Finite Group Databasebeta

← #31997040

Subgroups — #31997040

Order 18176.

1438 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556134C211yesmaximal, characteristic
2C2×#17986034C211yesmaximal, characteristic, complemented
3C2×#106×D71C211yesmaximal, characteristic, complemented
4#22590929C212yesmaximal
5#22588372C212yesmaximal, complemented
6#22588371C212yesmaximal, complemented
7#22588365C211yesmaximal, characteristic, complemented
8#22588362C211yesmaximal, characteristic, complemented
9#22588362C211yesmaximal, characteristic, complemented
10#22588058C212yesmaximal, complemented
11C71×#1807C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C2×#14556134—24no—
14C2×#14556134—24no—
15C2×#14556134C418yescomplemented
16C22×#11765849C2211yescharacteristic
17C22×#11765848C2211yescharacteristic
18C22×#11765848C2211yescharacteristic
19C22×C8×D71C2211yescharacteristic
20C22×C8×D71C2211yescharacteristic
21C2×#14556086C2212yes—
22C2×C4⋊C4×D71C2212yes—
23#17986034—24no—
24C2×#14556061C2212yes—
25C2×#14556060C2211yescharacteristic
26C2×#14556060C2211yescharacteristic
27C2×#14556059C2212yes—
28#106×D71—24no—
29#17985903—22no—
30#17985899—22no—
31#17985877—22no—
32#17985875—22no—
33#17985859C2212yes—
34#17985856C2212yes—
35#17985856C2212yes—
36#17985762C2212yes—
37#17985761C2212yes—
38#17985761C2212yes—
39C23×C8×C71C2211yescharacteristic
40C2×#106×C71C2211yescharacteristic
41C2×#106×C71C2211yescharacteristic
42#380×C71C2212yes—
43#340×C71C2212yes—
44C2×#11765865—22no—
45C2×#11765865—22no—
46C2×#11765865—22no—
47C2×#11765865—22no—
48C2×#11765865C2×C414yes—
49C2×#11765865Q814yes—
50C22×Dic142D411yescharacteristic
51C22×D284D411yescharacteristic
52C22×C22⋊D71D411yescharacteristic
53C22×C22⋊D71D411yescharacteristic
54C22×C4×D71—22no—
55C22×C4×D71C2311yescharacteristic
56C22×C4×D71C2311yescharacteristic
57#14556134—232no—
58C2×#11765849—22no—
59C2×#11765849—22no—
60C2×#11765849C2×C414yescomplemented
61C2×#11765848—22no—
62C2×#11765848—22no—
63C2×#11765848—22no—
64C2×#11765848—22no—
65C2×#11765848C2×C414yes—
66C2×#11765848C2×C414yescomplemented
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C2×C8×D71—22no—
70C2×C8×D71—22no—
71C2×C8×D71C2×C414yes—
72C2×C8×D71C2×C414yescomplemented
73C2×C8×D71D414yes—
74C2×C8×D71D414yes—
75#14556086—24no—
76#14556086—24no—
77C2×C4⋊Dic71C2312yes—
78C2×C4⋊Dic71C2312yes—
79C2×Dic71⋊C4—22no—
80C4⋊C4×D71—24no—
81C4⋊C4×D71—24no—
82C2×C22⋊Dic71—22no—
83C2×(C2×D71)⋊C4—22no—
84C2×(C2×D71)⋊C4—22no—
85C2×(C2×D71)⋊C4—22no—
86C2×C4×Dic71—22no—
87#14556061—24no—
88#14556060—22no—
89#14556060—22no—
90#14556059—24no—
91C71⋊#95—22no—
92C71⋊#95—22no—
93C71⋊#95—22no—
94C23×C4×C71C2311yescharacteristic
95C22×C8×C71—22no—
96C22×C8×C71—22no—
97C22×C8×C71C2×C414yescomplemented
98C22×C8×C71C2311yescharacteristic
99C22×C8×C71C2311yescharacteristic
100C2×C4⋊C4×C71C2312yes—
101C2×C4⋊C4×C71C2312yes—
102C2×C22⋊C4×C71—22no—
103#106×C71—22no—
104#106×C71—22no—
105#95×C71—22no—
106C23×D71C2×D411yescharacteristic
107C23×D71C2×D411yescharacteristic
108#11765865—28no—
109#11765865—28no—
110#11765865—28no—
111#11765865—28no—
112C2×Dic142—22no—
