Finite Group Databasebeta

← #31968148

Subgroups — #31968148

Order 18176.

592 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598904C211yesmaximal, characteristic
2#22597491C211yesmaximal, characteristic, complemented
3#22587252C211yesmaximal, characteristic, complemented
4#22587247C211yesmaximal, characteristic, complemented
5#22587230C211yesmaximal, characteristic, complemented
6#22587228C211yesmaximal, characteristic, complemented
7#22587162C211yesmaximal, characteristic, complemented
8#22587157C211yesmaximal, characteristic, complemented
9C8×#11765852C211yesmaximal, characteristic, complemented
10C8×#11765853C211yesmaximal, characteristic, complemented
11#22587095C211yesmaximal, characteristic, complemented
12#22586805C211yesmaximal, characteristic, complemented
13#22586803C211yesmaximal, characteristic, complemented
14#22586624C211yesmaximal, characteristic, complemented
15C71×#2993C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986398C411yescharacteristic
18#17986381C411yescharacteristic
19#17986356C411yescharacteristic
20#17986353C411yescharacteristic
21#17986238C411yescharacteristic
22#17986233C411yescharacteristic
23#17986230C411yescharacteristic
24#17986226C411yescharacteristic
25#17986129C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C4×#11765855C2211yescharacteristic
28C4×#11765852C2211yescharacteristic
29C4×#11765853C2211yescharacteristic
30C4×#11765850C2211yescharacteristic
31#17986078C2211yescharacteristic
32#17986074—21no—
33#17986073—21no—
34C8×D284C2211yescharacteristic
35#17986059—21no—
36#17986052C2211yescharacteristic
37#17986024—21no—
38#17986023C2211yescharacteristic
39C2×C4×C71⋊C8C2211yescharacteristic
40#17985749C2211yescharacteristic
41#17985749C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46#17985729C2211yescharacteristic
47#17985729C2211yescharacteristic
48D4⋊(C71⋊C8)C2211yescharacteristic
49D4⋊(C71⋊C8)C2211yescharacteristic
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×#11765855C2×C411yescharacteristic
68C2×#11765853C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765850C2×C411yescharacteristic
71C2×#11765849—21no—
72C2×#11765848—21no—
73C22×C71⋊C8C2×C411yescharacteristic
74C22×C71⋊C8C2×C411yescharacteristic
75C2×C8×D71—21no—
76#14556122—21no—
77#14556120C2×C411yescharacteristic
78#14556116C2×C411yescharacteristic
79#14556111—21no—
80#14556107C2×C411yescharacteristic
81#14556102C2×C411yescharacteristic
82C4×Dic142C2311yescharacteristic
83C4×D284C2311yescharacteristic
84C4×C22⋊D71—21no—
85#14556085—21no—
86#14556061C2×C411yescharacteristic
87#14556061C2×C411yescharacteristic
88#14556059C2×C411yescharacteristic
89#14556059C2×C411yescharacteristic
90C4⋊(C71⋊C8)—21no—
91C4⋊(C71⋊C8)—21no—
92C4⋊(C71⋊C8)C2311yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94Dic71⋊C8—21no—
95Dic71⋊C8—21no—
96#14556052C2×C411yescharacteristic
97#14556052C2×C411yescharacteristic
98#14556051C2×C411yescharacteristic
99#14556051C2×C411yescharacteristic
100#14556049C2×C411yescharacteristic
101#14556049C2×C411yescharacteristic
102#14556048C2×C411yescharacteristic
103#14556048C2×C411yescharacteristic
104C22⋊(C71⋊C8)—21no—
105(C2×D71)⋊C8—21no—
