Finite Group Databasebeta

← #31967523

Subgroups — #31967523

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#22598416C211yesmaximal, characteristic
2#22597420C211yesmaximal, characteristic, complemented
3#22587216C211yesmaximal, characteristic, complemented
4#22587209C211yesmaximal, characteristic, complemented
5#22587188C211yesmaximal, characteristic, complemented
6#22587181C211yesmaximal, characteristic, complemented
7#22587157C211yesmaximal, characteristic, complemented
8#22587152C211yesmaximal, characteristic, complemented
9C8×#11765855C211yesmaximal, characteristic, complemented
10C8×#11765853C211yesmaximal, characteristic, complemented
11#22587131C211yesmaximal, characteristic, complemented
12#22586768C211yesmaximal, characteristic, complemented
13#22586748C211yesmaximal, characteristic
14#22586640C211yesmaximal, characteristic, complemented
15C71×#2993C211yesmaximal, characteristic
16C2×C4×Dic142C2211yescharacteristic
17#17986327C411yescharacteristic
18#17986310C411yescharacteristic
19#17986303C411yescharacteristic
20#17986303C411yescharacteristic
21#17986218C411yescharacteristic
22#17986218C411yescharacteristic
23#17986154C411yescharacteristic
24#17986137C411yescharacteristic
25C8×Dic142C2211yescharacteristic
26C8×Dic142C2211yescharacteristic
27C8×Dic142C2211yescharacteristic
28C8×Dic142C2211yescharacteristic
29#17986123—21no—
30#17986123—21no—
31C4×#11765855C2211yescharacteristic
32C4×#11765855C2211yescharacteristic
33C4×#11765853C2211yescharacteristic
34C4×#11765853C2211yescharacteristic
35#17986052C2211yescharacteristic
36#17986052C2211yescharacteristic
37#17986052C2211yescharacteristic
38#17986049—21no—
39#17986049—21no—
40#17985751C2211yescharacteristic
41#17985751C2211yescharacteristic
42#17985749C2211yescharacteristic
43#17985749C2211yescharacteristic
44Q8⋊(C71⋊C8)C2211yescharacteristic
45Q8⋊(C71⋊C8)C2211yescharacteristic
46#17985729C2211yescharacteristic
47#17985729C2211yescharacteristic
48#17985729C2211yescharacteristic
49#17985729C2211yescharacteristic
50D4⋊(C71⋊C8)C2211yescharacteristic
51D4⋊(C71⋊C8)C2211yescharacteristic
52#17985721C2211yescharacteristic
53#17985721C2211yescharacteristic
54#17985720C2211yescharacteristic
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
67C22×Dic142C2×C411yescharacteristic
68C2×#11765855C2×C411yescharacteristic
69C2×#11765855C2×C411yescharacteristic
70C2×#11765853C2×C411yescharacteristic
71C2×#11765853C2×C411yescharacteristic
72C4×Dic142—21no—
73C4×Dic142—21no—
74C4×Dic142C2×C411yescharacteristic
75C4×Dic142C2×C411yescharacteristic
76C4×Dic142C2311yescharacteristic
77C4×Dic142C2311yescharacteristic
78C2×C4⋊Dic71C2×C411yescharacteristic
79C2×Dic71⋊C4—21no—
80C2×C4×Dic71—21no—
81#14556061C2×C411yescharacteristic
82#14556061C2×C411yescharacteristic
83#14556059C2×C411yescharacteristic
84#14556059C2×C411yescharacteristic
85C4⋊(C71⋊C8)—21no—
86C4⋊(C71⋊C8)C2×C411yescharacteristic
87C4⋊(C71⋊C8)C2×C411yescharacteristic
88C4⋊(C71⋊C8)C2311yescharacteristic
89C4⋊(C71⋊C8)C2311yescharacteristic
90Dic71⋊C8—21no—
91Dic71⋊C8—21no—
