Finite Group Databasebeta

← #34356371

Subgroups — #34356371

Order 20224.

748 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766403C211yesmaximal, characteristic
2#23764722C211yesmaximal, characteristic, complemented
3#23764596C211yesmaximal, characteristic
4#23764548C211yesmaximal, characteristic, complemented
5#23764408C211yesmaximal, characteristic
6#23764358C211yesmaximal, characteristic, complemented
7#23764079C211yesmaximal, characteristic, complemented
8#23764031C211yesmaximal, characteristic, complemented
9#23764008C211yesmaximal, characteristic
10#23763985C211yesmaximal, characteristic, complemented
11#23763744C211yesmaximal, characteristic, complemented
12#23763708C211yesmaximal, characteristic
13#23763484C211yesmaximal, characteristic
14C4×#14579084C211yesmaximal, characteristic, complemented
15C79×#2489C211yesmaximal, characteristic
16C22×#11929614C2211yescharacteristic
17C2×#14579140C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19C22×C4⋊Dic79C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
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×C4×C79⋊C8C2211yescharacteristic
31#19098469C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098460—21no—
34#19098450—21no—
35#19098435—21no—
36C4×C4⋊Dic79C2211yescharacteristic
37C22⋊C4×Dic79—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
60C23×Dic79—21no—
61C2×#11929614—22no—
62C2×#11929614—41no—
63C2×#11929614—41no—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579140—22no—
69#14579140—22no—
70C4×Dic158—22no—
71C4×Dic158—41no—
72C2×C4⋊Dic79—22no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79—41no—
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83C2×C4×Dic79—21no—
84#14579092—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—
94C4×C79⋊C8—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—
136C22×Dic79—42no—
137C22×Dic79—42no—
138C22×Dic79—81no—
139#11929614—44no—
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—
151C2×C79⋊C8—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—42no—
157C4⋊Dic79#4112yes—
158C4⋊Dic79#4112yes—
159Dic79⋊C4—24no—
160Dic79⋊C4—42no—
161Dic79⋊C4—81no—
162C22⋊Dic79—24no—
163C4×Dic79—42no—
164C4×Dic79—81no—
165C24×C79C2×D411yescharacteristic
166C2×Q8×C79—21no—
167C2×Q8×C79—21no—
168C2×Q8×C79—21no—
169C2×Q8×C79—21no—
170C2×Q8×C79—21no—
171C2×Q8×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×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2×D411yescharacteristic
184C22×C4×C79C2×D411yescharacteristic
185C22×C4×C79C2×D411yescharacteristic
186C22×C4×C79C2×D411yescharacteristic
187C22×C4×C79C2411yescharacteristic
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—42no—
194C4⋊C4×C79—42no—
195C4⋊C4×C79—42no—
196C4⋊C4×C79#4112yes—
197C22⋊C4×C79—22no—
198C22⋊C4×C79—22no—
199C22⋊C4×C79—42no—
200C22⋊C4×C79—42no—
201C42×C79—21no—
202C42×C79—21no—
203C42×C79—21no—
204C42×C79—21no—
205C42×C79—42no—
206C23×C79—21no—
207C23×C79—21no—
208C23×C79—21no—
209C23×C79—21no—
210C23×C79—21no—
211C23×C79—21no—
212C23×C79#13512yes—
213C23×C79C22×D411yescharacteristic
214Q8×C79—41no—
215Q8×C79—41no—
216Q8×C79—41no—
217Q8×C79—41no—
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—21no—
225C2×C4×C79—22no—
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—21no—
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×C79—41no—
248C2×C4×C79#13712yes—
249C2×C4×C79C2×#4111yescharacteristic
250C2×C4×C79C2×#4111yescharacteristic
251C2×C4×C79C2×#4111yescharacteristic
252C2×C4×C79C2×#4111yescharacteristic
253C2×C4×C79H(5,𝔽2)11yescharacteristic
254C2×C4×C79H(5,𝔽2)11yescharacteristic
255C2×Dic79—22no—
