Finite Group Databasebeta

← #34259424

Subgroups — #34259424

Order 20224.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773451C211yesmaximal, characteristic
2#23772559C211yesmaximal, characteristic, complemented
3#23772432C211yesmaximal, characteristic, complemented
4#23762291C211yesmaximal, characteristic, complemented
5#23762283C211yesmaximal, characteristic, complemented
6#23762267C211yesmaximal, characteristic, complemented
7#23762266C211yesmaximal, characteristic, complemented
8#23762260C211yesmaximal, characteristic, complemented
9C8×#11929611C211yesmaximal, characteristic, complemented
10#23762194C211yesmaximal, characteristic, complemented
11#23762184C211yesmaximal, characteristic, complemented
12#23761859C211yesmaximal, characteristic, complemented
13D4⋊C8×D79C211yesmaximal, characteristic, complemented
14#23761685C211yesmaximal, characteristic, complemented
15C79×#1588C211yesmaximal, characteristic, complemented
16#19099100C2211yescharacteristic
17C4×D4×D79C2211yescharacteristic
18#19098897C411yescharacteristic
19#19098883C411yescharacteristic
20#19098862C411yescharacteristic
21#19098861C411yescharacteristic
22#19098781C411yescharacteristic
23#19098780C411yescharacteristic
24#19098712C411yescharacteristic
25#19098698C411yescharacteristic
26#19098694C2211yescharacteristic, complemented
27C4×#11929611C2211yescharacteristic
28C4×D632C2211yescharacteristic
29C4×#11929609C2211yescharacteristic
30#19098642C2211yescharacteristic, complemented
31#19098641C2211yescharacteristic
32#19098637—21no—
33#19098633—21no—
34#19098632—21no—
35C8×D316—21no—
36C8×D316C2211yescharacteristic
37#19098628C2211yescharacteristic
38#19098628C2211yescharacteristic, complemented
39#19098622—21no—
40#19098619C2211yescharacteristic
41#19098614C2211yescharacteristic
42#19098614C2211yescharacteristic
43#19098588—21no—
44C4⋊C8×D79C2211yescharacteristic
45#19098526—21no—
46C22⋊C8×D79—21no—
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic, complemented
49#19098314C2211yescharacteristic, complemented
50#19098313C2211yescharacteristic, complemented
51Q8⋊(C79⋊C8)C2211yescharacteristic, complemented
52#19098292C2211yescharacteristic
53#19098292C2211yescharacteristic, complemented
54D4⋊(C79⋊C8)C2211yescharacteristic
55#19098290C2211yescharacteristic
56#19098290C2211yescharacteristic, complemented
57#19098284C2211yescharacteristic, complemented
58C8×C79⋊C8C2211yescharacteristic, complemented
59C8×C79⋊C8C2211yescharacteristic, complemented
60C8×Q8×C79C2211yescharacteristic, complemented
61C4×SD16×C79C2211yescharacteristic
62#436×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic, complemented
64Q8⋊C8×C79C2211yescharacteristic, complemented
65D4⋊C8×C79C2211yescharacteristic
66#322×C79C2211yescharacteristic, complemented
67C2×D4×D79C2×C411yescharacteristic
68C22×C4×D79—21no—
69C2×#11929611C2×C411yescharacteristic
70C2×D632C2×C411yescharacteristic
71C2×#11929609C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76C2×C8×D79—21no—
77C2×C8×D79—21no—
