Finite Group Databasebeta

← #1892×D71 #31999111

Subgroups — #1892×D71 #31999111

Order 18176.

2094 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C8×D71C211yesmaximal, characteristic
2C2×C22⋊Q8×D71C211yesmaximal, characteristic
3C2×#103×D71C212yesmaximal, complemented
4#380×D71C211yesmaximal, characteristic
5#22588764C211yesmaximal, characteristic, complemented
6#22588763C211yesmaximal, characteristic, complemented
7#22588762C212yesmaximal, complemented
8#22588761C211yesmaximal, characteristic, complemented
9#22588760C212yesmaximal, complemented
10#340×D71C212yesmaximal, complemented
11C71×#1892C211yesmaximal, characteristic, complemented
12C22×Q8×D71C2211yescharacteristic
13C23×C4×D71C2211yescharacteristic
14C23×C71⋊C8C2211yescharacteristic
15C22×C8×D71—28no—
16C22×C8×D71—24no—
17C22×C8×D71C2212yes—
18C22×C8×D71C2212yes—
19C2×#14556108C2211yescharacteristic
20C2×#14556107C2211yescharacteristic
21C2×#14556106—21no—
22C2×#14556105—21no—
23C2×#14556104C2211yescharacteristic
24C22⋊Q8×D71—24no—
25C22⋊Q8×D71C418yescomplemented
26C2×C4⋊C4×D71—21no—
27C2×C4⋊C4×D71C2212yes—
28C2×C4⋊C4×D71C2211yescharacteristic
29C2×C22⋊C4×D71—21no—
30C2×C22⋊C4×D71—21no—
31C2×#14556053C2212yescomplemented
32C2×#14556052C2212yescomplemented
33C2×#14556051C2212yescomplemented
34#103×D71—28no—
35#17985859C2211yescharacteristic
36#17985858—21no—
37#17985857—21no—
38#17985856C2212yes—
39#95×D71—21no—
40#17985762C2212yescomplemented
41#17985761C2212yescomplemented
42#17985761C2212yescomplemented
43C23×C8×C71C2211yescharacteristic
44C2×C22⋊Q8×C71C2211yescharacteristic
45C2×#103×C71C2212yescomplemented
46#380×C71C2211yescharacteristic
47#340×C71C2212yescomplemented
48C24×D71D411yescharacteristic
49C22×Dic142—21no—
50C22×Dic142C2311yescharacteristic
51C2×Q8×D71—22no—
52C2×Q8×D71—24no—
53C2×Q8×D71—24no—
54C2×Q8×D71C2×C414yes—
55C23×Dic71D411yescharacteristic
56C22×C4×D71—24no—
57C22×C4×D71—24no—
58C22×C4×D71—21no—
59C22×C4×D71—21no—
60C22×C4×D71—22no—
61C22×C4×D71C2×C414yes—
62C22×C4×D71D411yescharacteristic
63C22×C4×D71D411yescharacteristic
64C22×C4×D71D414yes—
65C22×C4×D71C2311yescharacteristic
66C22×C4×D71C2311yescharacteristic
67C22×C71⋊C8—24no—
68C22×C71⋊C8—22no—
69C22×C71⋊C8C2312yes—
70C2×C8×D71—232no—
71C2×C8×D71—24no—
72C2×C8×D71—24no—
73C2×C8×D71—24no—
74C2×C8×D71—24no—
75C2×C8×D71—24no—
76C2×C8×D71D418yes—
77#14556108—22no—
78#14556108C2×C414yescomplemented
79#14556107—22no—
80#14556107C2×C414yescomplemented
81#14556106—28no—
82#14556105—28no—
83#14556104—22no—
84#14556104C2×C414yescomplemented
85C2×C4⋊Dic71—21no—
86C2×C4⋊Dic71C2312yes—
87C2×C4⋊Dic71C2311yescharacteristic
88C2×Dic71⋊C4—21no—
89C2×Dic71⋊C4—21no—
90C2×Dic71⋊C4—22no—
91C2×Dic71⋊C4—21no—
92C4⋊C4×D71—28no—
93C4⋊C4×D71—22no—
94C4⋊C4×D71—24no—
95C4⋊C4×D71—24no—
96C4⋊C4×D71C2×C414yes—
97C2×C22⋊Dic71—21no—
98C2×C22⋊Dic71—21no—
99C2×(C2×D71)⋊C4—21no—
100C2×(C2×D71)⋊C4—21no—
101C2×(C2×D71)⋊C4—22no—
102C22⋊C4×D71—28no—
103C22⋊C4×D71—44no—
104#14556053—24no—
