Finite Group Databasebeta

← #2375×D71 #32160026

Subgroups — #2375×D71 #32160026

Order 18176.

4246 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#120×D71C211yesmaximal, characteristic
2#22591481C211yesmaximal, characteristic, complemented
3#22591480C216yesmaximal, complemented
4#22591479C212yesmaximal, complemented
5#22591478C213yesmaximal, complemented
6#22591477C213yesmaximal, complemented
7#390×D71C213yesmaximal, complemented
8#388×D71C216yesmaximal, complemented
9#386×D71C213yesmaximal, complemented
10#379×D71C212yesmaximal, complemented
11C71×#2375C211yesmaximal, characteristic, complemented
12C25×D71C2211yescharacteristic
13C22×D4×D71C2213yes—
14C23×C4×D71C2211yescharacteristic, complemented
15C2×#14556098C2213yes—
16C2×#14556097C2211yescharacteristic
17C2×#14556096C2213yes—
18#120×D71C4116yescomplemented
19C2×C4⋊C4×D71C2213yescomplemented
20C2×C22⋊C4×D71C2216yes—
21C2×C22⋊C4×D71C2216yes—
22C2×C22⋊C4×D71C2213yes—
23C2×C22⋊C4×D71C2213yescomplemented
24C2×C42×D71C2213yescomplemented
25#17985902C2213yescomplemented
26#17985901C2213yescomplemented
27#17985900C2213yescomplemented
28#17985899C2216yescomplemented
29#17985899C2216yescomplemented
30#17985893C2216yescomplemented
31#17985892C2216yescomplemented
32#17985891C2216yescomplemented
33#17985890C2216yescomplemented
34#17985889C2216yescomplemented
35#17985888C2216yescomplemented
36#17985887C2216yescomplemented
37#17985879C2213yescomplemented
38#17985878C2213yescomplemented
39#17985877C2216yescomplemented
40#17985876C2213yescomplemented
41#17985875C2216yescomplemented
42#17985855C2212yescomplemented
43#17985854C2216yescomplemented
44#17985854C2216yescomplemented
45#95×D71C2216yescomplemented
46C2×#120×C71C2211yescharacteristic
47#390×C71C2213yescomplemented
48#388×C71C2216yescomplemented
49#386×C71C2213yescomplemented
50#379×C71C2212yescomplemented
51C24×D71—46no—
52C24×D71—46no—
53C24×D71C2×C418yescomplemented
54C24×D71C2311yescharacteristic
55C24×D71C2313yes—
56C24×D71C2313yes—
57C22×D284C2313yes—
58C22×C22⋊D71C2313yes—
59C22×C22⋊D71C2313yes—
60C2×D4×D71—46no—
61C2×D4×D71—46no—
62C2×D4×D71C2×C4124yes—
63C23×Dic71C2311yescharacteristic, complemented
64C22×C4×D71—46no—
65C22×C4×D71C2312yes—
66C22×C4×D71C2316yes—
67C22×C4×D71C2313yes—
68C22×C4×D71C2312yescomplemented
69#14556098C2×C4124yescomplemented
70#14556097C2×C418yescomplemented
71#14556096C2×C4124yescomplemented
72C2×C4⋊Dic71C2313yescomplemented
73C2×Dic71⋊C4C2316yescomplemented
74C4⋊C4×D71—46no—
75C2×C22⋊Dic71C2316yes—
76C2×C22⋊Dic71C2316yes—
77C2×C22⋊Dic71C2313yes—
78C2×C22⋊Dic71C2313yescomplemented
79C2×(C2×D71)⋊C4C2316yes—
80C2×(C2×D71)⋊C4C2316yes—
81C2×(C2×D71)⋊C4C2316yes—
82C2×(C2×D71)⋊C4C2316yes—
83C2×(C2×D71)⋊C4C2316yes—
84C2×(C2×D71)⋊C4C2316yescomplemented
85C22⋊C4×D71—412no—
86C22⋊C4×D71—412no—
87C22⋊C4×D71—46no—
88C22⋊C4×D71C2×C4124yescomplemented
89C2×C4×Dic71C2316yescomplemented
90C2×C4×Dic71C2313yescomplemented
91C42×D71—46no—
