Finite Group Databasebeta

← #34258178

Subgroups — #34258178

Order 20224.

634 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773414C211yesmaximal, characteristic
2#23772378C211yesmaximal, characteristic, complemented
3C8×#11929623C211yesmaximal, characteristic, complemented
4#23762252C211yesmaximal, characteristic, complemented
5#23762241C211yesmaximal, characteristic, complemented
6#23762236C211yesmaximal, characteristic, complemented
7#23762206C211yesmaximal, characteristic, complemented
8#23762203C211yesmaximal, characteristic, complemented
9#23762203C211yesmaximal, characteristic, complemented
10D8×C79⋊C8C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761831C211yesmaximal, characteristic, complemented
13#23761805C211yesmaximal, characteristic, complemented
14#23761685C211yesmaximal, characteristic, complemented
15C79×#1581C211yesmaximal, characteristic, complemented
16#19099095C2211yescharacteristic
17C4×#11929623C2211yescharacteristic
18#19098906C411yescharacteristic
19#19098885C411yescharacteristic
20#19098869C411yescharacteristic
21#19098861C411yescharacteristic
22#19098784C411yescharacteristic
23#19098776C411yescharacteristic
24#19098726C411yescharacteristic
25#19098705C411yescharacteristic
26#19098693C2211yescharacteristic
27C8×Dic158C2211yescharacteristic
28#19098687C2211yescharacteristic
29#19098687C2211yescharacteristic
30C4×Dic316C2211yescharacteristic
31C4×#11929612C2211yescharacteristic
32C4×#11929612C2211yescharacteristic
33#19098632—21no—
34D4×C79⋊C8C2211yescharacteristic
35D4×C79⋊C8C2211yescharacteristic
36C8×C22⋊D79—21no—
37(C22⋊D79)⋊C8—21no—
38(C22⋊D79)⋊C8—21no—
39#19098621—21no—
40#19098619C2211yescharacteristic
41#19098615C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098613—21no—
45#19098521—21no—
46C2×C8×Dic79—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#19098284C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60C4×D8×C79C2211yescharacteristic
61#436×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64D4⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929623C2×C411yescharacteristic
68C2×Dic316C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C22×C79⋊C8—21no—
72C22×C79⋊C8—21no—
73C2×C8×D79D411yescharacteristic
74C2×C8×D79D411yescharacteristic
75#14579154—21no—
76#14579152C2×C411yescharacteristic
77#14579148C2×C411yescharacteristic
78#14579143—21no—
79#14579141C2×C411yescharacteristic
80#14579132C2×C411yescharacteristic
81C4×Dic158C2311yescharacteristic
82C4×Dic158C2311yescharacteristic
83#14579126C2×C411yescharacteristic
84D4×Dic79C2×C411yescharacteristic
85C4×C22⋊D79—21no—
86C4×C22⋊D79—21no—
87(C22⋊D79)⋊C4—21no—
88#14579120—21no—
89#14579119C2×C411yescharacteristic
90#14579117—21no—
91C2×C4×Dic79—21no—
92C42×D79C2311yescharacteristic
93#14579095C2×C411yescharacteristic
94#14579092C2×C411yescharacteristic
95#14579092C2×C411yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97C4⋊(C79⋊C8)C2311yescharacteristic
98Dic79⋊C8—21no—
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)⋊C8—21no—
116(C2×D79)⋊C8C2×C411yescharacteristic
117(C2×D79)⋊C8C2×C411yescharacteristic
118#14579071—21no—
119#14579071—21no—
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C8×Dic79—21no—
124C8×Dic79—21no—
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×D8×C79C2×C411yescharacteristic
130C2×M16×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—22no—
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—22no—
156#11929612—22no—
157#11929612—22no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—42no—
162C2×C79⋊C8—42no—
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C8×D79—22no—
168C8×D79—22no—
169C4⋊Dic79—21no—
170C4⋊Dic79—21no—
171C4⋊Dic79C22×C411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177C22⋊Dic79—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183C4×Dic79—21no—
