Finite Group Databasebeta

← #31968855

Subgroups — #31968855

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22598358C211yesmaximal, characteristic
2#22597496C211yesmaximal, characteristic, complemented
3#22597472C211yesmaximal, characteristic, complemented
4#22587239C211yesmaximal, characteristic, complemented
5#22587222C211yesmaximal, characteristic, complemented
6#22587221C211yesmaximal, characteristic, complemented
7#22587215C211yesmaximal, characteristic, complemented
8#22587187C211yesmaximal, characteristic, complemented
9#22587172C211yesmaximal, characteristic, complemented
10#22587155C211yesmaximal, characteristic, complemented
11#22587144C211yesmaximal, characteristic, complemented
12#22587114C211yesmaximal, characteristic, complemented
13#22586806C211yesmaximal, characteristic, complemented
14#22586640C211yesmaximal, characteristic, complemented
15C71×#1477C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17#17986340C411yescharacteristic
18#17986323C411yescharacteristic
19#17986306C411yescharacteristic
20#17986299C411yescharacteristic
21#17986221C411yescharacteristic
22#17986214C411yescharacteristic
23#17986161C411yescharacteristic
24#17986144C411yescharacteristic
25#17986126C2211yescharacteristic
26Q8×C71⋊C8C2211yescharacteristic
27Q8×C71⋊C8C2211yescharacteristic
28#17986123C2211yescharacteristic
29C4×#11765855C2211yescharacteristic
30C4×#11765852C2211yescharacteristic
31C4×#11765853C2211yescharacteristic
32C4×#11765850C2211yescharacteristic
33#17986077—21no—
34#17986077—21no—
35#17986077C2211yescharacteristic
36#17986077C2211yescharacteristic
37#17986076C2211yescharacteristic
38#17986070—21no—
39#17986064C2211yescharacteristic
40C8×C22⋊D71—21no—
41#17986059—21no—
42#17986052—21no—
43#17986052C2211yescharacteristic
44#17986052C2211yescharacteristic
45#17986024—21no—
46#17986023—21no—
47#17986023C2211yescharacteristic
48#17985751C2211yescharacteristic
49#17985751C2211yescharacteristic
50#17985751C2211yescharacteristic
51#17985751C2211yescharacteristic
52#17985731C2211yescharacteristic
53#17985731C2211yescharacteristic
54#17985728C2211yescharacteristic
55#17985728C2211yescharacteristic
56#17985721C2211yescharacteristic
57#17985721C2211yescharacteristic
58#17985720C2211yescharacteristic
59C8×C71⋊C8C2211yescharacteristic
60C4×#41×C71C2211yescharacteristic
61#431×C71C2211yescharacteristic
62C2×C4⋊C8×C71C2211yescharacteristic
63Q8⋊C8×C71C2211yescharacteristic
64Q8⋊C8×C71C2211yescharacteristic
65D4⋊C8×C71C2211yescharacteristic
66D4⋊C8×C71C2211yescharacteristic
67C2×#11765855C2×C411yescharacteristic
68C2×#11765853C2×C411yescharacteristic
69C2×#11765852C2×C411yescharacteristic
70C2×#11765850C2×C411yescharacteristic
71C2×#11765849—21no—
72C2×#11765849—21no—
73C2×#11765849—21no—
74C2×#11765848—21no—
75C2×C8×D71—21no—
76#14556122—21no—
77#14556120C2×C411yescharacteristic
78#14556116C2×C411yescharacteristic
79#14556111—21no—
80#14556107C2×C411yescharacteristic
81#14556102C2×C411yescharacteristic
82C4×Dic142C2311yescharacteristic
83C4×D284C2311yescharacteristic
84C4×C22⋊D71—21no—
85#14556085—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)C2×C411yescharacteristic
94C4⋊(C71⋊C8)C2×C411yescharacteristic
95C4⋊(C71⋊C8)D411yescharacteristic
96C4⋊(C71⋊C8)D411yescharacteristic
97C4⋊(C71⋊C8)C2311yescharacteristic
