Finite Group Databasebeta

← #34258505

Subgroups — #34258505

Order 20224.

604 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773412C211yesmaximal, characteristic
2#23772376C211yesmaximal, characteristic, complemented
3#23772361C211yesmaximal, characteristic, complemented
4#23762294C211yesmaximal, characteristic, complemented
5#23762293C211yesmaximal, characteristic, complemented
6#23762287C211yesmaximal, characteristic, complemented
7#23762282C211yesmaximal, characteristic, complemented
8#23762247C211yesmaximal, characteristic, complemented
9#23762240C211yesmaximal, characteristic, complemented
10#23762233C211yesmaximal, characteristic, complemented
11#23762226C211yesmaximal, characteristic, complemented
12#23762178C211yesmaximal, characteristic, complemented
13#23761835C211yesmaximal, characteristic, complemented
14#23761684C211yesmaximal, characteristic, complemented
15C79×#1482C211yesmaximal, characteristic, complemented
16C4×#11929624C2211yescharacteristic
17#19098901C411yescharacteristic
18#19098884C411yescharacteristic
19#19098867C411yescharacteristic
20#19098860C411yescharacteristic
21#19098782C411yescharacteristic
22#19098775C411yescharacteristic
23#19098723C411yescharacteristic
24#19098706C411yescharacteristic
25#19098694C2211yescharacteristic
26#19098690C2211yescharacteristic
27#19098690C2211yescharacteristic
28Q8×C79⋊C8C2211yescharacteristic
29C4×#11929614C2211yescharacteristic
30C4×#11929611C2211yescharacteristic
31C4×#11929612C2211yescharacteristic
32C4×#11929609C2211yescharacteristic
33#19098641C2211yescharacteristic
34#19098640C2211yescharacteristic
35#19098640C2211yescharacteristic
36#19098635—21no—
37#19098631—21no—
38#19098631—21no—
39D4×C79⋊C8—21no—
40D4×C79⋊C8C2211yescharacteristic
41#19098624—21no—
42#19098620C2211yescharacteristic
43#19098616C2211yescharacteristic
44#19098614—21no—
45#19098519—21no—
46C4×#11929607—21no—
47C2×C4×C79⋊C8C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098315C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098313C2211yescharacteristic
52#19098295C2211yescharacteristic
53#19098295C2211yescharacteristic
54#19098292C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098285C2211yescharacteristic
57#19098285C2211yescharacteristic
58C8×C79⋊C8C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#529×C79C2211yescharacteristic
61#431×C79C2211yescharacteristic
62#431×C79C2211yescharacteristic
63Q8⋊C8×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65D4⋊C8×C79C2211yescharacteristic
66D4⋊C8×C79C2211yescharacteristic
67C2×#11929624C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929611C2×C411yescharacteristic
71C2×#11929609C2×C411yescharacteristic
72C2×#11929608—21no—
73C2×#11929607—21no—
74C2×#11929607—21no—
75C22×C79⋊C8—21no—
76C22×C79⋊C8D411yescharacteristic
77C22×C79⋊C8D411yescharacteristic
78C4×Dic158C2×C411yescharacteristic
79C4×Dic158C2311yescharacteristic
80C4×D316C2×C411yescharacteristic
81C4×D316C2311yescharacteristic
82C4×C22⋊D79—21no—
83C4×C22⋊D79—21no—
84#14579120C2×C411yescharacteristic
85#14579118—21no—
86C42×D79—21no—
87#14579094C2×C411yescharacteristic
88#14579094C2×C411yescharacteristic
89#14579092C2×C411yescharacteristic
90#14579092C2×C411yescharacteristic
91C4⋊(C79⋊C8)—21no—
92C4⋊(C79⋊C8)—21no—
93C4⋊(C79⋊C8)C2×C411yescharacteristic
