Finite Group Databasebeta

← #34259708

Subgroups — #34259708

Order 20224.

716 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772352C211yesmaximal, characteristic, complemented
2C4×SD16×D79C211yesmaximal, characteristic
3#23772136C211yesmaximal, characteristic
4#23762291C211yesmaximal, characteristic, complemented
5#23762246C211yesmaximal, characteristic, complemented
6#23762241C211yesmaximal, characteristic, complemented
7#23762239C211yesmaximal, characteristic, complemented
8#23762234C211yesmaximal, characteristic, complemented
9C8×#11929613C211yesmaximal, characteristic, complemented
10#23762199C211yesmaximal, characteristic, complemented
11#23762154C211yesmaximal, characteristic, complemented
12#23761827C211yesmaximal, characteristic, complemented
13#23761823C211yesmaximal, characteristic
14C8×#11929607C211yesmaximal, characteristic, complemented
15C79×#1590C211yesmaximal, characteristic, complemented
16C2×SD16×D79C411yescharacteristic
17C2×M16×D79—21no—
18C4×Q8×D79C2211yescharacteristic
19C4×D4×D79C2211yescharacteristic
20#19098694C2211yescharacteristic, complemented
21#19098692—21no—
22#19098691—21no—
23C8×Dic158C2211yescharacteristic, complemented
24#19098687C2211yescharacteristic, complemented
25#19098654C411yescharacteristic
26C4×#11929613C2211yescharacteristic
27SD16×Dic79C411yescharacteristic
28C4×#11929611C2211yescharacteristic
29#19098650C411yescharacteristic
30C4×#11929612C2211yescharacteristic
31#19098648C411yescharacteristic
32#19098641C2211yescharacteristic
33#19098637—21no—
34#19098632—21no—
35C8×D316C2211yescharacteristic
36#19098628C2211yescharacteristic
37#19098614C2211yescharacteristic
38#106×D79C411yescharacteristic
39#19098585—21no—
40#19098585C2211yescharacteristic
41#19098585C2211yescharacteristic
42#103×D79C411yescharacteristic
43#102×D79C411yescharacteristic
44#19098525—21no—
45#19098515—21no—
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098314C2211yescharacteristic, complemented
49#19098313C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51#19098295C2211yescharacteristic, complemented
52Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
53#19098293C2211yescharacteristic, complemented
54#19098292C2211yescharacteristic
55D4⋊(C79⋊C8)C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic, complemented
58#19098284C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60#446×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#334×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic, complemented
67C2×Q8×D79C2×C411yescharacteristic
68C2×D4×D79C2×C411yescharacteristic
69C22×C4×D79—21no—
70C2×#11929613C2×C411yescharacteristic
71C2×#11929612C2×C411yescharacteristic
72C2×#11929611C2×C411yescharacteristic
73SD16×D79—24no—
74C2×#11929608—21no—
75C2×#11929607—21no—
76C2×#11929607—21no—
77C2×#11929607D411yescharacteristic
78C2×#11929607D411yescharacteristic
79M16×D79—44no—
80C2×C8×D79—21no—
81C2×C8×D79—21no—
82C2×C8×D79C2×C411yescharacteristic
83C2×C8×D79C2×C411yescharacteristic
84Q8×Dic79C2×C411yescharacteristic
85C4×Dic158—21no—
86C4×Dic158C2311yescharacteristic
87#14579126—21no—
88#14579126C2×C411yescharacteristic
