Finite Group Databasebeta

← #1809×D71 #31997021

Subgroups — #1809×D71 #31997021

Order 18176.

1662 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D71C211yesmaximal, characteristic
2C2×#107×D71C211yesmaximal, characteristic, complemented
3C2×#106×D71C211yesmaximal, characteristic, complemented
4#380×D71C212yesmaximal
5#22588374C211yesmaximal, characteristic, complemented
6#22588373C211yesmaximal, characteristic, complemented
7#22588372C212yesmaximal, complemented
8#22588371C212yesmaximal, complemented
9#22588370C211yesmaximal, characteristic, complemented
10#340×D71C212yesmaximal, 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×C4⋊C4×D71C2212yes—
22C2×C4⋊C4×D71C2212yes—
23C2×C22⋊C4×D71—22no—
24C2×#14556062C2211yescharacteristic, complemented
25C2×#14556061C2212yescomplemented
26#107×D71—24no—
27C2×#14556060C2211yescharacteristic, complemented
28C2×#14556059C2212yescomplemented
29#106×D71—24no—
30#17985859C2212yes—
31#17985858—22no—
32#17985857—22no—
33#17985856C2212yes—
34#17985856C2212yes—
35#95×D71—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×D71—22no—
51C22×C4×D71—22no—
52C22×C4×D71C2×C414yes—
53C22×C4×D71D411yescharacteristic
54C22×C4×D71D411yescharacteristic
55C22×C4×D71Q814yes—
56C22×C4×D71C2311yescharacteristic
57C22×C4×D71C2311yescharacteristic
58C2×#11765849—22no—
59C2×#11765849—22no—
60C2×#11765849C2×C414yescomplemented
61C2×#11765848—22no—
62C2×#11765848—22no—
63C2×#11765848—22no—
64C2×#11765848—22no—
65C2×#11765848C2×C414yescomplemented
66C2×#11765848C2×C414yescomplemented
67M16×D71—232no—
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C2×C8×D71—22no—
71C2×C8×D71—22no—
72C2×C8×D71C2×C414yes—
73C2×C8×D71C2×C414yes—
74C2×C8×D71D414yes—
75C2×C8×D71D414yes—
76C2×C4⋊Dic71C2312yes—
77C2×C4⋊Dic71C2312yes—
78C2×Dic71⋊C4—22no—
79C2×Dic71⋊C4—22no—
80C4⋊C4×D71—24no—
81C4⋊C4×D71—24no—
82C4⋊C4×D71—24no—
83C4⋊C4×D71—24no—
84C2×C22⋊Dic71—22no—
85C2×(C2×D71)⋊C4—22no—
86C2×(C2×D71)⋊C4—22no—
87C22⋊C4×D71—48no—
88#14556062—22no—
89#14556061—24no—
90#14556060—22no—
91#14556059—24no—
92C71⋊#95—22no—
93C71⋊#95—22no—
94C71⋊#95—22no—
95C23×C4×C71C2311yescharacteristic
96C2×M16×C71—22no—
97C2×M16×C71—22no—
98C2×M16×C71C2×C414yescomplemented
99C22×C8×C71C2311yescharacteristic
100C22×C8×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2312yes—
102C2×C4⋊C4×C71C2312yes—
103C2×C22⋊C4×C71—22no—
104#107×C71—22no—
105#106×C71—22no—
106#95×C71—22no—
107C23×D71—22no—
108C23×D71—22no—
109C23×D71—24no—
110C23×D71—24no—
111C23×D71C4⋊C414yes—
112C23×D71C2×D411yescharacteristic
113C23×D71C2×D411yescharacteristic
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71C4⋊C414yes—
119C22×Dic71C2×D411yescharacteristic
120C22×Dic71C2×D411yescharacteristic
121C2×C4×D71—22no—
122C2×C4×D71—22no—
123C2×C4×D71—21no—
124C2×C4×D71—21no—
125C2×C4×D71—21no—
126C2×C4×D71—21no—
127C2×C4×D71—22no—
128C2×C4×D71—22no—
129C2×C4×D71—28no—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71—28no—
133C2×C4×D71—44no—
134C2×C4×D71—44no—
135C2×C4×D71—44no—
136C2×C4×D71—44no—
137C2×C4×D71—44no—
138C2×C4×D71—44no—
139C2×C4×D71C4⋊C414yes—
140C2×C4×D71C4⋊C414yes—
141C2×C4×D71C22×C412yes—
142C2×C4×D71C22×C412yes—
143C2×C4×D71C2×D412yes—
144C2×C4×D71C2×D412yes—
