Finite Group Databasebeta

← #32004696

Subgroups — #32004696

Order 18176.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986341C211yesmaximal, characteristic
2C4×#14556107C211yesmaximal, characteristic
3C4×#14556087C211yesmaximal, characteristic
4#22588982C211yesmaximal, characteristic, complemented
5#22588975C211yesmaximal, characteristic, complemented
6#22588933C211yesmaximal, characteristic, complemented
7#22588926C211yesmaximal, characteristic, complemented
8#22588650C211yesmaximal, characteristic
9#22588541C211yesmaximal, characteristic
10#22588436C211yesmaximal, characteristic
11#22588237C211yesmaximal, characteristic, complemented
12#22588237C211yesmaximal, characteristic, complemented
13#22588171C211yesmaximal, characteristic, complemented
14#22588171C211yesmaximal, characteristic, complemented
15C71×#1749C211yesmaximal, characteristic
16C2×#14556107C2211yescharacteristic
17C2×C4×Dic142C2211yescharacteristic
18C2×#14556087C2211yescharacteristic
19C2×#14556087C2211yescharacteristic
20#17986341C418yescomplemented
21C2×#14556062C2211yescharacteristic
22C2×#14556062C2211yescharacteristic
23C2×#14556060C2211yescharacteristic
24C2×#14556060C2211yescharacteristic
25C2×#14556053C2211yescharacteristic
26C2×#14556053C2211yescharacteristic
27C2×#14556053C2211yescharacteristic
28C2×#14556053C2211yescharacteristic
29#17985905C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985883—21no—
32#17985872—21no—
33#17985859C2211yescharacteristic
34C4×Dic71⋊C4—21no—
35C4×C4⋊Dic71C2211yescharacteristic
36C4×C4⋊Dic71C2211yescharacteristic
37C4×C4⋊Dic71C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39C4×C4⋊Dic71C2211yescharacteristic
40C4×C22⋊Dic71—21no—
41C4×C22⋊Dic71—21no—
42C4×C22⋊Dic71—21no—
43#17985845—21no—
44C42×Dic71—21no—
45#17985762C2211yescharacteristic
46#17985762C2211yescharacteristic
47#17985762C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985762C2211yescharacteristic
50#17985762C2211yescharacteristic
51#17985762C2211yescharacteristic
52#17985762C2211yescharacteristic
53C22×C42×C71C2211yescharacteristic
54C2×C22⋊C8×C71C2211yescharacteristic
55C2×C22⋊C8×C71C2211yescharacteristic
56C2×#97×C71C2211yescharacteristic
57C2×#97×C71C2211yescharacteristic
58#336×C71C2211yescharacteristic
59#336×C71C2211yescharacteristic
60C22×Dic142C2311yescharacteristic
61#14556107—22no—
62#14556107C2×C414yes—
63C4×Dic142—22no—
64C4×Dic142—22no—
65#14556087—22no—
66#14556087—22no—
67#14556087—22no—
68#14556087C2×C414yes—
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71—21no—
81C2×C22⋊Dic71—21no—
82C2×C22⋊Dic71—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71—21no—
86C2×C4×Dic71—21no—
87C2×C4×Dic71—21no—
88C2×C4×Dic71—21no—
89C2×C4×Dic71—21no—
90C2×C4×Dic71—21no—
91#14556062—22no—
92#14556062C2×C414yescomplemented
93#14556060—22no—
94#14556060C2×C414yescomplemented
95#14556053—22no—
96#14556053—22no—
97#14556053C2×C414yescomplemented
98#14556053C2×C414yescomplemented
99C71⋊#95—21no—
100C71⋊#95—21no—
101C71⋊#95—21no—
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C71⋊#95—21no—
106C71⋊#95—21no—
107C71⋊#95—21no—
108C23×C4×C71D411yescharacteristic
109C23×C4×C71D411yescharacteristic
