Finite Group Databasebeta

← #31967563

Subgroups — #31967563

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598405C211yesmaximal, characteristic
2#22597449C211yesmaximal, characteristic
3#22597435C211yesmaximal, characteristic, complemented
4#22587214C211yesmaximal, characteristic, complemented
5#22587200C211yesmaximal, characteristic, complemented
6#22587193C211yesmaximal, characteristic, complemented
7#22587179C211yesmaximal, characteristic, complemented
8#22587156C211yesmaximal, characteristic, complemented
9#22587155C211yesmaximal, characteristic, complemented
10#22587144C211yesmaximal, characteristic, complemented
11#22587137C211yesmaximal, characteristic, complemented
12#22587123C211yesmaximal, characteristic, complemented
13#22586775C211yesmaximal, characteristic
14#22586640C211yesmaximal, characteristic, complemented
15C71×#1481C211yesmaximal, 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
29Q8×C71⋊C8C2211yescharacteristic
30Q8×C71⋊C8C2211yescharacteristic
31#17986123C2211yescharacteristic
32#17986123C2211yescharacteristic
33C4×#11765855C2211yescharacteristic
34C4×#11765855C2211yescharacteristic
35C4×#11765852C2211yescharacteristic
36C4×#11765852C2211yescharacteristic
37#17986064C2211yescharacteristic
38#17986064C2211yescharacteristic
39C8×C22⋊D71—21no—
40C8×C22⋊D71—21no—
41#17986060—21no—
42#17986060—21no—
43#17986056C2211yescharacteristic
44#17986052C2211yescharacteristic
45#17986052C2211yescharacteristic
46#17986051—21no—
47C4⋊C8×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#431×C71C2211yescharacteristic
62C2×C4⋊C8×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×C8×D71—21no—
73C2×C8×D71—21no—
74C4×Dic142C2×C411yescharacteristic
75C4×Dic142C2311yescharacteristic
76C4×D284C2×C411yescharacteristic
77C4×D284C2311yescharacteristic
78C4×C22⋊D71—21no—
79C4×C22⋊D71—21no—
80#14556087C2×C411yescharacteristic
81#14556085—21no—
82C42×D71—21no—
83#14556061C2×C411yescharacteristic
84#14556061C2×C411yescharacteristic
85#14556059C2×C411yescharacteristic
86#14556059C2×C411yescharacteristic
87C4⋊(C71⋊C8)—21no—
88C4⋊(C71⋊C8)—21no—
89C4⋊(C71⋊C8)—21no—
90C4⋊(C71⋊C8)—21no—
91C4⋊(C71⋊C8)C2×C411yescharacteristic
92C4⋊(C71⋊C8)C2×C411yescharacteristic
93C4⋊(C71⋊C8)D411yescharacteristic
94C4⋊(C71⋊C8)D411yescharacteristic
95C4⋊(C71⋊C8)C2311yescharacteristic
96C4⋊(C71⋊C8)C2311yescharacteristic
97Dic71⋊C8—21no—
98Dic71⋊C8—21no—
99#14556052C2×C411yescharacteristic
100#14556052C2×C411yescharacteristic
101#14556052C2×C411yescharacteristic
102#14556052C2×C411yescharacteristic
103#14556051C2×C411yescharacteristic
104#14556051C2×C411yescharacteristic
105#14556048C2×C411yescharacteristic
106#14556048C2×C411yescharacteristic
107C22⋊(C71⋊C8)—21no—
108C22⋊(C71⋊C8)C2×C411yescharacteristic
109C22⋊(C71⋊C8)C2×C411yescharacteristic
110C22⋊(C71⋊C8)D411yescharacteristic
111C22⋊(C71⋊C8)D411yescharacteristic
112(C2×D71)⋊C8—21no—
113(C2×D71)⋊C8—21no—
114#14556039—21no—
115#14556039—21no—
116#14556039—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—
125C22×C8×C71—21no—
