Finite Group Databasebeta

← #34388042

Subgroups — #34388042

Order 20224.

690 conjugacy classes of subgroups.

The subgroup lattice has 602 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#23764641C211yesmaximal, characteristic
5#23764641C211yesmaximal, characteristic
6#23764096C211yesmaximal, characteristic
7#23764089C211yesmaximal, characteristic, complemented
8#23764082C211yesmaximal, characteristic
9#23764075C211yesmaximal, characteristic, complemented
10#23763412C211yesmaximal, characteristic, complemented
11#23763407C211yesmaximal, characteristic
12#23763402C211yesmaximal, characteristic, complemented
13#23763397C211yesmaximal, characteristic
14#23763257C211yesmaximal, characteristic, complemented
15C79×#2647C211yesmaximal, characteristic, complemented
16C2×#14579149C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19C2×C4⋊(C79⋊C8)C2211yescharacteristic
20C2×C4⋊(C79⋊C8)C2211yescharacteristic
21C2×#14579084C2211yescharacteristic
22C2×#14579084C2211yescharacteristic
23C2×#14579084C2211yescharacteristic
24C2×#14579084C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579081C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31#19098469C2211yescharacteristic
32#19098465C2211yescharacteristic
33#19098455—21no—
34#19098455—21no—
35#19098437—21no—
36#19098437—21no—
37#19098433C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#400×C79—21no—
49#400×C79—21no—
50#398×C79—21no—
51#398×C79—21no—
52#389×C79C2211yescharacteristic
53#389×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55#389×C79C2211yescharacteristic
56#384×C79C2211yescharacteristic
57#382×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—
83C4⋊(C79⋊C8)—22no—
84C4⋊(C79⋊C8)—22no—
85#14579084—22no—
86#14579084—22no—
87#14579084—41no—
88#14579084—41no—
89#14579081—22no—
90#14579081—22no—
91#14579081—41no—
92#14579081—41no—
93#14579072—41no—
94C4×C79⋊C8—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
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—
164C22×C4×C79—21no—
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79C2×D411yescharacteristic
169C22×C4×C79C2×D411yescharacteristic
170C22×C4×C79C2×D411yescharacteristic
171C22×C4×C79C2×D411yescharacteristic
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2411yescharacteristic
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—21no—
178C4⋊C4×C79—21no—
179C4⋊C4×C79—21no—
180C4⋊C4×C79—21no—
181C4⋊C4×C79—21no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—81no—
184C4⋊C4×C79—81no—
185C4⋊C4×C79—81no—
186C4⋊C4×C79—81no—
187C4⋊C4×C79—81no—
188C4⋊C4×C79—81no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—21no—
193C42×C79#4112yes—
194C42×C79#4112yes—
195C23×C79C22×D411yescharacteristic
196C22×D79—22no—
197C22×D79—41no—
198C22×D79—41no—
199C22×D79—41no—
200C22×D79—41no—
201C22×D79—41no—
202C22×D79—41no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
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—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
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#12412yes—
232C2×C4×C79#12412yes—
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79C2×#4111yescharacteristic
237C2×C4×C79H(5,𝔽2)11yescharacteristic
238C2×C4×C792−1+411yescharacteristic
239C2×Dic79—22no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248D316—22no—
249D316—22no—
250D316—41no—
251D316—41no—
252C4×D79—84no—
253Dic158—22no—
254Dic158—22no—
255Dic158—41no—
256Dic158—41no—
257C79⋊C8—41no—
258C79⋊C8—41no—
