Finite Group Databasebeta

← #31968078

Subgroups — #31968078

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#22598936C211yesmaximal, characteristic
2#22597415C211yesmaximal, characteristic, complemented
3#22587252C211yesmaximal, characteristic, complemented
4#22587240C211yesmaximal, characteristic, complemented
5#22587230C211yesmaximal, characteristic, complemented
6#22587223C211yesmaximal, characteristic, complemented
7#22587162C211yesmaximal, characteristic, complemented
8#22587150C211yesmaximal, characteristic, complemented
9C8×#11765852C211yesmaximal, characteristic, complemented
10C8×#11765850C211yesmaximal, characteristic, complemented
11#22587105C211yesmaximal, characteristic, complemented
12#22586800C211yesmaximal, characteristic, complemented
13#22586790C211yesmaximal, characteristic, complemented
14#22586621C211yesmaximal, characteristic, complemented
15C71×#2993C211yesmaximal, characteristic, complemented
16#17986549C2211yescharacteristic
17#17986409C411yescharacteristic
18#17986389C411yescharacteristic
19#17986365C411yescharacteristic
20#17986358C411yescharacteristic
21#17986250C411yescharacteristic
22#17986250C411yescharacteristic
23#17986247C411yescharacteristic
24#17986247C411yescharacteristic
25C4×#11765852C2211yescharacteristic
26C4×#11765852C2211yescharacteristic
27C4×#11765850C2211yescharacteristic
28C4×#11765850C2211yescharacteristic
29#17986078C2211yescharacteristic
30#17986078C2211yescharacteristic
31#17986069—21no—
32#17986069—21no—
33C8×D284C2211yescharacteristic
34C8×D284C2211yescharacteristic
35#17986056C2211yescharacteristic
36#17985964—21no—
37#17985964—21no—
38C4×#11765849C2211yescharacteristic
39C2×C4×C71⋊C8C2211yescharacteristic
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46D4⋊(C71⋊C8)C2211yescharacteristic
47D4⋊(C71⋊C8)C2211yescharacteristic
48#17985726C2211yescharacteristic
49#17985726C2211yescharacteristic
50#17985726C2211yescharacteristic
51#17985726C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54C8×C71⋊C8C2211yescharacteristic
55C8×C71⋊C8C2211yescharacteristic
56#529×C71C2211yescharacteristic
57#446×C71C2211yescharacteristic
58C8×Q8×C71C2211yescharacteristic
59#436×C71—21no—
60#436×C71C2211yescharacteristic
61#435×C71—21no—
62C8×D4×C71C2211yescharacteristic
63#433×C71—21no—
64#431×C71C2211yescharacteristic
65#424×C71—21no—
66#423×C71C2211yescharacteristic
67C2×#11765865—21no—
68C2×#11765852C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765850C2×C411yescharacteristic
71C2×#11765850C2×C411yescharacteristic
72C2×#11765849C2×C411yescharacteristic
73C2×#11765849C2×C411yescharacteristic
74C2×#11765848—21no—
75C2×#11765848—21no—
76C22×C71⋊C8—21no—
77C2×C8×D71—21no—
78C2×C8×D71—21no—
79#14556128C2×C411yescharacteristic
80#14556125C2×C411yescharacteristic
81#14556116—21no—
82#14556102—21no—
83#14556102C2×C411yescharacteristic
84#14556102C2×C411yescharacteristic
85C4×D284—21no—
86C4×D284C2311yescharacteristic
87C4×D284C2311yescharacteristic
88#14556061C2×C411yescharacteristic
89#14556061C2×C411yescharacteristic
90#14556059C2×C411yescharacteristic
91#14556059C2×C411yescharacteristic
92C4⋊(C71⋊C8)C2311yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94#14556052C2×C411yescharacteristic
95#14556052C2×C411yescharacteristic
96#14556049C2×C411yescharacteristic
97#14556049C2×C411yescharacteristic
98#14556048C2×C411yescharacteristic
99#14556048C2×C411yescharacteristic
100#14556048C2×C411yescharacteristic
101#14556048C2×C411yescharacteristic
102C22⋊(C71⋊C8)—21no—
103(C2×D71)⋊C8—21no—
104(C2×D71)⋊C8—21no—
105(C2×D71)⋊C8—21no—
106(C2×D71)⋊C8—21no—
107#14556039—21no—
108#14556039C2×C411yescharacteristic
109#14556039C2×C411yescharacteristic
110C4×C71⋊C8—21no—
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C4×C71⋊C8C2311yescharacteristic
114C4×C71⋊C8C2311yescharacteristic
115C2×M16×C71—21no—
116C2×M16×C71—21no—
117C22×C8×C71—21no—
118#126×C71—21no—
119#125×C71C2×C411yescharacteristic
120#124×C71C2×C411yescharacteristic
