Finite Group Databasebeta

← #2058×D71 #32025475

Subgroups — #2058×D71 #32025475

Order 18176.

1454 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊Q8×D71C211yesmaximal, characteristic
2C2×#106×D71C212yesmaximal
3#389×D71C211yesmaximal, characteristic
4#22589674C211yesmaximal, characteristic, complemented
5#22589673C211yesmaximal, characteristic, complemented
6#22589672C212yesmaximal, complemented
7#22589671C212yesmaximal, complemented
8#22589670C211yesmaximal, characteristic, complemented
9#340×D71C212yesmaximal
10#336×D71C211yesmaximal, characteristic
11C71×#2058C211yesmaximal, characteristic, complemented
12C22×Q8×D71—21no—
13C22×C8×D71C2212yes—
14C2×#14556128C2211yescharacteristic
15C2×#14556127—21no—
16C2×#14556127C2212yes—
17C2×#14556126—21no—
18C4⋊Q8×D71—28no—
19C2×C4⋊C4×D71—21no—
20C2×C4⋊C4×D71—21no—
21C2×C4⋊C4×D71C2211yescharacteristic
22C2×C4⋊C4×D71C2212yes—
23C2×C4⋊C4×D71C2211yescharacteristic
24C2×C42×D71C2211yescharacteristic
25C2×#14556060C2212yes—
26C2×#14556059C2212yes—
27C2×#14556059C2212yes—
28#106×D71—28no—
29#17985898C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985896—21no—
32#17985895C2211yescharacteristic
33#17985894—21no—
34#95×D71—21no—
35#17985762C2212yes—
36#17985761C2212yes—
37#17985761C2212yes—
38#17985754C2211yescharacteristic
39#17985753C2212yes—
40C2×C4⋊Q8×C71C2211yescharacteristic
41C2×#106×C71C2212yes—
42#389×C71C2211yescharacteristic
43#340×C71C2212yes—
44#336×C71C2211yescharacteristic
45C22×Dic142—22no—
46C22×Dic142—21no—
47C2×Q8×D71—28no—
48C2×Q8×D71—44no—
49C2×Q8×D71—44no—
50C22×C4×D71—22no—
51C22×C4×D71—21no—
52C22×C4×D71—21no—
53C22×C4×D71D411yescharacteristic
54C22×C4×D71D411yescharacteristic
55C22×C4×D71C2311yescharacteristic
56C22×C71⋊C8C2312yes—
57C2×C8×D71—24no—
58C2×C8×D71—24no—
59C2×C8×D71Q818yescomplemented
60#14556128—24no—
61#14556127—28no—
62#14556127—24no—
63#14556127—24no—
64#14556126—28no—
65C2×C4⋊Dic71—21no—
66C2×C4⋊Dic71—21no—
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2312yes—
69C2×C4⋊Dic71C2311yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—22no—
72C2×Dic71⋊C4—22no—
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C4⋊C4×D71—28no—
77C4⋊C4×D71—24no—
78C4⋊C4×D71—22no—
79C4⋊C4×D71—24no—
80C4⋊C4×D71—44no—
81C4⋊C4×D71D418yes—
82C4⋊C4×D71Q814yescomplemented
83C2×C4×Dic71D411yescharacteristic
84C2×C4×Dic71D411yescharacteristic
85C2×C4×Dic71C2311yescharacteristic
86C42×D71—24no—
87#14556060—24no—
88#14556059—24no—
89#14556059—24no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C22×Q8×C71—21no—
94C22×C8×C71C2312yes—
95C4⋊Q8×C71—24no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71C2311yescharacteristic
99C2×C4⋊C4×C71C2312yes—
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C42×C71C2311yescharacteristic
102#106×C71—24no—
103#95×C71—21no—
104C23×D71C2×D411yescharacteristic
105C2×Dic142—24no—
106C2×Dic142—28no—
107C2×Dic142—44no—
108C2×Dic142—42no—
109C2×Dic142—42no—
110C2×Dic142—44no—
111Q8×D71—432no—
112C22×Dic71—22no—
113C22×Dic71—21no—
114C22×Dic71—21no—
115C22×Dic71C2×D411yescharacteristic
