Finite Group Databasebeta

← #34258804

Subgroups — #34258804

Order 20224.

630 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773964C211yesmaximal, characteristic
2#23772376C211yesmaximal, characteristic, complemented
3#23772369C211yesmaximal, characteristic, complemented
4#23762294C211yesmaximal, characteristic, complemented
5#23762293C211yesmaximal, characteristic, complemented
6#23762287C211yesmaximal, characteristic, complemented
7#23762282C211yesmaximal, characteristic, complemented
8#23762204C211yesmaximal, characteristic, complemented
9#23762203C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#23762192C211yesmaximal, characteristic, complemented
12#23762161C211yesmaximal, characteristic, complemented
13#23761861C211yesmaximal, characteristic, complemented
14C2×C8×C79⋊C8C211yesmaximal, characteristic, complemented
15C79×#1494C211yesmaximal, characteristic, complemented
16#19099097C2211yescharacteristic
17#19098962C411yescharacteristic
18#19098945C411yescharacteristic
19#19098920C411yescharacteristic
20#19098917C411yescharacteristic
21#19098802C411yescharacteristic
22#19098797C411yescharacteristic
23#19098794C411yescharacteristic
24#19098790C411yescharacteristic
25#19098694C2211yescharacteristic
26#19098690C2211yescharacteristic
27Q8×C79⋊C8C2211yescharacteristic
28#19098687C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929611C2211yescharacteristic
31C4×#11929612C2211yescharacteristic
32C4×#11929609C2211yescharacteristic
33#19098641C2211yescharacteristic
34#19098640C2211yescharacteristic
35#19098635—21no—
36#19098631—21no—
37D4×C79⋊C8—21no—
38D4×C79⋊C8C2211yescharacteristic
39#19098628C2211yescharacteristic
40(C22⋊D79)⋊C8—21no—
41#19098625—21no—
42#19098624—21no—
43#19098620C2211yescharacteristic
44#19098520—21no—
45#19098519—21no—
46C2×C4×C79⋊C8C2211yescharacteristic
47C2×C4×C79⋊C8C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56C8×C79⋊C8C2211yescharacteristic
57C8×C79⋊C8C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#529×C79C2211yescharacteristic
61#432×C79C2211yescharacteristic
62C2×C4×C8×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66D4⋊C8×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929612C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×#11929609C2×C411yescharacteristic
71C2×#11929608—21no—
72C2×#11929607—21no—
73C22×C79⋊C8—21no—
74C22×C79⋊C8C2×C411yescharacteristic
75C22×C79⋊C8C2×C411yescharacteristic
76C22×C79⋊C8D411yescharacteristic
77C22×C79⋊C8D411yescharacteristic
78C2×C8×D79—21no—
79#14579155—21no—
80#14579153C2×C411yescharacteristic
81#14579149C2×C411yescharacteristic
82#14579144—21no—
83#14579140C2×C411yescharacteristic
84#14579135C2×C411yescharacteristic
85C4×Dic158C2311yescharacteristic
86C4×D316C2311yescharacteristic
87C4×C22⋊D79—21no—
88#14579118—21no—
89#14579094C2×C411yescharacteristic
90#14579094C2×C411yescharacteristic
91#14579092C2×C411yescharacteristic
92#14579092C2×C411yescharacteristic
93C4⋊(C79⋊C8)—21no—
94C4⋊(C79⋊C8)—21no—
95C4⋊(C79⋊C8)C2311yescharacteristic
96C4⋊(C79⋊C8)C2311yescharacteristic
97Dic79⋊C8—21no—
98Dic79⋊C8—21no—
99#14579085C2×C411yescharacteristic
100#14579085C2×C411yescharacteristic
101#14579084C2×C411yescharacteristic
102#14579084C2×C411yescharacteristic
103#14579082C2×C411yescharacteristic
104#14579082C2×C411yescharacteristic
