Finite Group Databasebeta

← #31968865

Subgroups — #31968865

Order 18176.

598 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#22598360C211yesmaximal, characteristic
2#22597502C211yesmaximal, characteristic, complemented
3#22597469C211yesmaximal, characteristic, complemented
4#22587244C211yesmaximal, characteristic, complemented
5#22587226C211yesmaximal, characteristic, complemented
6#22587220C211yesmaximal, characteristic, complemented
7#22587220C211yesmaximal, characteristic, complemented
8#22587192C211yesmaximal, characteristic, complemented
9#22587171C211yesmaximal, characteristic, complemented
10#22587154C211yesmaximal, characteristic, complemented
11C8×Dic284C211yesmaximal, characteristic, complemented
12#22587115C211yesmaximal, characteristic, complemented
13#22586806C211yesmaximal, characteristic, complemented
14#22586728C211yesmaximal, characteristic, complemented
15C71×#1454C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17C4×#11765865C2211yescharacteristic
18#17986341C411yescharacteristic
19#17986324C411yescharacteristic
20#17986307C411yescharacteristic
21#17986300C411yescharacteristic
22#17986222C411yescharacteristic
23#17986215C411yescharacteristic
24#17986163C411yescharacteristic
25#17986146C411yescharacteristic
26#17986129C2211yescharacteristic
27C8×Dic142C2211yescharacteristic
28C8×Dic142C2211yescharacteristic
29C8×Dic142C2211yescharacteristic
30C4×Dic284C2211yescharacteristic
31C4×#11765854C2211yescharacteristic
32C4×#11765854C2211yescharacteristic
33C4×D568C2211yescharacteristic
34#17986078C2211yescharacteristic
35#17986078C2211yescharacteristic
36#17986078C2211yescharacteristic
37#17986074—21no—
38#17986074—21no—
39#17986067—21no—
40C8×D284C2211yescharacteristic
41(C22⋊D71)⋊C8—21no—
42#17986059—21no—
43#17986051—21no—
44#17986024—21no—
45#17986023C2211yescharacteristic
46#17986021—21no—
47C2×C4⋊(C71⋊C8)C2211yescharacteristic
48#17985752C2211yescharacteristic
49#17985752C2211yescharacteristic
50#17985752C2211yescharacteristic
51#17985752C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985731C2211yescharacteristic
55#17985731C2211yescharacteristic
56#17985728C2211yescharacteristic
57#17985728C2211yescharacteristic
58#17985728C2211yescharacteristic
59#17985728C2211yescharacteristic
60#431×C71C2211yescharacteristic
61#431×C71C2211yescharacteristic
62#431×C71C2211yescharacteristic
63#322×C71C2211yescharacteristic
64#322×C71C2211yescharacteristic
65#322×C71C2211yescharacteristic
66C82×C71C2211yescharacteristic
67C2×#11765865C2×C411yescharacteristic
68C2×Dic284C2×C411yescharacteristic
69C2×#11765854C2×C411yescharacteristic
70C2×#11765854C2×C411yescharacteristic
71C2×D568C2×C411yescharacteristic
72C2×#11765849—21no—
73C2×#11765848—21no—
74C2×#11765848—21no—
75C2×#11765848—21no—
76C22×C71⋊C8—21no—
77C2×C8×D71—21no—
78#14556122—21no—
79#14556120C2×C411yescharacteristic
