Finite Group Databasebeta

← #31968082

Subgroups — #31968082

Order 18176.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598929C211yesmaximal, characteristic
2#22597491C211yesmaximal, characteristic, complemented
3#22587252C211yesmaximal, characteristic, complemented
4#22587242C211yesmaximal, characteristic, complemented
5#22587228C211yesmaximal, characteristic, complemented
6#22587223C211yesmaximal, characteristic, complemented
7#22587162C211yesmaximal, characteristic, complemented
8#22587152C211yesmaximal, characteristic, complemented
9C8×#11765853C211yesmaximal, characteristic, complemented
10C8×#11765850C211yesmaximal, characteristic, complemented
11#22587108C211yesmaximal, characteristic, complemented
12#22586798C211yesmaximal, characteristic, complemented
13#22586791C211yesmaximal, characteristic, complemented
14#22586621C211yesmaximal, characteristic, complemented
15C71×#2981C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986405C411yescharacteristic
18#17986386C411yescharacteristic
19#17986363C411yescharacteristic
20#17986361C411yescharacteristic
21#17986250C411yescharacteristic
22#17986250C411yescharacteristic
23#17986243C411yescharacteristic
24#17986243C411yescharacteristic
25#17986129C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C4×#11765853C2211yescharacteristic
28C4×#11765853C2211yescharacteristic
29C4×#11765850C2211yescharacteristic
30C4×#11765850C2211yescharacteristic
31#17986078C2211yescharacteristic
32#17986074—21no—
33#17986073—21no—
34C8×D284C2211yescharacteristic
35#17986059—21no—
36#17986024—21no—
37#17986023C2211yescharacteristic
38C4×#11765849C2211yescharacteristic
39C2×C4×C71⋊C8C2211yescharacteristic
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44#17985729C2211yescharacteristic
45#17985729C2211yescharacteristic
46D4⋊(C71⋊C8)C2211yescharacteristic
47D4⋊(C71⋊C8)C2211yescharacteristic
48D4⋊(C71⋊C8)C2211yescharacteristic
49D4⋊(C71⋊C8)C2211yescharacteristic
50#17985726C2211yescharacteristic
51#17985726C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54C8×C71⋊C8C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56#532×C71C2211yescharacteristic
57#436×C71C2211yescharacteristic
58#436×C71C2211yescharacteristic
59#435×C71—21no—
60#435×C71—21no—
61C8×D4×C71C2211yescharacteristic
62C8×D4×C71C2211yescharacteristic
63#432×C71C2211yescharacteristic
64#431×C71C2211yescharacteristic
65#408×C71—21no—
66#408×C71—21no—
67C2×#11765853C2×C411yescharacteristic
68C2×#11765853C2×C411yescharacteristic
69C2×#11765850C2×C411yescharacteristic
70C2×#11765850C2×C411yescharacteristic
71C2×#11765849—21no—
72C2×#11765849C2×C411yescharacteristic
73C2×#11765849C2×C411yescharacteristic
74C2×#11765848—21no—
75C22×C71⋊C8—21no—
76C2×C8×D71—21no—
77#14556122—21no—
78#14556120C2×C411yescharacteristic
79#14556116C2×C411yescharacteristic
80#14556111—21no—
81#14556107C2×C411yescharacteristic
82#14556102C2×C411yescharacteristic
83C4×Dic142C2311yescharacteristic
84C4×D284C2311yescharacteristic
85C4×C22⋊D71—21no—
86#14556085—21no—
87#14556061C2×C411yescharacteristic
88#14556061C2×C411yescharacteristic
89#14556059C2×C411yescharacteristic
90#14556059C2×C411yescharacteristic
91C4⋊(C71⋊C8)—21no—
92C4⋊(C71⋊C8)C2311yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94Dic71⋊C8—21no—
95Dic71⋊C8—21no—
96#14556051C2×C411yescharacteristic
97#14556051C2×C411yescharacteristic
98#14556049C2×C411yescharacteristic
99#14556049C2×C411yescharacteristic
100#14556049C2×C411yescharacteristic
101#14556049C2×C411yescharacteristic