89C4×D316C2311yescharacteristic
90D4×Dic79C2×C411yescharacteristic
91#14579123C2×C411yescharacteristic
92C4×C22⋊D79—21no—
93(C22⋊D79)⋊C4—21no—
94C4⋊C4×D79—21no—
95C4⋊C4×D79C2×C411yescharacteristic
96C4⋊C4×D79C2×C411yescharacteristic
97C22⋊C4×D79—21no—
98C42×D79—21no—
99C42×D79C2311yescharacteristic
100#14579093C2×C411yescharacteristic
101#14579092C2×C411yescharacteristic
102#14579092C2×C411yescharacteristic
103C4⋊(C79⋊C8)—21no—
104C4⋊(C79⋊C8)C2311yescharacteristic
105C4⋊(C79⋊C8)C2311yescharacteristic
106Dic79⋊C8—21no—
107Dic79⋊C8—21no—
108Dic79⋊C8—21no—
109Dic79⋊C8—21no—
110Dic79⋊C8—21no—
111#14579086C2×C411yescharacteristic
112#14579085C2×C411yescharacteristic
113#14579084C2×C411yescharacteristic
114#14579083C2×C411yescharacteristic
115#14579082C2×C411yescharacteristic
116#14579081C2×C411yescharacteristic
117C22⋊(C79⋊C8)—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8—21no—
120(C2×D79)⋊C8—21no—
121#14579072—21no—
122#14579071D411yescharacteristic
123#14579071D411yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C8×Dic79C2×C411yescharacteristic
130C8×Dic79C2×C411yescharacteristic
131C2×SD16×C79C2×C411yescharacteristic
132C2×M16×C79—21no—
133C4×Q8×C79C2311yescharacteristic
134C4×D4×C79C2311yescharacteristic
135#106×C79C2×C411yescharacteristic
136C4⋊C8×C79—21no—
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139#103×C79C2×C411yescharacteristic
140#102×C79C2×C411yescharacteristic
141C22⋊C8×C79—21no—
142#97×C79—21no—
143C4×C8×C79C2311yescharacteristic
144C4×C8×C79C2311yescharacteristic
145C4×C8×C79C2311yescharacteristic
146C23×D79—21no—
147C2×Dic158—21no—
148C2×Dic158C22×C411yescharacteristic
149Q8×D79—22no—
150Q8×D79—41no—
151C2×D316C22×C411yescharacteristic
152C2×C22⋊D79—21no—
153D4×D79—22no—
154D4×D79—41no—
155C22×Dic79—21no—
156C2×C4×D79—21no—
157C2×C4×D79—21no—
158C2×C4×D79—21no—
159C2×C4×D79—21no—
160C2×C4×D79—21no—
161C2×C4×D79—42no—
162C2×C4×D79—42no—
163C2×C4×D79C22×C411yescharacteristic
164C2×C4×D79C2×D411yescharacteristic
165C2×C4×D79#4111yescharacteristic
166#11929613M1614yescomplemented
167#11929612—22no—
168#11929611—22no—
169#11929608—42no—
170#11929607—22no—
171#11929607—42no—
172#11929607—42no—
173#11929607SD1614yescomplemented
174C2×C79⋊C8—21no—
175C2×C79⋊C8—21no—
176C2×C79⋊C8—21no—
177C2×C79⋊C8C22×C411yescharacteristic
178C2×C79⋊C8C22×C411yescharacteristic
179C2×C79⋊C8C2×D411yescharacteristic
180C2×C79⋊C8C2×D411yescharacteristic
181C8×D79—22no—
182C8×D79—41no—
183C8×D79—42no—
184C8×D79—42no—
185C4⋊Dic79—21no—
186C4⋊Dic79C22×C411yescharacteristic
187C4⋊Dic79C22×C411yescharacteristic
188Dic79⋊C4—21no—
189Dic79⋊C4—21no—
190Dic79⋊C4—21no—
191Dic79⋊C4—21no—
192C22⋊Dic79—21no—
193(C2×D79)⋊C4—21no—
194(C2×D79)⋊C4—21no—
195C4×Dic79—21no—
196C4×Dic79—21no—
197C4×Dic79—21no—
198C4×Dic79C22×C411yescharacteristic
199C4×Dic79C2×D411yescharacteristic
200C4×Dic79#4111yescharacteristic
201C2×Q8×C79C22×C411yescharacteristic
202C2×D4×C79C22×C411yescharacteristic
203C22×C4×C79—21no—
204SD16×C79—22no—
205M16×C79—42no—
206C2×C8×C79—21no—
207C2×C8×C79—21no—
208C2×C8×C79—21no—
209C2×C8×C79C22×C411yescharacteristic
210C2×C8×C79C22×C411yescharacteristic