121#123×C71—21no—
122C22⋊Q8×C71C2×C411yescharacteristic
123#121×C71C2×C411yescharacteristic
124C4×Q8×C71C2311yescharacteristic
125C4×D4×C71—21no—
126C4×D4×C71C2311yescharacteristic
127#117×C71—21no—
128C2×C42×C71C2311yescharacteristic
129C4⋊C8×C71—21no—
130C4⋊C8×C71—21no—
131C4⋊C8×C71—21no—
132C4⋊C8×C71D411yescharacteristic
133C4⋊C8×C71D411yescharacteristic
134C4⋊C8×C71C2311yescharacteristic
135C4⋊C8×C71C2311yescharacteristic
136C22⋊C8×C71—21no—
137C22⋊C8×C71—21no—
138C22⋊C8×C71D411yescharacteristic
139C22⋊C8×C71D411yescharacteristic
140#97×C71—21no—
141C4×C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#11765865—81no—
145#11765865—81no—
146C2×Dic142—21no—
147C2×D284—21no—
148C2×D284—21no—
149C2×D284C22×C411yescharacteristic
150C2×D284C22×C411yescharacteristic
151C2×C22⋊D71—21no—
152C2×C22⋊D71—21no—
153C2×C4×D71—21no—
154C2×C4×D71—21no—
155#11765852—22no—
156#11765852—22no—
157#11765850—22no—
158#11765850—22no—
159#11765849—22no—
160#11765849—22no—
161#11765848—81no—
162#11765848—81no—
163C2×C71⋊C8—21no—
164C2×C71⋊C8—21no—
165C2×C71⋊C8—21no—
166C2×C71⋊C8—21no—
167C2×C71⋊C8—42no—
168C2×C71⋊C8C22×C411yescharacteristic
169C2×C71⋊C8C22×C411yescharacteristic
170C2×C71⋊C8C22×C411yescharacteristic
171C2×C71⋊C8C22×C411yescharacteristic
172C8×D71—81no—
173C8×D71—81no—
174C4⋊Dic71—21no—
175C4⋊Dic71C22×C411yescharacteristic
176C4⋊Dic71C22×C411yescharacteristic
177(C2×D71)⋊C4—21no—
178(C2×D71)⋊C4—21no—
179(C2×D71)⋊C4—21no—
180(C2×D71)⋊C4—21no—
181C2×Q8×C71C22×C411yescharacteristic
182C2×D4×C71—21no—
183C2×D4×C71C22×C411yescharacteristic
184C22×C4×C71—21no—
185C22×C4×C71C22×C411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71#4111yescharacteristic
188M16×C71—81no—
189M16×C71—81no—
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71—81no—
193C2×C8×C71C2×D411yescharacteristic
194C2×C8×C71C2×D411yescharacteristic
195C2×C8×C71C2×D411yescharacteristic
196C2×C8×C71C2×D411yescharacteristic
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71—21no—
200C4⋊C4×C71—21no—
201C4⋊C4×C71C22×C411yescharacteristic
202C4⋊C4×C71C22×C411yescharacteristic
203C22⋊C4×C71—21no—
204C22⋊C4×C71—21no—
205C22⋊C4×C71—21no—
206C22⋊C4×C71—21no—
207C22⋊C4×C71—21no—
208C42×C71—21no—
209C42×C71C22×C411yescharacteristic
210C42×C71C2×D411yescharacteristic
211C42×C71#4111yescharacteristic
212C42×C71C2411yescharacteristic
213C23×C71—21no—
214C23×C71C4×D411yescharacteristic
215C22×D71—21no—
216C22×D71—21no—
217Q8×C71—21no—
218Q8×C71—21no—
219D4×C71—21no—
220D4×C71—21no—
221D4×C71—81no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71C4×D411yescharacteristic
233C2×C4×C71C4×D411yescharacteristic
234C2×C4×C71C4×D411yescharacteristic
235C2×C4×C71C23×C411yescharacteristic
236C2×C4×C71C22×D411yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C22⋊D71—81no—
239C22⋊D71—81no—
240C2×Dic71—21no—
241C2×Dic71—21no—
242D284—21no—
243D284—21no—
244D284—21no—
245D284—21no—
246D284—81no—
247D284—81no—
248C4×D71—41no—
249C4×D71—41no—
250C4×D71—41no—
251C4×D71—41no—
252Dic142—81no—
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×C71C2×C4×D411yescharacteristic
274C2×D71—41no—
275C2×D71—41no—
276C2×D71—41no—
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×C71—41no—
287C4×C71C2×#13111yescharacteristic
288C4×C71C2×#13111yescharacteristic
289C4×C71#57611yescharacteristic
290C4×C71#57711yescharacteristic
291Dic71—41no—
292Dic71—41no—
293#17968—711nomaximal, Sylow
294C2×C71—41no—
295C2×C71—81no—
296C2×C71#293111yescharacteristic
297C2×C71#294611yescharacteristic
298C2×C71#295611yescharacteristic
299D71—81no—
300D71—81no—
301#3171—711no—
302#2993D7111yescharacteristic, complemented
303#2981—711no—
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#532—711no—
318#529C2×D7111yescharacteristic
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#446C2×D7111yescharacteristic
328C8×Q8C2×D7111yescharacteristic
329C4×SD16—711no—
330C4×SD16—711no—
331C4×D8—711no—
332C4×D8—711no—
333#436—21no—
