Finite Group Databasebeta

← #34296273

Subgroups — #34296273

Order 20224.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098920C211yesmaximal, characteristic
2#23766111C211yesmaximal, characteristic
3#23764370C211yesmaximal, characteristic, complemented
4#23764214C211yesmaximal, characteristic
5#23764096C211yesmaximal, characteristic
6#23764089C211yesmaximal, characteristic, complemented
7#23764082C211yesmaximal, characteristic
8#23764075C211yesmaximal, characteristic, complemented
9#23763527C211yesmaximal, characteristic, complemented
10#23763233C211yesmaximal, characteristic, complemented
11#23763228C211yesmaximal, characteristic
12C4×#14579084C211yesmaximal, characteristic, complemented
13C4×#14579081C211yesmaximal, characteristic
14#23763087C211yesmaximal, characteristic, complemented
15C79×#2330C211yesmaximal, characteristic, complemented
16C2×#14579149C2211yescharacteristic
17C2×C4×Dic158C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19#19098920C418yescomplemented
20C2×#14579084C2211yescharacteristic
21C2×#14579084C2211yescharacteristic
22C2×#14579084C2211yescharacteristic
23C2×#14579084C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579081C2211yescharacteristic
28C2×#14579072C2211yescharacteristic
29C2×#14579072C2211yescharacteristic
30C2×#14579072C2211yescharacteristic
31C2×C4×C79⋊C8C2211yescharacteristic
32C2×C4×C79⋊C8C2211yescharacteristic
33C2×C4×C79⋊C8C2211yescharacteristic
34#19098469C2211yescharacteristic
35#19098465C2211yescharacteristic
36#19098455—21no—
37#19098455—21no—
38#19098437—21no—
39#19098437—21no—
40#19098433C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49C2×#125×C79C2211yescharacteristic
50#389×C79—21no—
51#389×C79—21no—
52#389×C79C2211yescharacteristic
53#389×C79C2211yescharacteristic
54#387×C79—21no—
55#387×C79—21no—
56#384×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58C4×C4⋊C4×C79C2211yescharacteristic
59#376×C79C2211yescharacteristic
60C22×Dic158C2311yescharacteristic
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579149—22no—
68#14579149C2×C414yes—
69C4×Dic158—22no—
70C4×Dic158—22no—
71C4×D316—22no—
72C4×D316—22no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79—21no—
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×(C2×D79)⋊C4—21no—
79C2×(C2×D79)⋊C4—21no—
80C2×(C2×D79)⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×C4×Dic79—21no—
84#14579084—22no—
85#14579084—22no—
86#14579084C2×C414yescomplemented
87#14579084C2×C414yescomplemented
88#14579081—22no—
89#14579081—22no—
90#14579081C2×C414yes—
91#14579081C2×C414yes—
92#14579072—22no—
93#14579072—22no—
94#14579072—22no—
95C4×C79⋊C8—22no—
96C4×C79⋊C8—22no—
97C4×C79⋊C8C2×C414yescomplemented
98C79⋊#95—21no—
99C79⋊#95—21no—
100#125×C79—22no—
101#125×C79C2×C414yescomplemented
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79—21no—
104C2×C4⋊C4×C79—21no—
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×C4⋊C4×C79C2311yescharacteristic
112C2×C42×C79—21no—
113C2×C42×C79—21no—
114C2×C42×C79C2311yescharacteristic
115C2×C42×C79C2311yescharacteristic
116C2×C42×C79C2311yescharacteristic
117#95×C79—21no—
118#95×C79—21no—
119#95×C79—21no—
120#95×C79—21no—
121#95×C79D411yescharacteristic
122#95×C79D411yescharacteristic
123#95×C79D411yescharacteristic
124#95×C79D411yescharacteristic
125C23×D79—21no—
