Finite Group Databasebeta

← #31968584

Subgroups — #31968584

Order 18176.

592 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#22598416C211yesmaximal, characteristic
2#22597345C211yesmaximal, characteristic
3#22597291C211yesmaximal, characteristic, complemented
4#22587246C211yesmaximal, characteristic, complemented
5#22587233C211yesmaximal, characteristic, complemented
6#22587214C211yesmaximal, characteristic, complemented
7#22587194C211yesmaximal, characteristic, complemented
8#22587187C211yesmaximal, characteristic, complemented
9#22587186C211yesmaximal, characteristic, complemented
10#22587149C211yesmaximal, characteristic, complemented
11#22587144C211yesmaximal, characteristic, complemented
12#22587131C211yesmaximal, characteristic, complemented
13#22586736C211yesmaximal, characteristic, complemented
14#22586635C211yesmaximal, characteristic, complemented
15C71×#1495C211yesmaximal, characteristic
16C2×C4×Dic142C2211yescharacteristic
17#17986327C411yescharacteristic
18#17986310C411yescharacteristic
19#17986303C411yescharacteristic
20#17986303C411yescharacteristic
21#17986218C411yescharacteristic
22#17986218C411yescharacteristic
23#17986154C411yescharacteristic
24#17986137C411yescharacteristic
25#17986130C2211yescharacteristic
26#17986128—21no—
27#17986128—21no—
28#17986126C2211yescharacteristic
29#17986126C2211yescharacteristic
30Q8×C71⋊C8C2211yescharacteristic
31#17986123C2211yescharacteristic
32#17986123C2211yescharacteristic
33C4×#11765855C2211yescharacteristic
34C4×#11765855C2211yescharacteristic
35C4×#11765853C2211yescharacteristic
36C4×#11765853C2211yescharacteristic
37#17986077—21no—
38#17986077C2211yescharacteristic
39#17986071—21no—
40D4×C71⋊C8C2211yescharacteristic
41#17986060—21no—
42#17986052C2211yescharacteristic
43#17986052C2211yescharacteristic
44#17986022—21no—
45#17985955—21no—
46C4×#11765849C2211yescharacteristic
47M16×Dic71—21no—
48#17985751C2211yescharacteristic
49#17985751C2211yescharacteristic
50#17985749C2211yescharacteristic
51#17985749C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985731C2211yescharacteristic
55#17985731C2211yescharacteristic
56#17985721C2211yescharacteristic
57#17985721C2211yescharacteristic
58#17985720C2211yescharacteristic
59C8×C71⋊C8C2211yescharacteristic
60#529×C71C2211yescharacteristic
61#431×C71C2211yescharacteristic
62#423×C71C2211yescharacteristic
63Q8⋊C8×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65D4⋊C8×C71C2211yescharacteristic
66D4⋊C8×C71C2211yescharacteristic
67C22×Dic142C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765855C2×C411yescharacteristic
70C2×#11765853C2×C411yescharacteristic
71C2×#11765853C2×C411yescharacteristic
72C2×#11765849—21no—
73C2×#11765849D411yescharacteristic
74C2×#11765849D411yescharacteristic
75C22×C71⋊C8—21no—
76C4×Dic142—21no—
77C4×Dic142—21no—
78C4×Dic142C2×C411yescharacteristic
79C4×Dic142C2×C411yescharacteristic
80C4×Dic142C2311yescharacteristic
81C4×Dic142C2311yescharacteristic
82C2×C4⋊Dic71C2×C411yescharacteristic
83C2×Dic71⋊C4—21no—
84C2×C4×Dic71—21no—