126C4⋊Q8×C71C2×C411yescharacteristic
127C4⋊Q8×C71C2×C411yescharacteristic
128#125×C71—21no—
129C22⋊Q8×C71—21no—
130C22⋊Q8×C71C2×C411yescharacteristic
131C22⋊Q8×C71C2×C411yescharacteristic
132C4×Q8×C71—21no—
133C4×Q8×C71C2311yescharacteristic
134C4×Q8×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
144#11765865—22no—
145#11765865—22no—
146C2×Dic142C22×C411yescharacteristic
147C2×D284C22×C411yescharacteristic
148C2×C22⋊D71—21no—
149C2×C4×D71—21no—
150C2×C4×D71—21no—
151C2×C4×D71—21no—
152#11765855—22no—
153#11765855—22no—
154#11765852—22no—
155#11765852—22no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8C22×C411yescharacteristic
159C2×C71⋊C8C22×C411yescharacteristic
160C2×C71⋊C8C22×C411yescharacteristic
161C2×C71⋊C8C22×C411yescharacteristic
162C2×C71⋊C8C2×D411yescharacteristic
163C2×C71⋊C8C2×D411yescharacteristic
164C2×C71⋊C8C2×D411yescharacteristic
165C2×C71⋊C8C2×D411yescharacteristic
166C8×D71—81no—
167C8×D71—81no—
168C4⋊Dic71C22×C411yescharacteristic
169C4⋊Dic71C22×C411yescharacteristic
170Dic71⋊C4—21no—
171Dic71⋊C4—21no—
172C22⋊Dic71—21no—
173(C2×D71)⋊C4—21no—
174(C2×D71)⋊C4—21no—
175C4×Dic71—21no—
176C4×Dic71—21no—
177C4×Dic71—21no—
178C2×Q8×C71—21no—
179C2×Q8×C71C22×C411yescharacteristic
180C2×Q8×C71C22×C411yescharacteristic
181C22×C4×C71C22×C411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71#4111yescharacteristic
184C2×C8×C71—21no—
185C2×C8×C71—21no—
186C2×C8×C71—21no—
187C2×C8×C71—21no—
188C2×C8×C71—81no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×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—
200C42×C71—21no—
201C42×C71—21no—
202C42×C71C22×C411yescharacteristic
203C42×C71C2×D411yescharacteristic
204C42×C71#4111yescharacteristic
205C42×C71C2411yescharacteristic
206C23×C71C4×D411yescharacteristic
207C22×D71—21no—
208Q8×C71—21no—
209Q8×C71—21no—
210Q8×C71—21no—
211Q8×C71—21no—
212Q8×C71—81no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
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—41no—
222C2×C4×C71C4×D411yescharacteristic
223C2×C4×C71C4×D411yescharacteristic
224C2×C4×C71C4×D411yescharacteristic
225C2×C4×C71C23×C411yescharacteristic
226C2×C4×C71C22×D411yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C22⋊D71—24no—
229C2×Dic71—21no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232D284—21no—
233D284#13112yes—
234C4×D71—24no—
235C4×D71—41no—
236C4×D71—41no—
237C4×D71—81no—
238Dic142—21no—
239Dic142#13112yes—
240C8×C71—41no—
241C8×C71—41no—
242C8×C71—41no—
243C8×C71—41no—
244C71⋊C8—21no—
245C71⋊C8—21no—
246C71⋊C8—21no—
247C71⋊C8—21no—
248C71⋊C8—21no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C22×C71—21no—
255C22×C71—21no—
256C22×C71—21no—
257C22×C71C2×C4×D411yescharacteristic
258C2×D71—22no—
259C2×D71—41no—
260C2×D71—41no—
261C4×C71—21no—
262C4×C71—21no—
263C4×C71—21no—
264C4×C71—21no—
265C4×C71—41no—
266C4×C71—41no—
267C4×C71—41no—
268C4×C71—41no—
269C4×C71C2×#13111yescharacteristic
270C4×C71C2×#13111yescharacteristic
271C4×C71#57711yescharacteristic
272C4×C71#57811yescharacteristic