92Dic71⋊C8—21no—
93Dic71⋊C8—21no—
94#14556052C2×C411yescharacteristic
95#14556052C2×C411yescharacteristic
96#14556051C2×C411yescharacteristic
97#14556051C2×C411yescharacteristic
98#14556051C2×C411yescharacteristic
99#14556051C2×C411yescharacteristic
100#14556049C2×C411yescharacteristic
101#14556049C2×C411yescharacteristic
102C22⋊(C71⋊C8)—21no—
103C22⋊(C71⋊C8)—21no—
104C22⋊(C71⋊C8)C2×C411yescharacteristic
105C22⋊(C71⋊C8)C2×C411yescharacteristic
106C4×C71⋊C8—21no—
107C4×C71⋊C8C2311yescharacteristic
108C4×C71⋊C8C2311yescharacteristic
109C4×C71⋊C8C2311yescharacteristic
110C4×C71⋊C8C2311yescharacteristic
111C8×Dic71—21no—
112C8×Dic71—21no—
113C8×Dic71—21no—
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×Dic142—21no—
145C2×Dic142—22no—
146C2×Dic142—22no—
147C2×Dic142C22×C411yescharacteristic
148C2×Dic142C22×C411yescharacteristic
149C22×Dic71—21no—
150C22×Dic71—21no—
151#11765855—22no—
152#11765855—22no—
153#11765853—22no—
154#11765853—22no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8—21no—
159C2×C71⋊C8C22×C411yescharacteristic
160C2×C71⋊C8C22×C411yescharacteristic
161C2×C71⋊C8C22×C411yescharacteristic
162C2×C71⋊C8C22×C411yescharacteristic
163C4⋊Dic71—21no—
164C4⋊Dic71C22×C411yescharacteristic
165C4⋊Dic71C22×C411yescharacteristic
166Dic71⋊C4—21no—
167Dic71⋊C4—21no—
168Dic71⋊C4—21no—
169Dic71⋊C4—21no—
170C4×Dic71—21no—
171C4×Dic71—21no—
172C4×Dic71—21no—
173C4×Dic71—21no—
174C2×Q8×C71C22×C411yescharacteristic
175C2×D4×C71—21no—
176C2×D4×C71C22×C411yescharacteristic
177C22×C4×C71—21no—
178C22×C4×C71C22×C411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71#4111yescharacteristic
181M16×C71—81no—
182M16×C71—81no—
183C2×C8×C71—21no—
184C2×C8×C71—21no—
185C2×C8×C71—81no—
186C2×C8×C71C2×D411yescharacteristic
187C2×C8×C71C2×D411yescharacteristic
188C2×C8×C71C2×D411yescharacteristic
189C2×C8×C71C2×D411yescharacteristic
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71C22×C411yescharacteristic
195C4⋊C4×C71C22×C411yescharacteristic
196C22⋊C4×C71—21no—
197C22⋊C4×C71—21no—
198C22⋊C4×C71—21no—
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C42×C71—21no—
202C42×C71C22×C411yescharacteristic
203C42×C71C2×D411yescharacteristic
204C42×C71#4111yescharacteristic
205C42×C71C2411yescharacteristic
206C23×C71—21no—
207C23×C71C4×D411yescharacteristic
208Q8×C71—21no—
209Q8×C71—21no—
210D4×C71—21no—
211D4×C71—21no—
212D4×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—41no—
221C2×C4×C71—41no—
222C2×C4×C71—41no—
223C2×C4×C71C4×D411yescharacteristic
224C2×C4×C71C4×D411yescharacteristic
225C2×C4×C71C4×D411yescharacteristic
226C2×C4×C71C23×C411yescharacteristic
227C2×C4×C71C22×D411yescharacteristic
228C2×C4×C71C2×#4111yescharacteristic
229C2×Dic71—24no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—81no—
235Dic142—24no—
236Dic142—21no—
237Dic142—21no—
238Dic142#13112yes—
239Dic142#13112yes—
240C8×C71—21no—
241C8×C71—21no—
242C8×C71—21no—
243C8×C71—21no—
244C8×C71—41no—
245C8×C71—41no—
246C71⋊C8—21no—
247C71⋊C8—21no—
248C71⋊C8—21no—
249C71⋊C8—21no—
250C71⋊C8—41no—
