Finite Group Databasebeta

← #32005565

Subgroups — #32005565

Order 18176.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986358C211yesmaximal, characteristic
2#22591088C211yesmaximal, characteristic, complemented
3#22589300C211yesmaximal, characteristic, complemented
4#22589152C211yesmaximal, characteristic
5#22589040C211yesmaximal, characteristic
6#22589033C211yesmaximal, characteristic, complemented
7#22589026C211yesmaximal, characteristic
8#22589019C211yesmaximal, characteristic, complemented
9#22588475C211yesmaximal, characteristic, complemented
10#22588170C211yesmaximal, characteristic, complemented
11#22588165C211yesmaximal, characteristic
12C4×#14556052C211yesmaximal, characteristic, complemented
13C4×#14556049C211yesmaximal, characteristic
14#22588027C211yesmaximal, characteristic, complemented
15C71×#2326C211yesmaximal, characteristic
16C2×#14556128C2211yescharacteristic
17#17986358C418yescomplemented
18C2×#14556052C2211yescharacteristic
19C2×#14556052C2211yescharacteristic
20C2×#14556052C2211yescharacteristic
21C2×#14556052C2211yescharacteristic
22C2×#14556049C2211yescharacteristic
23C2×#14556049C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556039C2211yescharacteristic
27C2×#14556039C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×C4×C71⋊C8C2211yescharacteristic
30C2×C4×C71⋊C8C2211yescharacteristic
31C2×C4×C71⋊C8C2211yescharacteristic
32#17985903—21no—
33#17985903—21no—
34#17985897C2211yescharacteristic
35#17985897C2211yescharacteristic
36#17985882—21no—
37#17985882—21no—
38#17985865C2211yescharacteristic
39C4×C4⋊Dic71C2211yescharacteristic
40C4×C4⋊Dic71C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49C2×#124×C71C2211yescharacteristic
50C2×C4×Q8×C71C2211yescharacteristic
51C2×C4×D4×C71C2211yescharacteristic
52#391×C71C2211yescharacteristic
53#390×C71C2211yescharacteristic
54#388×C71—21no—
55#388×C71—21no—
56#385×C71—21no—
57#385×C71—21no—
58#384×C71C2211yescharacteristic
59#376×C71C2211yescharacteristic
60C22×Dic142—21no—
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65#14556128—22no—
66#14556128C2×C414yescomplemented
67C2×C4⋊Dic71—21no—
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76#14556052—22no—
77#14556052—22no—
78#14556052C2×C414yescomplemented
79#14556052C2×C414yescomplemented
80#14556049—22no—
81#14556049—22no—
82#14556049C2×C414yes—
83#14556049C2×C414yes—
84#14556039—22no—
85#14556039—22no—
86#14556039C2×C414yescomplemented
87C4×C71⋊C8—22no—
88C4×C71⋊C8—22no—
89C4×C71⋊C8—22no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C22×Q8×C71C2311yescharacteristic
97C22×D4×C71C2311yescharacteristic
98C23×C4×C71—21no—
99#124×C71—22no—
100#124×C71C2×C414yes—
101C4×Q8×C71—22no—
102C4×Q8×C71—22no—
103C4×D4×C71—22no—
104C4×D4×C71—22no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C22⋊C4×C71—21no—
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C22⋊C4×C71—21no—
115C2×C42×C71—21no—
116C2×C42×C71C2311yescharacteristic
117C2×C42×C71C2311yescharacteristic
118C2×C42×C71C2311yescharacteristic
119#95×C71—21no—
120#95×C71—21no—
121#95×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123#95×C71D411yescharacteristic
124#95×C71D411yescharacteristic
125C2×Dic142—24no—
126C2×Dic142—81no—
127C2×Dic142—81no—
128C22×Dic71—21no—
129C22×Dic71—21no—
130C22×Dic71—21no—
131C22×Dic71—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8C22×C412yes—
141C2×C71⋊C8C22×C412yes—
142C2×C71⋊C8#4112yes—
143C2×C71⋊C8#4112yes—
144C4⋊Dic71—24no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71C22×C412yes—
152C4⋊Dic71C22×C412yes—
153Dic71⋊C4—81no—
154Dic71⋊C4—81no—
155C4×Dic71—81no—
156C4×Dic71—81no—
157C24×C71—21no—
158C2×Q8×C71—21no—
159C2×Q8×C71—21no—
