Finite Group Databasebeta

← #34361460

Subgroups — #34361460

Order 20224.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767986C211yesmaximal, characteristic
2#23764943C211yesmaximal, characteristic, complemented
3#23764715C211yesmaximal, characteristic
4#23764643C211yesmaximal, characteristic, complemented
5#23764483C211yesmaximal, characteristic
6#23764257C211yesmaximal, characteristic
7#23764031C211yesmaximal, characteristic, complemented
8#23764027C211yesmaximal, characteristic
9#23763978C211yesmaximal, characteristic, complemented
10#23763959C211yesmaximal, characteristic, complemented
11#23763755C211yesmaximal, characteristic
12#23763755C211yesmaximal, characteristic
13#23763405C211yesmaximal, characteristic, complemented
14C4×#14579085C211yesmaximal, characteristic, complemented
15C79×#2365C211yesmaximal, characteristic, complemented
16C22×#11929611C2211yescharacteristic
17C22×#11929611C2211yescharacteristic
18C2×#14579156—21no—
19C2×#14579092C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×#14579085C2211yescharacteristic
27C2×#14579085C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579081C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31#19098490—21no—
32#19098485—21no—
33#19098465C2211yescharacteristic
34#19098465C2211yescharacteristic
35#19098465C2211yescharacteristic
36#19098461C2211yescharacteristic
37#19098455—21no—
38#19098433C2211yescharacteristic
39C4×C4⋊Dic79C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098318C2211yescharacteristic
45#19098318C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C4⋊Q8×C79C2211yescharacteristic
49C2×C4×Q8×C79C2211yescharacteristic
50C2×C4×Q8×C79C2211yescharacteristic
51#391×C79C2211yescharacteristic
52#391×C79C2211yescharacteristic
53#389×C79—21no—
54#387×C79—21no—
55#387×C79—21no—
56C4×C4⋊C4×C79—21no—
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59C22×D316C2311yescharacteristic
60C2×#11929611—22no—
61C2×#11929611—22no—
62C2×#11929611—22no—
63C2×#11929611—41no—
64C2×#11929611—41no—
65C2×#11929611—41no—
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70#14579156—44no—
71C2×C4⋊Dic79—21no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×C4×Dic79—21no—
84#14579092—22no—
85#14579092—41no—
86C4⋊(C79⋊C8)—41no—
87#14579085—22no—
88#14579085—22no—
89#14579085—22no—
90#14579085—41no—
91#14579085—41no—
92#14579085—41no—
93#14579081—22no—
94#14579081—41no—
95C4×C79⋊C8—41no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C22×Q8×C79C2311yescharacteristic
99C22×Q8×C79C2311yescharacteristic
100C4⋊Q8×C79—22no—
101C4⋊Q8×C79—22no—
102C4×Q8×C79—22no—
103C4×Q8×C79—22no—
104C4×Q8×C79—41no—
105C4×Q8×C79—41no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79—21no—
109C2×C4⋊C4×C79—21no—
110C2×C4⋊C4×C79D411yescharacteristic
111C2×C4⋊C4×C79D411yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C42×C79—21no—
115C2×C42×C79—21no—
116C2×C42×C79—21no—
117C2×C42×C79C2311yescharacteristic
118C2×C42×C79C2311yescharacteristic
119C2×C42×C79C2311yescharacteristic
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124#95×C79—21no—
125#95×C79D411yescharacteristic
126#95×C79D411yescharacteristic
127C23×D79—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316—21no—
133C2×D316—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C22×Dic79—21no—
137#11929611—44no—
138#11929611—44no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8#4112yes—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—42no—
158C4⋊Dic79—81no—
159Dic79⋊C4—42no—
160(C2×D79)⋊C4—42no—
161(C2×D79)⋊C4—42no—
162(C2×D79)⋊C4—42no—
163(C2×D79)⋊C4—42no—
164(C2×D79)⋊C4—81no—
165(C2×D79)⋊C4—81no—
166C4×Dic79—42no—
167C2×Q8×C79—21no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×C79—21no—
172C2×Q8×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79—21no—
175C2×Q8×C79—21no—
176C2×Q8×C79—21no—
177C2×Q8×C79#4112yes—
178C2×Q8×C79#4112yes—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—21no—
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79C2×D411yescharacteristic
189C22×C4×C79C2411yescharacteristic
190C4⋊C4×C79—24no—
191C4⋊C4×C79—24no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—41no—
196C4⋊C4×C79—42no—