256C2×Dic79—22no—
257C2×Dic79—22no—
258C2×Dic79—41no—
259C2×Dic79—41no—
260C2×Dic79—41no—
261C2×Dic79—41no—
262C2×Dic79—41no—
263C2×Dic79—41no—
264C2×Dic79—41no—
265C2×Dic79—41no—
266C2×Dic79—41no—
267C2×Dic79—84no—
268Dic158—22no—
269Dic158—22no—
270Dic158—41no—
271Dic158—41no—
272C79⋊C8—41no—
273C79⋊C8—41no—
274C79⋊C8—41no—
275C79⋊C8—41no—
276C22×C79—22no—
277C22×C79—22no—
278C22×C79—22no—
279C22×C79—41no—
280C22×C79—41no—
281C22×C79—41no—
282C22×C79—41no—
283C22×C79#52511yescharacteristic
284C22×C79#54611yescharacteristic
285C22×C79#54611yescharacteristic
286C22×C79C2×#13511yescharacteristic
287C22×C79C2×#13711yescharacteristic
288C22×C79#57711yescharacteristic
289C22×C79#57811yescharacteristic
290C4×C79—22no—
291C4×C79—41no—
292C4×C79—42no—
293C4×C79—41no—
294C4×C79—41no—
295C4×C79—41no—
296C4×C79—81no—
297C4×C79—81no—
298C4×C79—81no—
299C4×C79#52911yescharacteristic
300C4×C79#53811yescharacteristic
301C4×C79#55511yescharacteristic
302C4×C79#55911yescharacteristic
303Dic79—42no—
304Dic79—42no—
305Dic79—42no—
306Dic79—81no—
307#26725—791nomaximal, Sylow
308C2×C79—42no—
309C2×C79#245911yescharacteristic
310C2×C79#310811yescharacteristic
311C2×C79#312011yescharacteristic
312C2×C79#329611yescharacteristic
313C2×C79#330011yescharacteristic
314C2×C79#331211yescharacteristic
315C2×C79#333111yescharacteristic
316#2489D7911yescharacteristic
317#2360—791no—
318#2056—791no—
319#2035—791no—
320#2026—791no—
321#2001—791no—
322#1989—791no—
323#1933—791no—
324#1927—791no—
325#1923—791no—
326#1920—791no—
327#1878—791no—
328#1872—791no—
329#1823—791no—
330C4×#103—791no—
331C79#2672511yescharacteristic, complemented, Sylow
332C22×Q16—791no—
333C2×#125C2×D7911yescharacteristic
334C2×C22⋊Q8—791no—
335C2×C22⋊Q8C2×D7911yescharacteristic
336C2×C4×Q8—791no—
337C22×C4⋊C4—791no—
338C2×#106—791no—
339C2×C4⋊C8—791no—
340C2×#103—791no—
341C2×#103—791no—
342C2×#103—791no—
343C2×#103—791no—
344C2×#103—791no—
345C2×#103—791no—
346C2×C22⋊C8—791no—
347C2×C22⋊C8—791no—
348C2×C4×C8—791no—
349#400—21no—
350#397—21no—
351#391—791no—
352#391C2×D7911yescharacteristic
353#391C2×D7911yescharacteristic
354#389—791no—
355#389—1581no—
356#388—21no—
357#387—1581no—
358#385—1581no—
359#382—21no—
360#380C2×D7911yescharacteristic
361C4×C4⋊C4—791no—
362C4×C4⋊C4C2×D7911yescharacteristic
363C4×C22⋊C4—1581no—
364C4×C22⋊C4C2×D7911yescharacteristic
365#340—791no—
366#340—791no—
367#340—791no—
368#340—791no—
369#340—791no—
370#340—791no—
371#340—791no—
372#340—791no—
373#336—791no—
374#336—791no—
375C22×Q8—791no—
376C22×Q8C22×D7911yescharacteristic
377C23×C4—1581no—
378C23×C4C22×D7911yescharacteristic
379C2×Q16—3161no—
380C2×Q16—3161no—
381C2×Q16—1582no—
382C22×C8—791no—
383C22×C8—791no—
384C22×C8—791no—
385C22×C8—791no—
386#125—22no—
387#125—41no—
388C22⋊Q8—22no—
389C22⋊Q8—41no—
390C22⋊Q8—1582no—
391C22⋊Q8—1582no—
392C4×Q8—3161no—
393C4×Q8—1582no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—3161no—
398C2×C4⋊C4—791no—
399C2×C4⋊C4—791no—
400C2×C4⋊C4—791no—
401C2×C4⋊C4—1581no—
402C2×C4⋊C4—1581no—
403C2×C4⋊C4—1582no—
404C2×C4⋊C4—1581no—
405C2×C4⋊C4—1581no—
406C2×C4⋊C4C22×D7911yescharacteristic
407C2×C4⋊C4C22×D7911yescharacteristic
408C2×C4⋊C4C22×D7911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1581no—
412C2×C22⋊C4C22⋊D7911yescharacteristic
413C2×C22⋊C4C22⋊D7911yescharacteristic
414C2×C42—21no—
415C2×C42—1581no—
416C2×C42—1581no—
417C2×C42C22×D7911yescharacteristic
418C2×C42C22×D7911yescharacteristic
419#106—1582no—
420C4⋊C8—3161no—
421#103—3161no—
422#103—3161no—
423#103—1582no—
424#103—1582no—
425#103—1582no—
426#103—1582no—
427C22⋊C8—3161no—
428C22⋊C8—3161no—