80#14556116C2×C411yescharacteristic
81#14556111—21no—
82#14556107C2×C411yescharacteristic
83#14556102C2×C411yescharacteristic
84C4×Dic142C2×C411yescharacteristic
85C4×Dic142C2311yescharacteristic
86C4×Dic142C2311yescharacteristic
87C4×D284C2×C411yescharacteristic
88C4×D284C2311yescharacteristic
89C4×D284C2311yescharacteristic
90C4×C22⋊D71—21no—
91C4×C22⋊D71—21no—
92C4×C22⋊D71—21no—
93#14556087C2×C411yescharacteristic
94#14556085—21no—
95#14556085—21no—
96C42×D71—21no—
97#14556062C2×C411yescharacteristic
98#14556062C2×C411yescharacteristic
99#14556060C2×C411yescharacteristic
100#14556060C2×C411yescharacteristic
101C4⋊(C71⋊C8)—21no—
102C4⋊(C71⋊C8)—21no—
103C4⋊(C71⋊C8)C2311yescharacteristic
104C4⋊(C71⋊C8)C2311yescharacteristic
105C4⋊(C71⋊C8)C2311yescharacteristic
106C4⋊(C71⋊C8)C2311yescharacteristic
107Dic71⋊C8—21no—
108Dic71⋊C8—21no—
109Dic71⋊C8—21no—
110Dic71⋊C8—21no—
111#14556053C2×C411yescharacteristic
112#14556053C2×C411yescharacteristic
113#14556053C2×C411yescharacteristic
114#14556053C2×C411yescharacteristic
115#14556050C2×C411yescharacteristic
116#14556050C2×C411yescharacteristic
117#14556050C2×C411yescharacteristic
118#14556050C2×C411yescharacteristic
119(C2×D71)⋊C8—21no—
120(C2×D71)⋊C8—21no—
121(C2×D71)⋊C8—21no—
122(C2×D71)⋊C8—21no—
123#14556038—21no—
124C8×Dic71—21no—
125C8×Dic71—21no—
126C8×Dic71—21no—
127C2×C42×C71C2311yescharacteristic
128C4⋊C8×C71—21no—
129C4⋊C8×C71C2×C411yescharacteristic
130C4⋊C8×C71C2×C411yescharacteristic
131C4⋊C8×C71D411yescharacteristic
132C4⋊C8×C71D411yescharacteristic
133C22⋊C8×C71—21no—
134C22⋊C8×C71C2×C411yescharacteristic
135C22⋊C8×C71C2×C411yescharacteristic
136C22⋊C8×C71D411yescharacteristic
137C22⋊C8×C71D411yescharacteristic
138C4×C8×C71C2311yescharacteristic
139C4×C8×C71C2311yescharacteristic
140C4×C8×C71C2311yescharacteristic
141C4×C8×C71C2311yescharacteristic
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#11765865—22no—
145#11765865—41no—
146C2×Dic142C22×C411yescharacteristic
147C2×Dic142C22×C411yescharacteristic
148C2×D284C22×C411yescharacteristic
149C2×D284C22×C411yescharacteristic
150C2×C22⋊D71—21no—
151C2×C22⋊D71—21no—
152C2×C4×D71—21no—
153C2×C4×D71—21no—
154C2×C4×D71—21no—
155C2×C4×D71—21no—
156Dic284—22no—
157#11765854—22no—
158#11765854—41no—
159D568—41no—
160#11765849—81no—
161#11765848—81no—
162#11765848—81no—
163#11765848—81no—
164C2×C71⋊C8—21no—
165C2×C71⋊C8—21no—
166C2×C71⋊C8—21no—
167C2×C71⋊C8—21no—
168C2×C71⋊C8—81no—
169C8×D71—81no—
170C4⋊Dic71C22×C411yescharacteristic
171C4⋊Dic71C22×C411yescharacteristic
172C4⋊Dic71C22×C411yescharacteristic
173C4⋊Dic71C22×C411yescharacteristic
174Dic71⋊C4—21no—
175Dic71⋊C4—21no—
176Dic71⋊C4—21no—
177Dic71⋊C4—21no—
178Dic71⋊C4—21no—
179Dic71⋊C4—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—
186(C2×D71)⋊C4—21no—
187(C2×D71)⋊C4—21no—
188C4×Dic71—21no—
189C4×Dic71—21no—