94C4⋊(C79⋊C8)C2×C411yescharacteristic
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—
110C22⋊(C79⋊C8)C2×C411yescharacteristic
111C22⋊(C79⋊C8)C2×C411yescharacteristic
112(C2×D79)⋊C8—21no—
113(C2×D79)⋊C8—21no—
114#14579072—21no—
115#14579072—21no—
116#14579071—21no—
117#14579071—21no—
118C4×C79⋊C8—21no—
119C4×C79⋊C8D411yescharacteristic
120C4×C79⋊C8D411yescharacteristic
121C4×C79⋊C8C2311yescharacteristic
122C4×C79⋊C8C2311yescharacteristic
123C4×C79⋊C8C2311yescharacteristic
124C4×C79⋊C8C2311yescharacteristic
125#126×C79—21no—
126#125×C79C2×C411yescharacteristic
127#124×C79C2×C411yescharacteristic
128#123×C79—21no—
129C22⋊Q8×C79C2×C411yescharacteristic
130#121×C79C2×C411yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132C4×D4×C79—21no—
133C4×D4×C79C2311yescharacteristic
134#117×C79—21no—
135C2×C42×C79C2311yescharacteristic
136C4⋊C8×C79—21no—
137C4⋊C8×C79C2311yescharacteristic
138C4⋊C8×C79C2311yescharacteristic
139C22⋊C8×C79—21no—
140C22⋊C8×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144#11929624—41no—
145#11929624—41no—
146C2×Dic158C22×C411yescharacteristic
147C2×D316C22×C411yescharacteristic
148C2×C22⋊D79—21no—
149C2×C4×D79—21no—
150C2×C4×D79—21no—
151C2×C4×D79—21no—
152#11929614—41no—
153#11929612—41no—
154#11929611—41no—
155#11929609—41no—
156#11929608—42no—
157#11929607—42no—
158#11929607—42no—
159C2×C79⋊C8—22no—
160C2×C79⋊C8—22no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8—21no—
163C2×C79⋊C8—42no—
164C2×C79⋊C8C22×C411yescharacteristic
165C2×C79⋊C8C22×C411yescharacteristic
166C2×C79⋊C8C22×C411yescharacteristic
167C2×C79⋊C8C22×C411yescharacteristic
168C2×C79⋊C8C2×D411yescharacteristic
169C2×C79⋊C8C2×D411yescharacteristic
170C2×C79⋊C8C2×D411yescharacteristic
171C2×C79⋊C8C2×D411yescharacteristic
172C4⋊Dic79C22×C411yescharacteristic
173C4⋊Dic79C22×C411yescharacteristic
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176C22⋊Dic79—21no—
177(C2×D79)⋊C4—21no—
178(C2×D79)⋊C4—21no—
179C4×Dic79—21no—
180C4×Dic79—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—21no—
190C2×C8×C79—21no—
191C2×C8×C79—21no—
192C2×C8×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79—21no—
196C4⋊C4×C79—21no—
197C4⋊C4×C79C22×C411yescharacteristic
198C4⋊C4×C79C22×C411yescharacteristic
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C22⋊C4×C79—21no—
202C22⋊C4×C79—21no—
203C22⋊C4×C79—21no—
204C42×C79—21no—
205C42×C79C22×C411yescharacteristic
206C42×C79C2×D411yescharacteristic
207C42×C79#4111yescharacteristic
208C42×C79C2411yescharacteristic
209C23×C79—21no—
210C23×C79C4×D411yescharacteristic
211C22×D79—21no—
212Q8×C79—21no—
213Q8×C79—21no—
214D4×C79—21no—
215D4×C79—21no—
216D4×C79—42no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—21no—
222C2×C4×C79—21no—
223C2×C4×C79—21no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79C4×D411yescharacteristic
228C2×C4×C79C4×D411yescharacteristic
229C2×C4×C79C4×D411yescharacteristic
230C2×C4×C79C23×C411yescharacteristic
231C2×C4×C79C22×D411yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C22⋊D79—42no—
234C2×Dic79—21no—
235C2×Dic79—21no—
236C2×Dic79—21no—
237D316—21no—
238D316—21no—
239C4×D79—42no—
240C4×D79—42no—
241C4×D79—41no—
242C4×D79—41no—
243Dic158—21no—
244Dic158—21no—
245C8×C79—41no—
246C8×C79—41no—
247C8×C79—41no—