334#436—711no—
335#436—711no—
336#436C2×D7111yescharacteristic
337#435—21no—
338#435—1421no—
339#435—1421no—
340C8×D4—711no—
341C8×D4—711no—
342C8×D4C2×D7111yescharacteristic
343#433—21no—
344#432—711no—
345#431C2×D7111yescharacteristic
346#424—21no—
347#423C2×D7111yescharacteristic
348#408—1421no—
349#408—1421no—
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—1421no—
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—1421no—
381C22×C8—1421no—
382C22×C8—1421no—
383C4⋊Q8—711no—
384#127—711no—
385#126—21no—
386#125C4×D7111yescharacteristic
387#124—1421no—
388#124C4×D7111yescharacteristic
389#123—21no—
390C22⋊Q8C4×D7111yescharacteristic
391#121—711no—
392#121—711no—
393#121—1421no—
394#121C4×D7111yescharacteristic
395C4×Q8C22×D7111yescharacteristic
396C4×D4—21no—
397C4×D4—711no—
398C4×D4—711no—
399C4×D4—1421no—
400C4×D4C22×D7111yescharacteristic
401#117—21no—
402C2×C42C22×D7111yescharacteristic
403#107—711no—
404#107—711no—
405#106—711no—
406#106—711no—
407C4⋊C8—21no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—711no—
411C4⋊C8—711no—
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—1421no—
427C22⋊C8—1421no—
428C22⋊C8—1421no—
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—21no—
438C4×C8—711no—
439C4×C8—711no—
440C4×C8—711no—
441C4×C8—711no—
442C4×C8—1421no—
443C4×C8C22×D7111yescharacteristic
444C4×C8C22×D7111yescharacteristic
445#41—5681no—
446#41—5681no—
447C2×Q8—1421no—
448C2×Q8C2×C4×D7111yescharacteristic
449C2×D4—21no—
450C2×D4—711no—
451C2×D4—711no—
452C2×D4—1421no—
453C2×D4—1421no—
454C2×D4—1421no—
455C2×D4—1421no—
456C2×D4C2×C4×D7111yescharacteristic
457C22×C4—21no—
458C22×C4—1421no—
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—2842no—
477C2×C8—5681no—
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—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—711no—
496C4⋊C4—711no—
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—21no—
505C22⋊C4—1421no—
506C22⋊C4—1421no—
507C22⋊C4—1421no—
508C22⋊C4—1421no—
509C42—21no—
510C42C2×C4×D7111yescharacteristic
511C42C2×C22⋊D7111yescharacteristic
512C42#1176586511yescharacteristic
513C42C23×D7111yescharacteristic
514C23—21no—
515C23—1421no—
516C23—1421no—
517C23C4×C22⋊D7111yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—5681no—
521D4—21no—
522D4—21no—
523D4—81no—
524D4—5681no—
525D4—5681no—
526D4—5681no—
527D4—5681no—
528D4—1421no—
529D4—1421no—
530D4—1421no—
531D4—1421no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—2841no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—2841no—
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—2841no—
575C22—2841no—
576C22—2841no—
577C22—2841no—
578C22—2841no—
579C22—2841no—
580C22C2×C4×C22⋊D7111yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—2841no—
589C4—2841no—
590C4#131×D7111yescharacteristic
591C4#131×D7111yescharacteristic
592C4#1798680211yescharacteristic
593C4#1798680711yescharacteristic
594C2—41no—
595C2—81no—
596C2—5681no—
597C2—5681no—
598C2#2259705011yescharacteristic
599C2#2259716811yescharacteristic
600C2#2259724711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2993C2
socleC22×C71C2×C4×D4
prime radical#22598936C2
cosocleC42×C71C24
hypercenter#2993D71
hypocenterC71#17968
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #22597415C2semidirect
3. #22587252C2semidirect
4. #22587240C2semidirect
5. #22587230C2semidirect
6. #22587223C2semidirect
7. #22587162C2semidirect
8. #22587150C2semidirect
9. C8×#11765852C2semidirect
10. C8×#11765850C2semidirect
11. #22587105C2semidirect
12. #22586800C2semidirect
13. #22586790C2semidirect
14. #22586621C2semidirect
15. C71×#2993C2semidirect
302. #2993D71semidirect
316. C71#17968semidirect

Maximal subgroups

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

subgroupindex
#225989362
#225974152
#225872522
#225872402
#225872302
#225872232
#225871622
#225871502
C8×#117658522
C8×#117658502
#225871052
#225868002
#225867902
#225866212
C71×#29932
#1796871