105#14556052—24no—
106#14556051—24no—
107C71⋊#95—22no—
108C71⋊#95—21no—
109C22×Q8×C71C2311yescharacteristic
110C23×C4×C71C2311yescharacteristic
111C22×C8×C71—22no—
112C22×C8×C71—24no—
113C22×C8×C71C2312yes—
114C22⋊Q8×C71—22no—
115C22⋊Q8×C71C2×C414yescomplemented
116C2×C4⋊C4×C71—21no—
117C2×C4⋊C4×C71C2312yes—
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C22⋊C4×C71—21no—
120C2×C22⋊C4×C71—21no—
121#103×C71—24no—
122#95×C71—21no—
123C23×D71—24no—
124C23×D71—24no—
125C23×D71C22⋊C414yes—
126C23×D71SD1614yes—
127C23×D71Q1614yes—
128C23×D71C2×D411yescharacteristic
129C23×D71C2×D411yescharacteristic
130C2×Dic142—22no—
131C2×Dic142—22no—
132C2×Dic142—21no—
133C2×Dic142—24no—
134C2×Dic142—44no—
135C2×Dic142C22×C412yes—
136Q8×D71—216no—
137Q8×D71—44no—
138Q8×D71—44no—
139C22×Dic71—24no—
140C22×Dic71—21no—
141C22×Dic71—21no—
142C22×Dic71—22no—
143C22×Dic71C22⋊C414yes—
144C22×Dic71C2×D411yescharacteristic
145C22×Dic71C2×D411yescharacteristic
146C2×C4×D71—24no—
147C2×C4×D71—22no—
148C2×C4×D71—22no—
149C2×C4×D71—24no—
150C2×C4×D71—216no—
151C2×C4×D71—28no—
152C2×C4×D71—28no—
153C2×C4×D71—24no—
154C2×C4×D71—24no—
155C2×C4×D71—44no—
156C2×C4×D71—44no—
157C2×C4×D71—44no—
158C2×C4×D71—44no—
159C2×C4×D71—44no—
160C2×C4×D71C22⋊C414yes—
161C2×C4×D71C22⋊C414yes—
162C2×C4×D71C22×C412yes—
163C2×C4×D71C22×C412yes—
164C2×C4×D71C2×D412yes—
165C2×C4×D71C2×D414yes—
166C2×C4×D71C2×D412yes—
167C2×C4×D71#4114yes—
168C2×C71⋊C8—28no—
169C2×C71⋊C8—22no—
170C2×C71⋊C8—22no—
171C2×C71⋊C8C2×D414yes—
172C8×D71—216no—
173C8×D71—216no—
174C4⋊Dic71—21no—
175C4⋊Dic71—24no—
176C4⋊Dic71—22no—
177C4⋊Dic71—22no—
178C4⋊Dic71C22×C412yes—
179Dic71⋊C4—24no—
180Dic71⋊C4—24no—
181Dic71⋊C4—24no—
182Dic71⋊C4—44no—
183C22⋊Dic71—24no—
184C22⋊Dic71—42no—
185(C2×D71)⋊C4—24no—
186(C2×D71)⋊C4—24no—
187(C2×D71)⋊C4—44no—
188C24×C71C2×D411yescharacteristic
189C2×Q8×C71—21no—
190C2×Q8×C71—22no—
191C2×Q8×C71—22no—
192C2×Q8×C71C22×C412yes—
193C22×C4×C71—22no—
194C22×C4×C71—22no—
195C22×C4×C71—21no—
196C22×C4×C71—21no—
197C22×C4×C71—22no—
198C22×C4×C71C22×C412yes—
199C22×C4×C71C2×D411yescharacteristic
200C22×C4×C71C2×D412yes—
201C22×C4×C71C2411yescharacteristic
202C2×C8×C71—28no—
203C2×C8×C71—22no—
204C2×C8×C71—22no—
205C2×C8×C71C2×D414yes—
206C4⋊C4×C71—24no—
207C4⋊C4×C71—21no—
208C4⋊C4×C71—22no—
209C4⋊C4×C71—22no—
210C4⋊C4×C71C22×C412yes—
211C22⋊C4×C71—24no—
212C22⋊C4×C71—42no—
213C23×C71—22no—
214C23×C71—22no—
215C23×C71C2×C22⋊C412yes—
216C23×C71C2×SD1612yes—
217C23×C71C2×Q1612yes—
218C23×C71C22×D411yescharacteristic
219C22×D71—22no—
220C22×D71—22no—
221C22×D71—28no—
222C22×D71—28no—
223C22×D71—216no—
224C22×D71—216no—
225C22×D71#103116yescomplemented
226C22×D71C2×C22⋊C412yes—
227C22×D71C2×C22⋊C412yes—
228C22×D71#12114yes—
229C22×D71C2×SD1612yes—
230C22×D71C2×SD1612yes—
231C22×D71C2×Q1612yes—
232C22×D71C2×Q1612yes—
233C22×D71#13514yes—
234Q8×C71—24no—
235Q8×C71—41no—
236Q8×C71—41no—
237C2×C4×C71—22no—
238C2×C4×C71—24no—
239C2×C4×C71—22no—
240C2×C4×C71—22no—
