊D79—81no—
230C22⋊D79—81no—
231C22⋊D79—81no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79C4×D411yescharacteristic
235C2×Dic79C4×D411yescharacteristic
236C2×Dic79C4×D411yescharacteristic
237D316—21no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—81no—
242D316—81no—
243C4×D79—21no—
244C4×D79—21no—
245C4×D79—21no—
246C4×D79—21no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252Dic158—81no—
253C8×C79—22no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—41no—
258C8×C79—41no—
259C8×C79#13112yes—
260C79⋊C8—21no—
261C79⋊C8—21no—
262C79⋊C8—21no—
263C79⋊C8—41no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C79⋊C8#13112yes—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79C2×C4×D411yescharacteristic
271C2×D79—21no—
272C2×D79—21no—
273C2×D79—21no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C4×C79—21no—
281C4×C79—41no—
282C4×C79—41no—
283C4×C79—41no—
284C4×C79C2×#13111yescharacteristic
285C4×C79C2×#13111yescharacteristic
286C4×C79#57611yescharacteristic
287C4×C79#57711yescharacteristic
288Dic79—21no—
289Dic79—21no—
290Dic79—21no—
291Dic79—41no—
292Dic79—41no—
293#17968—791nomaximal, Sylow
294C2×C79—81no—
295C2×C79#293111yescharacteristic
296C2×C79#294611yescharacteristic
297C2×C79#295611yescharacteristic
298D79—41no—
299D79—81no—
300D79—81no—
301#3171—791no—
302#2993—791no—
303#2981—791no—
304#1594—791no—
305#1592—791no—
306#1590—791no—
307#1589—791no—
308#1580—791no—
309SD16⋊C8—791no—
310C8×SD16D7911yescharacteristic, complemented
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×Q8C2×D7911yescharacteristic
329C4×SD16—791no—
330C4×SD16C2×D7911yescharacteristic
331C4×D8—791no—
332C4×D8—791no—
333#436—791no—
334#436—791no—
335#436—791no—
336#436—1581no—
337#435—1581no—
338#435—1581no—
339#435—1581no—
340C8×D4—791no—
341C8×D4—791no—
342C8×D4C2×D7911yescharacteristic
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⋊C8C2×D7911yescharacteristic
358D4⋊C8—791no—
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8C2×D7911yescharacteristic
364#322—791no—
365#322—791no—
366C82—791no—
367C82C2×D7911yescharacteristic
368C2×#41—1581no—
369C2×SD16—791no—
370C2×SD16C4×D7911yescharacteristic
371C2×D8—791no—
372C2×D8—791no—
373C2×M16—791no—
374C2×M16—791no—
375C2×M16—1581no—
376C2×M16—1581no—
377C2×M16—1581no—
378C2×M16—1581no—
379C22×C8—21no—
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×Q8C22×D7911yescharacteristic
396C4×D4—791no—
397C4×D4—791no—
398C4×D4—1581no—
399C4×D4—1581no—
400C4×D4C22×D7911yescharacteristic
401#117—1581no—
402C2×C42—791no—
403#107—791no—
404#107—791no—
405#106—791no—
406#106C4×D7911yescharacteristic
407C4⋊C8—21no—
408C4⋊C8—791no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—791no—
412C4⋊C8—1581no—
413C4⋊C8—1581no—
414C4⋊C8C22×D7911yescharacteristic
415C4⋊C8C22×D7911yescharacteristic
416#103—791no—
417#103C4×D7911yescharacteristic
418#102—791no—
419#102—791no—
420#102—791no—
421#102—791no—
422#102—791no—
423#102C4×D7911yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—791no—
426C22⋊C8—791no—
427C22⋊C8—1581no—
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—21no—
438C4×C8—791no—
439C4×C8—791no—
440C4×C8—791no—
441C4×C8—1581no—
442C4×C8C22×D7911yescharacteristic
443C4×C8C22×D7911yescharacteristic
444C4×C8C22×D7911yescharacteristic
445#41—6321no—
446#41—6321no—
447C2×Q8—1581no—
448C2×Q8C2×C4×D7911yescharacteristic
449C2×D4—791no—
450C2×D4—791no—
451C2×D4—1581no—
452C2×D4—1581no—
453C2×D4—1581no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4C2×C4×D7911yescharacteristic
457C22×C4—21no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—1581no—
462C22×C4—1581no—
463SD16—22no—
464SD16—1582no—
465D8—1582no—
466D8—1582no—
467M16—6321no—
468M16—6321no—
469M16—6321no—
470M16—6321no—
471M16—1582no—
472M16—1582no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8—3162no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—791no—
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—21no—
492C4⋊C4—791no—
493C4⋊C4—791no—
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—1581no—
502C22⋊C4—1581no—
503C22⋊C4—1581no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C42—21no—
510C42—791no—
511C42—791no—
512C42—791no—
513C42C23×D7911yescharacteristic
514C23—21no—
515C23—791no—
516C23—1581no—
517C23—1581no—
518Q8—21no—
519Q8—21no—
520Q8—6321no—
521D4—21no—
522D4—21no—
523D4—6321no—
524D4—6321no—
525D4—6321no—
526D4—6321no—
527D4—6321no—
528D4—1581no—
529D4—1581no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—791no—
543C2×C4—791no—
544C2×C4—791no—
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—22no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—3161no—
561C8—3161no—
562C8—3161no—
563C8—792no—
564C8—1581no—
565C8—1581no—
566C8—1581no—
567C8#1457916712yes—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—3161no—
572C22—3161no—
573C22—3161no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—1581no—
578C22—1581no—
579C22—1581no—
580C22C4×D4×D7911yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—3161no—
586C4—3161no—
587C4—1581no—
588C4—1581no—
589C4—1581no—
590C4#131×D7911yescharacteristic
591C4C2×#1457916711yescharacteristic
592C4#1909936511yescharacteristic
593C4#1909936911yescharacteristic
594C2—81no—
595C2—3161no—
596C2—6321no—
597C2—6321no—
598C2#2377210711yescharacteristic
599C2#2377222711yescharacteristic
600C2#2377230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radical#23773991C2
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#17968
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
2. #23772548C2semidirect
3. #23772474C2semidirect
4. #23762313C2semidirect
5. #23762303C2semidirect
6. #23762287C2semidirect
7. #23762282C2semidirect
8. #23762222C2semidirect
9. #23762210C2semidirect
10. C8×D632C2semidirect
11. #23762164C2semidirect
12. #23761859C2semidirect
13. #23761849C2semidirect
14. C8×#11929607C2semidirect
15. C8×SD16×C79C2semidirect
310. C8×SD16D79semidirect
316. C79#17968semidirect

Maximal subgroups

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

subgroupindex
#237739912
#237725482
#237724742
#237623132
#237623032
#237622872
#237622822
#237622222
#237622102
C8×D6322
#237621642
#237618592
#237618492
C8×#119296072
C8×SD16×C792
#1796879