Finite Group Databasebeta

← #34356731

Subgroups — #34356731

Order 20224.

720 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766425C211yesmaximal, characteristic
2#23764722C211yesmaximal, characteristic, complemented
3#23764603C211yesmaximal, characteristic
4#23764548C211yesmaximal, characteristic, complemented
5#23764408C211yesmaximal, characteristic
6#23764358C211yesmaximal, characteristic, complemented
7#23764093C211yesmaximal, characteristic, complemented
8#23764038C211yesmaximal, characteristic, complemented
9#23764015C211yesmaximal, characteristic
10#23763992C211yesmaximal, characteristic, complemented
11#23763744C211yesmaximal, characteristic, complemented
12#23763713C211yesmaximal, characteristic
13#23763492C211yesmaximal, characteristic
14#23763233C211yesmaximal, characteristic, complemented
15C79×#2489C211yesmaximal, characteristic
16C22×#11929614C2211yescharacteristic
17C2×#14579140C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19C2×#14579120C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×#14579085C2211yescharacteristic
23C2×#14579085C2211yescharacteristic
24C2×#14579085C2211yescharacteristic
25C2×#14579084C2211yescharacteristic
26C2×#14579084C2211yescharacteristic
27C2×#14579084C2211yescharacteristic
28C2×C22⋊(C79⋊C8)C2211yescharacteristic
29C2×C22⋊(C79⋊C8)C2211yescharacteristic
30C2×#14579072C2211yescharacteristic
31#19098469C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098457—21no—
34#19098451—21no—
35#19098445—21no—
36C4×C4⋊Dic79C2211yescharacteristic
37#19098414—21no—
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#125×C79C2211yescharacteristic
49C2×C22⋊Q8×C79C2211yescharacteristic
50#400×C79—21no—
51#397×C79—21no—
52#391×C79C2211yescharacteristic
53#391×C79C2211yescharacteristic
54#388×C79—21no—
55#382×C79—21no—
56#380×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59C22×Dic158C2311yescharacteristic
60C2×#11929614—22no—
61C2×#11929614—41no—
62C2×#11929614—41no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579140—22no—
68#14579140—22no—
69C4×Dic158—22no—
70C4×Dic158—41no—
71#14579120—22no—
72#14579120—41no—
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C22⋊Dic79—21no—
80C2×C22⋊Dic79—21no—
81C2×C4×Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84#14579094—22no—
85C4⋊(C79⋊C8)—41no—
86#14579085—22no—
87#14579085—22no—
88#14579085—41no—
89#14579084—22no—
90#14579084—22no—
91#14579084—41no—
92C22⋊(C79⋊C8)—41no—
93C22⋊(C79⋊C8)—41no—
94#14579072—41no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C79⋊#95—21no—
98C79⋊#95—21no—
99C22×Q8×C79C2311yescharacteristic
100C23×C4×C79C2311yescharacteristic
101#125×C79—22no—
102#125×C79—41no—
103C22⋊Q8×C79—22no—
104C22⋊Q8×C79—41no—
105C2×C4⋊C4×C79—21no—
106C2×C4⋊C4×C79—21no—
107C2×C4⋊C4×C79—21no—
108C2×C4⋊C4×C79C2311yescharacteristic
109C2×C4⋊C4×C79C2311yescharacteristic
110C2×C4⋊C4×C79C2311yescharacteristic
111C2×C22⋊C4×C79—21no—
112C2×C22⋊C4×C79—21no—
113C2×C22⋊C4×C79D411yescharacteristic
114C2×C22⋊C4×C79D411yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79C2311yescharacteristic
117C2×C42×C79C2311yescharacteristic
118#95×C79—21no—
119#95×C79—21no—
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123#95×C79—21no—
124#95×C79D411yescharacteristic
125#95×C79D411yescharacteristic
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158#4112yes—
132C22×Dic79—21no—
133C22×Dic79—21no—
134C22×Dic79—21no—
135C22×Dic79—21no—
136#11929614—44no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
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—
149C4⋊Dic79—21no—
150C4⋊Dic79—21no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79#4112yes—
154C4⋊Dic79#4112yes—
155Dic79⋊C4—24no—
156Dic79⋊C4—42no—
157Dic79⋊C4—42no—
158C22⋊Dic79—24no—
159C22⋊Dic79—42no—
160C4×Dic79—42no—
161C4×Dic79—42no—
162C4×Dic79—42no—
163C24×C79C2×D411yescharacteristic
164C2×Q8×C79—21no—
165C2×Q8×C79—21no—
166C2×Q8×C79—21no—
167C2×Q8×C79—21no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79—21no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2411yescharacteristic
