Finite Group Databasebeta

← #34258910

Subgroups — #34258910

Order 20224.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774007C211yesmaximal, characteristic
2#23772494C211yesmaximal, characteristic, complemented
3#23762307C211yesmaximal, characteristic, complemented
4#23762302C211yesmaximal, characteristic, complemented
5#23762295C211yesmaximal, characteristic, complemented
6#23762288C211yesmaximal, characteristic, complemented
7#23762217C211yesmaximal, characteristic, complemented
8#23762212C211yesmaximal, characteristic, complemented
9C8×#11929614C211yesmaximal, characteristic, complemented
10C8×#11929612C211yesmaximal, characteristic, complemented
11#23762165C211yesmaximal, characteristic, complemented
12#23761860C211yesmaximal, characteristic
13#23761854C211yesmaximal, characteristic, complemented
14#23761681C211yesmaximal, characteristic, complemented
15C79×#2993C211yesmaximal, characteristic
16#19099120C2211yescharacteristic
17#19098973C411yescharacteristic
18#19098958C411yescharacteristic
19#19098933C411yescharacteristic
20#19098922C411yescharacteristic
21#19098819C411yescharacteristic
22#19098819C411yescharacteristic
23#19098807C411yescharacteristic
24#19098807C411yescharacteristic
25#19098693C2211yescharacteristic
26#19098693C2211yescharacteristic
27#19098692—21no—
28#19098692—21no—
29C8×Dic158C2211yescharacteristic
30C8×Dic158C2211yescharacteristic
31C4×#11929614C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929612C2211yescharacteristic
34C4×#11929612C2211yescharacteristic
35#19098621—21no—
36#19098621—21no—
37#19098620C2211yescharacteristic
38C4×#11929608C2211yescharacteristic
39C2×C4×C79⋊C8C2211yescharacteristic
40#19098315C2211yescharacteristic
41#19098315C2211yescharacteristic
42#19098313C2211yescharacteristic
43#19098313C2211yescharacteristic
44Q8⋊(C79⋊C8)C2211yescharacteristic
45Q8⋊(C79⋊C8)C2211yescharacteristic
46#19098293C2211yescharacteristic
47#19098293C2211yescharacteristic
48#19098293C2211yescharacteristic
49#19098293C2211yescharacteristic
50D4⋊(C79⋊C8)C2211yescharacteristic
51D4⋊(C79⋊C8)C2211yescharacteristic
52#19098285C2211yescharacteristic
53#19098285C2211yescharacteristic
54C8×C79⋊C8C2211yescharacteristic
55C8×C79⋊C8C2211yescharacteristic
56#529×C79C2211yescharacteristic
57#446×C79C2211yescharacteristic
58C8×Q8×C79C2211yescharacteristic
59#436×C79—21no—
60#436×C79C2211yescharacteristic
61#435×C79—21no—
62C8×D4×C79C2211yescharacteristic
63#433×C79—21no—
64#431×C79C2211yescharacteristic
65#424×C79—21no—
66#423×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929614C2×C411yescharacteristic
69C2×#11929612C2×C411yescharacteristic
70C2×#11929612C2×C411yescharacteristic
71C2×#11929608C2×C411yescharacteristic
72C2×#11929608C2×C411yescharacteristic
73C22×C79⋊C8—21no—
74#14579161C2×C411yescharacteristic
75#14579161C2×C411yescharacteristic
76#14579153—21no—
77#14579140—21no—
78#14579140C2×C411yescharacteristic
79#14579140C2×C411yescharacteristic
80C4×Dic158—21no—
81C4×Dic158C2311yescharacteristic
82C4×Dic158C2311yescharacteristic
83#14579120—21no—
84#14579094C2×C411yescharacteristic
85#14579094C2×C411yescharacteristic
86#14579092C2×C411yescharacteristic
87#14579092C2×C411yescharacteristic
88C4⋊(C79⋊C8)C2311yescharacteristic
89C4⋊(C79⋊C8)C2311yescharacteristic
90Dic79⋊C8—21no—
91Dic79⋊C8—21no—
92Dic79⋊C8—21no—
93Dic79⋊C8—21no—