116C22×Dic71C2×D411yescharacteristic
117C22×Dic71C2×D411yescharacteristic
118C2×C4×D71—22no—
119C2×C4×D71—22no—
120C2×C4×D71—22no—
121C2×C4×D71—22no—
122C2×C4×D71—28no—
123C2×C4×D71—24no—
124C2×C4×D71—44no—
125C2×C4×D71—42no—
126C2×C4×D71—42no—
127C2×C4×D71—44no—
128C2×C4×D71—42no—
129C2×C4×D71—42no—
130C2×C4×D71—42no—
131C2×C4×D71SD1614yes—
132C2×C4×D71SD1614yes—
133C2×C4×D71C2×D414yes—
134C2×C4×D71C2×Q814yes—
135C2×C4×D71C2×Q812yes—
136C2×C4×D71#4112yes—
137C2×C71⋊C8—22no—
138C2×C71⋊C8—22no—
139C2×C71⋊C8C2×Q814yescomplemented
140C8×D71—216no—
141C4⋊Dic71—24no—
142C4⋊Dic71—22no—
143C4⋊Dic71—21no—
144C4⋊Dic71—22no—
145C4⋊Dic71—42no—
146C4⋊Dic71C2×D414yes—
147C4⋊Dic71C2×Q812yescomplemented
148Dic71⋊C4—28no—
149Dic71⋊C4—28no—
150Dic71⋊C4—24no—
151Dic71⋊C4—42no—
152Dic71⋊C4—42no—
153Dic71⋊C4—42no—
154C4×Dic71—22no—
155C4×Dic71—22no—
156C4×Dic71—22no—
157C2×Q8×C71—24no—
158C2×Q8×C71—42no—
159C2×Q8×C71—42no—
160C22×C4×C71—22no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2411yescharacteristic
166C2×C8×C71—22no—
167C2×C8×C71—22no—
168C2×C8×C71C2×Q814yescomplemented
169C4⋊C4×C71—24no—
170C4⋊C4×C71—22no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—22no—
173C4⋊C4×C71—42no—
174C4⋊C4×C71C2×D414yes—
175C4⋊C4×C71C2×Q812yescomplemented
176C42×C71—22no—
177C23×C71C22×D411yescharacteristic
178C22×D71#12012yes—
179C22×D71C22⋊Q812yes—
180C22×D71C4⋊Q812yes—
181C22×D71C2×SD1612yes—
182C22×D71C2×SD1612yes—
183C22×D71AGL(1,ℤ/8)12yes—
184C22×D71#13712yes—
185Q8×C71—48no—
186C2×C4×C71—21no—
187C2×C4×C71—21no—
188C2×C4×C71—21no—
189C2×C4×C71—21no—
190C2×C4×C71—24no—
191C2×C4×C71—22no—
192C2×C4×C71—41no—
193C2×C4×C71—42no—
194C2×C4×C71—42no—
195C2×C4×C71—41no—
196C2×C4×C71—41no—
197C2×C4×C71—41no—
198C2×C4×C71—41no—
199C2×C4×C71C2×SD1612yes—
200C2×C4×C71C2×SD1612yes—
201C2×C4×C71C22×D412yes—
202C2×C4×C71C22×Q812yes—
203C2×C4×C71C22×Q811yescharacteristic
204C2×C4×C71C2×#4111yescharacteristic
205C2×Dic71—21no—
206C2×Dic71—21no—
207C2×Dic71—21no—
208C2×Dic71—21no—
209C2×Dic71—24no—
210C2×Dic71—22no—
211C2×Dic71—41no—
212C2×Dic71—42no—
213C2×Dic71—41no—
214C2×Dic71—42no—
215C2×Dic71—41no—
216C2×Dic71—41no—
217C2×Dic71—41no—
218C2×Dic71#12012yes—
219C2×Dic71C22⋊Q812yes—
220C2×Dic71C4⋊Q812yes—
221C2×Dic71C2×SD1612yes—
222C2×Dic71C2×SD1612yes—
223C4×D71—28no—
224C4×D71—28no—
225C4×D71—416no—
226C4×D71—48no—
227C4×D71—84no—
228C4×D71C22⋊Q818yes—
229C4×D71C4⋊Q818yes—
230Dic142—416no—
231Dic142—48no—
232C8×C71—24no—
233C71⋊C8—24no—
234C22×C71C2×#12011yescharacteristic
235C22×C71C2×C22⋊Q811yescharacteristic
236C22×C71C2×C4⋊Q811yescharacteristic
237C22×C71C22×SD1611yescharacteristic
238C22×C71C22×SD1611yescharacteristic
239C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
240C22×C71C2×#13711yescharacteristic
241C2×D71#39514yes—
242C2×D71#44814yes—
243C2×D71#45014yes—
244C2×D71#47514yes—
245C2×D71#47614yes—
246C2×D71#49814yes—
247C2×D71#50114yes—
248C4×C71—22no—
249C4×C71—22no—
250C4×C71—44no—
251C4×C71—42no—
252C4×C71—81no—
253C4×C71C2×C22⋊Q812yes—
254C4×C71C2×C4⋊Q812yes—
255Dic71—22no—
256Dic71—22no—
257Dic71—44no—
258Dic71—42no—
