Finite Group Databasebeta

← #34258903

Subgroups — #34258903

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#23773995C211yesmaximal, characteristic
2#23772551C211yesmaximal, characteristic, complemented
3#23772475C211yesmaximal, characteristic, complemented
4#23762315C211yesmaximal, characteristic, complemented
5#23762304C211yesmaximal, characteristic, complemented
6#23762292C211yesmaximal, characteristic, complemented
7#23762286C211yesmaximal, characteristic, complemented
8#23762225C211yesmaximal, characteristic, complemented
9#23762214C211yesmaximal, characteristic, complemented
10C8×#11929613C211yesmaximal, characteristic, complemented
11C8×D632C211yesmaximal, characteristic, complemented
12#23762167C211yesmaximal, characteristic, complemented
13#23761861C211yesmaximal, characteristic, complemented
14#23761852C211yesmaximal, characteristic, complemented
15C8×M16×C79C211yesmaximal, characteristic, complemented
16#19099113C2211yescharacteristic
17#19099097C2211yescharacteristic
18#19098972C411yescharacteristic
19#19098952C411yescharacteristic
20#19098928C411yescharacteristic
21#19098923C411yescharacteristic
22#19098816C411yescharacteristic
23#19098816C411yescharacteristic
24#19098809C411yescharacteristic
25#19098809C411yescharacteristic
26#19098693C2211yescharacteristic
27C8×Dic158C2211yescharacteristic
28C4×#11929613C2211yescharacteristic
29C4×#11929613C2211yescharacteristic
30C4×D632C2211yescharacteristic
31C4×D632C2211yescharacteristic
32#19098642C2211yescharacteristic
33#19098642C2211yescharacteristic
34#19098642C2211yescharacteristic
35#19098638—21no—
36#19098637—21no—
37#19098633—21no—
38#19098633—21no—
39C8×D316C2211yescharacteristic
40C8×D316C2211yescharacteristic
41C8×D316C2211yescharacteristic
42#19098623—21no—
43#19098620C2211yescharacteristic
44#19098588—21no—
45#19098587C2211yescharacteristic
46#19098528—21no—
47#19098528—21no—
48#19098316C2211yescharacteristic
49#19098316C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098314C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098292C2211yescharacteristic
57#19098292C2211yescharacteristic
58#19098292C2211yescharacteristic
59#19098292C2211yescharacteristic
60#431×C79C2211yescharacteristic
61C4×M16×C79C2211yescharacteristic
62C2×C4×C8×C79C2211yescharacteristic
63#322×C79C2211yescharacteristic
64#322×C79C2211yescharacteristic
65C82×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929624—21no—
68C2×#11929613C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×D632C2×C411yescharacteristic
71C2×D632C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C2×#11929607—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79#14579161C2×C411yescharacteristic
80#14579158C2×C411yescharacteristic
81#14579155—21no—
82#14579153C2×C411yescharacteristic
83#14579149—21no—
84#14579149C2×C411yescharacteristic
85#14579144—21no—
86#14579140C2×C411yescharacteristic
87#14579135—21no—
88#14579135C2×C411yescharacteristic
89#14579135C2×C411yescharacteristic
90#14579135C2×C411yescharacteristic
91C4×Dic158C2311yescharacteristic
92C4×D316—21no—
93C4×D316C2311yescharacteristic
94C4×D316C2311yescharacteristic
95C4×D316C2311yescharacteristic
96C4×C22⋊D79—21no—
97#14579118—21no—
98#14579095C2×C411yescharacteristic
99#14579095C2×C411yescharacteristic
100#14579093C2×C411yescharacteristic
101#14579093C2×C411yescharacteristic
102C4⋊(C79⋊C8)—21no—
103C4⋊(C79⋊C8)C2311yescharacteristic
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106C4⋊(C79⋊C8)C2311yescharacteristic
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109#14579086C2×C411yescharacteristic
110#14579086C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579083C2×C411yescharacteristic
113#14579083C2×C411yescharacteristic
114#14579083C2×C411yescharacteristic
115#14579083C2×C411yescharacteristic
116#14579083C2×C411yescharacteristic
117C22⋊(C79⋊C8)—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121(C2×D79)⋊C8—21no—
122(C2×D79)⋊C8—21no—
123(C2×D79)⋊C8—21no—
124#14579072—21no—
125#14579071—21no—
126C8×Dic79—21no—
127C2×M16×C79C2×C411yescharacteristic
128C2×M16×C79C2×C411yescharacteristic
129C22×C8×C79—21no—
130C2×C42×C79C2311yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79D411yescharacteristic
133C22⋊C8×C79D411yescharacteristic
134C22⋊C8×C79D411yescharacteristic
