Finite Group Databasebeta

← #34287844

Subgroups — #34287844

Order 20224.

1518 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D79C211yesmaximal, characteristic
2C2×#19098600C211yesmaximal, characteristic, complemented
3C2×#19098598C211yesmaximal, characteristic, complemented
4#23765782C212yesmaximal
5#23763434C211yesmaximal, characteristic, complemented
6#23763433C211yesmaximal, characteristic, complemented
7#23763432C212yesmaximal, complemented
8#23763431C212yesmaximal, complemented
9#23763430C211yesmaximal, characteristic, complemented
10#23763115C212yesmaximal, complemented
11C79×#1809C211yesmaximal, characteristic, complemented
12C23×C4×D79C2211yescharacteristic
13C22×#11929608C2211yescharacteristic
14C22×#11929607C2211yescharacteristic
15C22×#11929607C2211yescharacteristic
16C2×M16×D79—24no—
17C2×M16×D79—24no—
18C2×M16×D79C418yescomplemented
19C22×C8×D79C2211yescharacteristic
20C22×C8×D79C2211yescharacteristic
21C2×#14579119C2212yes—
22C2×#14579119C2212yes—
23#19098600—24no—
24#19098598—24no—
25C2×#14579095C2211yescharacteristic, complemented
26C2×#14579094C2212yescomplemented
27C2×#14579093C2211yescharacteristic, complemented
28C2×#14579092C2212yescomplemented
29#19098425—22no—
30#19098423C2212yes—
31#19098420C2212yes—
32#19098420C2212yes—
33#19098418—22no—
34C22⋊C4×Dic79—22no—
35#19098410—22no—
36#19098326C2212yescomplemented
37#19098325C2212yescomplemented
38#19098325C2212yescomplemented
39C22×M16×C79C2211yescharacteristic
40C2×#107×C79C2211yescharacteristic, complemented
41C2×#106×C79C2211yescharacteristic, complemented
42#380×C79C2212yes—
43#340×C79C2212yescomplemented
44C24×D79D411yescharacteristic
45C23×Dic79D411yescharacteristic
46C22×C4×D79—22no—
47C22×C4×D79—22no—
48C22×C4×D79—22no—
49C22×C4×D79—22no—
50C22×C4×D79C2×C414yes—
51C22×C4×D79D411yescharacteristic
52C22×C4×D79D411yescharacteristic
53C22×C4×D79Q814yes—
54C22×C4×D79C2311yescharacteristic
55C22×C4×D79C2311yescharacteristic
56C2×#11929608—22no—
57C2×#11929608—22no—
58C2×#11929608C2×C414yes—
59C2×#11929607—22no—
60C2×#11929607—22no—
61C2×#11929607—22no—
62C2×#11929607—22no—
63C2×#11929607C2×C414yes—
64C2×#11929607C2×C414yes—
65M16×D79—232no—
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C2×C8×D79—22no—
69C2×C8×D79—22no—
70C2×C8×D79C2×C414yes—
71C2×C8×D79C2×C414yes—
72C2×C8×D79D414yes—
73C2×C8×D79D414yes—
74#14579119—24no—
75#14579119—24no—
76#14579119—24no—
77#14579119—24no—
78C2×C4⋊Dic79C2312yes—
79C2×C4⋊Dic79C2312yes—
80C2×C22⋊Dic79—22no—
81C2×(C2×D79)⋊C4—22no—
82C2×(C2×D79)⋊C4—22no—
83C2×(C2×D79)⋊C4—22no—
84C2×(C2×D79)⋊C4—22no—
85C2×C4×Dic79—22no—
86C2×C4×Dic79—22no—
87#14579095—22no—
88#14579094—24no—
89#14579093—22no—
90#14579092—24no—
91C79⋊#95—22no—
92C79⋊#95—22no—
