Finite Group Databasebeta

← #31967521

Subgroups — #31967521

Order 18176.

596 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#22598420C211yesmaximal, characteristic
2#22597485C211yesmaximal, characteristic
3#22597421C211yesmaximal, characteristic, complemented
4#22587215C211yesmaximal, characteristic, complemented
5#22587208C211yesmaximal, characteristic, complemented
6#22587187C211yesmaximal, characteristic, complemented
7#22587180C211yesmaximal, characteristic, complemented
8#22587156C211yesmaximal, characteristic, complemented
9#22587155C211yesmaximal, characteristic, complemented
10#22587144C211yesmaximal, characteristic, complemented
11#22587137C211yesmaximal, characteristic, complemented
12#22587131C211yesmaximal, characteristic, complemented
13#22586749C211yesmaximal, characteristic
14#22586640C211yesmaximal, characteristic, complemented
15C71×#1487C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986329C411yescharacteristic
18#17986312C411yescharacteristic
19#17986306C411yescharacteristic
20#17986306C411yescharacteristic
21#17986221C411yescharacteristic
22#17986221C411yescharacteristic
23#17986155C411yescharacteristic
24#17986138C411yescharacteristic
25#17986126C2211yescharacteristic
26Q8×C71⋊C8—21no—
27Q8×C71⋊C8—21no—
28Q8×C71⋊C8C2211yescharacteristic
29Q8×C71⋊C8C2211yescharacteristic
30Q8×C71⋊C8C2211yescharacteristic
31Q8×C71⋊C8C2211yescharacteristic
32#17986123C2211yescharacteristic
33C4×#11765855C2211yescharacteristic
34C4×#11765855C2211yescharacteristic
35C4×#11765852C2211yescharacteristic
36C4×#11765852C2211yescharacteristic
37#17986076C2211yescharacteristic
38#17986075—21no—
39#17986068—21no—
40#17986064C2211yescharacteristic
41#17986059—21no—
42#17986052—21no—
43#17986052—21no—
44#17986052C2211yescharacteristic
45#17986052C2211yescharacteristic
46#17986052C2211yescharacteristic
47#17986024—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
60C2×C4×Q8×C71C2211yescharacteristic
61#431×C71C2211yescharacteristic
62#423×C71C2211yescharacteristic
63Q8⋊C8×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65Q8⋊C8×C71C2211yescharacteristic
66Q8⋊C8×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765852C2×C411yescharacteristic
71C2×#11765848—21no—
72C2×C8×D71—21no—
73#14556122—21no—
74#14556120C2×C411yescharacteristic
75#14556116C2×C411yescharacteristic
76#14556111—21no—
77#14556107C2×C411yescharacteristic
78#14556102C2×C411yescharacteristic
79C4×Dic142C2311yescharacteristic
80C4×D284C2311yescharacteristic
81C4×C22⋊D71—21no—
82#14556085—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#14556038—21no—
115C4×C71⋊C8—21no—
116C4×C71⋊C8—21no—
117C4×C71⋊C8—21no—
118C4×C71⋊C8—21no—
119C4×C71⋊C8—21no—
120C4×C71⋊C8C2311yescharacteristic
121C4×C71⋊C8C2311yescharacteristic
122C4×C71⋊C8C2311yescharacteristic
123C4×C71⋊C8C2311yescharacteristic
124C8×Dic71—21no—
125C22×Q8×C71C2×C411yescharacteristic
126C2×M16×C71—21no—
127C4×Q8×C71—21no—
128C4×Q8×C71—21no—
129C4×Q8×C71C2×C411yescharacteristic
130C4×Q8×C71C2×C411yescharacteristic
131C4×Q8×C71C2311yescharacteristic
132C4×Q8×C71C2311yescharacteristic
133C2×C4⋊C4×C71—21no—
134C2×C4⋊C4×C71C2×C411yescharacteristic
135C2×C42×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×Dic142C22×C411yescharacteristic
145C2×D284C22×C411yescharacteristic
146C2×C22⋊D71—21no—