184C4×Dic79—21no—
185C4×Dic79—21no—
186C4×Dic79—21no—
187C4×Dic79C22×C411yescharacteristic
188C4×Dic79C2×D411yescharacteristic
189C4×Dic79#4111yescharacteristic
190C2×D4×C79C22×C411yescharacteristic
191C2×D4×C79C22×C411yescharacteristic
192C22×C4×C79—21no—
193C22×C4×C79—21no—
194D8×C79—22no—
195M16×C79—42no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—21no—
199C2×C8×C79—42no—
200C2×C8×C79C22×C411yescharacteristic
201C2×C8×C79C22×C411yescharacteristic
202C2×C8×C79C2×D411yescharacteristic
203C2×C8×C79C2×D411yescharacteristic
204C4⋊C4×C79C22×C411yescharacteristic
205C4⋊C4×C79C22×C411yescharacteristic
206C22⋊C4×C79—21no—
207C22⋊C4×C79—21no—
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79—21no—
211C22×D79C4×D411yescharacteristic
212D4×C79—21no—
213D4×C79—21no—
214D4×C79—21no—
215D4×C79#13112yes—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C22⋊D79—24no—
226C22⋊D79—42no—
227C2×Dic79—24no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—42no—
233C2×Dic79C4×D411yescharacteristic
234C2×Dic79C4×D411yescharacteristic
235C2×Dic79C4×D411yescharacteristic
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—21no—
240C4×D79—41no—
241Dic158—21no—
242Dic158—21no—
243Dic158—21no—
244Dic158#13112yes—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—41no—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79#13512yes—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8#13512yes—
259C22×C79—22no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79C2×C4×D411yescharacteristic
266C2×D79—21no—
267C2×D79—21no—
268C2×D79—21no—
269C4×C79—21no—
270C4×C79—41no—
271C4×C79—41no—
272C4×C79C2×#13111yescharacteristic
273C4×C79C2×#13111yescharacteristic
274C4×C79C2×#13511yescharacteristic
275C4×C79C2×#13511yescharacteristic
276Dic79—22no—
277Dic79—21no—
278Dic79—21no—
279Dic79—21no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283#17904—791nomaximal, Sylow
284C2×C79—42no—
285C2×C79—81no—
286C2×C79#292911yescharacteristic
287C2×C79C4×#13511yescharacteristic
288C2×C79#295511yescharacteristic
289D79—41no—
290#3102—791no—
291#2966—791no—
292C8×#41—791no—
293#1584—791no—
294#1583—791no—
295#1582—791no—
296#1581D7911yescharacteristic, complemented
297C8×Q16—791no—
298C8×SD16—791no—
299C8×SD16—791no—
300C8×D8—791no—
301#1572—791no—
302#1488—791no—
303#1482—791no—
304#1450—791no—
305C79#1790411yescharacteristic, complemented, Sylow
306#529—791no—
307C4×#41—791no—
308#485—791no—
309#482—791no—
310#479—791no—
311#478—791no—
312#464—791no—
313#463—791no—
314#452—791no—
315#449—791no—
316#446—791no—
317C8×Q8—791no—
318C8×Q8—791no—
319C8×Q8—791no—
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16—791no—
323C4×D8C2×D7911yescharacteristic
324#436C2×D7911yescharacteristic
325#435—1581no—
326C8×D4—791no—
327C8×D4—791no—
328C8×D4—1581no—
329C8×D4—1581no—
330C8×D4—1581no—
331C8×D4C2×D7911yescharacteristic
332#433—1581no—
333#432—791no—
334#431—791no—
335#431—791no—
336#431—791no—
337#431—1581no—
338#405—1581no—
339C2×C4×C8—791no—
340C2×C4×C8—1581no—
341#335—791no—
342#335C2×D7911yescharacteristic
343#334—791no—
344#334—791no—
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8—791no—
348Q8⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8C2×D7911yescharacteristic
352D4⋊C8C2×D7911yescharacteristic
353#322—791no—
354#322C2×D7911yescharacteristic
355C82—791no—
356C82—791no—
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16—791no—
361C2×D8C4×D7911yescharacteristic
362C2×M16—21no—
363C22×C8—21no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—1581no—
367C22×C8—1581no—
368#126—1581no—
369#125—791no—
370#124—791no—
371#123—1581no—
372C22⋊Q8—791no—
373#121—791no—
374C4×Q8—791no—
375C4×Q8—791no—
376C4×Q8—791no—
377C4×D4—791no—
378C4×D4—1581no—
379C4×D4—1581no—
380C4×D4—1581no—
381C4×D4C22×D7911yescharacteristic
