Finite Group Databasebeta

← #34258807

Subgroups — #34258807

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773966C211yesmaximal, characteristic
2#23772377C211yesmaximal, characteristic, complemented
3#23772372C211yesmaximal, characteristic, complemented
4#23762294C211yesmaximal, characteristic, complemented
5#23762291C211yesmaximal, characteristic, complemented
6#23762289C211yesmaximal, characteristic, complemented
7#23762282C211yesmaximal, characteristic, complemented
8C8×#11929614C211yesmaximal, characteristic, complemented
9C8×#11929613C211yesmaximal, characteristic, complemented
10C8×#11929609C211yesmaximal, characteristic, complemented
11#23762160C211yesmaximal, characteristic, complemented
12#23761859C211yesmaximal, characteristic, complemented
13#23761857C211yesmaximal, characteristic, complemented
14C82×D79C211yesmaximal, characteristic, complemented
15C8×SD16×C79C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17#19099095C2211yescharacteristic
18#19098963C411yescharacteristic
19#19098944C411yescharacteristic
20#19098919C411yescharacteristic
21#19098916C411yescharacteristic
22#19098803C411yescharacteristic
23#19098798C411yescharacteristic
24#19098793C411yescharacteristic
25#19098789C411yescharacteristic
26#19098694C2211yescharacteristic
27#19098690C2211yescharacteristic
28C8×Dic158C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098641C2211yescharacteristic
33#19098640C2211yescharacteristic
34#19098637—21no—
35#19098633—21no—
36C8×D316—21no—
37C8×D316C2211yescharacteristic
38C8×C22⋊D79—21no—
39#19098623—21no—
40#19098621—21no—
41#19098619C2211yescharacteristic
42#19098619C2211yescharacteristic
43#19098518—21no—
44#19098517—21no—
45C4×C8×D79C2211yescharacteristic
46C4×C8×D79C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098314C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57C8×C79⋊C8C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929613C2×C411yescharacteristic
69C2×#11929609C2×C411yescharacteristic
70C2×#11929608—21no—
71C2×#11929607—21no—
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74C2×C8×D79C2×C411yescharacteristic
75C2×C8×D79C2×C411yescharacteristic
76C2×C8×D79D411yescharacteristic
77C2×C8×D79D411yescharacteristic
78#14579156—21no—
79#14579154—21no—
80#14579152C2×C411yescharacteristic
81#14579152C2×C411yescharacteristic
82#14579150C2×C411yescharacteristic
83#14579148C2×C411yescharacteristic
84#14579146—21no—
85#14579143—21no—
86#14579141C2×C411yescharacteristic
87#14579137C2×C411yescharacteristic
88#14579136C2×C411yescharacteristic
89#14579132C2×C411yescharacteristic
90C4×Dic158C2311yescharacteristic
91C4×D316—21no—
92C4×D316C2311yescharacteristic
93C4×C22⋊D79—21no—
94#14579120—21no—
95#14579118—21no—
96C42×D79C2311yescharacteristic
97#14579094C2×C411yescharacteristic
98#14579094C2×C411yescharacteristic
99#14579093C2×C411yescharacteristic
100C4⋊(C79⋊C8)—21no—
101C4⋊(C79⋊C8)C2311yescharacteristic
102C4⋊(C79⋊C8)C2311yescharacteristic
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105#14579086C2×C411yescharacteristic
106#14579085C2×C411yescharacteristic
107#14579084C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579082C2×C411yescharacteristic
110#14579081C2×C411yescharacteristic
111C22⋊(C79⋊C8)—21no—
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116#14579072—21no—
117#14579071—21no—
118#14579071—21no—
119#14579071—21no—
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C8×Dic79C2×C411yescharacteristic
126C8×Dic79C2×C411yescharacteristic
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
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
144C2×Dic158C22×C411yescharacteristic
145C2×D316—21no—
146C2×D316C22×C411yescharacteristic
147C2×C22⋊D79—21no—
148C2×C4×D79—21no—
149C2×C4×D79C22×C411yescharacteristic
150C2×C4×D79C2×D411yescharacteristic
151C2×C4×D79#4111yescharacteristic
152#11929614—22no—
153#11929613—22no—
154#11929609—22no—
155#11929608—42no—
