Finite Group Databasebeta

← #34319082

Subgroups — #34319082

Order 20224.

738 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#23766111C211yesmaximal, characteristic
2#23764963C211yesmaximal, characteristic, complemented
3#23764958C211yesmaximal, characteristic, complemented
4#23764819C211yesmaximal, characteristic, complemented
5#23764814C211yesmaximal, characteristic
6#23764641C211yesmaximal, characteristic
7#23764641C211yesmaximal, characteristic
8#23764538C211yesmaximal, characteristic, complemented
9#23764533C211yesmaximal, characteristic
10#23763233C211yesmaximal, characteristic, complemented
11#23763228C211yesmaximal, characteristic
12C4×#14579084C211yesmaximal, characteristic, complemented
13C4×#14579081C211yesmaximal, characteristic
14#23763152C211yesmaximal, characteristic, complemented
15C79×#2331C211yesmaximal, characteristic, complemented
16C2×#14579149C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19C2×#14579084C2211yescharacteristic
20C2×#14579084C2211yescharacteristic
21C2×#14579084C2211yescharacteristic
22C2×#14579084C2211yescharacteristic
23C2×#14579081C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579072C2211yescharacteristic
28C2×C4×C79⋊C8C2211yescharacteristic
29#19098469C2211yescharacteristic
30#19098465C2211yescharacteristic
31#19098455—21no—
32#19098455—21no—
33#19098437—21no—
34#19098437—21no—
35#19098433C2211yescharacteristic
36#19098325C2211yescharacteristic
37#19098325C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098318C2211yescharacteristic
45#19098318C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×C4⋊Q8×C79C2211yescharacteristic
49#391×C79—21no—
50#391×C79—21no—
51#389×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#387×C79—21no—
54#387×C79—21no—
55#382×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#376×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C22×D316C2311yescharacteristic
61C22×C4×D79—21no—
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66#14579149—22no—
67#14579149—22no—
68C4×Dic158—22no—
69C4×Dic158—22no—
70C4×D316—22no—
71C4×D316—22no—
72C2×C4⋊Dic79—21no—
73C2×C4⋊Dic79—21no—
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×C4×Dic79—21no—
83#14579084—22no—
84#14579084—22no—
85#14579084—22no—
86#14579084—22no—
87#14579081—22no—
88#14579081—22no—
89#14579081—22no—
90#14579081—22no—
91#14579072—22no—
92C4×C79⋊C8—22no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C22×Q8×C79—21no—
96C4⋊Q8×C79—22no—
97C4⋊Q8×C79—22no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79C2311yescharacteristic
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C42×C79—21no—
106C2×C42×C79—21no—
107C2×C42×C79C2311yescharacteristic
108C2×C42×C79C2311yescharacteristic
109C2×C42×C79C2311yescharacteristic
110#95×C79—21no—
111#95×C79—21no—
112#95×C79—21no—
113#95×C79—21no—
114#95×C79—21no—
115#95×C79—21no—
116#95×C79D411yescharacteristic
117#95×C79D411yescharacteristic
118#95×C79D411yescharacteristic
119#95×C79D411yescharacteristic
120C23×D79—21no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×Dic158—21no—
126C2×Dic158#4112yes—
127C2×D316—21no—
128C2×D316—21no—
129C2×D316—21no—
130C2×D316—21no—
131C2×D316—21no—
132C2×D316#4112yes—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136C2×C4×D79—42no—
137C2×C4×D79—81no—
138C2×C4×D79—81no—
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—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—81no—
156C4⋊Dic79—81no—
157Dic79⋊C4—42no—
158(C2×D79)⋊C4—24no—
159(C2×D79)⋊C4—24no—
160(C2×D79)⋊C4—42no—
161(C2×D79)⋊C4—81no—
162(C2×D79)⋊C4—81no—
163C4×Dic79—42no—
164C2×Q8×C79—24no—
165C2×Q8×C79—81no—
166C2×Q8×C79—81no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2411yescharacteristic
178C4⋊C4×C79—24no—
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—81no—
186C4⋊C4×C79—81no—
187C4⋊C4×C79#4112yes—
