Finite Group Databasebeta

← #31968539

Subgroups — #31968539

Order 18176.

594 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598405C211yesmaximal, characteristic
2#22597352C211yesmaximal, characteristic
3#22597289C211yesmaximal, characteristic, complemented
4#22587246C211yesmaximal, characteristic, complemented
5#22587234C211yesmaximal, characteristic, complemented
6#22587208C211yesmaximal, characteristic, complemented
7#22587200C211yesmaximal, characteristic, complemented
8#22587187C211yesmaximal, characteristic, complemented
9#22587179C211yesmaximal, characteristic, complemented
10#22587155C211yesmaximal, characteristic, complemented
11#22587137C211yesmaximal, characteristic, complemented
12#22587126C211yesmaximal, characteristic, complemented
13#22586728C211yesmaximal, characteristic
14#22586635C211yesmaximal, characteristic, complemented
15C71×#1497C211yesmaximal, characteristic, complemented
16C4×#11765865C2211yescharacteristic
17#17986337C411yescharacteristic
18#17986337C411yescharacteristic
19#17986281C411yescharacteristic
20#17986276C411yescharacteristic
21#17986206C411yescharacteristic
22#17986201C411yescharacteristic
23#17986159C411yescharacteristic
24#17986159C411yescharacteristic
25#17986130—21no—
26#17986130—21no—
27#17986130C2211yescharacteristic
28#17986130C2211yescharacteristic
29#17986126C2211yescharacteristic
30#17986126C2211yescharacteristic
31Q8×C71⋊C8C2211yescharacteristic
32Q8×C71⋊C8C2211yescharacteristic
33C4×#11765855C2211yescharacteristic
34C4×#11765855C2211yescharacteristic
35C4×#11765852C2211yescharacteristic
36C4×#11765852C2211yescharacteristic
37#17986073—21no—
38#17986073—21no—
39#17986064C2211yescharacteristic
40#17986064C2211yescharacteristic
41#17986054—21no—
42#17986052C2211yescharacteristic
43#17986026—21no—
44C2×C4⋊(C71⋊C8)C2211yescharacteristic
45#17985955—21no—
46C4×#11765849C2211yescharacteristic
47#97×D71—21no—
48#17985751C2211yescharacteristic
49#17985751C2211yescharacteristic
50#17985749C2211yescharacteristic
51#17985749C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985728C2211yescharacteristic
55#17985728C2211yescharacteristic
56#17985721C2211yescharacteristic
57#17985721C2211yescharacteristic
58#17985720C2211yescharacteristic
59C8×C71⋊C8C2211yescharacteristic
60#533×C71C2211yescharacteristic
61#432×C71C2211yescharacteristic
62#431×C71C2211yescharacteristic
63Q8⋊C8×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65Q8⋊C8×C71C2211yescharacteristic
66Q8⋊C8×C71C2211yescharacteristic
67C2×#11765865C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765855C2×C411yescharacteristic
70C2×#11765852C2×C411yescharacteristic
71C2×#11765852C2×C411yescharacteristic
72C2×#11765849D411yescharacteristic
73C2×#11765849D411yescharacteristic
74C22×C71⋊C8—21no—
75C2×C8×D71—21no—
76C2×C8×D71—21no—
77C4×Dic142C2×C411yescharacteristic
78C4×Dic142C2311yescharacteristic
79C4×D284C2×C411yescharacteristic
80C4×D284C2311yescharacteristic
81C4×C22⋊D71—21no—
82C4×C22⋊D71—21no—
83#14556087C2×C411yescharacteristic
84#14556085—21no—
85C42×D71—21no—
86#14556061C2×C411yescharacteristic
87#14556061C2×C411yescharacteristic
88#14556059C2×C411yescharacteristic
89#14556059C2×C411yescharacteristic
90C4⋊(C71⋊C8)—21no—
91C4⋊(C71⋊C8)—21no—
92C4⋊(C71⋊C8)—21no—
93C4⋊(C71⋊C8)—21no—
94C4⋊(C71⋊C8)—21no—
95C4⋊(C71⋊C8)C2×C411yescharacteristic
