Finite Group Databasebeta

← #34259172

Subgroups — #34259172

Order 20224.

618 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773414C211yesmaximal, characteristic
2#23772408C211yesmaximal, characteristic, complemented
3#23772401C211yesmaximal, characteristic, complemented
4#23762298C211yesmaximal, characteristic, complemented
5#23762293C211yesmaximal, characteristic, complemented
6#23762249C211yesmaximal, characteristic, complemented
7#23762236C211yesmaximal, characteristic, complemented
8#23762236C211yesmaximal, characteristic, complemented
9#23762230C211yesmaximal, characteristic, complemented
10#23762203C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761857C211yesmaximal, characteristic, complemented
13#23761779C211yesmaximal, characteristic, complemented
14C8×#11929607C211yesmaximal, characteristic, complemented
15C8×D8×C79C211yesmaximal, characteristic, complemented
16#19099095C2211yescharacteristic
17C4×#11929623C2211yescharacteristic
18#19098906C411yescharacteristic
19#19098885C411yescharacteristic
20#19098869C411yescharacteristic
21#19098861C411yescharacteristic
22#19098784C411yescharacteristic
23#19098776C411yescharacteristic
24#19098726C411yescharacteristic
25#19098705C411yescharacteristic
26#19098693C2211yescharacteristic
27#19098693C2211yescharacteristic
28#19098690C2211yescharacteristic
29#19098687C2211yescharacteristic
30C4×Dic316C2211yescharacteristic
31C4×#11929612C2211yescharacteristic
32C4×#11929612C2211yescharacteristic
33#19098641C2211yescharacteristic
34#19098638—21no—
35#19098634—21no—
36#19098634—21no—
37#19098631—21no—
38D4×C79⋊C8C2211yescharacteristic
39#19098621—21no—
40#19098619C2211yescharacteristic
41#19098617—21no—
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098518—21no—
45C4×#11929607C2211yescharacteristic
46C2×#14579071—21no—
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098293C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56D4⋊(C79⋊C8)C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C4×D8×C79C2211yescharacteristic
61C8×D4×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64D4⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607D411yescharacteristic
73C2×#11929607D411yescharacteristic
74C22×C79⋊C8—21no—
75C2×C8×D79—21no—
76#14579154—21no—
77#14579152C2×C411yescharacteristic
78#14579148C2×C411yescharacteristic
79#14579143—21no—
80#14579141C2×C411yescharacteristic
81#14579132C2×C411yescharacteristic
82C4×Dic158C2311yescharacteristic
83C4×Dic158C2311yescharacteristic
84#14579126C2×C411yescharacteristic
85D4×Dic79C2×C411yescharacteristic
86C4×C22⋊D79—21no—
87C4×C22⋊D79—21no—
88(C22⋊D79)⋊C4—21no—
89#14579120—21no—
90#14579119C2×C411yescharacteristic
91#14579117—21no—
92C2×C4×Dic79—21no—
93C42×D79C2311yescharacteristic
94#14579095C2×C411yescharacteristic
95#14579092C2×C411yescharacteristic
96#14579092C2×C411yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98C4⋊(C79⋊C8)C2311yescharacteristic
99Dic79⋊C8—21no—
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105#14579086C2×C411yescharacteristic
106#14579086C2×C411yescharacteristic
107#14579084C2×C411yescharacteristic
108#14579084C2×C411yescharacteristic
109#14579082C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111C22⋊(C79⋊C8)—21no—
112C22⋊(C79⋊C8)—21no—
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8C2×C411yescharacteristic
116(C2×D79)⋊C8C2×C411yescharacteristic
117#14579071—21no—
118#14579071—21no—
119#14579071—21no—
120#14579071—21no—
121#14579071D411yescharacteristic
122#14579071D411yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C8×Dic79—21no—
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×D8×C79C2×C411yescharacteristic
130C22×C8×C79—21no—
131C22×C8×C79—21no—
132C4×D4×C79C2311yescharacteristic
133C4×D4×C79C2311yescharacteristic
134#107×C79C2×C411yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#102×C79C2×C411yescharacteristic
138#102×C79C2×C411yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929623—22no—
145#11929623—41no—
146C2×Dic158C22×C411yescharacteristic
147C2×Dic158C22×C411yescharacteristic
148C2×C22⋊D79—21no—
149C2×C22⋊D79—21no—
150C22×Dic79—21no—
151C22×Dic79—21no—
152C2×C4×D79C22×C411yescharacteristic