241C2×C4×C71—22no—
242C2×C4×C71—22no—
243C2×C4×C71—42no—
244C2×C4×C71—42no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71C2×C22⋊C412yes—
249C2×C4×C71C23×C411yescharacteristic
250C2×C4×C71C22×D412yes—
251C2×C4×C71C22×D411yescharacteristic
252C2×C4×C71C2×#4112yes—
253C2×Dic71—22no—
254C2×Dic71—28no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—42no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71C2×C22⋊C412yes—
263C2×Dic71C2×C22⋊C412yes—
264C2×Dic71#12114yes—
265C4×D71—28no—
266C4×D71—24no—
267C4×D71—22no—
268C4×D71—22no—
269C4×D71—28no—
270C4×D71—48no—
271C4×D71—48no—
272C4×D71—88no—
273C4×D71C4×D418yes—
274C4×D71#12014yes—
275C4×D71#12314yes—
276Dic142—24no—
277Dic142—41no—
278Dic142—41no—
279Dic142—48no—
280C8×C71—24no—
281C71⋊C8—24no—
282C22×C71—22no—
283C22×C71—24no—
284C22×C71—24no—
285C22×C71—22no—
286C22×C71C2×#10314yescomplemented
287C22×C71C22×C22⋊C411yescharacteristic
288C22×C71C2×#12112yes—
289C22×C71C22×SD1611yescharacteristic
290C22×C71C22×Q1611yescharacteristic
291C22×C71C2×#13512yes—
292C2×D71—22no—
293C2×D71—24no—
294C2×D71—24no—
295C2×D71—22no—
296C2×D71—232no—
297C2×D71#38614yes—
298C2×D71C2×#10314yes—
299C2×D71C2×#10314yescomplemented
300C2×D71#41614yes—
301C2×D71#46518yes—
302C2×D71#46718yes—
303C4×C71—22no—
304C4×C71—42no—
305C4×C71—42no—
306C4×C71—82no—
307C4×C71C2×C4×D412yes—
308C4×C71C2×#12011yescharacteristic
309C4×C71C2×#12311yescharacteristic
310Dic71—22no—
311Dic71—42no—
312Dic71—42no—
313Dic71—82no—
314Dic71#38614yes—
315C2×#1892—711nomaximal, Sylow
316C2×C71—24no—
317C2×C71C2×#38611yescharacteristic
318C2×C71C22×#10311yescharacteristic
319C2×C71C2×#41611yescharacteristic
320C2×C71C2×#46512yes—
321C2×C71C2×#46712yes—
322D71—24no—
323D71#189218yesdirect product factor
324C24×C8—711no—
325C22×C22⋊Q8—711no—
326C22×#103—712no—
327C2×#380—711no—
328#1892—712no—
329#1892—711no—
330#1892—712no—
331#1892—711no—
332#1892—711no—
333#1892D7111yescharacteristic, direct product factor
334C2×#340—712no—
335C71C2×#189211yescharacteristic, complemented, Sylow
336C23×Q8—711no—
337C24×C4—711no—
338C23×C8—712no—
339C23×C8—711no—
340C23×C8—712no—
341C23×C8—1428no—
342C23×C8—1424no—
343C23×C8C2×D7111yescharacteristic
344C2×C22⋊Q8—711no—
345C2×C22⋊Q8—711no—
346C2×C22⋊Q8—718no—
347C2×C22⋊Q8—711no—
348C2×C22⋊Q8—1421no—
349C2×C22⋊Q8—1421no—
350C2×C22⋊Q8—1424no—
351C2×C22⋊Q8C2×D7111yescharacteristic
352C22×C4⋊C4—711no—
353C22×C4⋊C4—712no—
354C22×C4⋊C4—1421no—
355C22×C22⋊C4—1421no—
356C22×C22⋊C4—1421no—
357C2×#103—712no—
358C2×#103—712no—
359C2×#103—712no—
360C2×#103—1428no—
361C2×#103C2×D7112yescomplemented
362#380—711no—
363#380—712no—
364#380—1421no—
365#380—1421no—
366#380C2×D7111yescharacteristic
367C2×#95—1421no—
368#340—712no—
369#340—712no—
370#340—712no—
371#340C2×D7112yescomplemented
372C25—711no—
373C22×Q8—714no—
374C22×Q8—711no—
375C22×Q8—1421no—
376C22×Q8—1424no—
377C22×Q8—1424no—
378C22×Q8—1422no—
379C22×Q8C22×D7111yescharacteristic
380C23×C4—711no—