273Dic71—22no—
274Dic71—41no—
275Dic71—41no—
276#17921—711nomaximal, Sylow
277C2×C71—41no—
278C2×C71#292911yescharacteristic
279C2×C71#294711yescharacteristic
280C2×C71#295711yescharacteristic
281D71—42no—
282#3107—711no—
283#2988—711no—
284#2985—711no—
285#1587—711no—
286#1585—711no—
287#1584—711no—
288#1582—711no—
289#1579—711no—
290SD16⋊C8—711no—
291C8×Q16—711no—
292C8×SD16—711no—
293#1573—711no—
294#1488—711no—
295#1481D7111yescharacteristic, complemented
296#1454—711no—
297C71#1792111yescharacteristic, complemented, Sylow
298#533C2×D7111yescharacteristic
299C4×#41—711no—
300#485—711no—
301#485—711no—
302#475—711no—
303#474—711no—
304#462—711no—
305#461—711no—
306#452—711no—
307#452—711no—
308#446—711no—
309#446—711no—
310#446—1421no—
311#446—1421no—
312C8×Q8—711no—
313C8×Q8—711no—
314C8×Q8—711no—
315C8×Q8—711no—
316C4×Q16—711no—
317C4×Q16—711no—
318C4×SD16—711no—
319C4×SD16—711no—
320C8×D4—711no—
321C8×D4—711no—
322C8×D4—1421no—
323C8×D4—1421no—
324#433—1421no—
325#433—1421no—
326#432—711no—
327#431—711no—
328#431—711no—
329#431—1421no—
330#431C2×D7111yescharacteristic
331C2×C4⋊C8—1421no—
332C2×C4⋊C8C2×D7111yescharacteristic
333#335—711no—
334#335—711no—
335#334—711no—
336#334—711no—
337Q8⋊C8—711no—
338Q8⋊C8—711no—
339Q8⋊C8C2×D7111yescharacteristic
340Q8⋊C8C2×D7111yescharacteristic
341Q8⋊C8C2×D7111yescharacteristic
342Q8⋊C8C2×D7111yescharacteristic
343D4⋊C8—711no—
344D4⋊C8—711no—
345#322—711no—
346#322—711no—
347#322—711no—
348C82—711no—
349C2×#41—711no—
350C2×Q16—711no—
351C2×Q16—711no—
352C2×SD16—711no—
353C2×SD16—711no—
354C22×C8—21no—
355C22×C8—1421no—
356C22×C8—1421no—
357C4⋊Q8C4×D7111yescharacteristic
358C4⋊Q8C4×D7111yescharacteristic
359#125—21no—
360C22⋊Q8—21no—
361C22⋊Q8C4×D7111yescharacteristic
362C22⋊Q8C4×D7111yescharacteristic
363C4×Q8—21no—
364C4×Q8—711no—
365C4×Q8—711no—
366C4×Q8C22×D7111yescharacteristic
367C4×Q8C22×D7111yescharacteristic
368C4×D4—711no—
369C4×D4—711no—
370C4×D4—1421no—
371C4×D4—1421no—
372#117—21no—
373#117—711no—
374#117—1421no—
375C2×C42—1421no—
376C2×C42C22×D7111yescharacteristic
377#107—711no—
378#107—711no—
379#106—711no—
380#106—711no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—711no—
384C4⋊C8—711no—
385C4⋊C8—711no—
386C4⋊C8—711no—
387C4⋊C8—711no—
388C4⋊C8—711no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8—1421no—
394C4⋊C8—1421no—
395C4⋊C8C22×D7111yescharacteristic
396C4⋊C8C22×D7111yescharacteristic
397#103—711no—
398#103—711no—
399#103—711no—
400#103—711no—
401#103—711no—
402#103—711no—
403#102—711no—
404#102—711no—
405C22⋊C8—21no—
406C22⋊C8—711no—
407C22⋊C8—711no—
408C22⋊C8—711no—
409C22⋊C8—711no—
410C22⋊C8—1421no—
411C22⋊C8—1421no—
412C22⋊C8—1421no—
413#97—1421no—
414#97—1421no—
415#97—1421no—
416C4×C8—711no—
417C4×C8—711no—
418C4×C8—711no—
419C4×C8—711no—
420C4×C8—1421no—
421C4×C8—1421no—
422C4×C8—1421no—
423C4×C8—1421no—
424C4×C8C22×D7111yescharacteristic
425C4×C8C22×D7111yescharacteristic
426#41—1422no—
427#41—1422no—
428C2×Q8—21no—
429C2×Q8—711no—
430C2×Q8C2×C4×D7111yescharacteristic
