Finite Group Databasebeta

← #34259468

Subgroups — #34259468

Order 20224.

600 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773476C211yesmaximal, characteristic
2#23772560C211yesmaximal, characteristic
3#23772490C211yesmaximal, characteristic, complemented
4#23762294C211yesmaximal, characteristic, complemented
5#23762293C211yesmaximal, characteristic, complemented
6#23762281C211yesmaximal, characteristic, complemented
7#23762274C211yesmaximal, characteristic, complemented
8#23762268C211yesmaximal, characteristic, complemented
9#23762254C211yesmaximal, characteristic, complemented
10#23762204C211yesmaximal, characteristic, complemented
11#23762203C211yesmaximal, characteristic, complemented
12#23762186C211yesmaximal, characteristic, complemented
13C2×#19098295C211yesmaximal, characteristic, complemented
14#23761683C211yesmaximal, characteristic, complemented
15C79×#1494C211yesmaximal, characteristic
16C2×C4×Dic158C2211yescharacteristic
17#19098891C411yescharacteristic
18#19098874C411yescharacteristic
19#19098867C411yescharacteristic
20#19098867C411yescharacteristic
21#19098782C411yescharacteristic
22#19098782C411yescharacteristic
23#19098718C411yescharacteristic
24#19098701C411yescharacteristic
25#19098694C2211yescharacteristic
26#19098691—21no—
27#19098691—21no—
28#19098690C2211yescharacteristic
29#19098690C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31#19098687C2211yescharacteristic
32#19098687C2211yescharacteristic
33C4×#11929614C2211yescharacteristic
34C4×#11929614C2211yescharacteristic
35C4×#11929612C2211yescharacteristic
36C4×#11929612C2211yescharacteristic
37#19098641C2211yescharacteristic
38#19098635—21no—
39D4×C79⋊C8—21no—
40D4×C79⋊C8C2211yescharacteristic
41#19098624—21no—
42#19098617—21no—
43#19098616C2211yescharacteristic
44#19098616C2211yescharacteristic
45#19098613—21no—
46#19098590—21no—
47C2×C4⋊(C79⋊C8)C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098295C2211yescharacteristic
55#19098295C2211yescharacteristic
56#19098284C2211yescharacteristic
57#19098284C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#529×C79C2211yescharacteristic
61#432×C79C2211yescharacteristic
62C2×C4×C8×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66D4⋊C8×C79C2211yescharacteristic
67C22×Dic158C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929614C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929612C2×C411yescharacteristic
72C2×#11929608—21no—
73C22×C79⋊C8—21no—
74C22×C79⋊C8—21no—
75C4×Dic158—21no—
76C4×Dic158—21no—
77C4×Dic158C2×C411yescharacteristic
78C4×Dic158C2×C411yescharacteristic
79C4×Dic158C2311yescharacteristic
80C4×Dic158C2311yescharacteristic
81C2×C4⋊Dic79C2×C411yescharacteristic
82C2×Dic79⋊C4—21no—
83C2×C4×Dic79—21no—
84#14579094C2×C411yescharacteristic
85#14579094C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88C4⋊(C79⋊C8)—21no—
89C4⋊(C79⋊C8)—21no—
90C4⋊(C79⋊C8)—21no—
91C4⋊(C79⋊C8)C2×C411yescharacteristic
92C4⋊(C79⋊C8)C2×C411yescharacteristic
93C4⋊(C79⋊C8)D411yescharacteristic
94C4⋊(C79⋊C8)D411yescharacteristic
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101#14579085C2×C411yescharacteristic
102#14579085C2×C411yescharacteristic
103#14579084C2×C411yescharacteristic
104#14579084C2×C411yescharacteristic
105#14579084C2×C411yescharacteristic
106#14579084C2×C411yescharacteristic
107#14579082C2×C411yescharacteristic
108#14579082C2×C411yescharacteristic
109C22⋊(C79⋊C8)—21no—
110C22⋊(C79⋊C8)—21no—
111C22⋊(C79⋊C8)C2×C411yescharacteristic
112C22⋊(C79⋊C8)C2×C411yescharacteristic
113C22⋊(C79⋊C8)D411yescharacteristic
114C22⋊(C79⋊C8)D411yescharacteristic
115#14579072—21no—
116#14579071—21no—
117#14579071—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8C2311yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C22×C8×C79—21no—
126#126×C79—21no—
127#125×C79C2×C411yescharacteristic
128#124×C79C2×C411yescharacteristic
129#123×C79—21no—
130C22⋊Q8×C79C2×C411yescharacteristic
131#121×C79C2×C411yescharacteristic
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79—21no—
134C4×D4×C79C2311yescharacteristic
135#117×C79—21no—