381C23×C4—711no—
382C23×C4—714no—
383C23×C4—711no—
384C23×C4—714no—
385C23×C4—711no—
386C23×C4—711no—
387C23×C4—1422no—
388C23×C4—1421no—
389C23×C4—1421no—
390C23×C4—1424no—
391C23×C4—1424no—
392C23×C4C22×D7111yescharacteristic
393C22×C8—22no—
394C22×C8—24no—
395C22×C8—712no—
396C22×C8—718no—
397C22×C8—1424no—
398C22×C8—1424no—
399C22×C8—1424no—
400C22×C8—1424no—
401C22×C8—14232no—
402C22×C8—1424no—
403C22×C8—1422no—
404C22×C8—1424no—
405C22×C8C22×D7112yes—
406C22⋊Q8—22no—
407C22⋊Q8—714no—
408C22⋊Q8—714no—
409C22⋊Q8—714no—
410C22⋊Q8—1428no—
411C22⋊Q8—1428no—
412C22⋊Q8—1422no—
413C22⋊Q8—1422no—
414C22⋊Q8—1422no—
415C22⋊Q8C4×D7114yescomplemented
416C2×C4⋊C4—21no—
417C2×C4⋊C4—711no—
418C2×C4⋊C4—714no—
419C2×C4⋊C4—712no—
420C2×C4⋊C4—1424no—
421C2×C4⋊C4—1424no—
422C2×C4⋊C4—1422no—
423C2×C4⋊C4—1428no—
424C2×C4⋊C4—1422no—
425C2×C4⋊C4—1421no—
426C2×C4⋊C4—1421no—
427C2×C4⋊C4—1421no—
428C2×C4⋊C4—1421no—
429C2×C4⋊C4C22×D7112yes—
430C2×C4⋊C4C22×D7111yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—2844no—
434C2×C22⋊C4—1428no—
435C2×C22⋊C4—1422no—
436C2×C22⋊C4—1421no—
437C2×C22⋊C4—1421no—
438C2×C22⋊C4—1421no—
439C2×C22⋊C4—1421no—
440#103—24no—
441#103—1424no—
442#103—1424no—
443#103—1424no—
444#95—21no—
445#95—1422no—
446#95—1421no—
447C24—711no—
448C24—711no—
449C24—714no—
450C24—714no—
451C24—714no—
452C24—1424no—
453C24—1424no—
454C24D4×D7111yescharacteristic
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—2844no—
459C2×Q8—2844no—
460C2×Q8—2844no—
461C2×Q8—712no—
462C2×Q8—1424no—
463C2×Q8—14216no—
464C2×Q8—1422no—
465C2×Q8—1422no—
466C2×Q8—1421no—
467C2×Q8C2×C4×D7112yes—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—2844no—
474C22×C4—2844no—
475C22×C4—2844no—
476C22×C4—2844no—
477C22×C4—2844no—
478C22×C4—711no—
479C22×C4—712no—
480C22×C4—714no—
481C22×C4—712no—
482C22×C4—714no—
483C22×C4—711no—
484C22×C4—714no—
485C22×C4—712no—
486C22×C4—712no—
487C22×C4—714no—
488C22×C4—714no—
489C22×C4—1424no—
490C22×C4—1424no—
491C22×C4—1428no—
492C22×C4—1424no—
493C22×C4—14216no—
494C22×C4—1428no—
495C22×C4—1424no—
496C22×C4—1422no—
497C22×C4—1421no—
498C22×C4—1421no—
499C22×C4—1424no—
500C22×C4—1422no—
501C22×C4—1422no—
502C22×C4C2×C4×D7112yes—
503C22×C4D4×D7112yes—
504C22×C4D4×D7111yescharacteristic
505C22×C4C23×D7111yescharacteristic
506C2×C8—28no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—714no—
510C2×C8—14216no—
511C2×C8—14216no—
512C2×C8—1428no—
513C2×C8—1422no—
514C2×C8—1422no—
515C2×C8D4×D7114yes—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—2844no—
521C4⋊C4—712no—
522C4⋊C4—1424no—
523C4⋊C4—1424no—
524C4⋊C4—1424no—
525C4⋊C4—1422no—
526C4⋊C4—1422no—
527C4⋊C4—1421no—
528C4⋊C4—1424no—
529C4⋊C4C2×C4×D7112yes—
530C22⋊C4—24no—
531C22⋊C4—42no—
532C22⋊C4—2844no—
533C22⋊C4—2842no—
534C22⋊C4—1424no—
535C22⋊C4—1424no—
536C22⋊C4—1424no—
537C23—22no—
538C23—22no—
539C23—712no—
540C23—714no—
541C23—712no—
542C23—712no—
543C23—714no—
544C23—712no—
545C23—7116no—
546C23—712no—
547C23—712no—
