Finite Group Databasebeta

← C79×#19021 #34130710

Subgroups — C79×#19021 #34130710

Order 20224.

1012 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C79×#3558C211yesmaximal, characteristic
2C79×#3239C211yesmaximal, characteristic, complemented
3C2×#497×C79C211yesmaximal, characteristic, complemented
4C79×#3141C211yesmaximal, characteristic, complemented
5C79×#1661C214yesmaximal, complemented
6C79×#1659C214yesmaximal, complemented
7C79×#1530C211yesmaximal, characteristic, complemented
8C79×#1527C211yesmaximal, characteristic, complemented
9C79×#1524C211yesmaximal, characteristic, complemented
10C22×Q16×C79—21no—
11C22×SD16×C79—21no—
12#562×C79C2212yes—
13H(3,𝔽4)×C79C2212yes—
14#559×C79—21no—
15#548×C79—21no—
16#546×C79—22no—
17#546×C79—21no—
18D42×C79—21no—
19#542×C79—21no—
20#538×C79—22no—
21#538×C79—21no—
22#535×C79—21no—
23#534×C79—21no—
24#532×C79C2211yescharacteristic
25C2×C4⋊Q8×C79C2211yescharacteristic, complemented
26#529×C79C2211yescharacteristic
27C2×#124×C79C2211yescharacteristic
28#501×C79C2212yescomplemented
29#497×C79—22no—
30#497×C79C2214yescomplemented
31#489×C79C2214yescomplemented
32#489×C79C2212yescomplemented
33#487×C79—21no—
34#486×C79—21no—
35#484×C79—21no—
36#481×C79—21no—
37#475×C79—21no—
38#474×C79—21no—
39#432×C79C2211yescharacteristic, complemented
40#417×C79—22no—
41C4×M16×C79C2211yescharacteristic, complemented
42C2×#97×C79C2211yescharacteristic, complemented
43#330×C79C2212yescomplemented
44#329×C79C2212yescomplemented
45#329×C79C2214yescomplemented
46#329×C79C2214yescomplemented
47C2×#41×C79—21no—
48C2×#41×C79D412yes—
49C22×Q8×C79—21no—
50C22×Q8×C79D411yescharacteristic
51C22×D4×C79—22no—
52C22×D4×C79—21no—
53C22×D4×C79D411yescharacteristic
54C2×Q16×C79—24no—
55C2×Q16×C79—44no—
56C2×SD16×C79—24no—
57C2×SD16×C79—44no—
58C2×M16×C79—21no—
59C22×C8×C79—21no—
60C4⋊Q8×C79—21no—
61C4⋊Q8×C79—21no—
62C4⋊Q8×C79—21no—
63C4⋊Q8×C79D411yescharacteristic
64C4⋊Q8×C79C2312yes—
65#127×C79—21no—
66#127×C79D411yescharacteristic
67#126×C79—22no—
68#126×C79—21no—
69#125×C79C2311yescharacteristic
70#124×C79—21no—
71#124×C79—22no—
72#124×C79—22no—
73#124×C79—22no—
74#124×C79—21no—
75#124×C79—21no—
76#124×C79D412yes—
77#124×C79C2311yescharacteristic
78#124×C79C2312yes—
79#124×C79C2312yes—
80#123×C79—22no—
81#123×C79—22no—
82#123×C79—21no—
83#123×C79—22no—
84#123×C79—22no—
85C22⋊Q8×C79—21no—
86C22⋊Q8×C79—22no—
87C22⋊Q8×C79—22no—
88C22⋊Q8×C79D411yescharacteristic
89#121×C79—22no—
90#121×C79—22no—
91#121×C79—22no—
92#121×C79—21no—
93#121×C79—22no—
94#121×C79—22no—
95#121×C79—21no—
96#121×C79—21no—
97#121×C79D411yescharacteristic
98#120×C79—22no—
99#120×C79—22no—
100#120×C79—22no—
101#120×C79—22no—
102#120×C79—22no—
103C4×Q8×C79—21no—
104C4×Q8×C79D411yescharacteristic
105C4×D4×C79—22no—
106C4×D4×C79—22no—
107C4×D4×C79—21no—
108C4×D4×C79—21no—
109C4×D4×C79—21no—
110C4×D4×C79—21no—
111C4×D4×C79D411yescharacteristic
112#117×C79—21no—
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79—21no—
115C2×C22⋊C4×C79—22no—
116C2×C22⋊C4×C79—21no—
117C2×C22⋊C4×C79—21no—
118C2×C42×C79C2311yescharacteristic
119#107×C79—22no—
120#106×C79—22no—
121C4⋊C8×C79—21no—
122#103×C79—24no—
123#103×C79—22no—
124#102×C79—24no—
125#102×C79—22no—
126C22⋊C8×C79—21no—
127#97×C79—21no—
128#97×C79C2312yes—
129#97×C79C2312yes—
130#97×C79C2312yes—
131C4×C8×C79—21no—
132C24×C79—22no—
133C24×C79—21no—
134#41×C79—44no—
135#41×C79—82no—
136C2×Q8×C79—21no—
137C2×Q8×C79—22no—
138C2×Q8×C79—42no—
139C2×Q8×C79—42no—
140C2×Q8×C79—42no—
141C2×Q8×C79—41no—
142C2×Q8×C79C2×D411yescharacteristic
143C2×Q8×C79C2×D412yes—
144C2×Q8×C79C2×D412yes—
145C2×D4×C79—22no—
146C2×D4×C79—21no—
147C2×D4×C79—22no—
148C2×D4×C79—22no—
149C2×D4×C79—22no—
150C2×D4×C79—22no—
151C2×D4×C79—22no—
152C2×D4×C79—21no—
153C2×D4×C79—21no—
154C2×D4×C79—21no—
155C2×D4×C79—42no—
156C2×D4×C79—82no—
157C2×D4×C79—84no—
158C2×D4×C79C2×D411yescharacteristic
159C2×D4×C79C2×D412yes—
