Finite Group Databasebeta

← #1825×D71 #31997229

Subgroups — #1825×D71 #31997229

Order 18176.

1574 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D71C211yesmaximal, characteristic
2C2×C22⋊C8×D71C211yesmaximal, characteristic
3#380×D71C211yesmaximal, characteristic
4#22588436C211yesmaximal, characteristic, complemented
5#22588435C211yesmaximal, characteristic, complemented
6#22588434C212yesmaximal, complemented
7#22588433C212yesmaximal, complemented
8#22588432C211yesmaximal, characteristic, complemented
9#340×D71C212yesmaximal, complemented
10#340×D71C212yesmaximal, complemented
11C71×#1825C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C22×C8×D71C2212yes—
14C2×#14556087C2211yescharacteristic
15C2×#14556086C2211yescharacteristic
16C2×#14556086C2211yescharacteristic
17C2×#14556085—21no—
18C2×#14556084—21no—
19#117×D71—24no—
20#117×D71C418yescomplemented
21C2×C4⋊C4×D71C2212yes—
22C2×C4⋊C4×D71C2211yescharacteristic
23C2×C4⋊C4×D71C2211yescharacteristic
24C2×C22⋊C4×D71—21no—
25C2×C22⋊C4×D71—21no—
26C2×C42×D71—21no—
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×(C2×D71)⋊C8C2212yes—
29C22⋊C8×D71—24no—
30#17985859C2211yescharacteristic
31#17985858—21no—
32#17985857—21no—
33#17985856C2212yes—
34#95×D71—21no—
35#17985762C2212yescomplemented
36#17985762C2212yescomplemented
37#17985761C2212yescomplemented
38#17985761C2212yescomplemented
39#17985761C2212yescomplemented
40#17985761C2212yescomplemented
41C2×#117×C71C2211yescharacteristic
42C2×C22⋊C8×C71C2211yescharacteristic
43#380×C71C2211yescharacteristic
44#340×C71C2212yescomplemented
45#340×C71C2212yescomplemented
46C24×D71D411yescharacteristic
47C23×Dic71D411yescharacteristic
48C22×C4×D71—28no—
49C22×C4×D71—21no—
50C22×C4×D71—21no—
51C22×C4×D71—22no—
52C22×C4×D71C2×C414yes—
53C22×C4×D71D411yescharacteristic
54C22×C4×D71D411yescharacteristic
55C22×C4×D71D414yes—
56C22×C4×D71C2311yescharacteristic
57C22×C4×D71C2311yescharacteristic
58C22×C71⋊C8C2312yes—
59C2×C8×D71—24no—
60C2×C8×D71—28no—
61#14556087—22no—
62#14556087C2×C414yescomplemented
63#14556086—22no—
64#14556086—22no—
65#14556086C2×C414yescomplemented
66#14556086C2×C414yescomplemented
67#14556085—28no—
68#14556084—28no—
69C2×C4⋊Dic71C2312yes—
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—22no—
75C4⋊C4×D71—22no—
76C4⋊C4×D71—22no—
77C4⋊C4×D71—28no—
78C4⋊C4×D71C2×C414yes—
79C4⋊C4×D71C2×C414yes—
80C2×C22⋊Dic71—21no—
81C2×C22⋊Dic71—21no—
82C2×(C2×D71)⋊C4—21no—
83C2×(C2×D71)⋊C4—21no—
84C2×(C2×D71)⋊C4—22no—
85C22⋊C4×D71—28no—
86C22⋊C4×D71—44no—
87C2×C4×Dic71—21no—
88C2×C4×Dic71—21no—
89C2×C4×Dic71—21no—
90C42×D71—28no—
91C22⋊(C71⋊C8)—22no—
92(C2×D71)⋊C8—24no—
93C71⋊#95—22no—
94C71⋊#95—21no—
95C23×C4×C71C2311yescharacteristic
96C22×C8×C71C2312yes—
97#117×C71—22no—
98#117×C71C2×C414yescomplemented
99C2×C4⋊C4×C71C2312yes—
100C2×C4⋊C4×C71C2311yescharacteristic
101C2×C4⋊C4×C71C2311yescharacteristic
102C2×C22⋊C4×C71—21no—
103C2×C22⋊C4×C71—21no—
104C2×C42×C71—21no—
105C22⋊C8×C71—22no—
106#95×C71—21no—
107C23×D71—24no—
108C23×D71—24no—
109C23×D71—22no—
110C23×D71—22no—
111C23×D71C22⋊C414yes—
112C23×D71C2×D411yescharacteristic
113C23×D71C2×D411yescharacteristic
114C22×Dic71—24no—
115C22×Dic71—21no—
116C22×Dic71—21no—
