Finite Group Databasebeta

← #34298798

Subgroups — #34298798

Order 20224.

834 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
1C2×#19098615C211yesmaximal, characteristic
2#23765796C211yesmaximal, characteristic
3#382×D79C211yesmaximal, characteristic
4#23764170C211yesmaximal, characteristic, complemented
5#23764168C211yesmaximal, characteristic, complemented
6#23764166C211yesmaximal, characteristic, complemented
7#23763594C211yesmaximal, characteristic
8#23763591C211yesmaximal, characteristic
9#23763589C211yesmaximal, characteristic
10#23763586C211yesmaximal, characteristic
11#23763563C211yesmaximal, characteristic, complemented
12#23763561C211yesmaximal, characteristic, complemented
13#23763529C211yesmaximal, characteristic, complemented
14#23763529C211yesmaximal, characteristic, complemented
15C79×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D79C2211yescharacteristic
17C2×C4⋊C4×D79C2211yescharacteristic
18C2×C42×D79C2211yescharacteristic
19#19098615C418yescomplemented
20C2×#14579095C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579093C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×(C2×D79)⋊C8C2211yescharacteristic
28C2×(C2×D79)⋊C8C2211yescharacteristic
29C2×C8×Dic79C2211yescharacteristic
30C2×C8×Dic79C2211yescharacteristic
31#19098430—21no—
32#19098429C2211yescharacteristic
33#19098428C2211yescharacteristic
34#19098428C2211yescharacteristic
35#19098428C2211yescharacteristic
36#19098428C2211yescharacteristic
37#19098427—21no—
38#19098427—21no—
39#19098425—21no—
40#19098420C2211yescharacteristic
41#19098420C2211yescharacteristic
42C4×C4⋊Dic79C2211yescharacteristic
43C4×(C2×D79)⋊C4—21no—
44#19098409—21no—
45#95×D79—21no—
46#95×D79—21no—
47#19098326C2211yescharacteristic
48#19098326C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098325C2211yescharacteristic
51#19098325C2211yescharacteristic
52#19098325C2211yescharacteristic
53C2×#107×C79C2211yescharacteristic
54C2×#106×C79C2211yescharacteristic
55C2×C4⋊C8×C79C2211yescharacteristic
56#382×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#340×C79C2211yescharacteristic
60C22×C4×D79—21no—
61C22×C4×D79—21no—
62C22×C4×D79D411yescharacteristic
63C22×C4×D79D411yescharacteristic
64C22×C4×D79C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×Dic79⋊C4—21no—
72C2×Dic79⋊C4—21no—
73C4⋊C4×D79—41no—
74C4⋊C4×D79—41no—
75C4⋊C4×D79—41no—
76C4⋊C4×D79—41no—
77C2×(C2×D79)⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×C4×Dic79—21no—
80C2×C4×Dic79D411yescharacteristic
81C2×C4×Dic79D411yescharacteristic
82C2×C4×Dic79C2311yescharacteristic
83C42×D79C2×C414yes—
84C42×D79Q814yes—
85#14579095—41no—
86#14579094—41no—
87#14579094—41no—
88#14579093—41no—
89#14579092—41no—
90#14579092—41no—
91C4⋊(C79⋊C8)C2×C414yescomplemented
92(C2×D79)⋊C8C2×C414yes—
93(C2×D79)⋊C8C2×C414yes—
94C8×Dic79C2×C414yescomplemented
95C8×Dic79C2×C414yescomplemented
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—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C22×C8×C79C2311yescharacteristic
110C22×C8×C79C2311yescharacteristic
111C2×C4⋊C4×C79C2311yescharacteristic
112C2×C4⋊C4×C79C2311yescharacteristic
113C2×C4⋊C4×C79C2311yescharacteristic