110C23×C4×C71C2311yescharacteristic
111C22×C8×C71C2311yescharacteristic
112C22×C8×C71C2311yescharacteristic
113C22×C8×C71C2311yescharacteristic
114C22×C8×C71C2311yescharacteristic
115C2×C42×C71—22no—
116C2×C42×C71—22no—
117C2×C42×C71D411yescharacteristic
118C2×C42×C71D411yescharacteristic
119C2×C42×C71C2311yescharacteristic
120C2×C42×C71C2311yescharacteristic
121C22⋊C8×C71—22no—
122C22⋊C8×C71C2×C414yes—
123#97×C71—22no—
124#97×C71—22no—
125C2×Dic142—21no—
126C2×Dic142—21no—
127C2×Dic142—21no—
128C2×Dic142—21no—
129C2×Dic142—21no—
130C2×Dic142C22×C412yes—
131C22×Dic71—21no—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C22×Dic71—21no—
136C22×Dic71—21no—
137C22×Dic71—21no—
138C22×Dic71—21no—
139C4⋊Dic71—21no—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71C22×C412yes—
149C4⋊Dic71C22×C412yes—
150C4⋊Dic71C22×C412yes—
151C4⋊Dic71#4112yes—
152C4⋊Dic71#4112yes—
153Dic71⋊C4—24no—
154Dic71⋊C4—42no—
155Dic71⋊C4—42no—
156C22⋊Dic71—24no—
157C22⋊Dic71—24no—
158C22⋊Dic71—24no—
159C22⋊Dic71—42no—
160C4×Dic71—24no—
161C4×Dic71—24no—
162C4×Dic71—42no—
163C4×Dic71—42no—
164C4×Dic71—42no—
165C4×Dic71—42no—
166C4×Dic71—42no—
167C4×Dic71—42no—
168C24×C71C2×D411yescharacteristic
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—41no—
177C22×C4×C71—41no—
178C22×C4×C71—41no—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71#4112yes—
186C22×C4×C71C2411yescharacteristic
187C2×C8×C71—21no—
188C2×C8×C71—21no—
189C2×C8×C71—21no—
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71—21no—
193C2×C8×C71—21no—
194C2×C8×C71—21no—
195C2×C8×C71—21no—
196C2×C8×C71—21no—
197C2×C8×C71C22×C412yes—
198C2×C8×C71C22×C412yes—
199C42×C71—22no—
200C42×C71—22no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71#4112yes—
204C42×C71#4112yes—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71—21no—
209C23×C71—21no—
210C23×C71—21no—
211C23×C71C4×D412yes—
212C23×C71C22×D411yescharacteristic
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—21no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—44no—
229C2×C4×C71—44no—
230C2×C4×C71C4×D412yes—
231C2×C4×C71#12312yes—
232C2×C4×C71#12312yes—
233C2×C4×C71#13712yes—
234C2×C4×C71C23×C411yescharacteristic
235C2×C4×C71C2×#4111yescharacteristic
236C2×C4×C71C2×#4111yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C2×C4×C71C2×#4111yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
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—22no—
265Dic142—22no—
266Dic142—41no—
267Dic142—41no—
268C8×C71—22no—
269C8×C71—22no—
270C8×C71—41no—
271C8×C71—41no—
272C22×C71—22no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—41no—
276C22×C71—41no—
277C22×C71—41no—
278C22×C71—41no—
279C22×C71C2×C4×D411yescharacteristic
280C22×C71C2×#12311yescharacteristic
281C22×C71#52511yescharacteristic
282C22×C71C2×#13711yescharacteristic
283C22×C71#57511yescharacteristic
284C22×C71#57511yescharacteristic
285C22×C71#57511yescharacteristic
286C4×C71—22no—
287C4×C71—22no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—41no—
291C4×C71C4×#4111yescharacteristic
292C4×C71C4×#4111yescharacteristic
293C4×C71#52911yescharacteristic
294C4×C71#52911yescharacteristic
