Finite Group Databasebeta

← #34270750

Subgroups — #34270750

Order 20224.

620 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#23773982C211yesmaximal, characteristic, complemented
2#23773471C211yesmaximal, characteristic
3#41×C79⋊C8C211yesmaximal, characteristic
4#23762832C211yesmaximal, characteristic, complemented
5#23762828C211yesmaximal, characteristic, complemented
6#23762825C211yesmaximal, characteristic, complemented
7#23762821C211yesmaximal, characteristic, complemented
8#23762792C211yesmaximal, characteristic, complemented
9#23762787C211yesmaximal, characteristic, complemented
10#23762768C211yesmaximal, characteristic, complemented
11#23762763C211yesmaximal, characteristic, complemented
12#23762146C211yesmaximal, characteristic, complemented
13#23762065C211yesmaximal, characteristic, complemented
14#23762064C211yesmaximal, characteristic, complemented
15C79×#3107C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098966C2211yescharacteristic
18#19098942C2211yescharacteristic
19#19098915C2211yescharacteristic
20#19098913C2211yescharacteristic
21#19098904—21no—
22#19098848C2211yescharacteristic
23#19098843C2211yescharacteristic
24#19098802—21no—
25#19098802—21no—
26#19098773C2211yescharacteristic
27#19098768C2211yescharacteristic
28#19098725—21no—
29Q8×C79⋊C8C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31C4×#11929614—21no—
32C4×#11929614—21no—
33C4×#11929614—21no—
34C4×#11929612—21no—
35D4×C79⋊C8—21no—
36D4×C79⋊C8—21no—
37D4×C79⋊C8C2211yescharacteristic
38D4×C79⋊C8C2211yescharacteristic
39#19098616—21no—
40#19098616C2211yescharacteristic
41#19098616C2211yescharacteristic
42C2×C4×C79⋊C8—21no—
43C2×C4×C79⋊C8C2211yescharacteristic
44#19098315C2211yescharacteristic
45#19098315C2211yescharacteristic
46#19098313C2211yescharacteristic
47#19098313C2211yescharacteristic
48#19098310C2211yescharacteristic
49#19098310C2211yescharacteristic
50#19098309C2211yescharacteristic
51#19098309C2211yescharacteristic
52#19098309C2211yescharacteristic
53#19098309C2211yescharacteristic
54#19098308C2211yescharacteristic
55#19098308C2211yescharacteristic
56#533×C79C2211yescharacteristic
57C4×#41×C79C2211yescharacteristic
58#485×C79—21no—
59#475×C79C2211yescharacteristic
60#474×C79C2211yescharacteristic
61#462×C79C2211yescharacteristic
62#461×C79C2211yescharacteristic
63#452×C79—21no—
64C4×Q16×C79—21no—
65C4×SD16×C79—21no—
66#431×C79C2211yescharacteristic
67C2×#11929614—21no—
68C2×#11929614—21no—
69C2×#11929614—21no—
70C2×#11929612—21no—
71C22×C79⋊C8—21no—
72C22×C79⋊C8—21no—
73C22×C79⋊C8D411yescharacteristic
74C22×C79⋊C8D411yescharacteristic
75#14579161C2311yescharacteristic
76#14579161C2311yescharacteristic
77#14579153—21no—
78#14579140—21no—
79#14579140—21no—
80C4×Dic158—21no—
81C4×Dic158—21no—
82#14579120—21no—
83#14579094—21no—
84#14579094—21no—
85#14579094—21no—
86#14579092—21no—
87C4⋊(C79⋊C8)—21no—
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)C2311yescharacteristic
90C4⋊(C79⋊C8)C2311yescharacteristic
91C4⋊(C79⋊C8)C2311yescharacteristic
92C4⋊(C79⋊C8)C2311yescharacteristic
93#14579085—21no—
94#14579085—21no—
95#14579085—21no—
96#14579084—21no—
97#14579084—21no—
98#14579084—21no—
99#14579084—21no—
100#14579082—21no—
101C22⋊(C79⋊C8)—21no—
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)—21no—
105C4×C79⋊C8—21no—
106C4×C79⋊C8—21no—
107C4×C79⋊C8—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8D411yescharacteristic
110C4×C79⋊C8D411yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C2×#41×C79D411yescharacteristic
114C2×Q16×C79—21no—
115C2×SD16×C79—21no—
116C4⋊Q8×C79C2311yescharacteristic
117C4⋊Q8×C79C2311yescharacteristic
118#125×C79—21no—
119C22⋊Q8×C79—21no—
120C22⋊Q8×C79—21no—
121C4×Q8×C79—21no—
122C4×Q8×C79—21no—
123C4×Q8×C79D411yescharacteristic
124C4×Q8×C79C2311yescharacteristic
125C4×D4×C79—21no—
126C4×D4×C79—21no—
127C4×D4×C79D411yescharacteristic
128C4×D4×C79C2311yescharacteristic
129#117×C79—21no—
130#117×C79—21no—
131#117×C79D411yescharacteristic
132C2×C42×C79—21no—