94#14579085C2×C411yescharacteristic
95#14579085C2×C411yescharacteristic
96#14579084C2×C411yescharacteristic
97#14579084C2×C411yescharacteristic
98#14579084C2×C411yescharacteristic
99#14579084C2×C411yescharacteristic
100#14579082C2×C411yescharacteristic
101#14579082C2×C411yescharacteristic
102C22⋊(C79⋊C8)—21no—
103#14579072—21no—
104#14579072C2×C411yescharacteristic
105#14579072C2×C411yescharacteristic
106#14579071—21no—
107#14579071—21no—
108C4×C79⋊C8—21no—
109C4×C79⋊C8C2311yescharacteristic
110C4×C79⋊C8C2311yescharacteristic
111C4×C79⋊C8C2311yescharacteristic
112C4×C79⋊C8C2311yescharacteristic
113C8×Dic79—21no—
114C8×Dic79—21no—
115C2×M16×C79—21no—
116C2×M16×C79—21no—
117C22×C8×C79—21no—
118#126×C79—21no—
119#125×C79C2×C411yescharacteristic
120#124×C79C2×C411yescharacteristic
121#123×C79—21no—
122C22⋊Q8×C79C2×C411yescharacteristic
123#121×C79C2×C411yescharacteristic
124C4×Q8×C79C2311yescharacteristic
125C4×D4×C79—21no—
126C4×D4×C79C2311yescharacteristic
127#117×C79—21no—
128C2×C42×C79C2311yescharacteristic
129C4⋊C8×C79—21no—
130C4⋊C8×C79—21no—
131C4⋊C8×C79—21no—
132C4⋊C8×C79D411yescharacteristic
133C4⋊C8×C79D411yescharacteristic
134C4⋊C8×C79C2311yescharacteristic
135C4⋊C8×C79C2311yescharacteristic
136C22⋊C8×C79—21no—
137C22⋊C8×C79—21no—
138C22⋊C8×C79D411yescharacteristic
139C22⋊C8×C79D411yescharacteristic
140#97×C79—21no—
141C4×C8×C79—21no—
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×Dic158C22×C411yescharacteristic
147#11929614—22no—
148#11929614—22no—
149#11929612—22no—
150#11929612—22no—
151#11929608—22no—
152#11929608—22no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8—21no—
155C2×C79⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—42no—
158C2×C79⋊C8C22×C411yescharacteristic
159C2×C79⋊C8C22×C411yescharacteristic
160C2×C79⋊C8C22×C411yescharacteristic
161C2×C79⋊C8C22×C411yescharacteristic
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79C22×C411yescharacteristic
166C4⋊Dic79C22×C411yescharacteristic
167Dic79⋊C4—21no—
168Dic79⋊C4—21no—
169Dic79⋊C4—21no—
170Dic79⋊C4—21no—
171C22⋊Dic79—21no—
172C22⋊Dic79—21no—
173C4×Dic79—21no—
174C4×Dic79—21no—
175C2×Q8×C79C22×C411yescharacteristic
176C2×D4×C79—21no—
177C2×D4×C79C22×C411yescharacteristic
178C22×C4×C79—21no—
179C22×C4×C79C22×C411yescharacteristic
180C22×C4×C79C2×D411yescharacteristic
181C22×C4×C79#4111yescharacteristic
182M16×C79—81no—
183M16×C79—81no—
184C2×C8×C79—21no—
185C2×C8×C79—21no—
186C2×C8×C79—81no—
187C2×C8×C79C2×D411yescharacteristic
188C2×C8×C79C2×D411yescharacteristic
189C2×C8×C79C2×D411yescharacteristic
190C2×C8×C79C2×D411yescharacteristic
191C4⋊C4×C79—21no—
192C4⋊C4×C79—21no—
193C4⋊C4×C79—21no—
194C4⋊C4×C79—21no—
195C4⋊C4×C79C22×C411yescharacteristic
196C4⋊C4×C79C22×C411yescharacteristic
197C22⋊C4×C79—21no—
198C22⋊C4×C79—21no—
199C22⋊C4×C79—21no—
200C22⋊C4×C79—21no—
201C22⋊C4×C79—21no—
202C42×C79—21no—
203C42×C79C22×C411yescharacteristic
204C42×C79C2×D411yescharacteristic
205C42×C79#4111yescharacteristic
206C42×C79C2411yescharacteristic
207C23×C79—21no—
208C23×C79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211D4×C79—21no—
212D4×C79—21no—
213D4×C79—81no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—21no—
220C2×C4×C79—21no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79C4×D411yescharacteristic
225C2×C4×C79C4×D411yescharacteristic
226C2×C4×C79C4×D411yescharacteristic