102#14556048C2×C411yescharacteristic
103#14556048C2×C411yescharacteristic
104(C2×D71)⋊C8—21no—
105(C2×D71)⋊C8—21no—
106#14556039C2×C411yescharacteristic
107#14556039C2×C411yescharacteristic
108#14556038—21no—
109C4×C71⋊C8—21no—
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C4×C71⋊C8C2311yescharacteristic
114C8×Dic71—21no—
115C2×#41×C71—21no—
116C2×M16×C71—21no—
117C2×M16×C71—21no—
118C22×C8×C71—21no—
119C22×C8×C71—21no—
120C4⋊Q8×C71C2×C411yescharacteristic
121#127×C71C2×C411yescharacteristic
122#124×C71—21no—
123#121×C71—21no—
124#121×C71C2×C411yescharacteristic
125#121×C71C2×C411yescharacteristic
126C4×D4×C71—21no—
127C4×D4×C71C2311yescharacteristic
128C4×D4×C71C2311yescharacteristic
129C2×C42×C71C2311yescharacteristic
130C4⋊C8×C71D411yescharacteristic
131C4⋊C8×C71D411yescharacteristic
132C4⋊C8×C71C2311yescharacteristic
133C4⋊C8×C71C2311yescharacteristic
134C22⋊C8×C71—21no—
135C22⋊C8×C71—21no—
136C22⋊C8×C71—21no—
137C22⋊C8×C71—21no—
138C22⋊C8×C71—21no—
139C22⋊C8×C71D411yescharacteristic
140C22⋊C8×C71D411yescharacteristic
141#97×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142C22×C411yescharacteristic
145C2×D284C22×C411yescharacteristic
146C2×C22⋊D71—21no—
147C2×C4×D71—21no—
148#11765853—22no—
149#11765853—22no—
150#11765850—22no—
151#11765850—22no—
152#11765849—22no—
153#11765849—22no—
154#11765849—81no—
155#11765848—81no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8—21no—
160C2×C71⋊C8—42no—
161C2×C71⋊C8C22×C411yescharacteristic
162C2×C71⋊C8C22×C411yescharacteristic
163C2×C71⋊C8C22×C411yescharacteristic
164C2×C71⋊C8C22×C411yescharacteristic
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—
178#41×C71—81no—
179#41×C71—81no—
180C2×Q8×C71—21no—
181C2×D4×C71—21no—
182C2×D4×C71—21no—
183C2×D4×C71—21no—
184C2×D4×C71—21no—
185C2×D4×C71C22×C411yescharacteristic
186C2×D4×C71C22×C411yescharacteristic
187C22×C4×C71—21no—
188C22×C4×C71—21no—
189C22×C4×C71C22×C411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71#4111yescharacteristic
192M16×C71—81no—
193M16×C71—81no—
194C2×C8×C71—21no—
195C2×C8×C71—21no—
196C2×C8×C71—81no—
197C2×C8×C71—81no—
198C2×C8×C71C2×D411yescharacteristic
199C2×C8×C71C2×D411yescharacteristic
200C2×C8×C71C2×D411yescharacteristic
201C2×C8×C71C2×D411yescharacteristic
202C4⋊C4×C71—21no—
203C4⋊C4×C71C22×C411yescharacteristic
204C4⋊C4×C71C22×C411yescharacteristic
205C22⋊C4×C71—21no—
206C22⋊C4×C71—21no—
207C22⋊C4×C71—21no—
208C22⋊C4×C71—21no—
209C42×C71C22×C411yescharacteristic
210C42×C71C2×D411yescharacteristic
211C42×C71#4111yescharacteristic
212C42×C71C2411yescharacteristic
213C23×C71—21no—
214C23×C71—21no—
215C23×C71C4×D411yescharacteristic
216C22×D71—21no—
217Q8×C71—81no—
218D4×C71—21no—
219D4×C71—21no—
220D4×C71—21no—
221D4×C71—21no—
222D4×C71—81no—
223D4×C71—81no—
224D4×C71—81no—
225D4×C71—81no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71C4×D411yescharacteristic
238C2×C4×C71C4×D411yescharacteristic
239C2×C4×C71C4×D411yescharacteristic
240C2×C4×C71C23×C411yescharacteristic
241C2×C4×C71C22×D411yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C22⋊D71—81no—
244C2×Dic71—21no—
245C2×Dic71—21no—
246C2×Dic71—21no—
247D284—21no—
248D284—21no—
249C4×D71—41no—
250C4×D71—41no—
251Dic142—21no—
252Dic142—21no—
253C8×C71—21no—
254C8×C71—21no—
255C8×C71—21no—
256C8×C71—21no—
257C8×C71—41no—
258C8×C71—41no—
259C71⋊C8—22no—
260C71⋊C8—21no—
261C71⋊C8—21no—
262C71⋊C8—41no—
263C71⋊C8—41no—
264C71⋊C8—41no—
265C71⋊C8#13112yes—
266C71⋊C8#13112yes—
267C22×C71—21no—