126C2×Dic158—21no—
127C2×Dic158—21no—
128C2×Dic158—21no—
129C2×Dic158—21no—
130C2×Dic158—21no—
131C2×Dic158C22×C412yes—
132C2×D316—21no—
133C2×D316—21no—
134C2×D316—21no—
135C2×D316—21no—
136C2×D316—21no—
137C2×D316C22×C412yes—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C2×C4×D79—42no—
142C2×C4×D79—81no—
143C2×C4×D79—81no—
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—
152C2×C79⋊C8C22×C412yes—
153C2×C79⋊C8C22×C412yes—
154C2×C79⋊C8#4112yes—
155C2×C79⋊C8#4112yes—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—81no—
161C4⋊Dic79—81no—
162Dic79⋊C4—42no—
163(C2×D79)⋊C4—24no—
164(C2×D79)⋊C4—24no—
165(C2×D79)⋊C4—42no—
166(C2×D79)⋊C4—81no—
167(C2×D79)⋊C4—81no—
168C4×Dic79—42no—
169C22×C4×C79—21no—
170C22×C4×C79—21no—
171C22×C4×C79—21no—
172C22×C4×C79—21no—
173C22×C4×C79C2×D411yescharacteristic
174C22×C4×C79C2×D411yescharacteristic
175C22×C4×C79C2×D411yescharacteristic
176C22×C4×C79C2×D411yescharacteristic
177C22×C4×C79C2×D411yescharacteristic
178C22×C4×C79C2×D411yescharacteristic
179C22×C4×C79C2411yescharacteristic
180C4⋊C4×C79—24no—
181C4⋊C4×C79—24no—
182C4⋊C4×C79—21no—
183C4⋊C4×C79—21no—
184C4⋊C4×C79—21no—
185C4⋊C4×C79—21no—
186C4⋊C4×C79—21no—
187C4⋊C4×C79—21no—
188C4⋊C4×C79—81no—
189C4⋊C4×C79—81no—
190C4⋊C4×C79—81no—
191C4⋊C4×C79—81no—
192C4⋊C4×C79C22×C412yes—
193C4⋊C4×C79C22×C412yes—
194C42×C79—21no—
195C42×C79—21no—
196C42×C79—81no—
197C42×C79—81no—
198C42×C79C22×C412yes—
199C42×C79#4112yes—
200C42×C79#4112yes—
201C42×C79#4112yes—
202C23×C79C22×D411yescharacteristic
203C22×D79—22no—
204C22×D79—41no—
205C22×D79—41no—
206C22×D79—41no—
207C22×D79—41no—
208C22×D79—41no—
209C22×D79—41no—
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—22no—
219C2×C4×C79—22no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—21no—
225C2×C4×C79—21no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79—41no—
231C2×C4×C79—41no—
232C2×C4×C79—41no—
233C2×C4×C79—41no—
234C2×C4×C79—41no—
235C2×C4×C79—41no—
236C2×C4×C79C4×D412yes—
237C2×C4×C79C4×D412yes—
238C2×C4×C79#12412yes—
239C2×C4×C79#12412yes—
240C2×C4×C79C23×C411yescharacteristic
241C2×C4×C79C2×#4111yescharacteristic
242C2×C4×C79C2×#4111yescharacteristic
243C2×C4×C79C2×#4111yescharacteristic
244C2×C4×C79C2×#4111yescharacteristic
245C2×C4×C79C2×#4111yescharacteristic
246C2×Dic79—22no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C2×Dic79—41no—
252C2×Dic79—41no—
253C2×Dic79—41no—
254C2×Dic79—41no—
255D316—22no—
256D316—22no—
257D316—41no—
258D316—41no—
259C4×D79—84no—
260Dic158—22no—
261Dic158—22no—
262Dic158—41no—
263Dic158—41no—
264C79⋊C8—22no—
265C79⋊C8—22no—
266C79⋊C8—22no—
267C79⋊C8—22no—
268C22×C79C2×C4×D411yescharacteristic
269C22×C79C2×#12411yescharacteristic
270C22×C79#53211yescharacteristic
271C22×C79#57611yescharacteristic
272C22×C79#57711yescharacteristic
273C22×C79#57711yescharacteristic
274C22×C79#57811yescharacteristic
275C2×D79—42no—
276C2×D79—42no—
277C2×D79—42no—
278C2×D79—81no—
279C2×D79—81no—
280C2×D79—81no—
281C2×D79—81no—
282C4×C79—22no—
283C4×C79—22no—
284C4×C79—22no—
285C4×C79—22no—
286C4×C79—41no—
287C4×C79—41no—
288C4×C79—42no—
289C4×C79—42no—
290C4×C79—81no—
291C4×C79—81no—
292C4×C79C4×#4111yescharacteristic
293C4×C79C4×#4111yescharacteristic
294C4×C79#52911yescharacteristic
295C4×C79#52911yescharacteristic
296Dic79—42no—
297Dic79—81no—
298Dic79—81no—