85#14556061C2×C411yescharacteristic
86#14556061C2×C411yescharacteristic
87#14556059C2×C411yescharacteristic
88#14556059C2×C411yescharacteristic
89C4⋊(C71⋊C8)—21no—
90C4⋊(C71⋊C8)—21no—
91C4⋊(C71⋊C8)C2×C411yescharacteristic
92C4⋊(C71⋊C8)C2×C411yescharacteristic
93C4⋊(C71⋊C8)C2311yescharacteristic
94C4⋊(C71⋊C8)C2311yescharacteristic
95Dic71⋊C8—21no—
96Dic71⋊C8—21no—
97Dic71⋊C8—21no—
98Dic71⋊C8—21no—
99#14556052C2×C411yescharacteristic
100#14556052C2×C411yescharacteristic
101#14556051C2×C411yescharacteristic
102#14556051C2×C411yescharacteristic
103#14556051C2×C411yescharacteristic
104#14556051C2×C411yescharacteristic
105#14556049C2×C411yescharacteristic
106#14556049C2×C411yescharacteristic
107C22⋊(C71⋊C8)—21no—
108C22⋊(C71⋊C8)—21no—
109C22⋊(C71⋊C8)—21no—
110C22⋊(C71⋊C8)C2×C411yescharacteristic
111C22⋊(C71⋊C8)C2×C411yescharacteristic
112#14556039—21no—
113#14556039D411yescharacteristic
114#14556039D411yescharacteristic
115#14556038—21no—
116#14556038—21no—
117C4×C71⋊C8—21no—
118C4×C71⋊C8—21no—
119C4×C71⋊C8C2311yescharacteristic
120C4×C71⋊C8C2311yescharacteristic
121C4×C71⋊C8C2311yescharacteristic
122C4×C71⋊C8C2311yescharacteristic
123C8×Dic71—21no—
124C8×Dic71—21no—
125C2×M16×C71—21no—
126#126×C71—21no—
127#125×C71C2×C411yescharacteristic
128#124×C71C2×C411yescharacteristic
129#123×C71—21no—
130C22⋊Q8×C71C2×C411yescharacteristic
131#121×C71C2×C411yescharacteristic
132C4×Q8×C71C2311yescharacteristic
133C4×D4×C71—21no—
134C4×D4×C71C2311yescharacteristic
135#117×C71—21no—
136C2×C42×C71C2311yescharacteristic
137C4⋊C8×C71—21no—
138C4⋊C8×C71—21no—
139C4⋊C8×C71C2311yescharacteristic
140C4⋊C8×C71C2311yescharacteristic
141C22⋊C8×C71—21no—
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C2×Dic142—22no—
145C2×Dic142—21no—
146C2×Dic142—41no—
147C2×Dic142C22×C411yescharacteristic
148C2×Dic142C22×C411yescharacteristic
149C22×Dic71—21no—
150C22×Dic71—21no—
151#11765855—22no—
152#11765855—41no—
153#11765853—22no—
154#11765853—41no—
155#11765849—22no—
156#11765849—41no—
157#11765849—42no—
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
169C4⋊Dic71—21no—
170C4⋊Dic71C22×C411yescharacteristic
171C4⋊Dic71C22×C411yescharacteristic
172Dic71⋊C4—21no—
173Dic71⋊C4—21no—
174Dic71⋊C4—21no—
175Dic71⋊C4—21no—
176C4×Dic71—21no—
177C4×Dic71—21no—
178C4×Dic71—21no—
179C4×Dic71—21no—
180C2×Q8×C71C22×C411yescharacteristic
181C2×D4×C71—21no—
182C2×D4×C71C22×C411yescharacteristic
183C22×C4×C71—21no—
184C22×C4×C71C22×C411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71#4111yescharacteristic
187M16×C71—42no—
188C2×C8×C71—21no—
189C2×C8×C71—21no—
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71C22×C411yescharacteristic
197C4⋊C4×C71C22×C411yescharacteristic
198C22⋊C4×C71—21no—
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C22⋊C4×C71—21no—
203C42×C71—21no—
204C42×C71C22×C411yescharacteristic
205C42×C71C2×D411yescharacteristic
206C42×C71#4111yescharacteristic
207C42×C71C2411yescharacteristic
208C23×C71—21no—
209C23×C71C4×D411yescharacteristic
210Q8×C71—21no—
211Q8×C71—21no—
212D4×C71—21no—
213D4×C71—21no—
