Finite Group Databasebeta

← #34259603

Subgroups — #34259603

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#23773408C211yesmaximal, characteristic
2#23772501C211yesmaximal, characteristic, complemented
3#23762302C211yesmaximal, characteristic, complemented
4#23762288C211yesmaximal, characteristic, complemented
5#23762276C211yesmaximal, characteristic, complemented
6#23762257C211yesmaximal, characteristic, complemented
7#23762241C211yesmaximal, characteristic, complemented
8#23762236C211yesmaximal, characteristic, complemented
9#23762212C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762181C211yesmaximal, characteristic, complemented
12#23761872C211yesmaximal, characteristic, complemented
13#23761791C211yesmaximal, characteristic
14#23761700C211yesmaximal, characteristic, complemented
15C79×#2974C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098893C411yescharacteristic
18#19098893C411yescharacteristic
19#19098853C411yescharacteristic
20#19098853C411yescharacteristic
21#19098768C411yescharacteristic
22#19098768C411yescharacteristic
23#19098714C411yescharacteristic
24#19098714C411yescharacteristic
25#19098693C2211yescharacteristic, complemented
26#19098693C2211yescharacteristic, complemented
27#19098690—21no—
28C8×Dic158C2211yescharacteristic, complemented
29C8×Dic158C2211yescharacteristic, complemented
30#19098687—21no—
31C4×#11929612C2211yescharacteristic
32C4×#11929612C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098621—21no—
36#19098621—21no—
37#19098616C2211yescharacteristic
38#19098616C2211yescharacteristic
39#19098616C2211yescharacteristic
40#19098315C2211yescharacteristic, complemented
41#19098315C2211yescharacteristic, complemented
42#19098313C2211yescharacteristic, complemented
43#19098313C2211yescharacteristic, complemented
44#19098293C2211yescharacteristic, complemented
45#19098293C2211yescharacteristic, complemented
46#19098293C2211yescharacteristic, complemented
47#19098293C2211yescharacteristic, complemented
48D4⋊(C79⋊C8)C2211yescharacteristic
49D4⋊(C79⋊C8)C2211yescharacteristic
50D4⋊(C79⋊C8)C2211yescharacteristic
51D4⋊(C79⋊C8)C2211yescharacteristic
52#19098285C2211yescharacteristic, complemented
53#19098285C2211yescharacteristic, complemented
54#19098284C2211yescharacteristic, complemented
55C8×C79⋊C8C2211yescharacteristic, complemented
56C2×C4×D4×C79C2211yescharacteristic
57#436×C79C2211yescharacteristic
58#436×C79C2211yescharacteristic
59#435×C79—21no—
60#435×C79—21no—
61C8×D4×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#432×C79C2211yescharacteristic
64#431×C79C2211yescharacteristic
65#407×C79—21no—
66C2×C22⋊C8×C79—21no—
67C2×#11929612C2×C411yescharacteristic
68C2×#11929612C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71#14579161C2×C411yescharacteristic
72#14579161C2×C411yescharacteristic
73#14579153—21no—
74#14579140—21no—
75#14579140C2×C411yescharacteristic
76#14579140C2×C411yescharacteristic
77C4×Dic158—21no—
78C4×Dic158C2311yescharacteristic
79C4×Dic158C2311yescharacteristic
80#14579120—21no—
81#14579092C2×C411yescharacteristic
82#14579092C2×C411yescharacteristic
83#14579092C2×C411yescharacteristic
84#14579092C2×C411yescharacteristic
85C4⋊(C79⋊C8)—21no—
86C4⋊(C79⋊C8)C2×C411yescharacteristic
87C4⋊(C79⋊C8)C2×C411yescharacteristic
88C4⋊(C79⋊C8)C2311yescharacteristic
89C4⋊(C79⋊C8)C2311yescharacteristic
90Dic79⋊C8—21no—
91Dic79⋊C8—21no—
92Dic79⋊C8—21no—
93Dic79⋊C8—21no—