188C4⋊C4×C79#4112yes—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—21no—
193C42×C79—81no—
194C42×C79—81no—
195C42×C79#4112yes—
196C42×C79#4112yes—
197C23×C79C22×D411yescharacteristic
198C22×D79—22no—
199C22×D79—41no—
200C22×D79—41no—
201C22×D79—41no—
202C22×D79—41no—
203C22×D79—41no—
204C22×D79—41no—
205Q8×C79—84no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—22no—
215C2×C4×C79—22no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79#12412yes—
235C2×C4×C79#12412yes—
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79C2×#4111yescharacteristic
238C2×C4×C79C2×#4111yescharacteristic
239C2×C4×C79C2×#4111yescharacteristic
240C2×C4×C79C2×#4111yescharacteristic
241C2×C4×C79C2×#4111yescharacteristic
242C2×Dic79—22no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251D316—22no—
252D316—22no—
253D316—41no—
254D316—41no—
255C4×D79—84no—
256Dic158—22no—
257Dic158—22no—
258Dic158—41no—
259Dic158—41no—
260C79⋊C8—41no—
261C79⋊C8—41no—
262C79⋊C8—41no—
263C79⋊C8—41no—
264C22×C79#52511yescharacteristic
265C22×C79#52511yescharacteristic
266C22×C79C2×#12411yescharacteristic
267C22×C79#57611yescharacteristic
268C22×C79#57711yescharacteristic
269C22×C79#57711yescharacteristic
270C22×C79#57811yescharacteristic
271C2×D79—42no—
272C2×D79—42no—
273C2×D79—42no—
274C2×D79—81no—
275C2×D79—81no—
276C2×D79—81no—
277C2×D79—81no—
278C4×C79—22no—
279C4×C79—22no—
280C4×C79—41no—
281C4×C79—42no—
282C4×C79—42no—
283C4×C79—41no—
284C4×C79—41no—
285C4×C79—41no—
286C4×C79—81no—
287C4×C79—81no—
288C4×C79#52911yescharacteristic
289C4×C79#52911yescharacteristic
290C4×C79#52911yescharacteristic
291C4×C79#52911yescharacteristic
292Dic79—42no—
293Dic79—81no—
294Dic79—81no—
295#23827—791nomaximal, Sylow
296C2×C79#239811yescharacteristic
297C2×C79#311811yescharacteristic
298C2×C79#312111yescharacteristic
299C2×C79#312411yescharacteristic
300C2×C79#312511yescharacteristic
301C2×C79#313611yescharacteristic
302C2×C79#313711yescharacteristic
303D79—82no—
304#2331D7911yescharacteristic, complemented
305#2326—791no—
306#2100—791no—
307#2099—791no—
308#2074—791no—
309#2073—791no—
310#2042—791no—
311#2042—791no—
312#2024—791no—
313#2023—791no—
314#1763—791no—
315#1762—791no—
316C4×#103—791no—
317C4×#102—791no—
318#1746—791no—
319C79#2382711yescharacteristic, complemented, Sylow
320C2×C4⋊Q8C2×D7911yescharacteristic
321C2×#124—791no—
322C2×C4×Q8—791no—
323C2×C4×D4—791no—
324C2×#103—791no—
325C2×#103—791no—
326C2×#103—791no—
327C2×#103—791no—
328C2×#102—791no—
329C2×#102—791no—
330C2×#102—791no—
331C2×#102—791no—
332C2×#97—791no—
333C2×C4×C8—791no—
334#391—21no—
335#391—21no—
336#391—791no—
337#390—791no—
338#389C2×D7911yescharacteristic
339#389C2×D7911yescharacteristic
340#388—1581no—
341#388—1581no—
342#387—21no—
343#387—21no—
344#385—1581no—
345#385—1581no—
346#384—791no—
347#382C2×D7911yescharacteristic
348C4×C4⋊C4C2×D7911yescharacteristic
349C4×C4⋊C4C2×D7911yescharacteristic
350#376C2×D7911yescharacteristic
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#336—791no—
360#336—791no—
361#336—791no—
362#336—791no—
363C22×Q8—21no—
364C22×Q8—791no—
365C22×D4—791no—
366C23×C4—1581no—
367C22×C8—791no—
368C22×C8—791no—
369C22×C8—791no—
370C22×C8—791no—
371C4⋊Q8—22no—
372C4⋊Q8—22no—
373#124—1582no—
374#124—1582no—
375C4×Q8—1582no—
376C4×Q8—1582no—
377C4×D4—1582no—
378C4×D4—1582no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—791no—
383C2×C4⋊C4—791no—
384C2×C4⋊C4—1581no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4C22×D7911yescharacteristic
388C2×C4⋊C4C22×D7911yescharacteristic
389C2×C4⋊C4C22×D7911yescharacteristic
390C2×C4⋊C4C22×D7911yescharacteristic
391C2×C22⋊C4—1581no—
392C2×C22⋊C4—1581no—
393C2×C22⋊C4—1581no—
394C2×C22⋊C4—1581no—
395C2×C22⋊C4—1581no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—1581no—
399C2×C42C22×D7911yescharacteristic
400C2×C42C22×D7911yescharacteristic
