Finite Group Databasebeta

← #34258231

Subgroups — #34258231

Order 20224.

626 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#23773416C211yesmaximal, characteristic
2#23772372C211yesmaximal, characteristic, complemented
3C8×#11929625C211yesmaximal, characteristic, complemented
4#23762279C211yesmaximal, characteristic, complemented
5#23762268C211yesmaximal, characteristic, complemented
6#23762267C211yesmaximal, characteristic, complemented
7#23762256C211yesmaximal, characteristic, complemented
8#23762222C211yesmaximal, characteristic, complemented
9#23762217C211yesmaximal, characteristic, complemented
10#23762214C211yesmaximal, characteristic, complemented
11#23762179C211yesmaximal, characteristic, complemented
12#23761833C211yesmaximal, characteristic, complemented
13#23761805C211yesmaximal, characteristic, complemented
14#23761691C211yesmaximal, characteristic, complemented
15C79×SD16⋊C8C211yesmaximal, characteristic, complemented
16#19099095C2211yescharacteristic
17C4×#11929625C2211yescharacteristic
18#19098907C411yescharacteristic
19#19098886C411yescharacteristic
20#19098868C411yescharacteristic
21#19098862C411yescharacteristic
22#19098783C411yescharacteristic
23#19098777C411yescharacteristic
24#19098728C411yescharacteristic
25#19098707C411yescharacteristic
26#19098690C2211yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28C8×Dic158C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929613C2211yescharacteristic
31C4×#11929609C2211yescharacteristic
32#19098641C2211yescharacteristic
33#19098637—21no—
34C8×D316—21no—
35C8×D316C2211yescharacteristic
36#19098628—21no—
37#19098628C2211yescharacteristic
38C8×C22⋊D79—21no—
39#19098621—21no—
40#19098619C2211yescharacteristic
41#19098615—21no—
42#19098615C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098614C2211yescharacteristic
45#19098518—21no—
46C4×C8×D79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098316C2211yescharacteristic
49#19098313C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53#19098293C2211yescharacteristic
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098285C2211yescharacteristic
59#19098284C2211yescharacteristic
60C8×Q8×C79C2211yescharacteristic
61C4×SD16×C79C2211yescharacteristic
62C8×D4×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic
67C2×#11929625C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929613C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×C8×D79—21no—
73C2×C8×D79—21no—
74C2×C8×D79D411yescharacteristic
75C2×C8×D79D411yescharacteristic
76#14579154—21no—
77#14579152C2×C411yescharacteristic
78#14579148C2×C411yescharacteristic
79#14579143—21no—
80#14579141C2×C411yescharacteristic
81#14579132C2×C411yescharacteristic
82Q8×Dic79C2×C411yescharacteristic
83C4×Dic158C2311yescharacteristic
84C4×D316—21no—
85C4×D316C2311yescharacteristic
86#14579123—21no—
87#14579123C2×C411yescharacteristic
88C4×C22⋊D79—21no—
89#14579120—21no—
90#14579119—21no—
91#14579119C2×C411yescharacteristic
92C42×D79—21no—
93C42×D79C2311yescharacteristic
94#14579094C2×C411yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97C4⋊(C79⋊C8)—21no—
98C4⋊(C79⋊C8)C2311yescharacteristic
99C4⋊(C79⋊C8)C2311yescharacteristic
100Dic79⋊C8—21no—
101Dic79⋊C8—21no—
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
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#14579071—21no—
120#14579071—21no—
121C4×C79⋊C8—21no—
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C8×Dic79—21no—
126C8×Dic79—21no—
127C8×Dic79D411yescharacteristic
128C8×Dic79D411yescharacteristic
129C2×SD16×C79C2×C411yescharacteristic
130C22×C8×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—
140C4×C8×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929625—22no—