78C2×C8×D79—21no—
79#14579156—21no—
80#14579152C2×C411yescharacteristic
81#14579150C2×C411yescharacteristic
82#14579146—21no—
83#14579137C2×C411yescharacteristic
84#14579136C2×C411yescharacteristic
85C4×D316—21no—
86C4×D316C2311yescharacteristic
87C4×D316C2311yescharacteristic
88D4×Dic79C2×C411yescharacteristic
89#14579123C2×C411yescharacteristic
90C4×C22⋊D79—21no—
91(C22⋊D79)⋊C4—21no—
92#14579118—21no—
93C4⋊C4×D79C2×C411yescharacteristic
94C22⋊C4×D79—21no—
95C42×D79C2311yescharacteristic
96#14579095C2×C411yescharacteristic
97#14579094C2×C411yescharacteristic
98#14579092C2×C411yescharacteristic
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102Dic79⋊C8—21no—
103Dic79⋊C8C2×C411yescharacteristic
104Dic79⋊C8C2×C411yescharacteristic
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579085C2×C411yescharacteristic
108#14579083C2×C411yescharacteristic
109#14579083C2×C411yescharacteristic
110#14579082C2×C411yescharacteristic
111#14579081C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113C22⋊(C79⋊C8)—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8—21no—
117(C2×D79)⋊C8—21no—
118(C2×D79)⋊C8—21no—
119(C2×D79)⋊C8C2×C411yescharacteristic
120(C2×D79)⋊C8C2×C411yescharacteristic
121(C2×D79)⋊C8D411yescharacteristic
122(C2×D79)⋊C8D411yescharacteristic
123#14579072—21no—
124#14579071—21no—
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C4×C79⋊C8C2311yescharacteristic
128C8×Dic79—21no—
129C2×SD16×C79C2×C411yescharacteristic
130C2×M16×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
144C23×D79—21no—
145C2×D316—21no—
146C2×D316C22×C411yescharacteristic
147C2×D316C22×C411yescharacteristic
148C2×C22⋊D79—21no—
149D4×D79—22no—
150D4×D79—41no—
151C22×Dic79—21no—
152C2×C4×D79—21no—
153C2×C4×D79—21no—
154C2×C4×D79—21no—
155C2×C4×D79—21no—
156C2×C4×D79—81no—
157C2×C4×D79—81no—
158C2×C4×D79C22×C411yescharacteristic
159C2×C4×D79C2×D411yescharacteristic
160C2×C4×D79#4111yescharacteristic
161#11929611—22no—
162D632—22no—
163#11929609—41no—
164#11929608—81no—
165#11929607—81no—
166C2×C79⋊C8—21no—
167C2×C79⋊C8—21no—
168C2×C79⋊C8—21no—
169C2×C79⋊C8C22×C411yescharacteristic
170C2×C79⋊C8C22×C411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C2×C79⋊C8C2×D411yescharacteristic
173C8×D79—24no—
174C8×D79—81no—
175C8×D79—81no—
176C8×D79—81no—
177C8×D79—81no—
178C4⋊Dic79—21no—
179C4⋊Dic79C22×C411yescharacteristic
180C4⋊Dic79C22×C411yescharacteristic
181Dic79⋊C4—21no—
182Dic79⋊C4—21no—
183Dic79⋊C4—21no—
184C22⋊Dic79—21no—
185(C2×D79)⋊C4—21no—
186(C2×D79)⋊C4—21no—
187(C2×D79)⋊C4—21no—
188(C2×D79)⋊C4—21no—
189(C2×D79)⋊C4—21no—
190(C2×D79)⋊C4—21no—
191(C2×D79)⋊C4—21no—
192C4×Dic79C22×C411yescharacteristic
193C4×Dic79C2×D411yescharacteristic
194C4×Dic79#4111yescharacteristic
195C2×Q8×C79C22×C411yescharacteristic
196C2×D4×C79C22×C411yescharacteristic
197C22×C4×C79—21no—
198SD16×C79—41no—
199M16×C79—81no—
200C2×C8×C79—21no—
201C2×C8×C79—21no—
202C2×C8×C79—21no—
203C2×C8×C79C22×C411yescharacteristic
204C2×C8×C79C22×C411yescharacteristic