211C2×C8×C79C2×D411yescharacteristic
212C2×C8×C79C2×D411yescharacteristic
213C4⋊C4×C79—21no—
214C4⋊C4×C79C22×C411yescharacteristic
215C4⋊C4×C79C22×C411yescharacteristic
216C22⋊C4×C79—21no—
217C42×C79—21no—
218C42×C79C2411yescharacteristic
219C23×C79—21no—
220C22×D79—21no—
221C22×D79—42no—
222C22×D79—42no—
223C22×D79—81no—
224C22×D79C4×D411yescharacteristic
225Q8×C79—21no—
226Q8×C79—21no—
227D4×C79—21no—
228D4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79C23×C411yescharacteristic
235C2×C4×C79C22×D411yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C22⋊D79—42no—
238C2×Dic79—21no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—42no—
242C2×Dic79C4×D411yescharacteristic
243C2×Dic79C4×D411yescharacteristic
244C2×Dic79C4×D411yescharacteristic
245D316—21no—
246D316C2×M1612yes—
247C4×D79—21no—
248C4×D79—21no—
249C4×D79—41no—
250C4×D79—42no—
251C4×D79—42no—
252C4×D79—41no—
253C4×D79—41no—
254C4×D79—42no—
255C4×D79—81no—
256C4×D79C2×SD1612yes—
257C4×D79#13512yes—
258Dic158—21no—
259Dic158—42no—
260Dic158C2×M1612yescomplemented
261C8×C79—21no—
262C8×C79—21no—
263C8×C79—41no—
264C8×C79—41no—
265C8×C79—41no—
266C8×C79C2×M1612yescomplemented
267C8×C79C2×SD1612yescomplemented
268C79⋊C8—21no—
269C79⋊C8—21no—
270C79⋊C8—21no—
271C79⋊C8—41no—
272C79⋊C8—41no—
273C79⋊C8—41no—
274C79⋊C8C2×SD1612yescomplemented
275C22×C79—41no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79C2×C4×D411yescharacteristic
279C2×D79—21no—
280C2×D79—21no—
281C2×D79—22no—
282C2×D79—41no—
283C2×D79—41no—
284C2×D79—84no—
285C2×D79C4×SD1612yes—
286C4×C79—21no—
287C4×C79—41no—
288C4×C79—41no—
289C4×C79—41no—
290C4×C79C22×M1611yescharacteristic
291C4×C79#56811yescharacteristic
292C4×C79C22×SD1611yescharacteristic
293C4×C79C2×#13511yescharacteristic
294Dic79—21no—
295Dic79—22no—
296Dic79—21no—
297Dic79—41no—
298Dic79—41no—
299Dic79C4×SD1612yes—
300M16×SD16—791nomaximal, Sylow
301C2×C79—81no—
302C2×C79D4×M1611yescharacteristic
303C2×C79C2×C4×SD1611yescharacteristic
304C2×C79#295711yescharacteristic
305D79—22no—
306D79—42no—
307Q8×M16—791no—
308C2×C4×SD16—791no—
309D4×M16—791no—
310#1590—791no—
311#1590D7911yescharacteristic, complemented
312#1583—791no—
313#1583—791no—
314#1582—791no—
315#1582—791no—
316C8×SD16—791no—
317C8×SD16—791no—
318#1566—791no—
319#1487—791no—
320#1486—791no—
321C8×M16—791no—
322C79M16×SD1611yescharacteristic, complemented, Sylow
323C22×SD16—791no—
324C22×M16—1581no—
325C2×C4×Q8—791no—
326C2×C4×D4—791no—
327#446—791no—
328#446—1581no—
329#446—1581no—
330#446C2×D7911yescharacteristic, complemented
331C8×Q8—791no—
332C8×Q8—791no—
333C4×SD16—791no—
334C4×SD16—791no—
335C4×SD16—791no—
336C4×SD16—791no—
337C4×SD16—791no—
338C4×SD16—791no—
339C4×SD16—791no—
340C4×SD16C2×D7911yescharacteristic
341#436—791no—
342#436C2×D7911yescharacteristic
343#435—1581no—
344#435—1581no—
345C8×D4—791no—
346C8×D4—791no—
347#431—791no—
348C2×#106—791no—
349#423—791no—
350#423—791no—
351#423—1581no—
352C2×#103—791no—
353C2×#102—791no—
354#407—1581no—
355C4×M16—791no—
356C4×M16—1581no—
357C2×C4×C8—791no—
358#334—791no—
359#334—791no—
360#334—791no—
361#334C2×D7911yescharacteristic, complemented