160C2×Q8×C71—21no—
161C2×Q8×C71—21no—
162C2×Q8×C71—21no—
163C2×Q8×C71C22×C412yes—
164C2×D4×C71—21no—
165C2×D4×C71—21no—
166C2×D4×C71—21no—
167C2×D4×C71—21no—
168C2×D4×C71—21no—
169C2×D4×C71C22×C412yes—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—42no—
174C22×C4×C71—81no—
175C22×C4×C71—81no—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2411yescharacteristic
183C4⋊C4×C71—21no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—42no—
188C4⋊C4×C71—81no—
189C4⋊C4×C71—81no—
190C22⋊C4×C71—24no—
191C22⋊C4×C71—24no—
192C22⋊C4×C71—42no—
193C22⋊C4×C71—81no—
194C22⋊C4×C71—81no—
195C42×C71—21no—
196C42×C71—21no—
197C42×C71—42no—
198C42×C71C22×C412yes—
199C42×C71#4112yes—
200C42×C71#4112yes—
201C42×C71#4112yes—
202C23×C71—22no—
203C23×C71—41no—
204C23×C71—41no—
205C23×C71—41no—
206C23×C71—41no—
207C23×C71—41no—
208C23×C71—41no—
209C23×C71C22×D411yescharacteristic
210Q8×C71—22no—
211Q8×C71—22no—
212Q8×C71—41no—
213Q8×C71—41no—
214D4×C71—22no—
215D4×C71—22no—
216D4×C71—41no—
217D4×C71—41no—
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—21no—
225C2×C4×C71—21no—
226C2×C4×C71—22no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—84no—
242C2×C4×C71C4×D412yes—
243C2×C4×C71C4×D412yes—
244C2×C4×C71#12412yes—
245C2×C4×C71#12412yes—
246C2×C4×C71C23×C411yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264Dic142—84no—
265C71⋊C8—22no—
266C71⋊C8—22no—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71—42no—
272C22×C71—81no—
273C22×C71—81no—
274C22×C71—81no—
275C22×C71—81no—
276C22×C71C2×C4×D411yescharacteristic
277C22×C71C2×#12411yescharacteristic
278C22×C71#53211yescharacteristic
279C22×C71#57611yescharacteristic
280C22×C71#57711yescharacteristic
281C22×C71#57711yescharacteristic
282C22×C71#57811yescharacteristic
283C4×C71—22no—
284C4×C71—22no—
285C4×C71—22no—
286C4×C71—22no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—42no—
290C4×C71—81no—
291C4×C71—81no—
292C4×C71C4×#4111yescharacteristic
293C4×C71C4×#4111yescharacteristic
294C4×C71#52911yescharacteristic
295C4×C71#52911yescharacteristic
296Dic71—42no—
297Dic71—42no—
298Dic71—81no—
299Dic71—81no—
300#21654—711nomaximal, Sylow
301C2×C71—82no—
302C2×C71C4×#12411yescharacteristic
303C2×C71#294611yescharacteristic
304C2×C71#294711yescharacteristic
305C2×C71#313611yescharacteristic
306C2×C71#313711yescharacteristic
307C2×C71#317311yescharacteristic
308C2×C71#317511yescharacteristic
309C2×#489—711no—
310#2331—711no—
311#2326D7111yescharacteristic
312#1990—711no—
313#1958—711no—
314#1936—711no—
315#1935—711no—
316#1934—711no—
317#1933—711no—
318#1836—711no—
319#1763—711no—
320#1762—711no—
321C4×#103—711no—
322C4×#102—711no—
323#1726—711no—
324C71#2165411yescharacteristic, complemented, Sylow
325C2×C4⋊Q8—711no—
326C2×#124C2×D7111yescharacteristic
327C2×C4×Q8C2×D7111yescharacteristic
328C2×C4×D4C2×D7111yescharacteristic
329#489—718no—
330C2×#103—711no—
331C2×#103—711no—
332C2×#103—711no—
333C2×#103—711no—
334C2×#102—711no—
335C2×#102—711no—
336C2×#102—711no—
337C2×#102—711no—
338C2×#97—711no—
339C2×#97—711no—
340C2×#97—711no—
341C2×C4×C8—711no—
342C2×C4×C8—711no—
343C2×C4×C8—711no—
344#391—1421no—
345#391—1421no—
346#391C2×D7111yescharacteristic
347#390C2×D7111yescharacteristic
348#389—711no—
349#389—711no—
350#388—21no—
351#388—21no—
352#387—1421no—
353#387—1421no—
354#385—21no—
355#385—21no—
356#384C2×D7111yescharacteristic
357#382—711no—
358C4×C4⋊C4—711no—
359C4×C4⋊C4—711no—
360#376C2×D7111yescharacteristic
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369C22×Q8—1421no—
370C22×Q8C22×D7111yescharacteristic
371C22×D4C22×D7111yescharacteristic
372C23×C4—21no—
373C22×C8—711no—
374C22×C8—711no—
375C22×C8—711no—
