Finite Group Databasebeta

← #31965038

Subgroups — #31965038

Order 18176.

938 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597429C211yesmaximal, characteristic, complemented
2#22590991C211yesmaximal, characteristic, complemented
3#22586917C212yesmaximal, complemented
4#22586900C212yesmaximal, complemented
5#22586887C212yesmaximal, complemented
6#22586878C212yesmaximal, complemented
7#22586865C211yesmaximal, characteristic, complemented
8#22586848C211yesmaximal, characteristic, complemented
9#22586835C211yesmaximal, characteristic, complemented
10#22586826C211yesmaximal, characteristic, complemented
11C71×#3522C211yesmaximal, characteristic
12#17986130—21no—
13#17986130—22no—
14Q8×C71⋊C8—21no—
15#17986056C2212yes—
16#17986056C2211yescharacteristic
17#17986052C2212yes—
18#17986052C2211yescharacteristic
19#17985906—21no—
20#17985898—21no—
21#17985885—22no—
22#17985867C2212yes—
23#17985865C2212yes—
24C4×C4⋊Dic71C2211yescharacteristic
25#17985845C2211yescharacteristic
26#17985736C2211yescharacteristic
27#17985736C2212yes—
28#17985736C2211yescharacteristic
29#17985733C2211yescharacteristic
30#17985733C2212yes—
31#17985733C2212yes—
32#17985733C2212yes—
33#17985733C2211yescharacteristic
34#17985733C2212yes—
35#17985733C2212yes—
36#559×C71C411yescharacteristic, complemented
37#559×C71C412yescomplemented
38Q82×C71C411yescharacteristic, complemented
39#554×C71C412yescomplemented
40#554×C71C411yescharacteristic, complemented
41#550×C71C411yescharacteristic, complemented
42#548×C71—21no—
43#547×C71—22no—
44#547×C71—21no—
45#542×C71—21no—
46#542×C71—22no—
47#532×C71C2212yes—
48#532×C71C2211yescharacteristic
49#529×C71C2212yes—
50#529×C71C2211yescharacteristic
51#520×C71—21no—
52C2×C4×Q8×C71C2211yescharacteristic
53C2×C4⋊Dic71—21no—
54C2×C4⋊Dic71—22no—
55C2×C4×Dic71—21no—
56C2×C4×Dic71—21no—
57C4⋊(C71⋊C8)—22no—
58C4⋊(C71⋊C8)—21no—
59C4⋊(C71⋊C8)—21no—
60C4⋊(C71⋊C8)—22no—
61C22⋊(C71⋊C8)C2311yescharacteristic
62C22⋊(C71⋊C8)C2312yes—
63C22⋊(C71⋊C8)C2311yescharacteristic
64#14556039—22no—
65#14556039—21no—
66C4×C71⋊C8—21no—
67C4×C71⋊C8—22no—
68C71⋊#95—21no—
69C71⋊#95—22no—
70C71⋊#95—22no—
71C71⋊#95—22no—
72C71⋊#95C2311yescharacteristic
73C71⋊#95C2311yescharacteristic
74C71⋊#95C2312yes—
75C2×#41×C71—21no—
76C2×#41×C71—21no—
77C2×#41×C71—22no—
78C22×Q8×C71D411yescharacteristic
79C4⋊Q8×C71—22no—
80C4⋊Q8×C71—21no—
81C4⋊Q8×C71C2×C412yes—
82C4⋊Q8×C71C2×C411yescharacteristic
83#127×C71C2×C412yes—
84#127×C71C2×C411yescharacteristic
85#126×C71—22no—
86#126×C71—21no—
87#125×C71C2×C412yes—
88#125×C71C2×C411yescharacteristic
89#124×C71—21no—
90#124×C71—22no—
91#124×C71—22no—
92#124×C71—21no—
93#124×C71C2×C412yes—
94#124×C71C2×C411yescharacteristic
95#123×C71—21no—
96#123×C71—22no—
97#123×C71—22no—
98#123×C71—22no—
99#123×C71—22no—
100C22⋊Q8×C71—21no—