227C2×C4×C79C23×C411yescharacteristic
228C2×C4×C79C22×D411yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×Dic79—21no—
231C2×Dic79—21no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234Dic158—21no—
235Dic158—21no—
236Dic158—21no—
237Dic158—21no—
238Dic158—81no—
239C8×C79—21no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—41no—
244C8×C79—41no—
245C79⋊C8—22no—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—41no—
249C79⋊C8—41no—
250C79⋊C8—41no—
251C79⋊C8#13112yes—
252C79⋊C8#13112yes—
253C22×C79—21no—
254C22×C79—21no—
255C22×C79—21no—
256C22×C79—41no—
257C22×C79—41no—
258C22×C79—41no—
259C22×C79C2×C4×D411yescharacteristic
260C4×C79—21no—
261C4×C79—21no—
262C4×C79—21no—
263C4×C79—21no—
264C4×C79—41no—
265C4×C79—41no—
266C4×C79—41no—
267C4×C79C2×#13111yescharacteristic
268C4×C79C2×#13111yescharacteristic
269C4×C79#57711yescharacteristic
270C4×C79#57811yescharacteristic
271Dic79—41no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275#17969—791nomaximal, Sylow
276C2×C79—41no—
277C2×C79—81no—
278C2×C79#293111yescharacteristic
279C2×C79#294711yescharacteristic
280C2×C79#295611yescharacteristic
281#3172—791no—
282#2993D7911yescharacteristic
283#2985—791no—
284#1593—791no—
285#1592—791no—
286#1591—791no—
287#1590—791no—
288#1579—791no—
289SD16⋊C8—791no—
290C8×Q16—791no—
291C8×SD16—791no—
292#1569—791no—
293#1494—791no—
294#1493—791no—
295C8×M16—791no—
296C79#1796911yescharacteristic, complemented, Sylow
297#533—791no—
298#529C2×D7911yescharacteristic
299#501—791no—
300#497—791no—
301#491—791no—
302#489—791no—
303#470—791no—
304#470—791no—
305#468—791no—
306#468—791no—
307#446—791no—
308#446—791no—
309#446—1581no—
310#446—1581no—
311#446C2×D7911yescharacteristic
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8C2×D7911yescharacteristic
315C4×Q16—791no—
316C4×Q16—791no—
317C4×SD16—791no—
318C4×SD16—791no—
319#436—21no—
320#436C2×D7911yescharacteristic
321#435—21no—
322C8×D4C2×D7911yescharacteristic
323#433—21no—
324#433—1581no—
325#433—1581no—
326#432—791no—
327#431C2×D7911yescharacteristic
328#424—21no—
329#423C2×D7911yescharacteristic
330C4×M16—791no—
331C2×C4×C8—791no—
332#335—791no—
333#335—791no—
334#334—791no—
335#334—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8—791no—
341Q8⋊C8—791no—
342D4⋊C8—791no—
343D4⋊C8—791no—
344#322—791no—
345#322—791no—
346C82—791no—
347C82—791no—
348C2×Q16—791no—
349C2×Q16—791no—
350C2×SD16—791no—
351C2×SD16—791no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—791no—
355C2×M16—791no—
356C22×C8—21no—
357C22×C8—1581no—
358C4⋊Q8—791no—
359C4⋊Q8—791no—
360#126—21no—
361#125—1581no—
362#125C4×D7911yescharacteristic
363#124C4×D7911yescharacteristic
364#123—21no—
365C22⋊Q8—791no—
366C22⋊Q8—791no—
367C22⋊Q8—1581no—
368C22⋊Q8C4×D7911yescharacteristic
369#121C4×D7911yescharacteristic
370C4×Q8—791no—
371C4×Q8—791no—
372C4×Q8—1581no—
373C4×Q8C22×D7911yescharacteristic
374C4×D4—21no—
375C4×D4C22×D7911yescharacteristic
376#117—21no—
377#117—1581no—
378C2×C42C22×D7911yescharacteristic
379#107—791no—
380#107—791no—
381#106—791no—
382#106—791no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—1581no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8C22⋊D7911yescharacteristic
393C4⋊C8C22⋊D7911yescharacteristic
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103—791no—
401#103—791no—
402#102—791no—
403#102—791no—
404C22⋊C8—21no—