156#11929607—42no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C8×D79—24no—
165C8×D79—22no—
166C8×D79—22no—
167C8×D79—22no—
168C8×D79—22no—
169C8×D79—42no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79—21no—
173C4⋊Dic79C22×C411yescharacteristic
174C4⋊Dic79C22×C411yescharacteristic
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188C4×Dic79—21no—
189C4×Dic79C22×C411yescharacteristic
190C4×Dic79C2×D411yescharacteristic
191C4×Dic79#4111yescharacteristic
192C2×Q8×C79C22×C411yescharacteristic
193C2×D4×C79C22×C411yescharacteristic
194C22×C4×C79—21no—
195SD16×C79—22no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—42no—
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C4⋊C4×C79—21no—
205C4⋊C4×C79C22×C411yescharacteristic
206C4⋊C4×C79C22×C411yescharacteristic
207C22⋊C4×C79—21no—
208C42×C79—21no—
209C42×C79C2411yescharacteristic
210C23×C79—21no—
211C22×D79—21no—
212C22×D79C4×D411yescharacteristic
213Q8×C79—21no—
214Q8×C79—21no—
215D4×C79—21no—
216D4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C22⋊D79—42no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79C4×D411yescharacteristic
230C2×Dic79C4×D411yescharacteristic
231C2×Dic79C4×D411yescharacteristic
232D316—21no—
233D316—21no—
234D316—42no—
235C4×D79—22no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—41no—
241C4×D79—41no—
242Dic158—21no—
243Dic158—21no—
244C8×C79—22no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—41no—
248C8×C79—41no—
249C8×C79#13112yes—
250C8×C79#13512yes—
251C79⋊C8—22no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C79⋊C8#13112yes—
257C79⋊C8#13512yes—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79C2×C4×D411yescharacteristic
262C2×D79—21no—
263C2×D79—21no—
264C2×D79—41no—
265C2×D79—41no—
266C2×D79—41no—
267C2×D79#44312yes—
268C4×C79—21no—
269C4×C79—41no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79C2×#13111yescharacteristic
273C4×C79C2×#13111yescharacteristic
274C4×C79C2×#13511yescharacteristic
275C4×C79C2×#13511yescharacteristic
276Dic79—21no—
277Dic79—21no—
278Dic79—21no—
279Dic79—41no—
280Dic79—41no—
281Dic79—41no—
282#17959—791nomaximal, Sylow
283C2×C79—81no—
284C2×C79#293111yescharacteristic
285C2×C79C4×#13511yescharacteristic
286C2×C79C2×#44311yescharacteristic
287D79—22no—
288D79—81no—
289#3168—791no—
290#2966—791no—
291#2966—791no—
292#1591—791no—
293#1590—791no—
294#1590—791no—
295#1589—791no—
296C8×Q16—791no—
297C8×SD16—791no—
298C8×SD16D7911yescharacteristic, complemented
299C8×D8—791no—
300#1568—791no—
301#1494—791no—
302#1494—791no—
303C2×C82—791no—
304C79#1795911yescharacteristic, complemented, Sylow
305#529—791no—
306#529—791no—
307#499—791no—
308#495—791no—
309#488—791no—
310#488—791no—
311#467—791no—
312#466—791no—
313#465—791no—
314#465—791no—
315#446—791no—
316#446—791no—
317C8×Q8—791no—
318C8×Q8C2×D7911yescharacteristic
319C4×Q16—791no—
320C4×SD16—791no—
321C4×SD16C2×D7911yescharacteristic
322C4×D8—791no—
323#436—791no—
324#436—791no—
325#435—1581no—
326#435—1581no—
327C8×D4—791no—
328C8×D4—1581no—
329C8×D4—1581no—
330C8×D4C2×D7911yescharacteristic
331#433—1581no—
332#433—1581no—
333#432—791no—
334#432—791no—
335#405—1581no—
336#405—1581no—
337C2×C4×C8—791no—
338C2×C4×C8—791no—
339C2×C4×C8—791no—
340#335—791no—
341#335—791no—
342#334—791no—
343#334C2×D7911yescharacteristic
344Q8⋊C8—791no—
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8C2×D7911yescharacteristic
348D4⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8C2×D7911yescharacteristic
352C82—791no—
353C82—791no—
354C82—791no—
355C82C2×D7911yescharacteristic
356C2×Q16—791no—
357C2×SD16—791no—
358C2×SD16C4×D7911yescharacteristic
359C2×D8—791no—
360C2×M16—1581no—
361C2×M16—1581no—
362C22×C8—21no—
363C22×C8—791no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—791no—
367C22×C8—1581no—
368C22×C8—1581no—
369#126—1581no—
370#126—1581no—
371#125—791no—
372#125—791no—
373#124—791no—
374#124—791no—
375#123—1581no—
376#123—1581no—
377C22⋊Q8—791no—
378C22⋊Q8—791no—
379#121—791no—
