Finite Group Databasebeta

← #31997027

Subgroups — #31997027

Order 18176.

1518 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D71C211yesmaximal, characteristic
2C2×#17986036C211yesmaximal, characteristic, complemented
3C2×#17986034C211yesmaximal, characteristic, complemented
4#22590722C212yesmaximal
5#22588374C211yesmaximal, characteristic, complemented
6#22588373C211yesmaximal, characteristic, complemented
7#22588372C212yesmaximal, complemented
8#22588371C212yesmaximal, complemented
9#22588370C211yesmaximal, characteristic, complemented
10#22588055C212yesmaximal, complemented
11C71×#1809C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×#11765849C2211yescharacteristic
14C22×#11765848C2211yescharacteristic
15C22×#11765848C2211yescharacteristic
16C2×M16×D71—24no—
17C2×M16×D71—24no—
18C2×M16×D71C418yescomplemented
19C22×C8×D71C2211yescharacteristic
20C22×C8×D71C2211yescharacteristic
21C2×#14556086C2212yes—
22C2×#14556086C2212yes—
23#17986036—24no—
24#17986034—24no—
25C2×#14556062C2211yescharacteristic, complemented
26C2×#14556061C2212yescomplemented
27C2×#14556060C2211yescharacteristic, complemented
28C2×#14556059C2212yescomplemented
29#17985861—22no—
30#17985859C2212yes—
31#17985856C2212yes—
32#17985856C2212yes—
33#17985854—22no—
34C22⋊C4×Dic71—22no—
35#17985846—22no—
36#17985762C2212yescomplemented
37#17985761C2212yescomplemented
38#17985761C2212yescomplemented
39C22×M16×C71C2211yescharacteristic
40C2×#107×C71C2211yescharacteristic, complemented
41C2×#106×C71C2211yescharacteristic, complemented
42#380×C71C2212yes—
43#340×C71C2212yescomplemented
44C24×D71D411yescharacteristic
45C23×Dic71D411yescharacteristic
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—22no—
50C22×C4×D71C2×C414yes—
51C22×C4×D71D411yescharacteristic
52C22×C4×D71D411yescharacteristic
53C22×C4×D71Q814yes—
54C22×C4×D71C2311yescharacteristic
55C22×C4×D71C2311yescharacteristic
56C2×#11765849—22no—
57C2×#11765849—22no—
58C2×#11765849C2×C414yes—
59C2×#11765848—22no—
60C2×#11765848—22no—
61C2×#11765848—22no—
62C2×#11765848—22no—
63C2×#11765848C2×C414yes—
64C2×#11765848C2×C414yes—
65M16×D71—232no—
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C2×C8×D71—22no—
69C2×C8×D71—22no—
70C2×C8×D71C2×C414yes—
71C2×C8×D71C2×C414yes—
72C2×C8×D71D414yes—
73C2×C8×D71D414yes—
74#14556086—24no—
75#14556086—24no—
76#14556086—24no—
77#14556086—24no—
78C2×C4⋊Dic71C2312yes—
79C2×C4⋊Dic71C2312yes—
80C2×C22⋊Dic71—22no—
81C2×(C2×D71)⋊C4—22no—
82C2×(C2×D71)⋊C4—22no—
83C2×(C2×D71)⋊C4—22no—
84C2×(C2×D71)⋊C4—22no—
85C2×C4×Dic71—22no—
86C2×C4×Dic71—22no—
87#14556062—22no—
88#14556061—24no—
89#14556060—22no—
90#14556059—24no—
91C71⋊#95—22no—
92C71⋊#95—22no—
93C71⋊#95—22no—
94C23×C4×C71C2311yescharacteristic
95C2×M16×C71—22no—
96C2×M16×C71—22no—
97C2×M16×C71C2×C414yes—
98C22×C8×C71C2311yescharacteristic
99C22×C8×C71C2311yescharacteristic
100C2×C4⋊C4×C71C2312yes—
101C2×C4⋊C4×C71C2312yes—
102C2×C22⋊C4×C71—22no—
103#107×C71—22no—
104#106×C71—22no—
105#95×C71—22no—
106C23×D71—22no—
107C23×D71—22no—
108C23×D71—42no—
109C23×D71—42no—
110C23×D71C4⋊C414yes—
111C23×D71C2×D411yescharacteristic
112C23×D71C2×D411yescharacteristic
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71C4⋊C414yes—
118C22×Dic71C2×D411yescharacteristic