248C8×C79—41no—
249C79⋊C8—21no—
250C79⋊C8—21no—
251C79⋊C8—21no—
252C79⋊C8—21no—
253C79⋊C8—21no—
254C79⋊C8—21no—
255C79⋊C8—41no—
256C79⋊C8—41no—
257C79⋊C8#13512yes—
258C79⋊C8#13512yes—
259C22×C79—21no—
260C22×C79—21no—
261C22×C79—21no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79—41no—
265C22×C79C2×C4×D411yescharacteristic
266C2×D79—41no—
267C2×D79—41no—
268C2×D79—41no—
269C4×C79—21no—
270C4×C79—21no—
271C4×C79—21no—
272C4×C79—21no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—41no—
276C4×C79#56811yescharacteristic
277C4×C79#56811yescharacteristic
278C4×C79C2×#13511yescharacteristic
279C4×C79C2×#13511yescharacteristic
280Dic79—41no—
281Dic79—41no—
282Dic79—41no—
283#17932—791nomaximal, Sylow
284C2×C79—41no—
285C2×C79—81no—
286C2×C79#293011yescharacteristic
287C2×C79C4×#13511yescharacteristic
288C2×C79#295511yescharacteristic
289D79—81no—
290#3102—791no—
291#2966—791no—
292#2965—791no—
293#1591—791no—
294#1590—791no—
295#1590—791no—
296#1589—791no—
297#1584—791no—
298#1583—791no—
299#1582—791no—
300#1581—791no—
301#1572—791no—
302#1488—791no—
303#1482D7911yescharacteristic, complemented
304#1450—791no—
305C79#1793211yescharacteristic, complemented, Sylow
306#529C2×D7911yescharacteristic
307C4×#41—791no—
308#485—791no—
309#482—791no—
310#479—791no—
311#478—791no—
312#464—791no—
313#463—791no—
314#452—791no—
315#449—791no—
316#446—791no—
317#446—791no—
318#446—791no—
319C8×Q8—791no—
320C4×Q16—791no—
321C4×SD16—791no—
322C4×SD16—791no—
323C4×D8—791no—
324#436—791no—
325#436—791no—
326#436—791no—
327#435—1581no—
328#435—1581no—
329#435—1581no—
330C8×D4—791no—
331C8×D4—1581no—
332#433—1581no—
333#432—791no—
334#431—791no—
335#431—1581no—
336#431C2×D7911yescharacteristic
337#431C2×D7911yescharacteristic
338#405—1581no—
339C4×M16—1581no—
340C2×C4×C8—791no—
341#335—791no—
342#335—791no—
343#334—791no—
344#334—791no—
345Q8⋊C8—791no—
346Q8⋊C8—791no—
347Q8⋊C8C2×D7911yescharacteristic
348Q8⋊C8C2×D7911yescharacteristic
349D4⋊C8—791no—
350D4⋊C8—791no—
351D4⋊C8C2×D7911yescharacteristic
352D4⋊C8C2×D7911yescharacteristic
353#322—791no—
354#322—791no—
355C82—791no—
356C82—791no—
357C2×#41—791no—
358C2×Q16—791no—
359C2×SD16—791no—
360C2×SD16—791no—
361C2×D8—791no—
362C2×M16—1581no—
363C2×M16—1581no—
364C2×M16—1581no—
365C22×C8—791no—
366C22×C8—791no—
367C22×C8—1581no—
368#126—21no—
369#125C4×D7911yescharacteristic
370#124C4×D7911yescharacteristic
371#123—21no—
372C22⋊Q8C4×D7911yescharacteristic
373#121C4×D7911yescharacteristic
374C4×Q8—791no—
375C4×Q8—791no—
376C4×Q8C22×D7911yescharacteristic
377C4×D4—21no—
378C4×D4—791no—
379C4×D4—791no—
380C4×D4—1581no—
381C4×D4—1581no—
382C4×D4C22×D7911yescharacteristic
383#117—21no—
384#117—791no—
385#117—1581no—
386C2×C42—1581no—
387C2×C42C22×D7911yescharacteristic
388#107—791no—
389#107—791no—
390#106—791no—
391#106—791no—
392C4⋊C8—21no—
393C4⋊C8—791no—
394C4⋊C8—791no—
395C4⋊C8—791no—
396C4⋊C8—791no—
397C4⋊C8—1581no—
398C4⋊C8—1581no—
399C4⋊C8—1581no—
400C4⋊C8—1581no—
401C4⋊C8C22×D7911yescharacteristic
402C4⋊C8C22×D7911yescharacteristic
403#103—791no—
404#103—791no—
405#103—791no—
406#103—791no—
407#102—791no—
408#102—791no—
409#102—791no—
410#102—791no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—791no—
414C22⋊C8—791no—
415C22⋊C8—1581no—
416C22⋊C8—1581no—
417C22⋊C8—1581no—