135#97×C79C2×C411yescharacteristic
136#97×C79C2×C411yescharacteristic
137C4×C8×C79—21no—
138C4×C8×C79C2311yescharacteristic
139C4×C8×C79C2311yescharacteristic
140C4×C8×C79C2311yescharacteristic
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×D316—21no—
150C2×D316C22×C411yescharacteristic
151C2×D316C22×C411yescharacteristic
152C2×D316C22×C411yescharacteristic
153C2×C22⋊D79—21no—
154C2×C22⋊D79—21no—
155C2×C22⋊D79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159#11929613—22no—
160#11929613—22no—
161D632—22no—
162D632—22no—
163#11929608—81no—
164#11929607—81no—
165#11929607—81no—
166#11929607—81no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8—21no—
170C2×C79⋊C8—21no—
171C8×D79—81no—
172C8×D79—81no—
173C8×D79—81no—
174C4⋊Dic79—21no—
175C4⋊Dic79C22×C411yescharacteristic
176C4⋊Dic79C22×C411yescharacteristic
177C4⋊Dic79C22×C411yescharacteristic
178C4⋊Dic79C22×C411yescharacteristic
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183C22⋊Dic79—21no—
184(C2×D79)⋊C4—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×Dic79—21no—
193C22×C4×C79C22×C411yescharacteristic
194C22×C4×C79C2×D411yescharacteristic
195C22×C4×C79#4111yescharacteristic
196M16×C79—22no—
197M16×C79—22no—
198C2×C8×C79—21no—
199C2×C8×C79—21no—
200C2×C8×C79—42no—
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C2×C8×C79C2×D411yescharacteristic
208C2×C8×C79C2×D411yescharacteristic
209C42×C79C22×C411yescharacteristic
210C42×C79C2×D411yescharacteristic
211C42×C79#4111yescharacteristic
212C42×C79C2411yescharacteristic
213C23×C79C4×D411yescharacteristic
214C22×D79—21no—
215C22×D79—21no—
216C22×D79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—41no—
222C2×C4×C79C4×D411yescharacteristic
223C2×C4×C79C4×D411yescharacteristic
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C23×C411yescharacteristic
226C2×C4×C79C22×D411yescharacteristic
227C2×C4×C79C2×#4111yescharacteristic
228C22⋊D79—81no—
229C22⋊D79—81no—
230C22⋊D79—81no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236D316—21no—
237D316—21no—
238D316—21no—
239D316—21no—
240D316—21no—
241D316—21no—
242D316—81no—
243D316—81no—
244C4×D79—41no—
245C4×D79—41no—
246C4×D79—41no—
247C4×D79—41no—
248C4×D79—41no—
249C4×D79—41no—
250Dic158—21no—
251Dic158—21no—
252Dic158—81no—
253C8×C79—22no—
254C8×C79—21no—
255C8×C79—21no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—21no—
260C8×C79—41no—
261C8×C79#13112yes—
262C8×C79#13112yes—
263C79⋊C8—41no—
264C79⋊C8—41no—
265C79⋊C8—41no—
266C79⋊C8—41no—
267C22×C79—21no—
268C22×C79—21no—
269C22×C79—21no—
270C22×C79C2×C4×D411yescharacteristic
271C2×D79—41no—
272C2×D79—41no—
273C2×D79—41no—
274C2×D79—41no—
275C2×D79—41no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C4×C79—21no—
281C4×C79—21no—
282C4×C79—21no—
283C4×C79—21no—
284C4×C79C2×#13111yescharacteristic
285C4×C79C2×#13111yescharacteristic
286C4×C79#57611yescharacteristic
287C4×C79#57711yescharacteristic
288Dic79—41no—
289Dic79—41no—
290Dic79—41no—
291Dic79—41no—
292Dic79—41no—
293#17968—791nomaximal, Sylow
294C2×C79—41no—
295C2×C79#293111yescharacteristic
296C2×C79#294611yescharacteristic
297C2×C79#295611yescharacteristic
298D79—81no—
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×SD16—791no—
311C8×D8—791no—
312#1569—791no—
313#1494—791no—
314#1492—791no—
315C8×M16D7911yescharacteristic, complemented
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×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×D4—791no—
343#433—1581no—
344#432—791no—
345#431C2×D7911yescharacteristic
346#424—1581no—
347#423—791no—
348#408—1581no—
349#408—1581no—
350C4×M16C2×D7911yescharacteristic
351C2×C4×C8C2×D7911yescharacteristic
352#335—791no—
353#335—791no—
354#334—791no—
355#334—791no—
356Q8⋊C8—791no—
357Q8⋊C8—791no—
358D4⋊C8—791no—
359D4⋊C8—791no—
360D4⋊C8—791no—
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364#322C2×D7911yescharacteristic
365#322C2×D7911yescharacteristic
366C82C2×D7911yescharacteristic
367C82C2×D7911yescharacteristic
368C2×#41—1581no—
369C2×SD16—791no—
370C2×SD16—791no—
371C2×D8—791no—
372C2×D8—791no—
373C2×M16—1581no—
374C2×M16—1581no—
375C2×M16—1581no—
376C2×M16—1581no—
377C2×M16C4×D7911yescharacteristic
378C2×M16C4×D7911yescharacteristic
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×Q8—791no—
396C4×D4—791no—
