Finite Group Databasebeta

← #31968095

Subgroups — #31968095

Order 18176.

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#22598946C211yesmaximal, characteristic
2#22597485C211yesmaximal, characteristic
3#22597437C211yesmaximal, characteristic, complemented
4#22587248C211yesmaximal, characteristic, complemented
5#22587241C211yesmaximal, characteristic, complemented
6#22587236C211yesmaximal, characteristic, complemented
7#22587229C211yesmaximal, characteristic, complemented
8#22587158C211yesmaximal, characteristic, complemented
9#22587151C211yesmaximal, characteristic, complemented
10#22587146C211yesmaximal, characteristic, complemented
11#22587139C211yesmaximal, characteristic, complemented
12#22586798C211yesmaximal, characteristic
13#22586794C211yesmaximal, characteristic, complemented
14M16×C71⋊C8C211yesmaximal, characteristic, complemented
15C71×#1569C211yesmaximal, characteristic, complemented
16#17986556C2211yescharacteristic
17#17986533C2211yescharacteristic
18#17986408C411yescharacteristic
19#17986393C411yescharacteristic
20#17986368C411yescharacteristic
21#17986359C411yescharacteristic
22#17986257C411yescharacteristic
23#17986257C411yescharacteristic
24#17986245C411yescharacteristic
25#17986245C411yescharacteristic
26#17986127—21no—
27#17986127—21no—
28#17986126C2211yescharacteristic
29#17986126C2211yescharacteristic
30#17986126C2211yescharacteristic
31#17986123C2211yescharacteristic
32#17986123C2211yescharacteristic
33#17986123C2211yescharacteristic
34C4×Dic284C2211yescharacteristic
35C4×Dic284C2211yescharacteristic
36C4×#11765854C2211yescharacteristic
37C4×#11765854C2211yescharacteristic
38#17986076C2211yescharacteristic
39#17986075—21no—
40#17986068—21no—
41#17986064C2211yescharacteristic
42#17986059—21no—
43#17986057—21no—
44#17986057—21no—
45#17986052C2211yescharacteristic
46#17986024—21no—
47C2×C4×C71⋊C8C2211yescharacteristic
48#17985729C2211yescharacteristic
49#17985729C2211yescharacteristic
50#17985729C2211yescharacteristic
51#17985729C2211yescharacteristic
52#17985729C2211yescharacteristic
53#17985729C2211yescharacteristic
54#17985726C2211yescharacteristic
55#17985726C2211yescharacteristic
56#17985720C2211yescharacteristic
57#17985720C2211yescharacteristic
58C8×C71⋊C8C2211yescharacteristic
59C8×C71⋊C8C2211yescharacteristic
60#432×C71C2211yescharacteristic
61#423×C71C2211yescharacteristic
62C4×M16×C71C2211yescharacteristic
63#335×C71C2211yescharacteristic
64#335×C71C2211yescharacteristic
65#334×C71C2211yescharacteristic
66#334×C71C2211yescharacteristic
67C2×Dic284C2×C411yescharacteristic
68C2×Dic284C2×C411yescharacteristic
69C2×#11765854C2×C411yescharacteristic
70C2×#11765854C2×C411yescharacteristic
71C2×#11765848—21no—
72C22×C71⋊C8—21no—
73C2×C8×D71—21no—
74#14556128C2×C411yescharacteristic
75#14556128C2×C411yescharacteristic
76#14556122—21no—
77#14556120—21no—
78#14556120C2×C411yescharacteristic
79#14556116C2×C411yescharacteristic
80#14556111—21no—
81#14556107—21no—
82#14556107C2×C411yescharacteristic
83#14556107C2×C411yescharacteristic
84#14556107C2×C411yescharacteristic
85#14556102C2×C411yescharacteristic
86C4×Dic142—21no—
87C4×Dic142C2311yescharacteristic
88C4×Dic142C2311yescharacteristic
89C4×Dic142C2311yescharacteristic
90C4×D284C2311yescharacteristic
91C4×C22⋊D71—21no—
92#14556087—21no—
93#14556085—21no—
94#14556062C2×C411yescharacteristic
95#14556062C2×C411yescharacteristic
96#14556060C2×C411yescharacteristic
97#14556060C2×C411yescharacteristic