405C22⋊C8—21no—
406C22⋊C8—1581no—
407C22⋊C8C22⋊D7911yescharacteristic
408C22⋊C8C22⋊D7911yescharacteristic
409#97—21no—
410#97—791no—
411#97—791no—
412#97—1581no—
413#97—1581no—
414#97—1581no—
415C4×C8—21no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—791no—
419C4×C8—791no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8—1581no—
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C2×Q8—791no—
426C2×Q8—791no—
427C2×Q8—1581no—
428C2×Q8C2×C4×D7911yescharacteristic
429C2×D4—21no—
430C2×D4C2×C4×D7911yescharacteristic
431C22×C4—21no—
432C22×C4C2×C4×D7911yescharacteristic
433C22×C4C2×C22⋊D7911yescharacteristic
434C22×C4#1192962411yescharacteristic
435Q16—1582no—
436Q16—1582no—
437SD16—1582no—
438SD16—1582no—
439M16—81no—
440M16—81no—
441M16—1582no—
442M16—1582no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—81no—
446C2×C8—3162no—
447C2×C8—791no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—1581no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8C2×C22⋊D7911yescharacteristic
456C2×C8C2×C22⋊D7911yescharacteristic
457C2×C8C2×C22⋊D7911yescharacteristic
458C2×C8C2×C22⋊D7911yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—791no—
464C4⋊C4—791no—
465C4⋊C4—1581no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4C2×C4×D7911yescharacteristic
473C4⋊C4C2×C4×D7911yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—21no—
482C42—1581no—
483C42—1581no—
484C42C2×C4×D7911yescharacteristic
485C42C2×C22⋊D7911yescharacteristic
486C42#1192962411yescharacteristic
487C42C23×D7911yescharacteristic
488C23—21no—
489C23C4×C22⋊D7911yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—6321no—
493Q8—1581no—
494Q8—1581no—
495Q8—1581no—
496Q8—1581no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4C4×C22⋊D7911yescharacteristic
515C2×C4C4×C22⋊D7911yescharacteristic
516C2×C4C4×C22⋊D7911yescharacteristic
517C2×C4C22×C4×D7911yescharacteristic
518C2×C4C22×C22⋊D7911yescharacteristic
519C2×C4C2×#1192962411yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—3161no—
527C8—3161no—
528C8—3161no—
529C8—792no—
530C8—792no—
531C8—1582no—
532C8—1581no—
533C8—1581no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×C4×C22⋊D7911yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—3161no—
549C4—3161no—
550C4—3161no—
551C4—3161no—
552C4#131×D7911yescharacteristic
553C4#131×D7911yescharacteristic
554C4#1909937411yescharacteristic
555C4#1909937811yescharacteristic
556C2—41no—
557C2—81no—
558C2#2377211011yescharacteristic
559C2#2377223911yescharacteristic
560C2#2377230711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#2993C2
socleC22×C79C2×C4×D4
prime radical#121×C79C2×C4
cosocleC42×C79C24
hypercenter#2993D79
hypocenterC79#17969
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #23772494C2semidirect
3. #23762307C2semidirect
4. #23762302C2semidirect
5. #23762295C2semidirect
6. #23762288C2semidirect
7. #23762217C2semidirect
8. #23762212C2semidirect
9. C8×#11929614C2semidirect
10. C8×#11929612C2semidirect
11. #23762165C2semidirect
13. #23761854C2semidirect
14. #23761681C2semidirect
296. C79#17969semidirect

Maximal subgroups

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

subgroupindex
#237740072
#237724942
#237623072
#237623022
#237622952
#237622882
#237622172
#237622122
C8×#119296142
C8×#119296122
#237621652
#237618602
#237618542
#237616812
C79×#29932
#1796979