92C71⋊#95C2316yescomplemented
93C71⋊#95C2316yescomplemented
94C71⋊#95C2316yescomplemented
95C25×C71C2311yescharacteristic
96C22×D4×C71C2313yes—
97C23×C4×C71C2311yescharacteristic, complemented
98#120×C71C2×C418yescomplemented
99C2×C4⋊C4×C71C2313yescomplemented
100C2×C22⋊C4×C71C2316yes—
101C2×C22⋊C4×C71C2316yes—
102C2×C22⋊C4×C71C2313yes—
103C2×C22⋊C4×C71C2313yescomplemented
104C2×C42×C71C2313yescomplemented
105#95×C71C2316yescomplemented
106C23×D71—26no—
107C23×D71—26no—
108C23×D71—26no—
109C23×D71—26no—
110C23×D71—212no—
111C23×D71—212no—
112C23×D71—448no—
113C23×D71—43no—
114C23×D71—43no—
115C23×D71—46no—
116C23×D71—46no—
117C23×D71—448no—
118C23×D71—416no—
119C23×D71C22×C414yes—
120C23×D71C22×C4112yes—
121C23×D71C22×C4112yescomplemented
122C23×D71C2411yescharacteristic
123C23×D71C2413yes—
124C23×D71C2411yescharacteristic
125C2×D284—43no—
126C2×D284—43no—
127C2×D284C22×C4112yes—
128C2×C22⋊D71—43no—
129C2×C22⋊D71—43no—
130C2×C22⋊D71—43no—
131C2×C22⋊D71—43no—
132C2×C22⋊D71C22×C4112yes—
133C2×C22⋊D71C22×C4112yes—
134D4×D71—448no—
135C22×Dic71—43no—
136C22×Dic71C2412yes—
137C22×Dic71C2416yes—
138C22×Dic71C2413yes—
139C2×C4×D71—26no—
140C2×C4×D71—26no—
141C2×C4×D71—212no—
142C2×C4×D71—212no—
143C2×C4×D71—212no—
144C2×C4×D71—212no—
145C2×C4×D71—46no—
146C2×C4×D71—416no—
147C2×C4×D71C22×C4112yes—
148C4⋊Dic71—43no—
149Dic71⋊C4—46no—
150C22⋊Dic71—46no—
151C22⋊Dic71—46no—
152C22⋊Dic71—43no—
153C22⋊Dic71C22×C4112yescomplemented
154(C2×D71)⋊C4—46no—
155(C2×D71)⋊C4—46no—
156(C2×D71)⋊C4—46no—
157(C2×D71)⋊C4—46no—
158(C2×D71)⋊C4—46no—
159(C2×D71)⋊C4C22×C4124yescomplemented
160C4×Dic71—46no—
161C4×Dic71—43no—
162C24×C71—43no—
163C24×C71—43no—
164C24×C71C22×C414yescomplemented
165C24×C71C2411yescharacteristic
166C24×C71C2413yes—
167C2×D4×C71—43no—
168C2×D4×C71—43no—
169C2×D4×C71C22×C4112yes—
170C22×C4×C71—43no—
171C22×C4×C71C2412yes—
172C22×C4×C71C2416yes—
173C22×C4×C71C2413yes—
174C4⋊C4×C71—43no—
175C22⋊C4×C71—46no—
176C22⋊C4×C71—46no—
177C22⋊C4×C71—43no—
178C22⋊C4×C71C22×C4112yescomplemented
179C42×C71—43no—
180C23×C71—23no—
181C23×C71—26no—
182C23×C71—26no—
183C23×C71—23no—
184C23×C71—23no—
185C23×C71—23no—
186C23×C71—412no—
187C23×C71—412no—
188C23×C71—44no—
189C23×C71C23×C416yes—
190C23×C71C23×C412yes—
191C23×C71C2511yescharacteristic
192C22×D71—23no—
193C22×D71—26no—
194C22×D71—26no—
195C22×D71—23no—
196C22×D71—23no—
197C22×D71—23no—
198C22×D71—224no—
199C22×D71—224no—
200C22×D71—248no—
201C22×D71—424no—
202C22×D71—412no—
203C22×D71—424no—
204C22×D71—412no—
205C22×D71—412no—
206C22×D71—4128no—
207C22×D71—412no—
208C22×D71—424no—
209C22×D71—412no—
210C22×D71—44no—
211C22×D71—412no—
212C22×D71C23×C416yes—
213C22×D71C23×C412yes—
214C22×D71C23×C412yes—
215C22×D71H(5,𝔽2)16yes—
216C22×D71H(5,𝔽2)16yes—
217D4×C71—412no—
218C2×C4×C71—23no—
219C2×C4×C71—23no—
220C2×C4×C71—26no—
221C2×C4×C71—26no—
222C2×C4×C71—26no—
223C2×C4×C71—26no—
224C2×C4×C71—44no—
225C2×C4×C71C23×C416yes—
226C22⋊D71—412no—
227C22⋊D71—412no—