205C2×C8×C79C2×D411yescharacteristic
206C2×C8×C79C2×D411yescharacteristic
207C4⋊C4×C79—21no—
208C4⋊C4×C79C22×C411yescharacteristic
209C4⋊C4×C79C22×C411yescharacteristic
210C22⋊C4×C79—21no—
211C42×C79—21no—
212C42×C79C2411yescharacteristic
213C23×C79—21no—
214C22×D79—21no—
215C22×D79—21no—
216C22×D79—42no—
217C22×D79—42no—
218C22×D79—81no—
219C22×D79C4×D411yescharacteristic
220Q8×C79—21no—
221Q8×C79—21no—
222D4×C79—21no—
223D4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79C23×C411yescharacteristic
230C2×C4×C79C22×D411yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C22⋊D79—42no—
233C2×Dic79—21no—
234C2×Dic79—21no—
235C2×Dic79—81no—
236C2×Dic79C4×D411yescharacteristic
237C2×Dic79C4×D411yescharacteristic
238C2×Dic79C4×D411yescharacteristic
239D316—21no—
240D316—21no—
241D316—21no—
242D316—42no—
243D316#13112yes—
244C4×D79—22no—
245C4×D79—21no—
246C4×D79—21no—
247C4×D79—21no—
248C4×D79—21no—
249C4×D79—41no—
250C4×D79—41no—
251C4×D79—41no—
252C4×D79—41no—
253C4×D79—81no—
254C4×D79—81no—
255C8×C79—22no—
256C8×C79—21no—
257C8×C79—21no—
258C8×C79—21no—
259C8×C79—21no—
260C8×C79—41no—
261C8×C79—41no—
262C79⋊C8—22no—
263C79⋊C8—21no—
264C79⋊C8—21no—
265C79⋊C8—21no—
266C79⋊C8—21no—
267C79⋊C8—41no—
268C79⋊C8—41no—
269C22×C79—41no—
270C22×C79—41no—
271C22×C79—41no—
272C22×C79C2×C4×D411yescharacteristic
273C2×D79—21no—
274C2×D79—21no—
275C2×D79—22no—
276C2×D79—41no—
277C2×D79—41no—
278C2×D79—41no—
279C2×D79—41no—
280C2×D79—41no—
281C2×D79—84no—
282C2×D79#44312yes—
283C4×C79—21no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79—41no—
287C4×C79C2×#13111yescharacteristic
288C4×C79#56811yescharacteristic
289C4×C79#57611yescharacteristic
290C4×C79#57711yescharacteristic
291Dic79—21no—
292Dic79—21no—
293Dic79—21no—
294Dic79—41no—
295Dic79—41no—
296#18014—791nomaximal, Sylow
297C2×C79—81no—
298C2×C79#293311yescharacteristic
299C2×C79#294611yescharacteristic
300C2×C79C2×#44311yescharacteristic
301D79—22no—
302D79—42no—
303D79—81no—
304#3105—791no—
305#2994—791no—
306#2974—791no—
307#1590—791no—
308#1589—791no—
309#1588D7911yescharacteristic, complemented
310#1586—791no—
311#1586—791no—
312#1585—791no—
313C8×SD16—791no—
314C8×D8—791no—
315#1573—791no—
316#1494—791no—
317C2×D4⋊C8—791no—
318#1450—791no—
319C79#1801411yescharacteristic, complemented, Sylow
320#529—791no—
321C2×C4×D4—791no—
322#484—791no—
323#481—791no—
324#478—791no—
325#478—791no—
326#463—791no—
327#463—791no—
328#450—791no—
329#447—791no—
330#446—791no—
331C8×Q8C2×D7911yescharacteristic, complemented
332C4×SD16—791no—
333C4×SD16C2×D7911yescharacteristic
334C4×D8—791no—
335C4×D8—791no—
336#436—791no—
337#436—791no—
338#436C2×D7911yescharacteristic
339#435—1581no—
340#435—1581no—
341#435—1581no—
342C8×D4—791no—
343C8×D4—791no—
344C8×D4—791no—
345C8×D4—1581no—
346#433—1581no—
347#432—791no—
348#431—791no—
349#431—791no—
350#424—1581no—
351C2×C4⋊C8—791no—
352#407—1581no—
353C2×C22⋊C8—1581no—
354C2×C4×C8—791no—
355#335—791no—
356#335C2×D7911yescharacteristic, complemented
357#334—791no—