145C2×C4×D71C2×Q812yes—
146C2×C4×D71C2×Q812yes—
147C2×C4×D71#4112yes—
148C2×C4×D71#4112yes—
149#11765849—28no—
150#11765848—28no—
151#11765848—28no—
152C2×C71⋊C8—21no—
153C2×C71⋊C8—21no—
154C2×C71⋊C8C22×C412yes—
155C2×C71⋊C8C22×C412yes—
156C2×C71⋊C8C2×D412yes—
157C2×C71⋊C8C2×D412yes—
158C8×D71—28no—
159C8×D71C22⋊C4116yescomplemented
160C4⋊Dic71—22no—
161C4⋊Dic71—22no—
162C4⋊Dic71—22no—
163C4⋊Dic71—22no—
164Dic71⋊C4—44no—
165Dic71⋊C4—44no—
166C22⋊Dic71—44no—
167(C2×D71)⋊C4—44no—
168(C2×D71)⋊C4—44no—
169C24×C71C2×D411yescharacteristic
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71C22×C412yes—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×Q812yes—
179C22×C4×C71C2411yescharacteristic
180M16×C71—28no—
181C2×C8×C71—21no—
182C2×C8×C71—21no—
183C2×C8×C71C22×C412yes—
184C2×C8×C71C22×C412yes—
185C2×C8×C71C2×D412yes—
186C2×C8×C71C2×D412yes—
187C4⋊C4×C71—22no—
188C4⋊C4×C71—22no—
189C4⋊C4×C71—22no—
190C4⋊C4×C71—22no—
191C22⋊C4×C71—44no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—22no—
195C23×C71—22no—
196C23×C71C2×C4⋊C412yes—
197C23×C71C22×D411yescharacteristic
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—28no—
203C22×D71—216no—
204C22×D71—48no—
205C22×D71—44no—
206C22×D71—44no—
207C22×D71C2×C4⋊C412yes—
208C22×D71C2×C4⋊C412yes—
209C22×D71#12112yes—
210C22×D71#12112yes—
211C22×D71AGL(1,ℤ/8)14yes—
212C22×D71#13714yes—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—42no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71C2×C4⋊C412yes—
226C2×C4×C71C23×C411yescharacteristic
227C2×C4×C71C22×D411yescharacteristic
228C2×C4×C71C22×D411yescharacteristic
229C2×C4×C71C22×Q811yescharacteristic
230C2×C4×C71C2×#4111yescharacteristic
231C2×C4×C71C2×#4111yescharacteristic
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—28no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
241C2×Dic71C2×C4⋊C412yes—
242C2×Dic71C2×C4⋊C412yes—
243C2×Dic71#12112yes—
244C2×Dic71#12112yes—
245C4×D71—28no—
246C4×D71—22no—
247C4×D71—22no—
248C4×D71—22no—
249C4×D71—22no—
250C4×D71—28no—
251C4×D71—88no—
252C4×D71—88no—
253C4×D71C2×C22⋊C414yes—
254C4×D71#11714yes—
255C4×D71C22⋊Q814yes—
256C4×D71C22⋊Q814yes—
257C8×C71—22no—
258C8×C71C2×C22⋊C414yescomplemented
259C71⋊C8—22no—
260C71⋊C8C2×C22⋊C414yescomplemented
261C22×C71—24no—
262C22×C71—22no—
263C22×C71—42no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71C22×C4⋊C411yescharacteristic
267C22×C71C2×#12111yescharacteristic
268C22×C71C2×#12111yescharacteristic
269C22×C71C2×AGL(1,ℤ/8)12yes—
270C22×C71C2×#13712yes—
271C2×D71—24no—
272C2×D71—22no—
273C2×D71—42no—
274C2×D71—41no—
275C2×D71—41no—
276C2×D71—416no—
277C2×D71#38014yes—
278C2×D71#42818yes—
279C2×D71#46818yes—
280C2×D71#46914yes—
281C2×D71#47014yes—
282C4×C71—22no—
283C4×C71—82no—
284C4×C71—82no—
285C4×C71C22×C22⋊C411yescharacteristic
286C4×C71C2×#11711yescharacteristic
287C4×C71C2×C22⋊Q811yescharacteristic
288C4×C71C2×C22⋊Q811yescharacteristic
289Dic71—22no—
290Dic71—82no—
291Dic71—82no—
292Dic71#38014yes—
293C2×#1809—711nomaximal, Sylow
294C2×C71—42no—
295C2×C71C2×#38011yescharacteristic
296C2×C71C2×#42812yes—
297C2×C71C2×#46812yes—
298C2×C71C2×#46911yescharacteristic
299C2×C71C2×#47011yescharacteristic
300D71—42no—
301D71#180918yesdirect product factor