98C4⋊(C71⋊C8)D411yescharacteristic
99C4⋊(C71⋊C8)D411yescharacteristic
100Dic71⋊C8—21no—
101Dic71⋊C8—21no—
102Dic71⋊C8—21no—
103Dic71⋊C8—21no—
104Dic71⋊C8—21no—
105Dic71⋊C8—21no—
106#14556053C2×C411yescharacteristic
107#14556053C2×C411yescharacteristic
108#14556053C2×C411yescharacteristic
109#14556053C2×C411yescharacteristic
110#14556053C2×C411yescharacteristic
111#14556053C2×C411yescharacteristic
112#14556050C2×C411yescharacteristic
113#14556050C2×C411yescharacteristic
114C22⋊(C71⋊C8)D411yescharacteristic
115C22⋊(C71⋊C8)D411yescharacteristic
116(C2×D71)⋊C8—21no—
117(C2×D71)⋊C8—21no—
118#14556038—21no—
119#14556038—21no—
120#14556038—21no—
121C4×C71⋊C8—21no—
122C4×C71⋊C8C2311yescharacteristic
123C4×C71⋊C8C2311yescharacteristic
124C4×C71⋊C8C2311yescharacteristic
125C4×C71⋊C8C2311yescharacteristic
126C8×Dic71—21no—
127C8×Dic71—21no—
128C8×Dic71—21no—
129C2×M16×C71—21no—
130C2×M16×C71C2×C411yescharacteristic
131C2×M16×C71C2×C411yescharacteristic
132C2×C42×C71C2311yescharacteristic
133C4⋊C8×C71—21no—
134C4⋊C8×C71C2311yescharacteristic
135C4⋊C8×C71C2311yescharacteristic
136C4⋊C8×C71C2311yescharacteristic
137C4⋊C8×C71C2311yescharacteristic
138C22⋊C8×C71—21no—
139#97×C71—21no—
140#97×C71C2×C411yescharacteristic
141#97×C71C2×C411yescharacteristic
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142—21no—
145C2×Dic142C22×C411yescharacteristic
146C2×Dic142C22×C411yescharacteristic
147C2×Dic142C22×C411yescharacteristic
148C2×D284C22×C411yescharacteristic
149C2×C22⋊D71—21no—
150C2×C4×D71—21no—
151Dic284—22no—
152Dic284—22no—
153#11765854—22no—
154#11765854—22no—
155#11765848—81no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—42no—
159C2×C71⋊C8C2×D411yescharacteristic
160C2×C71⋊C8C2×D411yescharacteristic
161C2×C71⋊C8C2×D411yescharacteristic
162C2×C71⋊C8C2×D411yescharacteristic
163C8×D71—81no—
164C4⋊Dic71—21no—
165C4⋊Dic71—21no—
166C4⋊Dic71—21no—
167C4⋊Dic71C22×C411yescharacteristic
168C4⋊Dic71C22×C411yescharacteristic
169C4⋊Dic71C22×C411yescharacteristic
170C4⋊Dic71C22×C411yescharacteristic
171Dic71⋊C4—21no—
172Dic71⋊C4—21no—
173Dic71⋊C4—21no—
174Dic71⋊C4—21no—
175Dic71⋊C4—21no—
176Dic71⋊C4—21no—
177Dic71⋊C4—21no—
178Dic71⋊C4—21no—
179C22⋊Dic71—21no—
180C22⋊Dic71—21no—
181C22⋊Dic71—21no—
182(C2×D71)⋊C4—21no—
183(C2×D71)⋊C4—21no—
184(C2×D71)⋊C4—21no—
185(C2×D71)⋊C4—21no—
186C4×Dic71—21no—
187C4×Dic71—21no—
188C4×Dic71—21no—
189C22×C4×C71C22×C411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71#4111yescharacteristic
192M16×C71—22no—
193M16×C71—22no—
194M16×C71—81no—
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71—21no—
198C2×C8×C71—21no—
199C2×C8×C71C22×C411yescharacteristic
200C2×C8×C71C22×C411yescharacteristic
201C2×C8×C71C22×C411yescharacteristic
202C2×C8×C71C22×C411yescharacteristic
203C42×C71C22×C411yescharacteristic
204C42×C71C2×D411yescharacteristic
205C42×C71#4111yescharacteristic
206C42×C71C2411yescharacteristic
207C23×C71C4×D411yescharacteristic
208C22×D71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—41no—
214C2×C4×C71C4×D411yescharacteristic
215C2×C4×C71C4×D411yescharacteristic
216C2×C4×C71C4×D411yescharacteristic
217C2×C4×C71C23×C411yescharacteristic
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C2×#4111yescharacteristic
220C22⋊D71—81no—
221C2×Dic71—21no—
222C2×Dic71—21no—
223C2×Dic71—21no—
224C2×Dic71—21no—
225C2×Dic71—21no—
226C2×Dic71—21no—
227C2×Dic71—21no—
228D284—21no—
229D284—21no—
230C4×D71—41no—
231C4×D71—41no—
232Dic142—21no—