93C79⋊#95—22no—
94C23×C4×C79C2311yescharacteristic
95C2×M16×C79—22no—
96C2×M16×C79—22no—
97C2×M16×C79C2×C414yes—
98C22×C8×C79C2311yescharacteristic
99C22×C8×C79C2311yescharacteristic
100C2×C4⋊C4×C79C2312yes—
101C2×C4⋊C4×C79C2312yes—
102C2×C22⋊C4×C79—22no—
103#107×C79—22no—
104#106×C79—22no—
105#95×C79—22no—
106C23×D79—22no—
107C23×D79—22no—
108C23×D79—42no—
109C23×D79—42no—
110C23×D79C4⋊C414yes—
111C23×D79C2×D411yescharacteristic
112C23×D79C2×D411yescharacteristic
113C22×Dic79—22no—
114C22×Dic79—22no—
115C22×Dic79—22no—
116C22×Dic79—22no—
117C22×Dic79C4⋊C414yes—
118C22×Dic79C2×D411yescharacteristic
119C22×Dic79C2×D411yescharacteristic
120C2×C4×D79—22no—
121C2×C4×D79—22no—
122C2×C4×D79—28no—
123C2×C4×D79—28no—
124C2×C4×D79—28no—
125C2×C4×D79—28no—
126C2×C4×D79—22no—
127C2×C4×D79—22no—
128C2×C4×D79—21no—
129C2×C4×D79—21no—
130C2×C4×D79—21no—
131C2×C4×D79—21no—
132C2×C4×D79C4⋊C414yes—
133C2×C4×D79C4⋊C414yes—
134C2×C4×D79C22×C412yes—
135C2×C4×D79C22×C412yes—
136C2×C4×D79C2×D412yes—
137C2×C4×D79C2×D412yes—
138C2×C4×D79C2×Q812yes—
139C2×C4×D79C2×Q812yes—
140C2×C4×D79#4112yes—
141C2×C4×D79#4112yes—
142#11929608—28no—
143#11929607—28no—
144#11929607—28no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8C22×C412yes—
148C2×C79⋊C8C22×C412yes—
149C2×C79⋊C8C2×D412yes—
150C2×C79⋊C8C2×D412yes—
151C8×D79—28no—
152C8×D79C22⋊C4116yescomplemented
153C4⋊Dic79—22no—
154C4⋊Dic79—22no—
155C4⋊Dic79—22no—
156C4⋊Dic79—22no—
157C22⋊Dic79—44no—
158(C2×D79)⋊C4—44no—
159(C2×D79)⋊C4—44no—
160(C2×D79)⋊C4—44no—
161(C2×D79)⋊C4—44no—
162C4×Dic79—44no—
163C4×Dic79—44no—
164C24×C79C2×D411yescharacteristic
165C22×C4×C79—21no—
166C22×C4×C79—21no—
167C22×C4×C79—21no—
168C22×C4×C79—21no—
169C22×C4×C79—22no—
170C22×C4×C79—22no—
171C22×C4×C79C22×C412yes—
172C22×C4×C79C2×D411yescharacteristic
173C22×C4×C79C2×Q812yes—
174C22×C4×C79C2411yescharacteristic
175M16×C79—28no—
176C2×C8×C79—21no—
177C2×C8×C79—21no—
178C2×C8×C79C22×C412yes—
179C2×C8×C79C22×C412yes—
180C2×C8×C79C2×D412yes—
181C2×C8×C79C2×D412yes—
182C4⋊C4×C79—22no—
183C4⋊C4×C79—22no—
184C4⋊C4×C79—22no—
185C4⋊C4×C79—22no—
186C22⋊C4×C79—44no—
187C23×C79—21no—
188C23×C79—21no—
189C23×C79—22no—
190C23×C79—22no—
191C23×C79C2×C4⋊C412yes—
192C23×C79C22×D411yescharacteristic
193C22×D79—22no—
194C22×D79—22no—
195C22×D79—21no—
196C22×D79—21no—
197C22×D79—22no—
198C22×D79—22no—
199C22×D79—28no—
200C22×D79—48no—
201C22×D79—48no—
202C22×D79—44no—
203C22×D79—44no—
204C22×D79C2×C4⋊C412yes—
205C22×D79C2×C4⋊C412yes—
206C22×D79#12112yes—
207C22×D79#12112yes—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—22no—
211C2×C4×C79—22no—
212C2×C4×C79—22no—
213C2×C4×C79—22no—
214C2×C4×C79—42no—
215C2×C4×C79—42no—
216C2×C4×C79—42no—
217C2×C4×C79—42no—