160C2×D4×C79C2×D412yes—
161C22×C4×C79—22no—
162C22×C4×C79—22no—
163C22×C4×C79—21no—
164C22×C4×C79—21no—
165C22×C4×C79—22no—
166C22×C4×C79—41no—
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170Q16×C79—48no—
171SD16×C79—48no—
172M16×C79—42no—
173C2×C8×C79—24no—
174C2×C8×C79—22no—
175C2×C8×C79—22no—
176C2×C8×C79—22no—
177C4⋊C4×C79—22no—
178C4⋊C4×C79—22no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—22no—
186C4⋊C4×C79—22no—
187C4⋊C4×C79—41no—
188C4⋊C4×C79—41no—
189C4⋊C4×C79C2×D411yescharacteristic
190C4⋊C4×C79C2×D411yescharacteristic
191C22⋊C4×C79—22no—
192C22⋊C4×C79—22no—
193C22⋊C4×C79—22no—
194C22⋊C4×C79—22no—
195C22⋊C4×C79—22no—
196C22⋊C4×C79—21no—
197C22⋊C4×C79—22no—
198C22⋊C4×C79—21no—
199C22⋊C4×C79—22no—
200C22⋊C4×C79—21no—
201C22⋊C4×C79—22no—
202C22⋊C4×C79—22no—
203C22⋊C4×C79—21no—
204C22⋊C4×C79—21no—
205C22⋊C4×C79—22no—
206C22⋊C4×C79—22no—
207C42×C79—22no—
208C42×C79—21no—
209C42×C79C2×D411yescharacteristic
210C42×C79C2×D411yescharacteristic
211C42×C79C2×D411yescharacteristic
212C42×C79C2411yescharacteristic
213C23×C79—21no—
214C23×C79—22no—
215C23×C79—22no—
216C23×C79—42no—
217C23×C79—81no—
218C23×C79—81no—
219C23×C79—82no—
220C23×C79—82no—
221C23×C79—82no—
222C23×C79#12011yescharacteristic
223Q8×C79—42no—
224Q8×C79—41no—
225Q8×C79—42no—
226Q8×C79—42no—
227Q8×C79—42no—
228Q8×C79—42no—
229Q8×C79—82no—
230D4×C79—42no—
231D4×C79—41no—
232D4×C79—42no—
233D4×C79—42no—
234D4×C79—44no—
235D4×C79—42no—
236D4×C79—42no—
237D4×C79—81no—
238D4×C79—82no—
239D4×C79—82no—
240D4×C79—82no—
241D4×C79—82no—
242D4×C79—82no—
243C2×C4×C79—22no—
244C2×C4×C79—22no—
245C2×C4×C79—21no—
246C2×C4×C79—22no—
247C2×C4×C79—22no—
248C2×C4×C79—21no—
249C2×C4×C79—21no—
250C2×C4×C79—41no—
251C2×C4×C79—42no—
252C2×C4×C79—42no—
253C2×C4×C79—82no—
254C2×C4×C79—81no—
255C2×C4×C79—82no—
256C2×C4×C79—82no—
257C2×C4×C79—82no—
258C2×C4×C79#12011yescharacteristic
259C2×C4×C79#12011yescharacteristic
260C2×C4×C79#12011yescharacteristic
261C2×C4×C79C22×D411yescharacteristic
262C2×C4×C79C22×D411yescharacteristic
263C2×C4×C79C22×D411yescharacteristic
264C8×C79—24no—
265C8×C79—42no—
266C8×C79—42no—
267C22×C79—21no—
268C22×C79—21no—
269C22×C79—41no—
270C22×C79—42no—
271C22×C79—42no—
272C22×C79—42no—
273C22×C79—42no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79—42no—
277C22×C79—42no—
278C22×C79—162no—
279C22×C79—164no—
280C22×C79#45512yes—
281C22×C79C2×#12011yescharacteristic
282C4×C79—21no—
283C4×C79—21no—
284C4×C79—21no—
285C4×C79—21no—
286C4×C79—21no—
287C4×C79—41no—
288C4×C79—42no—
289C4×C79—41no—
290C4×C79—42no—
291C4×C79—41no—
292C4×C79—82no—
293C4×C79#45312yes—
294#19021C7911yesmaximal, characteristic, direct product factor, Sylow
295C2×C79—22no—
296C2×C79—82no—
297C2×C79—81no—
298C2×C79—82no—
299C2×C79C2×#45311yescharacteristic
300C2×C79C2×#45511yescharacteristic
301C2×C79#302211yescharacteristic
302#3558C2×C7911yescharacteristic
303#3239C2×C7911yescharacteristic, complemented
304C2×#497C2×C7911yescharacteristic, complemented
305#3141C2×C7911yescharacteristic, complemented
306#1661C2×C7914yescomplemented
307#1659C2×C7914yescomplemented
308#1530C2×C7911yescharacteristic, complemented
309#1527C2×C7911yescharacteristic, complemented
310#1524C2×C7911yescharacteristic, complemented
311C79#1902111yescharacteristic, direct product factor, Sylow
312C22×Q16—21no—
313C22×SD16—21no—
314#562C22×C7912yes—
315H(3,𝔽4)C22×C7912yes—
316#559—21no—
317#548—21no—
318#546—22no—
319#546—21no—
320D42—21no—
321#542—21no—
322#538—22no—
323#538—21no—
324#535—21no—
325#534—21no—
326#532C22×C7911yescharacteristic
327C2×C4⋊Q8C22×C7911yescharacteristic, complemented
328#529C22×C7911yescharacteristic
329C2×#124C22×C7911yescharacteristic
330#501C22×C7912yescomplemented
331#497—22no—
332#497C22×C7914yescomplemented
333#489C22×C7914yescomplemented
334#489C22×C7912yescomplemented
335#487—21no—
336#486—21no—
337#484—21no—
338#481—21no—
339#475—21no—