106(C2×D71)⋊C8—21no—
107#14556038—21no—
108C4×C71⋊C8C2×C411yescharacteristic
109C4×C71⋊C8C2×C411yescharacteristic
110C4×C71⋊C8C2311yescharacteristic
111C4×C71⋊C8C2311yescharacteristic
112C4×C71⋊C8C2311yescharacteristic
113C4×C71⋊C8C2311yescharacteristic
114C8×Dic71—21no—
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
144C2×Dic142C22×C411yescharacteristic
145C2×D284C22×C411yescharacteristic
146C2×C22⋊D71—21no—
147C2×C4×D71—21no—
148#11765855—22no—
149#11765853—22no—
150#11765852—22no—
151#11765850—22no—
152#11765849—81no—
153#11765848—81no—
154C2×C71⋊C8—22no—
155C2×C71⋊C8—22no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8—21no—
160C2×C71⋊C8C22×C411yescharacteristic
161C2×C71⋊C8C22×C411yescharacteristic
162C2×C71⋊C8C22×C411yescharacteristic
163C2×C71⋊C8C22×C411yescharacteristic
164C8×D71—81no—
165C4⋊Dic71C22×C411yescharacteristic
166C4⋊Dic71C22×C411yescharacteristic
167Dic71⋊C4—21no—
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170Dic71⋊C4—21no—
171C22⋊Dic71—21no—
172(C2×D71)⋊C4—21no—
173(C2×D71)⋊C4—21no—
174(C2×D71)⋊C4—21no—
175(C2×D71)⋊C4—21no—
176C4×Dic71—21no—
177C2×Q8×C71C22×C411yescharacteristic
178C2×D4×C71—21no—
179C2×D4×C71C22×C411yescharacteristic
180C22×C4×C71—21no—
181C22×C4×C71C22×C411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71#4111yescharacteristic
184M16×C71—81no—
185M16×C71—81no—
186C2×C8×C71—21no—
187C2×C8×C71—21no—
188C2×C8×C71—81no—
189C2×C8×C71C2×D411yescharacteristic
190C2×C8×C71C2×D411yescharacteristic
191C2×C8×C71C2×D411yescharacteristic
192C2×C8×C71C2×D411yescharacteristic
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—
201C22⋊C4×C71—21no—
202C22⋊C4×C71—21no—
203C22⋊C4×C71—21no—
204C42×C71—21no—
205C42×C71C22×C411yescharacteristic
206C42×C71C2×D411yescharacteristic
207C42×C71#4111yescharacteristic
208C42×C71C2411yescharacteristic
209C23×C71—21no—
210C23×C71C4×D411yescharacteristic
211C22×D71—21no—
212Q8×C71—21no—
213Q8×C71—21no—
214D4×C71—21no—
215D4×C71—21no—
216D4×C71—81no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—41no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71C4×D411yescharacteristic
228C2×C4×C71C4×D411yescharacteristic
229C2×C4×C71C4×D411yescharacteristic
230C2×C4×C71C23×C411yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C2×#4111yescharacteristic
233C22⋊D71—81no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237D284—21no—
238D284—21no—
239C4×D71—41no—
240C4×D71—41no—
241Dic142—21no—
242Dic142—21no—
243C8×C71—21no—
244C8×C71—21no—
245C8×C71—21no—
246C8×C71—21no—
247C8×C71—41no—
248C8×C71—41no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—41no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C71⋊C8#13112yes—
256C71⋊C8#13112yes—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71C2×C4×D411yescharacteristic
264C2×D71—41no—
265C2×D71—41no—
266C2×D71—41no—
267C4×C71—21no—
268C4×C71—21no—
269C4×C71—21no—
270C4×C71—21no—
271C4×C71—41no—
272C4×C71—41no—
273C4×C71—41no—
274C4×C71C2×#13111yescharacteristic
275C4×C71C2×#13111yescharacteristic
276C4×C71#57711yescharacteristic
277C4×C71#57711yescharacteristic
278Dic71—41no—
279Dic71—41no—
280Dic71—41no—
281#17974—711nomaximal, Sylow
282C2×C71—41no—