197C4⋊C4×C79—42no—
198C4⋊C4×C79SD1614yescomplemented
199C4⋊C4×C79#4112yes—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79—21no—
206C42×C79—42no—
207C42×C79—42no—
208C42×C79—42no—
209C23×C79C22×D411yescharacteristic
210C22×D79—41no—
211C22×D79—41no—
212C22×D79—41no—
213C22×D79—41no—
214C22×D79—41no—
215C22×D79—41no—
216C22×D79—41no—
217Q8×C79—22no—
218Q8×C79—22no—
219Q8×C79—22no—
220Q8×C79—22no—
221Q8×C79—41no—
222Q8×C79—41no—
223Q8×C79—41no—
224Q8×C79—41no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—22no—
233C2×C4×C79—22no—
234C2×C4×C79—21no—
235C2×C4×C79—21no—
236C2×C4×C79—22no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79—41no—
247C2×C4×C79—41no—
248C2×C4×C79—41no—
249C2×C4×C79—41no—
250C2×C4×C79—41no—
251C2×C4×C79—41no—
252C2×C4×C79C2×SD1612yes—
253C2×C4×C79C2×SD1612yes—
254C2×C4×C79C2×SD1612yes—
255C2×C4×C79C2×#4111yescharacteristic
256C2×C4×C79C2×#4111yescharacteristic
257C2×C4×C79C2×#4111yescharacteristic
258C2×C4×C79C2×#4111yescharacteristic
259C2×C4×C79H(5,𝔽2)11yescharacteristic
260C2×C4×C79H(5,𝔽2)11yescharacteristic
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—41no—
268C2×Dic79—41no—
269C2×Dic79—41no—
270D316—41no—
271D316—41no—
272D316—41no—
273D316—41no—
274C79⋊C8—22no—
275C79⋊C8—41no—
276C79⋊C8—41no—
277C79⋊C8—41no—
278C22×C79#52511yescharacteristic
279C22×C79#54611yescharacteristic
280C22×C79#55011yescharacteristic
281C22×C79C22×SD1611yescharacteristic
282C22×C79#57511yescharacteristic
283C22×C79#57711yescharacteristic
284C22×C79#57811yescharacteristic
285C2×D79—81no—
286C2×D79—81no—
287C2×D79—81no—
288C2×D79—81no—
289C2×D79—81no—
290C2×D79—81no—
291C2×D79—81no—
292C4×C79—22no—
293C4×C79—22no—
294C4×C79—22no—
295C4×C79—41no—
296C4×C79—42no—
297C4×C79—42no—
298C4×C79—41no—
299C4×C79—41no—
300C4×C79—42no—
301C4×C79—81no—
302C4×C79#52911yescharacteristic
303C4×C79#53811yescharacteristic
304C4×C79#55111yescharacteristic
305C4×C79#55311yescharacteristic
306Dic79—81no—
307Dic79—81no—
308Dic79—81no—
309#27066—791nomaximal, Sylow
310C2×C79#246711yescharacteristic
311C2×C79#310311yescharacteristic
312C2×C79#312111yescharacteristic
313C2×C79#329811yescharacteristic
314C2×C79#330711yescharacteristic
315C2×C79#339211yescharacteristic
316C2×C79#340211yescharacteristic
317D79—161no—
318#2512—791no—
319#2365D7911yescharacteristic, complemented
320#2096—791no—
321#2055—791no—
322#2043—791no—
323#2013—791no—
324#1968—791no—
325#1927—791no—
326#1926—791no—
327#1919—791no—
328#1916—791no—
329#1880—791no—
330#1880—791no—
331#1803—791no—
332C4×#103—791no—
333C79#2706611yescharacteristic, complemented, Sylow
334C22×SD16—791no—
335C22×SD16—791no—
336C2×C4⋊Q8C2×D7911yescharacteristic
337C2×#126—1581no—
338C2×C4×Q8C2×D7911yescharacteristic
339C2×C4×Q8C2×D7911yescharacteristic
340C2×#106—791no—
341C2×#106—791no—
342C2×C4⋊C8—791no—
343C2×#103—791no—
344C2×#103—791no—
345C2×#103—791no—
346C2×#103—791no—
347C2×#103—791no—
348C2×#103—791no—
349C2×#102—791no—
350C2×#102—791no—
351C2×C4×C8—791no—
352#397—1581no—
353#396—1581no—
354#391C2×D7911yescharacteristic
355#391C2×D7911yescharacteristic
356#390—791no—
357#390—791no—
358#390—791no—
359#389—21no—
360#389—791no—
361#388—1581no—
362#387—21no—
363#387—21no—
364#384—791no—
365C4×C4⋊C4—21no—
366C4×C4⋊C4—791no—
367C4×C4⋊C4C2×D7911yescharacteristic
368C4×C4⋊C4C2×D7911yescharacteristic
369#340—791no—
370#340—791no—
371#340—791no—
372#340—791no—
373#336—791no—
374#336—791no—
375#336—791no—
376#336—791no—
377C22×Q8C22×D7911yescharacteristic
378C22×Q8C22×D7911yescharacteristic
379C22×D4—791no—
380C2×SD16—3161no—
381C2×SD16—3161no—
382C2×SD16—3161no—
383C2×SD16—1582no—
384C2×SD16—1582no—
385C2×SD16—1582no—
386C22×C8—791no—
387C22×C8—791no—
388C22×C8—791no—
389C22×C8—791no—
390C4⋊Q8—22no—
391C4⋊Q8—22no—
392#126—3164no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—41no—
396C4×Q8—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—791no—
402C2×C4⋊C4—791no—
403C2×C4⋊C4—791no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4—1581no—
406C2×C4⋊C4—1581no—
407C2×C4⋊C4C22⋊D7911yescharacteristic
408C2×C4⋊C4C22⋊D7911yescharacteristic
409C2×C4⋊C4C22×D7911yescharacteristic