299#21643—791nomaximal, Sylow
300C2×C79C4×#12411yescharacteristic
301C2×C79#295111yescharacteristic
302C2×C79#295211yescharacteristic
303C2×C79#313611yescharacteristic
304C2×C79#313711yescharacteristic
305C2×C79#317811yescharacteristic
306C2×C79#317911yescharacteristic
307D79—82no—
308C2×#488—791no—
309#2330D7911yescharacteristic, complemented
310#2326—791no—
311#1992—791no—
312#1959—791no—
313#1936—791no—
314#1935—791no—
315#1934—791no—
316#1933—791no—
317#1834—791no—
318#1763—791no—
319#1762—791no—
320C4×#103—791no—
321C4×#102—791no—
322#1726—791no—
323C79#2164311yescharacteristic, complemented, Sylow
324C2×#125C2×D7911yescharacteristic
325C2×#124—791no—
326C2×C4×Q8—791no—
327C2×C4×D4—791no—
328#488—798no—
329C2×#103—791no—
330C2×#103—791no—
331C2×#103—791no—
332C2×#103—791no—
333C2×#102—791no—
334C2×#102—791no—
335C2×#102—791no—
336C2×#102—791no—
337C2×#97—791no—
338C2×#97—791no—
339C2×#97—791no—
340C2×C4×C8—791no—
341C2×C4×C8—791no—
342C2×C4×C8—791no—
343#391—791no—
344#390—791no—
345#389—21no—
346#389—21no—
347#389C2×D7911yescharacteristic
348#389C2×D7911yescharacteristic
349#388—1581no—
350#388—1581no—
351#387—21no—
352#387—21no—
353#385—1581no—
354#385—1581no—
355#384—791no—
356#384C2×D7911yescharacteristic
357C4×C4⋊C4C2×D7911yescharacteristic
358C4×C4⋊C4C2×D7911yescharacteristic
359#376C2×D7911yescharacteristic
360#340—791no—
361#340—791no—
362#340—791no—
363#340—791no—
364#340—791no—
365#340—791no—
366#340—791no—
367#340—791no—
368C22×Q8—791no—
369C22×D4—791no—
370C23×C4—1581no—
371C22×C8—791no—
372C22×C8—791no—
373C22×C8—791no—
374C22×C8—791no—
375#125—22no—
376#125C4×D7914yescomplemented
377#124—794no—
378#124—1582no—
379C4×Q8—1582no—
380C4×Q8—1582no—
381C4×D4—1582no—
382C4×D4—1582no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—791no—
390C2×C4⋊C4—791no—
391C2×C4⋊C4—1581no—
392C2×C4⋊C4—1581no—
393C2×C4⋊C4—1581no—
394C2×C4⋊C4C22×D7911yescharacteristic
395C2×C4⋊C4C22×D7911yescharacteristic
396C2×C4⋊C4C22×D7911yescharacteristic
397C2×C4⋊C4C22×D7911yescharacteristic
398C2×C22⋊C4—1581no—
399C2×C22⋊C4—1581no—
400C2×C22⋊C4—1581no—
401C2×C22⋊C4—1581no—
402C2×C22⋊C4—1581no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—1581no—
406C2×C42C22×D7911yescharacteristic
407C2×C42C22×D7911yescharacteristic
408C2×C42C22×D7911yescharacteristic
409#103—794no—
410#103—794no—
411#103—1582no—
412#103—1582no—
413#102—794no—
414#102—794no—
415#102—1582no—
416#102—1582no—
417#97—1582no—
418#97—1582no—
419#97—1582no—
420C4×C8—794no—
421C4×C8—1582no—
422C4×C8—1582no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1581no—
428#95—1581no—
429#95C22⋊D7911yescharacteristic
430#95C22⋊D7911yescharacteristic
431#95C22⋊D7911yescharacteristic
432#95C22⋊D7911yescharacteristic
433C24—1581no—
434C2×Q8—792no—
435C2×Q8—1581no—
436C2×Q8—1581no—
437C2×Q8—1581no—
438C2×Q8—1581no—
439C2×Q8—1581no—
440C2×D4—792no—
441C2×D4—1581no—
442C2×D4—1581no—
443C2×D4—1581no—
444C2×D4—1581no—
445C2×D4—1581no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—3162no—
451C22×C4—6321no—
452C22×C4—6321no—
453C22×C4—1581no—
454C22×C4—1581no—
455C22×C4—1581no—
456C22×C4C2×C22⋊D7911yescharacteristic
457C22×C4C2×C22⋊D7911yescharacteristic
458C22×C4C2×C22⋊D7911yescharacteristic
459C22×C4C2×C22⋊D7911yescharacteristic
460C22×C4C2×C22⋊D7911yescharacteristic
