Finite Group Databasebeta

← #2621×D71 #32164662

Subgroups — #2621×D71 #32164662

Order 18176.

3046 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D71C211yesmaximal, characteristic, complemented
2C2×C22⋊Q8×D71C212yesmaximal
3C2×#117×D71C211yesmaximal, characteristic
4#22594111C212yesmaximal, complemented
5#22594110C212yesmaximal, complemented
6#22594109C211yesmaximal, characteristic, complemented
7#22594108C214yesmaximal, complemented
8#22594107C211yesmaximal, characteristic, complemented
9#22594106C211yesmaximal, characteristic, complemented
10#22594105C214yesmaximal, complemented
11#398×D71C214yesmaximal, complemented
12#397×D71C214yesmaximal
13#391×D71C212yesmaximal, complemented
14#382×D71C211yesmaximal, characteristic, complemented
15C71×#2621C211yesmaximal, characteristic, complemented
16C22×Q8×D71C2212yes—
17C23×C4×D71C2211yescharacteristic
18C2×#14556128C2211yescharacteristic, complemented
19C2×#14556127C2214yescomplemented
20C2×#14556126C2212yescomplemented
21C4⋊Q8×D71—28no—
22C2×#14556108C2212yes—
23C2×#14556107C2212yes—
24C2×#14556106C2214yes—
25C2×#14556105C2214yes—
26C2×#14556104C2212yes—
27C22⋊Q8×D71—216no—
28C2×#14556087C2211yescharacteristic
29C2×#14556086C2212yes—
30C2×#14556085C2211yescharacteristic
31C2×#14556085C2211yescharacteristic
32C2×#14556084C2212yes—
33#117×D71—28no—
34C2×C4⋊C4×D71C2212yes—
35C2×C4⋊C4×D71C2212yes—
36C2×C4⋊C4×D71C2214yes—
37C2×C4⋊C4×D71C2214yes—
38C2×C22⋊C4×D71C2212yes—
39C2×C22⋊C4×D71C2214yes—
40C2×C42×D71C2211yescharacteristic
41C2×C42×D71C2211yescharacteristic
42#17985933C2218yescomplemented
43#17985932C2218yescomplemented
44#17985931C2214yescomplemented
45#17985930C2214yescomplemented
46#17985929C2214yescomplemented
47#17985928C2218yes—
48#17985927C2218yes—
49#17985926C2214yes—
50#17985925C2214yes—
51#17985924C2214yes—
52#17985906C2212yescomplemented
53#17985905C2212yescomplemented
54#17985904C2212yescomplemented
55#17985903C2218yescomplemented
56#17985865C2211yescharacteristic, complemented
57#17985864C2212yescomplemented
58#17985863C2214yescomplemented
59#95×D71C2214yes—
60#95×D71C2218yes—
61C2×C4⋊Q8×C71C2211yescharacteristic, complemented
62C2×C22⋊Q8×C71C2212yes—
63C2×#117×C71C2211yescharacteristic
64#398×C71C2214yescomplemented
65#397×C71C2214yes—
66#391×C71C2212yescomplemented
67#382×C71C2211yescharacteristic, complemented
68C24×D71C2311yescharacteristic
69C22×Dic142C2314yes—
70C22×Dic142C2312yes—
71C2×Q8×D71—28no—
72C2×Q8×D71—28no—
73C2×Q8×D71—28no—
74C23×Dic71C2311yescharacteristic
75C22×C4×D71—24no—
76C22×C4×D71—24no—
77C22×C4×D71D418yes—