190C4×Dic71—21no—
191C4×Dic71—21no—
192C22×C4×C71C22×C411yescharacteristic
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71#4111yescharacteristic
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71C22×C411yescharacteristic
198C2×C8×C71C22×C411yescharacteristic
199C2×C8×C71C22×C411yescharacteristic
200C2×C8×C71C22×C411yescharacteristic
201C2×C8×C71C2×D411yescharacteristic
202C2×C8×C71C2×D411yescharacteristic
203C2×C8×C71C2×D411yescharacteristic
204C2×C8×C71C2×D411yescharacteristic
205C42×C71C22×C411yescharacteristic
206C42×C71C2×D411yescharacteristic
207C42×C71#4111yescharacteristic
208C42×C71C2411yescharacteristic
209C23×C71C4×D411yescharacteristic
210C22×D71—21no—
211C22×D71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—41no—
217C2×C4×C71C4×D411yescharacteristic
218C2×C4×C71C4×D411yescharacteristic
219C2×C4×C71C4×D411yescharacteristic
220C2×C4×C71C23×C411yescharacteristic
221C2×C4×C71C22×D411yescharacteristic
222C2×C4×C71C2×#4111yescharacteristic
223C22⋊D71—42no—
224C22⋊D71—42no—
225C2×Dic71—21no—
226C2×Dic71—21no—
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231D284—21no—
232D284—21no—
233D284—21no—
234D284—21no—
235C4×D71—42no—
236C4×D71—42no—
237C4×D71—41no—
238C4×D71—41no—
239C4×D71—41no—
240C4×D71—41no—
241Dic142—21no—
242Dic142—21no—
243Dic142—21no—
244Dic142#13112yes—
245C8×C71—21no—
246C8×C71—21no—
247C8×C71—21no—
248C8×C71—21no—
249C8×C71—21no—
250C8×C71—21no—
251C8×C71—21no—
252C8×C71—21no—
253C8×C71—41no—
254C8×C71—41no—
255C71⋊C8—41no—
256C71⋊C8—41no—
257C71⋊C8—41no—
258C71⋊C8—41no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71C2×C4×D411yescharacteristic
263C2×D71—41no—
264C2×D71—41no—
265C2×D71—41no—
266C2×D71—41no—
267C2×D71—41no—
268C2×D71—41no—
269C4×C71—21no—
270C4×C71—21no—
271C4×C71—21no—
272C4×C71—21no—
273C4×C71C2×#13111yescharacteristic
274C4×C71#56811yescharacteristic
275C4×C71#57611yescharacteristic
276C4×C71#57811yescharacteristic
277Dic71—22no—
278Dic71—41no—
279Dic71—41no—
280Dic71—41no—
281Dic71—41no—
282Dic71—41no—
283#18033—711nomaximal, Sylow
284C2×C71—41no—
285C2×C71#293311yescharacteristic
286C2×C71#295111yescharacteristic
287C2×C71#295511yescharacteristic
288D71—81no—
289D71—81no—
290#3102—711no—
291#2994—711no—
292#2991—711no—
293#1592—711no—
294#1589—711no—
295#1588—711no—
296#1588—711no—
297#1584—711no—
298#1581—711no—
299SD16⋊C8—711no—
300C8×Q16—711no—
301#1571—711no—
302#1495—711no—
303#1477—711no—
304#1454D7111yescharacteristic, complemented
305C71#1803311yescharacteristic, complemented, Sylow
306#529—711no—
307C4×#41—711no—
308#485—711no—
309#482—711no—
310#479—711no—
311#478—711no—
312#464—711no—
313#463—711no—
314#452—711no—
315#449—711no—
316#446—711no—
317C8×Q8—711no—
318C8×Q8—711no—
319C8×Q8—711no—
320C4×Q16—711no—
321C4×SD16—711no—
322C4×SD16—711no—
323C4×D8—711no—
324#436—711no—
325#436—711no—