98C4⋊(C71⋊C8)C2311yescharacteristic
99Dic71⋊C8—21no—
100Dic71⋊C8—21no—
101#14556052C2×C411yescharacteristic
102#14556052C2×C411yescharacteristic
103#14556051C2×C411yescharacteristic
104#14556051C2×C411yescharacteristic
105#14556049C2×C411yescharacteristic
106#14556049C2×C411yescharacteristic
107#14556048C2×C411yescharacteristic
108#14556048C2×C411yescharacteristic
109C22⋊(C71⋊C8)—21no—
110C22⋊(C71⋊C8)—21no—
111C22⋊(C71⋊C8)C2×C411yescharacteristic
112C22⋊(C71⋊C8)C2×C411yescharacteristic
113C22⋊(C71⋊C8)D411yescharacteristic
114C22⋊(C71⋊C8)D411yescharacteristic
115(C2×D71)⋊C8—21no—
116(C2×D71)⋊C8—21no—
117#14556038—21no—
118C4×C71⋊C8—21no—
119C4×C71⋊C8—21no—
120C4×C71⋊C8C2311yescharacteristic
121C4×C71⋊C8C2311yescharacteristic
122C4×C71⋊C8C2311yescharacteristic
123C4×C71⋊C8C2311yescharacteristic
124C8×Dic71—21no—
125C2×#41×C71C2×C411yescharacteristic
126C22×C8×C71—21no—
127C4×Q8×C71C2×C411yescharacteristic
128C4×Q8×C71C2311yescharacteristic
129C4×D4×C71—21no—
130C4×D4×C71—21no—
131C4×D4×C71C2×C411yescharacteristic
132C4×D4×C71C2311yescharacteristic
133#117×C71—21no—
134#117×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#11765853—41no—
150#11765852—22no—
151#11765850—41no—
152#11765849—81no—
153#11765849—81no—
154#11765849—81no—
155#11765848—81no—
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—
167C4⋊Dic71C22×C411yescharacteristic
168C4⋊Dic71C22×C411yescharacteristic
169Dic71⋊C4—21no—
170Dic71⋊C4—21no—
171Dic71⋊C4—21no—
172Dic71⋊C4—21no—
173C22⋊Dic71—21no—
174(C2×D71)⋊C4—21no—
175(C2×D71)⋊C4—21no—
176(C2×D71)⋊C4—21no—
177(C2×D71)⋊C4—21no—
178C4×Dic71—21no—
179#41×C71—22no—
180#41×C71—41no—
181C2×Q8×C71C22×C411yescharacteristic
182C2×D4×C71—21no—
183C2×D4×C71C22×C411yescharacteristic
184C22×C4×C71—21no—
185C22×C4×C71—21no—
186C22×C4×C71—21no—
187C22×C4×C71C22×C411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71#4111yescharacteristic
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71—21no—
193C2×C8×C71—21no—
194C2×C8×C71—81no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71C22×C411yescharacteristic
198C4⋊C4×C71C22×C411yescharacteristic
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C22⋊C4×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71—21no—
205C42×C71C22×C411yescharacteristic
206C42×C71C2×D411yescharacteristic
207C42×C71#4111yescharacteristic
208C42×C71C2411yescharacteristic
209C23×C71—21no—
210C23×C71C4×D411yescharacteristic
211C22×D71—21no—
212Q8×C71—21no—
213Q8×C71#13112yes—
214D4×C71—21no—
215D4×C71—21no—
216D4×C71—42no—
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—21no—
224C2×C4×C71—41no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71C4×D411yescharacteristic
230C2×C4×C71C4×D411yescharacteristic
231C2×C4×C71C4×D411yescharacteristic
232C2×C4×C71C23×C411yescharacteristic
233C2×C4×C71C22×D411yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C22⋊D71—42no—
236C2×Dic71—21no—
237C2×Dic71—21no—
238C2×Dic71—21no—
239D284—21no—
240D284—21no—
241C4×D71—41no—
242C4×D71—41no—
243Dic142—21no—
244Dic142—21no—
245C8×C71—41no—
246C8×C71—41no—
247C8×C71—41no—
248C8×C71—41no—
249C71⋊C8—21no—
250C71⋊C8—21no—
251C71⋊C8—21no—
252C71⋊C8—21no—
253C71⋊C8—21no—
254C71⋊C8—21no—
255C71⋊C8—21no—
256C71⋊C8—21no—
257C71⋊C8—41no—
258C71⋊C8—41no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71—41no—