136C2×C42×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—22no—
145C2×Dic158—21no—
146C2×Dic158—41no—
147C2×Dic158C22×C411yescharacteristic
148C2×Dic158C22×C411yescharacteristic
149C22×Dic79—21no—
150C22×Dic79—21no—
151#11929614—22no—
152#11929614—41no—
153#11929612—22no—
154#11929612—41no—
155#11929608—81no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8—81no—
159C2×C79⋊C8—81no—
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C4⋊Dic79—21no—
169C4⋊Dic79C22×C411yescharacteristic
170C4⋊Dic79C22×C411yescharacteristic
171Dic79⋊C4—21no—
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175C4×Dic79—21no—
176C4×Dic79—21no—
177C4×Dic79—21no—
178C4×Dic79—21no—
179C2×Q8×C79C22×C411yescharacteristic
180C2×D4×C79—21no—
181C2×D4×C79C22×C411yescharacteristic
182C22×C4×C79—21no—
183C22×C4×C79C22×C411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79#4111yescharacteristic
186C2×C8×C79—24no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79—21no—
190C2×C8×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79C22×C411yescharacteristic
196C4⋊C4×C79C22×C411yescharacteristic
197C22⋊C4×C79—21no—
198C22⋊C4×C79—21no—
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C22⋊C4×C79—21no—
202C42×C79—21no—
203C42×C79C22×C411yescharacteristic
204C42×C79C2×D411yescharacteristic
205C42×C79#4111yescharacteristic
206C42×C79C2411yescharacteristic
207C23×C79—21no—
208C23×C79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211D4×C79—21no—
212D4×C79—21no—
213D4×C79—42no—
214C2×C4×C79—22no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—42no—
235C2×Dic79—42no—
236Dic158—21no—
237Dic158—21no—
238Dic158—21no—
239Dic158—42no—
240Dic158#13112yes—
241C8×C79—22no—
242C8×C79—22no—
243C8×C79—41no—
244C8×C79—41no—
245C79⋊C8—21no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C22×C79—21no—
256C22×C79—21no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79#44312yes—
261C22×C79C2×C4×D411yescharacteristic
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—21no—
265C4×C79—21no—
266C4×C79—41no—
267C4×C79—41no—
268C4×C79—41no—
269C4×C79C2×#13111yescharacteristic
270C4×C79#56811yescharacteristic
271C4×C79#57711yescharacteristic
272C4×C79#57811yescharacteristic
273Dic79—22no—
274Dic79—41no—
275Dic79—41no—
276Dic79—41no—
277#18017—791nomaximal, Sylow
278C2×C79—22no—
279C2×C79—81no—
280C2×C79#293311yescharacteristic
281C2×C79#294711yescharacteristic
282C2×C79C2×#44311yescharacteristic
283#3108—791no—
284#2994—791no—
285#2984—791no—
286#1591—791no—
287#1590—791no—
288#1588—791no—
289#1587—791no—
290#1587—791no—
291#1585—791no—
292C8×Q16—791no—
293C8×SD16—791no—
294#1573—791no—
295#1494D7911yescharacteristic
296C2×Q8⋊C8—791no—
297#1450—791no—
298C79#1801711yescharacteristic, complemented, Sylow
299#529C2×D7911yescharacteristic
300C2×C4×Q8—791no—
301#483—791no—
302#480—791no—
303#479—791no—
304#479—791no—
305#464—791no—
306#464—791no—
307#451—791no—
308#448—791no—
309#446—791no—
310#446—791no—
311#446—791no—
312#446—1581no—
313#446—1581no—
314C8×Q8—791no—
315C8×Q8—791no—
316C8×Q8—791no—
317C4×Q16—791no—
318C4×Q16—791no—
319C4×SD16—791no—
320C4×SD16—791no—
321#436—791no—
322#435—1581no—
323C8×D4—791no—
324C8×D4—1581no—
325#433—1581no—
326#432—1581no—
327#432C2×D7911yescharacteristic
328#431—791no—
329#431—791no—
330#431—1581no—
331#424—1581no—
332C2×C4⋊C8—791no—
333C2×C4×C8C2×D7911yescharacteristic
334#335—791no—
335#335—791no—
336#334—791no—
337#334—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342Q8⋊C8C2×D7911yescharacteristic
343Q8⋊C8C2×D7911yescharacteristic
344D4⋊C8C2×D7911yescharacteristic
345D4⋊C8C2×D7911yescharacteristic
346#322—791no—
347#322—791no—
348C82—791no—
349C82—791no—
350C22×Q8—791no—
351C2×Q16—791no—
352C2×Q16—791no—
353C2×SD16—791no—
354C2×SD16—791no—
355C2×M16—1581no—
356C22×C8—21no—
357C22×C8—1581no—
358C22×C8—1581no—
359#126—21no—
360#125C4×D7911yescharacteristic
361#124C4×D7911yescharacteristic
362#123—21no—
363C22⋊Q8C4×D7911yescharacteristic
364#121C4×D7911yescharacteristic
365C4×Q8—791no—
366C4×Q8—791no—
367C4×Q8—791no—
368C4×Q8—791no—
369C4×Q8—1581no—