96C4⋊(C71⋊C8)C2×C411yescharacteristic
97C4⋊(C71⋊C8)C2311yescharacteristic
98C4⋊(C71⋊C8)C2311yescharacteristic
99Dic71⋊C8—21no—
100Dic71⋊C8—21no—
101#14556052C2×C411yescharacteristic
102#14556052C2×C411yescharacteristic
103#14556052C2×C411yescharacteristic
104#14556052C2×C411yescharacteristic
105#14556051C2×C411yescharacteristic
106#14556051C2×C411yescharacteristic
107#14556048C2×C411yescharacteristic
108#14556048C2×C411yescharacteristic
109C22⋊(C71⋊C8)C2×C411yescharacteristic
110C22⋊(C71⋊C8)C2×C411yescharacteristic
111(C2×D71)⋊C8—21no—
112(C2×D71)⋊C8—21no—
113#14556039—21no—
114#14556039—21no—
115#14556039D411yescharacteristic
116#14556039D411yescharacteristic
117#14556038—21no—
118#14556038—21no—
119C4×C71⋊C8—21no—
120C4×C71⋊C8—21no—
121C4×C71⋊C8C2311yescharacteristic
122C4×C71⋊C8C2311yescharacteristic
123C4×C71⋊C8C2311yescharacteristic
124C4×C71⋊C8C2311yescharacteristic
125C4⋊Q8×C71C2×C411yescharacteristic
126C4⋊Q8×C71C2×C411yescharacteristic
127#125×C71—21no—
128C22⋊Q8×C71—21no—
129C22⋊Q8×C71C2×C411yescharacteristic
130C22⋊Q8×C71C2×C411yescharacteristic
131C4×Q8×C71—21no—
132C4×Q8×C71C2311yescharacteristic
133C4×Q8×C71C2311yescharacteristic
134#117×C71—21no—
135C2×C42×C71C2311yescharacteristic
136C4⋊C8×C71—21no—
137C4⋊C8×C71C2311yescharacteristic
138C4⋊C8×C71C2311yescharacteristic
139C22⋊C8×C71—21no—
140C22⋊C8×C71—21no—
141#97×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144#11765865—22no—
145#11765865—41no—
146C2×Dic142C22×C411yescharacteristic
147C2×D284C22×C411yescharacteristic
148C2×C22⋊D71—21no—
149C2×C4×D71—21no—
150C2×C4×D71—21no—
151C2×C4×D71—21no—
152#11765855—41no—
153#11765855—41no—
154#11765852—22no—
155#11765852—22no—
156#11765849—22no—
157#11765849—41no—
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
165C2×C71⋊C8C2×D411yescharacteristic
166C2×C71⋊C8C2×D411yescharacteristic
167C2×C71⋊C8C2×D411yescharacteristic
168C2×C71⋊C8C2×D411yescharacteristic
169C8×D71—42no—
170C8×D71—42no—
171C4⋊Dic71C22×C411yescharacteristic
172C4⋊Dic71C22×C411yescharacteristic
173Dic71⋊C4—21no—
174Dic71⋊C4—21no—
175C22⋊Dic71—21no—
176(C2×D71)⋊C4—21no—
177(C2×D71)⋊C4—21no—
178C4×Dic71—21no—
179C4×Dic71—21no—
180C4×Dic71—21no—
181C2×Q8×C71—21no—
182C2×Q8×C71C22×C411yescharacteristic
183C2×Q8×C71C22×C411yescharacteristic
184C22×C4×C71C22×C411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71#4111yescharacteristic
187C2×C8×C71—21no—
188C2×C8×C71—21no—
189C2×C8×C71—21no—
190C2×C8×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×C71—21no—
198C4⋊C4×C71C22×C411yescharacteristic
199C4⋊C4×C71C22×C411yescharacteristic
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71C22×C411yescharacteristic
205C42×C71C2×D411yescharacteristic
206C42×C71#4111yescharacteristic
207C42×C71C2411yescharacteristic
208C23×C71C4×D411yescharacteristic
209C22×D71—21no—
210Q8×C71—21no—
211Q8×C71—21no—
212Q8×C71—21no—
213Q8×C71—21no—
214Q8×C71—81no—
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—21no—
223C2×C4×C71—41no—
224C2×C4×C71C4×D411yescharacteristic
225C2×C4×C71C4×D411yescharacteristic
226C2×C4×C71C4×D411yescharacteristic
227C2×C4×C71C23×C411yescharacteristic
228C2×C4×C71C22×D411yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C22⋊D71—42no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234D284—21no—
235D284#13112yes—
236C4×D71—42no—
237C4×D71—41no—
238C4×D71—41no—