117C22×Dic71—22no—
118C22×Dic71C22⋊C414yes—
119C22×Dic71C2×D411yescharacteristic
120C22×Dic71C2×D411yescharacteristic
121C2×C4×D71—24no—
122C2×C4×D71—24no—
123C2×C4×D71—24no—
124C2×C4×D71—216no—
125C2×C4×D71—216no—
126C2×C4×D71—24no—
127C2×C4×D71—24no—
128C2×C4×D71—44no—
129C2×C4×D71—42no—
130C2×C4×D71—42no—
131C2×C4×D71—44no—
132C2×C4×D71—44no—
133C2×C4×D71—44no—
134C2×C4×D71C22⋊C414yes—
135C2×C4×D71C22⋊C414yes—
136C2×C4×D71C22×C412yes—
137C2×C4×D71C22×C412yes—
138C2×C4×D71C2×D412yes—
139C2×C4×D71C2×D412yes—
140C2×C4×D71#4118yes—
141C2×C71⋊C8—22no—
142C2×C71⋊C8—24no—
143C8×D71—48no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—24no—
147C4⋊Dic71C22×C412yes—
148C4⋊Dic71C22×C412yes—
149Dic71⋊C4—24no—
150Dic71⋊C4—24no—
151Dic71⋊C4—44no—
152C22⋊Dic71—24no—
153C22⋊Dic71—42no—
154(C2×D71)⋊C4—24no—
155(C2×D71)⋊C4—24no—
156(C2×D71)⋊C4—44no—
157C4×Dic71—24no—
158C4×Dic71—24no—
159C4×Dic71—24no—
160C24×C71C2×D411yescharacteristic
161C22×C4×C71—24no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—22no—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D412yes—
168C22×C4×C71C2411yescharacteristic
169C2×C8×C71—22no—
170C2×C8×C71—24no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—24no—
174C4⋊C4×C71C22×C412yes—
175C4⋊C4×C71C22×C412yes—
176C22⋊C4×C71—24no—
177C22⋊C4×C71—42no—
178C42×C71—24no—
179C23×C71—22no—
180C23×C71—22no—
181C23×C71—21no—
182C23×C71—21no—
183C23×C71C2×C22⋊C412yes—
184C23×C71C22×D411yescharacteristic
185C22×D71—22no—
186C22×D71—22no—
187C22×D71—21no—
188C22×D71—21no—
189C22×D71—28no—
190C22×D71—28no—
191C22×D71—28no—
192C22×D71—48no—
193C22×D71—48no—
194C22×D71C2×C22⋊C412yes—
195C22×D71C2×C22⋊C412yes—
196C22×D71#12314yes—
197C22×D71#13514yes—
198C22×D71AGL(1,ℤ/8)12yes—
199C22×D71#13712yes—
200C2×C4×C71—22no—
201C2×C4×C71—24no—
202C2×C4×C71—24no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—42no—
206C2×C4×C71—41no—
207C2×C4×C71—41no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—42no—
211C2×C4×C71C2×C22⋊C412yes—
212C2×C4×C71C23×C411yescharacteristic
213C2×C4×C71C22×D411yescharacteristic
214C2×C4×C71C2×#4114yes—
215C2×Dic71—22no—
216C2×Dic71—28no—
217C2×Dic71—22no—
218C2×Dic71—22no—
219C2×Dic71—42no—
220C2×Dic71—41no—
221C2×Dic71—41no—
222C2×Dic71—42no—
223C2×Dic71—42no—
224C2×Dic71—42no—
225C2×Dic71C2×C22⋊C412yes—
226C2×Dic71C2×C22⋊C412yes—
227C2×Dic71#12314yes—
228C4×D71—28no—
229C4×D71—24no—
230C4×D71—24no—
231C4×D71—28no—
232C4×D71—48no—
233C4×D71—48no—
234C4×D71—88no—
235C4×D71#11718yes—
236C4×D71#12318yes—
237C8×C71—42no—
238C71⋊C8—42no—
239C22×C71—22no—
240C22×C71—22no—
241C22×C71—22no—
242C22×C71—42no—
243C22×C71—42no—
244C22×C71C22×C22⋊C411yescharacteristic
245C22×C71C2×#12312yes—
246C22×C71C2×#13512yes—
247C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
248C22×C71C2×#13711yescharacteristic
249C2×D71—22no—
250C2×D71—22no—
251C2×D71—22no—
252C2×D71—42no—
253C2×D71—42no—
254C2×D71—416no—
255C2×D71#38114yes—
256C2×D71#41614yes—
257C2×D71#41714yes—
258C2×D71#48218yes—
259C2×D71#48518yes—
260C4×C71—22no—
261C4×C71—42no—
262C4×C71—42no—
263C4×C71—82no—
264C4×C71C2×#11712yes—
265C4×C71C2×#12312yes—
266Dic71—22no—
267Dic71—42no—
268Dic71—42no—
269Dic71—82no—
270Dic71#38114yes—