340#474—21no—
341#432C22×C7911yescharacteristic, complemented
342#417—22no—
343C4×M16C22×C7911yescharacteristic, complemented
344C2×#97C22×C7911yescharacteristic, complemented
345#330C22×C7912yescomplemented
346#329C22×C7912yescomplemented
347#329C22×C7914yescomplemented
348#329C22×C7914yescomplemented
349C2×#41—21no—
350C2×#41D4×C7912yes—
351C22×Q8—21no—
352C22×Q8D4×C7911yescharacteristic
353C22×D4—22no—
354C22×D4—21no—
355C22×D4D4×C7911yescharacteristic
356C2×Q16—24no—
357C2×Q16—44no—
358C2×SD16—24no—
359C2×SD16—44no—
360C2×M16—21no—
361C22×C8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—21no—
364C4⋊Q8—21no—
365C4⋊Q8D4×C7911yescharacteristic
366C4⋊Q8C23×C7912yes—
367#127—21no—
368#127D4×C7911yescharacteristic
369#126—22no—
370#126—21no—
371#125C23×C7911yescharacteristic
372#124—21no—
373#124—22no—
374#124—22no—
375#124—22no—
376#124—21no—
377#124—21no—
378#124D4×C7912yes—
379#124C23×C7912yes—
380#124C23×C7911yescharacteristic
381#124C23×C7912yes—
382#123—22no—
383#123—22no—
384#123—21no—
385#123—22no—
386#123—22no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8D4×C7911yescharacteristic
391#121—22no—
392#121—22no—
393#121—22no—
394#121—21no—
395#121—22no—
396#121—22no—
397#121—21no—
398#121—21no—
399#121D4×C7911yescharacteristic
400#120—22no—
401#120—22no—
402#120—22no—
403#120—22no—
404#120—22no—
405C4×Q8—21no—
406C4×Q8D4×C7911yescharacteristic
407C4×D4—22no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4D4×C7911yescharacteristic
414#117—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C42C23×C7911yescharacteristic
421#107—22no—
422#106—22no—
423C4⋊C8—21no—
424#103—24no—
425#103—22no—
426#102—24no—
427#102—22no—
428C22⋊C8—21no—
429#97—21no—
430#97C23×C7912yes—
431#97C23×C7912yes—
432#97C23×C7912yes—
433C4×C8—21no—
434C24—22no—
435C24—21no—
436#41—44no—
437#41—82no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8C2×D4×C7911yescharacteristic
445C2×Q8C2×D4×C7912yes—
446C2×Q8C2×D4×C7912yes—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—82no—
459C2×D4—84no—
460C2×D4C2×D4×C7911yescharacteristic
461C2×D4C2×D4×C7912yes—
462C2×D4C2×D4×C7912yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4C2×D4×C7911yescharacteristic
470C22×C4C2×D4×C7911yescharacteristic
471C22×C4C2×D4×C7911yescharacteristic
472Q16—48no—
473SD16—48no—
474M16—42no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4C2×D4×C7911yescharacteristic
492C4⋊C4C2×D4×C7911yescharacteristic
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C42—22no—
510C42—21no—
511C42C2×D4×C7911yescharacteristic
512C42C2×D4×C7911yescharacteristic
513C42C2×D4×C7911yescharacteristic
514C42C24×C7911yescharacteristic
515C23—21no—
516C23—22no—
517C23—22no—
518C23—42no—
519C23—81no—
520C23—81no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23#120×C7911yescharacteristic
525Q8—42no—
526Q8—41no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—82no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—42no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—81no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544D4—82no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4#120×C7911yescharacteristic
561C2×C4#120×C7911yescharacteristic
562C2×C4#120×C7911yescharacteristic
563C2×C4C22×D4×C7911yescharacteristic
564C2×C4C22×D4×C7911yescharacteristic
565C2×C4C22×D4×C7911yescharacteristic
566C8—24no—
567C8—42no—
568C8—42no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—162no—
581C22—164no—
582C22#455×C7912yes—
583C22C2×#120×C7911yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—82no—
595C4#453×C7912yes—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—82no—
600C2C2×#453×C7911yescharacteristic
601C2C2×#455×C7911yescharacteristic
602C2C79×#302211yescharacteristic
subgroupG/H
center, socleC22×C79C2×#120
commutator, Frattini, cosocleC42C24×C79
prime radicalC79×#3558C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G