214D4×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—41no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71C4×D411yescharacteristic
226C2×C4×C71C4×D411yescharacteristic
227C2×C4×C71C4×D411yescharacteristic
228C2×C4×C71C23×C411yescharacteristic
229C2×C4×C71C22×D411yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—42no—
236C2×Dic71—81no—
237Dic142—21no—
238Dic142—21no—
239Dic142—21no—
240Dic142—42no—
241Dic142#13112yes—
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×C71—41no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71C2×C4×D411yescharacteristic
263C4×C71—21no—
264C4×C71—21no—
265C4×C71—21no—
266C4×C71—21no—
267C4×C71—41no—
268C4×C71—41no—
269C4×C71—41no—
270C4×C71C2×#13111yescharacteristic
271C4×C71#56811yescharacteristic
272C4×C71C2×#13711yescharacteristic
273C4×C71#57511yescharacteristic
274Dic71—22no—
275Dic71—41no—
276Dic71—41no—
277Dic71—41no—
278#18012—711nomaximal, Sylow
279C2×C71—41no—
280C2×C71—81no—
281C2×C71#293311yescharacteristic
282C2×C71C4×#13711yescharacteristic
283C2×C71#295711yescharacteristic
284#3108—711no—
285#2969—711no—
286Q8×M16—711no—
287#1593—711no—
288#1590—711no—
289#1587—711no—
290#1585—711no—
291#1584—711no—
292#1583—711no—
293SD16⋊C8—711no—
294C8×Q16—711no—
295#1574—711no—
296#1495D7111yescharacteristic
297#1479—711no—
298#1453—711no—
299C71#1801211yescharacteristic, complemented, Sylow
300#529C2×D7111yescharacteristic
301C2×C4×Q8—711no—
302#483—711no—
303#480—711no—
304#479—711no—
305#479—711no—
306#464—711no—
307#464—711no—
308#451—711no—
309#448—711no—
310#446—711no—
311#446—711no—
312#446—711no—
313#446—1421no—
314#446—1421no—
315C8×Q8—711no—
316C8×Q8—711no—
317C8×Q8—711no—
318C4×Q16—711no—
319C4×Q16—711no—
320C4×SD16—711no—
321C4×SD16—711no—
322#436—711no—
323#436—1421no—
324#435—1421no—
325C8×D4—711no—
326#433—1421no—
327#431—711no—
328#431—711no—
329#431C2×D7111yescharacteristic
330#423—1421no—
331#423C2×D7111yescharacteristic
332#405—1421no—
333C4×M16—711no—
334C4×M16—1421no—
335#335—711no—
336#335—711no—
337#334—711no—
338#334—711no—
339Q8⋊C8—711no—
340Q8⋊C8—711no—
341Q8⋊C8—711no—
342Q8⋊C8—711no—
343Q8⋊C8C2×D7111yescharacteristic
344Q8⋊C8C2×D7111yescharacteristic
345D4⋊C8C2×D7111yescharacteristic
346D4⋊C8C2×D7111yescharacteristic
347#322—711no—
348#322—711no—
349#322—711no—
350C82—711no—
351C22×Q8—711no—
352C2×Q16—711no—
353C2×Q16—711no—
354C2×SD16—711no—
355C2×SD16—711no—
356C2×M16—21no—
357C2×M16—711no—
358C2×M16—711no—
359C2×M16—1421no—
360C22×C8—1421no—
361#126—21no—
362#125C4×D7111yescharacteristic
363#124C4×D7111yescharacteristic
364#123—21no—
365C22⋊Q8C4×D7111yescharacteristic
366#121C4×D7111yescharacteristic
367C4×Q8—711no—
368C4×Q8—711no—
369C4×Q8—711no—
370C4×Q8—711no—
371C4×Q8—1421no—
372C4×Q8—1421no—
373C4×Q8C22×D7111yescharacteristic
374C4×D4—21no—
375C4×D4C22×D7111yescharacteristic
376#117—21no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—1421no—
379C2×C42—1421no—
380C2×C42C22×D7111yescharacteristic
381#107—711no—
382#107—711no—
383#106—711no—
384#106—711no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—711no—