370C4×Q8—1581no—
371C4×Q8C22×D7911yescharacteristic
372C4×D4—21no—
373C4×D4C22×D7911yescharacteristic
374#117—21no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—1581no—
377C2×C42—1581no—
378C2×C42C22×D7911yescharacteristic
379#107—791no—
380#107—791no—
381#106—791no—
382#106—791no—
383C4⋊C8—791no—
384C4⋊C8—791no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8—1581no—
395C4⋊C8—1581no—
396C4⋊C8C22×D7911yescharacteristic
397C4⋊C8C22×D7911yescharacteristic
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#103—791no—
403#103—791no—
404#102—791no—
405#102—791no—
406C22⋊C8—21no—
407C22⋊C8—791no—
408C22⋊C8—791no—
409C22⋊C8—791no—
410C22⋊C8—791no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413#97—21no—
414#97—1581no—
415#97—1581no—
416#97—1581no—
417C4×C8—21no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—791no—
421C4×C8—791no—
422C4×C8—1581no—
423C4×C8—1581no—
424C4×C8—1581no—
425C4×C8C22×D7911yescharacteristic
426C4×C8C22×D7911yescharacteristic
427C2×Q8—3161no—
428C2×Q8—791no—
429C2×Q8—791no—
430C2×Q8—1582no—
431C2×Q8—1581no—
432C2×Q8C2×C4×D7911yescharacteristic
433C2×D4—21no—
434C2×D4C2×C4×D7911yescharacteristic
435C22×C4—21no—
436C22×C4—1581no—
437C22×C4—1581no—
438C22×C4C2×C4×D7911yescharacteristic
439C22×C4D4×D7911yescharacteristic
440C22×C4#1192962311yescharacteristic
441Q16—3161no—
442Q16—1582no—
443SD16—3161no—
444SD16—1582no—
445M16—6321no—
446C2×C8—24no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—791no—
452C2×C8—791no—
453C2×C8—791no—
454C2×C8—791no—
455C2×C8—791no—
456C2×C8—791no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—6321no—
460C2×C8—6321no—
461C2×C8—1581no—
462C2×C8—1581no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—791no—
468C4⋊C4—791no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4C2×C4×D7911yescharacteristic
475C4⋊C4C2×C4×D7911yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—1581no—
483C42—1581no—
484C42—1581no—
485C42—1581no—
486C42C2×C4×D7911yescharacteristic
487C42D4×D7911yescharacteristic
488C42#1192962311yescharacteristic
489C42C23×D7911yescharacteristic
490C23—21no—
491C23(C22⋊D79)⋊C411yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—3162no—
495Q8—792no—
496Q8—1581no—
497Q8—1581no—
498Q8—1581no—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—3162no—
513C2×C4—3162no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4—1581no—
518C2×C4(C22⋊D79)⋊C411yescharacteristic
519C2×C4(C22⋊D79)⋊C411yescharacteristic
520C2×C4(C22⋊D79)⋊C411yescharacteristic
521C2×C4C22×C4×D7911yescharacteristic
522C2×C4C2×D4×D7911yescharacteristic
523C2×C4C2×#1192962311yescharacteristic
524C8—22no—
525C8—22no—
526C8—41no—
527C8—41no—
528C8—3161no—
529C8—3161no—
530C8—1581no—
531C8—1581no—
532C8—1581no—
533C8—1581no—
534C8—1581no—
535C8—1581no—
536C8—1581no—
537C8—1581no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1909867712yes—
544C22C2×(C22⋊D79)⋊C411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—3161no—
553C4—3161no—
554C4—3161no—
555C4—1582no—
556C4#131×D7911yescharacteristic
557C4#1909932811yescharacteristic
558C4#1909937011yescharacteristic
559C4#1909937711yescharacteristic
560C2—22no—
561C2—81no—
562C2#2377213011yescharacteristic
563C2#2377224611yescharacteristic
564C2C2×#1909867711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1494C2
socleC22×C79C2×C4×D4
prime radical#121×C79C2×C4
cosocleC42×C79C24
hypercenter#1494D79
hypocenterC79#18017
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
3. #23772490C2semidirect
4. #23762294C2semidirect
5. #23762293C2semidirect
6. #23762281C2semidirect
7. #23762274C2semidirect
8. #23762268C2semidirect
9. #23762254C2semidirect
10. #23762204C2semidirect
11. #23762203C2semidirect
12. #23762186C2semidirect
13. C2×#19098295C2semidirect
14. #23761683C2semidirect
298. C79#18017semidirect

Maximal subgroups

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

subgroupindex
#237734762
#237725602
#237724902
#237622942
#237622932
#237622812
#237622742
#237622682
#237622542
#237622042
#237622032
#237621862
C2×#190982952
#237616832
C79×#14942
#1801779