Finite Group Databasebeta

← #34290427

Subgroups — #34290427

Order 20224.

926 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098598C211yesmaximal, characteristic
2#23765881C211yesmaximal, characteristic
3C4×#14579119C211yesmaximal, characteristic
4#23763992C211yesmaximal, characteristic, complemented
5#23763990C211yesmaximal, characteristic, complemented
6#23763986C211yesmaximal, characteristic, complemented
7#23763984C211yesmaximal, characteristic, complemented
8#23763980C211yesmaximal, characteristic, complemented
9#23763974C211yesmaximal, characteristic, complemented
10#23763968C211yesmaximal, characteristic, complemented
11#23763115C211yesmaximal, characteristic
12#23763115C211yesmaximal, characteristic
13#23763115C211yesmaximal, characteristic
14#336×D79C211yesmaximal, characteristic
15C79×#1919C211yesmaximal, characteristic, complemented
16C22×C8×D79C2211yescharacteristic
17C22×C8×D79C2211yescharacteristic
18C2×#14579119C2211yescharacteristic
19C2×#14579119C2211yescharacteristic
20C2×#14579119C2211yescharacteristic
21C2×#14579119C2211yescharacteristic
22C2×C42×D79C2211yescharacteristic
23#19098598C418yescomplemented
24C2×#14579094C2211yescharacteristic
25C2×#14579094C2211yescharacteristic
26C2×#14579093C2211yescharacteristic
27#19098462C2211yescharacteristic
28#19098461C2211yescharacteristic
29#19098459C2211yescharacteristic
30#19098437—21no—
31#19098434—21no—
32C4⋊C4×Dic79—21no—
33C4⋊C4×Dic79C2211yescharacteristic
34C4⋊C4×Dic79C2211yescharacteristic
35C4×C4⋊Dic79C2211yescharacteristic
36#19098414—21no—
37C4×(C2×D79)⋊C4—21no—
38#19098410—21no—
39#19098408—21no—
40C42×Dic79—21no—
41#19098326C2211yescharacteristic
42#19098326C2211yescharacteristic
43#19098326C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50#19098318C2211yescharacteristic
51#19098317C2211yescharacteristic
52#19098317C2211yescharacteristic
53C2×#106×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55C4×C4⋊C4×C79C2211yescharacteristic
56#340×C79C2211yescharacteristic
57#340×C79C2211yescharacteristic
58#340×C79C2211yescharacteristic
59#336×C79C2211yescharacteristic
60C22×C4×D79D411yescharacteristic
61C22×C4×D79D411yescharacteristic
62C22×C4×D79C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C2×C8×D79—22no—
66C2×C8×D79—22no—
67C2×C8×D79—22no—
68C2×C8×D79—22no—
69C2×C8×D79—22no—
70C2×C8×D79C2×C414yes—
71#14579119—22no—
72#14579119—22no—
73#14579119—22no—
74#14579119—22no—
75#14579119—22no—
76#14579119C2×C414yes—
77#14579119C2×C414yes—
78#14579119Q814yes—
79C2×C4⋊Dic79—21no—
80C2×C4⋊Dic79C2311yescharacteristic
81C2×C4⋊Dic79C2311yescharacteristic
82C2×C4⋊Dic79C2311yescharacteristic
83C2×C4⋊Dic79C2311yescharacteristic
84C2×Dic79⋊C4—21no—
85C2×Dic79⋊C4—21no—
86C2×(C2×D79)⋊C4—21no—
87C2×(C2×D79)⋊C4—21no—
88C2×(C2×D79)⋊C4—21no—
89C2×(C2×D79)⋊C4—21no—
90C2×C4×Dic79—21no—
91C2×C4×Dic79—21no—
92C2×C4×Dic79—21no—
93C2×C4×Dic79—21no—
94C2×C4×Dic79—21no—
95C2×C4×Dic79—21no—
96C2×C4×Dic79—21no—
97C2×C4×Dic79D411yescharacteristic
98C2×C4×Dic79D411yescharacteristic
99C2×C4×Dic79C2311yescharacteristic
100C42×D79—22no—
101C42×D79—22no—
102#14579094C2×C414yescomplemented
103#14579094C2×C414yescomplemented
104#14579093C2×C414yescomplemented
