Finite Group Databasebeta

← #34361014

Subgroups — #34361014

Order 20224.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23767833C211yesmaximal, characteristic
2#23766403C211yesmaximal, characteristic
3#23764810C211yesmaximal, characteristic, complemented
4#23764794C211yesmaximal, characteristic, complemented
5#23764604C211yesmaximal, characteristic
6#23764604C211yesmaximal, characteristic
7#23764159C211yesmaximal, characteristic, complemented
8#23764133C211yesmaximal, characteristic, complemented
9#23763993C211yesmaximal, characteristic, complemented
10#23763993C211yesmaximal, characteristic, complemented
11#23763657C211yesmaximal, characteristic, complemented
12#23763590C211yesmaximal, characteristic
13#23763491C211yesmaximal, characteristic
14C4×#14579086C211yesmaximal, characteristic, complemented
15C79×#1923C211yesmaximal, characteristic
16C2×#14579140C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C22×C4⋊Dic79C2211yescharacteristic
19C2×#14579095C2211yescharacteristic
20C2×#14579095C2211yescharacteristic
21C2×#14579095C2211yescharacteristic
22C2×#14579095C2211yescharacteristic
23C2×#14579086C2211yescharacteristic
24C2×#14579086C2211yescharacteristic
25C2×#14579086C2211yescharacteristic
26C2×#14579086C2211yescharacteristic
27#19098505—21no—
28#19098486—21no—
29#19098469C2211yescharacteristic
30#19098461C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098460—21no—
34#19098456—21no—
35#19098450—21no—
36#19098450—21no—
37#19098435—21no—
38#19098433C2211yescharacteristic
39#19098423C2211yescharacteristic
40#19098423C2211yescharacteristic
41C4×C4⋊Dic79C2211yescharacteristic
42C4×C4⋊Dic79C2211yescharacteristic
43C22⋊C4×Dic79—21no—
44#19098326C2211yescharacteristic
45#19098326C2211yescharacteristic
46#19098326C2211yescharacteristic
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098326C2211yescharacteristic
50#19098326C2211yescharacteristic
51#19098326C2211yescharacteristic
52C2×C4⋊C8×C79C2211yescharacteristic
53C2×C22⋊C8×C79C2211yescharacteristic
54C2×C22⋊C8×C79C2211yescharacteristic
55C2×C4×C8×C79C2211yescharacteristic
56C4×C22⋊C4×C79C2211yescharacteristic
57#336×C79C2211yescharacteristic
58#336×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C23×Dic79—21no—
61#14579140—22no—
62#14579140—22no—
63C4×Dic158—22no—
64C4×Dic158—41no—
65C2×C4⋊Dic79—22no—
66C2×C4⋊Dic79—21no—
67C2×C4⋊Dic79—21no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—41no—
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×C22⋊Dic79—21no—
83C2×C22⋊Dic79—21no—
84C2×C22⋊Dic79—21no—
85C2×C4×Dic79—21no—
86C2×C4×Dic79—21no—
87C2×C4×Dic79—21no—
88#14579095—22no—
89#14579095—22no—
90#14579095—22no—
91#14579095—41no—
92#14579086—22no—
93#14579086—22no—
94#14579086—22no—
95#14579086—41no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C23×C4×C79C2311yescharacteristic
106C22×C8×C79C2311yescharacteristic
107C22×C8×C79C2311yescharacteristic
108C22×C8×C79C2311yescharacteristic
109C22×C8×C79C2311yescharacteristic
110C2×C22⋊C4×C79D411yescharacteristic
111C2×C22⋊C4×C79D411yescharacteristic
112C2×C42×C79C2311yescharacteristic
113C2×C42×C79C2311yescharacteristic
114C4⋊C8×C79—22no—
115C22⋊C8×C79—22no—
116C22⋊C8×C79—41no—
117C4×C8×C79—41no—
118#95×C79D411yescharacteristic
119#95×C79D411yescharacteristic
120C2×Dic158—21no—
121C2×Dic158—21no—
122C2×Dic158—21no—
123C2×Dic158—21no—
124C2×Dic158—21no—
125C2×Dic158#4112yes—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79—21no—
129C22×Dic79—21no—
130C22×Dic79—21no—
131C22×Dic79—21no—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—42no—
135C22×Dic79—42no—
136C22×Dic79—81no—
137C4⋊Dic79—21no—
138C4⋊Dic79—21no—
139C4⋊Dic79—21no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79—21no—
149C4⋊Dic79—42no—
150C4⋊Dic79—81no—
151C4⋊Dic79—81no—
152C4⋊Dic79#4112yes—
153C4⋊Dic79#4112yes—
154Dic79⋊C4—24no—
155Dic79⋊C4—42no—
156Dic79⋊C4—42no—
157Dic79⋊C4—81no—