271C2×#1825—711nomaximal, Sylow
272C2×C71—42no—
273C2×C71C2×#38111yescharacteristic
274C2×C71C2×#41611yescharacteristic
275C2×C71C2×#41711yescharacteristic
276C2×C71C2×#48212yes—
277C2×C71C2×#48512yes—
278D71—42no—
279D71#182518yesdirect product factor
280C22×#117—711no—
281C22×C22⋊C8—711no—
282C2×#380—711no—
283#1825—711no—
284#1825—711no—
285#1825—712no—
286#1825—712no—
287#1825—711no—
288#1825D7111yescharacteristic, direct product factor
289C2×#340—712no—
290C2×#340—712no—
291C71C2×#182511yescharacteristic, complemented, Sylow
292C24×C4—711no—
293C23×C8—712no—
294C2×#117—711no—
295C2×#117—718no—
296C2×#117—711no—
297C2×#117—711no—
298C2×#117—1421no—
299C2×#117—1421no—
300C2×#117—1424no—
301C2×#117C2×D7111yescharacteristic
302C22×C4⋊C4—711no—
303C22×C4⋊C4—711no—
304C22×C4⋊C4—712no—
305C22×C22⋊C4—1421no—
306C22×C22⋊C4—1421no—
307C22×C42—1421no—
308C2×C22⋊C8—711no—
309C2×C22⋊C8—712no—
310C2×C22⋊C8—1424no—
311C2×C22⋊C8C2×D7111yescharacteristic
312#380—711no—
313#380—712no—
314#380—1421no—
315#380—1421no—
316#380C2×D7111yescharacteristic
317C2×#95—1421no—
318#340—712no—
319#340—712no—
320#340—712no—
321#340—712no—
322#340—712no—
323#340—712no—
324#340C2×D7112yescomplemented
325#340C2×D7112yescomplemented
326C25—711no—
327C23×C4—711no—
328C23×C4—711no—
329C23×C4—714no—
330C23×C4—711no—
331C23×C4—714no—
332C23×C4—711no—
333C23×C4—711no—
334C23×C4—1422no—
335C23×C4—1421no—
336C23×C4—1421no—
337C23×C4—1428no—
338C23×C4C22×D7111yescharacteristic
339C22×C8—712no—
340C22×C8—1424no—
341C22×C8—1428no—
342C22×C8C22×D7112yes—
343#117—22no—
344#117—714no—
345#117—714no—
346#117—714no—
347#117—1428no—
348#117—1428no—
349#117—1422no—
350#117—1422no—
351#117—1422no—
352#117C4×D7114yescomplemented
353C2×C4⋊C4—711no—
354C2×C4⋊C4—711no—
355C2×C4⋊C4—714no—
356C2×C4⋊C4—714no—
357C2×C4⋊C4—712no—
358C2×C4⋊C4—1428no—
359C2×C4⋊C4—1422no—
360C2×C4⋊C4—1422no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1421no—
363C2×C4⋊C4—1421no—
364C2×C4⋊C4C22×D7112yes—
365C2×C4⋊C4C22×D7111yescharacteristic
366C2×C4⋊C4C22×D7111yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—2844no—
370C2×C22⋊C4—1428no—
371C2×C22⋊C4—1422no—
372C2×C22⋊C4—1421no—
373C2×C22⋊C4—1421no—
374C2×C22⋊C4—1421no—
375C2×C22⋊C4—1421no—
376C2×C42—21no—
377C2×C42—1428no—
378C2×C42—1421no—
379C2×C42—1421no—
380C2×C42—1421no—
381C22⋊C8—22no—
382C22⋊C8—1424no—
383C22⋊C8—1422no—
384#95—21no—
385#95—1422no—
386#95—1421no—
387C24—711no—
388C24—711no—
389C24—714no—
390C24—1422no—
391C24—1424no—
392C24—1424no—
393C24—1422no—
394C24D4×D7111yescharacteristic
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—2844no—
400C22×C4—2842no—
401C22×C4—2842no—
402C22×C4—2844no—
403C22×C4—2844no—
404C22×C4—2844no—
405C22×C4—711no—
406C22×C4—712no—
407C22×C4—712no—
408C22×C4—718no—
409C22×C4—711no—
410C22×C4—714no—
411C22×C4—712no—
412C22×C4—712no—
413C22×C4—714no—
414C22×C4—714no—
415C22×C4—1424no—
416C22×C4—1424no—
417C22×C4—1424no—
418C22×C4—14216no—
419C22×C4—14216no—
420C22×C4—1424no—
421C22×C4—1422no—
422C22×C4—1421no—
423C22×C4—1421no—
424C22×C4—1424no—
425C22×C4—1424no—
426C22×C4C2×C4×D7112yes—
427C22×C4D4×D7112yes—
428C22×C4D4×D7111yescharacteristic
429C22×C4C23×D7111yescharacteristic