218C2×C4×C79—42no—
219C2×C4×C79—42no—
220C2×C4×C79C2×C4⋊C412yes—
221C2×C4×C79C23×C411yescharacteristic
222C2×C4×C79C22×D411yescharacteristic
223C2×C4×C79C22×D411yescharacteristic
224C2×C4×C79C22×Q811yescharacteristic
225C2×C4×C79C2×#4111yescharacteristic
226C2×C4×C79C2×#4111yescharacteristic
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—28no—
230C2×Dic79—42no—
231C2×Dic79—42no—
232C2×Dic79—42no—
233C2×Dic79—42no—
234C2×Dic79—42no—
235C2×Dic79—42no—
236C2×Dic79C2×C4⋊C412yes—
237C2×Dic79C2×C4⋊C412yes—
238C2×Dic79#12112yes—
239C2×Dic79#12112yes—
240C4×D79—28no—
241C4×D79—22no—
242C4×D79—22no—
243C4×D79—22no—
244C4×D79—22no—
245C4×D79—28no—
246C4×D79C2×C22⋊C414yes—
247C4×D79#11714yes—
248C4×D79C22⋊Q814yes—
249C4×D79C22⋊Q814yes—
250C8×C79—22no—
251C8×C79C2×C22⋊C414yes—
252C79⋊C8—22no—
253C79⋊C8C2×C22⋊C414yes—
254C22×C79—24no—
255C22×C79—22no—
256C22×C79—42no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79C22×C4⋊C411yescharacteristic
260C22×C79C2×#12111yescharacteristic
261C22×C79C2×#12111yescharacteristic
262C22×C79#57512yes—
263C22×C79#57512yes—
264C2×D79—24no—
265C2×D79—24no—
266C2×D79—22no—
267C2×D79—22no—
268C2×D79—24no—
269C2×D79—22no—
270C2×D79—416no—
271C2×D79—42no—
272C2×D79—41no—
273C2×D79—41no—
274C2×D79#38014yes—
275C4×C79—22no—
276C4×C79—82no—
277C4×C79—82no—
278C4×C79C22×C22⋊C411yescharacteristic
279C4×C79C2×#11711yescharacteristic
280C4×C79C2×C22⋊Q811yescharacteristic
281C4×C79C2×C22⋊Q811yescharacteristic
282Dic79—22no—
283Dic79—82no—
284Dic79—82no—
285Dic79#38014yes—
286#20660—791nomaximal, Sylow
287C2×C79—42no—
288C2×C79C2×#38011yescharacteristic
289C2×C79#291112yes—
290C2×C79#305412yes—
291C2×C79#305411yescharacteristic
292C2×C79#305411yescharacteristic
293D79—24no—
294D79—42no—
295C23×M16—791no—
296C2×#427—791no—
297C2×#427—791no—
298#2280—792no—
299#1809—791no—
300#1809—791no—
301#1809—792no—
302#1809—792no—
303#1809—791no—
304#1809D7911yescharacteristic, complemented
305#1735—792no—
306C79#2066011yescharacteristic, complemented, Sylow
307C24×C4—791no—
308C22×M16—791no—
309C22×M16—798no—
310C22×M16—791no—
311C22×M16—791no—
312C22×M16—1584no—
313C22×M16—1584no—
314C22×M16C2×D7911yescharacteristic
315C23×C8—791no—
316C23×C8—791no—
317C2×#117—792no—
318C2×#117—792no—
319#427—1584no—
320#427—1584no—
321C2×#107—791no—
322C2×#107—792no—
323C2×#107C2×D7911yescharacteristic, complemented
324C2×#106—791no—
325C2×#106—792no—
326C2×#106C2×D7911yescharacteristic, complemented
327#381—1582no—
328#380—792no—
329#380—792no—
330#380—792no—
331#380C2×D7912yes—
332#379—1582no—
333C4×C22⋊C4—1582no—
334C4×C22⋊C4—1582no—
335#340—792no—
336#340—792no—
337#340—792no—
338#340C2×D7912yescomplemented
339C25—791no—
340C23×C4—791no—
341C23×C4—791no—
342C23×C4—794no—