268C22×C71—21no—
269C22×C71—21no—
270C22×C71—41no—
271C22×C71—41no—
272C22×C71—41no—
273C22×C71—41no—
274C22×C71—41no—
275C22×C71—41no—
276C22×C71C2×C4×D411yescharacteristic
277C2×D71—41no—
278C2×D71—41no—
279C2×D71—41no—
280C4×C71—21no—
281C4×C71—21no—
282C4×C71—21no—
283C4×C71—21no—
284C4×C71—41no—
285C4×C71—41no—
286C4×C71C2×#13111yescharacteristic
287C4×C71C2×#13111yescharacteristic
288C4×C71#57611yescharacteristic
289C4×C71#57711yescharacteristic
290Dic71—41no—
291Dic71—41no—
292Dic71—41no—
293#17968—711nomaximal, Sylow
294C2×C71—41no—
295C2×C71—81no—
296C2×C71—81no—
297C2×C71#293111yescharacteristic
298C2×C71#294611yescharacteristic
299C2×C71#295611yescharacteristic
300D71—81no—
301#3171—711no—
302#2993—711no—
303#2981D7111yescharacteristic, complemented
304#1594—711no—
305#1592—711no—
306#1590—711no—
307#1589—711no—
308#1580—711no—
309SD16⋊C8—711no—
310C8×SD16—711no—
311C8×D8—711no—
312#1569—711no—
313#1494—711no—
314#1492—711no—
315C8×M16—711no—
316C71#1796811yescharacteristic, complemented, Sylow
317#532C2×D7111yescharacteristic
318#529—711no—
319#501—711no—
320#496—711no—
321#490—711no—
322#489—711no—
323#469—711no—
324#469—711no—
325#468—711no—
326#468—711no—
327#446—711no—
328C8×Q8—711no—
329C4×SD16—711no—
330C4×SD16—711no—
331C4×D8—711no—
332C4×D8—711no—
333#436—711no—
334#436—1421no—
335#436C2×D7111yescharacteristic
336#436C2×D7111yescharacteristic
337#435—21no—
338#435—21no—
339#435—1421no—
340C8×D4—711no—
341C8×D4C2×D7111yescharacteristic
342C8×D4C2×D7111yescharacteristic
343#433—1421no—
344#432C2×D7111yescharacteristic
345#431C2×D7111yescharacteristic
346#424—1421no—
347#423—711no—
348#408—21no—
349#408—21no—
350C4×M16—711no—
351C2×C4×C8—711no—
352#335—711no—
353#335—711no—
354#334—711no—
355#334—711no—
356Q8⋊C8—711no—
357Q8⋊C8—711no—
358D4⋊C8—711no—
359D4⋊C8—711no—
360D4⋊C8—711no—
361D4⋊C8—711no—
362D4⋊C8—711no—
363D4⋊C8—711no—
364#322—711no—
365#322—711no—
366C82—711no—
367C82—711no—
368C2×#41—21no—
369C2×SD16—711no—
370C2×SD16—711no—
371C2×D8—711no—
372C2×D8—711no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—711no—
376C2×M16—711no—
377C2×M16—1421no—
378C2×M16—1421no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—1421no—
382C22×C8—1421no—
383C4⋊Q8C4×D7111yescharacteristic
384#127C4×D7111yescharacteristic
385#126—1421no—
386#125—711no—
387#124—21no—
388#124—711no—
389#123—1421no—
390C22⋊Q8—711no—
391#121—21no—
392#121—711no—
393#121C4×D7111yescharacteristic
394#121C4×D7111yescharacteristic
395C4×Q8—711no—
396C4×D4—21no—
397C4×D4—711no—
398C4×D4—1421no—
399C4×D4C22×D7111yescharacteristic
400C4×D4C22×D7111yescharacteristic
401#117—1421no—
402C2×C42C22×D7111yescharacteristic
403#107—711no—
404#107—711no—
405#106—711no—
406#106—711no—
407C4⋊C8—711no—
408C4⋊C8—711no—
409C4⋊C8—1421no—
410C4⋊C8—1421no—
411C4⋊C8—1421no—
412C4⋊C8C22⋊D7111yescharacteristic
413C4⋊C8C22⋊D7111yescharacteristic
414C4⋊C8C22×D7111yescharacteristic
415C4⋊C8C22×D7111yescharacteristic
416#103—711no—
417#103—711no—
418#102—711no—
419#102—711no—
420#102—711no—
421#102—711no—
422#102—711no—
423#102—711no—
424C22⋊C8—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—1421no—
430C22⋊C8—1421no—
431C22⋊C8C22⋊D7111yescharacteristic
432C22⋊C8C22⋊D7111yescharacteristic
433#97—21no—
434#97—711no—
435#97—711no—
436#97—1421no—
437C4×C8—711no—
438C4×C8—711no—
439C4×C8—711no—
440C4×C8—711no—
441C4×C8—1421no—
442C4×C8—1421no—
443C4×C8C22×D7111yescharacteristic
444C4×C8C22×D7111yescharacteristic
445#41—81no—
446#41—81no—
447C2×Q8—21no—