228C2×Dic71—23no—
229C2×Dic71—23no—
230C2×Dic71—26no—
231C2×Dic71—26no—
232C2×Dic71—26no—
233C2×Dic71—26no—
234C2×Dic71—44no—
235C2×Dic71C23×C416yes—
236D284—412no—
237C4×D71—224no—
238C4×D71—48no—
239C4×D71—424no—
240C4×D71—48no—
241C22×C71—26no—
242C22×C71—26no—
243C22×C71—212no—
244C22×C71—416no—
245C22×C71—43no—
246C22×C71—46no—
247C22×C71—43no—
248C22×C71—41no—
249C22×C71—43no—
250C22×C71C24×C411yescharacteristic
251C22×C71C2×H(5,𝔽2)13yes—
252C22×C71C2×H(5,𝔽2)13yes—
253C2×D71—26no—
254C2×D71—26no—
255C2×D71—212no—
256C2×D71—448no—
257C2×D71—43no—
258C2×D71—43no—
259C2×D71—46no—
260C2×D71—41no—
261C2×D71—43no—
262C2×D71—416no—
263C2×D71—448no—
264C2×D71#518112yes—
265C2×D71#560112yes—
266C2×D71H(3,𝔽4)14yes—
267C4×C71—26no—
268C4×C71—46no—
269C4×C71—42no—
270Dic71—26no—
271Dic71—46no—
272Dic71—42no—
273C2×#2375—711nomaximal, Sylow
274C2×C71—46no—
275C2×C71—42no—
276C2×C71C2×#51813yes—
277C2×C71C2×#56013yes—
278C2×C71C2×H(3,𝔽4)11yescharacteristic
279D71—42no—
280D71—46no—
281D71#237518yesdirect product factor
282C22×#120—711no—
283#2375—713no—
284#2375—716no—
285#2375—713no—
286#2375—712no—
287#2375—711no—
288#2375D7111yescharacteristic, direct product factor
289C2×#390—713no—
290C2×#388—716no—
291C2×#386—713no—
292C2×#379—712no—
293C71C2×#237511yescharacteristic, complemented, Sylow
294C26—711no—
295C23×D4—713no—
296C24×C4—711no—
297C2×#120—711no—
298C2×#120—713no—
299C2×#120—7116no—
300C2×#120—713no—
301C2×#120C2×D7111yescharacteristic
302C22×C4⋊C4—713no—
303C22×C22⋊C4—716no—
304C22×C22⋊C4—716no—
305C22×C22⋊C4—713no—
306C22×C22⋊C4—713no—
307C22×C42—713no—
308#390—713no—
309#390—713no—
310#390—716no—
311#390—716no—
312#390—713no—
313#390C2×D7113yescomplemented
314#388—716no—
315#388—716no—
316#388—716no—
317#388—716no—
318#388—716no—
319#388—716no—
320#388—716no—
321#388C2×D7116yescomplemented
322#386—713no—
323#386—713no—
324#386—716no—
325#386—713no—
326#386—716no—
327#386C2×D7113yescomplemented
328#379—712no—
329#379—716no—
330#379—716no—
331#379C2×D7112yescomplemented
332C2×#95—716no—
333C25—2846no—
334C25—2846no—
335C25—718no—
336C25—713no—
337C25—713no—
338C25—711no—
339C25C22×D7111yescharacteristic
340C22×D4—2846no—
341C22×D4—2846no—
342C22×D4—713no—
343C22×D4—713no—
344C22×D4—7124no—
345C22×D4—713no—
346C22×D4C22×D7113yes—
347C23×C4—2846no—
348C23×C4—711no—
349C23×C4—712no—
350C23×C4—713no—
351C23×C4—716no—
352C23×C4—712no—
353C23×C4C22×D7111yescharacteristic, complemented
354#120—718no—
355#120—7124no—
356#120—7124no—
357#120C4×D7118yescomplemented
358C2×C4⋊C4—2846no—
359C2×C4⋊C4—716no—
360C2×C4⋊C4—713no—
361C2×C4⋊C4C22×D7113yescomplemented
362C2×C22⋊C4—28412no—
363C2×C22⋊C4—28412no—
364C2×C22⋊C4—2846no—
365C2×C22⋊C4—716no—
366C2×C22⋊C4—716no—
367C2×C22⋊C4—713no—
368C2×C22⋊C4—716no—
369C2×C22⋊C4—716no—
370C2×C22⋊C4—716no—
371C2×C22⋊C4—716no—
372C2×C22⋊C4—716no—
373C2×C22⋊C4—713no—
374C2×C22⋊C4—716no—
375C2×C22⋊C4—7124no—
376C2×C22⋊C4C22×D7116yes—