461C22×C4C2×C22⋊D7911yescharacteristic
462C22×C4C23×D7911yescharacteristic
463C2×C8—792no—
464C2×C8—792no—
465C2×C8—792no—
466C2×C8—792no—
467C2×C8—1581no—
468C2×C8—1581no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1581no—
472C2×C8—1581no—
473C2×C8—1581no—
474C2×C8—1581no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—3162no—
488C4⋊C4—6321no—
489C4⋊C4—6321no—
490C4⋊C4—1581no—
491C4⋊C4—1581no—
492C4⋊C4—1581no—
493C4⋊C4—1581no—
494C4⋊C4C2×C4×D7912yes—
495C4⋊C4C2×C4×D7912yes—
496C22⋊C4—3162no—
497C22⋊C4—6321no—
498C22⋊C4—6321no—
499C22⋊C4—1584no—
500C22⋊C4—1584no—
501C42—21no—
502C42—21no—
503C42—81no—
504C42—81no—
505C42—3162no—
506C42C2×C4×D7912yes—
507C42#1192962412yes—
508C42#1192962512yes—
509C42#1192962512yes—
510C23—3161no—
511C23—3161no—
512C23—3161no—
513C23—3161no—
514C23—3161no—
515C23—3161no—
516C23—1582no—
517C23C22×C22⋊D7911yescharacteristic
518Q8—3161no—
519Q8—3161no—
520Q8—1582no—
521Q8—1582no—
522D4—3161no—
523D4—3161no—
524D4—1582no—
525D4—1582no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3161no—
553C2×C4—3161no—
554C2×C4—3161no—
555C2×C4—3161no—
556C2×C4—3161no—
557C2×C4—3161no—
558C2×C4—3161no—
559C2×C4—3161no—
560C2×C4—6324no—
561C2×C4—1582no—
562C2×C4C4×C22⋊D7912yes—
563C2×C4C4×C22⋊D7912yes—
564C2×C4#1457915012yes—
565C2×C4#1457915012yes—
566C2×C4C22×C4×D7911yescharacteristic
567C2×C4#41×D7911yescharacteristic
568C2×C4#41×D7911yescharacteristic
569C2×C4C2×#1192962411yescharacteristic
570C2×C4C2×#1192962511yescharacteristic
571C2×C4C2×#1192962511yescharacteristic
572C8—1582no—
573C8—1582no—
574C8—1582no—
575C8—1582no—
576C22—3162no—
577C22—3162no—
578C22—3162no—
579C22—6321no—
580C22—6321no—
581C22—6321no—
582C22—6321no—
583C22C2×C4×C22⋊D7911yescharacteristic
584C22C2×#1457915011yescharacteristic
585C22#1909910911yescharacteristic
586C22#1909936611yescharacteristic
587C22#1909937111yescharacteristic
588C22#1909937411yescharacteristic
589C22#1909937811yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—42no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—3162no—
601C4—6321no—
602C4—6321no—
603C4C4×#1192962511yescharacteristic
604C4C4×#1192962511yescharacteristic
605C4#1909910011yescharacteristic
606C4#1909910011yescharacteristic
607C2—6322no—
608C2C4×#1457915011yescharacteristic
609C2#2377227311yescharacteristic
610C2#2377228111yescharacteristic
611C2#2377375911yescharacteristic
612C2#2377377011yescharacteristic
613C2#2377407011yescharacteristic
614C2#2377408511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2330C2
socleC23×C79C22×D4
prime radicalC22×D316C23
cosocleC22×C4×C79C24
hypercenter#2330D79
hypocenterC79#21643
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
3. #23764370C2semidirect
6. #23764089C2semidirect
8. #23764075C2semidirect
9. #23763527C2semidirect
10. #23763233C2semidirect
12. C4×#14579084C2semidirect
14. #23763087C2semidirect
15. C79×#2330C2semidirect
19. #19098920C4semidirect
86. #14579084C2×C4semidirect
87. #14579084C2×C4semidirect
97. C4×C79⋊C8C2×C4semidirect
101. #125×C79C2×C4semidirect
309. #2330D79semidirect
323. C79#21643semidirect
376. #125C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190989202
#237661112
#237643702
#237642142
#237640962
#237640892
#237640822
#237640752
#237635272
#237632332
#237632282
C4×#145790842
C4×#145790812
#237630872
C79×#23302
#2164379