158Dic79⋊C4—81no—
159C22⋊Dic79—24no—
160C22⋊Dic79—81no—
161C22⋊Dic79—81no—
162C4×Dic79—42no—
163C4×Dic79—81no—
164C4×Dic79—81no—
165C24×C79C2×D411yescharacteristic
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2411yescharacteristic
178C2×C8×C79—21no—
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79—21no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79#4112yes—
190C22⋊C4×C79—41no—
191C22⋊C4×C79Q1614yes—
192C42×C79—21no—
193C42×C79—21no—
194C42×C79—21no—
195C42×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79—21no—
200C23×C79—21no—
201C23×C79—21no—
202C23×C79C2×Q1612yes—
203C23×C79C22×D411yescharacteristic
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—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79C2×Q1612yes—
222C2×C4×C79C2×Q1612yes—
223C2×C4×C79C2×#4111yescharacteristic
224C2×C4×C79C2×#4111yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C2×C4×C79H(5,𝔽2)11yescharacteristic
228C2×C4×C792−1+411yescharacteristic
229C2×Dic79—22no—
230C2×Dic79—22no—
231C2×Dic79—22no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—84no—
254Dic158—22no—
255Dic158—22no—
256Dic158—41no—
257Dic158—41no—
258C8×C79—22no—
259C8×C79—41no—
260C8×C79—41no—
261C8×C79—41no—
262C22×C79—22no—
263C22×C79—22no—
264C22×C79—22no—
265C22×C79—41no—
266C22×C79—41no—
267C22×C79—41no—
268C22×C79—41no—
269C22×C79#52511yescharacteristic
270C22×C79#54611yescharacteristic
271C22×C79#54711yescharacteristic
272C22×C79C22×Q1611yescharacteristic
273C22×C79#57511yescharacteristic
274C22×C79#57611yescharacteristic
275C22×C79#57711yescharacteristic
276C4×C79—22no—
277C4×C79—22no—
278C4×C79—41no—
279C4×C79—41no—
280C4×C79—41no—
281C4×C79#52911yescharacteristic
282C4×C79#53911yescharacteristic
283C4×C79#55311yescharacteristic
284C4×C79#55611yescharacteristic
285Dic79—42no—
286Dic79—42no—
287Dic79—42no—
288Dic79—81no—
289Dic79—81no—
290Dic79—81no—
291Dic79—81no—
292Dic79—81no—
293#27036—791nomaximal, Sylow
294C2×C79—42no—
295C2×C79#246611yescharacteristic
296C2×C79#310411yescharacteristic
297C2×C79#311911yescharacteristic
298C2×C79#329911yescharacteristic
299C2×C79#331911yescharacteristic
300C2×C79#333811yescharacteristic
301C2×C79#335111yescharacteristic
302#2497—791no—
303#2360—791no—
304#2072—791no—
305#2068—791no—
306#2036—791no—
307#2036—791no—
308#1947—791no—
309#1942—791no—
310#1923D7911yescharacteristic
311#1921—791no—
312#1921—791no—
313#1863—791no—
314#1852—791no—
315#1824—791no—
316C4×#103—791no—
317C79#2703611yescharacteristic, complemented, Sylow
318C2×C22⋊Q8—791no—
319C2×C4×Q8—791no—
320C22×C4⋊C4—791no—
321C2×#107—791no—
322C2×#107—791no—
323C2×#107—791no—
324C2×#107—791no—
325C2×C4⋊C8C2×D7911yescharacteristic
326C2×#103—791no—
327C2×#103—791no—
328C2×#103—791no—
329C2×#103—791no—
330C2×C22⋊C8C2×D7911yescharacteristic
331C2×C22⋊C8C2×D7911yescharacteristic
332C2×C4×C8C2×D7911yescharacteristic
333#400—1581no—
334#396—1581no—
335#391—791no—
336#389—791no—
337#389—791no—
338#389—791no—
339#389—1581no—
340#388—1581no—
341#387—1581no—
342#387—1581no—
343#385—1581no—
344#384—791no—
345#380—791no—
346#380—791no—
347C4×C4⋊C4—791no—
348C4×C4⋊C4—791no—
349C4×C22⋊C4—1581no—
350C4×C22⋊C4C2×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#336C2×D7911yescharacteristic
360#336C2×D7911yescharacteristic
361C22×Q8—791no—
362C23×C4—1581no—
363C23×C4C22×D7911yescharacteristic
364C22×C8C22×D7911yescharacteristic
365C22×C8C22×D7911yescharacteristic
366C22×C8C22×D7911yescharacteristic
367C22×C8C22×D7911yescharacteristic
368C22⋊Q8—1582no—
369C22⋊Q8—1582no—
370C4×Q8—3161no—
371C4×Q8—1582no—
372C2×C4⋊C4—3161no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—791no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4—1581no—
383C2×C4⋊C4—1582no—
384C2×C4⋊C4—1581no—
385C2×C4⋊C4—1581no—
386C2×C4⋊C4—1581no—
387C2×C4⋊C4—1581no—
388C2×C4⋊C4—1581no—
389C2×C22⋊C4—1581no—
390C2×C22⋊C4—1581no—
391C2×C22⋊C4—1581no—