358#334—791no—
359Q8⋊C8—791no—
360Q8⋊C8C2×D7911yescharacteristic, complemented
361D4⋊C8—791no—
362D4⋊C8—791no—
363D4⋊C8—791no—
364D4⋊C8—791no—
365D4⋊C8—791no—
366D4⋊C8C2×D7911yescharacteristic
367#322—791no—
368#322C2×D7911yescharacteristic, complemented
369C82—791no—
370C82—791no—
371C22×D4—791no—
372C23×C4—1581no—
373C2×SD16—791no—
374C2×SD16C4×D7911yescharacteristic
375C2×D8—791no—
376C2×D8—791no—
377C2×M16—21no—
378C2×M16—1581no—
379C2×M16—1581no—
380C22×C8—1581no—
381C22×C8—1581no—
382C22×C8—1581no—
383C22×C8—1581no—
384C22×C8—1581no—
385#126—1581no—
386#125—791no—
387#124—791no—
388#123—1581no—
389C22⋊Q8—791no—
390#121—791no—
391C4×Q8C22×D7911yescharacteristic
392C4×D4—791no—
393C4×D4—791no—
394C4×D4—791no—
395C4×D4—791no—
396C4×D4—1581no—
397C4×D4—1581no—
398C4×D4—1581no—
399C4×D4C22×D7911yescharacteristic
400#117—1581no—
401C2×C4⋊C4—791no—
402C2×C22⋊C4—1581no—
403C2×C42—791no—
404#107—791no—
405#107—791no—
406#106—791no—
407#106C4×D7911yescharacteristic
408C4⋊C8—21no—
409C4⋊C8—791no—
410C4⋊C8—791no—
411C4⋊C8—791no—
412C4⋊C8—791no—
413C4⋊C8—791no—
414C4⋊C8—791no—
415C4⋊C8—1581no—
416C4⋊C8—1581no—
417C4⋊C8C22×D7911yescharacteristic
418C4⋊C8C22×D7911yescharacteristic
419#103—791no—
420#103C4×D7911yescharacteristic
421#102—791no—
422#102—791no—
423#102—791no—
424#102—791no—
425#102—791no—
426#102C4×D7911yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—791no—
429C22⋊C8—791no—
430C22⋊C8—791no—
431C22⋊C8—791no—
432C22⋊C8—1581no—
433C22⋊C8—1581no—
434C22⋊C8—1581no—
435C22⋊C8—1581no—
436C22⋊C8—1581no—
437C22⋊C8—1581no—
438#97—1581no—
439#97—1581no—
440C4×C8—21no—
441C4×C8—791no—
442C4×C8—791no—
443C4×C8—791no—
444C4×C8—1581no—
445C4×C8C22×D7911yescharacteristic
446C4×C8C22×D7911yescharacteristic
447C4×C8C22×D7911yescharacteristic
448C24—1581no—
449C2×Q8C2×C4×D7911yescharacteristic
450C2×D4—3161no—
451C2×D4—791no—
452C2×D4—791no—
453C2×D4—1581no—
454C2×D4—1581no—
455C2×D4—1582no—
456C2×D4C2×C4×D7911yescharacteristic
457C22×C4—21no—
458C22×C4—791no—
459C22×C4—791no—
460C22×C4—791no—
461C22×C4—6321no—
462C22×C4—6321no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4—1581no—
467C22×C4—1581no—
468SD16—41no—
469SD16—1582no—
470D8—3161no—
471D8—1582no—
472M16—81no—
473M16—6321no—
474M16—6321no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—791no—
479C2×C8—791no—
480C2×C8—791no—
481C2×C8—791no—
482C2×C8—6321no—
483C2×C8—6321no—
484C2×C8—6321no—
485C2×C8—6321no—
486C2×C8—1584no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8C2×C4×D7911yescharacteristic
491C2×C8C2×C4×D7911yescharacteristic
492C2×C8D4×D7911yescharacteristic
493C2×C8D4×D7911yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—791no—
496C4⋊C4—791no—
497C4⋊C4—1581no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4C2×C4×D7911yescharacteristic
502C4⋊C4C2×C4×D7911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1581no—
505C22⋊C4—1581no—
506C22⋊C4—1581no—
507C22⋊C4—1581no—
508C22⋊C4—1581no—
509C22⋊C4—1581no—
510C22⋊C4—1581no—
511C22⋊C4—1581no—
512C42—21no—