Complements

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

NH ≅ G/Ntype
2. C79×#3239C2semidirect
3. C2×#497×C79C2semidirect
4. C79×#3141C2semidirect
5. C79×#1661C2semidirect
6. C79×#1659C2semidirect
7. C79×#1530C2semidirect
8. C79×#1527C2semidirect
9. C79×#1524C2semidirect
25. C2×C4⋊Q8×C79C22semidirect
28. #501×C79C22semidirect
30. #497×C79C22semidirect
31. #489×C79C22semidirect
32. #489×C79C22semidirect
39. #432×C79C22semidirect
41. C4×M16×C79C22semidirect
42. C2×#97×C79C22semidirect
43. #330×C79C22semidirect
44. #329×C79C22semidirect
45. #329×C79C22semidirect
46. #329×C79C22semidirect
294. #19021C79direct (N × H)
303. #3239C2×C79semidirect
304. C2×#497C2×C79semidirect
305. #3141C2×C79semidirect
306. #1661C2×C79semidirect
307. #1659C2×C79semidirect
308. #1530C2×C79semidirect
309. #1527C2×C79semidirect
310. #1524C2×C79semidirect
311. C79#19021direct (N × H)
327. C2×C4⋊Q8C22×C79semidirect
330. #501C22×C79semidirect
332. #497C22×C79semidirect
333. #489C22×C79semidirect
334. #489C22×C79semidirect
341. #432C22×C79semidirect
343. C4×M16C22×C79semidirect
344. C2×#97C22×C79semidirect
345. #330C22×C79semidirect
346. #329C22×C79semidirect
347. #329C22×C79semidirect
348. #329C22×C79semidirect

Maximal subgroups

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

subgroupindex
C79×#35582
C79×#32392
C2×#497×C792
C79×#31412
C79×#16612
C79×#16592
C79×#15302
C79×#15272
C79×#15242
#1902179