101C22⋊Q8×C71—22no—
102C22⋊Q8×C71—22no—
103C22⋊Q8×C71—22no—
104C22⋊Q8×C71—22no—
105C22⋊Q8×C71C2311yescharacteristic
106#121×C71—21no—
107#121×C71—22no—
108#121×C71—22no—
109#121×C71—21no—
110#121×C71C2312yes—
111#121×C71C2311yescharacteristic
112C4×Q8×C71—21no—
113C4×Q8×C71C2×C412yes—
114C4×Q8×C71C2×C411yescharacteristic
115C4×Q8×C71C2×C411yescharacteristic
116C4×Q8×C71C2×C411yescharacteristic
117C4×Q8×C71C2×C412yes—
118C4×Q8×C71D411yescharacteristic
119C4×Q8×C71D411yescharacteristic
120C4×Q8×C71D412yes—
121C4×D4×C71—21no—
122C4×D4×C71—22no—
123C4×D4×C71—22no—
124C4×D4×C71—21no—
125C4×D4×C71—21no—
126C4×D4×C71—22no—
127C4×D4×C71C2×C412yes—
128C4×D4×C71C2×C411yescharacteristic
129C4×D4×C71C2×C412yes—
130C4×D4×C71C2×C412yes—
131C4×D4×C71C2×C412yes—
132#117×C71—21no—
133#117×C71—22no—
134C2×C4⋊C4×C71—22no—
135C2×C4⋊C4×C71—21no—
136C2×C4⋊C4×C71D412yes—
137C2×C4⋊C4×C71D411yescharacteristic
138C2×C42×C71C2311yescharacteristic
139C2×C42×C71C2312yes—
140C22×Dic71—21no—
141C22×Dic71—21no—
142C22×Dic71—22no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—22no—
146C4⋊Dic71—42no—
147C4⋊Dic71—82no—
148C4×Dic71—42no—
149C4×Dic71—81no—
150#41×C71—44no—
151#41×C71—81no—
152#41×C71—82no—
153#41×C71—81no—
154#41×C71—82no—
155C2×Q8×C71—22no—
156C2×Q8×C71—21no—
157C2×Q8×C71—42no—
158C2×Q8×C71C22⋊C411yescharacteristic
159C2×Q8×C71C22⋊C411yescharacteristic
160C2×Q8×C71C22⋊C412yes—
161C2×Q8×C71C22×C411yescharacteristic
162C2×D4×C71—21no—
163C2×D4×C71—22no—
164C2×D4×C71—22no—
165C2×D4×C71—21no—
166C2×D4×C71—22no—
167C2×D4×C71—21no—
168C2×D4×C71C22×C412yes—
169C2×D4×C71C22×C411yescharacteristic
170C22×C4×C71—22no—
171C22×C4×C71—21no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D412yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D412yes—
179C22×C4×C71C2411yescharacteristic
180C4⋊C4×C71—21no—
181C4⋊C4×C71—22no—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—22no—
186C4⋊C4×C71—22no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71C22⋊C412yes—
190C4⋊C4×C71C22⋊C411yescharacteristic
191C4⋊C4×C71C22⋊C412yes—
192C4⋊C4×C71C22⋊C411yescharacteristic
193C4⋊C4×C71C22⋊C412yes—
194C4⋊C4×C71C22⋊C412yes—
195C4⋊C4×C71C22⋊C412yes—
196C4⋊C4×C71C22×C411yescharacteristic
197C4⋊C4×C71C22×C411yescharacteristic
198C4⋊C4×C71C22×C412yes—
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C22⋊C4×C71—22no—
202C22⋊C4×C71—21no—
203C22⋊C4×C71—22no—
204C22⋊C4×C71—22no—
205C22⋊C4×C71—21no—
206C22⋊C4×C71—22no—
207C22⋊C4×C71—22no—
208C22⋊C4×C71—22no—
209C22⋊C4×C71—21no—
210C22⋊C4×C71—22no—
211C42×C71—22no—
212C42×C71—21no—
213C42×C71C22×C411yescharacteristic
214C42×C71C22×C412yes—
215C42×C71C22×C412yes—
216C42×C71C22×C411yescharacteristic
217C42×C71C2×D411yescharacteristic
218C42×C71C2×D412yes—
219C42×C71C2×D412yes—
220C42×C71C2×D411yescharacteristic
221C23×C71—22no—