376C22×C8—711no—
377C4⋊Q8—714no—
378C4⋊Q8—1422no—
379#124—22no—
380#124C4×D7114yes—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×D4—22no—
384C4×D4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—711no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C42—21no—
403C2×C42—1421no—
404C2×C42—1421no—
405C2×C42C22×D7111yescharacteristic
406C2×C42C22×D7111yescharacteristic
407C2×C42C22×D7111yescharacteristic
408#103—714no—
409#103—714no—
410#103—1422no—
411#103—1422no—
412#102—714no—
413#102—714no—
414#102—1422no—
415#102—1422no—
416#97—714no—
417#97—1422no—
418#97—1422no—
419C4×C8—1422no—
420C4×C8—1422no—
421C4×C8—1422no—
422#95—21no—
423#95—21no—
424#95—1421no—
425#95—1421no—
426#95—1421no—
427#95—1421no—
428#95—1421no—
429#95—1421no—
430#95C22⋊D7111yescharacteristic
431#95C22⋊D7111yescharacteristic
432#95C22⋊D7111yescharacteristic
433#95C22⋊D7111yescharacteristic
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—5681no—
441C2×Q8—5681no—
442C2×Q8—1424no—
443C2×Q8C2×C4×D7112yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D7112yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C23×D7111yescharacteristic
467C2×C8—712no—
468C2×C8—712no—
469C2×C8—712no—
470C2×C8—712no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—5681no—
487C4⋊C4—5681no—
488C4⋊C4—712no—
489C4⋊C4—712no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1424no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—5681no—
506C42—5681no—
507C42C2×C4×D7112yes—
508C42#1176586412yes—
509C42#1176586412yes—
510C42#1176586512yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C22×C22⋊D7111yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—5684no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—1422no—
563C2×C4—1422no—
564C2×C4C4×C22⋊D7112yes—
565C2×C4C4×C22⋊D7112yes—
566C2×C4#1455611512yes—
567C2×C4#1455611512yes—
568C2×C4C22×C4×D7111yescharacteristic
569C2×C4#41×D7111yescharacteristic
570C2×C4#41×D7111yescharacteristic
571C2×C4C2×#1176586411yescharacteristic
572C2×C4C2×#1176586411yescharacteristic
573C2×C4C2×#1176586511yescharacteristic
574C8—1422no—
575C8—1422no—
576C8—1422no—
577C8—1422no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×C4×C22⋊D7111yescharacteristic
586C22C2×#1455611511yescharacteristic
587C22#1798654511yescharacteristic
588C22#1798680211yescharacteristic
589C22#1798680711yescharacteristic
590C22#1798681011yescharacteristic
591C22#1798681411yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—2842no—
602C4—2842no—
603C4—5681no—
604C4—5681no—
605C4C4×#1176586411yescharacteristic
606C4C4×#1176586411yescharacteristic
607C4#1798653111yescharacteristic
608C4#1798653111yescharacteristic
609C2—82no—
610C2C4×#1455611511yescharacteristic
611C2#2259716811yescharacteristic
612C2#2259717911yescharacteristic
613C2#2259869411yescharacteristic
614C2#2259870511yescharacteristic
615C2#2259895111yescharacteristic
616C2#2259897311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2326C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2326D71
hypocenterC71#21654
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #22591088C2semidirect
3. #22589300C2semidirect
6. #22589033C2semidirect
8. #22589019C2semidirect
9. #22588475C2semidirect
10. #22588170C2semidirect
12. C4×#14556052C2semidirect
14. #22588027C2semidirect
17. #17986358C4semidirect
66. #14556128C2×C4semidirect
78. #14556052C2×C4semidirect
79. #14556052C2×C4semidirect
86. #14556039C2×C4semidirect
324. C71#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#179863582
#225910882
#225893002
#225891522
#225890402
#225890332
#225890262
#225890192
#225884752
#225881702
#225881652
C4×#145560522
C4×#145560492
#225880272
C71×#23262
#2165471