Finite Group Databasebeta

← #34317196

Subgroups — #34317196

Order 20224.

892 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098581C211yesmaximal, characteristic
2#23771920C211yesmaximal, characteristic
3C2×#19098545C211yesmaximal, characteristic
4#23764851C218yesmaximal, complemented
5C2×#19098335C211yesmaximal, characteristic, complemented
6#23763127C211yesmaximal, characteristic, complemented
7C2×#19098319C211yesmaximal, characteristic, complemented
8C79×#2905C211yesmaximal, characteristic, complemented
9C22×#11929624C2211yescharacteristic
10C22×#11929608C2211yescharacteristic
11C22×#11929608C2211yescharacteristic
12C2×#14579120C2211yescharacteristic
13#19098581—22no—
14#19098581C2214yes—
15C2×#14579089—22no—
16#19098573—21no—
17#19098569—21no—
18#19098565—22no—
19#19098560C2214yes—
20#19098545—22no—
21#19098545C2214yes—
22C2×#14579079—21no—
23C2×#14579077—21no—
24#19098335C2214yescomplemented
25#19098328C2214yescomplemented
26#19098319C2214yescomplemented
27C22×M16×C79C2211yescharacteristic
28C22×M16×C79C2211yescharacteristic
29C2×#117×C79C2211yescharacteristic
30#424×C79—22no—
31#424×C79C2214yescomplemented
32#423×C79—22no—
33C2×#11929624—21no—
34C2×#11929624—21no—
35C2×#11929624—21no—
36C2×#11929624—21no—
37C2×#11929624—21no—
38C2×#11929624—21no—
39C2×#11929624—21no—
40C2×#11929624—21no—
41C2×#11929624—21no—
42C2×#11929624—21no—
43C2×#11929624D412yes—
44C2×#11929624C2312yes—
45C22×Dic158D411yescharacteristic
46C22×D316D411yescharacteristic
47C22×C22⋊D79—21no—
48C22×C4×D79—21no—
49C2×#11929608—21no—
50C2×#11929608—21no—
51C2×#11929608—21no—
52C2×#11929608—21no—
53C2×#11929608—21no—
54C2×#11929608—21no—
55C2×#11929608—21no—
56C2×#11929608—21no—
57C2×#11929608—21no—
58C2×#11929608—21no—
59C2×#11929608C2312yes—
60C2×#11929608C2312yes—
61C22×C79⋊C8—21no—
62C22×C79⋊C8—21no—
63#14579120—21no—
64#14579120—21no—
65#14579120—21no—
66#14579120C2312yes—
67C2×C4⋊Dic79—21no—
68C2×C22⋊Dic79—21no—
69C2×C4×Dic79—21no—
70#14579089—28no—
71#14579086—24no—
72#14579083—24no—
73#14579079—24no—
74#14579077—24no—
75C23×C4×C79C2311yescharacteristic
76C2×M16×C79—21no—
77C2×M16×C79—21no—
78C2×M16×C79—21no—
79C2×M16×C79—21no—
80C2×M16×C79—21no—
81C2×M16×C79—21no—
82C2×M16×C79—21no—
83C2×M16×C79—21no—
84C2×M16×C79—21no—
85C2×M16×C79—21no—
86C2×M16×C79—21no—
87C2×M16×C79C2312yes—
88C22×C8×C79—21no—
89C22×C8×C79C2312yes—
90#117×C79—21no—
91#117×C79—21no—
92#117×C79D412yes—
93#117×C79C2312yes—
94C2×C4⋊C4×C79—21no—
95C2×C22⋊C4×C79—21no—
96C2×C42×C79D412yes—
97C4⋊C8×C79—28no—
98C23×D79—21no—
99#11929624—22no—
100#11929624—22no—
101#11929624—22no—
102#11929624—22no—
103#11929624—22no—
104#11929624—22no—
105#11929624—22no—
106#11929624—22no—
107#11929624—41no—
108#11929624—41no—
109#11929624—41no—
110#11929624—41no—
111#11929624—41no—
112#11929624—41no—
113#11929624—41no—
114#11929624—41no—
115C2×Dic158—21no—
116C2×Dic158—21no—
117C2×Dic158—21no—
118C2×Dic158—21no—
119C2×Dic158—21no—
120C2×Dic158C2×D412yes—
121C2×D316—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316—21no—
125C2×D316—21no—
126C2×D316C2×D412yes—
127C2×C22⋊D79—22no—
128C2×C22⋊D79—22no—
129C2×C22⋊D79—41no—
130C2×C22⋊D79—41no—
131C2×C22⋊D79—41no—
132C2×C22⋊D79—41no—
133C22×Dic79—21no—
134C22×Dic79—41no—
135C2×C4×D79—22no—
136C2×C4×D79—22no—
137C2×C4×D79—41no—
138C2×C4×D79—41no—
139C2×C4×D79—41no—
140C2×C4×D79—41no—
141#11929608—22no—
142#11929608—22no—
143#11929608—22no—
144#11929608—22no—
145#11929608—41no—
146#11929608—41no—
147#11929608—41no—
148#11929608—41no—