548C23—1428no—
549C23—14216no—
550C23—14216no—
551C23—1428no—
552C23—1422no—
553C23—1422no—
554C23C22⋊C4×D7112yes—
555C23SD16×D7112yes—
556C23Q16×D7112yes—
557C23C2×D4×D7111yescharacteristic
558Q8—24no—
559Q8—41no—
560Q8—41no—
561Q8—2848no—
562Q8—2841no—
563Q8—2841no—
564Q8—1424no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—2848no—
577C2×C4—2848no—
578C2×C4—2842no—
579C2×C4—2842no—
580C2×C4—2842no—
581C2×C4—2842no—
582C2×C4—2842no—
583C2×C4—5688no—
584C2×C4—712no—
585C2×C4—714no—
586C2×C4—718no—
587C2×C4—714no—
588C2×C4—714no—
589C2×C4—712no—
590C2×C4—1428no—
591C2×C4—1428no—
592C2×C4—1422no—
593C2×C4—1422no—
594C2×C4—1428no—
595C2×C4—1422no—
596C2×C4—1424no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4C22⋊C4×D7112yes—
600C2×C4C22×C4×D7111yescharacteristic
601C2×C4C2×D4×D7112yes—
602C2×C4C2×D4×D7111yescharacteristic
603C2×C4#41×D7112yes—
604C8—24no—
605C8—1424no—
606C22—24no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—718no—
611C22—718no—
612C22—714no—
613C22—714no—
614C22—714no—
615C22—714no—
616C22—14232no—
617C22—1424no—
618C22—1424no—
619C22—1422no—
620C22—1422no—
621C22#103×D7114yescomplemented
622C22C2×C22⋊C4×D7111yescharacteristic
623C22#121×D7112yes—
624C22C2×SD16×D7111yescharacteristic
625C22C2×Q16×D7111yescharacteristic
626C22#135×D7112yes—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—2842no—
632C4—2842no—
633C4—5682no—
634C4—714no—
635C4—1422no—
636C4C4×D4×D7112yes—
637C4#120×D7111yescharacteristic
638C4#123×D7111yescharacteristic
639C2—24no—
640C2—718no—
641C2—1424no—
642C2#386×D7111yescharacteristic
643C2C2×#103×D7111yescharacteristic
644C2#416×D7111yescharacteristic
645C2#465×D7112yes—
646C2#467×D7112yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1892C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1892D71
hypocenterC71C2×#1892
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. C2×#103×D71C2semidirect
5. #22588764C2semidirect
6. #22588763C2semidirect
7. #22588762C2semidirect
8. #22588761C2semidirect
9. #22588760C2semidirect
10. #340×D71C2semidirect
11. C71×#1892C2semidirect
25. C22⋊Q8×D71C4semidirect
31. C2×#14556053C22semidirect
32. C2×#14556052C22semidirect
33. C2×#14556051C22semidirect
40. #17985762C22semidirect
41. #17985761C22semidirect
42. #17985761C22semidirect
45. C2×#103×C71C22semidirect
47. #340×C71C22semidirect
78. #14556108C2×C4semidirect
80. #14556107C2×C4semidirect
84. #14556104C2×C4semidirect
115. C22⋊Q8×C71C2×C4semidirect
225. C22×D71#103semidirect
286. C22×C71C2×#103semidirect
299. C2×D71C2×#103semidirect
323. D71#1892direct (N × H)
333. #1892D71direct (N × H)
335. C71C2×#1892semidirect
361. C2×#103C2×D71semidirect
371. #340C2×D71semidirect
415. C22⋊Q8C4×D71semidirect
621. C22#103×D71semidirect

Maximal subgroups

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

subgroupindex
C23×C8×D712
C2×C22⋊Q8×D712
C2×#103×D712
#380×D712
#225887642
#225887632
#225887622
#225887612
#225887602
#340×D712
C71×#18922
C2×#189271