113C2×Dic142—22no—
114C2×Dic142C4⋊C414yes—
115C2×D284—22no—
116C2×D284—22no—
117C2×D284C4⋊C414yes—
118C2×C22⋊D71—22no—
119C2×C22⋊D71—22no—
120C2×C22⋊D71—22no—
121C2×C22⋊D71—22no—
122C2×C22⋊D71C4⋊C414yes—
123C2×C22⋊D71C4⋊C414yes—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71C2×D411yescharacteristic
127C22×Dic71C2×D411yescharacteristic
128C2×C4×D71—21no—
129C2×C4×D71—21no—
130C2×C4×D71—21no—
131C2×C4×D71—21no—
132C2×C4×D71—44no—
133C2×C4×D71—82no—
134C2×C4×D71—82no—
135C2×C4×D71C22×C412yes—
136C2×C4×D71C22×C412yes—
137C2×C4×D71C2×D412yes—
138C2×C4×D71C2×D412yes—
139C2×C4×D71C2×Q812yes—
140C2×C4×D71C2×Q812yes—
141C2×C4×D71#4112yes—
142C2×C4×D71#4112yes—
143#11765849—28no—
144#11765848—28no—
145#11765848—28no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8C22×C412yes—
149C2×C71⋊C8C22×C412yes—
150C2×C71⋊C8C2×D412yes—
151C2×C71⋊C8C2×D412yes—
152C8×D71—28no—
153C8×D71C22⋊C4116yescomplemented
154C4⋊Dic71—22no—
155C4⋊Dic71—22no—
156C4⋊Dic71—22no—
157C4⋊Dic71—22no—
158Dic71⋊C4—44no—
159C22⋊Dic71—82no—
160(C2×D71)⋊C4—44no—
161(C2×D71)⋊C4—82no—
162(C2×D71)⋊C4—82no—
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
175C2×C8×C71—21no—
176C2×C8×C71—21no—
177C2×C8×C71—28no—
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—82no—
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×D71C2×C4⋊C412yes—
200C22×D71C2×C4⋊C412yes—
201C22×D71#12112yes—
202C22×D71#12112yes—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—22no—
206C2×C4×C71—22no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71—42no—
212C2×C4×C71—42no—
213C2×C4×C71—42no—
214C2×C4×C71—42no—
215C2×C4×C71C2×C4⋊C412yes—
216C2×C4×C71C23×C411yescharacteristic
217C2×C4×C71C22×D411yescharacteristic
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C22×Q811yescharacteristic
220C2×C4×C71C2×#4111yescharacteristic
221C2×C4×C71C2×#4111yescharacteristic
222C22⋊D71—28no—
223C22⋊D71—28no—
224C2×Dic71—21no—
225C2×Dic71—21no—
226C2×Dic71—42no—
227C2×Dic71—42no—
228C2×Dic71—42no—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71C2×C4⋊C412yes—
233C2×Dic71C2×C4⋊C412yes—
234C2×Dic71#12112yes—
235C2×Dic71#12112yes—
236D284—28no—
237C4×D71—22no—
238C4×D71—22no—
239C4×D71—22no—
240C4×D71—22no—
241C4×D71—88no—
242C4×D71C2×C22⋊C414yes—
243C4×D71#11714yes—
244C4×D71C22⋊Q814yes—
245C4×D71C22⋊Q814yes—
246Dic142—28no—
247C8×C71—22no—
248C8×C71C2×C22⋊C414yes—
249C71⋊C8—22no—
250C71⋊C8C2×C22⋊C414yes—
251C22×C71—22no—
252C22×C71—22no—
253C22×C71—22no—
254C22×C71—42no—
255C22×C71—42no—
256C22×C71C22×C4⋊C411yescharacteristic
257C22×C71C2×#12111yescharacteristic
258C22×C71C2×#12111yescharacteristic
259C22×C71#57712yes—
260C22×C71#57712yes—
261C2×D71—22no—
262C2×D71—22no—
263C2×D71—24no—
264C2×D71—24no—
265C2×D71—22no—
266C2×D71—22no—
267C2×D71—22no—
268C2×D71—42no—
269C2×D71—42no—
270C2×D71#38014yes—
271C4×C71—22no—
272C4×C71—82no—
273C4×C71—82no—
274C4×C71C22×C22⋊C411yescharacteristic
275C4×C71C2×#11711yescharacteristic
276C4×C71C2×C22⋊Q811yescharacteristic
277C4×C71C2×C22⋊Q811yescharacteristic
278Dic71—22no—
279Dic71—82no—
280Dic71—82no—
281Dic71#38014yes—
282#20662—711nomaximal, Sylow
283C2×C71—42no—