94#14579084C2×C411yescharacteristic
95#14579084C2×C411yescharacteristic
96#14579084C2×C411yescharacteristic
97#14579084C2×C411yescharacteristic
98#14579082C2×C411yescharacteristic
99#14579082C2×C411yescharacteristic
100#14579082C2×C411yescharacteristic
101#14579082C2×C411yescharacteristic
102C22⋊(C79⋊C8)—21no—
103C22⋊(C79⋊C8)—21no—
104C22⋊(C79⋊C8)C2×C411yescharacteristic
105C22⋊(C79⋊C8)C2×C411yescharacteristic
106#14579071—21no—
107#14579071—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8C2311yescharacteristic
110C4×C79⋊C8C2311yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C8×Dic79—21no—
114C8×Dic79—21no—
115C22×D4×C79C2×C411yescharacteristic
116C23×C4×C79—21no—
117C2×M16×C79—21no—
118C2×M16×C79—21no—
119C22×C8×C79—21no—
120C22×C8×C79—21no—
121C4×D4×C79—21no—
122C4×D4×C79—21no—
123C4×D4×C79C2×C411yescharacteristic
124C4×D4×C79C2×C411yescharacteristic
125C4×D4×C79C2311yescharacteristic
126C4×D4×C79C2311yescharacteristic
127C2×C4⋊C4×C79C2×C411yescharacteristic
128C2×C22⋊C4×C79—21no—
129C2×C42×C79C2311yescharacteristic
130C4⋊C8×C79D411yescharacteristic
131C4⋊C8×C79D411yescharacteristic
132C4⋊C8×C79C2311yescharacteristic
133C4⋊C8×C79C2311yescharacteristic
134C22⋊C8×C79—21no—
135C22⋊C8×C79—21no—
136C22⋊C8×C79—21no—
137C22⋊C8×C79—21no—
138C22⋊C8×C79—21no—
139C22⋊C8×C79D411yescharacteristic
140C22⋊C8×C79D411yescharacteristic
141#97×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147#11929612—22no—
148#11929612—22no—
149#11929612—41no—
150#11929612—41no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8C22×C411yescharacteristic
156C2×C79⋊C8C22×C411yescharacteristic
157C2×C79⋊C8C22×C411yescharacteristic
158C2×C79⋊C8C22×C411yescharacteristic
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79C22×C411yescharacteristic
163C4⋊Dic79C22×C411yescharacteristic
164Dic79⋊C4—21no—
165Dic79⋊C4—21no—
166Dic79⋊C4—21no—
167Dic79⋊C4—21no—
168C22⋊Dic79—21no—
169C22⋊Dic79—21no—
170C4×Dic79—21no—
171C4×Dic79—21no—
172C24×C79—21no—
173C2×D4×C79—22no—
174C2×D4×C79—21no—
175C2×D4×C79—41no—
176C2×D4×C79C22×C411yescharacteristic
177C2×D4×C79C22×C411yescharacteristic
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79—81no—
184C22×C4×C79—81no—
185C22×C4×C79C22×C411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79#4111yescharacteristic
188M16×C79—81no—
189M16×C79—81no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—81no—
193C2×C8×C79—81no—
194C2×C8×C79C2×D411yescharacteristic
195C2×C8×C79C2×D411yescharacteristic
196C2×C8×C79C2×D411yescharacteristic
197C2×C8×C79C2×D411yescharacteristic
198C4⋊C4×C79—21no—
199C4⋊C4×C79C22×C411yescharacteristic
200C4⋊C4×C79C22×C411yescharacteristic
201C22⋊C4×C79—21no—
202C22⋊C4×C79—21no—
203C22⋊C4×C79—21no—
204C22⋊C4×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79—21no—
211C23×C79—42no—
212C23×C79—42no—
213C23×C79—81no—
214C23×C79C4×D411yescharacteristic
215D4×C79—21no—
216D4×C79—21no—
217D4×C79—21no—
218D4×C79—42no—
219D4×C79#13112yes—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—81no—
232C2×C4×C79—81no—
233C2×C4×C79—81no—
234C2×C4×C79C4×D411yescharacteristic
235C2×C4×C79C4×D411yescharacteristic
236C2×C4×C79C4×D411yescharacteristic
237C2×C4×C79C23×C411yescharacteristic
238C2×C4×C79C22×D411yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×Dic79—21no—
241C2×Dic79—21no—
242C2×Dic79—21no—
243C2×Dic79—21no—
244Dic158—21no—
245Dic158—21no—
246Dic158—21no—
247Dic158—21no—
248Dic158—42no—