429C4×C8—3161no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1581no—
437#95—1581no—
438#95—1581no—
439#95—1581no—
440#95C22⋊D7911yescharacteristic
441#95C22⋊D7911yescharacteristic
442C24C2×C22⋊D7911yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—792no—
450C2×Q8—1581no—
451C2×Q8—1581no—
452C2×Q8—1581no—
453C2×Q8—1581no—
454C2×Q8—1581no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—3162no—
466C22×C4—3162no—
467C22×C4—6321no—
468C22×C4—1581no—
469C22×C4—1581no—
470C22×C4—1581no—
471C22×C4—1581no—
472C22×C4C2×C22⋊D7911yescharacteristic
473C22×C4C2×C22⋊D7911yescharacteristic
474C22×C4C2×C22⋊D7911yescharacteristic
475C22×C4C2×C22⋊D7911yescharacteristic
476C22×C4C2×C22⋊D7911yescharacteristic
477C22×C4C23×D7911yescharacteristic
478Q16—3164no—
479C2×C8—1581no—
480C2×C8—1581no—
481C2×C8—1581no—
482C2×C8—1581no—
483C2×C8—1581no—
484C2×C8—1581no—
485C2×C8—1581no—
486C2×C8—1581no—
487C2×C8—1581no—
488C2×C8—1581no—
489C2×C8—1581no—
490C2×C8—1581no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—3162no—
500C4⋊C4—3162no—
501C4⋊C4—792no—
502C4⋊C4—792no—
503C4⋊C4—6321no—
504C4⋊C4—1584no—
505C4⋊C4—1581no—
506C4⋊C4—1581no—
507C4⋊C4—1581no—
508C4⋊C4—1581no—
509C4⋊C4#1192962312yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—1584no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—3162no—
521C42—6321no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1457918112yes—
529C23C22×C22⋊D7911yescharacteristic
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534Q8—3161no—
535Q8—3161no—
536Q8—1582no—
537Q8—1582no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
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—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—3161no—
569C2×C4—3161no—
570C2×C4—3161no—
571C2×C4—3161no—
572C2×C4—3161no—
573C2×C4—3161no—
574C2×C4—3161no—
575C2×C4—3161no—
576C2×C4—3161no—
577C2×C4—6324no—
578C2×C4—1582no—
579C2×C4—1582no—
580C2×C4—1582no—
581C2×C4#1457919112yes—
582C2×C4#41×D7911yescharacteristic
583C2×C4#41×D7911yescharacteristic
584C2×C4#41×D7911yescharacteristic
585C2×C4C2×#1192962311yescharacteristic
586C2×C4#1457920611yescharacteristic
587C2×C4#1457920711yescharacteristic
588C8—3161no—
589C8—3161no—
590C8—3161no—
591C8—3161no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22#1909907011yescharacteristic
600C22#1909920111yescharacteristic
601C22#1909920711yescharacteristic
602C22C2×#1457918111yescharacteristic
603C22C2×#1457919111yescharacteristic
604C22#1909937111yescharacteristic
605C22#1909937811yescharacteristic
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—3162no—
616C4—3162no—
617C4—3162no—
618C4—6321no—
619C4#1909909211yescharacteristic
620C4#1909914711yescharacteristic
621C4#1909927511yescharacteristic
622C4#1909929511yescharacteristic
623C2—42no—
624C2#2376733311yescharacteristic
625C2#2377347611yescharacteristic
626C2#2377360811yescharacteristic
627C2#2377522911yescharacteristic
628C2#2377528711yescharacteristic
629C2#2377544511yescharacteristic
630C2#2377571611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2489C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2489D79
hypocenterC79#26725
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. #23764079C2semidirect
8. #23764031C2semidirect
10. #23763985C2semidirect
11. #23763744C2semidirect
14. C4×#14579084C2semidirect
331. C79#26725semidirect

Maximal subgroups

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

subgroupindex
#237664032
#237647222
#237645962
#237645482
#237644082
#237643582
#237640792
#237640312
#237640082
#237639852
#237637442
#237637082
#237634842
C4×#145790842
C79×#24892
#2672579