105C79⋊#95—21no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C79⋊#95—21no—
109C79⋊#95—21no—
110C79⋊#95—21no—
111C22×C8×C79C2311yescharacteristic
112C22×C8×C79C2311yescharacteristic
113C2×C4⋊C4×C79—21no—
114C2×C4⋊C4×C79C2311yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C42×C79—21no—
119C2×C42×C79C2311yescharacteristic
120#106×C79C2×C414yescomplemented
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—22no—
134C2×C4×D79—22no—
135C2×C4×D79—41no—
136C2×C4×D79—41no—
137C2×C4×D79C22×C412yes—
138C2×C4×D79C2×Q812yes—
139C2×C4×D79#4112yes—
140C2×C4×D79#4112yes—
141C2×C4×D79#4112yes—
142C2×C4×D79#4112yes—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8C22×C412yes—
149C8×D79—28no—
150C8×D79—44no—
151C4⋊Dic79—21no—
152C4⋊Dic79—21no—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—42no—
157C4⋊Dic79C22×C412yes—
158C4⋊Dic79C22×C412yes—
159C4⋊Dic79C2×Q812yes—
160Dic79⋊C4—42no—
161Dic79⋊C4—42no—
162(C2×D79)⋊C4—24no—
163(C2×D79)⋊C4—24no—
164(C2×D79)⋊C4—24no—
165(C2×D79)⋊C4—42no—
166C4×Dic79—24no—
167C4×Dic79—24no—
168C4×Dic79—24no—
169C4×Dic79—22no—
170C4×Dic79—22no—
171C4×Dic79—21no—
172C4×Dic79—21no—
173C4×Dic79—42no—
174C4×Dic79—42no—
175C4×Dic79—42no—
176C4×Dic79—42no—
177C22×C4×C79—21no—
178C22×C4×C79—21no—
179C22×C4×C79—21no—
180C22×C4×C79—21no—
181C22×C4×C79C2×D411yescharacteristic
182C22×C4×C79C2×D411yescharacteristic
183C22×C4×C79C2411yescharacteristic
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—21no—
187C2×C8×C79—21no—
188C2×C8×C79—21no—
189C2×C8×C79C22×C412yes—
190C4⋊C4×C79—21no—
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—42no—
196C4⋊C4×C79C22×C412yes—
197C4⋊C4×C79C22×C412yes—
198C4⋊C4×C79C2×Q812yes—
199C42×C79—21no—
200C42×C79—21no—
201C42×C79—42no—
202C23×C79C22×D411yescharacteristic
203C22×D79—21no—
204C22×D79—21no—
205C22×D79—21no—
206C22×D79—21no—
207C22×D79C4×D412yes—
208C22×D79C22⋊Q812yes—
209C22×D79#12312yes—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—22no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79#13512yes—
228C2×C4×C79C23×C411yescharacteristic
229C2×C4×C79C22×Q811yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—22no—
237C2×Dic79—22no—
238C2×Dic79—22no—
239C2×Dic79—21no—
240C2×Dic79—21no—
241C2×Dic79—21no—
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×Dic79C4×D412yes—
252C2×Dic79C22⋊Q812yes—
253C2×Dic79#12312yes—
254C2×Dic79#13512yes—
255C4×D79—44no—
256C4×D79—44no—
257C4×D79#11714yes—
258C4×D79C4×Q814yes—
259C4×D79#12514yes—
260C4×D79#12614yes—
261C8×C79—22no—
262C8×C79—41no—
263C79⋊C8—22no—
264C79⋊C8—41no—
265C22×C79C2×C4×D411yescharacteristic
266C22×C79C2×C22⋊Q811yescharacteristic
267C22×C79C2×#12311yescharacteristic
268C22×C79C2×#13511yescharacteristic
269C22×C79C2×#13511yescharacteristic
270C22×C79#57511yescharacteristic
271C22×C79#57511yescharacteristic
272C2×D79—22no—
273C2×D79—22no—
274C2×D79—22no—
275C2×D79—22no—
276C2×D79—22no—
277C2×D79—22no—
278C2×D79#38714yes—
279C4×C79—22no—
280C4×C79—42no—
281C4×C79—41no—
282C4×C79—42no—
283C4×C79—42no—
284C4×C79—81no—
285C4×C79C2×#11711yescharacteristic
286C4×C79C2×C4×Q811yescharacteristic
287C4×C79C2×#12511yescharacteristic
288C4×C79C2×#12611yescharacteristic