302C23×M16—711no—
303C22×#107—711no—
304C22×#106—711no—
305C2×#380—712no—
306#1809—711no—
307#1809—711no—
308#1809—712no—
309#1809—712no—
310#1809—711no—
311#1809D7111yescharacteristic, direct product factor
312C2×#340—712no—
313C71C2×#180911yescharacteristic, complemented, Sylow
314C24×C4—711no—
315C22×M16—711no—
316C22×M16—718no—
317C22×M16—711no—
318C22×M16—711no—
319C22×M16—1424no—
320C22×M16—1424no—
321C22×M16C2×D7111yescharacteristic
322C23×C8—711no—
323C23×C8—711no—
324C22×C4⋊C4—712no—
325C22×C4⋊C4—712no—
326C22×C22⋊C4—1422no—
327C2×#107—711no—
328C2×#107—712no—
329C2×#107—1424no—
330C2×#107C2×D7111yescharacteristic, complemented
331C2×#106—711no—
332C2×#106—712no—
333C2×#106—1424no—
334C2×#106C2×D7111yescharacteristic, complemented
335#380—712no—
336#380—712no—
337#380—712no—
338#380—1422no—
339#380—1422no—
340#380C2×D7112yes—
341C2×#95—1422no—
342#340—712no—
343#340—712no—
344#340—712no—
345#340C2×D7112yescomplemented
346C25—711no—
347C23×C4—711no—
348C23×C4—711no—
349C23×C4—714no—
350C23×C4—711no—
351C23×C4—711no—
352C23×C4—714no—
353C23×C4—711no—
354C23×C4—1422no—
355C23×C4—1422no—
356C23×C4—1422no—
357C23×C4—1422no—
358C23×C4—1422no—
359C23×C4—1422no—
360C23×C4C22×D7111yescharacteristic
361C2×M16—22no—
362C2×M16—22no—
363C2×M16—714no—
364C2×M16—714no—
365C2×M16—714no—
366C2×M16—1422no—
367C2×M16—1422no—
368C2×M16—14232no—
369C2×M16—1422no—
370C2×M16—1422no—
371C2×M16—1422no—
372C2×M16—1422no—
373C2×M16C4×D7114yescomplemented
374C22×C8—711no—
375C22×C8—714no—
376C22×C8—714no—
377C22×C8—714no—
378C22×C8—711no—
379C22×C8—714no—
380C22×C8—1422no—
381C22×C8—1422no—
382C22×C8C22×D7111yescharacteristic
383C22×C8C22×D7111yescharacteristic
384C2×C4⋊C4—712no—
385C2×C4⋊C4—712no—
386C2×C4⋊C4—1424no—
387C2×C4⋊C4—1424no—
388C2×C4⋊C4—1424no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4C22×D7112yes—
393C2×C4⋊C4C22×D7112yes—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—2848no—
396C2×C22⋊C4—1422no—
397C2×C22⋊C4—1422no—
398C2×C22⋊C4—1422no—
399#107—22no—
400#107—1424no—
401#107—1422no—
402#106—22no—
403#106—1424no—
404#106—1422no—
405#95—22no—
406#95—1422no—
407#95—1422no—
408#95—1422no—
409C24—711no—
410C24—711no—
411C24—714no—
412C24—1424no—
413C24—1422no—
414C24—1422no—
415C24—1424no—
416C24D4×D7111yescharacteristic
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—2844no—
424C22×C4—2844no—
425C22×C4—2844no—
426C22×C4—2844no—
427C22×C4—2844no—
428C22×C4—2844no—
429C22×C4—711no—
430C22×C4—712no—
431C22×C4—712no—
432C22×C4—712no—
433C22×C4—712no—
434C22×C4—712no—
435C22×C4—712no—
436C22×C4—711no—
437C22×C4—714no—
438C22×C4—712no—
439C22×C4—712no—
440C22×C4—714no—
441C22×C4—714no—
442C22×C4—1428no—
443C22×C4—1422no—
444C22×C4—1422no—
445C22×C4—1428no—
446C22×C4—1428no—
447C22×C4—1428no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4C2×C4×D7112yes—
459C22×C4D4×D7111yescharacteristic
460C22×C4Q8×D7112yes—
461C22×C4C23×D7111yescharacteristic
462M16—28no—
463M16—1428no—
464M16—1428no—
465M16—1428no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—712no—
469C2×C8—712no—
470C2×C8—712no—
471C2×C8—7116no—
472C2×C8—712no—
473C2×C8—1428no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8C2×C4×D7112yes—
477C2×C8C2×C4×D7112yes—