377C2×C22⋊C4C22×D7116yes—
378C2×C22⋊C4C22×D7113yes—
379C2×C22⋊C4C22×D7113yescomplemented
380C2×C42—2846no—
381C2×C42—716no—
382C2×C42—713no—
383C2×C42C22×D7113yescomplemented
384#95—716no—
385#95—716no—
386#95—716no—
387#95C22×D7116yescomplemented
388C24—43no—
389C24—43no—
390C24—2846no—
391C24—2846no—
392C24—2843no—
393C24—2843no—
394C24—28448no—
395C24—28416no—
396C24—28448no—
397C24—7112no—
398C24—711no—
399C24—7112no—
400C24—714no—
401C24—713no—
402C24—711no—
403C24—1426no—
404C24—14212no—
405C24—14212no—
406C24—1426no—
407C24—1426no—
408C24—1426no—
409C24C2×C4×D7114yescomplemented
410C24C23×D7113yes—
411C24C23×D7111yescharacteristic
412C2×D4—43no—
413C2×D4—43no—
414C2×D4—2843no—
415C2×D4—2843no—
416C2×D4—28448no—
417C2×D4—2843no—
418C2×D4—2843no—
419C2×D4—2843no—
420C2×D4—2843no—
421C2×D4—7112no—
422C2×D4—7112no—
423C2×D4—7112no—
424C2×D4C2×C4×D71112yes—
425C22×C4—43no—
426C22×C4—2846no—
427C22×C4—28416no—
428C22×C4—2843no—
429C22×C4—713no—
430C22×C4—7112no—
431C22×C4—712no—
432C22×C4—716no—
433C22×C4—14212no—
434C22×C4—14212no—
435C22×C4—14212no—
436C22×C4—14212no—
437C22×C4—1426no—
438C22×C4—1426no—
439C22×C4C23×D7112yes—
440C22×C4C23×D7116yes—
441C22×C4C23×D7113yes—
442C4⋊C4—43no—
443C4⋊C4—2846no—
444C4⋊C4—2843no—
445C22⋊C4—46no—
446C22⋊C4—46no—
447C22⋊C4—43no—
448C22⋊C4—2846no—
449C22⋊C4—2846no—
450C22⋊C4—2846no—
451C22⋊C4—2846no—
452C22⋊C4—2846no—
453C22⋊C4—2846no—
454C22⋊C4—2843no—
455C22⋊C4—2846no—
456C22⋊C4—7112no—
457C22⋊C4—7124no—
458C22⋊C4C2×C4×D71112yescomplemented
459C42—43no—
460C42—2843no—
461C42—2846no—
462C23—23no—
463C23—26no—
464C23—26no—
465C23—23no—
466C23—23no—
467C23—23no—
468C23—412no—
469C23—412no—
470C23—44no—
471C23—28412no—
472C23—28424no—
473C23—284128no—
474C23—28412no—
475C23—28424no—
476C23—28412no—
477C23—2844no—
478C23—28412no—
479C23—28424no—
480C23—28412no—
481C23—28412no—
482C23—712no—
483C23—716no—
484C23—716no—
485C23—716no—
486C23—712no—
487C23—14224no—
488C23—14224no—
489C23—14248no—
490C23—1423no—
491C23—1426no—
492C23—1426no—
493C23—1423no—
494C23—1423no—
495C23—1423no—
496C23C22×C4×D7116yes—
497C23C22×C4×D7112yes—
498C23C24×D7111yescharacteristic
499D4—412no—
500D4—28412no—
501D4—28412no—
502D4—28412no—
503C2×C4—23no—
504C2×C4—23no—
505C2×C4—26no—
506C2×C4—26no—
507C2×C4—26no—
508C2×C4—26no—
509C2×C4—44no—
510C2×C4—2848no—
511C2×C4—2848no—
512C2×C4—28424no—
513C2×C4—2844no—
514C2×C4—716no—
515C2×C4—14224no—
516C2×C4—1426no—
517C2×C4—1426no—
518C2×C4—1426no—
519C2×C4—1426no—
520C2×C4—1423no—
521C2×C4—1423no—
522C2×C4C22×C4×D7116yes—
523C22—26no—
524C22—26no—
525C22—212no—
526C22—416no—
527C22—43no—
528C22—46no—
529C22—43no—
530C22—41no—
531C22—43no—
532C22—28448no—
533C22—28448no—
534C22—28416no—
535C22—2843no—
536C22—2846no—
537C22—2843no—
538C22—2841no—
539C22—2843no—
540C22—7112no—
541C22—7112no—
542C22—714no—
543C22—1426no—
544C22—1426no—
545C22—14212no—
546C22C23×C4×D7111yescharacteristic
547C22H(5,𝔽2)×D7113yes—
548C22H(5,𝔽2)×D7113yes—