388C4⋊C8—711no—
389C4⋊C8—711no—
390C4⋊C8—711no—
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—711no—
409C22⋊C8—711no—
410C22⋊C8—1421no—
411C22⋊C8—1421no—
412C22⋊C8—1421no—
413#97—711no—
414#97—711no—
415#97—1421no—
416#97—1421no—
417#97—1421no—
418C4×C8—711no—
419C4×C8—711no—
420C4×C8—711no—
421C4×C8—711no—
422C4×C8—1421no—
423C4×C8—1421no—
424C4×C8—1421no—
425C4×C8—1421no—
426C4×C8C22×D7111yescharacteristic
427C4×C8C22×D7111yescharacteristic
428C2×Q8—2841no—
429C2×Q8—711no—
430C2×Q8—711no—
431C2×Q8—1422no—
432C2×Q8—1421no—
433C2×Q8C2×C4×D7111yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D7111yescharacteristic
436C22×C4—21no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4C2×C4×D7111yescharacteristic
440C22×C4D4×D7111yescharacteristic
441C22×C4#1176586411yescharacteristic
442Q16—2841no—
443Q16—1422no—
444SD16—2841no—
445SD16—1422no—
446M16—42no—
447M16—2842no—
448M16—2841no—
449M16—1422no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
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—711no—
470C4⋊C4—711no—
471C4⋊C4—1421no—
472C4⋊C4—1421no—
473C4⋊C4—1421no—
474C4⋊C4—1421no—
475C4⋊C4—1421no—
476C4⋊C4C2×C4×D7111yescharacteristic
477C4⋊C4C2×C4×D7111yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C42—21no—
484C42—1421no—
485C42—1421no—
486C42—1421no—
487C42—1421no—
488C42C2×C4×D7111yescharacteristic
489C42D4×D7111yescharacteristic
490C42#1176586411yescharacteristic
491C42C23×D7111yescharacteristic
492C23—21no—
493C23(C22⋊D71)⋊C411yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—2842no—
497Q8—712no—
498Q8—1421no—
499Q8—1421no—
500Q8—1421no—
501D4—21no—
502D4—21no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—2842no—
515C2×C4—5681no—
516C2×C4—1421no—
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—41no—
544C22—41no—
545C22—41no—
546C22C2×(C22⋊D71)⋊C411yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—2841no—
555C4—2841no—
556C4—2841no—
557C4—1422no—
558C4#131×D7111yescharacteristic
559C4#1798676411yescharacteristic
560C4#137×D7111yescharacteristic
561C4#1798679511yescharacteristic
562C2—41no—
563C2—81no—
564C2#2259707011yescharacteristic
565C2#2259716411yescharacteristic
566C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1495C2
socleC22×C71C2×C4×D4
prime radical#121×C71C2×C4
cosocleC42×C71C24
hypercenter#1495D71
hypocenterC71#18012
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1495

Complements

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

NH ≅ G/Ntype
3. #22597291C2semidirect
4. #22587246C2semidirect
5. #22587233C2semidirect
6. #22587214C2semidirect
7. #22587194C2semidirect
8. #22587187C2semidirect
9. #22587186C2semidirect
10. #22587149C2semidirect
11. #22587144C2semidirect
12. #22587131C2semidirect
13. #22586736C2semidirect
14. #22586635C2semidirect
299. C71#18012semidirect

Maximal subgroups

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

subgroupindex
#225984162
#225973452
#225972912
#225872462
#225872332
#225872142
#225871942
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