289Dic79—22no—
290Dic79—42no—
291Dic79—41no—
292Dic79—42no—
293Dic79—42no—
294Dic79—81no—
295Dic79#38714yes—
296#21032—791nomaximal, Sylow
297C2×C79C2×#38711yescharacteristic
298C2×C79C4×#13511yescharacteristic
299C2×C79#294311yescharacteristic
300C2×C79#307611yescharacteristic
301C2×C79#307611yescharacteristic
302C2×C79#309111yescharacteristic
303C2×C79#309111yescharacteristic
304D79—24no—
305C2×#427—791no—
306#2296—791no—
307C4×#117—791no—
308#1921—791no—
309#1921—791no—
310#1920—791no—
311#1920—791no—
312#1919—791no—
313#1919D7911yescharacteristic, complemented
314#1918—791no—
315#1918—791no—
316#1735—791no—
317#1735—791no—
318#1735—791no—
319C2×#336—791no—
320C79#2103211yescharacteristic, complemented, Sylow
321C23×C8—791no—
322C23×C8—791no—
323C2×#117—791no—
324C2×#117—791no—
325C2×#117—791no—
326C2×#117—791no—
327C22×C42—791no—
328#427—798no—
329C2×#107—791no—
330C2×#107—791no—
331C2×#106—791no—
332C2×#106C2×D7911yescharacteristic
333#389—791no—
334#389—791no—
335#389—791no—
336#389C2×D7911yescharacteristic
337#385—1581no—
338#385—1581no—
339C4×C4⋊C4—791no—
340C4×C4⋊C4—791no—
341C4×C4⋊C4—791no—
342C4×C4⋊C4—1581no—
343C4×C4⋊C4—1581no—
344C4×C4⋊C4C2×D7911yescharacteristic
345C4×C22⋊C4—1581no—
346C4×C22⋊C4—1581no—
347#376—1581no—
348C43—1581no—
349#340—791no—
350#340—791no—
351#340—791no—
352#340—791no—
353#340—791no—
354#340—791no—
355#340—791no—
356#340—791no—
357#340—791no—
358#340C2×D7911yescharacteristic
359#340C2×D7911yescharacteristic
360#340C2×D7911yescharacteristic
361#336—791no—
362#336—791no—
363#336—791no—
364#336C2×D7911yescharacteristic
365C23×C4—791no—
366C23×C4—791no—
367C23×C4—791no—
368C22×C8—791no—
369C22×C8—794no—
370C22×C8—791no—
371C22×C8—1582no—
372C22×C8—1582no—
373C22×C8—1582no—
374C22×C8—1582no—
375C22×C8—1582no—
376C22×C8C22×D7911yescharacteristic
377C22×C8C22×D7911yescharacteristic
378#117—794no—
379#117—794no—
380#117—794no—
381#117—1582no—
382#117—1582no—
383#117—1582no—
384#117—1582no—
385#117—1582no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—791no—
388C2×C4⋊C4—791no—
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×C42—21no—
403C2×C42—791no—
404C2×C42—791no—
405C2×C42—791no—
406C2×C42—1582no—
407C2×C42—1582no—
408C2×C42—1581no—
409C2×C42—1581no—
410C2×C42—1581no—
411C2×C42—1581no—
412C2×C42—1581no—
413C2×C42—1581no—
414C2×C42—1581no—
415C2×C42C22×D7911yescharacteristic
416#107—794no—
417#107—794no—
418#106—794no—
419#106C4×D7914yescomplemented
420#95—21no—
421#95—21no—
422#95—1581no—
423#95—1581no—
424#95—1581no—
425#95—1581no—
426#95—1581no—
427#95—1581no—
428C24—791no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—3161no—
434C22×C4—3161no—
435C22×C4—791no—
436C22×C4—792no—
437C22×C4—792no—
438C22×C4—792no—
439C22×C4—792no—
440C22×C4—792no—
441C22×C4—792no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1582no—
447C22×C4—1581no—
448C22×C4—1581no—
449C22×C4—1581no—
450C22×C4—1581no—
451C22×C4—1582no—
452C22×C4D4×D7911yescharacteristic
453C22×C4D4×D7911yescharacteristic
454C22×C4C23×D7911yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—3164no—
461C2×C8—792no—
462C2×C8—1588no—
463C2×C8—1581no—
