Finite Group Databasebeta

← #31968091

Subgroups — #31968091

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#22598940C211yesmaximal, characteristic
2#22597485C211yesmaximal, characteristic
3#22597435C211yesmaximal, characteristic, complemented
4#22587246C211yesmaximal, characteristic, complemented
5#22587245C211yesmaximal, characteristic, complemented
6#22587234C211yesmaximal, characteristic, complemented
7#22587227C211yesmaximal, characteristic, complemented
8#22587156C211yesmaximal, characteristic, complemented
9#22587155C211yesmaximal, characteristic, complemented
10#22587144C211yesmaximal, characteristic, complemented
11#22587137C211yesmaximal, characteristic, complemented
12#22587105C211yesmaximal, characteristic, complemented
13#22586801C211yesmaximal, characteristic
14C8×#11765849C211yesmaximal, characteristic, complemented
15C71×#1493C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986405C411yescharacteristic
18#17986391C411yescharacteristic
19#17986367C411yescharacteristic
20#17986361C411yescharacteristic
21#17986255C411yescharacteristic
22#17986255C411yescharacteristic
23#17986247C411yescharacteristic
24#17986247C411yescharacteristic
25#17986130—21no—
26#17986130—21no—
27#17986130C2211yescharacteristic
28#17986130C2211yescharacteristic
29#17986126C2211yescharacteristic
30Q8×C71⋊C8C2211yescharacteristic
31Q8×C71⋊C8C2211yescharacteristic
32#17986123C2211yescharacteristic
33C4×#11765855C2211yescharacteristic
34C4×#11765855C2211yescharacteristic
35C4×#11765852C2211yescharacteristic
36C4×#11765852C2211yescharacteristic
37#17986076C2211yescharacteristic
38#17986075—21no—
39#17986068—21no—
40#17986064C2211yescharacteristic
41#17986060—21no—
42#17986060—21no—
43#17986059—21no—
44#17986056C2211yescharacteristic
45#17986052C2211yescharacteristic
46#17986024—21no—
47C4×#11765849C2211yescharacteristic
48#17985751C2211yescharacteristic
49#17985751C2211yescharacteristic
50#17985749C2211yescharacteristic
51#17985749C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985728C2211yescharacteristic
55#17985728C2211yescharacteristic
56#17985721C2211yescharacteristic
57#17985721C2211yescharacteristic
58C8×C71⋊C8C2211yescharacteristic
59C8×C71⋊C8C2211yescharacteristic
60#533×C71C2211yescharacteristic
61#423×C71C2211yescharacteristic
62C2×C4×C8×C71C2211yescharacteristic
63Q8⋊C8×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65Q8⋊C8×C71C2211yescharacteristic
66Q8⋊C8×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765852C2×C411yescharacteristic
71C2×#11765849C2×C411yescharacteristic
72C2×#11765849C2×C411yescharacteristic
73C2×#11765848—21no—
74C2×C8×D71—21no—
75#14556122—21no—
76#14556120C2×C411yescharacteristic
77#14556116C2×C411yescharacteristic
78#14556111—21no—
79#14556107C2×C411yescharacteristic
80#14556102C2×C411yescharacteristic
81C4×Dic142C2311yescharacteristic
82C4×D284C2311yescharacteristic
83C4×C22⋊D71—21no—
84#14556085—21no—
85#14556061C2×C411yescharacteristic
86#14556061C2×C411yescharacteristic
87#14556059C2×C411yescharacteristic
88#14556059C2×C411yescharacteristic
89C4⋊(C71⋊C8)—21no—
90C4⋊(C71⋊C8)—21no—
91C4⋊(C71⋊C8)—21no—
92C4⋊(C71⋊C8)—21no—
93C4⋊(C71⋊C8)D411yescharacteristic
94C4⋊(C71⋊C8)D411yescharacteristic
95C4⋊(C71⋊C8)C2311yescharacteristic
96C4⋊(C71⋊C8)C2311yescharacteristic
97Dic71⋊C8—21no—
98Dic71⋊C8—21no—
99#14556052C2×C411yescharacteristic
100#14556052C2×C411yescharacteristic
101#14556052C2×C411yescharacteristic
102#14556052C2×C411yescharacteristic
103#14556051C2×C411yescharacteristic
104#14556051C2×C411yescharacteristic
105#14556048C2×C411yescharacteristic
106#14556048C2×C411yescharacteristic
107C22⋊(C71⋊C8)—21no—
108C22⋊(C71⋊C8)D411yescharacteristic
109C22⋊(C71⋊C8)D411yescharacteristic
110(C2×D71)⋊C8—21no—
111(C2×D71)⋊C8—21no—
112#14556039—21no—
113#14556039—21no—
114#14556039—21no—
115#14556039C2×C411yescharacteristic
116#14556039C2×C411yescharacteristic
117#14556038—21no—
118C4×C71⋊C8—21no—
119C4×C71⋊C8—21no—
120C4×C71⋊C8C2311yescharacteristic
121C4×C71⋊C8C2311yescharacteristic
122C4×C71⋊C8C2311yescharacteristic