78C22×C4×D71C2314yes—
79C22×C4×D71C2312yes—
80C22×C4×D71C2314yes—
81C22×C4×D71C2311yescharacteristic
82C22×C4×D71C2311yescharacteristic
83C22×C4×D71C2311yescharacteristic
84C22×C4×D71C2312yes—
85C22×C4×D71C2311yescharacteristic
86#14556128—24no—
87#14556127—216no—
88#14556126—28no—
89#14556108—28no—
90#14556107—28no—
91#14556106—216no—
92#14556105—216no—
93#14556104—28no—
94#14556087—24no—
95#14556086—28no—
96#14556085—24no—
97#14556085—24no—
98#14556084—28no—
99C2×C4⋊Dic71C2312yes—
100C2×C4⋊Dic71C2314yes—
101C2×C4⋊Dic71C2314yes—
102C2×C4⋊Dic71C2312yes—
103C2×Dic71⋊C4C2314yes—
104C2×Dic71⋊C4C2318yes—
105C2×Dic71⋊C4C2314yes—
106C2×Dic71⋊C4C2312yes—
107C2×Dic71⋊C4C2314yes—
108C2×Dic71⋊C4C2312yes—
109C4⋊C4×D71—28no—
110C4⋊C4×D71—216no—
111C4⋊C4×D71—28no—
112C4⋊C4×D71—216no—
113C2×C22⋊Dic71C2314yes—
114C2×C22⋊Dic71C2312yes—
115C2×(C2×D71)⋊C4C2312yes—
116C2×(C2×D71)⋊C4C2314yes—
117C2×(C2×D71)⋊C4C2314yes—
118C2×(C2×D71)⋊C4C2312yes—
119C22⋊C4×D71—216no—
120C22⋊C4×D71—28no—
121C2×C4×Dic71C2312yes—
122C2×C4×Dic71C2311yescharacteristic
123C2×C4×Dic71C2312yes—
124C2×C4×Dic71C2311yescharacteristic
125C42×D71—24no—
126C42×D71D418yes—
127C71⋊#95C2318yes—
128C71⋊#95C2314yes—
129C71⋊#95C2318yes—
130C71⋊#95C2318yes—
131C71⋊#95C2318yes—
132C22×Q8×C71C2312yes—
133C23×C4×C71C2311yescharacteristic
134C4⋊Q8×C71—24no—
135C22⋊Q8×C71—28no—
136#117×C71—24no—
137C2×C4⋊C4×C71C2312yes—
138C2×C4⋊C4×C71C2312yes—
139C2×C4⋊C4×C71C2314yes—
140C2×C4⋊C4×C71C2314yes—
141C2×C22⋊C4×C71C2312yes—
142C2×C22⋊C4×C71C2314yes—
143C2×C42×C71C2311yescharacteristic
144C2×C42×C71C2311yescharacteristic
145#95×C71C2314yes—
146#95×C71C2318yes—
147C23×D71—22no—
148C23×D71—22no—
149C23×D71—28no—
150C23×D71C2×D414yes—
151C23×D71C2411yescharacteristic
152C23×D71C2411yescharacteristic
153C2×Dic142—24no—
154C2×Dic142—24no—
155C2×Dic142—24no—
156C2×Dic142—28no—
157C2×Dic142—28no—
158C2×Dic142—28no—
159Q8×D71—432no—
160C22×Dic71—22no—
161C22×Dic71—22no—
162C22×Dic71C2×D414yes—
163C22×Dic71C2414yes—
164C22×Dic71C2412yes—
165C22×Dic71C2414yes—
166C22×Dic71C2411yescharacteristic
167C22×Dic71C2412yes—
168C22×Dic71C2411yescharacteristic
169C2×C4×D71—22no—
170C2×C4×D71—22no—
171C2×C4×D71—28no—
172C2×C4×D71—22no—
173C2×C4×D71—22no—
174C2×C4×D71—22no—
175C2×C4×D71—22no—
176C2×C4×D71—232no—
177C2×C4×D71—28no—
178C2×C4×D71—28no—
179C2×C4×D71—24no—
180C2×C4×D71—28no—
181C2×C4×D71—28no—
182C2×C4×D71—28no—
183C2×C4×D71—216no—
184C2×C4×D71C2×D414yes—
185C2×C4×D71C2×D414yes—
186C2×C4×D71C2×D414yes—
187C2×C4×D71C2×D418yes—
188C2×C4×D71C2×D414yes—
189C2×C4×D71C2×D414yes—