239C4×D71—81no—
240Dic142—21no—
241Dic142—21no—
242C8×C71—41no—
243C8×C71—41no—
244C8×C71—41no—
245C8×C71—41no—
246C71⋊C8—21no—
247C71⋊C8—21no—
248C71⋊C8—21no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—21no—
253C71⋊C8—41no—
254C71⋊C8—41no—
255C71⋊C8#13712yes—
256C22×C71—21no—
257C22×C71—21no—
258C22×C71—21no—
259C22×C71C2×C4×D411yescharacteristic
260C2×D71—22no—
261C2×D71—41no—
262C2×D71—41no—
263C4×C71—21no—
264C4×C71—21no—
265C4×C71—21no—
266C4×C71—21no—
267C4×C71—41no—
268C4×C71—41no—
269C4×C71—41no—
270C4×C71—41no—
271C4×C71C2×#13111yescharacteristic
272C4×C71#56811yescharacteristic
273C4×C71C2×#13711yescharacteristic
274C4×C71#57511yescharacteristic
275Dic71—41no—
276Dic71—41no—
277Dic71—41no—
278#18009—711nomaximal, Sylow
279C2×C71—41no—
280C2×C71#293311yescharacteristic
281C2×C71C4×#13711yescharacteristic
282C2×C71#295711yescharacteristic
283D71—42no—
284#3107—711no—
285#2970—711no—
286#2962—711no—
287#1593—711no—
288#1591—711no—
289#1587—711no—
290#1585—711no—
291#1584—711no—
292#1582—711no—
293SD16⋊C8—711no—
294C8×SD16—711no—
295#1573—711no—
296#1497D7111yescharacteristic, complemented
297#1477—711no—
298#1453—711no—
299C71#1800911yescharacteristic, complemented, Sylow
300#533C2×D7111yescharacteristic
301C4×#41—711no—
302#485—711no—
303#485—711no—
304#475—711no—
305#474—711no—
306#462—711no—
307#461—711no—
308#452—711no—
309#452—711no—
310#446—711no—
311#446—711no—
312#446—711no—
313#446—711no—
314#446—1421no—
315#446—1421no—
316C8×Q8—711no—
317C8×Q8—711no—
318C4×Q16—711no—
319C4×Q16—711no—
320C4×SD16—711no—
321C4×SD16—711no—
322#435—1421no—
323#435—1421no—
324C8×D4—711no—
325C8×D4—711no—
326#432—1421no—
327#432C2×D7111yescharacteristic
328#431—711no—
329#431C2×D7111yescharacteristic
330#424—1421no—
331C2×C4⋊C8—711no—
332#405—1421no—
333C4×M16—711no—
334C2×#97—1421no—
335#335—711no—
336#335—711no—
337#334—711no—
338#334—711no—
339Q8⋊C8—711no—
340Q8⋊C8—711no—
341Q8⋊C8C2×D7111yescharacteristic
342Q8⋊C8C2×D7111yescharacteristic
343Q8⋊C8C2×D7111yescharacteristic
344Q8⋊C8C2×D7111yescharacteristic
345D4⋊C8—711no—
346D4⋊C8—711no—
347#322—711no—
348#322—711no—
349#322—711no—
350C82—711no—
351C2×#41—711no—
352C2×Q16—711no—
353C2×Q16—711no—
354C2×SD16—711no—
355C2×SD16—711no—
356C2×M16—711no—
357C2×M16—711no—
358C22×C8—1421no—
359C22×C8—1421no—
360C22×C8—1421no—
361C4⋊Q8C4×D7111yescharacteristic
362C4⋊Q8C4×D7111yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C4×D7111yescharacteristic
366C22⋊Q8C4×D7111yescharacteristic
367C4×Q8—21no—
368C4×Q8—711no—
369C4×Q8—711no—
370C4×Q8C22×D7111yescharacteristic
371C4×Q8C22×D7111yescharacteristic
372C4×D4—711no—
373C4×D4—711no—
374C4×D4—1421no—
375C4×D4—1421no—
376#117—21no—
377#117—711no—
378#117—1421no—
379C2×C42—1421no—
380C2×C42C22×D7111yescharacteristic
381#107—711no—
382#107—711no—
383#106—711no—
384#106—711no—
385C4⋊C8—21no—
386C4⋊C8—711no—
387C4⋊C8—711no—
388C4⋊C8—711no—
389C4⋊C8—711no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8—1421no—
394C4⋊C8—1421no—
395C4⋊C8—1421no—
396C4⋊C8—1421no—
397C4⋊C8C22×D7111yescharacteristic
398C4⋊C8C22×D7111yescharacteristic
399#103—711no—
400#103—711no—
401#103—711no—
402#103—711no—
403#103—711no—
404#103—711no—
405#102—711no—
406#102—711no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8—711no—
410C22⋊C8—711no—
411C22⋊C8—1421no—
412C22⋊C8—1421no—
413#97—21no—