478C2×C8D4×D7112yes—
479C2×C8D4×D7112yes—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—2844no—
485C4⋊C4—2844no—
486C4⋊C4—1422no—
487C4⋊C4—1422no—
488C4⋊C4—1422no—
489C4⋊C4—1422no—
490C22⋊C4—44no—
491C22⋊C4—2844no—
492C22⋊C4—2844no—
493C22⋊C4—2844no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—2848no—
499C23—2844no—
500C23—2844no—
501C23—712no—
502C23—712no—
503C23—712no—
504C23—714no—
505C23—714no—
506C23—712no—
507C23—1428no—
508C23—14216no—
509C23—1421no—
510C23—1421no—
511C23—1422no—
512C23—1422no—
513C23C4⋊C4×D7112yes—
514C23C2×D4×D7111yescharacteristic
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—2842no—
528C2×C4—2842no—
529C2×C4—2842no—
530C2×C4—2842no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—5688no—
534C2×C4—5688no—
535C2×C4—712no—
536C2×C4—712no—
537C2×C4—712no—
538C2×C4—714no—
539C2×C4—714no—
540C2×C4—714no—
541C2×C4—714no—
542C2×C4—712no—
543C2×C4—1428no—
544C2×C4—1428no—
545C2×C4—1428no—
546C2×C4—1421no—
547C2×C4—1421no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4C4⋊C4×D7112yes—
553C2×C4C22×C4×D7111yescharacteristic
554C2×C4C2×D4×D7111yescharacteristic
555C2×C4C2×D4×D7111yescharacteristic
556C2×C4C2×Q8×D7111yescharacteristic
557C2×C4#41×D7111yescharacteristic
558C2×C4#41×D7111yescharacteristic
559C8—22no—
560C8—714no—
561C8—1422no—
562C8C22⋊C4×D7114yescomplemented
563C22—24no—
564C22—22no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—28416no—
569C22—2842no—
570C22—2841no—
571C22—2841no—
572C22—718no—
573C22—714no—
574C22—714no—
575C22—718no—
576C22—714no—
577C22—1424no—
578C22—1422no—
579C22C2×C4⋊C4×D7111yescharacteristic
580C22#121×D7111yescharacteristic
581C22#121×D7111yescharacteristic
582C22AGL(1,ℤ/8)×D7112yes—
583C22#137×D7112yes—
584C4—22no—
585C4—82no—
586C4—82no—
587C4—5682no—
588C4—5682no—
589C4—714no—
590C4—1422no—
591C4C2×C22⋊C4×D7111yescharacteristic
592C4#117×D7111yescharacteristic
593C4C22⋊Q8×D7111yescharacteristic
594C4C22⋊Q8×D7111yescharacteristic
595C2—42no—
596C2—2842no—
597C2—718no—
598C2#380×D7111yescharacteristic
599C2#428×D7112yes—
600C2#468×D7112yes—
601C2#469×D7111yescharacteristic
602C2#470×D7111yescharacteristic
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
hypocenterC71C2×#1809
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×#107×D71C2semidirect
3. C2×#106×D71C2semidirect
5. #22588374C2semidirect
6. #22588373C2semidirect
7. #22588372C2semidirect
8. #22588371C2semidirect
9. #22588370C2semidirect
10. #340×D71C2semidirect
11. C71×#1809C2semidirect
18. C2×M16×D71C4semidirect
24. C2×#14556062C22semidirect
25. 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
60. C2×#11765849C2×C4semidirect
65. C2×#11765848C2×C4semidirect
66. C2×#11765848C2×C4semidirect
98. C2×M16×C71C2×C4semidirect
159. C8×D71C22⋊C4semidirect
258. C8×C71C2×C22⋊C4semidirect
260. C71⋊C8C2×C22⋊C4semidirect
301. D71#1809direct (N × H)
311. #1809D71direct (N × H)
313. C71C2×#1809semidirect
330. C2×#107C2×D71semidirect
334. C2×#106C2×D71semidirect
345. #340C2×D71semidirect
373. C2×M16C4×D71semidirect
562. C8C22⋊C4×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×#107×D712
C2×#106×D712
#380×D712
#225883742
#225883732
#225883722
#225883712
#225883702
#340×D712
C71×#18092
C2×#180971