362Q8⋊C8—791no—
363Q8⋊C8—791no—
364Q8⋊C8—791no—
365Q8⋊C8C2×D7911yescharacteristic, complemented
366D4⋊C8—791no—
367D4⋊C8—791no—
368D4⋊C8—791no—
369D4⋊C8C2×D7911yescharacteristic
370#322—791no—
371#322—791no—
372C82—791no—
373C82C2×D7911yescharacteristic, complemented
374C22×Q8—791no—
375C22×D4—791no—
376C23×C4—1581no—
377C2×SD16—791no—
378C2×SD16—791no—
379C2×SD16—791no—
380C2×SD16—1584no—
381C2×SD16C4×D7911yescharacteristic
382C2×M16—21no—
383C2×M16—3164no—
384C2×M16—791no—
385C2×M16—791no—
386C2×M16—1581no—
387C2×M16—1581no—
388C2×M16—1581no—
389C22×C8—791no—
390C22×C8—791no—
391C22×C8—1581no—
392C22×C8—1581no—
393C4×Q8—791no—
394C4×Q8—791no—
395C4×Q8—791no—
396C4×Q8—1581no—
397C4×Q8—1581no—
398C4×Q8C22×D7911yescharacteristic
399C4×D4—791no—
400C4×D4—791no—
401C4×D4—791no—
402C4×D4—1581no—
403C4×D4—1581no—
404C4×D4C22×D7911yescharacteristic
405C2×C4⋊C4—791no—
406C2×C4⋊C4—791no—
407C2×C4⋊C4—1581no—
408C2×C22⋊C4—1581no—
409C2×C42—791no—
410C2×C42—1581no—
411#106—791no—
412#106—791no—
413#106—791no—
414#106C4×D7911yescharacteristic
415C4⋊C8—21no—
416C4⋊C8—791no—
417C4⋊C8—791no—
418C4⋊C8—1581no—
419C4⋊C8—1581no—
420C4⋊C8—1581no—
421C4⋊C8—1581no—
422C4⋊C8—1581no—
423C4⋊C8—1581no—
424C4⋊C8C22×D7911yescharacteristic
425C4⋊C8C22×D7911yescharacteristic
426#103—791no—
427#103—791no—
428#103—791no—
429#103C4×D7911yescharacteristic
430#102—791no—
431#102—791no—
432#102—791no—
433#102C4×D7911yescharacteristic
434C22⋊C8—21no—
435C22⋊C8—1581no—
436C22⋊C8—1581no—
437C22⋊C8—1581no—
438C22⋊C8—1581no—
439#97—21no—
440#97—791no—
441#97—791no—
442#97—1581no—
443C4×C8—791no—
444C4×C8—791no—
445C4×C8—791no—
446C4×C8—791no—
447C4×C8—791no—
448C4×C8—1581no—
449C4×C8—1581no—
450C4×C8C22×D7911yescharacteristic
451C4×C8C22×D7911yescharacteristic
452C4×C8C22×D7911yescharacteristic
453C24—1581no—
454C2×Q8—3161no—
455C2×Q8—791no—
456C2×Q8—1581no—
457C2×Q8—1582no—
458C2×Q8C2×C4×D7911yescharacteristic
459C2×D4—3161no—
460C2×D4—791no—
461C2×D4—1581no—
462C2×D4—1582no—
463C2×D4C2×C4×D7911yescharacteristic
464C22×C4—21no—
465C22×C4—3162no—
466C22×C4—3162no—
467C22×C4—791no—
468C22×C4—791no—
469C22×C4—791no—
470C22×C4—1581no—
471C22×C4—1581no—
472C22×C4—1581no—
473C22×C4—1581no—
474C22×C4—1581no—
475C22×C4—1581no—
476SD16—22no—
477SD16—794no—
478SD16—1582no—
479SD16—1582no—
480M16—42no—
481M16—3162no—
482M16—3162no—
483M16—3162no—
484M16—794no—
485M16—1582no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—3161no—
490C2×C8—3162no—
491C2×C8—3162no—
492C2×C8—791no—
493C2×C8—791no—
494C2×C8—791no—
495C2×C8—791no—
496C2×C8—1581no—
497C2×C8—1581no—
498C2×C8—1581no—
499C2×C8—1582no—
500C2×C8C2×C4×D7911yescharacteristic
501C2×C8C2×C4×D7911yescharacteristic
502C2×C8D4×D7911yescharacteristic
503C2×C8D4×D7911yescharacteristic
504C4⋊C4—21no—
505C4⋊C4—791no—
506C4⋊C4—791no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4—1581no—
510C4⋊C4—1581no—
511C4⋊C4—1581no—
512C4⋊C4C2×C4×D7911yescharacteristic
513C4⋊C4C2×C4×D7911yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—1581no—
516C22⋊C4—1581no—
517C22⋊C4—1581no—
518C42—21no—
519C42—791no—
520C42—791no—
521C42—791no—
522C42—1581no—
523C42—1581no—