145#11929625—22no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316—21no—
148C2×D316C22×C411yescharacteristic
149C2×C22⋊D79—21no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152C2×C4×D79—21no—
153C2×C4×D79C22×C411yescharacteristic
154C2×C4×D79C2×D411yescharacteristic
155C2×C4×D79#4111yescharacteristic
156#11929614—22no—
157#11929613—22no—
158#11929609—22no—
159#11929608—42no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C8×D79—41no—
168C8×D79—41no—
169C8×D79—42no—
170C8×D79—42no—
171C4⋊Dic79—21no—
172C4⋊Dic79—21no—
173C4⋊Dic79—21no—
174C4⋊Dic79C22×C411yescharacteristic
175C4⋊Dic79C22×C411yescharacteristic
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178C22⋊Dic79—21no—
179C22⋊Dic79—21no—
180C22⋊Dic79—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185C4×Dic79—21no—
186C4×Dic79—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—22no—
195C2×C8×C79—21no—
196C2×C8×C79—21no—
197C2×C8×C79—21no—
198C2×C8×C79—42no—
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×D79C4×D411yescharacteristic
212Q8×C79—21no—
213Q8×C79#13112yes—
214D4×C79—21no—
215D4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×D411yescharacteristic
223C2×C4×C79C2×#4111yescharacteristic
224C22⋊D79—42no—
225C2×Dic79—21no—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79C4×D411yescharacteristic
229C2×Dic79C4×D411yescharacteristic
230C2×Dic79C4×D411yescharacteristic
231D316—24no—
232D316—21no—
233D316#13112yes—
234C4×D79—24no—
235C4×D79—21no—
236C4×D79—21no—
237C4×D79—21no—
238C4×D79—21no—
239C4×D79—41no—
240C4×D79—42no—
241C4×D79—41no—
242C4×D79—41no—
243Dic158—21no—
244Dic158—21no—
245C8×C79—21no—
246C8×C79—21no—
247C8×C79—21no—
248C8×C79—21no—
249C8×C79—41no—
250C8×C79—41no—
251C8×C79—41no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—21no—
256C79⋊C8—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×C4×D411yescharacteristic
263C2×D79—22no—
264C2×D79—21no—
265C2×D79—21no—
266C2×D79—21no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—22no—
270C4×C79—21no—
271C4×C79—41no—
272C4×C79—41no—
273C4×C79C2×#13111yescharacteristic
274C4×C79C2×#13111yescharacteristic
275C4×C79#57511yescharacteristic
276C4×C79#57511yescharacteristic
277Dic79—21no—
278Dic79—21no—
279Dic79—21no—
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283#17909—791nomaximal, Sylow
284C2×C79—81no—
285C2×C79#292911yescharacteristic
286C2×C79#294311yescharacteristic
287C2×C79#295511yescharacteristic
288D79—41no—
289D79—42no—
290#3102—791no—
291#2966—791no—
292C8×#41—791no—
293#1588—791no—
294#1587—791no—
295#1586—791no—
296#1585—791no—
297#1580—791no—
298#1579—791no—
299SD16⋊C8—791no—
300SD16⋊C8D7911yescharacteristic, complemented
301#1572—791no—
302#1488—791no—
303#1482—791no—
304#1452—791no—
305C79#1790911yescharacteristic, 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×Q8C2×D7911yescharacteristic
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16C2×D7911yescharacteristic
323C4×D8—791no—
324#436—791no—
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⋊C8C2×D7911yescharacteristic
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8—791no—
352D4⋊C8C2×D7911yescharacteristic
353#322—791no—
354#322—791no—
355#322—791no—
356#322C2×D7911yescharacteristic
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16C4×D7911yescharacteristic
361C2×D8—791no—
362C2×M16—1581no—
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×Q8C22×D7911yescharacteristic
377C4×D4—791no—
378C4×D4—791no—
379C4×D4—1581no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4C22×D7911yescharacteristic
383#117—791no—
384#117—1581no—
385#117—1581no—
386C2×C42—791no—
387C2×C42—1581no—
388#107—791no—
389#107—791no—
390#106—791no—