105#14579081C2×C411yescharacteristic
106#14579081C2×C411yescharacteristic
107C22⋊(C79⋊C8)—21no—
108C22⋊(C79⋊C8)—21no—
109C22⋊(C79⋊C8)—21no—
110(C2×D79)⋊C8—21no—
111(C2×D79)⋊C8—21no—
112#14579072—21no—
113#14579072—21no—
114#14579071—21no—
115C4×C79⋊C8—21no—
116C4×C79⋊C8C2×C411yescharacteristic
117C4×C79⋊C8C2×C411yescharacteristic
118C4×C79⋊C8D411yescharacteristic
119C4×C79⋊C8D411yescharacteristic
120C4×C79⋊C8C2311yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C8×Dic79—21no—
125C22×C8×C79—21no—
126#126×C79—21no—
127#125×C79C2×C411yescharacteristic
128#124×C79C2×C411yescharacteristic
129#123×C79—21no—
130C22⋊Q8×C79C2×C411yescharacteristic
131#121×C79C2×C411yescharacteristic
132C4×Q8×C79C2311yescharacteristic
133C4×D4×C79—21no—
134C4×D4×C79C2311yescharacteristic
135#117×C79—21no—
136C2×C42×C79C2311yescharacteristic
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158C22×C411yescharacteristic
145C2×D316C22×C411yescharacteristic
146C2×C22⋊D79—21no—
147C2×C4×D79—21no—
148#11929614—22no—
149#11929612—22no—
150#11929611—22no—
151#11929609—22no—
152#11929608—42no—
153#11929607—42no—
154C2×C79⋊C8—22no—
155C2×C79⋊C8—22no—
156C2×C79⋊C8—22no—
157C2×C79⋊C8—22no—
158C2×C79⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—42no—
161C2×C79⋊C8C22×C411yescharacteristic
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C2×C79⋊C8C2×D411yescharacteristic
167C2×C79⋊C8C2×D411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C8×D79—42no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176C22⋊Dic79—21no—
177(C2×D79)⋊C4—21no—
178(C2×D79)⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181C4×Dic79—21no—
182C2×Q8×C79C22×C411yescharacteristic
183C2×D4×C79—21no—
184C2×D4×C79C22×C411yescharacteristic
185C22×C4×C79—21no—
186C22×C4×C79C22×C411yescharacteristic
187C22×C4×C79C2×D411yescharacteristic
188C22×C4×C79#4111yescharacteristic
189C2×C8×C79—24no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C2×C8×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79—21no—
198C4⋊C4×C79C22×C411yescharacteristic
199C4⋊C4×C79C22×C411yescharacteristic
200C22⋊C4×C79—21no—
201C22⋊C4×C79—21no—
202C22⋊C4×C79—21no—
203C22⋊C4×C79—21no—
204C22⋊C4×C79—21no—
205C42×C79—21no—
206C42×C79C22×C411yescharacteristic
207C42×C79C2×D411yescharacteristic
208C42×C79#4111yescharacteristic
209C42×C79C2411yescharacteristic
210C23×C79—21no—
211C23×C79C4×D411yescharacteristic
212C22×D79—21no—
213Q8×C79—21no—
214Q8×C79—21no—
215D4×C79—21no—
216D4×C79—21no—
217D4×C79—42no—
218C2×C4×C79—22no—
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—21no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79C4×D411yescharacteristic
229C2×C4×C79C4×D411yescharacteristic
230C2×C4×C79C4×D411yescharacteristic
231C2×C4×C79C23×C411yescharacteristic
232C2×C4×C79C22×D411yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C22⋊D79—42no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237C2×Dic79—21no—
238D316—21no—
239D316—21no—
240C4×D79—41no—
241C4×D79—41no—
242Dic158—21no—
243Dic158—21no—
244C8×C79—22no—
245C8×C79—22no—
246C8×C79—41no—
247C8×C79—41no—
248C79⋊C8—21no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C79⋊C8#13112yes—
255C79⋊C8#13112yes—
256C79⋊C8#13512yes—
257C79⋊C8#13512yes—
258C22×C79—21no—
259C22×C79—21no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79—41no—