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—42no—
192C4⋊C4×C79—42no—
193C4⋊C4×C79—42no—
194C4⋊C4×C79#4112yes—
195C22⋊C4×C79—41no—
196C22⋊C4×C79—41no—
197C22⋊C4×C79—42no—
198C22⋊C4×C79—42no—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—42no—
204C23×C79—21no—
205C23×C79—21no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79C22×D411yescharacteristic
212Q8×C79—41no—
213Q8×C79—41no—
214Q8×C79—41no—
215Q8×C79—41no—
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—22no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—21no—
227C2×C4×C79—21no—
228C2×C4×C79—21no—
229C2×C4×C79—21no—
230C2×C4×C79—21no—
231C2×C4×C79—21no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79—41no—
237C2×C4×C79—41no—
238C2×C4×C79—41no—
239C2×C4×C79—41no—
240C2×C4×C79—41no—
241C2×C4×C79—41no—
242C2×C4×C79—41no—
243C2×C4×C79—41no—
244C2×C4×C79—41no—
245C2×C4×C79—41no—
246C2×C4×C79—41no—
247C2×C4×C79C2×#4111yescharacteristic
248C2×C4×C79C2×#4111yescharacteristic
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×C4×C79H(5,𝔽2)11yescharacteristic
252C2×C4×C79H(5,𝔽2)11yescharacteristic
253C2×Dic79—22no—
254C2×Dic79—22no—
255C2×Dic79—22no—
256C2×Dic79—41no—
257C2×Dic79—41no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265Dic158—22no—
266Dic158—22no—
267Dic158—41no—
268Dic158—41no—
269C79⋊C8—41no—
270C79⋊C8—41no—
271C79⋊C8—41no—
272C79⋊C8—41no—
273C22×C79—41no—
274C22×C79—41no—
275C22×C79—41no—
276C22×C79—41no—
277C22×C79—41no—
278C22×C79—41no—
279C22×C79—41no—
280C22×C79#52511yescharacteristic
281C22×C79#54611yescharacteristic
282C22×C79#54611yescharacteristic
283C22×C79#57511yescharacteristic
284C22×C79#57511yescharacteristic
285C22×C79#57811yescharacteristic
286C22×C79#57811yescharacteristic
287C4×C79—41no—
288C4×C79—41no—
289C4×C79—41no—
290C4×C79—41no—
291C4×C79—42no—
292C4×C79—41no—
293C4×C79—81no—
294C4×C79—81no—
295C4×C79—81no—
296C4×C79#52911yescharacteristic
297C4×C79#53811yescharacteristic
298C4×C79#55511yescharacteristic
299C4×C79#55911yescharacteristic
300Dic79—42no—
301Dic79—42no—
302Dic79—42no—
303Dic79—81no—
304#26749—791nomaximal, Sylow
305C2×C79—81no—
306C2×C79#245911yescharacteristic
307C2×C79#311111yescharacteristic
308C2×C79#312711yescharacteristic
309C2×C79#331311yescharacteristic
310C2×C79#331511yescharacteristic
311C2×C79#331511yescharacteristic
312C2×C79#332711yescharacteristic
313#2489D7911yescharacteristic
314#2362—791no—
315#2056—791no—
316#2036—791no—
317#2026—791no—
318#2001—791no—
319#1989—791no—
320#1935—791no—
321#1928—791no—
322#1924—791no—
323#1921—791no—
324#1878—791no—
325#1873—791no—
326#1825—791no—
327#1763—791no—
328C79#2674911yescharacteristic, complemented, Sylow
329C22×Q16—791no—
330C2×#125C2×D7911yescharacteristic
331C2×C22⋊Q8—791no—
332C2×C22⋊Q8C2×D7911yescharacteristic
333C2×C4×Q8—791no—
334C2×#117—791no—
335C2×#107—791no—
336C2×C4⋊C8—791no—
337C2×#103—791no—
338C2×#103—791no—
339C2×#103—791no—
340C2×#103—791no—
341C2×#103—791no—
342C2×#103—791no—
343C2×C22⋊C8—791no—
344C2×C22⋊C8—791no—
345C2×#97—791no—
346#400—21no—
347#397—21no—
348#391—791no—
349#391C2×D7911yescharacteristic
350#391C2×D7911yescharacteristic
351#389—791no—
352#388—21no—
353#388—1581no—
354#388—1581no—
355#387—1581no—
356#382—21no—
357#380C2×D7911yescharacteristic
358C4×C4⋊C4—791no—
359C4×C4⋊C4—1581no—
360C4×C4⋊C4C2×D7911yescharacteristic
361C4×C22⋊C4C2×D7911yescharacteristic
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#336—791no—
371#336—791no—
372C22×Q8—791no—
373C22×Q8C22×D7911yescharacteristic
374C23×C4C22×D7911yescharacteristic
375C2×Q16—3161no—
376C2×Q16—3161no—
377C2×Q16—1582no—
378C22×C8—791no—
379C22×C8—791no—
380C22×C8—791no—
381C22×C8—791no—
382#125—22no—
383#125—41no—
384C22⋊Q8—22no—
385C22⋊Q8—41no—
386C22⋊Q8—1582no—
387C22⋊Q8—1582no—
388C4×Q8—3161no—
389C4×Q8—1582no—
390#117—3161no—
391#117—1582no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—791no—
396C2×C4⋊C4—791no—
397C2×C4⋊C4—791no—
398C2×C4⋊C4—1581no—
