Finite Group Databasebeta

← #34258909

Subgroups — #34258909

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#23774004C211yesmaximal, characteristic
2#23772552C211yesmaximal, characteristic
3#23772496C211yesmaximal, characteristic, complemented
4#23762307C211yesmaximal, characteristic, complemented
5#23762300C211yesmaximal, characteristic, complemented
6#23762292C211yesmaximal, characteristic, complemented
7#23762219C211yesmaximal, characteristic, complemented
8#23762211C211yesmaximal, characteristic, complemented
9#23762204C211yesmaximal, characteristic, complemented
10#23762197C211yesmaximal, characteristic, complemented
11#23762166C211yesmaximal, characteristic, complemented
12#23761859C211yesmaximal, characteristic
13#23761853C211yesmaximal, characteristic, complemented
14C8×#11929607C211yesmaximal, characteristic, complemented
15C79×#1591C211yesmaximal, characteristic, complemented
16#19099117C2211yescharacteristic
17#19099100C2211yescharacteristic
18#19098971C411yescharacteristic
19#19098956C411yescharacteristic
20#19098930C411yescharacteristic
21#19098924C411yescharacteristic
22#19098820C411yescharacteristic
23#19098818C411yescharacteristic
24#19098812C411yescharacteristic
25#19098810C411yescharacteristic
26#19098693C2211yescharacteristic
27#19098691—21no—
28#19098690—21no—
29Q8×C79⋊C8C2211yescharacteristic
30Q8×C79⋊C8C2211yescharacteristic
31#19098687C2211yescharacteristic
32C4×#11929614C2211yescharacteristic
33C4×#11929613C2211yescharacteristic
34C4×#11929611C2211yescharacteristic
35#19098642C2211yescharacteristic
36#19098640—21no—
37#19098636—21no—
38#19098628C2211yescharacteristic
39#19098625—21no—
40#19098625—21no—
41#19098623—21no—
42#19098619C2211yescharacteristic
43#19098614C2211yescharacteristic
44#19098588—21no—
45#19098585C2211yescharacteristic
46C4×#11929607C2211yescharacteristic
47C4×C8×D79C2211yescharacteristic
48#19098315C2211yescharacteristic
49#19098314C2211yescharacteristic
50#19098313C2211yescharacteristic
51#19098295C2211yescharacteristic
52Q8⋊(C79⋊C8)C2211yescharacteristic
53Q8⋊(C79⋊C8)C2211yescharacteristic
54#19098293C2211yescharacteristic
55#19098292C2211yescharacteristic
56#19098290C2211yescharacteristic
57#19098285C2211yescharacteristic
58#19098284C2211yescharacteristic
59C8×C79⋊C8C2211yescharacteristic
60#446×C79C2211yescharacteristic
61#446×C79C2211yescharacteristic
62C4×Q16×C79C2211yescharacteristic
63#335×C79C2211yescharacteristic
64Q8⋊C8×C79C2211yescharacteristic
65Q8⋊C8×C79C2211yescharacteristic
66C82×C79C2211yescharacteristic
67C2×#11929614C2×C411yescharacteristic
68C2×#11929613C2×C411yescharacteristic
69C2×#11929611C2×C411yescharacteristic
70C2×#11929607—21no—
71C2×#11929607—21no—
72C2×#11929607C2×C411yescharacteristic
73C2×#11929607C2×C411yescharacteristic
74C2×C8×D79—21no—
75C2×C8×D79—21no—
76#14579160C2×C411yescharacteristic
77#14579159C2×C411yescharacteristic
78#14579156—21no—
79#14579152C2×C411yescharacteristic
80#14579151—21no—
81#14579150C2×C411yescharacteristic
82#14579146—21no—
83#14579141C2×C411yescharacteristic
84#14579139—21no—
85#14579137C2×C411yescharacteristic
86#14579137C2×C411yescharacteristic
87#14579136C2×C411yescharacteristic
88C4×Dic158—21no—
89C4×Dic158C2311yescharacteristic
90C4×D316—21no—
91C4×D316C2311yescharacteristic
92#14579118—21no—
93#14579118—21no—
94C42×D79C2311yescharacteristic
95#14579094C2×C411yescharacteristic
96#14579093C2×C411yescharacteristic