222C23×C71—21no—
223C23×C71C22×D411yescharacteristic
224Q8×C71—22no—
225Q8×C71—22no—
226Q8×C71—42no—
227Q8×C71—41no—
228Q8×C71—81no—
229Q8×C71—82no—
230D4×C71—42no—
231D4×C71—41no—
232D4×C71—44no—
233D4×C71—42no—
234D4×C71—41no—
235D4×C71—41no—
236D4×C71—42no—
237D4×C71—42no—
238D4×C71—81no—
239D4×C71—82no—
240C2×C4×C71—21no—
241C2×C4×C71—22no—
242C2×C4×C71—21no—
243C2×C4×C71—22no—
244C2×C4×C71—22no—
245C2×C4×C71—21no—
246C2×C4×C71—22no—
247C2×C4×C71—21no—
248C2×C4×C71—21no—
249C2×C4×C71—42no—
250C2×C4×C71—41no—
251C2×C4×C71—44no—
252C2×C4×C71—42no—
253C2×C4×C71—82no—
254C2×C4×C71—82no—
255C2×C4×C71—82no—
256C2×C4×C71C2×C22⋊C412yes—
257C2×C4×C71C2×C22⋊C411yescharacteristic
258C2×C4×C71C2×C22⋊C412yes—
259C2×C4×C71C2×C22⋊C411yescharacteristic
260C2×C4×C71C2×C22⋊C412yes—
261C2×C4×C71C2×C22⋊C412yes—
262C2×C4×C71C2×C22⋊C411yescharacteristic
263C2×C4×C71C2×C22⋊C411yescharacteristic
264C2×C4×C71C23×C411yescharacteristic
265C2×C4×C71C22×D411yescharacteristic
266C2×Dic71—41no—
267C2×Dic71—41no—
268C2×Dic71—42no—
269C2×Dic71—41no—
270C2×Dic71—41no—
271C2×Dic71—42no—
272C2×Dic71—42no—
273C2×Dic71—41no—
274C2×Dic71—41no—
275C71⋊C8—41no—
276C71⋊C8—41no—
277C71⋊C8—42no—
278C22×C71—21no—
279C22×C71—21no—
280C22×C71—42no—
281C22×C71—41no—
282C22×C71—42no—
283C22×C71—41no—
284C22×C71—42no—
285C22×C71—41no—
286C22×C71C22×C22⋊C411yescharacteristic
287C22×C71#57611yescharacteristic
288C22×C71#57711yescharacteristic
289C4×C71—21no—
290C4×C71—21no—
291C4×C71—21no—
292C4×C71—22no—
293C4×C71—21no—
294C4×C71—22no—
295C4×C71—21no—
296C4×C71—22no—
297C4×C71—22no—
298C4×C71—41no—
299C4×C71—42no—
300C4×C71—41no—
301C4×C71—41no—
302C4×C71#42112yes—
303Dic71—81no—
304Dic71—82no—
305Dic71—81no—
306#17602—711nomaximal, Sylow
307C2×C71—21no—
308C2×C71—21no—
309C2×C71—82no—
310C2×C71—81no—
311C2×C71#288411yescharacteristic
312C2×C71#289711yescharacteristic
313C2×C71C2×#42111yescharacteristic
314#3522D7111yescharacteristic
315#2984—711no—
316#2320—711no—
317#1521—712no—
318#1517—712no—
319#1514—712no—
320#1512—712no—
321#1509—711no—
322#1505—711no—
323#1502—711no—
324#1500—711no—
325C71#1760211yescharacteristic, complemented, Sylow
326#559Dic7112yescomplemented
327#559Dic7111yescharacteristic, complemented
328Q82Dic7111yescharacteristic, complemented
329#554Dic7111yescharacteristic, complemented
330#554Dic7112yescomplemented
331#550Dic7111yescharacteristic, complemented
332#548—21no—
333#547—22no—
334#547—21no—
335#542—21no—
336#542—22no—
337#532C2×D7111yescharacteristic
338#532C2×D7112yes—
339#529C2×D7111yescharacteristic
340#529C2×D7112yes—
341#520—21no—
342C2×C4×Q8C2×D7111yescharacteristic
343#446—1421no—
344#446—1422no—
345C8×Q8—1421no—
346#432—711no—
347#432—712no—
348#431—711no—
349#431—712no—
350#391—1421no—
351#389—1421no—
352#387—1422no—
353#383—712no—
354#382—712no—