249C8×C79—21no—
250C8×C79—21no—
251C8×C79—21no—
252C8×C79—21no—
253C8×C79—41no—
254C8×C79—41no—
255C79⋊C8—21no—
256C79⋊C8—21no—
257C79⋊C8—21no—
258C79⋊C8—21no—
259C79⋊C8—41no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C22×C79—21no—
264C22×C79—22no—
265C22×C79—21no—
266C22×C79—21no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79—84no—
273C22×C79C2×C4×D411yescharacteristic
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—21no—
277C4×C79—21no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79C2×#13111yescharacteristic
281C4×C79#56811yescharacteristic
282C4×C79#57711yescharacteristic
283C4×C79#57711yescharacteristic
284Dic79—41no—
285Dic79—41no—
286Dic79—41no—
287Dic79—41no—
288#18028—791nomaximal, Sylow
289C2×C79—41no—
290C2×C79—42no—
291C2×C79—81no—
292C2×C79#293311yescharacteristic
293C2×C79#294911yescharacteristic
294C2×C79#295511yescharacteristic
295#3101—791no—
296#2986—791no—
297#2974D7911yescharacteristic
298#1592—791no—
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×D4C2×D7911yescharacteristic
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—791no—
323#446—1581no—
324C8×Q8—791no—
325C8×Q8—791no—
326C8×Q8—1581no—
327C4×SD16—791no—
328C4×SD16—791no—
329C4×SD16—791no—
330C4×SD16—791no—
331#436C2×D7911yescharacteristic
332#436C2×D7911yescharacteristic
333#435—21no—
334#435—21no—
335C8×D4C2×D7911yescharacteristic
336C8×D4C2×D7911yescharacteristic
337#433—1581no—
338#433—1581no—
339#432C2×D7911yescharacteristic
340#431—791no—
341#431—791no—
342#431—791no—
343#431C2×D7911yescharacteristic
344#407—21no—
345C2×C22⋊C8—21no—
346#335—791no—
347#335—791no—
348#334—791no—
349#334—791no—
350Q8⋊C8—791no—
351Q8⋊C8—791no—
352Q8⋊C8—791no—
353Q8⋊C8—791no—
354D4⋊C8—791no—
355D4⋊C8—791no—
356D4⋊C8—791no—
357D4⋊C8—791no—
358#322—791no—
359#322—791no—
360#322—791no—
361C82—791no—
362C22×D4C4×D7911yescharacteristic
363C23×C4—21no—
364C2×SD16—791no—
365C2×SD16—791no—
366C2×SD16—791no—
367C2×SD16—791no—
368C2×M16—21no—
369C2×M16—21no—
370C22×C8—21no—
371C22×C8—21no—
372C4⋊Q8—791no—
373C4⋊Q8—791no—
374#125—1581no—
375C22⋊Q8—791no—
376C22⋊Q8—791no—
377C22⋊Q8—1581no—
378C4×Q8—791no—
379C4×Q8—791no—
380C4×Q8—1581no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4C4×D7911yescharacteristic
384C4×D4C4×D7911yescharacteristic
385C4×D4C22×D7911yescharacteristic
386C4×D4C22×D7911yescharacteristic
387#117—1581no—
388C2×C4⋊C4C4×D7911yescharacteristic
389C2×C22⋊C4—21no—
390C2×C42C22×D7911yescharacteristic
391#106—791no—
392#106—791no—
393#106—791no—
394#106—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—791no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8—1581no—
403C4⋊C8—1581no—
404C4⋊C8C22⋊D7911yescharacteristic
405C4⋊C8C22⋊D7911yescharacteristic
406C4⋊C8C22×D7911yescharacteristic
407C4⋊C8C22×D7911yescharacteristic
408#103—791no—
409#103—791no—
410#103—791no—
411#103—791no—
412#102—791no—
413#102—791no—
414#102—791no—
415#102—791no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420C22⋊C8—21no—
421C22⋊C8—791no—
422C22⋊C8—791no—
423C22⋊C8—1581no—
424C22⋊C8—1581no—
425C22⋊C8C22⋊D7911yescharacteristic
426C22⋊C8C22⋊D7911yescharacteristic
427#97—21no—
428#97—1581no—
429#97—1581no—
430C4×C8—791no—
431C4×C8—791no—
432C4×C8—791no—
433C4×C8—791no—
434C4×C8—1581no—
435C4×C8—1581no—
436C4×C8—1581no—
437C4×C8C22×D7911yescharacteristic
438C4×C8C22×D7911yescharacteristic
439C24—21no—
440C2×Q8—791no—
441C2×Q8—791no—