259C79⋊C8—41no—
260C79⋊C8—41no—
261C22×C79#52511yescharacteristic
262C22×C79C2×#12411yescharacteristic
263C22×C79#53711yescharacteristic
264C22×C79#57611yescharacteristic
265C22×C79#57711yescharacteristic
266C22×C79#57711yescharacteristic
267C22×C79#57811yescharacteristic
268C2×D79—42no—
269C2×D79—42no—
270C2×D79—42no—
271C2×D79—81no—
272C2×D79—81no—
273C2×D79—81no—
274C2×D79—81no—
275C4×C79—22no—
276C4×C79—22no—
277C4×C79—41no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79—81no—
282C4×C79—81no—
283C4×C79—81no—
284C4×C79—81no—
285C4×C79#55211yescharacteristic
286C4×C79#55211yescharacteristic
287C4×C79#55311yescharacteristic
288C4×C79#55311yescharacteristic
289Dic79—42no—
290Dic79—81no—
291Dic79—81no—
292#29134—791nomaximal, Sylow
293C2×C79#256211yescharacteristic
294C2×C79#311911yescharacteristic
295C2×C79#312011yescharacteristic
296C2×C79#313611yescharacteristic
297C2×C79#313711yescharacteristic
298C2×C79#326211yescharacteristic
299C2×C79#326311yescharacteristic
300D79—82no—
301#2647D7911yescharacteristic, complemented
302#2326—791no—
303#2100—791no—
304#2099—791no—
305#2042—791no—
306#2042—791no—
307#1936—791no—
308#1935—791no—
309#1934—791no—
310#1933—791no—
311#1805—791no—
312#1804—791no—
313#1803—791no—
314#1802—791no—
315#1768—791no—
316C79#2913411yescharacteristic, complemented, Sylow
317C2×#124—791no—
318C2×C4×Q8—791no—
319C2×C4×D4—791no—
320C2×C4⋊C8—791no—
321C2×C4⋊C8—791no—
322C2×#103—791no—
323C2×#103—791no—
324C2×#103—791no—
325C2×#103—791no—
326C2×#102—791no—
327C2×#102—791no—
328C2×#102—791no—
329C2×#102—791no—
330C2×#97—791no—
331C2×C4×C8—791no—
332#400—21no—
333#400—21no—
334#398—21no—
335#398—21no—
336#391—791no—
337#390—791no—
338#389C2×D7911yescharacteristic
339#389C2×D7911yescharacteristic
340#389C2×D7911yescharacteristic
341#389C2×D7911yescharacteristic
342#388—1581no—
343#388—1581no—
344#385—1581no—
345#385—1581no—
346#384—791no—
347#384C2×D7911yescharacteristic
348#382C2×D7911yescharacteristic
349#376C2×D7911yescharacteristic
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#336—791no—
359#336—791no—
360C22×Q8—791no—
361C22×D4—791no—
362C23×C4—1581no—
363C22×C8—791no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—791no—
367#124—1582no—
368#124—1582no—
369C4×Q8—1582no—
370C4×Q8—1582no—
371C4×D4—1582no—
372C4×D4—1582no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—791no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4—1581no—
383C2×C4⋊C4—1581no—
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C4⋊C4C22×D7911yescharacteristic
386C2×C4⋊C4C22×D7911yescharacteristic
387C2×C4⋊C4C22×D7911yescharacteristic
388C2×C22⋊C4—1581no—
389C2×C22⋊C4—1581no—
390C2×C22⋊C4—1581no—
391C2×C22⋊C4—1581no—
392C2×C22⋊C4—1581no—
393C2×C42—1581no—
394C2×C42C22×D7911yescharacteristic
395C2×C42C22×D7911yescharacteristic
396C2×C42C22×D7911yescharacteristic
397C4⋊C8—1582no—
398C4⋊C8—1582no—
399#103—3161no—
400#103—3161no—
401#103—1582no—
402#103—1582no—
403#102—3161no—
404#102—3161no—
405#102—1582no—
406#102—1582no—
407#97—3161no—
408C4×C8—3161no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1581no—
416#95—1581no—
417#95C22⋊D7911yescharacteristic
418#95C22⋊D7911yescharacteristic
419#95C22⋊D7911yescharacteristic
420#95C22⋊D7911yescharacteristic
421C24—1581no—
422C2×Q8—792no—
423C2×Q8—1581no—
424C2×Q8—1581no—
425C2×Q8—1581no—
426C2×Q8—1581no—
427C2×Q8—1581no—
428C2×D4—792no—
429C2×D4—1581no—
430C2×D4—1581no—
431C2×D4—1581no—
432C2×D4—1581no—