382C4×D4C22×D7911yescharacteristic
383#117—791no—
384#117—1581no—
385#117—1581no—
386C2×C42—791no—
387C2×C42—1581no—
388#107—791no—
389#107C4×D7911yescharacteristic
390#106—791no—
391#106—791no—
392C4⋊C8—791no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—1581no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8C22×D7911yescharacteristic
402C4⋊C8C22×D7911yescharacteristic
403#103—791no—
404#103—791no—
405#103—791no—
406#103—791no—
407#102—791no—
408#102—791no—
409#102C4×D7911yescharacteristic
410#102C4×D7911yescharacteristic
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—791no—
414C22⋊C8—791no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—1581no—
421#97—1581no—
422C4×C8—791no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—1581no—
428C4×C8—1581no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434#41—1582no—
435#41—1582no—
436C2×Q8—791no—
437C2×Q8—791no—
438C2×D4—1581no—
439C2×D4—1581no—
440C2×D4C2×C4×D7911yescharacteristic
441C2×D4C2×C4×D7911yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—791no—
447C22×C4—1581no—
448C22×C4—1581no—
449Q16—1582no—
450SD16—1582no—
451SD16—1582no—
452D8—22no—
453M16—42no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—42no—
458C2×C8—3162no—
459C2×C8—3162no—
460C2×C8—791no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—1582no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1582no—
469C2×C8C2×C4×D7911yescharacteristic
470C2×C8C2×C4×D7911yescharacteristic
471C2×C8D4×D7911yescharacteristic
472C2×C8D4×D7911yescharacteristic
473C4⋊C4—791no—
474C4⋊C4—791no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—791no—
492C42—791no—
493C42—791no—
494C42—1581no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—21no—
501C23—791no—
502Q8—792no—
503Q8—1581no—
504Q8—1581no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—3162no—
510D4—1584no—
511D4#1457916912yes—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—3161no—
519C2×C4—3162no—
520C2×C4—791no—
521C2×C4—791no—
522C2×C4—791no—
523C2×C4—1581no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1584no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4—1581no—
530C2×C4—1581no—
531C2×C4—1581no—
532C2×C4C22×C4×D7911yescharacteristic
533C2×C4C2×D4×D7911yescharacteristic
534C2×C4#41×D7911yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—3161no—
542C8—3161no—
543C8—3161no—
544C8—792no—
545C8—1581no—
546C8—1581no—
547C8—1581no—
548C8#1457918012yes—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—1581no—
556C22—1581no—
557C22—1581no—
558C22C4×D4×D7911yescharacteristic
559C4—21no—
560C4—41no—
561C4—41no—
562C4—3161no—
563C4—3161no—
564C4—3161no—
565C4—1581no—
566C4—1581no—
567C4—1581no—
568C4—1582no—
569C4#131×D7911yescharacteristic
570C4C2×#1457916911yescharacteristic
571C4#135×D7911yescharacteristic
572C4C2×#1457918011yescharacteristic
573C2—42no—
574C2—81no—
575C2—3161no—
576C2#2377209111yescharacteristic
577C2C4×#1457918011yescharacteristic
578C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1581C2
socleC22×C79C2×C4×D4
prime radical#19098705C4
cosocleC42×C79C24
hypercenter#1581D79
hypocenterC79#17904
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #23772378C2semidirect
3. C8×#11929623C2semidirect
4. #23762252C2semidirect
5. #23762241C2semidirect
6. #23762236C2semidirect
7. #23762206C2semidirect
8. #23762203C2semidirect
9. #23762203C2semidirect
10. D8×C79⋊C8C2semidirect
11. #23762179C2semidirect
12. #23761831C2semidirect
13. #23761805C2semidirect
14. #23761685C2semidirect
15. C79×#1581C2semidirect
296. #1581D79semidirect
305. C79#17904semidirect

Maximal subgroups

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

subgroupindex
#237734142
#237723782
C8×#119296232
#237622522
#237622412
#237622362
#237622062
#237622032
D8×C79⋊C82
#237621792
#237618312
#237618052
#237616852
C79×#15812
#1790479