343C23×C4—791no—
344C23×C4—791no—
345C23×C4—794no—
346C23×C4—791no—
347C23×C4—1582no—
348C23×C4—1582no—
349C23×C4—1582no—
350C23×C4—1582no—
351C23×C4C22×D7911yescharacteristic
352C2×M16—22no—
353C2×M16—22no—
354C2×M16—794no—
355C2×M16—794no—
356C2×M16—794no—
357C2×M16—1582no—
358C2×M16—1582no—
359C2×M16—15832no—
360C2×M16—1582no—
361C2×M16—1582no—
362C2×M16—1582no—
363C2×M16—1582no—
364C2×M16C4×D7914yes—
365C22×C8—791no—
366C22×C8—794no—
367C22×C8—794no—
368C22×C8—794no—
369C22×C8—791no—
370C22×C8—794no—
371C22×C8—1582no—
372C22×C8—1582no—
373C22×C8C22×D7911yescharacteristic
374C22×C8C22×D7911yescharacteristic
375#117—1584no—
376#117—1584no—
377#117—1584no—
378#117—1584no—
379C2×C4⋊C4—792no—
380C2×C4⋊C4—792no—
381C2×C4⋊C4C22×D7912yes—
382C2×C4⋊C4C22×D7912yes—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1582no—
385C2×C22⋊C4—1582no—
386C2×C22⋊C4—1582no—
387C2×C22⋊C4—1582no—
388C2×C22⋊C4—1582no—
389C2×C42—1582no—
390C2×C42—1582no—
391#107—22no—
392#107—1584no—
393#107—1582no—
394#106—22no—
395#106—1584no—
396#106—1582no—
397#95—22no—
398#95—1582no—
399#95—1582no—
400#95—1582no—
401C24—3162no—
402C24—3162no—
403C24—794no—
404C24—791no—
405C24—791no—
406C24—1582no—
407C24—1582no—
408C24D4×D7911yescharacteristic
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—791no—
416C22×C4—792no—
417C22×C4—792no—
418C22×C4—792no—
419C22×C4—792no—
420C22×C4—792no—
421C22×C4—792no—
422C22×C4—794no—
423C22×C4—791no—
424C22×C4—792no—
425C22×C4—794no—
426C22×C4—794no—
427C22×C4—792no—
428C22×C4—1588no—
429C22×C4—1582no—
430C22×C4—1582no—
431C22×C4—1588no—
432C22×C4—1588no—
433C22×C4—1588no—
434C22×C4—1582no—
435C22×C4—1582no—
436C22×C4—1582no—
437C22×C4—1582no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1581no—
441C22×C4—1581no—
442C22×C4—1581no—
443C22×C4—1581no—
444C22×C4C2×C4×D7912yes—
445C22×C4D4×D7911yescharacteristic
446C22×C4Q8×D7912yes—
447C22×C4C23×D7911yescharacteristic
448M16—28no—
449M16—1588no—
450M16—1588no—
451M16—1588no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—792no—
455C2×C8—792no—
456C2×C8—792no—
457C2×C8—7916no—
458C2×C8—792no—
459C2×C8—1588no—
460C2×C8—1581no—
461C2×C8—1581no—
462C2×C8C2×C4×D7912yes—
463C2×C8C2×C4×D7912yes—
464C2×C8D4×D7912yes—
465C2×C8D4×D7912yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—1582no—
471C4⋊C4—1582no—
472C4⋊C4—1582no—
473C4⋊C4—1582no—
474C22⋊C4—44no—
475C22⋊C4—3164no—
476C22⋊C4—3164no—
477C22⋊C4—3164no—
478C22⋊C4—3164no—
479C22⋊C4—3164no—
480C42—3164no—
481C42—3164no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—3168no—
487C23—3168no—
488C23—3164no—
489C23—3164no—
490C23—792no—
491C23—792no—
492C23—792no—
493C23—792no—
494C23—1588no—
495C23—1581no—
496C23—1581no—
497C23—1582no—