133C2×C42×C79C2311yescharacteristic
134#107×C79—21no—
135#106×C79—21no—
136C4⋊C8×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138#103×C79—21no—
139#103×C79—21no—
140#103×C79—21no—
141#102×C79—21no—
142C22⋊C8×C79—21no—
143C4×C8×C79—21no—
144C2×Dic158—21no—
145C2×Dic158—21no—
146#11929614—42no—
147#11929614—42no—
148#11929614—42no—
149#11929612—42no—
150C2×C79⋊C8—22no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—41no—
156C2×C79⋊C8—42no—
157C2×C79⋊C8—42no—
158C2×C79⋊C8C2×D411yescharacteristic
159C2×C79⋊C8C2×D411yescharacteristic
160C2×C79⋊C8C2×D411yescharacteristic
161C2×C79⋊C8C2×D411yescharacteristic
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166Dic79⋊C4—41no—
167Dic79⋊C4—41no—
168C22⋊Dic79—41no—
169C4×Dic79—41no—
170#41×C79—22no—
171#41×C79—22no—
172C2×Q8×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79C2×D411yescharacteristic
175C2×D4×C79—21no—
176C2×D4×C79C2×D411yescharacteristic
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79#4111yescharacteristic
183Q16×C79—42no—
184SD16×C79—42no—
185C2×C8×C79—21no—
186C2×C8×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—41no—
201C42×C79—21no—
202C42×C79—21no—
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—41no—
213Q8×C79—41no—
214Q8×C79—41no—
215Q8×C79—41no—
216Q8×C79#13512yes—
217D4×C79—24no—
218D4×C79—21no—
219D4×C79#13512yes—
220C2×C4×C79—24no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—22no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—42no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79#12111yescharacteristic
235C2×C4×C79#12111yescharacteristic
236C2×C4×C79#12111yescharacteristic
237C2×C4×C79C22×D411yescharacteristic
238C2×C4×C79C22×D411yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×Dic79—41no—
241C2×Dic79—41no—
242Dic158—41no—
243Dic158—41no—
244Dic158—41no—
245Dic158—41no—
246C8×C79—41no—
247C8×C79—41no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—41no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C79⋊C8#13512yes—
256C22×C79—21no—
257C22×C79—21no—
258C22×C79—22no—
259C22×C79—21no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×#12111yescharacteristic
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—22no—
266C4×C79—21no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79—81no—
270C4×C79—81no—
271C4×C79#47312yes—
272C4×C79C2×#13511yescharacteristic
273C4×C79C2×#13511yescharacteristic
274C4×C79C2×#13511yescharacteristic
275C4×C79#57511yescharacteristic
276Dic79—81no—
277Dic79—81no—
278#19129—791nomaximal, Sylow
279C2×C79—41no—
280C2×C79—42no—
281C2×C79#303511yescharacteristic
282C2×C79#305011yescharacteristic
283C2×C79C2×#47311yescharacteristic
284#3170—791no—
285#3107—791no—
286#3107D7911yescharacteristic
287C8×#41—791no—
288#1680—791no—
289#1680—791no—
290#1679—791no—
291#1679—791no—
292#1674—791no—
293#1673—791no—
294#1670—791no—
295#1669—791no—
296#1564—791no—
297#1544—791no—
298#1544—791no—
299C79#1912911yescharacteristic, complemented, Sylow
300#533—791no—
301#533C2×D7911yescharacteristic
302C4×#41C2×D7911yescharacteristic
303#500—791no—
304#494—791no—
305#487—791no—
306#487—791no—
307#485—21no—
308#485—1581no—
309#475—791no—
310#475C2×D7911yescharacteristic
311#474—791no—
312#474C2×D7911yescharacteristic
313#467—1581no—
314#467—1581no—
315#462—791no—
316#462C2×D7911yescharacteristic
317#461—791no—
318#461C2×D7911yescharacteristic
319#452—21no—
320#452—1581no—
321C8×Q8—791no—
322C8×Q8—791no—
323C4×Q16—21no—
324C4×Q16—1581no—
325C4×Q16—1581no—
326C4×Q16—1581no—
327C4×SD16—21no—
328C4×SD16—1581no—
329C8×D4—791no—
330C8×D4—791no—
331C8×D4—1581no—
332C8×D4—1581no—
333#431—791no—
334#431—791no—
335#431—1581no—
336#431C2×D7911yescharacteristic
337C2×C4×C8—791no—
338C2×C4×C8—1581no—
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—
351C2×#41C22⋊D7911yescharacteristic
352C2×Q16—21no—
353C2×Q16—1581no—
354C2×Q16—1581no—
355C2×Q16—1581no—
356C2×SD16—21no—
357C2×SD16—1581no—
358C22×C8—791no—
359C22×C8—791no—
360C22×C8—1581no—
361C22×C8—1581no—
362C4⋊Q8—791no—