153C2×C4×D79C2×D411yescharacteristic
154C2×C4×D79#4111yescharacteristic
155Dic316—41no—
156#11929612—22no—
157#11929612—41no—
158#11929608—42no—
159#11929607—22no—
160#11929607—22no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—21no—
164C2×C79⋊C8—42no—
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C8×D79—42no—
170C4⋊Dic79—21no—
171C4⋊Dic79—21no—
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181C22⋊Dic79—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79—21no—
188C4×Dic79C22×C411yescharacteristic
189C4×Dic79C2×D411yescharacteristic
190C4×Dic79#4111yescharacteristic
191C2×D4×C79C22×C411yescharacteristic
192C2×D4×C79C22×C411yescharacteristic
193C22×C4×C79—21no—
194C22×C4×C79—21no—
195D8×C79—22no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—42no—
200C2×C8×C79—42no—
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C22×C411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C2×C8×C79C2×D411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C4⋊C4×C79C22×C411yescharacteristic
207C22⋊C4×C79—21no—
208C22⋊C4×C79—21no—
209C42×C79C2411yescharacteristic
210C23×C79—21no—
211C23×C79—21no—
212C22×D79C4×D411yescharacteristic
213D4×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216D4×C79#13112yes—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79C23×C411yescharacteristic
224C2×C4×C79C22×D411yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C22⋊D79—42no—
227C22⋊D79—81no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—42no—
233C2×Dic79—81no—
234C2×Dic79C4×D411yescharacteristic
235C2×Dic79C4×D411yescharacteristic
236C2×Dic79C4×D411yescharacteristic
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—21no—
241C4×D79—41no—
242Dic158—21no—
243Dic158—21no—
244Dic158—21no—
245Dic158—21no—
246C8×C79—22no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—21no—
250C8×C79—41no—
251C8×C79—41no—
252C8×C79#13512yes—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8#13512yes—
260C22×C79—22no—
261C22×C79—41no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79C2×C4×D411yescharacteristic
267C2×D79—21no—
268C2×D79—21no—
269C2×D79—21no—
270C4×C79—21no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79C2×#13111yescharacteristic
274C4×C79#56811yescharacteristic
275C4×C79C2×#13511yescharacteristic
276C4×C79C2×#13511yescharacteristic
277Dic79—21no—
278Dic79—21no—
279Dic79—21no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283Dic79—41no—
284#17996—791nomaximal, Sylow
285C2×C79—42no—
286C2×C79—81no—
287C2×C79#293311yescharacteristic
288C2×C79C4×#13511yescharacteristic
289C2×C79#295611yescharacteristic
290D79—41no—
291#3102—791no—
292#2970—791no—
293#2969—791no—
294#1591—791no—
295#1590—791no—
296#1584—791no—
297#1582—791no—
298#1582—791no—
299#1581—791no—
300C8×SD16—791no—
301C8×D8D7911yescharacteristic, complemented
302#1572—791no—
303#1494—791no—
304#1476—791no—
305C8×M16—791no—
306C79#1799611yescharacteristic, complemented, Sylow
307#529—791no—
308C4×#41—791no—
309#485—791no—
310#482—791no—
311#479—791no—
312#478—791no—
313#464—791no—
314#463—791no—
315#452—791no—
316#449—791no—
317#446—791no—
318#446—791no—
319#446—791no—
320C8×Q8—791no—
321C4×Q16—791no—
322C4×SD16—791no—
323C4×SD16—791no—
324C4×D8C2×D7911yescharacteristic
325#436—791no—
326#436—1581no—
327#435—1581no—
328#435—1581no—
329#435—1581no—
330C8×D4—791no—
331C8×D4C2×D7911yescharacteristic
332C8×D4C2×D7911yescharacteristic
333#433—1581no—
334#432—791no—
335#432—1581no—
336#431—791no—
337#431—791no—
338#405—1581no—
339C4×M16—791no—
340C2×#97—1581no—
341C2×C4×C8—791no—
342#335—791no—
343#335C2×D7911yescharacteristic
344#334—791no—
345#334—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8—791no—
349Q8⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8C2×D7911yescharacteristic
353D4⋊C8C2×D7911yescharacteristic
354#322—791no—
355#322—791no—
356C82—791no—
357C82C2×D7911yescharacteristic
358C2×#41—791no—
359C2×Q16—791no—
360C2×SD16—791no—
361C2×SD16—791no—
362C2×D8C4×D7911yescharacteristic
363C2×M16—791no—
364C2×M16—791no—
365C2×M16—1581no—
366C22×C8—21no—
367C22×C8—21no—
368C22×C8—1581no—
369C22×C8—1581no—
370#126—1581no—
371#125—791no—
372#124—791no—
373#123—1581no—
374C22⋊Q8—791no—
375#121—791no—