97#14579092C2×C411yescharacteristic
98C4⋊(C79⋊C8)—21no—
99C4⋊(C79⋊C8)—21no—
100C4⋊(C79⋊C8)C2311yescharacteristic
101C4⋊(C79⋊C8)C2311yescharacteristic
102Dic79⋊C8—21no—
103Dic79⋊C8—21no—
104Dic79⋊C8—21no—
105Dic79⋊C8D411yescharacteristic
106Dic79⋊C8D411yescharacteristic
107#14579086C2×C411yescharacteristic
108#14579085C2×C411yescharacteristic
109#14579085C2×C411yescharacteristic
110#14579084C2×C411yescharacteristic
111#14579083C2×C411yescharacteristic
112#14579081C2×C411yescharacteristic
113(C2×D79)⋊C8—21no—
114(C2×D79)⋊C8—21no—
115(C2×D79)⋊C8—21no—
116(C2×D79)⋊C8D411yescharacteristic
117(C2×D79)⋊C8D411yescharacteristic
118#14579071—21no—
119#14579071—21no—
120#14579071C2×C411yescharacteristic
121#14579071C2×C411yescharacteristic
122C4×C79⋊C8—21no—
123C4×C79⋊C8—21no—
124C4×C79⋊C8C2311yescharacteristic
125C4×C79⋊C8C2311yescharacteristic
126C4×C79⋊C8C2311yescharacteristic
127C8×Dic79—21no—
128C8×Dic79—21no—
129C2×Q16×C79C2×C411yescharacteristic
130C4×Q8×C79C2311yescharacteristic
131C4×Q8×C79C2311yescharacteristic
132#107×C79C2×C411yescharacteristic
133C4⋊C8×C79—21no—
134C4⋊C8×C79—21no—
135C4⋊C8×C79C2311yescharacteristic
136C4⋊C8×C79C2311yescharacteristic
137#103×C79C2×C411yescharacteristic
138#103×C79C2×C411yescharacteristic
139#97×C79—21no—
140#97×C79—21no—
141C4×C8×C79C2311yescharacteristic
142C4×C8×C79C2311yescharacteristic
143C4×C8×C79C2311yescharacteristic
144C2×Dic158—21no—
145C2×Dic158C22×C411yescharacteristic
146C2×D316—21no—
147C2×D316C22×C411yescharacteristic
148C2×C4×D79—21no—
149C2×C4×D79C22×C411yescharacteristic
150C2×C4×D79C2×D411yescharacteristic
151C2×C4×D79#4111yescharacteristic
152#11929614—22no—
153#11929613—22no—
154#11929611—22no—
155#11929607—22no—
156#11929607—22no—
157#11929607—81no—
158#11929607—81no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8—21no—
161C2×C79⋊C8—21no—
162C2×C79⋊C8C22×C411yescharacteristic
163C2×C79⋊C8C22×C411yescharacteristic
164C2×C79⋊C8C2×D411yescharacteristic
165C2×C79⋊C8C2×D411yescharacteristic
166C8×D79—42no—
167C8×D79—81no—
168C4⋊Dic79—21no—
169C4⋊Dic79—21no—
170C4⋊Dic79C22×C411yescharacteristic
171C4⋊Dic79C22×C411yescharacteristic
172Dic79⋊C4—21no—
173Dic79⋊C4—21no—
174Dic79⋊C4—21no—
175Dic79⋊C4—21no—
176Dic79⋊C4—21no—
177Dic79⋊C4—21no—
178Dic79⋊C4—21no—
179(C2×D79)⋊C4—21no—
180(C2×D79)⋊C4—21no—
181(C2×D79)⋊C4—21no—
182(C2×D79)⋊C4—21no—
183(C2×D79)⋊C4—21no—
184(C2×D79)⋊C4—21no—
185(C2×D79)⋊C4—21no—
186C4×Dic79—21no—
187C4×Dic79C22×C411yescharacteristic
188C4×Dic79C2×D411yescharacteristic
189C4×Dic79#4111yescharacteristic
190C2×Q8×C79C22×C411yescharacteristic
191C2×Q8×C79C22×C411yescharacteristic
192Q16×C79—22no—
193C2×C8×C79—21no—
194C2×C8×C79—21no—
195C2×C8×C79—21no—
196C2×C8×C79C22×C411yescharacteristic
197C2×C8×C79C22×C411yescharacteristic
198C2×C8×C79C2×D411yescharacteristic
199C2×C8×C79C2×D411yescharacteristic
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79C22×C411yescharacteristic
203C4⋊C4×C79C22×C411yescharacteristic
204C42×C79—21no—
205C42×C79—21no—
206C42×C79C2411yescharacteristic
207C22×D79—21no—
208C22×D79C4×D411yescharacteristic
209Q8×C79—21no—
210Q8×C79—21no—
211Q8×C79—21no—
212Q8×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79C23×C411yescharacteristic
218C2×C4×C79C22×D411yescharacteristic
219C2×C4×C79C2×#4111yescharacteristic
220C2×Dic79—21no—
221C2×Dic79—21no—
222C2×Dic79—21no—
223C2×Dic79C4×D411yescharacteristic