399C2×C4⋊C4—1581no—
400C2×C4⋊C4—1581no—
401C2×C4⋊C4C22×D7911yescharacteristic
402C2×C4⋊C4C22×D7911yescharacteristic
403C2×C4⋊C4C22×D7911yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1581no—
407C2×C22⋊C4—1581no—
408C2×C22⋊C4C22⋊D7911yescharacteristic
409C2×C22⋊C4C22⋊D7911yescharacteristic
410C2×C42—21no—
411C2×C42—1581no—
412C2×C42—1581no—
413C2×C42—1581no—
414C2×C42C22×D7911yescharacteristic
415C2×C42C22×D7911yescharacteristic
416#107—1582no—
417C4⋊C8—3161no—
418#103—3161no—
419#103—3161no—
420#103—1582no—
421#103—1582no—
422#103—1582no—
423#103—1582no—
424C22⋊C8—3161no—
425C22⋊C8—3161no—
426#97—3161no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1581no—
434#95—1581no—
435#95—1581no—
436#95—1581no—
437#95C22⋊D7911yescharacteristic
438#95C22⋊D7911yescharacteristic
439C24C2×C22⋊D7911yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—792no—
447C2×Q8—1581no—
448C2×Q8—1581no—
449C2×Q8—1581no—
450C2×Q8—1581no—
451C2×Q8—1581no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1581no—
463C22×C4—1581no—
464C22×C4—1581no—
465C22×C4—1581no—
466C22×C4C2×C22⋊D7911yescharacteristic
467C22×C4C2×C22⋊D7911yescharacteristic
468C22×C4C2×C22⋊D7911yescharacteristic
469C22×C4C2×C22⋊D7911yescharacteristic
470C22×C4C2×C22⋊D7911yescharacteristic
471C22×C4C23×D7911yescharacteristic
472Q16—3164no—
473C2×C8—1581no—
474C2×C8—1581no—
475C2×C8—1581no—
476C2×C8—1581no—
477C2×C8—1581no—
478C2×C8—1581no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—3162no—
494C4⋊C4—3162no—
495C4⋊C4—792no—
496C4⋊C4—792no—
497C4⋊C4—1584no—
498C4⋊C4—1581no—
499C4⋊C4—1581no—
500C4⋊C4—1581no—
501C4⋊C4—1581no—
502C4⋊C4#1192962312yes—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—3162no—
508C22⋊C4—1584no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—3162no—
515C42—3162no—
516C42—3162no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×C22⋊D7911yescharacteristic
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—3161no—
530Q8—3161no—
531Q8—1582no—
532Q8—1582no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—3161no—
565C2×C4—3161no—
566C2×C4—3161no—
567C2×C4—3161no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—1582no—
574C2×C4—1582no—
575C2×C4—1582no—
576C2×C4#41×D7911yescharacteristic
577C2×C4#41×D7911yescharacteristic
578C2×C4#41×D7911yescharacteristic
579C2×C4C2×#1192962311yescharacteristic
580C2×C4#1457920611yescharacteristic
581C2×C4#1457920711yescharacteristic
582C8—3161no—
583C8—3161no—
584C8—3161no—
585C8—3161no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1909907011yescharacteristic
594C22#1909920111yescharacteristic
595C22#1909920711yescharacteristic
596C22#1909936111yescharacteristic
597C22#1909936111yescharacteristic
598C22#1909937811yescharacteristic
599C22#1909937811yescharacteristic
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—3162no—
610C4—3162no—
611C4—3162no—
612C4—6321no—
613C4#1909909211yescharacteristic
614C4#1909914711yescharacteristic
615C4#1909927511yescharacteristic
616C4#1909929511yescharacteristic
617C2—81no—
618C2#2376733311yescharacteristic
619C2#2377350311yescharacteristic
620C2#2377369511yescharacteristic
621C2#2377546611yescharacteristic
622C2#2377549011yescharacteristic
623C2#2377549611yescharacteristic
624C2#2377566211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2489C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2489D79
hypocenterC79#26749
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2489

Complements

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

NH ≅ G/Ntype
2. #23764722C2semidirect
4. #23764548C2semidirect
6. #23764358C2semidirect
7. #23764093C2semidirect
8. #23764038C2semidirect
10. #23763992C2semidirect
11. #23763744C2semidirect
14. #23763233C2semidirect
328. C79#26749semidirect

Maximal subgroups

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

subgroupindex
#237664252
#237647222
#237646032
#237645482
#237644082
#237643582
#237640932
#237640382
#237640152
#237639922
#237637442
#237637132
#237634922
#237632332
C79×#24892
#2674979