283C2×C71—81no—
284C2×C71#293111yescharacteristic
285C2×C71#295211yescharacteristic
286C2×C71#295711yescharacteristic
287D71—81no—
288#3168—711no—
289#2993—711no—
290#2993D7111yescharacteristic, complemented
291#1594—711no—
292#1593—711no—
293#1590—711no—
294#1590—711no—
295#1580—711no—
296#1579—711no—
297C8×SD16—711no—
298C8×SD16—711no—
299#1566—711no—
300#1495—711no—
301#1495—711no—
302#1450—711no—
303C71#1797411yescharacteristic, complemented, Sylow
304#529—711no—
305#529C2×D7111yescharacteristic
306#499—711no—
307#495—711no—
308#488—711no—
309#488—711no—
310#467—711no—
311#466—711no—
312#465—711no—
313#465—711no—
314#446—711no—
315#446C2×D7111yescharacteristic
316C8×Q8—711no—
317C8×Q8C2×D7111yescharacteristic
318C4×Q16—711no—
319C4×SD16—711no—
320C4×SD16—711no—
321C4×D8—711no—
322#436—21no—
323#436—711no—
324#436—1421no—
325#436C2×D7111yescharacteristic
326#435—21no—
327#435—1421no—
328C8×D4—711no—
329C8×D4C2×D7111yescharacteristic
330#433—21no—
331#433—1421no—
332#431—711no—
333#431C2×D7111yescharacteristic
334#424—21no—
335#424—1421no—
336#423—711no—
337#423C2×D7111yescharacteristic
338C2×C4×C8—711no—
339#334—711no—
340#334—711no—
341#334—711no—
342#334—711no—
343Q8⋊C8—711no—
344Q8⋊C8—711no—
345Q8⋊C8—711no—
346Q8⋊C8—711no—
347D4⋊C8—711no—
348D4⋊C8—711no—
349D4⋊C8—711no—
350D4⋊C8—711no—
351#322—711no—
352#322—711no—
353C82—711no—
354C82—711no—
355C2×Q16—711no—
356C2×SD16—711no—
357C2×SD16—711no—
358C2×D8—711no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—1421no—
362C2×M16—1421no—
363C22×C8—21no—
364C22×C8—711no—
365C22×C8—711no—
366C22×C8—1421no—
367#126—21no—
368#126—1421no—
369#125—711no—
370#125C4×D7111yescharacteristic
371#124—711no—
372#124C4×D7111yescharacteristic
373#123—21no—
374#123—1421no—
375C22⋊Q8—711no—
376C22⋊Q8C4×D7111yescharacteristic
377#121—711no—
378#121C4×D7111yescharacteristic
379C4×Q8—711no—
380C4×Q8C22×D7111yescharacteristic
381C4×D4—21no—
382C4×D4—711no—
383C4×D4—1421no—
384C4×D4C22×D7111yescharacteristic
385#117—21no—
386#117—1421no—
387C2×C42C22×D7111yescharacteristic
388#107—711no—
389#107—711no—
390#106—711no—
391#106—711no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395C4⋊C8—711no—
396C4⋊C8—711no—
397C4⋊C8—1421no—
398C4⋊C8—1421no—
399C4⋊C8—1421no—
400C4⋊C8—1421no—
401C4⋊C8C22⋊D7111yescharacteristic
402C4⋊C8C22⋊D7111yescharacteristic
403C4⋊C8C22×D7111yescharacteristic
404C4⋊C8C22×D7111yescharacteristic
405#103—711no—
406#103—711no—
407#103—711no—
408#103—711no—
409#102—711no—
410#102—711no—
411#102—711no—
412#102—711no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—1421no—
416C22⋊C8—1421no—
417C22⋊C8—1421no—
418C22⋊C8C22⋊D7111yescharacteristic
419C22⋊C8C22⋊D7111yescharacteristic
420#97—21no—
421#97—1421no—
422C4×C8—21no—
423C4×C8—711no—
424C4×C8—711no—
425C4×C8—711no—
426C4×C8—711no—
427C4×C8—711no—
428C4×C8—711no—
429C4×C8—1421no—
430C4×C8C22×D7111yescharacteristic
431C4×C8C22×D7111yescharacteristic
432C2×Q8—711no—
433C2×Q8C2×C4×D7111yescharacteristic
434C2×D4—21no—
435C2×D4—711no—
436C2×D4—1421no—
437C2×D4C2×C4×D7111yescharacteristic
438C22×C4—21no—
439C22×C4—1421no—
440C22×C4C2×C4×D7111yescharacteristic
441C22×C4C2×C22⋊D7111yescharacteristic