149C2×C79⋊C8—22no—
150C2×C79⋊C8—22no—
151C2×C79⋊C8—41no—
152C2×C79⋊C8—41no—
153C2×C79⋊C8—41no—
154C2×C79⋊C8—41no—
155C4⋊Dic79—22no—
156C4⋊Dic79—41no—
157C22⋊Dic79—22no—
158C22⋊Dic79—41no—
159C4×Dic79—22no—
160C4×Dic79—41no—
161C24×C79#4111yescharacteristic
162C22×C4×C79—22no—
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79#4111yescharacteristic
168C22×C4×C79#4111yescharacteristic
169C22×C4×C79#4111yescharacteristic
170C22×C4×C79#4111yescharacteristic
171C22×C4×C79#4111yescharacteristic
172C22×C4×C79#4111yescharacteristic
173C22×C4×C79#4111yescharacteristic
174C22×C4×C79#4111yescharacteristic
175C22×C4×C79#4111yescharacteristic
176C22×C4×C79C2411yescharacteristic
177M16×C79—22no—
178M16×C79—22no—
179M16×C79—41no—
180M16×C79—41no—
181M16×C79—41no—
182M16×C79—41no—
183M16×C79—41no—
184M16×C79—41no—
185C2×C8×C79—22no—
186C2×C8×C79—22no—
187C2×C8×C79—22no—
188C2×C8×C79—22no—
189C2×C8×C79—41no—
190C2×C8×C79—41no—
191C4⋊C4×C79—22no—
192C4⋊C4×C79—22no—
193C22⋊C4×C79—22no—
194C22⋊C4×C79—22no—
195C42×C79—22no—
196C42×C79C2×D414yes—
197C23×C79—81no—
198C23×C79#12311yescharacteristic
199C23×C79#12311yescharacteristic
200C23×C79#12311yescharacteristic
201C23×C79#12311yescharacteristic
202C23×C79#12611yescharacteristic
203C23×C79#12611yescharacteristic
204C23×C79C2×#4111yescharacteristic
205C22×D79—22no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—82no—
209C2×C4×C79—24no—
210C2×C4×C79—82no—
211C2×C4×C79#12111yescharacteristic
212C2×C4×C79#12111yescharacteristic
213C2×C4×C79#12111yescharacteristic
214C2×C4×C79#12111yescharacteristic
215C2×C4×C79#12311yescharacteristic
216C2×C4×C79#12311yescharacteristic
217C2×C4×C79#12311yescharacteristic
218C2×C4×C79#12311yescharacteristic
219C2×C4×C79#12311yescharacteristic
220C2×C4×C79#12311yescharacteristic
221C2×C4×C79#12311yescharacteristic
222C2×C4×C79#12311yescharacteristic
223C2×C4×C79#12411yescharacteristic
224C2×C4×C79#12411yescharacteristic
225C2×C4×C79#12411yescharacteristic
226C2×C4×C79#12411yescharacteristic
227C2×C4×C79#12611yescharacteristic
228C2×C4×C79#12611yescharacteristic
229C2×C4×C79#12611yescharacteristic
230C2×C4×C79#12611yescharacteristic
231C2×C4×C79#12611yescharacteristic
232C2×C4×C79#12611yescharacteristic
233C2×C4×C79C22×D411yescharacteristic
234C2×C4×C79C22×D411yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C22⋊D79—42no—
240C22⋊D79—42no—
241C22⋊D79—42no—
242C22⋊D79—42no—
243C2×Dic79—22no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—42no—
247C2×Dic79—82no—
248D316—22no—
249D316—22no—
250D316—41no—
251D316—41no—
252C4×D79—42no—
253C4×D79—42no—
254C4×D79—42no—
255C4×D79—42no—
256Dic158—22no—
257Dic158—22no—
258Dic158—41no—
259Dic158—41no—
260C8×C79—42no—
261C8×C79—42no—
262C79⋊C8—42no—
263C79⋊C8—42no—
264C22×C79—41no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79#39711yescharacteristic
272C22×C79#39711yescharacteristic
273C22×C79#39711yescharacteristic
274C22×C79#39711yescharacteristic
275C22×C79C2×#12311yescharacteristic
276C22×C79C2×#12311yescharacteristic
277C22×C79C2×#12611yescharacteristic
278C2×D79—42no—
279C2×D79—82no—
280C2×D79—82no—
281C2×D79—82no—
282C4×C79—44no—
283C4×C79#39711yescharacteristic
284C4×C79#39711yescharacteristic
285C4×C79#39711yescharacteristic
286C4×C79#39711yescharacteristic
287C4×C79C2×#12111yescharacteristic
288C4×C79C2×#12311yescharacteristic
289C4×C79C2×#12411yescharacteristic
290C4×C79C2×#12611yescharacteristic
291Dic79—42no—
292Dic79—82no—
293#23649—791nomaximal, Sylow
294C2×C79—21no—
295C2×C79—21no—