391#106C4×D7911yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
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#103C4×D7911yescharacteristic
407#102—791no—
408#102—791no—
409#102—791no—
410#102C4×D7911yescharacteristic
411C22⋊C8—21no—
412C22⋊C8—791no—
413C22⋊C8—791no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—1581no—
421#97—1581no—
422C4×C8—21no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
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×Q8C2×C4×D7911yescharacteristic
438C2×D4—791no—
439C2×D4—1581no—
440C2×D4—1581no—
441C2×D4C2×C4×D7911yescharacteristic
442C22×C4—21no—
443C22×C4—791no—
444C22×C4—791no—
445C22×C4—791no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449Q16—1582no—
450SD16—22no—
451SD16—1582no—
452D8—1582no—
453M16—3162no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—42no—
458C2×C8—3162no—
459C2×C8—3161no—
460C2×C8—3161no—
461C2×C8—3162no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—791no—
465C2×C8—791no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8C2×C4×D7911yescharacteristic
470C2×C8C2×C4×D7911yescharacteristic
471C2×C8D4×D7911yescharacteristic
472C2×C8D4×D7911yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—791no—
475C4⋊C4—791no—
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—1581no—
485C22⋊C4—1581no—
486C22⋊C4—1581no—
487C22⋊C4—1581no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C42—21no—
492C42—791no—
493C42—791no—
494C42—791no—
495C42—1581no—
496C42—1581no—
497C42—1581no—
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—791no—
501C23—1581no—
502Q8—21no—
503Q8—1581no—
504Q8—1581no—
505Q8#1457916712yes—
506D4—21no—
507D4—21no—
508D4—3162no—
509D4—792no—
510D4—1584no—
511D4—1581no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—3161no—
518C2×C4—3162no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—791no—
522C2×C4—791no—
523C2×C4—791no—
524C2×C4—1584no—
525C2×C4—1581no—
526C2×C4—1581no—
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—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—3161no—
543C8—3161no—
544C8—3161no—
545C8—1581no—
546C8—1581no—
547C8—1581no—
548C8—1581no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—3161no—
553C22—3161no—
554C22—1581no—
555C22—1581no—
556C22—1582no—
557C22—1581no—
558C22C4×D4×D7911yescharacteristic
559C4—21no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—3161no—
564C4—3161no—
565C4—3161no—
566C4—1581no—
567C4—1581no—
568C4—1581no—
569C4#131×D7911yescharacteristic
570C4C2×#1457916711yescharacteristic
571C4#1909935911yescharacteristic
572C4#1909936211yescharacteristic
573C2—81no—
574C2—3162no—
575C2—3161no—
576C2#2377208911yescharacteristic
577C2#2377219511yescharacteristic
578C2#2377230011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×SD16⋊C8C2
socleC22×C79C2×C4×D4
prime radical#19098707C4
cosocleC42×C79C24
hypercenterSD16⋊C8D79
hypocenterC79#17909
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ SD16⋊C8

Complements

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

NH ≅ G/Ntype
2. #23772372C2semidirect
3. C8×#11929625C2semidirect
4. #23762279C2semidirect
5. #23762268C2semidirect
6. #23762267C2semidirect
7. #23762256C2semidirect
8. #23762222C2semidirect
9. #23762217C2semidirect
10. #23762214C2semidirect
11. #23762179C2semidirect
12. #23761833C2semidirect
13. #23761805C2semidirect
14. #23761691C2semidirect
15. C79×SD16⋊C8C2semidirect
300. SD16⋊C8D79semidirect
305. C79#17909semidirect

Maximal subgroups

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

subgroupindex
#237734162
#237723722
C8×#119296252
#237622792
#237622682
#237622672
#237622562
#237622222
#237622172
#237622142
#237621792
#237618332
#237618052
#237616912
C79×SD16⋊C82
#1790979