442C22×C4#1176586511yescharacteristic
443Q16—1422no—
444SD16—1422no—
445SD16—1422no—
446D8—1422no—
447M16—81no—
448M16—81no—
449M16—5681no—
450M16—5681no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—81no—
454C2×C8—5681no—
455C2×C8—711no—
456C2×C8—711no—
457C2×C8—711no—
458C2×C8—711no—
459C2×C8—1421no—
460C2×C8—1421no—
461C2×C8—1421no—
462C2×C8—1421no—
463C2×C8—1422no—
464C2×C8—1422no—
465C2×C8C2×C22⋊D7111yescharacteristic
466C2×C8C2×C22⋊D7111yescharacteristic
467C2×C8C2×C22⋊D7111yescharacteristic
468C2×C8C2×C22⋊D7111yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—711no—
474C4⋊C4—711no—
475C4⋊C4—1421no—
476C4⋊C4—1421no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4C2×C4×D7111yescharacteristic
480C4⋊C4C2×C4×D7111yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1421no—
487C22⋊C4—1421no—
488C22⋊C4—1421no—
489C22⋊C4—1421no—
490C22⋊C4—1421no—
491C42—21no—
492C42—1421no—
493C42C2×C4×D7111yescharacteristic
494C42C2×C22⋊D7111yescharacteristic
495C42#1176586511yescharacteristic
496C42C23×D7111yescharacteristic
497C23—21no—
498C23—1421no—
499C23C4×C22⋊D7111yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—1421no—
503Q8—1421no—
504D4—21no—
505D4—21no—
506D4—81no—
507D4—5681no—
508D4—1421no—
509D4—1421no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—2841no—
521C2×C4—2841no—
522C2×C4—1421no—
523C2×C4—1421no—
524C2×C4—1421no—
525C2×C4C4×C22⋊D7111yescharacteristic
526C2×C4C4×C22⋊D7111yescharacteristic
527C2×C4C4×C22⋊D7111yescharacteristic
528C2×C4C22×C4×D7111yescharacteristic
529C2×C4C22×C22⋊D7111yescharacteristic
530C2×C4C2×#1176586511yescharacteristic
531C8—21no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—41no—
536C8—41no—
537C8—2841no—
538C8—2841no—
539C8—2841no—
540C8—2841no—
541C8—712no—
542C8—712no—
543C8—1421no—
544C8—1421no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—2841no—
552C22—2841no—
553C22—2841no—
554C22C2×C4×C22⋊D7111yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—2841no—
563C4—2841no—
564C4—2841no—
565C4#131×D7111yescharacteristic
566C4#131×D7111yescharacteristic
567C4#1798680711yescharacteristic
568C4#1798681011yescharacteristic
569C2—41no—
570C2—81no—
571C2—5681no—
572C2#2259705011yescharacteristic
573C2#2259722111yescharacteristic
574C2#2259725511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2993C2
socleC22×C71C2×C4×D4
prime radical#17986233C4
cosocleC42×C71C24
hypercenter#2993D71
hypocenterC71#17974
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. #22597491C2semidirect
3. #22587252C2semidirect
4. #22587247C2semidirect
5. #22587230C2semidirect
6. #22587228C2semidirect
7. #22587162C2semidirect
8. #22587157C2semidirect
9. C8×#11765852C2semidirect
10. C8×#11765853C2semidirect
11. #22587095C2semidirect
12. #22586805C2semidirect
13. #22586803C2semidirect
14. #22586624C2semidirect
15. C71×#2993C2semidirect
290. #2993D71semidirect
303. C71#17974semidirect

Maximal subgroups

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

subgroupindex
#225989042
#225974912
#225872522
#225872472
#225872302
#225872282
#225871622
#225871572
C8×#117658522
C8×#117658532
#225870952
#225868052
#225868032
#225866242
C71×#29932
#1797471