433C2×D4—1581no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—3162no—
439C22×C4—6321no—
440C22×C4—6321no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4C2×C22⋊D7911yescharacteristic
445C22×C4C2×C22⋊D7911yescharacteristic
446C22×C4C2×C22⋊D7911yescharacteristic
447C22×C4C2×C22⋊D7911yescharacteristic
448C22×C4C2×C22⋊D7911yescharacteristic
449C22×C4C2×C22⋊D7911yescharacteristic
450C22×C4C23×D7911yescharacteristic
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8—1581no—
456C2×C8—1581no—
457C2×C8—1581no—
458C2×C8—1581no—
459C2×C8—1581no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8—1581no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—3162no—
478C4⋊C4—6321no—
479C4⋊C4—6321no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C22⋊C4—3162no—
485C22⋊C4—6321no—
486C22⋊C4—6321no—
487C22⋊C4—1584no—
488C22⋊C4—1584no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—3162no—
494C42#1192962512yes—
495C42#1192962512yes—
496C23—3161no—
497C23—3161no—
498C23—3161no—
499C23—3161no—
500C23—3161no—
501C23—3161no—
502C23—1582no—
503C23C22×C22⋊D7911yescharacteristic
504Q8—3161no—
505Q8—3161no—
506Q8—1582no—
507Q8—1582no—
508D4—3161no—
509D4—3161no—
510D4—1582no—
511D4—1582no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
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—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—3161no—
541C2×C4—3161no—
542C2×C4—3161no—
543C2×C4—3161no—
544C2×C4—3161no—
545C2×C4—3161no—
546C2×C4—3161no—
547C2×C4—3161no—
548C2×C4—6324no—
549C2×C4—1582no—
550C2×C4#1457915012yes—
551C2×C4#1457915012yes—
552C2×C4#41×D7911yescharacteristic
553C2×C4#41×D7911yescharacteristic
554C2×C4C2×#1192962511yescharacteristic
555C2×C4C2×#1192962511yescharacteristic
556C2×C4#1457920711yescharacteristic
557C2×C4#1457920811yescharacteristic
558C8—3161no—
559C8—3161no—
560C8—3161no—
561C8—3161no—
562C22—3162no—
563C22—3162no—
564C22—3162no—
565C22—6321no—
566C22—6321no—
567C22—6321no—
568C22—6321no—
569C22#1909907011yescharacteristic
570C22C2×#1457915011yescharacteristic
571C22#1909914111yescharacteristic
572C22#1909936611yescharacteristic
573C22#1909937111yescharacteristic
574C22#1909937411yescharacteristic
575C22#1909937811yescharacteristic
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—3162no—
587C4—6321no—
588C4—6321no—
589C4#1909925111yescharacteristic
590C4#1909925111yescharacteristic
591C4#1909926611yescharacteristic
592C4#1909926611yescharacteristic
593C2—6322no—
594C2#2376856711yescharacteristic
595C2#2377359311yescharacteristic
596C2#2377360811yescharacteristic
597C2#2377375911yescharacteristic
598C2#2377377011yescharacteristic
599C2#2377485611yescharacteristic
600C2#2377486511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2647C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2647D79
hypocenterC79#29134
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #2647

Complements

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

NH ≅ G/Ntype
2. #23764963C2semidirect
3. #23764958C2semidirect
7. #23764089C2semidirect
9. #23764075C2semidirect
10. #23763412C2semidirect
12. #23763402C2semidirect
14. #23763257C2semidirect
15. C79×#2647C2semidirect
301. #2647D79semidirect
316. C79#29134semidirect

Maximal subgroups

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

subgroupindex
#237661112
#237649632
#237649582
#237646412
#237640962
#237640892
#237640822
#237640752
#237634122
#237634072
#237634022
#237633972
#237632572
C79×#26472
#2913479