549C4—26no—
550C4—46no—
551C4—42no—
552C4—2846no—
553C4—2842no—
554C4—1426no—
555C2—46no—
556C2—42no—
557C2—2846no—
558C2—2842no—
559C2—718no—
560C2#518×D7113yes—
561C2#560×D7113yes—
562C2H(3,𝔽4)×D7111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2375C2
socle, cosocleC23×C71C25
prime radicalC2×#120×D71C2
hypercenter#2375D71
hypocenterC71C2×#2375
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2375

Complements

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

NH ≅ G/Ntype
2. #22591481C2semidirect
3. #22591480C2semidirect
4. #22591479C2semidirect
5. #22591478C2semidirect
6. #22591477C2semidirect
7. #390×D71C2semidirect
8. #388×D71C2semidirect
9. #386×D71C2semidirect
10. #379×D71C2semidirect
11. C71×#2375C2semidirect
14. C23×C4×D71C22semidirect
18. #120×D71C4semidirect
19. C2×C4⋊C4×D71C22semidirect
23. C2×C22⋊C4×D71C22semidirect
24. C2×C42×D71C22semidirect
25. #17985902C22semidirect
26. #17985901C22semidirect
27. #17985900C22semidirect
28. #17985899C22semidirect
29. #17985899C22semidirect
30. #17985893C22semidirect
31. #17985892C22semidirect
32. #17985891C22semidirect
33. #17985890C22semidirect
34. #17985889C22semidirect
35. #17985888C22semidirect
36. #17985887C22semidirect
37. #17985879C22semidirect
38. #17985878C22semidirect
39. #17985877C22semidirect
40. #17985876C22semidirect
41. #17985875C22semidirect
42. #17985855C22semidirect
43. #17985854C22semidirect
44. #17985854C22semidirect
45. #95×D71C22semidirect
47. #390×C71C22semidirect
48. #388×C71C22semidirect
49. #386×C71C22semidirect
50. #379×C71C22semidirect
53. C24×D71C2×C4semidirect
63. C23×Dic71C23semidirect
68. C22×C4×D71C23semidirect
69. #14556098C2×C4semidirect
70. #14556097C2×C4semidirect
71. #14556096C2×C4semidirect
72. C2×C4⋊Dic71C23semidirect
73. C2×Dic71⋊C4C23semidirect
78. C2×C22⋊Dic71C23semidirect
84. C2×(C2×D71)⋊C4C23semidirect
88. C22⋊C4×D71C2×C4semidirect
89. C2×C4×Dic71C23semidirect
90. C2×C4×Dic71C23semidirect
92. C71⋊#95C23semidirect
93. C71⋊#95C23semidirect
94. C71⋊#95C23semidirect
97. C23×C4×C71C23semidirect
98. #120×C71C2×C4semidirect
99. C2×C4⋊C4×C71C23semidirect
103. C2×C22⋊C4×C71C23semidirect
104. C2×C42×C71C23semidirect
105. #95×C71C23semidirect
121. C23×D71C22×C4semidirect
153. C22⋊Dic71C22×C4semidirect
159. (C2×D71)⋊C4C22×C4semidirect
164. C24×C71C22×C4semidirect
178. C22⋊C4×C71C22×C4semidirect
281. D71#2375direct (N × H)
288. #2375D71direct (N × H)
293. C71C2×#2375semidirect
313. #390C2×D71semidirect
321. #388C2×D71semidirect
327. #386C2×D71semidirect
331. #379C2×D71semidirect
353. C23×C4C22×D71semidirect
357. #120C4×D71semidirect
361. C2×C4⋊C4C22×D71semidirect
379. C2×C22⋊C4C22×D71semidirect
383. C2×C42C22×D71semidirect
387. #95C22×D71semidirect
409. C24C2×C4×D71semidirect
458. C22⋊C4C2×C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#120×D712
#225914812
#225914802
#225914792
#225914782
#225914772
#390×D712
#388×D712
#386×D712
#379×D712
C71×#23752
C2×#237571