524C42—1581no—
525C42C23×D7911yescharacteristic
526C23—21no—
527C23—3162no—
528C23—3162no—
529C23—791no—
530C23—6321no—
531C23—1581no—
532Q8—21no—
533Q8—21no—
534Q8—3162no—
535Q8—792no—
536Q8—1581no—
537D4—21no—
538D4—21no—
539D4—3162no—
540D4—792no—
541D4—1581no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—3161no—
548C2×C4—3162no—
549C2×C4—3162no—
550C2×C4—3162no—
551C2×C4—3162no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—791no—
555C2×C4—792no—
556C2×C4—792no—
557C2×C4—791no—
558C2×C4—791no—
559C2×C4—6321no—
560C2×C4—1581no—
561C2×C4—1581no—
562C2×C4—1581no—
563C2×C4—1581no—
564C2×C4—1581no—
565C2×C4C22×C4×D7911yescharacteristic
566C2×C4C2×D4×D7911yescharacteristic
567C2×C4#41×D7911yescharacteristic
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—3161no—
574C8—3161no—
575C8—3161no—
576C8—792no—
577C8—1581no—
578C8—1581no—
579C8—1581no—
580C8M16×D7912yescomplemented
581C8SD16×D7912yescomplemented
582C22—41no—
583C22—41no—
584C22—41no—
585C22—3161no—
586C22—3161no—
587C22—792no—
588C22—6324no—
589C22—1582no—
590C22—1581no—
591C22—1581no—
592C22C4×D4×D7911yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—3161no—
598C4—3161no—
599C4—792no—
600C4—1581no—
601C4—1582no—
602C4—1581no—
603C4C2×M16×D7911yescharacteristic
604C4#1909932811yescharacteristic
605C4C2×SD16×D7911yescharacteristic
606C4#135×D7911yescharacteristic
607C2—81no—
608C2—3162no—
609C2—1582no—
610C2#2377213611yescharacteristic
611C2C4×SD16×D7911yescharacteristic
612C2#2377231211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1590C2
socleC22×C79C2×C4×D4
prime radicalC2×D4×D79C2×C4
cosocleC42×C79C24
hypercenter#1590D79
hypocenterC79M16×SD16
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
1. #23772352C2semidirect
4. #23762291C2semidirect
5. #23762246C2semidirect
6. #23762241C2semidirect
7. #23762239C2semidirect
8. #23762234C2semidirect
9. C8×#11929613C2semidirect
10. #23762199C2semidirect
11. #23762154C2semidirect
12. #23761827C2semidirect
14. C8×#11929607C2semidirect
15. C79×#1590C2semidirect
20. #19098694C22semidirect
23. C8×Dic158C22semidirect
24. #19098687C22semidirect
48. #19098314C22semidirect
49. #19098313C22semidirect
50. #19098313C22semidirect
51. #19098295C22semidirect
52. Q8⋊(C79⋊C8)C22semidirect
53. #19098293C22semidirect
57. #19098285C22semidirect
58. #19098284C22semidirect
59. C8×C79⋊C8C22semidirect
60. #446×C79C22semidirect
63. #334×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. C82×C79C22semidirect
166. #11929613M16semidirect
173. #11929607SD16semidirect
260. Dic158C2×M16semidirect
266. C8×C79C2×M16semidirect
267. C8×C79C2×SD16semidirect
274. C79⋊C8C2×SD16semidirect
311. #1590D79semidirect
322. C79M16×SD16semidirect
330. #446C2×D79semidirect
361. #334C2×D79semidirect
365. Q8⋊C8C2×D79semidirect
373. C82C2×D79semidirect
580. C8M16×D79semidirect
581. C8SD16×D79semidirect

Maximal subgroups

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

subgroupindex
#237723522
C4×SD16×D792
#237721362
#237622912
#237622462
#237622412
#237622392
#237622342
C8×#119296132
#237621992
#237621542
#237618272
#237618232
C8×#119296072
C79×#15902
M16×SD1679