448C2×Q8—711no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—711no—
454C2×D4—1421no—
455C2×D4C2×C4×D7111yescharacteristic
456C2×D4C2×C4×D7111yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—1421no—
460C22×C4C2×C4×D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4#1176586511yescharacteristic
463SD16—1422no—
464SD16—1422no—
465D8—1422no—
466D8—1422no—
467M16—81no—
468M16—81no—
469M16—5681no—
470M16—5681no—
471M16—1422no—
472M16—1422no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—81no—
477C2×C8—2842no—
478C2×C8—5681no—
479C2×C8—711no—
480C2×C8—711no—
481C2×C8—711no—
482C2×C8—711no—
483C2×C8—1421no—
484C2×C8—1421no—
485C2×C8—1421no—
486C2×C8—1421no—
487C2×C8C2×C22⋊D7111yescharacteristic
488C2×C8C2×C22⋊D7111yescharacteristic
489C2×C8C2×C22⋊D7111yescharacteristic
490C2×C8C2×C22⋊D7111yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—711no—
493C4⋊C4—711no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4C2×C4×D7111yescharacteristic
499C4⋊C4C2×C4×D7111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—1421no—
505C22⋊C4—1421no—
506C22⋊C4—1421no—
507C22⋊C4—1421no—
508C22⋊C4—1421no—
509C42—1421no—
510C42C2×C4×D7111yescharacteristic
511C42C2×C22⋊D7111yescharacteristic
512C42#1176586511yescharacteristic
513C42C23×D7111yescharacteristic
514C23—21no—
515C23—21no—
516C23—1421no—
517C23C4×C22⋊D7111yescharacteristic
518Q8—81no—
519Q8—1421no—
520Q8—1421no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—81no—
526D4—81no—
527D4—81no—
528D4—81no—
529D4—5681no—
530D4—1421no—
531D4—1421no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—1421no—
546C2×C4—1421no—
547C2×C4—1421no—
548C2×C4C4×C22⋊D7111yescharacteristic
549C2×C4C4×C22⋊D7111yescharacteristic
550C2×C4C4×C22⋊D7111yescharacteristic
551C2×C4C22×C4×D7111yescharacteristic
552C2×C4C22×C22⋊D7111yescharacteristic
553C2×C4C2×#1176586511yescharacteristic
554C8—21no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—2841no—
561C8—2841no—
562C8—2841no—
563C8—712no—
564C8—712no—
565C8—1422no—
566C8—1421no—
567C8—1421no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—2841no—
578C22—2841no—
579C22—2841no—
580C22C2×C4×C22⋊D7111yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—2841no—
588C4—2841no—
589C4—2841no—
590C4#131×D7111yescharacteristic
591C4#131×D7111yescharacteristic
592C4#1798680211yescharacteristic
593C4#1798681011yescharacteristic
594C2—41no—
595C2—81no—
596C2—81no—
597C2—5681no—
598C2#2259705011yescharacteristic
599C2#2259717211yescharacteristic
600C2#2259724711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2981C2
socleC22×C71C2×C4×D4
prime radical#22598929C2
cosocleC42×C71C24
hypercenter#2981D71
hypocenterC71#17968
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2981

Complements

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

NH ≅ G/Ntype
2. #22597491C2semidirect
3. #22587252C2semidirect
4. #22587242C2semidirect
5. #22587228C2semidirect
6. #22587223C2semidirect
7. #22587162C2semidirect
8. #22587152C2semidirect
9. C8×#11765853C2semidirect
10. C8×#11765850C2semidirect
11. #22587108C2semidirect
12. #22586798C2semidirect
13. #22586791C2semidirect
14. #22586621C2semidirect
15. C71×#2981C2semidirect
303. #2981D71semidirect
316. C71#17968semidirect

Maximal subgroups

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

subgroupindex
#225989292
#225974912
#225872522
#225872422
#225872282
#225872232
#225871622
#225871522
C8×#117658532
C8×#117658502
#225871082
#225867982
#225867912
#225866212
C71×#29812
#1796871