513C42—791no—
514C42—791no—
515C42—791no—
516C42C23×D7911yescharacteristic
517C23—21no—
518C23—3162no—
519C23—3162no—
520C23—791no—
521C23—6321no—
522C23—1581no—
523C23—1581no—
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—3162no—
529D4—3162no—
530D4—792no—
531D4—1581no—
532D4—1581no—
533D4—1581no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—3161no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—791no—
544C2×C4—791no—
545C2×C4—791no—
546C2×C4—6321no—
547C2×C4—6321no—
548C2×C4—6321no—
549C2×C4—1582no—
550C2×C4—1581no—
551C2×C4—1581no—
552C2×C4—1581no—
553C2×C4—1581no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4C22×C4×D7911yescharacteristic
557C2×C4C2×D4×D7911yescharacteristic
558C2×C4#41×D7911yescharacteristic
559C8—22no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—41no—
565C8—41no—
566C8—3161no—
567C8—3161no—
568C8—1582no—
569C8—1581no—
570C8—1581no—
571C8—1581no—
572C8—1581no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—3161no—
577C22—3161no—
578C22—3161no—
579C22—3161no—
580C22—3161no—
581C22—792no—
582C22—6324no—
583C22—1582no—
584C22—1581no—
585C22—1581no—
586C22C4×D4×D7911yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—3161no—
592C4—3161no—
593C4—1581no—
594C4—1581no—
595C4—1581no—
596C4#131×D7911yescharacteristic
597C4#1909932911yescharacteristic
598C4#1909936711yescharacteristic
599C4#1909937211yescharacteristic
600C2—81no—
601C2—3162no—
602C2—6321no—
603C2—1582no—
604C2#2377212311yescharacteristic
605C2#2377223111yescharacteristic
606C2#443×D7911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1588C2
socleC22×C79C2×C4×D4
prime radical#19098698C4
cosocleC42×C79C24
hypercenter#1588D79
hypocenterC79#18014
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1588

Complements

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

NH ≅ G/Ntype
2. #23772559C2semidirect
3. #23772432C2semidirect
4. #23762291C2semidirect
5. #23762283C2semidirect
6. #23762267C2semidirect
7. #23762266C2semidirect
8. #23762260C2semidirect
9. C8×#11929611C2semidirect
10. #23762194C2semidirect
11. #23762184C2semidirect
12. #23761859C2semidirect
13. D4⋊C8×D79C2semidirect
14. #23761685C2semidirect
15. C79×#1588C2semidirect
26. #19098694C22semidirect
30. #19098642C22semidirect
38. #19098628C22semidirect
48. #19098315C22semidirect
49. #19098314C22semidirect
50. #19098313C22semidirect
51. Q8⋊(C79⋊C8)C22semidirect
53. #19098292C22semidirect
56. #19098290C22semidirect
57. #19098284C22semidirect
58. C8×C79⋊C8C22semidirect
59. C8×C79⋊C8C22semidirect
60. C8×Q8×C79C22semidirect
63. #335×C79C22semidirect
64. Q8⋊C8×C79C22semidirect
66. #322×C79C22semidirect
309. #1588D79semidirect
319. C79#18014semidirect
331. C8×Q8C2×D79semidirect
356. #335C2×D79semidirect
360. Q8⋊C8C2×D79semidirect
368. #322C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734512
#237725592
#237724322
#237622912
#237622832
#237622672
#237622662
#237622602
C8×#119296112
#237621942
#237621842
#237618592
D4⋊C8×D792
#237616852
C79×#15882
#1801479