380#121—791no—
381C4×Q8—791no—
382C4×Q8C22×D7911yescharacteristic
383C4×D4—791no—
384C4×D4—1581no—
385C4×D4—1581no—
386C4×D4C22×D7911yescharacteristic
387#117—1581no—
388#117—1581no—
389C2×C42—791no—
390#107—791no—
391#107—791no—
392#106—791no—
393#106C4×D7911yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8C22×D7911yescharacteristic
401C4⋊C8C22×D7911yescharacteristic
402#103—791no—
403#103—791no—
404#103—791no—
405#103C4×D7911yescharacteristic
406#102—791no—
407#102—791no—
408#102—791no—
409#102C4×D7911yescharacteristic
410C22⋊C8—21no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413C22⋊C8—1581no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416#97—1581no—
417#97—1581no—
418#97—1581no—
419#97—1581no—
420C4×C8—21no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8C22×D7911yescharacteristic
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C2×Q8—791no—
434C2×Q8C2×C4×D7911yescharacteristic
435C2×D4—791no—
436C2×D4—1581no—
437C2×D4—1581no—
438C2×D4C2×C4×D7911yescharacteristic
439C22×C4—21no—
440C22×C4—791no—
441C22×C4—791no—
442C22×C4—791no—
443C22×C4—1581no—
444Q16—1582no—
445SD16—22no—
446SD16—1582no—
447D8—1582no—
448M16—3162no—
449M16—3162no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—3162no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—1582no—
460C2×C8—1582no—
461C2×C8—1584no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1582no—
466C2×C8—1582no—
467C2×C8C2×C4×D7911yescharacteristic
468C2×C8C2×C4×D7911yescharacteristic
469C2×C8D4×D7911yescharacteristic
470C2×C8D4×D7911yescharacteristic
471C4⋊C4—21no—
472C4⋊C4—791no—
473C4⋊C4—791no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C42—21no—
494C42—791no—
495C42—791no—
496C42—791no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—791no—
501C23—1581no—
502Q8—21no—
503Q8—21no—
504Q8—1581no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—3162no—
509D4—3162no—
510D4—1581no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—3161no—
518C2×C4—3161no—
519C2×C4—791no—
520C2×C4—791no—
521C2×C4—791no—
522C2×C4—1582no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4C22×C4×D7911yescharacteristic
531C2×C4C2×D4×D7911yescharacteristic
532C2×C4#41×D7911yescharacteristic
533C8—22no—
534C8—21no—
535C8—21no—
536C8—41no—
537C8—41no—
538C8—3161no—
539C8—3161no—
540C8—792no—
541C8—792no—
542C8—1582no—
543C8—1581no—
544C8—1581no—
545C8#1457916712yes—
546C8#1457918212yes—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—3161no—
551C22—3161no—
552C22—3161no—
553C22—792no—
554C22—1581no—
555C22—1581no—
556C22C4×D4×D7911yescharacteristic
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—3161no—
562C4—3161no—
563C4—3161no—
564C4—1581no—
565C4—1581no—
566C4—1581no—
567C4#131×D7911yescharacteristic
568C4C2×#1457916711yescharacteristic
569C4#135×D7911yescharacteristic
570C4C2×#1457918211yescharacteristic
571C2—81no—
572C2—6321no—
573C2—1582no—
574C2#2377210711yescharacteristic
575C2C4×#1457918211yescharacteristic
576C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C8D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×SD16×C79C2
socleC22×C79C2×C4×D4
prime radical#19098798C4
cosocleC42×C79C24
hypercenterC8×SD16D79
hypocenterC79#17959
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #23772377C2semidirect
3. #23772372C2semidirect
4. #23762294C2semidirect
5. #23762291C2semidirect
6. #23762289C2semidirect
7. #23762282C2semidirect
8. C8×#11929614C2semidirect
9. C8×#11929613C2semidirect
10. C8×#11929609C2semidirect
11. #23762160C2semidirect
12. #23761859C2semidirect
13. #23761857C2semidirect
14. C82×D79C2semidirect
15. C8×SD16×C79C2semidirect
298. C8×SD16D79semidirect
304. C79#17959semidirect

Maximal subgroups

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

subgroupindex
#237739662
#237723772
#237723722
#237622942
#237622912
#237622892
#237622822
C8×#119296142
C8×#119296132
C8×#119296092
#237621602
#237618592
#237618572
C82×D792
C8×SD16×C792
#1795979