119C22×Dic71C2×D411yescharacteristic
120C2×C4×D71—22no—
121C2×C4×D71—22no—
122C2×C4×D71—21no—
123C2×C4×D71—21no—
124C2×C4×D71—21no—
125C2×C4×D71—21no—
126C2×C4×D71—22no—
127C2×C4×D71—22no—
128C2×C4×D71—28no—
129C2×C4×D71—28no—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71C4⋊C414yes—
133C2×C4×D71C4⋊C414yes—
134C2×C4×D71C22×C412yes—
135C2×C4×D71C22×C412yes—
136C2×C4×D71C2×D412yes—
137C2×C4×D71C2×D412yes—
138C2×C4×D71C2×Q812yes—
139C2×C4×D71C2×Q812yes—
140C2×C4×D71#4112yes—
141C2×C4×D71#4112yes—
142#11765849—28no—
143#11765848—28no—
144#11765848—28no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8C22×C412yes—
148C2×C71⋊C8C22×C412yes—
149C2×C71⋊C8C2×D412yes—
150C2×C71⋊C8C2×D412yes—
151C8×D71—28no—
152C8×D71C22⋊C4116yescomplemented
153C4⋊Dic71—22no—
154C4⋊Dic71—22no—
155C4⋊Dic71—22no—
156C4⋊Dic71—22no—
157C22⋊Dic71—44no—
158(C2×D71)⋊C4—44no—
159(C2×D71)⋊C4—44no—
160(C2×D71)⋊C4—44no—
161(C2×D71)⋊C4—44no—
162C4×Dic71—44no—
163C4×Dic71—44no—
164C24×C71C2×D411yescharacteristic
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×Q812yes—
174C22×C4×C71C2411yescharacteristic
175M16×C71—28no—
176C2×C8×C71—21no—
177C2×C8×C71—21no—
178C2×C8×C71C22×C412yes—
179C2×C8×C71C22×C412yes—
180C2×C8×C71C2×D412yes—
181C2×C8×C71C2×D412yes—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—22no—
186C22⋊C4×C71—44no—
187C23×C71—21no—
188C23×C71—21no—
189C23×C71—22no—
190C23×C71—22no—
191C23×C71C2×C4⋊C412yes—
192C23×C71C22×D411yescharacteristic
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—21no—
196C22×D71—21no—
197C22×D71—22no—
198C22×D71—22no—
199C22×D71—28no—
200C22×D71—48no—
201C22×D71—48no—
202C22×D71—44no—
203C22×D71—44no—
204C22×D71C2×C4⋊C412yes—
205C22×D71C2×C4⋊C412yes—
206C22×D71#12112yes—
207C22×D71#12112yes—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71C2×C4⋊C412yes—
221C2×C4×C71C23×C411yescharacteristic
222C2×C4×C71C22×D411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×Q811yescharacteristic
225C2×C4×C71C2×#4111yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×Dic71—21no—
228C2×Dic71—21no—
229C2×Dic71—28no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71C2×C4⋊C412yes—
237C2×Dic71C2×C4⋊C412yes—
238C2×Dic71#12112yes—
239C2×Dic71#12112yes—
240C4×D71—28no—
241C4×D71—22no—
242C4×D71—22no—
243C4×D71—22no—
244C4×D71—22no—
245C4×D71—28no—
246C4×D71C2×C22⋊C414yes—
247C4×D71#11714yes—
248C4×D71C22⋊Q814yes—
249C4×D71C22⋊Q814yes—
250C8×C71—22no—
251C8×C71C2×C22⋊C414yes—
252C71⋊C8—22no—
253C71⋊C8C2×C22⋊C414yes—
254C22×C71—24no—
255C22×C71—22no—
256C22×C71—42no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71C22×C4⋊C411yescharacteristic
260C22×C71C2×#12111yescharacteristic
261C22×C71C2×#12111yescharacteristic
262C22×C71#57512yes—
263C22×C71#57512yes—
264C2×D71—24no—
265C2×D71—24no—
266C2×D71—22no—
267C2×D71—22no—
268C2×D71—24no—
269C2×D71—22no—
270C2×D71—416no—
271C2×D71—42no—
272C2×D71—41no—
273C2×D71—41no—
274C2×D71#38014yes—
275C4×C71—22no—
276C4×C71—82no—
277C4×C71—82no—
278C4×C71C22×C22⋊C411yescharacteristic
279C4×C71C2×#11711yescharacteristic
280C4×C71C2×C22⋊Q811yescharacteristic
281C4×C71C2×C22⋊Q811yescharacteristic
282Dic71—22no—
283Dic71—82no—
284Dic71—82no—
285Dic71#38014yes—
286#20660—711nomaximal, Sylow
287C2×C71—42no—
288C2×C71C2×#38011yescharacteristic