123C4×C71⋊C8C2311yescharacteristic
124C8×Dic71—21no—
125C2×M16×C71—21no—
126C22×C8×C71—21no—
127C4⋊Q8×C71C2×C411yescharacteristic
128C4⋊Q8×C71C2×C411yescharacteristic
129#125×C71—21no—
130C22⋊Q8×C71—21no—
131C22⋊Q8×C71C2×C411yescharacteristic
132C22⋊Q8×C71C2×C411yescharacteristic
133C4×Q8×C71—21no—
134C4×Q8×C71C2311yescharacteristic
135C4×Q8×C71C2311yescharacteristic
136#117×C71—21no—
137C2×C42×C71C2311yescharacteristic
138C4⋊C8×C71—21no—
139C4⋊C8×C71C2311yescharacteristic
140C4⋊C8×C71C2311yescharacteristic
141C4×C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142C22×C411yescharacteristic
145C2×D284C22×C411yescharacteristic
146C2×C22⋊D71—21no—
147C2×C4×D71—21no—
148#11765855—22no—
149#11765855—22no—
150#11765852—22no—
151#11765852—22no—
152#11765849—22no—
153#11765849—22no—
154#11765848—81no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8C22×C411yescharacteristic
158C2×C71⋊C8C22×C411yescharacteristic
159C2×C71⋊C8C22×C411yescharacteristic
160C2×C71⋊C8C22×C411yescharacteristic
161C2×C71⋊C8C2×D411yescharacteristic
162C2×C71⋊C8C2×D411yescharacteristic
163C2×C71⋊C8C2×D411yescharacteristic
164C2×C71⋊C8C2×D411yescharacteristic
165C8×D71—81no—
166C4⋊Dic71C22×C411yescharacteristic
167C4⋊Dic71C22×C411yescharacteristic
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170Dic71⋊C4—21no—
171Dic71⋊C4—21no—
172C22⋊Dic71—21no—
173(C2×D71)⋊C4—21no—
174(C2×D71)⋊C4—21no—
175(C2×D71)⋊C4—21no—
176(C2×D71)⋊C4—21no—
177C4×Dic71—21no—
178C2×Q8×C71—21no—
179C2×Q8×C71C22×C411yescharacteristic
180C2×Q8×C71C22×C411yescharacteristic
181C22×C4×C71C22×C411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71#4111yescharacteristic
184M16×C71—81no—
185C2×C8×C71—21no—
186C2×C8×C71—21no—
187C2×C8×C71—21no—
188C2×C8×C71—21no—
189C2×C8×C71—42no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71C22×C411yescharacteristic
198C4⋊C4×C71C22×C411yescharacteristic
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71C22×C411yescharacteristic
204C42×C71C2×D411yescharacteristic
205C42×C71#4111yescharacteristic
206C42×C71C2411yescharacteristic
207C23×C71C4×D411yescharacteristic
208C22×D71—21no—
209Q8×C71—21no—
210Q8×C71—21no—
211Q8×C71—21no—
212Q8×C71—21no—
213Q8×C71—81no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—41no—
223C2×C4×C71C4×D411yescharacteristic
224C2×C4×C71C4×D411yescharacteristic
225C2×C4×C71C4×D411yescharacteristic
226C2×C4×C71C23×C411yescharacteristic
227C2×C4×C71C22×D411yescharacteristic
228C2×C4×C71C2×#4111yescharacteristic
229C22⋊D71—81no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233D284—21no—
234D284—21no—
235C4×D71—41no—
236C4×D71—41no—
237Dic142—21no—
238Dic142—21no—
239C8×C71—22no—
240C8×C71—41no—
241C8×C71—41no—
242C8×C71—41no—
243C71⋊C8—21no—
244C71⋊C8—21no—
245C71⋊C8—21no—
246C71⋊C8—21no—
247C71⋊C8—21no—
248C71⋊C8—21no—
249C71⋊C8—41no—
250C71⋊C8—41no—
251C71⋊C8#13112yes—
252C71⋊C8#13112yes—
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×C71—41no—
265C4×C71—41no—
266C4×C71—41no—
267C4×C71—41no—
268C4×C71C2×#13111yescharacteristic
269C4×C71C2×#13111yescharacteristic
270C4×C71#57711yescharacteristic
271C4×C71#57811yescharacteristic
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#1569—711no—
293#1494—711no—
294#1493D7111yescharacteristic, complemented
295C8×M16—711no—
296C71#1796911yescharacteristic, complemented, Sylow
297#533C2×D7111yescharacteristic
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#432—711no—
327#431—711no—
328#424—1421no—
329#423C2×D7111yescharacteristic
330C4×M16—711no—
331C2×C4×C8C2×D7111yescharacteristic
332#335—711no—
333#335—711no—
334#334—711no—
335#334—711no—
336Q8⋊C8—711no—
337Q8⋊C8—711no—
338Q8⋊C8C2×D7111yescharacteristic
339Q8⋊C8C2×D7111yescharacteristic
340Q8⋊C8C2×D7111yescharacteristic
341Q8⋊C8C2×D7111yescharacteristic
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—711no—
354C2×M16—711no—
355C2×M16—1421no—
356C22×C8—21no—
357C22×C8—1421no—
358C4⋊Q8C4×D7111yescharacteristic
359C4⋊Q8C4×D7111yescharacteristic
360#126—1421no—
361#125—21no—
362#125—711no—