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C42×C79—21no—
116C2×C42×C79C2311yescharacteristic
117#107×C79—41no—
118#106×C79—41no—
119C4⋊C8×C79C2×C414yescomplemented
120#95×C79—21no—
121#95×C79—21no—
122#95×C79—21no—
123C23×D79C2×D411yescharacteristic
124C22×Dic79—21no—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C22×Dic79C2×D411yescharacteristic
129C22×Dic79C2×D411yescharacteristic
130C22×Dic79C2×D411yescharacteristic
131C2×C4×D79—22no—
132C2×C4×D79—22no—
133C2×C4×D79—21no—
134C2×C4×D79—21no—
135C2×C4×D79—21no—
136C2×C4×D79—21no—
137C2×C4×D79—41no—
138C2×C4×D79—41no—
139C2×C4×D79—42no—
140C2×C4×D79—42no—
141C2×C4×D79—42no—
142C2×C4×D79—42no—
143C2×C4×D79—42no—
144C2×C4×D79—42no—
145C2×C4×D79C4⋊C414yes—
146C2×C4×D79C4⋊C414yes—
147C2×C4×D79C22×C412yes—
148C2×C4×D79C2×Q812yes—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8C22×C412yes—
154C2×C79⋊C8C22×C412yes—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163Dic79⋊C4—42no—
164Dic79⋊C4—42no—
165(C2×D79)⋊C4—42no—
166(C2×D79)⋊C4—42no—
167C4×Dic79—42no—
168C4×Dic79C4⋊C414yes—
169C4×Dic79C4⋊C414yes—
170C4×Dic79C22×C412yes—
171C4×Dic79C2×Q812yes—
172C22×C4×C79—21no—
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—21no—
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2411yescharacteristic
179C2×C8×C79—21no—
180C2×C8×C79—21no—
181C2×C8×C79—21no—
182C2×C8×C79—21no—
183C2×C8×C79C22×C412yes—
184C2×C8×C79C22×C412yes—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—21no—
189C4⋊C4×C79—21no—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C42×C79—42no—
194C42×C79C22×C412yes—
195C42×C79C2×Q812yes—
196C23×C79C22×D411yescharacteristic
197C22×D79—21no—
198C22×D79—21no—
199C22×D79—21no—
200C22×D79—21no—
201C22×D79C2×C4⋊C412yes—
202C22×D79#13512yes—
203C22×D79#13512yes—
204C2×C4×C79—21no—
205C2×C4×C79—21no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—41no—
209C2×C4×C79—41no—
210C2×C4×C79—41no—
211C2×C4×C79—41no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—41no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79C2×C4⋊C412yes—
221C2×C4×C79C2×C4⋊C412yes—
222C2×C4×C79C23×C411yescharacteristic
223C2×C4×C79C22×Q811yescharacteristic
224C2×C4×C79H(5,𝔽2)11yescharacteristic
225C2×C4×C79H(5,𝔽2)11yescharacteristic
226C2×C4×C792−1+411yescharacteristic
227C2×C4×C792−1+411yescharacteristic
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
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×Dic79C2×C4⋊C412yes—
247C2×Dic79C2×C4⋊C412yes—
248C2×Dic79C2×C4⋊C412yes—
249C4×D79—28no—
250C4×D79—22no—
251C4×D79—22no—
252C4×D79—22no—
253C4×D79—22no—
254C4×D79—44no—
255C4×D79—84no—
256C4×D79—84no—
257C8×C79—22no—
258C8×C79—22no—
259C79⋊C8—22no—
260C79⋊C8—22no—
261C22×C79C22×C4⋊C411yescharacteristic
262C22×C79#53511yescharacteristic
263C22×C79#53611yescharacteristic
264C22×C79C2×#13511yescharacteristic
265C22×C79C2×#13511yescharacteristic
266C22×C79#57511yescharacteristic
267C22×C79#57511yescharacteristic
268C2×D79—22no—
269C2×D79—22no—
270C2×D79—22no—
271C2×D79—22no—
272C2×D79—22no—
273C2×D79—22no—
274C2×D79#42714yes—
275C4×C79—22no—
276C4×C79—22no—
277C4×C79—81no—
278C4×C79—81no—
279C4×C79—81no—
280C4×C79—81no—