251C71⋊C8—41no—
252C71⋊C8—41no—
253C71⋊C8—41no—
254C22×C71—21no—
255C22×C71—21no—
256C22×C71—21no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71C2×C4×D411yescharacteristic
261C4×C71—21no—
262C4×C71—21no—
263C4×C71—21no—
264C4×C71—21no—
265C4×C71—41no—
266C4×C71—41no—
267C4×C71—41no—
268C4×C71C2×#13111yescharacteristic
269C4×C71C2×#13111yescharacteristic
270C4×C71#57711yescharacteristic
271C4×C71#57811yescharacteristic
272Dic71—22no—
273Dic71—22no—
274Dic71—41no—
275Dic71—41no—
276#17919—711nomaximal, Sylow
277C2×C71—41no—
278C2×C71—81no—
279C2×C71#292911yescharacteristic
280C2×C71#294711yescharacteristic
281C2×C71#295711yescharacteristic
282#3108—711no—
283#2993D7111yescharacteristic
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#1487—711no—
295#1482—711no—
296#1454—711no—
297C71#1791911yescharacteristic, complemented, Sylow
298#529C2×D7111yescharacteristic
299C2×C4×Q8—711no—
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#446C2×D7111yescharacteristic
309C8×Q8—711no—
310C8×Q8—711no—
311C8×Q8—711no—
312C8×Q8—711no—
313C8×Q8—1421no—
314C8×Q8—1421no—
315C8×Q8C2×D7111yescharacteristic
316C4×Q16—711no—
317C4×Q16—711no—
318C4×SD16—711no—
319C4×SD16—711no—
320#436—21no—
321#436C2×D7111yescharacteristic
322#435—21no—
323C8×D4C2×D7111yescharacteristic
324#433—21no—
325#431—711no—
326#431—711no—
327#431—711no—
328#431—1421no—
329#431—1421no—
330#431C2×D7111yescharacteristic
331#424—21no—
332#423C2×D7111yescharacteristic
333#335—711no—
334#335—711no—
335#334—711no—
336#334—711no—
337Q8⋊C8—711no—
338Q8⋊C8—711no—
339Q8⋊C8—711no—
340Q8⋊C8—711no—
341Q8⋊C8—711no—
342Q8⋊C8—711no—
343D4⋊C8—711no—
344D4⋊C8—711no—
345#322—711no—
346#322—711no—
347#322—711no—
348C82—711no—
349C22×Q8—711no—
350C2×Q16—711no—
351C2×Q16—711no—
352C2×SD16—711no—
353C2×SD16—711no—
354C2×M16—21no—
355C2×M16—21no—
356C22×C8—21no—
357#126—21no—
358#125C4×D7111yescharacteristic
359#124C4×D7111yescharacteristic
360#123—21no—
361C22⋊Q8C4×D7111yescharacteristic
362#121C4×D7111yescharacteristic
363C4×Q8—711no—
364C4×Q8—711no—
365C4×Q8—711no—
366C4×Q8—711no—
367C4×Q8—1421no—
368C4×Q8—1421no—
369C4×Q8C22×D7111yescharacteristic
370C4×D4—21no—
371C4×D4C22×D7111yescharacteristic
372#117—21no—
373C2×C4⋊C4—711no—
374C2×C4⋊C4—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—21no—
384C4⋊C8—711no—
385C4⋊C8—711no—
386C4⋊C8—711no—
387C4⋊C8—711no—
388C4⋊C8—1421no—
389C4⋊C8—1421no—
390C4⋊C8—1421no—
391C4⋊C8—1421no—
392C4⋊C8—1421no—
393C4⋊C8C22⋊D7111yescharacteristic
394C4⋊C8C22⋊D7111yescharacteristic
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—21no—
407C22⋊C8—711no—
408C22⋊C8—711no—
409C22⋊C8—1421no—
410C22⋊C8—1421no—
411C22⋊C8C22⋊D7111yescharacteristic
412C22⋊C8C22⋊D7111yescharacteristic
413#97—21no—
414C4×C8—21no—
415C4×C8—711no—
416C4×C8—711no—
417C4×C8—711no—
418C4×C8—711no—
419C4×C8—1421no—
420C4×C8—1421no—
421C4×C8—1421no—
422C4×C8—1421no—
423C4×C8—1421no—
424C4×C8C22×D7111yescharacteristic
425C4×C8C22×D7111yescharacteristic
426C2×Q8—711no—
427C2×Q8—711no—
428C2×Q8—1422no—
429C2×Q8—1422no—
430C2×Q8—1421no—