190C2×C4×D71#4114yes—
191C4⋊Dic71—24no—
192C4⋊Dic71—28no—
193C4⋊Dic71—24no—
194C4⋊Dic71—28no—
195Dic71⋊C4—24no—
196Dic71⋊C4—28no—
197Dic71⋊C4—24no—
198Dic71⋊C4—28no—
199Dic71⋊C4—28no—
200Dic71⋊C4—216no—
201C22⋊Dic71—28no—
202C22⋊Dic71—24no—
203(C2×D71)⋊C4—28no—
204(C2×D71)⋊C4—28no—
205(C2×D71)⋊C4—24no—
206(C2×D71)⋊C4—24no—
207C4×Dic71—22no—
208C4×Dic71—24no—
209C4×Dic71C2×D418yes—
210C4×Dic71C2×D414yes—
211C24×C71C2411yescharacteristic
212C2×Q8×C71—24no—
213C2×Q8×C71—24no—
214C2×Q8×C71—24no—
215C22×C4×C71—22no—
216C22×C4×C71—22no—
217C22×C4×C71C2×D414yes—
218C22×C4×C71C2414yes—
219C22×C4×C71C2412yes—
220C22×C4×C71C2414yes—
221C22×C4×C71C2411yescharacteristic
222C22×C4×C71C2412yes—
223C22×C4×C71C2411yescharacteristic
224C4⋊C4×C71—24no—
225C4⋊C4×C71—28no—
226C4⋊C4×C71—24no—
227C4⋊C4×C71—28no—
228C22⋊C4×C71—28no—
229C22⋊C4×C71—24no—
230C42×C71—22no—
231C42×C71C2×D414yes—
232C23×C71—21no—
233C23×C71—21no—
234C23×C71—24no—
235C23×C71C22×D412yes—
236C23×C71C2511yescharacteristic
237C22×D71—21no—
238C22×D71—21no—
239C22×D71—24no—
240C22×D71—28no—
241C22×D71—216no—
242C22×D71—48no—
243C22×D71—48no—
244C22×D71C22×D412yes—
245C22×D71C22×D412yes—
246C22×D71C22×D412yes—
247C22×D71C2×#4112yes—
248C22×D712−1+418yes—
249Q8×C71—48no—
250C2×C4×C71—21no—
251C2×C4×C71—21no—
252C2×C4×C71—24no—
253C2×C4×C71—28no—
254C2×C4×C71—22no—
255C2×C4×C71—24no—
256C2×C4×C71—24no—
257C2×C4×C71—24no—
258C2×C4×C71—24no—
259C2×C4×C71—24no—
260C2×C4×C71—28no—
261C2×C4×C71C22×D414yes—
262C2×C4×C71C22×D412yes—
263C2×C4×C71C22×D412yes—
264C2×C4×C71C22×D412yes—
265C2×C4×C71C2×#4112yes—
266C2×Dic71—21no—
267C2×Dic71—21no—
268C2×Dic71—24no—
269C2×Dic71—28no—
270C2×Dic71—22no—
271C2×Dic71—24no—
272C2×Dic71—24no—
273C2×Dic71—24no—
274C2×Dic71—24no—
275C2×Dic71—24no—
276C2×Dic71—28no—
277C2×Dic71C22×D414yes—
278C2×Dic71C22×D412yes—
279C2×Dic71C22×D412yes—
280C2×Dic71C22×D412yes—
281C2×Dic71C2×#4112yes—
282C4×D71—28no—
283C4×D71—216no—
284C4×D71—28no—
285C4×D71—28no—
286C4×D71—416no—
287C4×D71—48no—
288C4×D71—416no—
289C4×D71#121116yes—
290Dic142—48no—
291Dic142—416no—
292C22×C71—22no—
293C22×C71—24no—
294C22×C71—42no—
295C22×C71—42no—
296C22×C71C23×D411yescharacteristic
297C22×C71C23×D411yescharacteristic
298C22×C71C22×#4111yescharacteristic
299C22×C71C2×2−1+414yes—
300C2×D71—22no—
301C2×D71—24no—
302C2×D71—42no—
303C2×D71—42no—
304C2×D71—416no—
305C2×D71C2×#12114yes—
306C2×D71#53618yes—
307C2×D71#53618yes—
308C2×D71#54118yes—
309C4×C71—22no—
310C4×C71—24no—
311C4×C71—44no—
312C4×C71—42no—
313C4×C71—44no—
314C4×C71C2×#12114yes—
315Dic71—22no—