265C22×C71C2×C4×D411yescharacteristic
266C2×D71—41no—
267C2×D71—41no—
268C2×D71—41no—
269C4×C71—21no—
270C4×C71—22no—
271C4×C71—21no—
272C4×C71—21no—
273C4×C71—21no—
274C4×C71—41no—
275C4×C71—41no—
276C4×C71C2×#13111yescharacteristic
277C4×C71#56811yescharacteristic
278C4×C71#57611yescharacteristic
279C4×C71#57811yescharacteristic
280Dic71—41no—
281Dic71—41no—
282Dic71—41no—
283#18033—711nomaximal, Sylow
284C2×C71—41no—
285C2×C71—81no—
286C2×C71#293311yescharacteristic
287C2×C71#295111yescharacteristic
288C2×C71#295511yescharacteristic
289D71—81no—
290#3102—711no—
291#2994—711no—
292#2991—711no—
293#1592—711no—
294#1589—711no—
295#1588—711no—
296#1588—711no—
297#1584—711no—
298#1581—711no—
299SD16⋊C8—711no—
300C8×Q16—711no—
301#1571—711no—
302#1495—711no—
303#1477D7111yescharacteristic, complemented
304#1454—711no—
305C71#1803311yescharacteristic, complemented, Sylow
306#529—711no—
307C4×#41C2×D7111yescharacteristic
308#485—711no—
309#482—711no—
310#479—711no—
311#478—711no—
312#464—711no—
313#463—711no—
314#452—711no—
315#449—711no—
316#446—711no—
317C8×Q8—711no—
318C8×Q8—711no—
319C8×Q8—711no—
320C4×Q16—711no—
321C4×SD16—711no—
322C4×SD16—711no—
323C4×D8—711no—
324#436—711no—
325#436—711no—
326#436—711no—
327#436—1421no—
328#436—1421no—
329#435—1421no—
330C8×D4—711no—
331C8×D4—1421no—
332#433—1421no—
333#431—711no—
334#431—711no—
335#431—1421no—
336#431C2×D7111yescharacteristic
337#424—1421no—
338#423—711no—
339#423—1421no—
340C2×C4⋊C8C2×D7111yescharacteristic
341#335—711no—
342#335—711no—
343#335—711no—
344#335—711no—
345Q8⋊C8—711no—
346Q8⋊C8—711no—
347Q8⋊C8C2×D7111yescharacteristic
348Q8⋊C8C2×D7111yescharacteristic
349D4⋊C8—711no—
350D4⋊C8—711no—
351D4⋊C8C2×D7111yescharacteristic
352D4⋊C8C2×D7111yescharacteristic
353#322—711no—
354#322—711no—
355#322—711no—
356C82—711no—
357C2×#41C4×D7111yescharacteristic
358C2×Q16—711no—
359C2×SD16—711no—
360C2×SD16—711no—
361C2×D8—711no—
362C2×M16—1421no—
363C2×M16—1421no—
364C2×M16—1421no—
365C2×M16—1421no—
366C22×C8—21no—
367C22×C8—1421no—
368#126—1421no—
369#125—711no—
370#124—711no—
371#123—1421no—
372C22⋊Q8—711no—
373#121—711no—
374C4×Q8—711no—
375C4×Q8C4×D7111yescharacteristic
376C4×Q8C22×D7111yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—711no—
380C4×D4—1421no—
381C4×D4C4×D7111yescharacteristic
382C4×D4C22×D7111yescharacteristic
383#117—21no—
384#117—1421no—
385#117C4×D7111yescharacteristic
386C2×C42—21no—
387C2×C42C22×D7111yescharacteristic
388#107—711no—
389#107—711no—
390#106—711no—
391#106—711no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—711no—
395C4⋊C8—711no—
396C4⋊C8—711no—
397C4⋊C8—711no—
398C4⋊C8—711no—
399C4⋊C8—711no—
400C4⋊C8—1421no—
401C4⋊C8—1421no—
402C4⋊C8—1421no—
403C4⋊C8—1421no—
404C4⋊C8—1421no—
405C4⋊C8C22×D7111yescharacteristic
406C4⋊C8C22×D7111yescharacteristic
407#103—711no—
408#103—711no—
409#103—711no—
410#103—711no—
411#102—711no—
412#102—711no—
413#102—711no—
414#102—711no—
415C22⋊C8—21no—
416C22⋊C8—711no—
417C22⋊C8—711no—
418C22⋊C8—711no—
419C22⋊C8—711no—
420C22⋊C8—1421no—
421C22⋊C8—1421no—
422C22⋊C8—1421no—
423C22⋊C8—1421no—
424#97—1421no—
425C4×C8—711no—
426C4×C8—711no—
427C4×C8—711no—
428C4×C8—711no—
429C4×C8—1421no—
430C4×C8—1421no—
431C4×C8—1421no—
432C4×C8C22×D7111yescharacteristic
433C4×C8C22×D7111yescharacteristic
434#41—22no—
435#41—41no—
436C2×Q8—711no—