259Dic71—81no—
260Dic71#39514yes—
261C2×#2058—711nomaximal, Sylow
262C2×C71C2×#39511yescharacteristic
263C2×C71C2×#45011yescharacteristic
264C2×C71C2×#44811yescharacteristic
265C2×C71C2×#47611yescharacteristic
266C2×C71C2×#47511yescharacteristic
267C2×C71C2×#49811yescharacteristic
268C2×C71C2×#50111yescharacteristic
269D71#205818yesdirect product factor
270C22×C4⋊Q8—711no—
271C22×#106—712no—
272C2×#389—711no—
273#2058—712no—
274#2058—711no—
275#2058—712no—
276#2058—711no—
277#2058—711no—
278#2058D7111yescharacteristic, direct product factor
279C2×#340—712no—
280C2×#336—711no—
281C71C2×#205811yescharacteristic, complemented, Sylow
282C23×Q8—1421no—
283C23×C8—712no—
284C2×C4⋊Q8—712no—
285C2×C4⋊Q8—711no—
286C2×C4⋊Q8—1421no—
287C2×C4⋊Q8—1421no—
288C2×C4⋊Q8—1428no—
289C2×C4⋊Q8C2×D7111yescharacteristic
290C22×C4⋊C4—712no—
291C22×C4⋊C4—711no—
292C22×C4⋊C4—711no—
293C22×C4⋊C4—1421no—
294C22×C4⋊C4—1421no—
295C22×C42—711no—
296C2×#106—712no—
297C2×#106—712no—
298C2×#106—712no—
299C2×#106—1428no—
300C2×#106C2×D7112yes—
301#389—711no—
302#389—711no—
303#389—711no—
304#389—1421no—
305#389—1421no—
306#389C2×D7111yescharacteristic
307C2×#95—1421no—
308#340—712no—
309#340—712no—
310#340—712no—
311#340C2×D7112yes—
312#336—712no—
313#336—711no—
314#336C2×D7111yescharacteristic
315C22×Q8—21no—
316C22×Q8—2844no—
317C22×Q8—2844no—
318C22×Q8—1422no—
319C22×Q8—1428no—
320C22×Q8—1421no—
321C23×C4—711no—
322C23×C4—711no—
323C23×C4—711no—
324C23×C4—1421no—
325C23×C4—1422no—
326C23×C4—1421no—
327C22×C8—712no—
328C22×C8—718no—
329C22×C8—1424no—
330C22×C8—1424no—
331C22×C8C22×D7112yes—
332C4⋊Q8—24no—
333C4⋊Q8—1424no—
334C4⋊Q8—1424no—
335C4⋊Q8—1428no—
336C4⋊Q8—1428no—
337C4⋊Q8—1424no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—2844no—
341C2×C4⋊C4—712no—
342C2×C4⋊C4—711no—
343C2×C4⋊C4—718no—
344C2×C4⋊C4—714no—
345C2×C4⋊C4—711no—
346C2×C4⋊C4—1424no—
347C2×C4⋊C4—1424no—
348C2×C4⋊C4—1422no—
349C2×C4⋊C4—1428no—
350C2×C4⋊C4—1421no—
351C2×C4⋊C4—1421no—
352C2×C4⋊C4—1421no—
353C2×C4⋊C4—1421no—
354C2×C4⋊C4—1422no—
355C2×C4⋊C4—1421no—
356C2×C4⋊C4—1422no—
357C2×C4⋊C4—1421no—
358C2×C4⋊C4C22×D7111yescharacteristic
359C2×C4⋊C4C22×D7112yes—
360C2×C4⋊C4C22×D7111yescharacteristic
361C2×C42—711no—
362C2×C42—711no—
363C2×C42—711no—
364C2×C42—1424no—
365C2×C42C22×D7111yescharacteristic
366#106—24no—
367#106—1424no—
368#106—1424no—
369#106—1424no—
370#95—21no—
371#95—1421no—
372#95—1421no—
373#95—1421no—
374C24—711no—
375C2×Q8—24no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—2842no—
379C2×Q8—2844no—
380C2×Q8—2844no—
381C2×Q8—28432no—
382C2×Q8—2842no—
383C2×Q8—1424no—
384C2×Q8—1428no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—2842no—
389C22×C4—2842no—
390C22×C4—2844no—
391C22×C4—2844no—
392C22×C4—2842no—
393C22×C4—2842no—
394C22×C4—2842no—
395C22×C4—711no—
396C22×C4—714no—
397C22×C4—712no—
398C22×C4—714no—
399C22×C4—712no—
400C22×C4—711no—
401C22×C4—711no—
402C22×C4—714no—
403C22×C4—714no—
404C22×C4—1428no—
405C22×C4—1424no—
406C22×C4—1422no—
407C22×C4—1422no—
408C22×C4—1422no—
409C22×C4—1422no—
410C22×C4—1421no—
411C22×C4—1421no—