376C4×Q8—791no—
377C4×Q8—791no—
378C4×Q8—791no—
379C4×D4—791no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4—1581no—
383C4×D4C22×D7911yescharacteristic
384C4×D4C22×D7911yescharacteristic
385#117—791no—
386#117—1581no—
387#117—1581no—
388C2×C42—791no—
389C2×C42—1581no—
390#107—791no—
391#107C4×D7911yescharacteristic
392#106—791no—
393#106—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—791no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8—1581no—
402C4⋊C8C22×D7911yescharacteristic
403C4⋊C8C22×D7911yescharacteristic
404#103—791no—
405#103—791no—
406#103—791no—
407#103—791no—
408#102—791no—
409#102—791no—
410#102C4×D7911yescharacteristic
411#102C4×D7911yescharacteristic
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—791no—
415C22⋊C8—791no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—791no—
421#97—791no—
422#97—1581no—
423#97—1581no—
424#97—1581no—
425#97—1581no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—791no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8—1581no—
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434C4×C8C22×D7911yescharacteristic
435#41—3161no—
436#41—1582no—
437C2×Q8—791no—
438C2×Q8—791no—
439C2×D4—1581no—
440C2×D4—1581no—
441C2×D4C2×C4×D7911yescharacteristic
442C2×D4C2×C4×D7911yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—791no—
446C22×C4—791no—
447C22×C4—791no—
448C22×C4—1581no—
449C22×C4—1581no—
450Q16—3161no—
451SD16—3161no—
452SD16—1582no—
453D8—22no—
454M16—3162no—
455M16—1582no—
456M16—1582no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—3162no—
463C2×C8—3162no—
464C2×C8—791no—
465C2×C8—791no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8C2×C4×D7911yescharacteristic
472C2×C8C2×C4×D7911yescharacteristic
473C2×C8D4×D7911yescharacteristic
474C2×C8D4×D7911yescharacteristic
475C4⋊C4—791no—
476C4⋊C4—791no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4C2×C4×D7911yescharacteristic
484C4⋊C4C2×C4×D7911yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C42—791no—
494C42—791no—
495C42—791no—
496C42—1581no—
497C42—1581no—
498C42—1581no—
499C42—1581no—
500C42C23×D7911yescharacteristic
501C23—21no—
502C23—21no—
503C23—791no—
504Q8—1581no—
505Q8—1581no—
506Q8—1581no—
507Q8—1581no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—3162no—
512D4—6321no—
513D4#1457916712yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—3161no—
521C2×C4—3162no—
522C2×C4—791no—
523C2×C4—791no—
524C2×C4—791no—
525C2×C4—6321no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4C22×C4×D7911yescharacteristic
535C2×C4C2×D4×D7911yescharacteristic
536C2×C4#41×D7911yescharacteristic
537C8—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—3161no—
544C8—3161no—
545C8—3161no—
546C8—792no—
547C8—1581no—
548C8—1581no—
549C8—1581no—
550C8#1457918012yes—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—1581no—
558C22—1581no—
559C22—1581no—
560C22C4×D4×D7911yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—3161no—
565C4—3161no—
566C4—3161no—
567C4—3161no—
568C4—1581no—
569C4—1581no—
570C4—1581no—
571C4C2×#1457916711yescharacteristic
572C4#1909932811yescharacteristic
573C4#135×D7911yescharacteristic
574C4C2×#1457918011yescharacteristic
575C2—42no—
576C2—81no—
577C2—3161no—
578C2#2377212011yescharacteristic
579C2C4×#1457918011yescharacteristic
580C2#2377230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC8×D8×C79C2
socleC22×C79C2×C4×D4
prime radical#19098705C4
cosocleC42×C79C24
hypercenterC8×D8D79
hypocenterC79#17996
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #23772408C2semidirect
3. #23772401C2semidirect
4. #23762298C2semidirect
5. #23762293C2semidirect
6. #23762249C2semidirect
7. #23762236C2semidirect
8. #23762236C2semidirect
9. #23762230C2semidirect
10. #23762203C2semidirect
11. #23762179C2semidirect
12. #23761857C2semidirect
13. #23761779C2semidirect
14. C8×#11929607C2semidirect
15. C8×D8×C79C2semidirect
301. C8×D8D79semidirect
306. C79#17996semidirect

Maximal subgroups

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

subgroupindex
#237734142
#237724082
#237724012
#237622982
#237622932
#237622492
#237622362
#237622302
#237622032
#237621792
#237618572
#237617792
C8×#119296072
C8×D8×C792
#1799679