355C4×C4⋊C4—711no—
356#376—711no—
357#328—711no—
358#328—711no—
359#328—712no—
360#327—711no—
361#327—711no—
362#327—712no—
363#327—712no—
364#327—712no—
365#327—712no—
366#327—712no—
367C2×#41—21no—
368C2×#41—21no—
369C2×#41—22no—
370C22×Q8C22⋊D7111yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8C2×Dic7111yescharacteristic
374C4⋊Q8C2×Dic7112yes—
375#127C2×Dic7111yescharacteristic
376#127C2×Dic7112yes—
377#126—22no—
378#126—21no—
379#125C2×Dic7111yescharacteristic
380#125C2×Dic7112yes—
381#124—21no—
382#124—22no—
383#124—22no—
384#124—21no—
385#124C2×Dic7111yescharacteristic
386#124C2×Dic7112yes—
387#123—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8C22×D7111yescharacteristic
398#121—21no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121C22×D7111yescharacteristic
403#121C22×D7112yes—
404C4×Q8—21no—
405C4×Q8C2×Dic7111yescharacteristic
406C4×Q8C2×Dic7112yes—
407C4×Q8C2×Dic7111yescharacteristic
408C4×Q8C2×Dic7111yescharacteristic
409C4×Q8C2×Dic7112yes—
410C4×Q8C22⋊D7111yescharacteristic
411C4×Q8C22⋊D7111yescharacteristic
412C4×Q8C22⋊D7112yes—
413C4×D4—21no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—22no—
419C4×D4C2×Dic7111yescharacteristic
420C4×D4C2×Dic7112yes—
421C4×D4C2×Dic7112yes—
422C4×D4C2×Dic7112yes—
423C4×D4C2×Dic7112yes—
424#117—21no—
425#117—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—1421no—
429C2×C4⋊C4—1422no—
430C2×C4⋊C4C22⋊D7112yes—
431C2×C4⋊C4C22⋊D7111yescharacteristic
432C2×C42—1421no—
433C2×C42—1421no—
434C2×C42C22×D7111yescharacteristic
435C2×C42C22×D7112yes—
436C4⋊C8—1422no—
437C4⋊C8—1421no—
438C4⋊C8—1421no—
439C4⋊C8—1422no—
440C22⋊C8—711no—
441C22⋊C8—711no—
442C22⋊C8—712no—
443#97—1422no—
444#97—1421no—
445C4×C8—1421no—
446C4×C8—1422no—
447#95—711no—
448#95—711no—
449#95—712no—
450#95—1421no—
451#95—1422no—
452#95—1422no—
453#95—1422no—
454#41—44no—
455#41—81no—
456#41—82no—
457#41—81no—
458#41—82no—
459C2×Q8—22no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C22⋊Dic7111yescharacteristic
463C2×Q8C22⋊Dic7111yescharacteristic
464C2×Q8C22⋊Dic7112yes—
465C2×Q8C22×Dic7111yescharacteristic
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4C22×Dic7111yescharacteristic
473C2×D4C22×Dic7112yes—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—1421no—
480C22×C4—1421no—
481C22×C4—1422no—
482C22×C4C2×C22⋊D7111yescharacteristic
483C22×C4C2×C22⋊D7112yes—
484C22×C4C2×C22⋊D7111yescharacteristic
485C22×C4C2×C22⋊D7112yes—
486C22×C4C23×D7111yescharacteristic
487C2×C8—1421no—
488C2×C8—1421no—
489C2×C8—1422no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—2842no—
500C4⋊C4—5682no—
501C4⋊C4C22⋊Dic7112yes—
502C4⋊C4C22⋊Dic7111yescharacteristic
503C4⋊C4C22⋊Dic7112yes—
504C4⋊C4C22⋊Dic7111yescharacteristic
505C4⋊C4C22⋊Dic7112yes—
506C4⋊C4C22⋊Dic7112yes—
507C4⋊C4C22⋊Dic7112yes—