224C2×Dic79C4×D411yescharacteristic
225C2×Dic79C4×D411yescharacteristic
226D316—21no—
227D316—21no—
228D316—81no—
229C4×D79—21no—
230C4×D79—21no—
231C4×D79—21no—
232C4×D79—21no—
233C4×D79—41no—
234C4×D79—41no—
235C4×D79—41no—
236Dic158—21no—
237Dic158—21no—
238Dic158—81no—
239C8×C79—22no—
240C8×C79—21no—
241C8×C79—21no—
242C8×C79—21no—
243C8×C79—41no—
244C8×C79—41no—
245C8×C79#13112yes—
246C79⋊C8—21no—
247C79⋊C8—21no—
248C79⋊C8—21no—
249C79⋊C8—41no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C79⋊C8#13112yes—
253C22×C79C2×C4×D411yescharacteristic
254C2×D79—21no—
255C2×D79—21no—
256C2×D79—21no—
257C2×D79—41no—
258C2×D79—41no—
259C2×D79—41no—
260C4×C79—21no—
261C4×C79—41no—
262C4×C79—41no—
263C4×C79—41no—
264C4×C79—41no—
265C4×C79C2×#13111yescharacteristic
266C4×C79C2×#13111yescharacteristic
267C4×C79#57711yescharacteristic
268C4×C79#57811yescharacteristic
269Dic79—21no—
270Dic79—21no—
271Dic79—21no—
272Dic79—41no—
273Dic79—41no—
274Dic79—41no—
275#17969—791nomaximal, Sylow
276C2×C79#293111yescharacteristic
277C2×C79#294711yescharacteristic
278C2×C79#295611yescharacteristic
279D79—41no—
280D79—81no—
281#3172—791no—
282#2993—791no—
283#2985—791no—
284#1593—791no—
285#1592—791no—
286#1591D7911yescharacteristic, complemented
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#529—791no—
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—1581no—
309#446—1581no—
310#446C2×D7911yescharacteristic
311#446C2×D7911yescharacteristic
312C8×Q8—791no—
313C8×Q8—791no—
314C8×Q8—791no—
315C4×Q16—791no—
316C4×Q16C2×D7911yescharacteristic
317C4×SD16—791no—
318C4×SD16—791no—
319#436—791no—
320#436—1581no—
321#435—1581no—
322C8×D4—791no—
323#433—1581no—
324#433—1581no—
325#433—1581no—
326#432—791no—
327#431—791no—
328#424—1581no—
329#423—791no—
330C4×M16—791no—
331C2×C4×C8—791no—
332#335—791no—
333#335C2×D7911yescharacteristic
334#334—791no—
335#334—791no—
336Q8⋊C8—791no—
337Q8⋊C8—791no—
338Q8⋊C8—791no—
339Q8⋊C8—791no—
340Q8⋊C8C2×D7911yescharacteristic
341Q8⋊C8C2×D7911yescharacteristic
342D4⋊C8—791no—
343D4⋊C8—791no—
344#322—791no—
345#322—791no—
346C82—791no—
347C82C2×D7911yescharacteristic
348C2×Q16—791no—
349C2×Q16C4×D7911yescharacteristic
350C2×SD16—791no—
351C2×SD16—791no—
352C2×M16—791no—
353C2×M16—791no—
354C2×M16—1581no—
355C2×M16—1581no—
356C22×C8—1581no—
357C22×C8—1581no—
358C4⋊Q8—791no—
359C4⋊Q8—791no—
360#126—1581no—
361#125—791no—
362#125—1581no—
363#124—791no—
364#123—1581no—
365C22⋊Q8—791no—
366C22⋊Q8—791no—
367C22⋊Q8—791no—
368C22⋊Q8—1581no—
369#121—791no—
370C4×Q8—791no—
371C4×Q8—1581no—
372C4×Q8C22×D7911yescharacteristic
373C4×Q8C22×D7911yescharacteristic
374C4×D4—791no—
375C4×D4—1581no—
376#117—1581no—
377#117—1581no—
378C2×C42—791no—
379#107—791no—
380#107C4×D7911yescharacteristic
381#106—791no—
382#106—791no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—791no—
386C4⋊C8—791no—
387C4⋊C8—791no—
388C4⋊C8—791no—
389C4⋊C8—1581no—
390C4⋊C8—1581no—
391C4⋊C8—1581no—
392C4⋊C8—1581no—
393C4⋊C8—1581no—
394C4⋊C8C22×D7911yescharacteristic
395C4⋊C8C22×D7911yescharacteristic
396#103—791no—
397#103—791no—
398#103—791no—
399#103—791no—
400#103C4×D7911yescharacteristic
401#103C4×D7911yescharacteristic
402#102—791no—
403#102—791no—
404C22⋊C8—791no—
405C22⋊C8—791no—
406C22⋊C8—1581no—
407C22⋊C8—1581no—
408C22⋊C8—1581no—
409#97—21no—
410#97—21no—
411#97—791no—
412#97—791no—
413#97—1581no—
414#97—1581no—
415C4×C8—791no—
416C4×C8—791no—