281C4×C79#52011yescharacteristic
282C4×C79#52011yescharacteristic
283C4×C79#54411yescharacteristic
284C4×C79#54411yescharacteristic
285Dic79—22no—
286Dic79—22no—
287Dic79—22no—
288Dic79—81no—
289Dic79—81no—
290Dic79—81no—
291Dic79—81no—
292#21983—791nomaximal, Sylow
293C2×C79#231711yescharacteristic
294C2×C79C2×#42711yescharacteristic
295C2×C79#291111yescharacteristic
296C2×C79#321111yescharacteristic
297C2×C79#321111yescharacteristic
298C2×C79#323211yescharacteristic
299C2×C79#323211yescharacteristic
300D79—24no—
301C2×#431—791no—
302#2283—791no—
303C2×#382—791no—
304#1949—791no—
305#1949—791no—
306#1949—791no—
307#1949D7911yescharacteristic, complemented
308#1853—791no—
309#1853—791no—
310#1852—791no—
311#1852—791no—
312#1846—791no—
313#1846—791no—
314#1835—791no—
315#1835—791no—
316C79#2198311yescharacteristic, complemented, Sylow
317C22×C4⋊C4—791no—
318C22×C4⋊C4—791no—
319C22×C42—791no—
320#431—798no—
321C2×#107—791no—
322C2×#107—791no—
323C2×#107—791no—
324C2×#107C2×D7911yescharacteristic
325C2×#106—791no—
326C2×#106—791no—
327C2×#106—791no—
328C2×#106C2×D7911yescharacteristic
329C2×C4⋊C8—791no—
330C2×C4⋊C8C2×D7911yescharacteristic
331C2×C22⋊C8—791no—
332C2×C22⋊C8—791no—
333C2×C4×C8—791no—
334C2×C4×C8—791no—
335#383—1581no—
336#382—791no—
337#382—791no—
338#382—791no—
339#382—791no—
340#382—791no—
341#382—1581no—
342#382—1581no—
343#382C2×D7911yescharacteristic
344#381—1581no—
345#380—791no—
346#380—791no—
347C4×C4⋊C4—791no—
348C4×C4⋊C4C2×D7911yescharacteristic
349C4×C22⋊C4—1581no—
350#376—1581no—
351C2×#95—1581no—
352C2×#95—1581no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340—791no—
359#340C2×D7911yescharacteristic
360#340C2×D7911yescharacteristic
361C23×C4—791no—
362C23×C4—791no—
363C23×C4—791no—
364C23×C4—1581no—
365C23×C4—1581no—
366C22×C8—791no—
367C22×C8—791no—
368C22×C8C22×D7911yescharacteristic
369C22×C8C22×D7911yescharacteristic
370C2×C4⋊C4—3161no—
371C2×C4⋊C4—3161no—
372C2×C4⋊C4—3161no—
373C2×C4⋊C4—3161no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C4⋊C4C22×D7911yescharacteristic
382C2×C4⋊C4C22×D7911yescharacteristic
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C22⋊C4—1581no—
385C2×C22⋊C4—1581no—
386C2×C42—21no—
387C2×C42—791no—
388C2×C42—791no—
389C2×C42—791no—
390C2×C42—794no—
391C2×C42—794no—
392C2×C42—1581no—
393C2×C42C22×D7911yescharacteristic
394#107—41no—
395#107—3161no—
396#107—3161no—
397#107—3161no—
398#106—41no—
399#106—3161no—
400#106—3161no—
401#106—3161no—
402C4⋊C8—794no—
403C4⋊C8C4×D7914yescomplemented
404C22⋊C8—794no—
405C22⋊C8—794no—
406C4×C8—794no—
407C4×C8—794no—
408#95—21no—
409#95—21no—
410#95—21no—
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#95—1581no—
421#95—1581no—
422#95—1581no—
423#95—1581no—
424C24—791no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—3161no—
430C22×C4—3161no—
431C22×C4—3162no—
432C22×C4—3162no—
433C22×C4—3162no—
434C22×C4—3162no—
435C22×C4—3162no—
436C22×C4—3162no—
437C22×C4—794no—
438C22×C4—791no—
439C22×C4—791no—
440C22×C4—791no—
441C22×C4—792no—
442C22×C4—794no—
443C22×C4—792no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1581no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1581no—