431C2×Q8C2×C4×D7111yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D7111yescharacteristic
434C22×C4—21no—
435C22×C4—1421no—
436C22×C4—1421no—
437C22×C4C2×C4×D7111yescharacteristic
438C22×C4C2×C22⋊D7111yescharacteristic
439C22×C4#1176586511yescharacteristic
440Q16—1422no—
441Q16—1422no—
442SD16—1422no—
443SD16—1422no—
444M16—81no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—711no—
450C2×C8—711no—
451C2×C8—711no—
452C2×C8—711no—
453C2×C8—1421no—
454C2×C8—1421no—
455C2×C8—1421no—
456C2×C8—1421no—
457C2×C8C2×C22⋊D7111yescharacteristic
458C2×C8C2×C22⋊D7111yescharacteristic
459C2×C8C2×C22⋊D7111yescharacteristic
460C2×C8C2×C22⋊D7111yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—711no—
466C4⋊C4—711no—
467C4⋊C4—1421no—
468C4⋊C4—1421no—
469C4⋊C4—1421no—
470C4⋊C4—1421no—
471C4⋊C4—1421no—
472C4⋊C4C2×C4×D7111yescharacteristic
473C4⋊C4C2×C4×D7111yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C42—21no—
480C42—1421no—
481C42—1421no—
482C42—1421no—
483C42—1421no—
484C42C2×C4×D7111yescharacteristic
485C42C2×C22⋊D7111yescharacteristic
486C42#1176586511yescharacteristic
487C42C23×D7111yescharacteristic
488C23—21no—
489C23C4×C22⋊D7111yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—712no—
493Q8—712no—
494Q8—1424no—
495Q8—1421no—
496Q8—1421no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—5681no—
511C2×C4—1421no—
512C2×C4—1421no—
513C2×C4—1424no—
514C2×C4—1421no—
515C2×C4—1421no—
516C2×C4C4×C22⋊D7111yescharacteristic
517C2×C4C4×C22⋊D7111yescharacteristic
518C2×C4C4×C22⋊D7111yescharacteristic
519C2×C4C22×C4×D7111yescharacteristic
520C2×C4C22×C22⋊D7111yescharacteristic
521C2×C4C2×#1176586511yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—2841no—
529C8—2841no—
530C8—2841no—
531C8—2841no—
532C8—1421no—
533C8—1421no—
534C8—1421no—
535C8—1421no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D7111yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—2841no—
551C4—2841no—
552C4—1422no—
553C4—1422no—
554C4#131×D7111yescharacteristic
555C4#131×D7111yescharacteristic
556C4#1798681011yescharacteristic
557C4#1798681411yescharacteristic
558C2—41no—
559C2—81no—
560C2#2259703211yescharacteristic
561C2#2259717911yescharacteristic
562C2#2259725511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#2993C2
socleC22×C71C2×C4×D4
prime radical#121×C71C2×C4
cosocleC42×C71C24
hypercenter#2993D71
hypocenterC71#17919
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. #22597420C2semidirect
3. #22587216C2semidirect
4. #22587209C2semidirect
5. #22587188C2semidirect
6. #22587181C2semidirect
7. #22587157C2semidirect
8. #22587152C2semidirect
9. C8×#11765855C2semidirect
10. C8×#11765853C2semidirect
11. #22587131C2semidirect
12. #22586768C2semidirect
14. #22586640C2semidirect
297. C71#17919semidirect

Maximal subgroups

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

subgroupindex
#225984162
#225974202
#225872162
#225872092
#225871882
#225871812
#225871572
#225871522
C8×#117658552
C8×#117658532
#225871312
#225867682
#225867482
#225866402
C71×#29932
#1791971