498C23—1582no—
499C23—1582no—
500C23—1582no—
501C23C4⋊C4×D7912yes—
502C23C2×D4×D7911yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—3162no—
516C2×C4—3162no—
517C2×C4—3162no—
518C2×C4—3162no—
519C2×C4—3162no—
520C2×C4—3162no—
521C2×C4—792no—
522C2×C4—792no—
523C2×C4—792no—
524C2×C4—794no—
525C2×C4—794no—
526C2×C4—794no—
527C2×C4—794no—
528C2×C4—792no—
529C2×C4—1588no—
530C2×C4—1588no—
531C2×C4—1588no—
532C2×C4—1581no—
533C2×C4—1581no—
534C2×C4—1582no—
535C2×C4—1582no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4C4⋊C4×D7912yes—
539C2×C4C22×C4×D7911yescharacteristic
540C2×C4C2×D4×D7911yescharacteristic
541C2×C4C2×D4×D7911yescharacteristic
542C2×C4C2×Q8×D7911yescharacteristic
543C2×C4#41×D7911yescharacteristic
544C2×C4#41×D7911yescharacteristic
545C8—22no—
546C8—794no—
547C8—1582no—
548C8C22⋊C4×D7914yes—
549C22—22no—
550C22—24no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—31616no—
555C22—3162no—
556C22—3161no—
557C22—3161no—
558C22—794no—
559C22—1582no—
560C22—1584no—
561C22—1584no—
562C22—1582no—
563C22—1584no—
564C22—1582no—
565C22C2×C4⋊C4×D7911yescharacteristic
566C22#121×D7911yescharacteristic
567C22#121×D7911yescharacteristic
568C22#1909935912yes—
569C22#1909936212yes—
570C4—22no—
571C4—82no—
572C4—82no—
573C4—794no—
574C4—6322no—
575C4—6322no—
576C4—1582no—
577C4C2×C22⋊C4×D7911yescharacteristic
578C4#117×D7911yescharacteristic
579C4C22⋊Q8×D7911yescharacteristic
580C4C22⋊Q8×D7911yescharacteristic
581C2—42no—
582C2—3162no—
583C2—1584no—
584C2#380×D7911yescharacteristic
585C2#2377198712yes—
586C2#2377300412yes—
587C2#2377300711yescharacteristic
588C2#2377301111yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1809C2
socleC23×C79C22×D4
prime radicalC24×D79D4
cosocleC22×C4×C79C24
hypercenter#1809D79
hypocenterC79#20660
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×#19098600C2semidirect
3. C2×#19098598C2semidirect
5. #23763434C2semidirect
6. #23763433C2semidirect
7. #23763432C2semidirect
8. #23763431C2semidirect
9. #23763430C2semidirect
10. #23763115C2semidirect
11. C79×#1809C2semidirect
18. C2×M16×D79C4semidirect
25. C2×#14579095C22semidirect
26. C2×#14579094C22semidirect
27. C2×#14579093C22semidirect
28. C2×#14579092C22semidirect
36. #19098326C22semidirect
37. #19098325C22semidirect
38. #19098325C22semidirect
40. C2×#107×C79C22semidirect
41. C2×#106×C79C22semidirect
43. #340×C79C22semidirect
152. C8×D79C22⋊C4semidirect
304. #1809D79semidirect
306. C79#20660semidirect
323. C2×#107C2×D79semidirect
326. C2×#106C2×D79semidirect
338. #340C2×D79semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D792
C2×#190986002
C2×#190985982
#237657822
#237634342
#237634332
#237634322
#237634312
#237634302
#237631152
C79×#18092
#2066079