414#97—711no—
415#97—711no—
416#97—1421no—
417#97—1421no—
418#97—1421no—
419#97—1421no—
420C4×C8—711no—
421C4×C8—711no—
422C4×C8—711no—
423C4×C8—711no—
424C4×C8—1421no—
425C4×C8—1421no—
426C4×C8C22×D7111yescharacteristic
427C4×C8C22×D7111yescharacteristic
428#41—2841no—
429#41—1422no—
430C2×Q8—21no—
431C2×Q8—711no—
432C2×Q8C2×C4×D7111yescharacteristic
433C2×Q8C2×C4×D7111yescharacteristic
434C2×D4—711no—
435C2×D4—1421no—
436C22×C4—1421no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4C2×C4×D7111yescharacteristic
440C22×C4D4×D7111yescharacteristic
441C22×C4#1176586411yescharacteristic
442Q16—2841no—
443Q16—2841no—
444SD16—1422no—
445SD16—1422no—
446M16—2841no—
447M16—1422no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—2842no—
453C2×C8—2842no—
454C2×C8—2842no—
455C2×C8—711no—
456C2×C8—711no—
457C2×C8—711no—
458C2×C8—711no—
459C2×C8—711no—
460C2×C8—711no—
461C2×C8—711no—
462C2×C8—711no—
463C2×C8—1421no—
464C2×C8—1421no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—711no—
473C4⋊C4—711no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4C2×C4×D7111yescharacteristic
477C4⋊C4C2×C4×D7111yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1421no—
481C22⋊C4—1421no—
482C22⋊C4—1421no—
483C42—21no—
484C42—21no—
485C42—1421no—
486C42—1421no—
487C42—1421no—
488C42C2×C4×D7111yescharacteristic
489C42D4×D7111yescharacteristic
490C42#1176586411yescharacteristic
491C42C23×D7111yescharacteristic
492C23—1421no—
493C23(C22⋊D71)⋊C411yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—1421no—
500Q8—1421no—
501D4—2842no—
502D4—712no—
503D4—1421no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—2842no—
514C2×C4—2841no—
515C2×C4—2841no—
516C2×C4—5681no—
517C2×C4—1421no—
518C2×C4—1421no—
519C2×C4—1421no—
520C2×C4(C22⋊D71)⋊C411yescharacteristic
521C2×C4(C22⋊D71)⋊C411yescharacteristic
522C2×C4(C22⋊D71)⋊C411yescharacteristic
523C2×C4C22×C4×D7111yescharacteristic
524C2×C4C2×D4×D7111yescharacteristic
525C2×C4C2×#1176586411yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—2841no—
531C8—2841no—
532C8—712no—
533C8—1421no—
534C8—1421no—
535C8—1421no—
536C8—1421no—
537C8—1421no—
538C8—1421no—
539C8—1421no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2841no—
544C22—2841no—
545C22—1422no—
546C22C2×(C22⋊D71)⋊C411yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—2841no—
556C4—2841no—
557C4—2841no—
558C4#131×D7111yescharacteristic
559C4#1798676411yescharacteristic
560C4#137×D7111yescharacteristic
561C4#1798679811yescharacteristic
562C2—41no—
563C2—2842no—
564C2#2259707011yescharacteristic
565C2#2259715511yescharacteristic
566C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1497C2
socleC22×C71C2×C4×D4
prime radicalC2×#11765865C2×C4
cosocleC42×C71C24
hypercenter#1497D71
hypocenterC71#18009
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
3. #22597289C2semidirect
4. #22587246C2semidirect
5. #22587234C2semidirect
6. #22587208C2semidirect
7. #22587200C2semidirect
8. #22587187C2semidirect
9. #22587179C2semidirect
10. #22587155C2semidirect
11. #22587137C2semidirect
12. #22587126C2semidirect
14. #22586635C2semidirect
15. C71×#1497C2semidirect
296. #1497D71semidirect
299. C71#18009semidirect

Maximal subgroups

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

subgroupindex
#225984052
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C71×#14972
#1800971