233Dic142—21no—
234Dic142—21no—
235Dic142—21no—
236Dic142—21no—
237Dic142—21no—
238Dic142—81no—
239C8×C71—21no—
240C8×C71—21no—
241C8×C71—41no—
242C8×C71—41no—
243C8×C71—41no—
244C8×C71—41no—
245C8×C71#13112yes—
246C8×C71#13112yes—
247C71⋊C8—22no—
248C71⋊C8—21no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—41no—
253C22×C71—21no—
254C22×C71—21no—
255C22×C71—21no—
256C22×C71C2×C4×D411yescharacteristic
257C2×D71—41no—
258C2×D71—41no—
259C2×D71—41no—
260C4×C71—21no—
261C4×C71—21no—
262C4×C71—21no—
263C4×C71—21no—
264C4×C71C2×#13111yescharacteristic
265C4×C71C2×#13111yescharacteristic
266C4×C71#57711yescharacteristic
267C4×C71#57811yescharacteristic
268Dic71—41no—
269Dic71—41no—
270Dic71—41no—
271Dic71—41no—
272Dic71—41no—
273Dic71—41no—
274Dic71—41no—
275#17969—711nomaximal, Sylow
276C2×C71—41no—
277C2×C71#293111yescharacteristic
278C2×C71#294711yescharacteristic
279C2×C71#295611yescharacteristic
280D71—81no—
281#3172—711no—
282#2993—711no—
283#2985—711no—
284#1593—711no—
285#1592—711no—
286#1591—711no—
287#1590—711no—
288#1579—711no—
289SD16⋊C8—711no—
290C8×Q16—711no—
291C8×SD16—711no—
292#1569D7111yescharacteristic, complemented
293#1494—711no—
294#1493—711no—
295C8×M16—711no—
296C71#1796911yescharacteristic, complemented, Sylow
297#533—711no—
298#529—711no—
299#501—711no—
300#497—711no—
301#491—711no—
302#489—711no—
303#470—711no—
304#470—711no—
305#468—711no—
306#468—711no—
307#446—711no—
308#446—711no—
309#446—711no—
310#446—1421no—
311#446—1421no—
312C8×Q8—711no—
313C8×Q8—711no—
314C8×Q8—711no—
315C4×Q16—711no—
316C4×Q16—711no—
317C4×SD16—711no—
318C4×SD16—711no—
319#436—711no—
320#436—1421no—
321#435—1421no—
322C8×D4—711no—
323#433—1421no—
324#433—1421no—
325#433—1421no—
326#432C2×D7111yescharacteristic
327#431—711no—
328#424—1421no—
329#423C2×D7111yescharacteristic
330C4×M16C2×D7111yescharacteristic
331C2×C4×C8—711no—
332#335C2×D7111yescharacteristic
333#335C2×D7111yescharacteristic
334#334C2×D7111yescharacteristic
335#334C2×D7111yescharacteristic
336Q8⋊C8—711no—
337Q8⋊C8—711no—
338Q8⋊C8—711no—
339Q8⋊C8—711no—
340Q8⋊C8—711no—
341Q8⋊C8—711no—
342D4⋊C8—711no—
343D4⋊C8—711no—
344#322—711no—
345#322—711no—
346C82—711no—
347C82—711no—
348C2×Q16—711no—
349C2×Q16—711no—
350C2×SD16—711no—
351C2×SD16—711no—
352C2×M16—21no—
353C2×M16—1421no—
354C2×M16C4×D7111yescharacteristic
355C2×M16C4×D7111yescharacteristic
356C22×C8—1421no—
357C22×C8—1421no—
358C4⋊Q8—711no—
359C4⋊Q8—711no—
360#126—1421no—
361#125—711no—
362#125—1421no—
363#124—711no—
364#123—1421no—
365C22⋊Q8—711no—
366C22⋊Q8—711no—
367C22⋊Q8—711no—
368C22⋊Q8—1421no—
369#121—711no—
370C4×Q8—711no—
371C4×Q8—711no—
372C4×Q8—711no—
373C4×Q8—1421no—
374C4×D4—711no—
375C4×D4—1421no—
376#117—1421no—
377#117—1421no—
378C2×C42C22×D7111yescharacteristic
379#107—711no—
380#107—711no—
381#106—711no—
382#106—711no—
383C4⋊C8—21no—
384C4⋊C8—711no—
385C4⋊C8—711no—
386C4⋊C8—1421no—
387C4⋊C8—1421no—
388C4⋊C8—1421no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8C22×D7111yescharacteristic
393C4⋊C8C22×D7111yescharacteristic
394C4⋊C8C22×D7111yescharacteristic
395C4⋊C8C22×D7111yescharacteristic
396#103—711no—
397#103—711no—
398#103—711no—
399#103—711no—
400#103—711no—
401#103—711no—
402#102—711no—
403#102—711no—
404C22⋊C8—21no—
405C22⋊C8—711no—
406C22⋊C8—711no—
407C22⋊C8—1421no—
408C22⋊C8—1421no—
409#97—21no—
410#97—1421no—
411#97—1421no—
412#97—1421no—
413#97C4×D7111yescharacteristic
414#97C4×D7111yescharacteristic
415C4×C8—711no—