289C2×C71#291112yes—
290C2×C71#305412yes—
291C2×C71#305411yescharacteristic
292C2×C71#305411yescharacteristic
293D71—24no—
294D71—42no—
295C23×M16—711no—
296C2×#427—711no—
297C2×#427—711no—
298#2280—712no—
299#1809—711no—
300#1809—711no—
301#1809—712no—
302#1809—712no—
303#1809—711no—
304#1809D7111yescharacteristic, complemented
305#1735—712no—
306C71#2066011yescharacteristic, complemented, Sylow
307C24×C4—711no—
308C22×M16—711no—
309C22×M16—718no—
310C22×M16—711no—
311C22×M16—711no—
312C22×M16—1424no—
313C22×M16—1424no—
314C22×M16C2×D7111yescharacteristic
315C23×C8—711no—
316C23×C8—711no—
317C2×#117—712no—
318C2×#117—712no—
319#427—1424no—
320#427—1424no—
321C2×#107—711no—
322C2×#107—712no—
323C2×#107C2×D7111yescharacteristic, complemented
324C2×#106—711no—
325C2×#106—712no—
326C2×#106C2×D7111yescharacteristic, complemented
327#381—1422no—
328#380—712no—
329#380—712no—
330#380—712no—
331#380C2×D7112yes—
332#379—1422no—
333C4×C22⋊C4—1422no—
334C4×C22⋊C4—1422no—
335#340—712no—
336#340—712no—
337#340—712no—
338#340C2×D7112yescomplemented
339C25—711no—
340C23×C4—711no—
341C23×C4—711no—
342C23×C4—714no—
343C23×C4—711no—
344C23×C4—711no—
345C23×C4—714no—
346C23×C4—711no—
347C23×C4—1422no—
348C23×C4—1422no—
349C23×C4—1422no—
350C23×C4—1422no—
351C23×C4C22×D7111yescharacteristic
352C2×M16—22no—
353C2×M16—22no—
354C2×M16—714no—
355C2×M16—714no—
356C2×M16—714no—
357C2×M16—1422no—
358C2×M16—1422no—
359C2×M16—14232no—
360C2×M16—1422no—
361C2×M16—1422no—
362C2×M16—1422no—
363C2×M16—1422no—
364C2×M16C4×D7114yes—
365C22×C8—711no—
366C22×C8—714no—
367C22×C8—714no—
368C22×C8—714no—
369C22×C8—711no—
370C22×C8—714no—
371C22×C8—1422no—
372C22×C8—1422no—
373C22×C8C22×D7111yescharacteristic
374C22×C8C22×D7111yescharacteristic
375#117—1424no—
376#117—1424no—
377#117—1424no—
378#117—1424no—
379C2×C4⋊C4—712no—
380C2×C4⋊C4—712no—
381C2×C4⋊C4C22×D7112yes—
382C2×C4⋊C4C22×D7112yes—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—1422no—
385C2×C22⋊C4—1422no—
386C2×C22⋊C4—1422no—
387C2×C22⋊C4—1422no—
388C2×C22⋊C4—1422no—
389C2×C42—1422no—
390C2×C42—1422no—
391#107—22no—
392#107—1424no—
393#107—1422no—
394#106—22no—
395#106—1424no—
396#106—1422no—
397#95—22no—
398#95—1422no—
399#95—1422no—
400#95—1422no—
401C24—2842no—
402C24—2842no—
403C24—714no—
404C24—711no—
405C24—711no—
406C24—1422no—
407C24—1422no—
408C24D4×D7111yescharacteristic
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—711no—
416C22×C4—712no—
417C22×C4—712no—
418C22×C4—712no—
419C22×C4—712no—
420C22×C4—712no—
421C22×C4—712no—
422C22×C4—711no—
423C22×C4—714no—
424C22×C4—712no—
425C22×C4—712no—
426C22×C4—714no—
427C22×C4—714no—
428C22×C4—1428no—
429C22×C4—1422no—
430C22×C4—1422no—
431C22×C4—1428no—
432C22×C4—1428no—
433C22×C4—1428no—
434C22×C4—1422no—
435C22×C4—1422no—
436C22×C4—1422no—
437C22×C4—1422no—
438C22×C4—1422no—
439C22×C4—1422no—
440C22×C4—1421no—
441C22×C4—1421no—
442C22×C4—1421no—
443C22×C4—1421no—
444C22×C4C2×C4×D7112yes—
445C22×C4D4×D7111yescharacteristic
446C22×C4Q8×D7112yes—
447C22×C4C23×D7111yescharacteristic
448M16—28no—
449M16—1428no—
450M16—1428no—
451M16—1428no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—712no—
455C2×C8—712no—
456C2×C8—712no—
457C2×C8—7116no—
458C2×C8—712no—
459C2×C8—1428no—
460C2×C8—1421no—