392C2×C22⋊C4D31611yescharacteristic
393C2×C22⋊C4D31611yescharacteristic
394C2×C42—1581no—
395C2×C42—1581no—
396C2×C42—1581no—
397C2×C42C22×D7911yescharacteristic
398C2×C42C22×D7911yescharacteristic
399#107—3161no—
400#107—1582no—
401#107—1582no—
402#107—1582no—
403C4⋊C8—22no—
404#103—3161no—
405#103—1582no—
406#103—1582no—
407#103—1582no—
408C22⋊C8—22no—
409C22⋊C8—41no—
410C4×C8—41no—
411#95—1581no—
412#95—1581no—
413#95—1581no—
414#95—1581no—
415#95—1581no—
416#95—1581no—
417#95—1581no—
418#95—1581no—
419#95—1581no—
420#95D31611yescharacteristic
421#95D31611yescharacteristic
422C24C2×D31611yescharacteristic
423C2×Q8—792no—
424C2×Q8—1581no—
425C2×Q8—1581no—
426C2×Q8—1581no—
427C2×Q8—1581no—
428C2×Q8—1581no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—3162no—
436C22×C4—3162no—
437C22×C4—6321no—
438C22×C4—1581no—
439C22×C4—1581no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4C2×D31611yescharacteristic
447C22×C4C2×D31611yescharacteristic
448C22×C4C2×D31611yescharacteristic
449C22×C4C2×D31611yescharacteristic
450C22×C4C2×D31611yescharacteristic
451C22×C4C23×D7911yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8#1192962312yes—
464C4⋊C4—3162no—
465C4⋊C4—3162no—
466C4⋊C4—3162no—
467C4⋊C4—792no—
468C4⋊C4—792no—
469C4⋊C4—6321no—
470C4⋊C4—6321no—
471C4⋊C4—6321no—
472C4⋊C4—6321no—
473C4⋊C4—1584no—
474C4⋊C4—1581no—
475C4⋊C4—1581no—
476C4⋊C4—1581no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C22⋊C4—41no—
487C22⋊C4—6321no—
488C22⋊C4—6321no—
489C22⋊C4—1584no—
490C22⋊C4Dic31614yes—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—3162no—
496C42—6321no—
497C42—6321no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C2×Dic31612yes—
505C23C22×D31611yescharacteristic
506Q8—3161no—
507Q8—3161no—
508Q8—1582no—
509Q8—1582no—
510C2×C4—21no—
511C2×C4—21no—
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—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—3161no—
537C2×C4—3161no—
538C2×C4—3161no—
539C2×C4—3161no—
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—1582no—
551C2×C4—1582no—
552C2×C4C2×Dic31612yes—
553C2×C4C2×Dic31612yes—
554C2×C4#41×D7911yescharacteristic
555C2×C4#41×D7911yescharacteristic
556C2×C4#41×D7911yescharacteristic
557C2×C4C2×#1192962311yescharacteristic
558C2×C4#1457920611yescharacteristic
559C2×C4#1457920811yescharacteristic
560C8—22no—
561C8—41no—
562C8—41no—
563C8—41no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1909907511yescharacteristic
572C22#1909920811yescharacteristic
573C22#1909921711yescharacteristic
574C22C22×Dic31611yescharacteristic
575C22#1909936311yescharacteristic
576C22#1909936811yescharacteristic
577C22#1909937511yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—3162no—
584C4—3162no—
585C4—3162no—
586C4—6321no—
587C4—6321no—
588C4—6321no—
589C4—6321no—
590C4—6321no—
591C4#1909909211yescharacteristic
592C4#1909915611yescharacteristic
593C4#1909925611yescharacteristic
594C4#1909928011yescharacteristic
595C2—42no—
596C2#2376743411yescharacteristic
597C2#2377344011yescharacteristic
598C2#2377360111yescharacteristic
599C2#2377527711yescharacteristic
600C2#2377555811yescharacteristic
601C2#2377579311yescharacteristic
602C2#2377596211yescharacteristic
subgroupG/H
centerC23C22×D316
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1923C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#1923D79
hypocenterC79#27036
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #23764810C2semidirect
4. #23764794C2semidirect
7. #23764159C2semidirect
8. #23764133C2semidirect
9. #23763993C2semidirect
10. #23763993C2semidirect
11. #23763657C2semidirect
14. C4×#14579086C2semidirect
317. C79#27036semidirect

Maximal subgroups

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

subgroupindex
#237678332
#237664032
#237648102
#237647942
#237646042
#237641592
#237641332
#237639932
#237636572
#237635902
#237634912
C4×#145790862
C79×#19232
#2703679