Finite Group Databasebeta

← #34259610

Subgroups — #34259610

Order 20224.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773406C211yesmaximal, characteristic
2#23772498C211yesmaximal, characteristic, complemented
3#23772432C211yesmaximal, characteristic
4#23762293C211yesmaximal, characteristic, complemented
5#23762279C211yesmaximal, characteristic, complemented
6#23762256C211yesmaximal, characteristic, complemented
7#23762241C211yesmaximal, characteristic, complemented
8#23762234C211yesmaximal, characteristic, complemented
9#23762214C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761871C211yesmaximal, characteristic, complemented
13#23761789C211yesmaximal, characteristic
14#23761701C211yesmaximal, characteristic, complemented
15C79×#1592C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17C4×D4×D79C2211yescharacteristic
18#19098895C411yescharacteristic
19#19098892C411yescharacteristic
20#19098852C411yescharacteristic
21#19098851C411yescharacteristic
22#19098767C411yescharacteristic
23#19098766C411yescharacteristic
24#19098715C411yescharacteristic
25#19098712C411yescharacteristic
26#19098691—21no—
27#19098690C2211yescharacteristic, complemented
28Q8×C79⋊C8C2211yescharacteristic, complemented
29C8×Dic158C2211yescharacteristic, complemented
30#19098687—21no—
31C4×#11929613C2211yescharacteristic
32C4×#11929611C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34#19098641C2211yescharacteristic
35#19098637—21no—
36#19098632—21no—
37C8×D316C2211yescharacteristic
38#19098628C2211yescharacteristic
39#19098625—21no—
40#19098623—21no—
41#19098619C2211yescharacteristic
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098614C2211yescharacteristic
46#19098526—21no—
47C22⋊C8×D79—21no—
48#19098316C2211yescharacteristic, complemented
49#19098313C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51#19098295C2211yescharacteristic, complemented
52Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
53#19098293C2211yescharacteristic, complemented
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic, complemented
58#19098284C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#446×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic, complemented
67C2×D4×D79C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929613C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929611C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×C8×D79—21no—
74C2×C8×D79—21no—
75#14579160C2×C411yescharacteristic
76#14579159C2×C411yescharacteristic
77#14579151—21no—
78#14579141C2×C411yescharacteristic
79#14579139—21no—
80#14579137C2×C411yescharacteristic
81C4×Dic158—21no—
82C4×Dic158C2311yescharacteristic
83C4×D316C2311yescharacteristic
84D4×Dic79C2×C411yescharacteristic
85#14579123C2×C411yescharacteristic
86C4×C22⋊D79—21no—
87(C22⋊D79)⋊C4—21no—
88#14579118—21no—
89C4⋊C4×D79C2×C411yescharacteristic
90C22⋊C4×D79—21no—
91C42×D79C2311yescharacteristic
92#14579093C2×C411yescharacteristic
93#14579092C2×C411yescharacteristic
94#14579092C2×C411yescharacteristic
95C4⋊(C79⋊C8)—21no—
96C4⋊(C79⋊C8)C2311yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98Dic79⋊C8—21no—
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8C2×C411yescharacteristic
102Dic79⋊C8C2×C411yescharacteristic
103Dic79⋊C8D411yescharacteristic
104Dic79⋊C8D411yescharacteristic
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(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8C2×C411yescharacteristic
118(C2×D79)⋊C8C2×C411yescharacteristic
119(C2×D79)⋊C8D411yescharacteristic
120(C2×D79)⋊C8D411yescharacteristic
121#14579071—21no—
122#14579071—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×SD16×C79C2×C411yescharacteristic
130C2×M16×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—
140#97×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C23×D79—21no—
145C2×Dic158—21no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316C22×C411yescharacteristic
148C2×C22⋊D79—21no—
149D4×D79—22no—
150D4×D79—41no—
151C22×Dic79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—81no—
156C2×C4×D79—81no—
157C2×C4×D79C22×C411yescharacteristic
158C2×C4×D79C2×D411yescharacteristic
159C2×C4×D79#4111yescharacteristic
160#11929613—22no—
161#11929612—41no—
162#11929611—22no—
163#11929608—81no—
164C2×C79⋊C8—21no—
165C2×C79⋊C8—21no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C22×C411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C8×D79—81no—