508C4⋊C4C22×Dic7111yescharacteristic
509C4⋊C4C22×Dic7112yes—
510C4⋊C4C22×Dic7111yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C42—22no—
524C42—21no—
525C42—2842no—
526C42—5681no—
527C42C22×Dic7111yescharacteristic
528C42C22×Dic7112yes—
529C42C22×Dic7112yes—
530C42C22×Dic7111yescharacteristic
531C42C2×C22⋊D7111yescharacteristic
532C42C2×C22⋊D7112yes—
533C42C2×C22⋊D7112yes—
534C42C2×C22⋊D7111yescharacteristic
535C23—22no—
536C23—21no—
537C23C22×C22⋊D7111yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—41no—
542Q8—81no—
543Q8—82no—
544D4—41no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—41no—
549D4—41no—
550D4—42no—
551D4—42no—
552D4—81no—
553D4—82no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—2841no—
571C2×C4—2841no—
572C2×C4—2842no—
573C2×C4—2841no—
574C2×C4—2841no—
575C2×C4—2842no—
576C2×C4—2842no—
577C2×C4—2841no—
578C2×C4—2841no—
579C2×C4C2×C22⋊Dic7112yes—
580C2×C4C2×C22⋊Dic7111yescharacteristic
581C2×C4C2×C22⋊Dic7112yes—
582C2×C4C2×C22⋊Dic7111yescharacteristic
583C2×C4C2×C22⋊Dic7112yes—
584C2×C4C2×C22⋊Dic7112yes—
585C2×C4C2×C22⋊Dic7111yescharacteristic
586C2×C4C2×C22⋊Dic7111yescharacteristic
587C2×C4C23×Dic7111yescharacteristic
588C2×C4C22×C22⋊D7111yescharacteristic
589C8—2841no—
590C8—2841no—
591C8—2842no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22C22×C22⋊Dic7111yescharacteristic
601C22#1798680211yescharacteristic
602C22#1798681011yescharacteristic
603C4—21no—
604C4—22no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—5681no—
617C4—5682no—
618C4—5681no—
619C4#1798601612yes—
620C2—21no—
621C2—21no—
622C2—82no—
623C2—81no—
624C2#2259680411yescharacteristic
625C2#2259686611yescharacteristic
626C2C2×#1798601611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
Fitting, prime radicalC71×#3522C2
socleC22×C71C22×C22⋊C4
cosocleC22×C4×C71C24
hypercenter#3522D71
hypocenterC71#17602
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3522

Complements

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

NH ≅ G/Ntype
1. #22597429C2semidirect
2. #22590991C2semidirect
3. #22586917C2semidirect
4. #22586900C2semidirect
5. #22586887C2semidirect
6. #22586878C2semidirect
7. #22586865C2semidirect
8. #22586848C2semidirect
9. #22586835C2semidirect
10. #22586826C2semidirect
36. #559×C71C4semidirect
37. #559×C71C4semidirect
38. Q82×C71C4semidirect
39. #554×C71C4semidirect
40. #554×C71C4semidirect
41. #550×C71C4semidirect
325. C71#17602semidirect
326. #559Dic71semidirect
327. #559Dic71semidirect
328. Q82Dic71semidirect
329. #554Dic71semidirect
330. #554Dic71semidirect
331. #550Dic71semidirect

Maximal subgroups

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

subgroupindex
#225974292
#225909912
#225869172
#225869002
#225868872
#225868782
#225868652
#225868482
#225868352
#225868262
C71×#35222
#1760271