Finite Group Databasebeta

← #34275877

Subgroups — #34275877

Order 20224.

580 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774548C211yesmaximal, characteristic, complemented
2#23774424C211yesmaximal, characteristic
3#23772560C211yesmaximal, characteristic
4#23762952C211yesmaximal, characteristic, complemented
5#23762945C211yesmaximal, characteristic, complemented
6#23762916C211yesmaximal, characteristic, complemented
7#23762911C211yesmaximal, characteristic, complemented
8#23762828C211yesmaximal, characteristic, complemented
9#23762821C211yesmaximal, characteristic, complemented
10#23762792C211yesmaximal, characteristic, complemented
11#23762787C211yesmaximal, characteristic, complemented
12#23762183C211yesmaximal, characteristic, complemented
13#23762100C211yesmaximal, characteristic, complemented
14C2×#19098310C211yesmaximal, characteristic, complemented
15C79×#3248C211yesmaximal, characteristic
16C2×#14579161C2211yescharacteristic
17#19098966C2211yescharacteristic
18#19098966C2211yescharacteristic
19#19098958—21no—
20#19098933—21no—
21#19098913C2211yescharacteristic
22#19098913C2211yescharacteristic
23#19098896—21no—
24#19098896—21no—
25#19098858—21no—
26#19098858—21no—
27#19098773C2211yescharacteristic
28#19098773C2211yescharacteristic
29#19098768C2211yescharacteristic
30#19098768C2211yescharacteristic
31#19098718—21no—
32#19098701—21no—
33#19098694C2211yescharacteristic
34Q8×C79⋊C8C2211yescharacteristic
35#19098641C2211yescharacteristic
36#19098635—21no—
37D4×C79⋊C8—21no—
38D4×C79⋊C8C2211yescharacteristic
39#19098624—21no—
40#19098620C2211yescharacteristic
41#19098616C2211yescharacteristic
42#19098590—21no—
43C2×C4⋊(C79⋊C8)C2211yescharacteristic
44#19098315C2211yescharacteristic
45#19098315C2211yescharacteristic
46#19098313C2211yescharacteristic
47#19098313C2211yescharacteristic
48#19098310C2211yescharacteristic
49#19098310C2211yescharacteristic
50#19098310C2211yescharacteristic
51#19098310C2211yescharacteristic
52#19098309C2211yescharacteristic
53#19098309C2211yescharacteristic
54#19098309C2211yescharacteristic
55#19098309C2211yescharacteristic
56#533×C79C2211yescharacteristic
57#529×C79C2211yescharacteristic
58#479×C79—21no—
59#475×C79C2211yescharacteristic
60#474×C79C2211yescharacteristic
61#467×C79—21no—
62#465×C79—21no—
63#464×C79—21no—
64#462×C79C2211yescharacteristic
65#461×C79C2211yescharacteristic
66C2×C4⋊C8×C79C2211yescharacteristic
67C22×Dic158—21no—
68C2×#11929614—21no—
69C2×#11929614—21no—
70C2×#11929612—21no—
71C2×#11929612—21no—
72C2×#11929608—21no—
73C22×C79⋊C8—21no—
74C22×C79⋊C8—21no—
75#14579161—21no—
76#14579161—21no—
77#14579161—21no—
78#14579161C2311yescharacteristic
79#14579161C2311yescharacteristic
80C2×C4⋊Dic79—21no—
81C2×C4⋊Dic79—21no—
82#14579094—21no—
83#14579094—21no—
84#14579094—21no—
85#14579094—21no—
86C4⋊(C79⋊C8)—21no—
87C4⋊(C79⋊C8)—21no—
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)D411yescharacteristic
90C4⋊(C79⋊C8)D411yescharacteristic
91C4⋊(C79⋊C8)C2311yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93C4⋊(C79⋊C8)C2311yescharacteristic
94C4⋊(C79⋊C8)C2311yescharacteristic
95#14579085—21no—
96#14579085—21no—
97#14579085—21no—
98#14579085—21no—
99#14579084—21no—
100#14579084—21no—
101#14579084—21no—
102#14579084—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)—21no—
105C22⋊(C79⋊C8)—21no—
106C22⋊(C79⋊C8)D411yescharacteristic
107C22⋊(C79⋊C8)D411yescharacteristic
108#14579072—21no—
109#14579072—21no—
110C4×C79⋊C8—21no—
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C2×Q16×C79—21no—
114C2×SD16×C79—21no—
115C22×C8×C79—21no—
116C4⋊Q8×C79C2311yescharacteristic
117C4⋊Q8×C79C2311yescharacteristic
118#126×C79—21no—
119#125×C79—21no—
120#125×C79D411yescharacteristic
121#124×C79D411yescharacteristic
122#123×C79—21no—
123C22⋊Q8×C79—21no—
124C22⋊Q8×C79—21no—
125C22⋊Q8×C79D411yescharacteristic
126#121×C79D411yescharacteristic
127C4×Q8×C79—21no—
128C4×Q8×C79—21no—
129C4×Q8×C79C2311yescharacteristic
130C4×D4×C79—21no—
131C4×D4×C79C2311yescharacteristic
132#117×C79—21no—
133#117×C79—21no—
134C2×C42×C79C2311yescharacteristic
135#107×C79—21no—
136#106×C79—21no—
137C4⋊C8×C79—21no—