296C2×C79—21no—
297C2×C79—21no—
298C2×C79—21no—
299C2×C79—21no—
300C2×C79—21no—
301C2×C79C2×#39711yescharacteristic
302D79—82no—
303#2905D7911yescharacteristic, complemented
304C2×#421—791no—
305#2896—791no—
306C2×#413—791no—
307#2080—798no—
308C2×#344—791no—
309#1738—791no—
310C2×H(3,ℤ/4)—791no—
311C79#2364911yescharacteristic, complemented, Sylow
312C22×#41—791no—
313C22×M16—791no—
314C22×M16—791no—
315C22×M16C2×D7911yescharacteristic
316C22×M16C2×D7911yescharacteristic
317C2×#117—791no—
318C2×#117C2×D7911yescharacteristic
319#424—22no—
320#424C2×D7914yescomplemented
321#423—22no—
322#421—794no—
323#421—1582no—
324C2×#104—1582no—
325#419—1581no—
326#418—1581no—
327#417—1582no—
328#416—794no—
329#413—794no—
330#413—1582no—
331C2×#101—1581no—
332C2×#100—1581no—
333#344—794no—
334#341—794no—
335H(3,ℤ/4)—794no—
336C2×#41—792no—
337C2×#41—792no—
338C2×#41—1581no—
339C2×#41—1581no—
340C2×#41—1581no—
341C2×#41—1581no—
342C2×#41—1581no—
343C2×#41—1581no—
344C2×#41—1581no—
345C2×#41—1581no—
346C2×#41—1581no—
347C2×#41—1581no—
348C22×Q8—791no—
349C22×D4—791no—
350C22×D4—1581no—
351C23×C4—1581no—
352C23×C4C22×D7911yescharacteristic
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—792no—
365C2×M16—792no—
366C2×M16—1581no—
367C2×M16—1581no—
368C2×M16—1581no—
369C2×M16—1581no—
370C2×M16—1581no—
371C2×M16—1581no—
372C2×M16—1581no—
373C2×M16—1581no—
374C2×M16—1581no—
375C2×M16—1581no—
376C2×M16C22×D7912yes—
377C22×C8—21no—
378C22×C8—1581no—
379C22×C8—1581no—
380C22×C8C22×D7912yes—
381#117—21no—
382#117—21no—
383#117—792no—
384#117—1581no—
385#117—1581no—
386#117—1581no—
387#117C22⋊D7912yes—
388#117C22×D7912yes—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—1581no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1581no—
393C2×C42—1581no—
394C2×C42C22⋊D7912yes—
395C4⋊C8—28no—
396#104—1588no—
397#103—1584no—
398#102—1584no—
399#101—1584no—
400#100—1584no—
401C24—1581no—
402C24#1192962411yescharacteristic
403#41—3161no—
404#41—3161no—
405#41—3161no—
406#41—3161no—
407#41—3161no—
408#41—3161no—
409#41—3161no—
410#41—3161no—
411#41—1582no—
412#41—1582no—
413#41—1582no—
414#41—1582no—
415#41—1582no—
416#41—1582no—
417#41—1582no—
418#41—1582no—
419C2×Q8—792no—
420C2×Q8—1581no—
421C2×Q8—1581no—
422C2×Q8—1581no—
423C2×Q8—1581no—
424C2×Q8—1581no—
425C2×D4—3161no—
426C2×D4—3161no—
427C2×D4—3161no—
428C2×D4—3161no—
429C2×D4—792no—
430C2×D4—1582no—
431C2×D4—1582no—
432C2×D4—1581no—
433C2×D4—1581no—
434C2×D4—1581no—
435C2×D4—1581no—
436C2×D4—1581no—
437C22×C4—22no—
438C22×C4—3161no—
439C22×C4—3161no—
440C22×C4—3161no—
441C22×C4—3161no—
442C22×C4—3161no—
443C22×C4—1581no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4D4×D7911yescharacteristic
447C22×C4D4×D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C2×C22⋊D7911yescharacteristic
450C22×C4#1192962311yescharacteristic
451C22×C4#1192962311yescharacteristic
452C22×C4#1192962411yescharacteristic
453C22×C4#1192962411yescharacteristic
454C22×C4#1192962411yescharacteristic
455C22×C4#1192962511yescharacteristic
456C22×C4#1192962511yescharacteristic
457C22×C4#1192962511yescharacteristic
458C22×C4#1192962511yescharacteristic
459C22×C4C23×D7911yescharacteristic
460M16—22no—
461M16—22no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—3161no—
469M16—3161no—
470M16—3161no—
471M16—3161no—
472M16—1582no—
473M16—1582no—
474M16—1582no—
475M16—1582no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—3161no—
483C2×C8—3161no—
484C2×C8—3161no—
485C2×C8—3161no—
486C2×C8—1582no—