363C4⋊Q8—791no—
364C4⋊Q8C22×D7911yescharacteristic
365C4⋊Q8C22×D7911yescharacteristic
366#125—21no—
367#125—1581no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8—1581no—
371C22⋊Q8—1581no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—1581no—
375C4×Q8—1581no—
376C4×Q8C22⋊D7911yescharacteristic
377C4×Q8C22×D7911yescharacteristic
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22⋊D7911yescharacteristic
381C4×D4C22×D7911yescharacteristic
382#117—21no—
383#117—21no—
384#117—1581no—
385#117C22⋊D7911yescharacteristic
386C2×C42—21no—
387C2×C42C22×D7911yescharacteristic
388#107—21no—
389#107—1581no—
390#107—1581no—
391#107—1581no—
392#106—21no—
393#106—1581no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8C22×D7911yescharacteristic
401C4⋊C8C22×D7911yescharacteristic
402#103—21no—
403#103—21no—
404#103—21no—
405#103—1581no—
406#103—1581no—
407#103—1581no—
408#103—1581no—
409#103—1581no—
410#103—1581no—
411#103—1581no—
412#102—21no—
413#102—1581no—
414C22⋊C8—21no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C4×C8—21no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—791no—
424C4×C8—1581no—
425C4×C8—1581no—
426C4×C8—1581no—
427C4×C8—1581no—
428#41—22no—
429#41—22no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—1581no—
433C2×Q8—1581no—
434C2×Q8C2×C22⋊D7911yescharacteristic
435C2×D4—21no—
436C2×D4C2×C22⋊D7911yescharacteristic
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4D4×D7911yescharacteristic
441C22×C4C2×C22⋊D7911yescharacteristic
442C22×C4#1192962311yescharacteristic
443Q16—42no—
444Q16—3162no—
445Q16—3162no—
446Q16—3162no—
447SD16—42no—
448SD16—3162no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—3162no—
452C2×C8—3162no—
453C2×C8—3161no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1582no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—3161no—
472C4⋊C4—3161no—
473C4⋊C4—1581no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4C2×C22⋊D7911yescharacteristic
478C4⋊C4C2×C22⋊D7911yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—41no—
483C22⋊C4—3161no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—41no—
488C42—3161no—
489C42D4×D7911yescharacteristic
490C42C2×C22⋊D7911yescharacteristic
491C42#1192962311yescharacteristic
492C42C23×D7911yescharacteristic
493C23—21no—
494C23#1457913311yescharacteristic
495Q8—21no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—3161no—
501Q8—3161no—
502Q8—3161no—
503Q8—3161no—
504Q8#1457918112yes—
505D4—24no—
506D4—21no—
507D4#1457918112yes—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—3161no—
523C2×C4—3161no—
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—792no—
537C8—1581no—
538C8—1581no—
539C8—1581no—
540C22—21no—
541C22—22no—
542C22—21no—
543C22—21no—
544C22—41no—
545C22—41no—
546C22C2×#1457913311yescharacteristic
547C4—21no—
548C4—22no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—81no—
554C4—81no—
555C4—6321no—
556C4—6321no—
557C4#1909883612yes—
558C4#135×D7911yescharacteristic
559C4C2×#1457918111yescharacteristic
560C4C2×#1457918111yescharacteristic
561C4#1909936011yescharacteristic
562C2—41no—
563C2—42no—
564C2#2377285111yescharacteristic
565C2#2377297811yescharacteristic
566C2C2×#1909883611yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC79×#3107C2
socleC22×C79C2×#121
prime radicalC2×#41×C79D4
hypercenter#3107D79
hypocenterC79#19129
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
1. #23773982C2semidirect
4. #23762832C2semidirect
5. #23762828C2semidirect
6. #23762825C2semidirect
7. #23762821C2semidirect
8. #23762792C2semidirect
9. #23762787C2semidirect
10. #23762768C2semidirect
11. #23762763C2semidirect
12. #23762146C2semidirect
13. #23762065C2semidirect
14. #23762064C2semidirect
299. C79#19129semidirect

Maximal subgroups

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

subgroupindex
#237739822
#237734712
#41×C79⋊C82
#237628322
#237628282
#237628252
#237628212
#237627922
#237627872
#237627682
#237627632
#237621462
#237620652
#237620642
C79×#31072
#1912979