316Dic71—24no—
317Dic71—44no—
318Dic71—42no—
319Dic71—44no—
320Dic71C2×#12114yes—
321C2×#2621—711nomaximal, Sylow
322C2×C71—42no—
323C2×C71C22×#12111yescharacteristic
324C2×C71C2×#53612yes—
325C2×C71C2×#53612yes—
326C2×C71C2×#54112yes—
327D71—42no—
328D71#262118yesdirect product factor
329C22×C4⋊Q8—711no—
330C22×C22⋊Q8—712no—
331C22×#117—711no—
332#2621—714no—
333#2621—714no—
334#2621—712no—
335#2621—711no—
336#2621—711no—
337#2621—712no—
338#2621—711no—
339#2621D7111yescharacteristic, direct product factor
340C2×#398—714no—
341C2×#397—714no—
342C2×#391—712no—
343C2×#382—711no—
344C71C2×#262111yescharacteristic, complemented, Sylow
345C23×Q8—712no—
346C24×C4—711no—
347C2×C4⋊Q8—712no—
348C2×C4⋊Q8—714no—
349C2×C4⋊Q8—711no—
350C2×C4⋊Q8—1428no—
351C2×C4⋊Q8C2×D7111yescharacteristic, complemented
352C2×C22⋊Q8—712no—
353C2×C22⋊Q8—712no—
354C2×C22⋊Q8—714no—
355C2×C22⋊Q8—712no—
356C2×C22⋊Q8—714no—
357C2×C22⋊Q8—14216no—
358C2×C22⋊Q8C2×D7112yes—
359C2×#117—711no—
360C2×#117—712no—
361C2×#117—712no—
362C2×#117—711no—
363C2×#117—711no—
364C2×#117—1428no—
365C2×#117C2×D7111yescharacteristic
366C22×C4⋊C4—714no—
367C22×C4⋊C4—712no—
368C22×C4⋊C4—714no—
369C22×C4⋊C4—712no—
370C22×C22⋊C4—712no—
371C22×C22⋊C4—714no—
372C22×C42—711no—
373C22×C42—711no—
374#398—714no—
375#398—714no—
376#398—718no—
377#398—714no—
378#398—718no—
379#398C2×D7114yescomplemented
380#397—714no—
381#397—714no—
382#397—718no—
383#397—714no—
384#397—718no—
385#397C2×D7114yes—
386#391—712no—
387#391—712no—
388#391—712no—
389#391—718no—
390#391C2×D7112yescomplemented
391#382—712no—
392#382—711no—
393#382—714no—
394#382C2×D7111yescharacteristic, complemented
395C2×#95—714no—
396C2×#95—718no—
397C25—711no—
398C22×Q8—714no—
399C22×Q8—712no—
400C22×Q8—1428no—
401C22×Q8—1428no—
402C22×Q8—1428no—
403C22×Q8C22×D7112yes—
404C23×C4—711no—
405C23×C4—711no—
406C23×C4—718no—
407C23×C4—712no—
408C23×C4—711no—
409C23×C4—711no—
410C23×C4—711no—
411C23×C4—714no—
412C23×C4—714no—
413C23×C4—712no—
414C23×C4—1424no—
415C23×C4—1424no—
416C23×C4C22×D7111yescharacteristic
417C4⋊Q8—24no—
418C4⋊Q8—14216no—
419C4⋊Q8—1428no—
420C4⋊Q8—1424no—
421C22⋊Q8—28no—
422C22⋊Q8—14216no—
423C22⋊Q8—14216no—
424C22⋊Q8—1428no—
425C22⋊Q8—1428no—
426C22⋊Q8—1428no—
427#117—24no—
428#117—1424no—
429#117—1428no—
430#117—1428no—
431#117—1424no—
432#117—1424no—
433C2×C4⋊C4—712no—
434C2×C4⋊C4—712no—
435C2×C4⋊C4—712no—
436C2×C4⋊C4—714no—
437C2×C4⋊C4—712no—
438C2×C4⋊C4—714no—
439C2×C4⋊C4—714no—
440C2×C4⋊C4—714no—
441C2×C4⋊C4—718no—
442C2×C4⋊C4—714no—
443C2×C4⋊C4—1428no—
444C2×C4⋊C4—1428no—