437C2×Q8C2×C4×D7111yescharacteristic
438C2×D4—21no—
439C2×D4—711no—
440C2×D4—1421no—
441C2×D4C2×C4×D7111yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—1421no—
446C22×C4C2×C4×D7111yescharacteristic
447C22×C4D4×D7111yescharacteristic
448C22×C4#1176586411yescharacteristic
449Q16—1422no—
450SD16—2841no—
451SD16—1422no—
452D8—2841no—
453M16—5681no—
454M16—5681no—
455M16—5681no—
456M16—5681no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—81no—
462C2×C8—5681no—
463C2×C8—711no—
464C2×C8—711no—
465C2×C8—711no—
466C2×C8—711no—
467C2×C8—711no—
468C2×C8—711no—
469C2×C8—711no—
470C2×C8—711no—
471C2×C8—1421no—
472C2×C8—1421no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—711no—
476C4⋊C4—711no—
477C4⋊C4—1421no—
478C4⋊C4—1421no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4C2×C4×D7111yescharacteristic
482C4⋊C4C2×C4×D7111yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1421no—
487C22⋊C4—1421no—
488C22⋊C4—1421no—
489C22⋊C4—1421no—
490C22⋊C4—1421no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—1421no—
495C42C2×C4×D7111yescharacteristic
496C42D4×D7111yescharacteristic
497C42#1176586411yescharacteristic
498C42C23×D7111yescharacteristic
499C23—21no—
500C23—1421no—
501C23(C22⋊D71)⋊C411yescharacteristic
502Q8—21no—
503Q8—1421no—
504Q8—1421no—
505Q8#1455613412yes—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—2842no—
510D4—1421no—
511D4—1421no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—2841no—
525C2×C4—2841no—
526C2×C4—1421no—
527C2×C4—1421no—
528C2×C4—1421no—
529C2×C4(C22⋊D71)⋊C411yescharacteristic
530C2×C4(C22⋊D71)⋊C411yescharacteristic
531C2×C4(C22⋊D71)⋊C411yescharacteristic
532C2×C4C22×C4×D7111yescharacteristic
533C2×C4C2×D4×D7111yescharacteristic
534C2×C4C2×#1176586411yescharacteristic
535C8—41no—
536C8—41no—
537C8—41no—
538C8—41no—
539C8—2841no—
540C8—2841no—
541C8—1421no—
542C8—1421no—
543C8—1421no—
544C8—1421no—
545C8—1421no—
546C8—1421no—
547C8—1421no—
548C8—1421no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—2841no—
556C22—2841no—
557C22—2841no—
558C22C2×(C22⋊D71)⋊C411yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—2841no—
567C4—2841no—
568C4—2841no—
569C4C2×#1455613411yescharacteristic
570C4#1798676511yescharacteristic
571C4#1798680311yescharacteristic
572C4#1798681511yescharacteristic
573C2—41no—
574C2—81no—
575C2—5681no—
576C2#2259706111yescharacteristic
577C2#2259721111yescharacteristic
578C2#2259723911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1477C2
socleC22×C71C2×C4×D4
prime radical#17986144C4
cosocleC42×C71C24
hypercenter#1477D71
hypocenterC71#18033
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #22597496C2semidirect
3. #22597472C2semidirect
4. #22587239C2semidirect
5. #22587222C2semidirect
6. #22587221C2semidirect
7. #22587215C2semidirect
8. #22587187C2semidirect
9. #22587172C2semidirect
10. #22587155C2semidirect
11. #22587144C2semidirect
12. #22587114C2semidirect
13. #22586806C2semidirect
14. #22586640C2semidirect
15. C71×#1477C2semidirect
303. #1477D71semidirect
305. C71#18033semidirect

Maximal subgroups

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

subgroupindex
#225983582
#225974962
#225974722
#225872392
#225872222
#225872212
#225872152
#225871872
#225871722
#225871552
#225871442
#225871142
#225868062
#225866402
C71×#14772
#1803371