442C2×Q8—1581no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—41no—
446C2×D4C2×C4×D7911yescharacteristic
447C2×D4C2×C4×D7911yescharacteristic
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4C2×C4×D7911yescharacteristic
456C22×C4C2×C22⋊D7911yescharacteristic
457C22×C4#1192962411yescharacteristic
458SD16—3161no—
459SD16—3161no—
460SD16—1582no—
461SD16—1582no—
462M16—81no—
463M16—81no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—81no—
467C2×C8—81no—
468C2×C8—791no—
469C2×C8—791no—
470C2×C8—791no—
471C2×C8—791no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8C2×C22⋊D7911yescharacteristic
477C2×C8C2×C22⋊D7911yescharacteristic
478C2×C8C2×C22⋊D7911yescharacteristic
479C2×C8C2×C22⋊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—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—1581no—
497C22⋊C4—1581no—
498C42—1581no—
499C42—1581no—
500C42C2×C4×D7911yescharacteristic
501C42C2×C22⋊D7911yescharacteristic
502C42#1192962411yescharacteristic
503C42C23×D7911yescharacteristic
504C23—21no—
505C23—21no—
506C23—42no—
507C23—42no—
508C23—81no—
509C23C4×C22⋊D7911yescharacteristic
510Q8—3162no—
511Q8—1581no—
512Q8—1581no—
513Q8—1581no—
514Q8—1581no—
515D4—21no—
516D4—21no—
517D4—21no—
518D4—42no—
519D4#1457916912yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1581no—
537C2×C4—1581no—
538C2×C4C4×C22⋊D7911yescharacteristic
539C2×C4C4×C22⋊D7911yescharacteristic
540C2×C4C4×C22⋊D7911yescharacteristic
541C2×C4C22×C4×D7911yescharacteristic
542C2×C4C22×C22⋊D7911yescharacteristic
543C2×C4C2×#1192962411yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—3161no—
551C8—3161no—
552C8—3161no—
553C8—3161no—
554C8—1581no—
555C8—1581no—
556C8—1581no—
557C8—1581no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—84no—
568C22C2×C4×C22⋊D7911yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—3161no—
576C4—3161no—
577C4—3161no—
578C4—3161no—
579C4C2×#1457916911yescharacteristic
580C4#1909932911yescharacteristic
581C4#1909937411yescharacteristic
582C4#1909937411yescharacteristic
583C2—41no—
584C2—42no—
585C2—81no—
586C2#2377212511yescharacteristic
587C2#2377225711yescharacteristic
588C2#2377230111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2974C2
socleC22×C79C2×C4×D4
prime radicalC22×D4×C79C2×C4
cosocleC42×C79C24
hypercenter#2974D79
hypocenterC79#18028
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #23772501C2semidirect
3. #23762302C2semidirect
4. #23762288C2semidirect
5. #23762276C2semidirect
6. #23762257C2semidirect
7. #23762241C2semidirect
8. #23762236C2semidirect
9. #23762212C2semidirect
10. C8×#11929612C2semidirect
11. #23762181C2semidirect
12. #23761872C2semidirect
14. #23761700C2semidirect
25. #19098693C22semidirect
26. #19098693C22semidirect
28. C8×Dic158C22semidirect
29. C8×Dic158C22semidirect
40. #19098315C22semidirect
41. #19098315C22semidirect
42. #19098313C22semidirect
43. #19098313C22semidirect
44. #19098293C22semidirect
45. #19098293C22semidirect
46. #19098293C22semidirect
47. #19098293C22semidirect
52. #19098285C22semidirect
53. #19098285C22semidirect
54. #19098284C22semidirect
55. C8×C79⋊C8C22semidirect
310. C79#18028semidirect

Maximal subgroups

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

subgroupindex
#237734082
#237725012
#237623022
#237622882
#237622762
#237622572
#237622412
#237622362
#237622122
C8×#119296122
#237621812
#237618722
#237617912
#237617002
C79×#29742
#1802879