147C2×C4×D71—21no—
148#11765855—22no—
149#11765855—22no—
150#11765852—22no—
151#11765852—22no—
152#11765848—81no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8C22×C411yescharacteristic
156C2×C71⋊C8C22×C411yescharacteristic
157C2×C71⋊C8C22×C411yescharacteristic
158C2×C71⋊C8C22×C411yescharacteristic
159C2×C71⋊C8C2×D411yescharacteristic
160C2×C71⋊C8C2×D411yescharacteristic
161C2×C71⋊C8C2×D411yescharacteristic
162C2×C71⋊C8C2×D411yescharacteristic
163C8×D71—81no—
164C4⋊Dic71C22×C411yescharacteristic
165C4⋊Dic71C22×C411yescharacteristic
166Dic71⋊C4—21no—
167Dic71⋊C4—21no—
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170C22⋊Dic71—21no—
171(C2×D71)⋊C4—21no—
172(C2×D71)⋊C4—21no—
173(C2×D71)⋊C4—21no—
174(C2×D71)⋊C4—21no—
175C4×Dic71—21no—
176C2×Q8×C71—22no—
177C2×Q8×C71—22no—
178C2×Q8×C71—21no—
179C2×Q8×C71C22×C411yescharacteristic
180C2×Q8×C71C22×C411yescharacteristic
181C22×C4×C71—21no—
182C22×C4×C71—21no—
183C22×C4×C71C22×C411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71#4111yescharacteristic
186M16×C71—81no—
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×C71C22×C411yescharacteristic
197C4⋊C4×C71C22×C411yescharacteristic
198C42×C71—21no—
199C42×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—24no—
209Q8×C71—21no—
210Q8×C71—21no—
211Q8×C71#13112yes—
212Q8×C71#13112yes—
213C2×C4×C71—24no—
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—21no—
222C2×C4×C71—41no—
223C2×C4×C71—81no—
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—81no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234D284—21no—
235D284—21no—
236C4×D71—41no—
237C4×D71—41no—
238Dic142—21no—
239Dic142—21no—
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—41no—
259C2×D71—41no—
260C2×D71—41no—
261C4×C71—22no—
262C4×C71—22no—
263C4×C71—21no—
264C4×C71—21no—
265C4×C71—21no—
266C4×C71—21no—
267C4×C71—41no—
268C4×C71—41no—
269C4×C71C2×#13111yescharacteristic
270C4×C71C2×#13111yescharacteristic
271C4×C71#57711yescharacteristic
272C4×C71#57811yescharacteristic
273Dic71—41no—
274Dic71—41no—
275Dic71—41no—
276#17919—711nomaximal, Sylow
277C2×C71—41no—
278C2×C71#292911yescharacteristic
279C2×C71#294711yescharacteristic
280C2×C71#295711yescharacteristic
281D71—81no—
282#3108—711no—
283#2993—711no—
284#2982—711no—
285#1588—711no—
286#1587—711no—
287#1584—711no—
288#1583—711no—
289#1579—711no—
290SD16⋊C8—711no—
291C8×Q16—711no—
292C8×SD16—711no—
293#1574—711no—
294#1487D7111yescharacteristic, complemented
295#1482—711no—
296#1454—711no—
297C71#1791911yescharacteristic, complemented, Sylow
298#529—711no—
299C2×C4×Q8C2×D7111yescharacteristic
300#483—711no—
301#480—711no—
302#479—711no—
303#479—711no—
304#464—711no—
305#464—711no—
306#451—711no—
307#448—711no—
308#446—711no—
309C8×Q8—711no—
310C8×Q8—711no—
311C8×Q8—711no—
312C8×Q8—711no—
313C8×Q8—711no—
314C8×Q8—1421no—
315C8×Q8—1421no—
316C4×Q16—711no—
317C4×Q16—711no—
318C4×SD16—711no—
319C4×SD16—711no—
320#436—711no—
321#436—1421no—
322#435—1421no—
323C8×D4—711no—
324#433—1421no—
325#431—711no—
326#431—711no—
327#431—711no—
328#431—1421no—
329#431—1421no—
330#431C2×D7111yescharacteristic
331#424—1421no—
332#423C2×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—
349C22×Q8C4×D7111yescharacteristic
350C2×Q16—711no—
351C2×Q16—711no—
352C2×SD16—711no—
353C2×SD16—711no—