445C2×C4⋊C4—14216no—
446C2×C4⋊C4—14216no—
447C2×C4⋊C4C22×D7112yes—
448C2×C4⋊C4C22×D7112yes—
449C2×C4⋊C4C22×D7114yes—
450C2×C4⋊C4C22×D7114yes—
451C2×C22⋊C4—712no—
452C2×C22⋊C4—714no—
453C2×C22⋊C4—712no—
454C2×C22⋊C4—712no—
455C2×C22⋊C4—714no—
456C2×C22⋊C4—714no—
457C2×C22⋊C4—1428no—
458C2×C22⋊C4—14216no—
459C2×C22⋊C4C22×D7112yes—
460C2×C22⋊C4C22×D7114yes—
461C2×C42—712no—
462C2×C42—711no—
463C2×C42—718no—
464C2×C42—712no—
465C2×C42—711no—
466C2×C42—1424no—
467C2×C42C22×D7111yescharacteristic
468C2×C42C22×D7111yescharacteristic
469#95—718no—
470#95—714no—
471#95—718no—
472#95—718no—
473#95—718no—
474#95C22×D7114yes—
475#95C22×D7118yes—
476C24—711no—
477C24—714no—
478C24—711no—
479C24—1422no—
480C24—1422no—
481C24—1428no—
482C24C23×D7111yescharacteristic
483C2×Q8—24no—
484C2×Q8—24no—
485C2×Q8—24no—
486C2×Q8—28432no—
487C2×Q8—1428no—
488C2×Q8—1428no—
489C2×Q8—1428no—
490C2×Q8—1424no—
491C2×Q8—1424no—
492C2×Q8—1424no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—711no—
496C22×C4—714no—
497C22×C4—714no—
498C22×C4—714no—
499C22×C4—714no—
500C22×C4—712no—
501C22×C4—711no—
502C22×C4—718no—
503C22×C4—714no—
504C22×C4—714no—
505C22×C4—714no—
506C22×C4—712no—
507C22×C4—714no—
508C22×C4—714no—
509C22×C4—1424no—
510C22×C4—1428no—
511C22×C4—1428no—
512C22×C4—1428no—
513C22×C4—14216no—
514C22×C4—1428no—
515C22×C4—1428no—
516C22×C4—1422no—
517C22×C4—1422no—
518C22×C4—14232no—
519C22×C4—1422no—
520C22×C4—1428no—
521C22×C4—1422no—
522C22×C4—1422no—
523C22×C4—1422no—
524C22×C4—1422no—
525C22×C4—1422no—
526C22×C4D4×D7114yes—
527C22×C4C23×D7112yes—
528C22×C4C23×D7114yes—
529C22×C4C23×D7114yes—
530C22×C4C23×D7112yes—
531C22×C4C23×D7111yescharacteristic
532C22×C4C23×D7111yescharacteristic
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—1424no—
538C4⋊C4—1428no—
539C4⋊C4—1428no—
540C4⋊C4—14216no—
541C4⋊C4—1424no—
542C4⋊C4—1428no—
543C4⋊C4—1424no—
544C4⋊C4—1428no—
545C4⋊C4—1424no—
546C4⋊C4—1428no—
547C22⋊C4—28no—
548C22⋊C4—24no—
549C22⋊C4—1424no—
550C22⋊C4—1428no—
551C22⋊C4—1428no—
552C22⋊C4—1424no—
553C22⋊C4—1428no—
554C22⋊C4—1424no—
555C42—22no—
556C42—718no—
557C42—714no—
558C42—1424no—
559C42—1422no—
560C42D4×D7114yes—
561C23—21no—
562C23—21no—
563C23—24no—
564C23—2848no—
565C23—2848no—
566C23—712no—
567C23—712no—
568C23—712no—
569C23—718no—
570C23—712no—
571C23—1428no—
572C23—14216no—
573C23—1421no—
574C23—1421no—
575C23—1424no—
576C23C2×D4×D7112yes—
577C23C24×D7111yescharacteristic
578Q8—48no—
579Q8—28416no—
580Q8—2848no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28416no—
593C2×C4—28416no—
594C2×C4—2848no—