416C4×C8—711no—
417C4×C8—711no—
418C4×C8—711no—
419C4×C8—1421no—
420C4×C8—1421no—
421C4×C8—1421no—
422C4×C8—1421no—
423C4×C8C22×D7111yescharacteristic
424C4×C8C22×D7111yescharacteristic
425C2×Q8—711no—
426C2×Q8—711no—
427C2×Q8—711no—
428C2×Q8—1421no—
429C2×D4—711no—
430C2×D4—1421no—
431C22×C4—1421no—
432C22×C4C2×C4×D7111yescharacteristic
433C22×C4D4×D7111yescharacteristic
434C22×C4#1176586611yescharacteristic
435Q16—1422no—
436Q16—1422no—
437SD16—1422no—
438SD16—1422no—
439M16—22no—
440M16—22no—
441M16—81no—
442M16—5681no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—2842no—
448C2×C8—5681no—
449C2×C8—711no—
450C2×C8—711no—
451C2×C8—711no—
452C2×C8—711no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8C2×C4×D7111yescharacteristic
456C2×C8C2×C4×D7111yescharacteristic
457C2×C8C2×C4×D7111yescharacteristic
458C2×C8C2×C4×D7111yescharacteristic
459C4⋊C4—711no—
460C4⋊C4—711no—
461C4⋊C4—711no—
462C4⋊C4—711no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C22⋊C4—1421no—
475C22⋊C4—1421no—
476C22⋊C4—1421no—
477C22⋊C4—1421no—
478C22⋊C4—1421no—
479C22⋊C4—1421no—
480C22⋊C4—1421no—
481C42—1421no—
482C42—1421no—
483C42—1421no—
484C42C2×C4×D7111yescharacteristic
485C42D4×D7111yescharacteristic
486C42#1176586611yescharacteristic
487C42C23×D7111yescharacteristic
488C23—1421no—
489C23#1455609011yescharacteristic
490Q8—5681no—
491Q8—1421no—
492Q8—1421no—
493Q8—1421no—
494Q8—1421no—
495Q8—1421no—
496Q8—1421no—
497D4—5681no—
498D4—1421no—
499D4—1421no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—2841no—
506C2×C4—2841no—
507C2×C4—1421no—
508C2×C4—1421no—
509C2×C4—1421no—
510C2×C4—1421no—
511C2×C4—1421no—
512C2×C4—1421no—
513C2×C4—1421no—
514C2×C4#1455609011yescharacteristic
515C2×C4#1455609011yescharacteristic
516C2×C4#1455609011yescharacteristic
517C2×C4C22×C4×D7111yescharacteristic
518C2×C4C2×D4×D7111yescharacteristic
519C2×C4C2×#1176586611yescharacteristic
520C8—21no—
521C8—21no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—2841no—
527C8—1422no—
528C8—1421no—
529C8—1421no—
530C8—1421no—
531C8—1421no—
532C8#1455613612yes—
533C8#1455613612yes—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2841no—
538C22—2841no—
539C22—2841no—
540C22C2×#1455609011yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—2841no—
546C4—2841no—
547C4—2841no—
548C4—2841no—
549C4—2841no—
550C4—2841no—
551C4—2841no—
552C4C2×#1455613611yescharacteristic
553C4C2×#1455613611yescharacteristic
554C4#1798680911yescharacteristic
555C4#1798681511yescharacteristic
556C2—41no—
557C2—5681no—
558C2#2259705111yescharacteristic
559C2#2259718111yescharacteristic
560C2#2259724911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1569C2
socleC22×C71C2×C4×D4
prime radical#14556102C2×C4
cosocleC42×C71C24
hypercenter#1569D71
hypocenterC71#17969
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
3. #22597437C2semidirect
4. #22587248C2semidirect
5. #22587241C2semidirect
6. #22587236C2semidirect
7. #22587229C2semidirect
8. #22587158C2semidirect
9. #22587151C2semidirect
10. #22587146C2semidirect
11. #22587139C2semidirect
13. #22586794C2semidirect
14. M16×C71⋊C8C2semidirect
15. C71×#1569C2semidirect
292. #1569D71semidirect
296. C71#17969semidirect

Maximal subgroups

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

subgroupindex
#225989462
#225974852
#225974372
#225872482
#225872412
#225872362
#225872292
#225871582
#225871512
#225871462
#225871392
#225867982
#225867942
M16×C71⋊C82
C71×#15692
#1796971