326#436—711no—
327#436—1421no—
328#436—1421no—
329#435—1421no—
330C8×D4—711no—
331C8×D4—1421no—
332#433—1421no—
333#431—1421no—
334#431C2×D7111yescharacteristic
335#431C2×D7111yescharacteristic
336#431C2×D7111yescharacteristic
337#424—1421no—
338#423—711no—
339#423—1421no—
340C2×C4⋊C8—711no—
341#335—711no—
342#335—711no—
343#335—711no—
344#335—711no—
345Q8⋊C8—711no—
346Q8⋊C8—711no—
347Q8⋊C8—711no—
348Q8⋊C8—711no—
349D4⋊C8—711no—
350D4⋊C8—711no—
351D4⋊C8—711no—
352D4⋊C8—711no—
353#322C2×D7111yescharacteristic
354#322C2×D7111yescharacteristic
355#322C2×D7111yescharacteristic
356C82C2×D7111yescharacteristic
357C2×#41—711no—
358C2×Q16—711no—
359C2×SD16—711no—
360C2×SD16—711no—
361C2×D8—711no—
362C2×M16—1421no—
363C2×M16—1421no—
364C2×M16—1421no—
365C2×M16—1421no—
366C22×C8—1421no—
367C22×C8—1421no—
368#126—1421no—
369#125—711no—
370#124—711no—
371#123—1421no—
372C22⋊Q8—711no—
373#121—711no—
374C4×Q8—711no—
375C4×Q8—711no—
376C4×Q8—711no—
377C4×D4—711no—
378C4×D4—711no—
379C4×D4—711no—
380C4×D4—1421no—
381C4×D4—1421no—
382C4×D4—1421no—
383#117—711no—
384#117—1421no—
385#117—1421no—
386C2×C42—1421no—
387C2×C42C22×D7111yescharacteristic
388#107—711no—
389#107—711no—
390#106—711no—
391#106—711no—
392C4⋊C8—21no—
393C4⋊C8—711no—
394C4⋊C8—711no—
395C4⋊C8—711no—
396C4⋊C8—711no—
397C4⋊C8—1421no—
398C4⋊C8—1421no—
399C4⋊C8—1421no—
400C4⋊C8—1421no—
401C4⋊C8—1421no—
402C4⋊C8—1421no—
403C4⋊C8C4×D7111yescharacteristic
404C4⋊C8C4×D7111yescharacteristic
405C4⋊C8D28411yescharacteristic
406C4⋊C8D28411yescharacteristic
407#103—711no—
408#103—711no—
409#103—711no—
410#103—711no—
411#102—711no—
412#102—711no—
413#102—711no—
414#102—711no—
415C22⋊C8—21no—
416C22⋊C8—1421no—
417C22⋊C8—1421no—
418C22⋊C8—1421no—
419C22⋊C8—1421no—
420C22⋊C8C4×D7111yescharacteristic
421C22⋊C8C4×D7111yescharacteristic
422C22⋊C8D28411yescharacteristic
423C22⋊C8D28411yescharacteristic
424#97—1421no—
425C4×C8—1421no—
426C4×C8—1421no—
427C4×C8—1421no—
428C4×C8C22×D7111yescharacteristic
429C4×C8C22×D7111yescharacteristic
430C4×C8C22×D7111yescharacteristic
431C4×C8C22×D7111yescharacteristic
432C4×C8C22×D7111yescharacteristic
433C4×C8C22×D7111yescharacteristic
434#41—2841no—
435#41—1422no—
436C2×Q8—711no—
437C2×Q8—711no—
438C2×D4—711no—
439C2×D4—711no—
440C2×D4—1421no—
441C2×D4—1421no—
442C22×C4—1421no—
443C22×C4—1421no—
444C22×C4—1421no—
445C22×C4—1421no—
446C22×C4C2×C4×D7111yescharacteristic
447C22×C4C2×D28411yescharacteristic
448C22×C4#1176586511yescharacteristic
449Q16—1422no—
450SD16—2841no—
451SD16—1422no—
452D8—2841no—
453M16—5681no—
454M16—5681no—
455M16—5681no—
456M16—5681no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—5681no—
460C2×C8—5681no—
461C2×C8—1421no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8C2×C4×D7111yescharacteristic
466C2×C8C2×C4×D7111yescharacteristic
467C2×C8C2×C4×D7111yescharacteristic