138C4⋊C8×C79C2311yescharacteristic
139C4⋊C8×C79C2311yescharacteristic
140#103×C79—21no—
141#103×C79—21no—
142#103×C79—21no—
143#102×C79—21no—
144C2×Dic158—21no—
145C2×Dic158—21no—
146C2×Dic158—41no—
147C2×Dic158—81no—
148C2×Dic158—81no—
149C22×Dic79—41no—
150#11929614—81no—
151#11929614—81no—
152#11929612—81no—
153#11929612—81no—
154#11929608—81no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—81no—
160C2×C79⋊C8—81no—
161C2×C79⋊C8C2×D411yescharacteristic
162C2×C79⋊C8C2×D411yescharacteristic
163C2×C79⋊C8C2×D411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79—21no—
168C4⋊Dic79—21no—
169C4⋊Dic79—41no—
170C4⋊Dic79—41no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79C2×D411yescharacteristic
174C2×D4×C79—21no—
175C2×D4×C79C2×D411yescharacteristic
176C22×C4×C79—21no—
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79#4111yescharacteristic
180Q16×C79—81no—
181SD16×C79—81no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—81no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—41no—
194C4⋊C4×C79—41no—
195C4⋊C4×C79C2×D411yescharacteristic
196C4⋊C4×C79C2×D411yescharacteristic
197C22⋊C4×C79—21no—
198C22⋊C4×C79—21no—
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C22⋊C4×C79—21no—
202C22⋊C4×C79—41no—
203C42×C79—21no—
204C42×C79—41no—
205C42×C79C2×D411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79#12111yescharacteristic
211Q8×C79—21no—
212Q8×C79—21no—
213Q8×C79—41no—
214Q8×C79—41no—
215Q8×C79—41no—
216Q8×C79—41no—
217D4×C79—21no—
218D4×C79—21no—
219D4×C79—42no—
220C2×C4×C79—22no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79#12111yescharacteristic
233C2×C4×C79#12111yescharacteristic
234C2×C4×C79#12111yescharacteristic
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C22×D411yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—82no—
241Dic158—41no—
242Dic158—41no—
243Dic158—41no—
244Dic158—41no—
245Dic158—82no—
246C8×C79—41no—
247C8×C79—41no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8—41no—
256C22×C79—21no—
257C22×C79—21no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79#47312yes—
262C22×C79C2×#12111yescharacteristic
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—21no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79—41no—
270C4×C79—81no—
271C4×C79—81no—
272C4×C79#57611yescharacteristic
273C4×C79#57711yescharacteristic
274C4×C79#57711yescharacteristic
275C4×C79#57811yescharacteristic
276Dic79—81no—
277Dic79—81no—
278#19590—791nomaximal, Sylow
279C2×C79—22no—
280C2×C79—81no—
281C2×C79#304611yescharacteristic
282C2×C79#306211yescharacteristic
283C2×C79C2×#47311yescharacteristic
284#3248D7911yescharacteristic
285#3230—791no—
286#3218—791no—
287#2994—791no—
288#1700—791no—
289#1699—791no—
290#1694—791no—
291#1693—791no—
292#1680—791no—
293#1679—791no—
294#1674—791no—
295#1673—791no—
296#1573—791no—
297#1552—791no—
298C2×#332—791no—
299C79#1959011yescharacteristic, complemented, Sylow
300#533C2×D7911yescharacteristic
301C2×C4⋊Q8—791no—
302#529C2×D7911yescharacteristic
303#500—791no—
304#500—791no—
305#497—1581no—
306#491—1581no—
307#487—791no—
308#487—791no—
309#484—1581no—
310#484—1581no—
311#479—21no—
312#477—1581no—
313#477—1581no—
314#475C2×D7911yescharacteristic
315#474C2×D7911yescharacteristic
316#467—21no—
317#465—21no—
318#464—21no—
319#462—791no—
320#462—791no—
321#462C2×D7911yescharacteristic
322#461—791no—
323#461—791no—
324#461C2×D7911yescharacteristic
325#451—1581no—
326#448—1581no—
327#446—791no—
328C8×Q8—791no—
329#436—791no—
330#435—1581no—
331C8×D4—791no—
332C8×D4—1581no—
333#433—1581no—
334#432—791no—
335#431—791no—
336#424—1581no—
337C2×C4⋊C8—791no—
338C2×C4⋊C8C2×D7911yescharacteristic
339#335—791no—
340#335—791no—
341#334—791no—
342#334—791no—
343#332—791no—
344#332—791no—
345#332—791no—
346#332—791no—
347#332—791no—
348#332—791no—
349#332—791no—
350#332—791no—
351C22×Q8—1581no—
352C2×Q16—21no—
353C2×Q16—1581no—
354C2×Q16—1581no—
355C2×SD16—21no—
356C2×SD16—1581no—
357C2×SD16—1581no—
358C2×M16—1581no—
359C22×C8—21no—
360C22×C8—1581no—
361C22×C8—1581no—
362C4⋊Q8—791no—
363C4⋊Q8—791no—
364C4⋊Q8—1581no—
365C4⋊Q8—1581no—
366C4⋊Q8—1581no—