172C8×D79—81no—
173C4⋊Dic79—21no—
174C4⋊Dic79C22×C411yescharacteristic
175C4⋊Dic79C22×C411yescharacteristic
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179Dic79⋊C4—21no—
180Dic79⋊C4—21no—
181Dic79⋊C4—21no—
182C22⋊Dic79—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192C2×D4×C79C22×C411yescharacteristic
193C22×C4×C79—21no—
194SD16×C79—41no—
195M16×C79—81no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79C22×C411yescharacteristic
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C2×D411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C4⋊C4×C79—21no—
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C22⋊C4×C79—21no—
207C42×C79—21no—
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C22×D79—21no—
211C22×D79—42no—
212C22×D79—42no—
213C22×D79—81no—
214C22×D79C4×D411yescharacteristic
215Q8×C79—21no—
216Q8×C79—21no—
217D4×C79—21no—
218D4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79C23×C411yescharacteristic
225C2×C4×C79C22×D411yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C22⋊D79—42no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—81no—
232C2×Dic79C4×D411yescharacteristic
233C2×Dic79C4×D411yescharacteristic
234C2×Dic79C4×D411yescharacteristic
235D316—21no—
236D316#13112yes—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—21no—
241C4×D79—41no—
242C4×D79—41no—
243C4×D79—41no—
244C4×D79—81no—
245C4×D79—81no—
246Dic158—21no—
247Dic158—21no—
248Dic158—42no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—41no—
254C8×C79—41no—
255C8×C79—41no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—21no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79C2×C4×D411yescharacteristic
267C2×D79—22no—
268C2×D79—21no—
269C2×D79—21no—
270C2×D79—21no—
271C2×D79—41no—
272C2×D79—41no—
273C2×D79—84no—
274C4×C79—21no—
275C4×C79—41no—
276C4×C79—41no—
277C4×C79—41no—
278C4×C79C2×#13111yescharacteristic
279C4×C79#56811yescharacteristic
280C4×C79#57711yescharacteristic
281C4×C79#57711yescharacteristic
282Dic79—21no—
283Dic79—21no—
284Dic79—21no—
285Dic79—41no—
286Dic79—41no—
287Dic79—41no—
288#18028—791nomaximal, Sylow
289C2×C79—81no—
290C2×C79#293311yescharacteristic
291C2×C79#294911yescharacteristic
292C2×C79#295511yescharacteristic
293D79—41no—
294D79—42no—
295#3101—791no—
296#2986—791no—
297#2974—791no—
298#1592D7911yescharacteristic, complemented
299#1590—791no—
300#1588—791no—
301#1585—791no—
302#1583—791no—
303#1582—791no—
304SD16⋊C8—791no—
305C8×SD16—791no—
306#1572—791no—
307#1497—791no—
308#1478—791no—
309#1454—791no—
310C79#1802811yescharacteristic, complemented, Sylow
311#533—791no—
312C2×C4×D4—791no—
313#483—791no—
314#483—791no—
315#476—791no—
316#476—791no—
317#461—791no—
318#461—791no—
319#450—791no—
320#450—791no—
321#446—791no—
322#446—1581no—
323#446C2×D7911yescharacteristic, complemented
324C8×Q8—791no—
325C8×Q8—791no—
326C8×Q8—1581no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330C4×SD16C2×D7911yescharacteristic
331#436—791no—
332#436C2×D7911yescharacteristic
333#435—1581no—
334#435—1581no—
335C8×D4—791no—
336C8×D4—791no—
337#433—1581no—
338#433—1581no—
339#432—791no—
340#431—791no—
341#431—791no—
342#431—791no—
343#431—791no—
344#407—1581no—
345C2×C22⋊C8—1581no—
346#335—791no—
347#335C2×D7911yescharacteristic, complemented
348#334—791no—
349#334—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353Q8⋊C8C2×D7911yescharacteristic, complemented
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8C2×D7911yescharacteristic
358#322—791no—
359#322—791no—
360#322C2×D7911yescharacteristic, complemented
361C82—791no—
362C22×D4—791no—
363C23×C4—1581no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16C4×D7911yescharacteristic
368C2×M16—21no—
369C2×M16—1581no—
370C22×C8—1581no—
371C22×C8—1581no—
372C4⋊Q8—791no—
373C4⋊Q8—791no—
374#125—1581no—
375C22⋊Q8—791no—
376C22⋊Q8—791no—
377C22⋊Q8—1581no—
378C4×Q8—791no—
379C4×Q8—1581no—
380C4×Q8C22×D7911yescharacteristic
381C4×D4—791no—
382C4×D4—791no—
383C4×D4—791no—
384C4×D4—1581no—
385C4×D4—1581no—
386C4×D4C22×D7911yescharacteristic
387#117—1581no—
388C2×C4⋊C4—791no—
389C2×C22⋊C4—1581no—
390C2×C42—791no—
391#106—791no—
392#106—791no—
393#106—791no—
394#106C4×D7911yescharacteristic
395C4⋊C8—21no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—791no—
400C4⋊C8—791no—
401C4⋊C8—791no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8—1581no—
405C4⋊C8—1581no—
406C4⋊C8C22×D7911yescharacteristic