468C2×C8C2×C4×D7111yescharacteristic
469C2×C8C2×D28411yescharacteristic
470C2×C8C2×D28411yescharacteristic
471C2×C8C2×D28411yescharacteristic
472C2×C8C2×D28411yescharacteristic
473C4⋊C4—711no—
474C4⋊C4—711no—
475C4⋊C4—711no—
476C4⋊C4—711no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C22⋊C4—1421no—
484C22⋊C4—1421no—
485C22⋊C4—1421no—
486C22⋊C4—1421no—
487C22⋊C4—1421no—
488C22⋊C4—1421no—
489C22⋊C4—1421no—
490C22⋊C4—1421no—
491C42—1421no—
492C42—1421no—
493C42—1421no—
494C42—1421no—
495C42C2×C4×D7111yescharacteristic
496C42C2×D28411yescharacteristic
497C42#1176586511yescharacteristic
498C42C23×D7111yescharacteristic
499C23—1421no—
500C23—1421no—
501C23C4×D28411yescharacteristic
502Q8—712no—
503Q8—1421no—
504Q8—1421no—
505Q8—1421no—
506D4—2842no—
507D4—2842no—
508D4—1421no—
509D4—1421no—
510D4—1421no—
511D4—1421no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—2842no—
518C2×C4—2842no—
519C2×C4—2841no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—2841no—
523C2×C4—1421no—
524C2×C4—1421no—
525C2×C4—1421no—
526C2×C4—1421no—
527C2×C4—1421no—
528C2×C4—1421no—
529C2×C4C4×D28411yescharacteristic
530C2×C4C4×D28411yescharacteristic
531C2×C4C4×D28411yescharacteristic
532C2×C4C22×C4×D7111yescharacteristic
533C2×C4C22×D28411yescharacteristic
534C2×C4C2×#1176586511yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—41no—
544C8—41no—
545C8—2841no—
546C8—2841no—
547C8—2841no—
548C8—2841no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—2841no—
553C22—2841no—
554C22—2841no—
555C22—2841no—
556C22—2841no—
557C22—2841no—
558C22C2×C4×D28411yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—2841no—
564C4—2841no—
565C4—2841no—
566C4—2841no—
567C4—2841no—
568C4—1422no—
569C4#131×D7111yescharacteristic
570C4#1798676411yescharacteristic
571C4#1798680411yescharacteristic
572C4#1798681611yescharacteristic
573C2—41no—
574C2—5681no—
575C2—5681no—
576C2#2259706811yescharacteristic
577C2#2259721711yescharacteristic
578C2#2259724411yescharacteristic
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1454C2
socleC22×C71C2×C4×D4
prime radical#17986146C4
cosocleC42×C71C24
hypercenter#1454D71
hypocenterC71#18033
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #22597502C2semidirect
3. #22597469C2semidirect
4. #22587244C2semidirect
5. #22587226C2semidirect
6. #22587220C2semidirect
7. #22587220C2semidirect
8. #22587192C2semidirect
9. #22587171C2semidirect
10. #22587154C2semidirect
11. C8×Dic284C2semidirect
12. #22587115C2semidirect
13. #22586806C2semidirect
14. #22586728C2semidirect
15. C71×#1454C2semidirect
304. #1454D71semidirect
305. C71#18033semidirect

Maximal subgroups

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

subgroupindex
#225983602
#225975022
#225974692
#225872442
#225872262
#225872202
#225871922
#225871712
#225871542
C8×Dic2842
#225871152
#225868062
#225867282
C71×#14542
#1803371