431C2×Q8C2×C4×D7111yescharacteristic
432C2×D4—711no—
433C2×D4—1421no—
434C22×C4—1421no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4C2×C4×D7111yescharacteristic
438C22×C4D4×D7111yescharacteristic
439C22×C4#1176586411yescharacteristic
440Q16—1422no—
441Q16—1422no—
442SD16—1422no—
443SD16—1422no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—5681no—
450C2×C8—5681no—
451C2×C8—711no—
452C2×C8—711no—
453C2×C8—711no—
454C2×C8—711no—
455C2×C8—711no—
456C2×C8—711no—
457C2×C8—711no—
458C2×C8—711no—
459C2×C8—1421no—
460C2×C8—1421no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—711no—
469C4⋊C4—711no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4C2×C4×D7111yescharacteristic
473C4⋊C4C2×C4×D7111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1421no—
477C22⋊C4—1421no—
478C22⋊C4—1421no—
479C42—21no—
480C42—21no—
481C42—1421no—
482C42—1421no—
483C42—1421no—
484C42C2×C4×D7111yescharacteristic
485C42D4×D7111yescharacteristic
486C42#1176586411yescharacteristic
487C42C23×D7111yescharacteristic
488C23—1421no—
489C23(C22⋊D71)⋊C411yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—712no—
496Q8—1421no—
497D4—712no—
498D4—1424no—
499D4—1421no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—2841no—
510C2×C4—2841no—
511C2×C4—5681no—
512C2×C4—1424no—
513C2×C4—1421no—
514C2×C4—1421no—
515C2×C4—1421no—
516C2×C4(C22⋊D71)⋊C411yescharacteristic
517C2×C4(C22⋊D71)⋊C411yescharacteristic
518C2×C4(C22⋊D71)⋊C411yescharacteristic
519C2×C4C22×C4×D7111yescharacteristic
520C2×C4C2×D4×D7111yescharacteristic
521C2×C4C2×#1176586411yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—2841no—
527C8—2841no—
528C8—1421no—
529C8—1421no—
530C8—1421no—
531C8—1421no—
532C8—1421no—
533C8—1421no—
534C8—1421no—
535C8—1421no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—2841no—
540C22—2841no—
541C22—1422no—
542C22C2×(C22⋊D71)⋊C411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—2841no—
552C4—2841no—
553C4—1422no—
554C4#131×D7111yescharacteristic
555C4#131×D7111yescharacteristic
556C4#1798680511yescharacteristic
557C4#1798681211yescharacteristic
558C2—41no—
559C2—2842no—
560C2#2259703711yescharacteristic
561C2#2259717711yescharacteristic
562C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1481C2
socleC22×C71C2×C4×D4
prime radicalC2×#11765865C2×C4
cosocleC42×C71C24
hypercenter#1481D71
hypocenterC71#17921
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. #22597435C2semidirect
4. #22587214C2semidirect
5. #22587200C2semidirect
6. #22587193C2semidirect
7. #22587179C2semidirect
8. #22587156C2semidirect
9. #22587155C2semidirect
10. #22587144C2semidirect
11. #22587137C2semidirect
12. #22587123C2semidirect
14. #22586640C2semidirect
15. C71×#1481C2semidirect
295. #1481D71semidirect
297. C71#17921semidirect

Maximal subgroups

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

subgroupindex
#225984052
#225974492
#225974352
#225872142
#225872002
#225871932
#225871792
#225871562
#225871552
#225871442
#225871372
#225871232
#225867752
#225866402
C71×#14812
#1792171