487C2×C8—1582no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—3161no—
491C4⋊C4—1582no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—3161no—
495C22⋊C4—1582no—
496C42—22no—
497C42—3161no—
498C42—1582no—
499C42C2×C22⋊D7914yes—
500C23—81no—
501C23—3161no—
502C23—3161no—
503C23—6322no—
504C23—1582no—
505C23#1457914411yescharacteristic
506C23#1457914411yescharacteristic
507C23#1457914611yescharacteristic
508C23#1457914611yescharacteristic
509C23#1457915611yescharacteristic
510C23#1457915611yescharacteristic
511C23C2×#1192962411yescharacteristic
512Q8—3161no—
513Q8—3161no—
514Q8—1582no—
515Q8—1582no—
516D4—3162no—
517D4—3162no—
518D4—3162no—
519D4—3162no—
520D4—3161no—
521D4—3161no—
522D4—1582no—
523D4—1582no—
524C2×C4—24no—
525C2×C4—82no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3162no—
529C2×C4—3162no—
530C2×C4—3162no—
531C2×C4—3162no—
532C2×C4—3162no—
533C2×C4—6322no—
534C2×C4—1582no—
535C2×C4#1457913211yescharacteristic
536C2×C4#1457913211yescharacteristic
537C2×C4#1457913211yescharacteristic
538C2×C4#1457913211yescharacteristic
539C2×C4#1457914411yescharacteristic
540C2×C4#1457914411yescharacteristic
541C2×C4#1457914611yescharacteristic
542C2×C4#1457914611yescharacteristic
543C2×C4#1457914611yescharacteristic
544C2×C4#1457914611yescharacteristic
545C2×C4#1457914611yescharacteristic
546C2×C4#1457914611yescharacteristic
547C2×C4#1457915011yescharacteristic
548C2×C4#1457915011yescharacteristic
549C2×C4#1457915011yescharacteristic
550C2×C4#1457915011yescharacteristic
551C2×C4#1457915611yescharacteristic
552C2×C4#1457915611yescharacteristic
553C2×C4#1457915611yescharacteristic
554C2×C4#1457915611yescharacteristic
555C2×C4#1457915611yescharacteristic
556C2×C4#1457915611yescharacteristic
557C2×C4C2×D4×D7911yescharacteristic
558C2×C4C22×C22⋊D7911yescharacteristic
559C2×C4C2×#1192962311yescharacteristic
560C2×C4C2×#1192962411yescharacteristic
561C2×C4C2×#1192962511yescharacteristic
562C2×C4C2×#1192962511yescharacteristic
563C8—42no—
564C8—42no—
565C8—3162no—
566C8—3162no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—3162no—
575C22—6322no—
576C22—6322no—
577C22—6322no—
578C22#1909848911yescharacteristic
579C22#1909848911yescharacteristic
580C22#1909848911yescharacteristic
581C22#1909848911yescharacteristic
582C22C2×#1457914411yescharacteristic
583C22C2×#1457914611yescharacteristic
584C22C2×#1457915611yescharacteristic
585C4—44no—
586C4—3162no—
587C4—6322no—
588C4#1909848911yescharacteristic
589C4#1909848911yescharacteristic
590C4#1909848911yescharacteristic
591C4#1909848911yescharacteristic
592C4C2×#1457913211yescharacteristic
593C4C2×#1457914611yescharacteristic
594C4C2×#1457915011yescharacteristic
595C4C2×#1457915611yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—6322no—
604C2C2×#1909848911yescharacteristic
subgroupG/H
centerC4C2×#14579150
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2905C2
socleC2×C79C2×#397
prime radicalC22×#11929624C22
hypercenter#2905D79
hypocenterC79#23649
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
4. #23764851C2semidirect
5. C2×#19098335C2semidirect
6. #23763127C2semidirect
7. C2×#19098319C2semidirect
8. C79×#2905C2semidirect
24. #19098335C22semidirect
25. #19098328C22semidirect
26. #19098319C22semidirect
31. #424×C79C22semidirect
303. #2905D79semidirect
311. C79#23649semidirect
320. #424C2×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190985812
#237719202
C2×#190985452
#237648512
C2×#190983352
#237631272
C2×#190983192
C79×#29052
#2364979