412C22×C4—1422no—
413C22×C4D4×D7111yescharacteristic
414C22×C4D4×D7111yescharacteristic
415C22×C4C23×D7111yescharacteristic
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—714no—
419C2×C8—14216no—
420C2×C8—1422no—
421C2×C8—1422no—
422C2×C8Q8×D7114yescomplemented
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—42no—
428C4⋊C4—2842no—
429C4⋊C4—2842no—
430C4⋊C4—2842no—
431C4⋊C4—2842no—
432C4⋊C4—714no—
433C4⋊C4—712no—
434C4⋊C4—1424no—
435C4⋊C4—1428no—
436C4⋊C4—1428no—
437C4⋊C4—1424no—
438C4⋊C4—1422no—
439C4⋊C4—1422no—
440C4⋊C4—1421no—
441C4⋊C4D4×D7114yes—
442C4⋊C4Q8×D7112yescomplemented
443C42—22no—
444C42—1422no—
445C42—1422no—
446C42—1422no—
447C23—712no—
448C23—712no—
449C23—712no—
450C23—712no—
451C23—712no—
452C23—712no—
453C23—712no—
454C23C2×D4×D7111yescharacteristic
455Q8—48no—
456Q8—2848no—
457Q8—28416no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—28416no—
472C2×C4—2848no—
473C2×C4—2841no—
474C2×C4—2842no—
475C2×C4—2842no—
476C2×C4—2841no—
477C2×C4—2841no—
478C2×C4—2841no—
479C2×C4—2841no—
480C2×C4—5684no—
481C2×C4—712no—
482C2×C4—712no—
483C2×C4—712no—
484C2×C4—718no—
485C2×C4—718no—
486C2×C4—712no—
487C2×C4—712no—
488C2×C4—1428no—
489C2×C4—1428no—
490C2×C4—1424no—
491C2×C4—1422no—
492C2×C4—1421no—
493C2×C4—1421no—
494C2×C4—1421no—
495C2×C4—1421no—
496C2×C4SD16×D7112yes—
497C2×C4SD16×D7112yes—
498C2×C4C2×D4×D7112yes—
499C2×C4C2×Q8×D7112yes—
500C2×C4C2×Q8×D7111yescharacteristic
501C2×C4#41×D7111yescharacteristic
502C8—24no—
503C8—1424no—
504C22—714no—
505C22—714no—
506C22—714no—
507C22—714no—
508C22—714no—
509C22—714no—
510C22—714no—
511C22#120×D7111yescharacteristic
512C22C22⋊Q8×D7111yescharacteristic
513C22C4⋊Q8×D7111yescharacteristic
514C22C2×SD16×D7111yescharacteristic
515C22C2×SD16×D7111yescharacteristic
516C22AGL(1,ℤ/8)×D7111yescharacteristic
517C22#137×D7111yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—81no—
523C4—2844no—
524C4—2842no—
525C4—5681no—
526C4—714no—
527C4—1422no—
528C4—1422no—
529C4C22⋊Q8×D7112yes—
530C4C4⋊Q8×D7112yes—
531C2—718no—
532C2#395×D7111yescharacteristic
533C2#450×D7111yescharacteristic
534C2#448×D7111yescharacteristic
535C2#476×D7111yescharacteristic
536C2#475×D7111yescharacteristic
537C2#498×D7111yescharacteristic
538C2#501×D7111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2058C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
hypercenter#2058D71
hypocenterC71C2×#2058
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
4. #22589674C2semidirect
5. #22589673C2semidirect
6. #22589672C2semidirect
7. #22589671C2semidirect
8. #22589670C2semidirect
11. C71×#2058C2semidirect
59. C2×C8×D71Q8semidirect
82. C4⋊C4×D71Q8semidirect
139. C2×C71⋊C8C2×Q8semidirect
147. C4⋊Dic71C2×Q8semidirect
168. C2×C8×C71C2×Q8semidirect
175. C4⋊C4×C71C2×Q8semidirect
269. D71#2058direct (N × H)
278. #2058D71direct (N × H)
281. C71C2×#2058semidirect
422. C2×C8Q8×D71semidirect
442. C4⋊C4Q8×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×#106×D712
#389×D712
#225896742
#225896732
#225896722
#225896712
#225896702
#340×D712
#336×D712
C71×#20582
C2×#205871