284C2×C71C2×#38011yescharacteristic
285C2×C71#291211yescharacteristic
286C2×C71#291211yescharacteristic
287C2×C71#305712yes—
288C2×C71#306012yes—
289D71—24no—
290D71—42no—
291C22×#131—711no—
292C2×#427—711no—
293C22×#106—711no—
294#2311—712no—
295#1809—712no—
296#1809—712no—
297#1807—711no—
298#1807—711no—
299#1807—711no—
300#1807D7111yescharacteristic, complemented
301#1736—712no—
302C71#2066211yescharacteristic, complemented, Sylow
303C22×#41—711no—
304C2×#131—718no—
305C2×#131—1424no—
306C2×#131—1424no—
307C22×M16—711no—
308C22×M16—711no—
309C22×M16—711no—
310C23×C8—711no—
311C23×C8—711no—
312C23×C8C2×D7111yescharacteristic
313C2×#117—712no—
314C22×C4⋊C4—712no—
315#427—1424no—
316C2×#107—712no—
317C2×#106—711no—
318C2×#106—711no—
319C2×#106—712no—
320C2×#106—1424no—
321C2×#106C2×D7111yescharacteristic
322C2×#106C2×D7111yescharacteristic
323#391—1422no—
324#390—1422no—
325#386—1422no—
326#386—1422no—
327#380—712no—
328#380—712no—
329#380—712no—
330#380C2×D7112yes—
331#340—712no—
332#340—712no—
333#340—712no—
334#340C2×D7112yes—
335C2×#41—714no—
336C2×#41—714no—
337C2×#41—1422no—
338C2×#41—1422no—
339C2×#41—1422no—
340C2×#41—1422no—
341C22×Q8—711no—
342C22×D4—711no—
343C22×D4—711no—
344C22×D4—711no—
345C23×C4—711no—
346C23×C4—711no—
347C23×C4—1422no—
348C23×C4C22×D7111yescharacteristic
349#131—14232no—
350C2×M16—714no—
351C2×M16—714no—
352C2×M16—714no—
353C2×M16—1422no—
354C2×M16—1422no—
355C2×M16—1422no—
356C2×M16—1422no—
357C2×M16—1422no—
358C2×M16—1422no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—711no—
362C22×C8—714no—
363C22×C8—714no—
364C22×C8—714no—
365C22×C8—711no—
366C22×C8—714no—
367C22×C8—1422no—
368C22×C8—1422no—
369C22×C8C4×D7114yescomplemented
370C22×C8C22×D7111yescharacteristic
371C22×C8C22×D7111yescharacteristic
372#117—1424no—
373#117—1424no—
374C2×C4⋊C4—712no—
375C2×C4⋊C4—712no—
376C2×C4⋊C4—1424no—
377C2×C4⋊C4—1424no—
378C2×C4⋊C4—1422no—
379C2×C4⋊C4C22×D7112yes—
380C2×C4⋊C4C22×D7112yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—1422no—
383C2×C22⋊C4—1422no—
384C2×C22⋊C4—1422no—
385C2×C22⋊C4—1422no—
386C2×C42—1422no—
387#107—1424no—
388#106—22no—
389#106—22no—
390#106—1424no—
391#106—1422no—
392#106—1422no—
393#95—22no—
394#95—1422no—
395#95—1422no—
396#95—1422no—
397C24—711no—
398C24—711no—
399C24D4×D7111yescharacteristic
400#41—1428no—
401#41—1428no—
402#41—1428no—
403#41—1428no—
404C2×Q8—714no—
405C2×Q8—1422no—
406C2×Q8—1422no—
407C2×D4—714no—
408C2×D4—714no—
409C2×D4—714no—
410C2×D4—1422no—
411C2×D4—1422no—
412C2×D4—1422no—
413C2×D4—1422no—
414C2×D4—1422no—
415C2×D4—1422no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—2844no—
423C22×C4—5682no—
424C22×C4—5682no—
425C22×C4—711no—
426C22×C4—712no—
427C22×C4—712no—
428C22×C4—712no—
429C22×C4—712no—
430C22×C4—712no—
431C22×C4—712no—
432C22×C4—711no—
433C22×C4—712no—
434C22×C4—712no—
435C22×C4—1422no—
436C22×C4—1422no—
437C22×C4—1421no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4C2×C4×D7112yes—
442C22×C4D4×D7111yescharacteristic
443C22×C4Q8×D7112yes—
444C22×C4C23×D7111yescharacteristic
445M16—1428no—
446M16—1428no—