407C4⋊C8C22×D7911yescharacteristic
408#103—791no—
409#103—791no—
410#103—791no—
411#103C4×D7911yescharacteristic
412#102—791no—
413#102—791no—
414#102—791no—
415#102C4×D7911yescharacteristic
416C22⋊C8—21no—
417C22⋊C8—791no—
418C22⋊C8—791no—
419C22⋊C8—791no—
420C22⋊C8—791no—
421C22⋊C8—1581no—
422C22⋊C8—1581no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8—1581no—
426C22⋊C8—1581no—
427#97—21no—
428#97—1581no—
429#97—1581no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—1581no—
434C4×C8—1581no—
435C4×C8—1581no—
436C4×C8C22×D7911yescharacteristic
437C4×C8C22×D7911yescharacteristic
438C4×C8C22×D7911yescharacteristic
439C24—1581no—
440C2×Q8—791no—
441C2×Q8—1581no—
442C2×Q8C2×C4×D7911yescharacteristic
443C2×D4—3161no—
444C2×D4—791no—
445C2×D4—1581no—
446C2×D4—1582no—
447C2×D4C2×C4×D7911yescharacteristic
448C22×C4—21no—
449C22×C4—791no—
450C22×C4—791no—
451C22×C4—791no—
452C22×C4—6321no—
453C22×C4—6321no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4—1581no—
457C22×C4—1581no—
458SD16—41no—
459SD16—3161no—
460SD16—1582no—
461SD16—1582no—
462M16—81no—
463M16—6321no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—6321no—
472C2×C8—6321no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8C2×C4×D7911yescharacteristic
477C2×C8C2×C4×D7911yescharacteristic
478C2×C8D4×D7911yescharacteristic
479C2×C8D4×D7911yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—791no—
482C4⋊C4—791no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C4⋊C4—1581no—
488C4⋊C4—1581no—
489C4⋊C4—1581no—
490C4⋊C4C2×C4×D7911yescharacteristic
491C4⋊C4C2×C4×D7911yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—1581no—
494C22⋊C4—1581no—
495C22⋊C4—1581no—
496C22⋊C4—1581no—
497C22⋊C4—1581no—
498C42—21no—
499C42—791no—
500C42—791no—
501C42—791no—
502C42—1581no—
503C42C23×D7911yescharacteristic
504C23—21no—
505C23—3162no—
506C23—3162no—
507C23—791no—
508C23—6321no—
509C23—1581no—
510Q8—21no—
511Q8—21no—
512Q8—3162no—
513Q8—1581no—
514Q8—1581no—
515D4—21no—
516D4—21no—
517D4—3162no—
518D4—792no—
519D4—1581no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—791no—
529C2×C4—791no—
530C2×C4—791no—
531C2×C4—6321no—
532C2×C4—6321no—
533C2×C4—6321no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1581no—
540C2×C4—1581no—
541C2×C4C22×C4×D7911yescharacteristic
542C2×C4C2×D4×D7911yescharacteristic
543C2×C4#41×D7911yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—3161no—
552C8—3161no—
553C8—3161no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C8—1581no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—3161no—
562C22—3161no—
563C22—6324no—
564C22—1581no—
565C22—1581no—
566C22—1581no—
567C22—1582no—
568C22C4×D4×D7911yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—3161no—
574C4—3161no—
575C4—3161no—
576C4—1581no—
577C4—1581no—
578C4—1581no—
579C4#131×D7911yescharacteristic
580C4#1909932911yescharacteristic
581C4#1909937211yescharacteristic
582C4#1909937311yescharacteristic
583C2—81no—
584C2—3162no—
585C2—3161no—
586C2#2377212911yescharacteristic
587C2#2377225611yescharacteristic
588C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1592C2
socleC22×C79C2×C4×D4
prime radicalC2×D4×D79C2×C4
cosocleC42×C79C24
hypercenter#1592D79
hypocenterC79#18028
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
2. #23772498C2semidirect
4. #23762293C2semidirect
5. #23762279C2semidirect
6. #23762256C2semidirect
7. #23762241C2semidirect
8. #23762234C2semidirect
9. #23762214C2semidirect
10. #23762197C2semidirect
11. #23762179C2semidirect
12. #23761871C2semidirect
14. #23761701C2semidirect
15. C79×#1592C2semidirect
27. #19098690C22semidirect
28. Q8×C79⋊C8C22semidirect
29. C8×Dic158C22semidirect
48. #19098316C22semidirect
49. #19098313C22semidirect
50. #19098313C22semidirect
51. #19098295C22semidirect
52. Q8⋊(C79⋊C8)C22semidirect
53. #19098293C22semidirect
57. #19098285C22semidirect
58. #19098284C22semidirect
59. C8×C79⋊C8C22semidirect
60. #446×C79C22semidirect
63. #335×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. #322×C79C22semidirect
298. #1592D79semidirect
310. C79#18028semidirect
323. #446C2×D79semidirect
347. #335C2×D79semidirect
353. Q8⋊C8C2×D79semidirect
360. #322C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734062
#237724982
#237724322
#237622932
#237622792
#237622562
#237622412
#237622342
#237622142
#237621972
#237621792
#237618712
#237617892
#237617012
C79×#15922
#1802879