417C4×C8—791no—
418C4×C8—1581no—
419C4×C8—1581no—
420C4×C8—1581no—
421C4×C8—1581no—
422C4×C8C22×D7911yescharacteristic
423C4×C8C22×D7911yescharacteristic
424C4×C8C22×D7911yescharacteristic
425C2×Q8—791no—
426C2×Q8—1581no—
427C2×Q8C2×C4×D7911yescharacteristic
428C2×Q8C2×C4×D7911yescharacteristic
429C2×D4—791no—
430C2×D4—1581no—
431C22×C4—791no—
432C22×C4—791no—
433C22×C4—791no—
434C22×C4—1581no—
435Q16—22no—
436Q16—1582no—
437SD16—1582no—
438SD16—1582no—
439M16—6321no—
440M16—6321no—
441M16—1582no—
442M16—1582no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—3162no—
447C2×C8—791no—
448C2×C8—791no—
449C2×C8—791no—
450C2×C8—791no—
451C2×C8—6321no—
452C2×C8—1581no—
453C2×C8—1581no—
454C2×C8—1581no—
455C2×C8C2×C4×D7911yescharacteristic
456C2×C8C2×C4×D7911yescharacteristic
457C2×C8D4×D7911yescharacteristic
458C2×C8D4×D7911yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—791no—
462C4⋊C4—791no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
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—1581no—
475C22⋊C4—1581no—
476C22⋊C4—1581no—
477C22⋊C4—1581no—
478C22⋊C4—1581no—
479C22⋊C4—1581no—
480C22⋊C4—1581no—
481C42—21no—
482C42—21no—
483C42—791no—
484C42—791no—
485C42—791no—
486C42—1581no—
487C42C23×D7911yescharacteristic
488C23—791no—
489C23—1581no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—6321no—
495Q8—1581no—
496Q8—1581no—
497D4—6321no—
498D4—1581no—
499D4—1581no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—3161no—
505C2×C4—3161no—
506C2×C4—3161no—
507C2×C4—791no—
508C2×C4—791no—
509C2×C4—791no—
510C2×C4—1581no—
511C2×C4—1581no—
512C2×C4—1581no—
513C2×C4—1581no—
514C2×C4—1581no—
515C2×C4—1581no—
516C2×C4—1581no—
517C2×C4C22×C4×D7911yescharacteristic
518C2×C4C2×D4×D7911yescharacteristic
519C2×C4#41×D7911yescharacteristic
520C8—22no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—3161no—
527C8—3161no—
528C8—3161no—
529C8—792no—
530C8—1581no—
531C8—1581no—
532C8—1581no—
533C8#1457916912yes—
534C22—3161no—
535C22—3161no—
536C22—3161no—
537C22—1581no—
538C22—1581no—
539C22—1581no—
540C22C4×D4×D7911yescharacteristic
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—3161no—
547C4—3161no—
548C4—3161no—
549C4—1581no—
550C4—1581no—
551C4—1581no—
552C4#131×D7911yescharacteristic
553C4C2×#1457916911yescharacteristic
554C4#1909937011yescharacteristic
555C4#1909937611yescharacteristic
556C2—3161no—
557C2—6321no—
558C2#2377210911yescharacteristic
559C2#2377223811yescharacteristic
560C2#2377230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1591C2
socleC22×C79C2×C4×D4
prime radical#14579136C2×C4
cosocleC42×C79C24
hypercenter#1591D79
hypocenterC79#17969
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1591

Complements

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

NH ≅ G/Ntype
3. #23772496C2semidirect
4. #23762307C2semidirect
5. #23762300C2semidirect
6. #23762292C2semidirect
7. #23762219C2semidirect
8. #23762211C2semidirect
9. #23762204C2semidirect
10. #23762197C2semidirect
11. #23762166C2semidirect
13. #23761853C2semidirect
14. C8×#11929607C2semidirect
15. C79×#1591C2semidirect
286. #1591D79semidirect
296. C79#17969semidirect

Maximal subgroups

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

subgroupindex
#237740042
#237725522
#237724962
#237623072
#237623002
#237622922
#237622192
#237622112
#237622042
#237621972
#237621662
#237618592
#237618532
C8×#119296072
C79×#15912
#1796979