595C2×C4—712no—
596C2×C4—712no—
597C2×C4—712no—
598C2×C4—7116no—
599C2×C4—714no—
600C2×C4—712no—
601C2×C4—1428no—
602C2×C4—1428no—
603C2×C4—14216no—
604C2×C4—1422no—
605C2×C4—1424no—
606C2×C4—1424no—
607C2×C4—1424no—
608C2×C4—1424no—
609C2×C4—1424no—
610C2×C4—1428no—
611C2×C4—1428no—
612C2×C4—1421no—
613C2×C4—1421no—
614C2×C4—1424no—
615C2×C4—1428no—
616C2×C4C2×D4×D7114yes—
617C2×C4C2×D4×D7112yes—
618C2×C4C2×D4×D7112yes—
619C2×C4C2×D4×D7112yes—
620C2×C4#41×D7112yes—
621C22—22no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—28416no—
626C22—2842no—
627C22—2842no—
628C22—714no—
629C22—718no—
630C22—718no—
631C22—718no—
632C22—1422no—
633C22—1424no—
634C22C22×D4×D7111yescharacteristic
635C22C22×D4×D7111yescharacteristic
636C22C2×#41×D7111yescharacteristic
637C222−1+4×D7114yes—
638C4—22no—
639C4—24no—
640C4—44no—
641C4—42no—
642C4—44no—
643C4—2844no—
644C4—2842no—
645C4—2844no—
646C4—714no—
647C4—1422no—
648C4—1424no—
649C4#121×D7114yes—
650C2—42no—
651C2—2842no—
652C2—718no—
653C2C2×#121×D7111yescharacteristic
654C2#536×D7112yes—
655C2#536×D7112yes—
656C2#541×D7112yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2621C2
prime radicalC24×D71C23
hypercenter#2621D71
hypocenterC71C2×#2621
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2621

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊Q8×D71C2semidirect
4. #22594111C2semidirect
5. #22594110C2semidirect
6. #22594109C2semidirect
7. #22594108C2semidirect
8. #22594107C2semidirect
9. #22594106C2semidirect
10. #22594105C2semidirect
11. #398×D71C2semidirect
13. #391×D71C2semidirect
14. #382×D71C2semidirect
15. C71×#2621C2semidirect
18. C2×#14556128C22semidirect
19. C2×#14556127C22semidirect
20. C2×#14556126C22semidirect
42. #17985933C22semidirect
43. #17985932C22semidirect
44. #17985931C22semidirect
45. #17985930C22semidirect
46. #17985929C22semidirect
52. #17985906C22semidirect
53. #17985905C22semidirect
54. #17985904C22semidirect
55. #17985903C22semidirect
56. #17985865C22semidirect
57. #17985864C22semidirect
58. #17985863C22semidirect
61. C2×C4⋊Q8×C71C22semidirect
64. #398×C71C22semidirect
66. #391×C71C22semidirect
67. #382×C71C22semidirect
328. D71#2621direct (N × H)
339. #2621D71direct (N × H)
344. C71C2×#2621semidirect
351. C2×C4⋊Q8C2×D71semidirect
379. #398C2×D71semidirect
390. #391C2×D71semidirect
394. #382C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×D712
C2×C22⋊Q8×D712
C2×#117×D712
#225941112
#225941102
#225941092
#225941082
#225941072
#225941062
#225941052
#398×D712
#397×D712
#391×D712
#382×D712
C71×#26212
C2×#262171