354C2×M16—21no—
355C2×M16—1421no—
356C22×C8—1421no—
357#126—1421no—
358#125—711no—
359#124—711no—
360#123—1421no—
361C22⋊Q8—711no—
362#121—711no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—711no—
366C4×Q8C4×D7111yescharacteristic
367C4×Q8C4×D7111yescharacteristic
368C4×Q8C22×D7111yescharacteristic
369C4×Q8C22×D7111yescharacteristic
370C4×D4—711no—
371C4×D4—1421no—
372#117—1421no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C4×D7111yescharacteristic
375C2×C42—21no—
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—
414C4×C8—711no—
415C4×C8—711no—
416C4×C8—711no—
417C4×C8—711no—
418C4×C8—1421no—
419C4×C8—1421no—
420C4×C8—1421no—
421C4×C8—1421no—
422C4×C8—1421no—
423C4×C8—1421no—
424C4×C8C22×D7111yescharacteristic
425C4×C8C22×D7111yescharacteristic
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—711no—
430C2×Q8C2×C4×D7111yescharacteristic
431C2×Q8C2×C4×D7111yescharacteristic
432C2×D4—711no—
433C2×D4—1421no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1421no—
437C22×C4C2×C4×D7111yescharacteristic
438C22×C4D4×D7111yescharacteristic
439C22×C4#1176586411yescharacteristic
440Q16—1422no—
441Q16—1422no—
442SD16—1422no—
443SD16—1422no—
444M16—81no—
445M16—5681no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
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—711no—
467C4⋊C4—711no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4C2×C4×D7111yescharacteristic
473C4⋊C4C2×C4×D7111yescharacteristic
474C22⋊C4—1421no—
475C22⋊C4—1421no—
476C22⋊C4—1421no—
477C22⋊C4—1421no—
478C22⋊C4—1421no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—1421no—
484C42C2×C4×D7111yescharacteristic
485C42D4×D7111yescharacteristic
486C42#1176586411yescharacteristic
487C42C23×D7111yescharacteristic
488C23—1421no—
489C23(C22⋊D71)⋊C411yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—1421no—
494Q8—1421no—
495Q8#1455613612yes—
496Q8#1455613612yes—
497D4—5681no—
498D4—1421no—
499D4—1421no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—81no—
511C2×C4—2841no—
512C2×C4—2841no—
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—2841no—
542C22C2×(C22⋊D71)⋊C411yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—2841no—
552C4—2841no—
553C4—2841no—
554C4C2×#1455613611yescharacteristic
555C4C2×#1455613611yescharacteristic
556C4#1798680511yescharacteristic
557C4#1798681211yescharacteristic
558C2—41no—
559C2—5681no—
560C2#2259703011yescharacteristic
561C2#2259717711yescharacteristic
562C2#2259725311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1487C2
socleC22×C71C2×C4×D4
prime radical#14556102C2×C4
cosocleC42×C71C24
hypercenter#1487D71
hypocenterC71#17919
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
3. #22597421C2semidirect
4. #22587215C2semidirect
5. #22587208C2semidirect
6. #22587187C2semidirect
7. #22587180C2semidirect
8. #22587156C2semidirect
9. #22587155C2semidirect
10. #22587144C2semidirect
11. #22587137C2semidirect
12. #22587131C2semidirect
14. #22586640C2semidirect
15. C71×#1487C2semidirect
294. #1487D71semidirect
297. C71#17919semidirect

Maximal subgroups

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

subgroupindex
#225984202
#225974852
#225974212
#225872152
#225872082
#225871872
#225871802
#225871562
#225871552
#225871442
#225871372
#225871312
#225867492
#225866402
C71×#14872
#1791971