363#124—711no—
364#123—1421no—
365C22⋊Q8—21no—
366C22⋊Q8—711no—
367C22⋊Q8C4×D7111yescharacteristic
368C22⋊Q8C4×D7111yescharacteristic
369#121—711no—
370C4×Q8—21no—
371C4×Q8—711no—
372C4×Q8C22×D7111yescharacteristic
373C4×Q8C22×D7111yescharacteristic
374C4×D4—711no—
375C4×D4—1421no—
376#117—21no—
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—711no—
387C4⋊C8—711no—
388C4⋊C8—1421no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8—1421no—
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—711no—
405C22⋊C8—711no—
406C22⋊C8—1421no—
407C22⋊C8—1421no—
408C22⋊C8—1421no—
409#97—711no—
410#97—711no—
411#97—1421no—
412#97—1421no—
413#97—1421no—
414#97—1421no—
415C4×C8—21no—
416C4×C8—711no—
417C4×C8—711no—
418C4×C8—711no—
419C4×C8—711no—
420C4×C8—1421no—
421C4×C8—1421no—
422C4×C8—1421no—
423C4×C8C22×D7111yescharacteristic
424C4×C8C22×D7111yescharacteristic
425C2×Q8—21no—
426C2×Q8—711no—
427C2×Q8C2×C4×D7111yescharacteristic
428C2×Q8C2×C4×D7111yescharacteristic
429C2×D4—711no—
430C2×D4—1421no—
431C22×C4—1421no—
432C22×C4C2×C4×D7111yescharacteristic
433C22×C4D4×D7111yescharacteristic
434C22×C4#1176586411yescharacteristic
435Q16—1422no—
436Q16—1422no—
437SD16—1422no—
438SD16—1422no—
439M16—81no—
440M16—5681no—
441M16—1422no—
442M16—1422no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—5681no—
449C2×C8—711no—
450C2×C8—711no—
451C2×C8—711no—
452C2×C8—711no—
453C2×C8—711no—
454C2×C8—711no—
455C2×C8—711no—
456C2×C8—711no—
457C2×C8—1421no—
458C2×C8—1421no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—711no—
467C4⋊C4—711no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4C2×C4×D7111yescharacteristic
473C4⋊C4C2×C4×D7111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1421no—
477C22⋊C4—1421no—
478C22⋊C4—1421no—
479C22⋊C4—1421no—
480C22⋊C4—1421no—
481C42—21no—
482C42—21no—
483C42—1421no—
484C42C2×C4×D7111yescharacteristic
485C42D4×D7111yescharacteristic
486C42#1176586411yescharacteristic
487C42C23×D7111yescharacteristic
488C23—1421no—
489C23(C22⋊D71)⋊C411yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—1421no—
496Q8—1421no—
497D4—5681no—
498D4—1421no—
499D4—1421no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—2841no—
510C2×C4—2841no—
511C2×C4—1421no—
512C2×C4—1421no—
513C2×C4—1421no—
514C2×C4(C22⋊D71)⋊C411yescharacteristic
515C2×C4(C22⋊D71)⋊C411yescharacteristic
516C2×C4(C22⋊D71)⋊C411yescharacteristic
517C2×C4C22×C4×D7111yescharacteristic
518C2×C4C2×D4×D7111yescharacteristic
519C2×C4C2×#1176586411yescharacteristic
520C8—22no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—2841no—
525C8—2841no—
526C8—712no—
527C8—712no—
528C8—1421no—
529C8—1421no—
530C8—1421no—
531C8—1421no—
532C8—1421no—
533C8—1421no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2841no—
538C22—2841no—
539C22—2841no—
540C22C2×(C22⋊D71)⋊C411yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—2841no—
550C4—2841no—
551C4—2841no—
552C4#131×D7111yescharacteristic
553C4#131×D7111yescharacteristic
554C4#1798680511yescharacteristic
555C4#1798681211yescharacteristic
556C2—41no—
557C2—5681no—
558C2#2259704811yescharacteristic
559C2#2259717711yescharacteristic
560C2#2259724611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1493C2
socleC22×C71C2×C4×D4
prime radical#14556102C2×C4
cosocleC42×C71C24
hypercenter#1493D71
hypocenterC71#17969
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
3. #22597435C2semidirect
4. #22587246C2semidirect
5. #22587245C2semidirect
6. #22587234C2semidirect
7. #22587227C2semidirect
8. #22587156C2semidirect
9. #22587155C2semidirect
10. #22587144C2semidirect
11. #22587137C2semidirect
12. #22587105C2semidirect
14. C8×#11765849C2semidirect
15. C71×#1493C2semidirect
294. #1493D71semidirect
296. C71#17969semidirect

Maximal subgroups

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

subgroupindex
#225989402
#225974852
#225974352
#225872462
#225872452
#225872342
#225872272
#225871562
#225871552
#225871442
#225871372
#225871052
#225868012
C8×#117658492
C71×#14932
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