397C4×D4—791no—
398C4×D4—791no—
399C4×D4—1581no—
400C4×D4—1581no—
401#117—1581no—
402C2×C42C22×D7911yescharacteristic
403#107—791no—
404#107—791no—
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⋊C8D31611yescharacteristic
415C4⋊C8D31611yescharacteristic
416#103—791no—
417#103—791no—
418#102—791no—
419#102—791no—
420#102—791no—
421#102—791no—
422#102—791no—
423#102—791no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427C22⋊C8—1581no—
428C22⋊C8—1581no—
429C22⋊C8—1581no—
430C22⋊C8—1581no—
431C22⋊C8D31611yescharacteristic
432C22⋊C8D31611yescharacteristic
433#97—1581no—
434#97—1581no—
435#97C4×D7911yescharacteristic
436#97C4×D7911yescharacteristic
437C4×C8—21no—
438C4×C8—1581no—
439C4×C8C22×D7911yescharacteristic
440C4×C8C22×D7911yescharacteristic
441C4×C8C22×D7911yescharacteristic
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—791no—
451C2×D4—791no—
452C2×D4—1581no—
453C2×D4—1581no—
454C2×D4—1581no—
455C2×D4—1581no—
456C2×D4—1581no—
457C22×C4—1581no—
458C22×C4—1581no—
459C22×C4—1581no—
460C22×C4C2×C4×D7911yescharacteristic
461C22×C4C2×D31611yescharacteristic
462C22×C4#1192962411yescharacteristic
463SD16—1582no—
464SD16—1582no—
465D8—1582no—
466D8—1582no—
467M16—22no—
468M16—22no—
469M16—6321no—
470M16—6321no—
471M16—6321no—
472M16—6321no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—42no—
476C2×C8—6321no—
477C2×C8—6321no—
478C2×C8—6321no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8C2×C4×D7911yescharacteristic
484C2×C8C2×C4×D7911yescharacteristic
485C2×C8C2×C4×D7911yescharacteristic
486C2×C8C2×C4×D7911yescharacteristic
487C2×C8C2×D31611yescharacteristic
488C2×C8C2×D31611yescharacteristic
489C2×C8C2×D31611yescharacteristic
490C2×C8C2×D31611yescharacteristic
491C4⋊C4—791no—
492C4⋊C4—791no—
493C4⋊C4—791no—
494C4⋊C4—791no—
495C4⋊C4—1581no—
496C4⋊C4—1581no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C22⋊C4—1581no—
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—1581no—
510C42C2×C4×D7911yescharacteristic
511C42C2×D31611yescharacteristic
512C42#1192962411yescharacteristic
513C42C23×D7911yescharacteristic
514C23—1581no—
515C23—1581no—
516C23—1581no—
517C23C4×D31611yescharacteristic
518Q8—6321no—
519Q8—1581no—
520Q8—1581no—
521D4—6321no—
522D4—6321no—
523D4—6321no—
524D4—6321no—
525D4—6321no—
526D4—1581no—
527D4—1581no—
528D4—1581no—
529D4—1581no—
530D4—1581no—
531D4—1581no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4C4×D31611yescharacteristic
549C2×C4C4×D31611yescharacteristic
550C2×C4C4×D31611yescharacteristic
551C2×C4C22×C4×D7911yescharacteristic
552C2×C4C22×D31611yescharacteristic
553C2×C4C2×#1192962411yescharacteristic
554C8—22no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—3161no—
563C8—3161no—
564C8—3161no—
565C8—3161no—
566C8#1457916712yes—
567C8#1457916712yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—3161no—
572C22—3161no—
573C22—3161no—
574C22—3161no—
575C22—3161no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—3161no—
580C22C2×C4×D31611yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—3161no—
586C4—3161no—
587C4—3161no—
588C4—3161no—
589C4—3161no—
590C4C2×#1457916711yescharacteristic
591C4C2×#1457916711yescharacteristic
592C4#1909936811yescharacteristic
593C4#1909937511yescharacteristic
594C2—41no—
595C2—6321no—
596C2—6321no—
597C2—6321no—
598C2#2377211311yescharacteristic
599C2#2377223411yescharacteristic
600C2#2377231111yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×M16×C79C2
socleC22×C79C2×C4×D4
prime radical#23773995C2
cosocleC42×C79C24
hypercenterC8×M16D79
hypocenterC79#17968
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #23772551C2semidirect
3. #23772475C2semidirect
4. #23762315C2semidirect
5. #23762304C2semidirect
6. #23762292C2semidirect
7. #23762286C2semidirect
8. #23762225C2semidirect
9. #23762214C2semidirect
10. C8×#11929613C2semidirect
11. C8×D632C2semidirect
12. #23762167C2semidirect
13. #23761861C2semidirect
14. #23761852C2semidirect
15. C8×M16×C79C2semidirect
315. C8×M16D79semidirect
316. C79#17968semidirect

Maximal subgroups

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

subgroupindex
#237739952
#237725512
#237724752
#237623152
#237623042
#237622922
#237622862
#237622252
#237622142
C8×#119296132
C8×D6322
#237621672
#237618612
#237618522
C8×M16×C792
#1796879