295Dic71—42no—
296Dic71—42no—
297Dic71—42no—
298Dic71—42no—
299Dic71—42no—
300Dic71—42no—
301Dic71—81no—
302Dic71—81no—
303#21583—711nomaximal, Sylow
304C2×C71—42no—
305C2×C71C4×#12311yescharacteristic
306C2×C71#294311yescharacteristic
307C2×C71C4×#13711yescharacteristic
308C2×C71#309311yescharacteristic
309C2×C71#309511yescharacteristic
310C2×C71#311111yescharacteristic
311C2×C71#311611yescharacteristic
312C2×#485—711no—
313C4×C22⋊Q8—711no—
314C4×#117—711no—
315#1928—711no—
316#1927—711no—
317#1921—711no—
318#1920—711no—
319#1873—711no—
320#1854—711no—
321#1825—711no—
322#1776—711no—
323#1776—711no—
324#1763—711no—
325#1763—711no—
326#1749D7111yescharacteristic
327C71#2158311yescharacteristic, complemented, Sylow
328C2×C22⋊Q8—711no—
329C2×C4×Q8—711no—
330C2×#117—711no—
331C2×#117—711no—
332C22×C42C2×D7111yescharacteristic
333#485—718no—
334C2×#107—711no—
335C2×#107—711no—
336C2×#106—711no—
337C2×#106—711no—
338C2×#103—711no—
339C2×#103—711no—
340C2×#103—711no—
341C2×#103—711no—
342C2×C22⋊C8C2×D7111yescharacteristic
343C2×C22⋊C8C2×D7111yescharacteristic
344C2×#97C2×D7111yescharacteristic
345C2×#97C2×D7111yescharacteristic
346#391—711no—
347#389—711no—
348#387—1421no—
349#385—1421no—
350#380—711no—
351C4×C4⋊C4—711no—
352C4×C4⋊C4—711no—
353C4×C4⋊C4—711no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4—711no—
356C4×C4⋊C4—1421no—
357C4×C22⋊C4—1421no—
358C4×C22⋊C4—1421no—
359C4×C22⋊C4—1421no—
360#376—1421no—
361C43—1421no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#336C2×D7111yescharacteristic
371#336C2×D7111yescharacteristic
372C22×Q8—711no—
373C23×C4D28411yescharacteristic
374C23×C4D28411yescharacteristic
375C23×C4C22×D7111yescharacteristic
376C22×C8C22×D7111yescharacteristic
377C22×C8C22×D7111yescharacteristic
378C22×C8C22×D7111yescharacteristic
379C22×C8C22×D7111yescharacteristic
380C22⋊Q8—714no—
381C22⋊Q8—1422no—
382C4×Q8—1422no—
383C4×Q8—1422no—
384#117—714no—
385#117—1422no—
386#117—1422no—
387#117—1422no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—711no—
392C2×C4⋊C4—711no—
393C2×C4⋊C4—711no—
394C2×C4⋊C4—711no—
395C2×C4⋊C4—1421no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4—1421no—
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1421no—
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1421no—
405C2×C42—1421no—
406C2×C42—1421no—
407C2×C42—1421no—
408C2×C42—1421no—
409C2×C42—1421no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42D28411yescharacteristic
413C2×C42D28411yescharacteristic
414C2×C42C22×D7111yescharacteristic
415C2×C42C22×D7111yescharacteristic
416#107—714no—
417#107—1422no—
418#106—714no—
419#106—1422no—
420#103—714no—
421#103—714no—
422#103—1422no—
423#103—1422no—
424C22⋊C8—22no—
425C22⋊C8C4×D7114yes—
426#97—22no—
427#97—22no—
428#95—1421no—
429#95—1421no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436#95—1421no—
437C24C2×D28411yescharacteristic
438C2×Q8—712no—
439C2×Q8—1421no—
440C2×Q8—1421no—
441C2×Q8—1421no—
442C2×Q8—1421no—
443C2×Q8—1421no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4C2×C4×D7112yes—
463C22×C4C2×D28411yescharacteristic
464C22×C4C2×D28411yescharacteristic
465C22×C4C2×D28411yescharacteristic