367C4⋊Q8C22×D7911yescharacteristic
368C4⋊Q8C22×D7911yescharacteristic
369#126—21no—
370#125—21no—
371#125C22⋊D7911yescharacteristic
372#124C22⋊D7911yescharacteristic
373#123—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8C22⋊D7911yescharacteristic
377#121C22⋊D7911yescharacteristic
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22×D7911yescharacteristic
381C4×D4—21no—
382C4×D4C22×D7911yescharacteristic
383#117—21no—
384#117—21no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C42C22×D7911yescharacteristic
388#107—21no—
389#107—1581no—
390#107—1581no—
391#107—1581no—
392#107—1581no—
393#106—21no—
394C4⋊C8—21no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8C22×D7911yescharacteristic
405C4⋊C8C22×D7911yescharacteristic
406#103—21no—
407#103—21no—
408#103—21no—
409#103—1581no—
410#103—1581no—
411#103—1581no—
412#103—1581no—
413#103—1581no—
414#103—1581no—
415#103—1581no—
416#103—1581no—
417#102—21no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—1581no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423#97—1581no—
424#97—1581no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—1581no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—3161no—
431C2×Q8—6321no—
432C2×Q8—6321no—
433C2×Q8—1581no—
434C2×Q8—1581no—
435C2×Q8C2×C22⋊D7911yescharacteristic
436C2×D4—21no—
437C2×D4C2×C22⋊D7911yescharacteristic
438C22×C4—21no—
439C22×C4—3161no—
440C22×C4D4×D7911yescharacteristic
441C22×C4C2×C22⋊D7911yescharacteristic
442C22×C4#1192962311yescharacteristic
443Q16—81no—
444Q16—6321no—
445Q16—6321no—
446SD16—81no—
447SD16—6321no—
448SD16—6321no—
449M16—6321no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—81no—
453C2×C8—791no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—6321no—
458C2×C8—6321no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—3161no—
474C4⋊C4—3161no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4C2×C22⋊D7911yescharacteristic
480C4⋊C4C2×C22⋊D7911yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—41no—
487C42—21no—
488C42—41no—
489C42D4×D7911yescharacteristic
490C42C2×C22⋊D7911yescharacteristic
491C42#1192962311yescharacteristic
492C42C23×D7911yescharacteristic
493C23—21no—
494C23#1457913311yescharacteristic
495Q8—21no—
496Q8—21no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—3161no—
502Q8—3161no—
503Q8—3161no—
504Q8—3161no—
505Q8—6322no—
506D4—21no—
507D4—21no—
508D4—42no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—6322no—
524C2×C4#1457913311yescharacteristic
525C2×C4#1457913311yescharacteristic
526C2×C4#1457913311yescharacteristic
527C2×C4C2×D4×D7911yescharacteristic
528C2×C4C22×C22⋊D7911yescharacteristic
529C2×C4C2×#1192962311yescharacteristic
530C8—41no—
531C8—41no—
532C8—3161no—
533C8—3161no—
534C8—3161no—
535C8—3161no—
536C8—1581no—
537C8—1581no—
538C8—1581no—
539C8—1581no—
540C22—21no—
541C22—21no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#1909883612yes—
546C22C2×#1457913311yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4—6321no—
557C4—6321no—
558C4#1909936611yescharacteristic
559C4#1909937011yescharacteristic
560C4#1909937111yescharacteristic
561C4#1909937711yescharacteristic
562C2—22no—
563C2—81no—
564C2#2377294811yescharacteristic
565C2#2377308211yescharacteristic
566C2C2×#1909883611yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3248C2
socleC22×C79C2×#121
prime radical#121×C79D4
hypercenter#3248D79
hypocenterC79#19590
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3248

Complements

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

NH ≅ G/Ntype
1. #23774548C2semidirect
4. #23762952C2semidirect
5. #23762945C2semidirect
6. #23762916C2semidirect
7. #23762911C2semidirect
8. #23762828C2semidirect
9. #23762821C2semidirect
10. #23762792C2semidirect
11. #23762787C2semidirect
12. #23762183C2semidirect
13. #23762100C2semidirect
14. C2×#19098310C2semidirect
299. C79#19590semidirect

Maximal subgroups

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

subgroupindex
#237745482
#237744242
#237725602
#237629522
#237629452
#237629162
#237629112
#237628282
#237628212
#237627922
#237627872
#237621832
#237621002
C2×#190983102
C79×#32482
#1959079