452C22×C4—1581no—
453C22×C4—1581no—
454C22×C4D4×D7911yescharacteristic
455C22×C4D4×D7911yescharacteristic
456C22×C4C23×D7911yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—792no—
462C2×C8—792no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8C2×C4×D7912yes—
468C2×C8C2×C4×D7912yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—3162no—
478C4⋊C4—3162no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4—1581no—
487C22⋊C4—3162no—
488C22⋊C4—3162no—
489C42—42no—
490C42—3162no—
491C42—792no—
492C42—794no—
493C42—794no—
494C42—792no—
495C42C2×C4×D7912yes—
496C42Q8×D7912yes—
497C23—792no—
498C23—792no—
499C23—792no—
500C23—1581no—
501C23—1581no—
502C23—1581no—
503C23—1581no—
504C23C2×D4×D7911yescharacteristic
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—3164no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—792no—
535C2×C4—792no—
536C2×C4—792no—
537C2×C4—6324no—
538C2×C4—6324no—
539C2×C4—1588no—
540C2×C4—1581no—
541C2×C4—1581no—
542C2×C4—1581no—
543C2×C4—1581no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1582no—
547C2×C4—1582no—
548C2×C4—1582no—
549C2×C4—1582no—
550C2×C4C4⋊C4×D7912yes—
551C2×C4C4⋊C4×D7912yes—
552C2×C4C22×C4×D7911yescharacteristic
553C2×C4C2×Q8×D7911yescharacteristic
554C2×C4#1457920611yescharacteristic
555C2×C4#1457920711yescharacteristic
556C2×C4#1457920811yescharacteristic
557C2×C4#1457920911yescharacteristic
558C8—22no—
559C8—22no—
560C8—1582no—
561C8—1582no—
562C22—794no—
563C22—1582no—
564C22—1582no—
565C22—1582no—
566C22—1582no—
567C22—1582no—
568C22—1582no—
569C22C2×C4⋊C4×D7911yescharacteristic
570C22#1909912911yescharacteristic
571C22#1909913611yescharacteristic
572C22#135×D7911yescharacteristic
573C22#135×D7911yescharacteristic
574C22#1909935911yescharacteristic
575C22#1909936211yescharacteristic
576C4—22no—
577C4—22no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—6321no—
583C4—6321no—
584C4—6321no—
585C4—6321no—
586C4—1582no—
587C4—1582no—
588C4—1582no—
589C4#1909904211yescharacteristic
590C4#1909904511yescharacteristic
591C4#1909919011yescharacteristic
592C4#1909919311yescharacteristic
593C2—1584no—
594C2#2376602211yescharacteristic
595C2#427×D7911yescharacteristic
596C2#2377198711yescharacteristic
597C2#2377436311yescharacteristic
598C2#2377436911yescharacteristic
599C2#2377456611yescharacteristic
600C2#2377457211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1949C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1949D79
hypocenterC79#21983
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #23764170C2semidirect
5. #23764168C2semidirect
6. #23764166C2semidirect
11. #23763563C2semidirect
12. #23763561C2semidirect
13. #23763529C2semidirect
14. #23763529C2semidirect
15. C79×#1949C2semidirect
19. #19098615C4semidirect
91. C4⋊(C79⋊C8)C2×C4semidirect
94. C8×Dic79C2×C4semidirect
95. C8×Dic79C2×C4semidirect
119. C4⋊C8×C79C2×C4semidirect
307. #1949D79semidirect
316. C79#21983semidirect
403. C4⋊C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190986152
#237657962
#382×D792
#237641702
#237641682
#237641662
#237635942
#237635912
#237635892
#237635862
#237635632
#237635612
#237635292
C79×#19492
#2198379