461C2×C8—1421no—
462C2×C8C2×C4×D7112yes—
463C2×C8C2×C4×D7112yes—
464C2×C8D4×D7112yes—
465C2×C8D4×D7112yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—1422no—
471C4⋊C4—1422no—
472C4⋊C4—1422no—
473C4⋊C4—1422no—
474C22⋊C4—44no—
475C22⋊C4—2844no—
476C22⋊C4—2844no—
477C22⋊C4—2844no—
478C22⋊C4—2844no—
479C22⋊C4—2844no—
480C42—2844no—
481C42—2844no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—2848no—
487C23—2848no—
488C23—2844no—
489C23—2844no—
490C23—712no—
491C23—712no—
492C23—712no—
493C23—712no—
494C23—1428no—
495C23—1421no—
496C23—1421no—
497C23—1422no—
498C23—1422no—
499C23—1422no—
500C23—1422no—
501C23C4⋊C4×D7112yes—
502C23C2×D4×D7111yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—2842no—
516C2×C4—2842no—
517C2×C4—2842no—
518C2×C4—2842no—
519C2×C4—2842no—
520C2×C4—2842no—
521C2×C4—712no—
522C2×C4—712no—
523C2×C4—712no—
524C2×C4—714no—
525C2×C4—714no—
526C2×C4—714no—
527C2×C4—714no—
528C2×C4—712no—
529C2×C4—1428no—
530C2×C4—1428no—
531C2×C4—1428no—
532C2×C4—1421no—
533C2×C4—1421no—
534C2×C4—1422no—
535C2×C4—1422no—
536C2×C4—1422no—
537C2×C4—1422no—
538C2×C4C4⋊C4×D7112yes—
539C2×C4C22×C4×D7111yescharacteristic
540C2×C4C2×D4×D7111yescharacteristic
541C2×C4C2×D4×D7111yescharacteristic
542C2×C4C2×Q8×D7111yescharacteristic
543C2×C4#41×D7111yescharacteristic
544C2×C4#41×D7111yescharacteristic
545C8—22no—
546C8—714no—
547C8—1422no—
548C8C22⋊C4×D7114yes—
549C22—24no—
550C22—22no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—28416no—
555C22—2842no—
556C22—2841no—
557C22—2841no—
558C22—714no—
559C22—1424no—
560C22—1422no—
561C22—1424no—
562C22—1422no—
563C22—1424no—
564C22—1422no—
565C22C2×C4⋊C4×D7111yescharacteristic
566C22#121×D7111yescharacteristic
567C22#121×D7111yescharacteristic
568C22#1798679512yes—
569C22#1798679812yes—
570C4—22no—
571C4—82no—
572C4—82no—
573C4—5682no—
574C4—5682no—
575C4—714no—
576C4—1422no—
577C4C2×C22⋊C4×D7111yescharacteristic
578C4#117×D7111yescharacteristic
579C4C22⋊Q8×D7111yescharacteristic
580C4C22⋊Q8×D7111yescharacteristic
581C2—42no—
582C2—2842no—
583C2—1424no—
584C2#380×D7111yescharacteristic
585C2#2259692712yes—
586C2#2259794412yes—
587C2#2259794711yescharacteristic
588C2#2259795111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1809C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1809D71
hypocenterC71#20660
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×#17986036C2semidirect
3. C2×#17986034C2semidirect
5. #22588374C2semidirect
6. #22588373C2semidirect
7. #22588372C2semidirect
8. #22588371C2semidirect
9. #22588370C2semidirect
10. #22588055C2semidirect
11. C71×#1809C2semidirect
18. C2×M16×D71C4semidirect
25. C2×#14556062C22semidirect
26. C2×#14556061C22semidirect
27. C2×#14556060C22semidirect
28. C2×#14556059C22semidirect
36. #17985762C22semidirect
37. #17985761C22semidirect
38. #17985761C22semidirect
40. C2×#107×C71C22semidirect
41. C2×#106×C71C22semidirect
43. #340×C71C22semidirect
152. C8×D71C22⋊C4semidirect
304. #1809D71semidirect
306. C71#20660semidirect
323. C2×#107C2×D71semidirect
326. C2×#106C2×D71semidirect
338. #340C2×D71semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D712
C2×#179860362
C2×#179860342
#225907222
#225883742
#225883732
#225883722
#225883712
#225883702
#225880552
C71×#18092
#2066071