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—2848no—
433C2×C8—1422no—
434C2×C8—1424no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—2844no—
439C4⋊C4—712no—
440C4⋊C4—712no—
441C4⋊C4—1424no—
442C4⋊C4—1424no—
443C4⋊C4—1424no—
444C4⋊C4—1421no—
445C4⋊C4—1421no—
446C4⋊C4C2×C4×D7112yes—
447C4⋊C4C2×C4×D7112yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—2844no—
451C22⋊C4—2842no—
452C22⋊C4—1424no—
453C22⋊C4—1424no—
454C22⋊C4—1424no—
455C42—24no—
456C42—1424no—
457C42—1424no—
458C42—1424no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—2848no—
464C23—2848no—
465C23—712no—
466C23—714no—
467C23—714no—
468C23—712no—
469C23—712no—
470C23—712no—
471C23—1428no—
472C23—1428no—
473C23—1428no—
474C23—1422no—
475C23—1422no—
476C23—1421no—
477C23—1421no—
478C23C22⋊C4×D7112yes—
479C23C2×D4×D7111yescharacteristic
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—2848no—
492C2×C4—2848no—
493C2×C4—2842no—
494C2×C4—2841no—
495C2×C4—2841no—
496C2×C4—2842no—
497C2×C4—2842no—
498C2×C4—2842no—
499C2×C4—5688no—
500C2×C4—712no—
501C2×C4—714no—
502C2×C4—718no—
503C2×C4—718no—
504C2×C4—712no—
505C2×C4—1428no—
506C2×C4—1428no—
507C2×C4—1422no—
508C2×C4—1422no—
509C2×C4—1428no—
510C2×C4—1422no—
511C2×C4—1424no—
512C2×C4—1424no—
513C2×C4C22⋊C4×D7112yes—
514C2×C4C22×C4×D7111yescharacteristic
515C2×C4C2×D4×D7111yescharacteristic
516C2×C4#41×D7114yes—
517C8—42no—
518C8—2842no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—28416no—
525C22—2842no—
526C22—2842no—
527C22—718no—
528C22—718no—
529C22—714no—
530C22—714no—
531C22—714no—
532C22—1422no—
533C22—1422no—
534C22—1422no—
535C22C2×C22⋊C4×D7111yescharacteristic
536C22#123×D7112yes—
537C22#135×D7112yes—
538C22AGL(1,ℤ/8)×D7111yescharacteristic
539C22#137×D7111yescharacteristic
540C4—22no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—2842no—
545C4—2842no—
546C4—5682no—
547C4—714no—
548C4—1422no—
549C4#117×D7112yes—
550C4#123×D7112yes—
551C2—42no—
552C2—2842no—
553C2—718no—
554C2#381×D7111yescharacteristic
555C2#416×D7111yescharacteristic
556C2#417×D7111yescharacteristic
557C2#482×D7112yes—
558C2#485×D7112yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1825C2
socleC23×C71C22×D4
prime radicalC24×D71D4
cosocleC22×C4×C71C24
hypercenter#1825D71
hypocenterC71C2×#1825
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #22588436C2semidirect
5. #22588435C2semidirect
6. #22588434C2semidirect
7. #22588433C2semidirect
8. #22588432C2semidirect
9. #340×D71C2semidirect
10. #340×D71C2semidirect
11. C71×#1825C2semidirect
20. #117×D71C4semidirect
35. #17985762C22semidirect
36. #17985762C22semidirect
37. #17985761C22semidirect
38. #17985761C22semidirect
39. #17985761C22semidirect
40. #17985761C22semidirect
44. #340×C71C22semidirect
45. #340×C71C22semidirect
62. #14556087C2×C4semidirect
65. #14556086C2×C4semidirect
66. #14556086C2×C4semidirect
98. #117×C71C2×C4semidirect
279. D71#1825direct (N × H)
288. #1825D71direct (N × H)
291. C71C2×#1825semidirect
324. #340C2×D71semidirect
325. #340C2×D71semidirect
352. #117C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D712
C2×C22⋊C8×D712
#380×D712
#225884362
#225884352
#225884342
#225884332
#225884322
#340×D712
C71×#18252
C2×#182571