466C22×C4C2×D28411yescharacteristic
467C22×C4C2×D28411yescharacteristic
468C22×C4#1176586512yes—
469C22×C4C23×D7111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8C2×C4×D7112yes—
481C2×C8C2×C4×D7112yes—
482C4⋊C4—2842no—
483C4⋊C4—2842no—
484C4⋊C4—712no—
485C4⋊C4—712no—
486C4⋊C4—712no—
487C4⋊C4—712no—
488C4⋊C4—712no—
489C4⋊C4—1421no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1424no—
499C22⋊C4—2842no—
500C22⋊C4—1424no—
501C22⋊C4—1424no—
502C22⋊C4—1424no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—2842no—
508C42—2842no—
509C42—2842no—
510C42—2842no—
511C42—2842no—
512C42—2842no—
513C42—1424no—
514C42—1424no—
515C42#1176586412yes—
516C42#1176586412yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23C4×D28412yes—
524C23C22×D28411yescharacteristic
525Q8—2841no—
526Q8—2841no—
527Q8—1422no—
528Q8—1422no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
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—2841no—
563C2×C4—2841no—
564C2×C4—1422no—
565C2×C4—1422no—
566C2×C4—1422no—
567C2×C4—1422no—
568C2×C4—1422no—
569C2×C4—1422no—
570C2×C4C4×D28412yes—
571C2×C4#1455611212yes—
572C2×C4#1455611212yes—
573C2×C4#1455616112yes—
574C2×C4C22×C4×D7111yescharacteristic
575C2×C4#41×D7111yescharacteristic
576C2×C4#41×D7111yescharacteristic
577C2×C4C2×#1176586411yescharacteristic
578C2×C4C2×#1176586411yescharacteristic
579C2×C4C2×#1176586511yescharacteristic
580C8—22no—
581C8—22no—
582C8—41no—
583C8—41no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×C4×D28411yescharacteristic
592C22C2×#1455611211yescharacteristic
593C22#1798651111yescharacteristic
594C22C2×#1455616111yescharacteristic
595C22#1798679911yescharacteristic
596C22#1798679911yescharacteristic
597C22#1798679911yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—41no—
603C4—2842no—
604C4—2842no—
605C4—2842no—
606C4—2842no—
607C4—2842no—
608C4—2842no—
609C4—5681no—
610C4—5681no—
611C4C4×#1176586411yescharacteristic
612C4C4×#1176586411yescharacteristic
613C4#1798652911yescharacteristic
614C4#1798652911yescharacteristic
615C2—42no—
616C2C4×#1455611211yescharacteristic
617C2#2259714211yescharacteristic
618C2C4×#1455616111yescharacteristic
619C2#2259827311yescharacteristic
620C2#2259828911yescharacteristic
621C2#2259845111yescharacteristic
622C2#2259850311yescharacteristic
subgroupG/H
centerC23C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1749C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#1749D71
hypocenterC71#21583
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
4. #22588982C2semidirect
5. #22588975C2semidirect
6. #22588933C2semidirect
7. #22588926C2semidirect
11. #22588237C2semidirect
12. #22588237C2semidirect
13. #22588171C2semidirect
14. #22588171C2semidirect
20. #17986341C4semidirect
92. #14556062C2×C4semidirect
94. #14556060C2×C4semidirect
97. #14556053C2×C4semidirect
98. #14556053C2×C4semidirect
327. C71#21583semidirect

Maximal subgroups

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

subgroupindex
C2×#179863412
C4×#145561072
C4×#145560872
#225889822
#225889752
#225889332
#225889262
#225886502
#225885412
#225884362
#225882372
#225881712
C71×#17492
#2158371