418C22⋊C8—1581no—
419C22⋊C8—1581no—
420#97—1581no—
421#97—1581no—
422#97—1581no—
423#97—1581no—
424C4×C8—21no—
425C4×C8—791no—
426C4×C8—791no—
427C4×C8—791no—
428C4×C8—791no—
429C4×C8—791no—
430C4×C8—791no—
431C4×C8—1581no—
432C4×C8C22×D7911yescharacteristic
433C4×C8C22×D7911yescharacteristic
434#41—3161no—
435#41—3161no—
436C2×Q8—791no—
437C2×Q8C2×C4×D7911yescharacteristic
438C2×D4—21no—
439C2×D4—791no—
440C2×D4—1581no—
441C2×D4C2×C4×D7911yescharacteristic
442C22×C4—21no—
443C22×C4—1581no—
444C22×C4—1581no—
445C22×C4—1581no—
446C22×C4C2×C4×D7911yescharacteristic
447C22×C4D4×D7911yescharacteristic
448C22×C4#1192962311yescharacteristic
449Q16—3161no—
450SD16—3161no—
451SD16—3161no—
452D8—3161no—
453M16—3162no—
454M16—3162no—
455M16—3162no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—3162no—
461C2×C8—791no—
462C2×C8—791no—
463C2×C8—791no—
464C2×C8—791no—
465C2×C8—791no—
466C2×C8—791no—
467C2×C8—791no—
468C2×C8—791no—
469C2×C8—1581no—
470C2×C8—1581no—
471C2×C8—1582no—
472C2×C8—1582no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—791no—
478C4⋊C4—791no—
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—
491C42—21no—
492C42—1581no—
493C42—1581no—
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—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—3162no—
523C2×C4—3162no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—1581no—
527C2×C4—1581no—
528C2×C4—1581no—
529C2×C4(C22⋊D79)⋊C411yescharacteristic
530C2×C4(C22⋊D79)⋊C411yescharacteristic
531C2×C4(C22⋊D79)⋊C411yescharacteristic
532C2×C4C22×C4×D7911yescharacteristic
533C2×C4C2×D4×D7911yescharacteristic
534C2×C4C2×#1192962311yescharacteristic
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—3161no—
540C8—3161no—
541C8—792no—
542C8—792no—
543C8—1581no—
544C8—1581no—
545C8—1581no—
546C8—1581no—
547C8—1581no—
548C8—1581no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—3161no—
556C22—3161no—
557C22—3161no—
558C22C2×(C22⋊D79)⋊C411yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—3161no—
567C4—3161no—
568C4—3161no—
569C4#1909932811yescharacteristic
570C4#1909932811yescharacteristic
571C4#135×D7911yescharacteristic
572C4#135×D7911yescharacteristic
573C2—41no—
574C2—81no—
575C2—6321no—
576C2#2377210611yescharacteristic
577C2#2377216811yescharacteristic
578C2#2377229911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1482C2
socleC22×C79C2×C4×D4
prime radical#19098706C4
cosocleC42×C79C24
hypercenter#1482D79
hypocenterC79#17932
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1482

Complements

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

NH ≅ G/Ntype
2. #23772376C2semidirect
3. #23772361C2semidirect
4. #23762294C2semidirect
5. #23762293C2semidirect
6. #23762287C2semidirect
7. #23762282C2semidirect
8. #23762247C2semidirect
9. #23762240C2semidirect
10. #23762233C2semidirect
11. #23762226C2semidirect
12. #23762178C2semidirect
13. #23761835C2semidirect
14. #23761684C2semidirect
15. C79×#1482C2semidirect
303. #1482D79semidirect
305. C79#17932semidirect

Maximal subgroups

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

subgroupindex
#237734122
#237723762
#237723612
#237622942
#237622932
#237622872
#237622822
#237622472
#237622402
#237622332
#237622262
#237621782
#237618352
#237616842
C79×#14822
#1793279