447M16—1428no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—28no—
451C2×C8—712no—
452C2×C8—712no—
453C2×C8—712no—
454C2×C8—7116no—
455C2×C8—712no—
456C2×C8—1421no—
457C2×C8—1421no—
458C2×C8—1428no—
459C2×C8C2×C4×D7112yes—
460C2×C8C2×C4×D7112yes—
461C2×C8D4×D7112yes—
462C2×C8D4×D7112yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—2844no—
468C4⋊C4—1422no—
469C4⋊C4—1422no—
470C4⋊C4—1422no—
471C4⋊C4—1422no—
472C22⋊C4—82no—
473C22⋊C4—2844no—
474C22⋊C4—5682no—
475C22⋊C4—5682no—
476C22⋊C4—5682no—
477C42—2844no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—712no—
483C23—712no—
484C23—712no—
485C23—712no—
486C23—1421no—
487C23—1421no—
488C23—1422no—
489C23—1422no—
490C23—1422no—
491C23—1422no—
492C23C4⋊C4×D7112yes—
493C23C2×D4×D7111yescharacteristic
494Q8—1428no—
495D4—1428no—
496D4—1428no—
497D4—1428no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—2842no—
511C2×C4—2842no—
512C2×C4—2842no—
513C2×C4—2842no—
514C2×C4—2842no—
515C2×C4—2842no—
516C2×C4—5688no—
517C2×C4—712no—
518C2×C4—712no—
519C2×C4—712no—
520C2×C4—714no—
521C2×C4—714no—
522C2×C4—714no—
523C2×C4—714no—
524C2×C4—712no—
525C2×C4—1421no—
526C2×C4—1421no—
527C2×C4—1422no—
528C2×C4—1422no—
529C2×C4—1422no—
530C2×C4—1422no—
531C2×C4C4⋊C4×D7112yes—
532C2×C4C22×C4×D7111yescharacteristic
533C2×C4C2×D4×D7111yescharacteristic
534C2×C4C2×D4×D7111yescharacteristic
535C2×C4C2×Q8×D7111yescharacteristic
536C2×C4#41×D7111yescharacteristic
537C2×C4#41×D7111yescharacteristic
538C8—22no—
539C8—714no—
540C8—1422no—
541C8C22⋊C4×D7114yes—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—2842no—
548C22—2842no—
549C22—714no—
550C22—1422no—
551C22—1422no—
552C22—1422no—
553C22—1422no—
554C22—1422no—
555C22—1424no—
556C22—1424no—
557C22C2×C4⋊C4×D7111yescharacteristic
558C22#121×D7111yescharacteristic
559C22#121×D7111yescharacteristic
560C22#1798680512yes—
561C22#1798680612yes—
562C4—22no—
563C4—82no—
564C4—82no—
565C4—5682no—
566C4—5682no—
567C4—714no—
568C4—1422no—
569C4C2×C22⋊C4×D7111yescharacteristic
570C4#117×D7111yescharacteristic
571C4C22⋊Q8×D7111yescharacteristic
572C4C22⋊Q8×D7111yescharacteristic
573C2—42no—
574C2—2842no—
575C2—1424no—
576C2#380×D7111yescharacteristic
577C2#2259693111yescharacteristic
578C2#2259693211yescharacteristic
579C2#2259797412yes—
580C2#2259799712yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1807C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenter#1807D71
hypocenterC71#20662
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. C2×#17986034C2semidirect
3. C2×#106×D71C2semidirect
5. #22588372C2semidirect
6. #22588371C2semidirect
7. #22588365C2semidirect
8. #22588362C2semidirect
9. #22588362C2semidirect
10. #22588058C2semidirect
11. C71×#1807C2semidirect
15. C2×#14556134C4semidirect
60. C2×#11765849C2×C4semidirect
66. C2×#11765848C2×C4semidirect
72. C2×C8×D71C2×C4semidirect
97. C22×C8×C71C2×C4semidirect
153. C8×D71C22⋊C4semidirect
300. #1807D71semidirect
302. C71#20662semidirect
369. C22×C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×#145561342
C2×#179860342
C2×#106×D712
#225909292
#225883722
#225883712
#225883652
#225883622
#225880582
C71×#18072
#2066271