263C22×C79#44312yes—
264C22×C79C2×C4×D411yescharacteristic
265C2×D79—41no—
266C2×D79—41no—
267C2×D79—41no—
268C4×C79—21no—
269C4×C79—21no—
270C4×C79—21no—
271C4×C79—21no—
272C4×C79—41no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79C2×#13111yescharacteristic
276C4×C79C2×#13111yescharacteristic
277C4×C79C2×#13511yescharacteristic
278C4×C79C2×#13511yescharacteristic
279Dic79—41no—
280Dic79—41no—
281Dic79—41no—
282#17959—791nomaximal, Sylow
283C2×C79—22no—
284C2×C79—81no—
285C2×C79#293111yescharacteristic
286C2×C79C4×#13511yescharacteristic
287C2×C79C2×#44311yescharacteristic
288D79—81no—
289#3168—791no—
290#2966—791no—
291#2966—791no—
292#1591—791no—
293#1590—791no—
294#1590—791no—
295#1589—791no—
296C8×Q16—791no—
297C8×SD16—791no—
298C8×SD16—791no—
299C8×D8—791no—
300#1568—791no—
301#1494—791no—
302#1494D7911yescharacteristic, complemented
303C2×C82—791no—
304C79#1795911yescharacteristic, complemented, Sylow
305#529—791no—
306#529C2×D7911yescharacteristic
307#499—791no—
308#495—791no—
309#488—791no—
310#488—791no—
311#467—791no—
312#466—791no—
313#465—791no—
314#465—791no—
315#446—791no—
316#446—791no—
317C8×Q8—791no—
318C8×Q8—791no—
319C4×Q16—791no—
320C4×SD16—791no—
321C4×SD16—791no—
322C4×D8—791no—
323#436—791no—
324#436—791no—
325#435—1581no—
326#435—1581no—
327C8×D4—791no—
328C8×D4—791no—
329C8×D4—1581no—
330C8×D4—1581no—
331#433—1581no—
332#433—1581no—
333#432—791no—
334#432C2×D7911yescharacteristic
335#405—1581no—
336#405—1581no—
337C2×C4×C8—791no—
338C2×C4×C8—791no—
339C2×C4×C8C2×D7911yescharacteristic
340#335—791no—
341#335—791no—
342#334—791no—
343#334—791no—
344Q8⋊C8—791no—
345Q8⋊C8—791no—
346Q8⋊C8C2×D7911yescharacteristic
347Q8⋊C8C2×D7911yescharacteristic
348D4⋊C8—791no—
349D4⋊C8—791no—
350D4⋊C8C2×D7911yescharacteristic
351D4⋊C8C2×D7911yescharacteristic
352C82—791no—
353C82—791no—
354C82—791no—
355C82—791no—
356C2×Q16—791no—
357C2×SD16—791no—
358C2×SD16—791no—
359C2×D8—791no—
360C2×M16—1581no—
361C2×M16—1581no—
362C22×C8—21no—
363C22×C8—791no—
364C22×C8—791no—
365C22×C8—791no—
366C22×C8—791no—
367C22×C8—1581no—
368C22×C8—1581no—
369#126—21no—
370#126—1581no—
371#125—791no—
372#125C4×D7911yescharacteristic
373#124—791no—
374#124C4×D7911yescharacteristic
375#123—21no—
376#123—1581no—
377C22⋊Q8—791no—
378C22⋊Q8C4×D7911yescharacteristic
379#121—791no—
380#121C4×D7911yescharacteristic
381C4×Q8—791no—
382C4×Q8C22×D7911yescharacteristic
383C4×D4—21no—
384C4×D4—791no—
385C4×D4—1581no—
386C4×D4C22×D7911yescharacteristic
387#117—21no—
388#117—1581no—
389C2×C42C22×D7911yescharacteristic
390#107—791no—
391#107—791no—
392#106—791no—
393#106—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—1581no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8C22×D7911yescharacteristic
401C4⋊C8C22×D7911yescharacteristic
402#103—791no—
403#103—791no—
404#103—791no—
405#103—791no—
406#102—791no—
407#102—791no—
408#102—791no—
409#102—791no—
410C22⋊C8—21no—
411C22⋊C8—1581no—
412C22⋊C8—1581no—
413C22⋊C8—1581no—
414C22⋊C8—1581no—
415C22⋊C8—1581no—
416#97—21no—
417#97—1581no—
418#97—1581no—
419#97—1581no—
420C4×C8—21no—
421C4×C8—791no—
422C4×C8—791no—
423C4×C8—791no—
424C4×C8—791no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—791no—
429C4×C8—1581no—
430C4×C8—1581no—
431C4×C8C22×D7911yescharacteristic
432C4×C8C22×D7911yescharacteristic