401C2×C42C22×D7911yescharacteristic
402#103—1582no—
403#103—1582no—
404#103—1582no—
405#103—1582no—
406#102—1582no—
407#102—1582no—
408#102—1582no—
409#102—1582no—
410#97—1582no—
411C4×C8—1582no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1581no—
419#95—1581no—
420#95C22⋊D7911yescharacteristic
421#95C22⋊D7911yescharacteristic
422#95C22⋊D7911yescharacteristic
423#95C22⋊D7911yescharacteristic
424C24—1581no—
425C2×Q8—24no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8—792no—
429C2×Q8—1581no—
430C2×Q8—1581no—
431C2×Q8—1581no—
432C2×Q8—1581no—
433C2×Q8—1581no—
434C2×D4—792no—
435C2×D4—1581no—
436C2×D4—1581no—
437C2×D4—1581no—
438C2×D4—1581no—
439C2×D4—1581no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—3162no—
445C22×C4—6321no—
446C22×C4—6321no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4C2×C22⋊D7911yescharacteristic
451C22×C4C2×C22⋊D7911yescharacteristic
452C22×C4C2×C22⋊D7911yescharacteristic
453C22×C4C2×C22⋊D7911yescharacteristic
454C22×C4C2×C22⋊D7911yescharacteristic
455C22×C4C2×C22⋊D7911yescharacteristic
456C22×C4C23×D7911yescharacteristic
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8—1581no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—3162no—
479C4⋊C4—6321no—
480C4⋊C4—6321no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4#1192962412yes—
486C4⋊C4#1192962412yes—
487C22⋊C4—3162no—
488C22⋊C4—6321no—
489C22⋊C4—6321no—
490C22⋊C4—1584no—
491C22⋊C4—1584no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—81no—
497C42—81no—
498C42—3162no—
499C42#1192962512yes—
500C42#1192962512yes—
501C23—3161no—
502C23—3161no—
503C23—3161no—
504C23—3161no—
505C23—3161no—
506C23—3161no—
507C23—1582no—
508C23C22×C22⋊D7911yescharacteristic
509Q8—84no—
510Q8—3161no—
511Q8—3161no—
512Q8—1582no—
513Q8—1582no—
514D4—3161no—
515D4—3161no—
516D4—1582no—
517D4—1582no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—3161no—
549C2×C4—3161no—
550C2×C4—3161no—
551C2×C4—3161no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—6324no—
555C2×C4—1582no—
556C2×C4#1457915012yes—
557C2×C4#1457915012yes—
558C2×C4#41×D7911yescharacteristic
559C2×C4#41×D7911yescharacteristic
560C2×C4C2×#1192962411yescharacteristic
561C2×C4C2×#1192962411yescharacteristic
562C2×C4C2×#1192962511yescharacteristic
563C2×C4C2×#1192962511yescharacteristic
564C8—3161no—
565C8—3161no—
566C8—3161no—
567C8—3161no—
568C22—3162no—
569C22—3162no—
570C22—3162no—
571C22—6321no—
572C22—6321no—
573C22—6321no—
574C22—6321no—
575C22#1909907011yescharacteristic
576C22#1909907411yescharacteristic
577C22C2×#1457915011yescharacteristic
578C22#1909936611yescharacteristic
579C22#1909937111yescharacteristic
580C22#1909937411yescharacteristic
581C22#1909937811yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—3162no—
593C4—6321no—
594C4—6321no—
595C4#1909910011yescharacteristic
596C4#1909910011yescharacteristic
597C4#1909910011yescharacteristic
598C4#1909910011yescharacteristic
599C2—6322no—
600C2#2376668311yescharacteristic
601C2#2377358411yescharacteristic
602C2#2377362911yescharacteristic
603C2#2377365611yescharacteristic
604C2#2377367111yescharacteristic
605C2#2377375911yescharacteristic
606C2#2377377011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2331C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2331D79
hypocenterC79#23827
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
2. #23764963C2semidirect
3. #23764958C2semidirect
4. #23764819C2semidirect
8. #23764538C2semidirect
10. #23763233C2semidirect
12. C4×#14579084C2semidirect
14. #23763152C2semidirect
15. C79×#2331C2semidirect
304. #2331D79semidirect
319. C79#23827semidirect

Maximal subgroups

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

subgroupindex
#237661112
#237649632
#237649582
#237648192
#237648142
#237646412
#237645382
#237645332
#237632332
#237632282
C4×#145790842
C4×#145790812
#237631522
C79×#23312
#2382779