410C2×C4⋊C4C22×D7911yescharacteristic
411C2×C22⋊C4—1581no—
412C2×C22⋊C4—1581no—
413C2×C22⋊C4—1581no—
414C2×C22⋊C4—1581no—
415C2×C22⋊C4—1581no—
416C2×C22⋊C4—1581no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—1581no—
421C2×C42C22×D7911yescharacteristic
422C2×C42C22×D7911yescharacteristic
423C2×C42C22×D7911yescharacteristic
424#106—3161no—
425#106—1582no—
426C4⋊C8—3161no—
427#103—3161no—
428#103—3161no—
429#103—3161no—
430#103—1582no—
431#103—1582no—
432#103—1582no—
433#102—3161no—
434#102—1582no—
435C4×C8—3161no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1581no—
442#95—1581no—
443#95C22⋊D7911yescharacteristic
444#95C22⋊D7911yescharacteristic
445C24—1581no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8#1192962412yes—
457C2×Q8#1192962412yes—
458C2×D4—1581no—
459C2×D4—1581no—
460C2×D4—1581no—
461C2×D4—1581no—
462C2×D4—1581no—
463C2×D4—1581no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1581no—
469C22×C4—1581no—
470C22×C4—1581no—
471C22×C4C2×C22⋊D7911yescharacteristic
472C22×C4C2×C22⋊D7911yescharacteristic
473C22×C4C2×C22⋊D7911yescharacteristic
474C22×C4C2×C22⋊D7911yescharacteristic
475C22×C4C2×C22⋊D7911yescharacteristic
476C22×C4C2×C22⋊D7911yescharacteristic
477C22×C4C23×D7911yescharacteristic
478SD16—3164no—
479SD16—3164no—
480C2×C8—792no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8—1581no—
491C2×C8—1581no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—3162no—
501C4⋊C4—3162no—
502C4⋊C4—6321no—
503C4⋊C4—1581no—
504C4⋊C4—1581no—
505C4⋊C4—1581no—
506C4⋊C4—1581no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4#1192961114yescomplemented
510C4⋊C4#1192962312yes—
511C22⋊C4—3162no—
512C22⋊C4—3162no—
513C22⋊C4—3162no—
514C22⋊C4—3162no—
515C22⋊C4—6321no—
516C22⋊C4—6321no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—3162no—
527C23—3161no—
528C23—3161no—
529C23—3161no—
530C23—3161no—
531C23—3161no—
532C23—3161no—
533C23—3161no—
534C23C22×C22⋊D7911yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543D4—3161no—
544D4—3161no—
545D4—3161no—
546D4—3161no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—3161no—
575C2×C4—3161no—
576C2×C4—3161no—
577C2×C4—3161no—
578C2×C4—3161no—
579C2×C4—3161no—
580C2×C4—3161no—
581C2×C4—3161no—
582C2×C4—3161no—
583C2×C4C2×#1192961112yes—
584C2×C4C2×#1192961112yes—
585C2×C4C2×#1192961112yes—
586C2×C4#41×D7911yescharacteristic
587C2×C4C2×#1192962311yescharacteristic
588C2×C4C2×#1192962411yescharacteristic
589C2×C4C2×#1192962411yescharacteristic
590C2×C4#1457920611yescharacteristic
591C2×C4#1457920711yescharacteristic
592C8—3161no—
593C8—3161no—
594C8—3161no—
595C8—1582no—
596C22—6321no—
597C22—6321no—
598C22—6321no—
599C22—6321no—
600C22—6321no—
601C22—6321no—
602C22—6321no—
603C22#1909907411yescharacteristic
604C22#1909920711yescharacteristic
605C22#1909923211yescharacteristic
606C22C22×#1192961111yescharacteristic
607C22#1909936111yescharacteristic
608C22#1909937111yescharacteristic
609C22#1909937811yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—6321no—
621C4—6321no—
622C4—6321no—
623C4#1909909311yescharacteristic
624C4#1909914611yescharacteristic
625C4#1909924211yescharacteristic
626C4#1909925911yescharacteristic
627C2—12641no—
628C2#2376744511yescharacteristic
629C2#2377343111yescharacteristic
630C2#2377362911yescharacteristic
631C2#2377526511yescharacteristic
632C2#2377537611yescharacteristic
633C2#2377637711yescharacteristic
634C2#2377647511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2365C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2365D79
hypocenterC79#27066
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #23764943C2semidirect
4. #23764643C2semidirect
7. #23764031C2semidirect
9. #23763978C2semidirect
10. #23763959C2semidirect
13. #23763405C2semidirect
14. C4×#14579085C2semidirect
15. C79×#2365C2semidirect
198. C4⋊C4×C79SD16semidirect
319. #2365D79semidirect
333. C79#27066semidirect
509. C4⋊C4#11929611semidirect

Maximal subgroups

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

subgroupindex
#237679862
#237649432
#237647152
#237646432
#237644832
#237642572
#237640312
#237640272
#237639782
#237639592
#237637552
#237634052
C4×#145790852
C79×#23652
#2706679