433C2×Q8—791no—
434C2×Q8C2×C4×D7911yescharacteristic
435C2×D4—21no—
436C2×D4—791no—
437C2×D4—1581no—
438C2×D4C2×C4×D7911yescharacteristic
439C22×C4—21no—
440C22×C4—1581no—
441C22×C4C2×C4×D7911yescharacteristic
442C22×C4D4×D7911yescharacteristic
443C22×C4#1192962311yescharacteristic
444Q16—1582no—
445SD16—1582no—
446SD16—1582no—
447D8—1582no—
448M16—3162no—
449M16—3162no—
450C2×C8—24no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—3162no—
456C2×C8—3162no—
457C2×C8—791no—
458C2×C8—791no—
459C2×C8—791no—
460C2×C8—791no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—791no—
465C2×C8—1581no—
466C2×C8—1581no—
467C2×C8—1582no—
468C2×C8—1582no—
469C2×C8—1582no—
470C2×C8—1582no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—791no—
476C4⋊C4—791no—
477C4⋊C4—1581no—
478C4⋊C4—1581no—
479C4⋊C4—1581no—
480C4⋊C4—1581no—
481C4⋊C4C2×C4×D7911yescharacteristic
482C4⋊C4C2×C4×D7911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1581no—
489C22⋊C4—1581no—
490C22⋊C4—1581no—
491C22⋊C4—1581no—
492C22⋊C4—1581no—
493C42—21no—
494C42—1581no—
495C42C2×C4×D7911yescharacteristic
496C42D4×D7911yescharacteristic
497C42#1192962311yescharacteristic
498C42C23×D7911yescharacteristic
499C23—21no—
500C23—1581no—
501C23(C22⋊D79)⋊C411yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—1581no—
505Q8—1581no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—3162no—
510D4—1581no—
511D4—1581no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—1581no—
525C2×C4—1581no—
526C2×C4—1581no—
527C2×C4(C22⋊D79)⋊C411yescharacteristic
528C2×C4(C22⋊D79)⋊C411yescharacteristic
529C2×C4(C22⋊D79)⋊C411yescharacteristic
530C2×C4C22×C4×D7911yescharacteristic
531C2×C4C2×D4×D7911yescharacteristic
532C2×C4C2×#1192962311yescharacteristic
533C8—22no—
534C8—22no—
535C8—41no—
536C8—41no—
537C8—3161no—
538C8—3161no—
539C8—792no—
540C8—792no—
541C8—792no—
542C8—792no—
543C8—1581no—
544C8—1581no—
545C8—1581no—
546C8—1581no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—3161no—
553C22—3161no—
554C22—3161no—
555C22#1909867712yes—
556C22C2×(C22⋊D79)⋊C411yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—3161no—
565C4—3161no—
566C4—3161no—
567C4#131×D7911yescharacteristic
568C4#131×D7911yescharacteristic
569C4#135×D7911yescharacteristic
570C4#135×D7911yescharacteristic
571C2—22no—
572C2—81no—
573C2—6321no—
574C2#2377210811yescharacteristic
575C2#2377216811yescharacteristic
576C2C2×#1909867711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1494C2
socleC22×C79C2×C4×D4
prime radical#19098797C4
cosocleC42×C79C24
hypercenter#1494D79
hypocenterC79#17959
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #23772376C2semidirect
3. #23772369C2semidirect
4. #23762294C2semidirect
5. #23762293C2semidirect
6. #23762287C2semidirect
7. #23762282C2semidirect
8. #23762204C2semidirect
9. #23762203C2semidirect
10. #23762197C2semidirect
11. #23762192C2semidirect
12. #23762161C2semidirect
13. #23761861C2semidirect
14. C2×C8×C79⋊C8C2semidirect
15. C79×#1494C2semidirect
302. #1494D79semidirect
304. C79#17959semidirect

Maximal subgroups

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

subgroupindex
#237739642
#237723762
#237723692
#237622942
#237622932
#237622872
#237622822
#237622042
#237622032
#237621972
#237621922
#237621612
#237618612
C2×C8×C79⋊C82
C79×#14942
#1795979