464C2×C8—1581no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1581no—
468C2×C8C2×C4×D7912yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—3162no—
476C4⋊C4—3162no—
477C4⋊C4—3162no—
478C4⋊C4—792no—
479C4⋊C4—792no—
480C4⋊C4—792no—
481C4⋊C4—1581no—
482C4⋊C4—1581no—
483C4⋊C4—1581no—
484C4⋊C4—1581no—
485C4⋊C4—1581no—
486C4⋊C4C2×C4×D7912yes—
487C4⋊C4C2×C4×D7912yes—
488C4⋊C4Q8×D7912yes—
489C22⋊C4—3162no—
490C22⋊C4—1584no—
491C22⋊C4—1584no—
492C22⋊C4—1584no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—3162no—
497C42—3162no—
498C42—3162no—
499C42—3162no—
500C42—1582no—
501C42—1582no—
502C42—1584no—
503C42—1584no—
504C42—1584no—
505C42—1581no—
506C42—1581no—
507C23—792no—
508C23—792no—
509C23—792no—
510C23—1581no—
511C23—1581no—
512C23—1581no—
513C23—1581no—
514C23C2×D4×D7911yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—3164no—
533C2×C4—3164no—
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—792no—
544C2×C4—792no—
545C2×C4—792no—
546C2×C4—794no—
547C2×C4—794no—
548C2×C4—794no—
549C2×C4—794no—
550C2×C4—792no—
551C2×C4—1582no—
552C2×C4—1582no—
553C2×C4—1582no—
554C2×C4—1581no—
555C2×C4—1581no—
556C2×C4—1581no—
557C2×C4—1581no—
558C2×C4—1581no—
559C2×C4#1457918212yes—
560C2×C4C22×C4×D7911yescharacteristic
561C2×C4C2×Q8×D7911yescharacteristic
562C2×C4#41×D7911yescharacteristic
563C2×C4#41×D7911yescharacteristic
564C2×C4#41×D7911yescharacteristic
565C2×C4#41×D7911yescharacteristic
566C8—22no—
567C8—41no—
568C8—3161no—
569C8—1582no—
570C22—794no—
571C22—1582no—
572C22—1582no—
573C22—1582no—
574C22—1582no—
575C22—1582no—
576C22—1582no—
577C22C4×D4×D7911yescharacteristic
578C22C22⋊Q8×D7911yescharacteristic
579C22#123×D7911yescharacteristic
580C22#135×D7911yescharacteristic
581C22C2×#1457918211yescharacteristic
582C22#1909935911yescharacteristic
583C22#1909936211yescharacteristic
584C4—22no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—81no—
590C4—3162no—
591C4—3162no—
592C4—3161no—
593C4—3162no—
594C4—794no—
595C4—6321no—
596C4—1582no—
597C4#117×D7911yescharacteristic
598C4C4×Q8×D7911yescharacteristic
599C4#125×D7911yescharacteristic
600C4#126×D7911yescharacteristic
601C2—1584no—
602C2#387×D7911yescharacteristic
603C2C4×#1457918211yescharacteristic
604C2#2377219511yescharacteristic
605C2#2377318311yescharacteristic
606C2#2377318511yescharacteristic
607C2#2377331011yescharacteristic
608C2#2377331811yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1919C2
socleC23×C79C22×D4
prime radicalC23×D79C2×D4
cosocleC22×C4×C79C24
hypercenter#1919D79
hypocenterC79#21032
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #23763992C2semidirect
5. #23763990C2semidirect
6. #23763986C2semidirect
7. #23763984C2semidirect
8. #23763980C2semidirect
9. #23763974C2semidirect
10. #23763968C2semidirect
15. C79×#1919C2semidirect
23. #19098598C4semidirect
102. #14579094C2×C4semidirect
103. #14579094C2×C4semidirect
104. #14579093C2×C4semidirect
120. #106×C79C2×C4semidirect
313. #1919D79semidirect
320. C79#21032semidirect
419. #106C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190985982
#237658812
C4×#145791192
#237639922
#237639902
#237639862
#237639842
#237639802
#237639742
#237639682
#237631152
#336×D792
C79×#19192
#2103279