Finite Group Databasebeta

← C127×#17620 #44581293

Subgroups — C127×#17620 #44581293

Order 32512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3496C211yesmaximal, characteristic
2C127×#2973C211yesmaximal, characteristic, complemented
3C127×#2353C211yesmaximal, characteristic, complemented
4C127×#1519C212yesmaximal, complemented
5C127×#1514C212yesmaximal, complemented
6C127×#1510C212yesmaximal, complemented
7C127×#1507C212yesmaximal, complemented
8C127×#1505C212yesmaximal, complemented
9C2×#327×C127C212yesmaximal, complemented
10C2×H(5,𝔽2)×C127—21no—
11#547×C127C414yescomplemented
12#546×C127—24no—
13D42×C127C414yescomplemented
14#537×C127—21no—
15#534×C127—21no—
16#534×C127—21no—
17#532×C127C2212yes—
18C2×#121×C127C2212yes—
19C2×#121×C127C2212yes—
20C2×C4×D4×C127C2211yescharacteristic
21#435×C127—24no—
22#432×C127C2212yescomplemented
23#407×C127C2212yescomplemented
24C2×C22⋊C8×C127C2212yescomplemented
25#388×C127—21no—
26#387×C127—21no—
27#386×C127—21no—
28#383×C127C2211yescharacteristic, complemented
29#381×C127C2211yescharacteristic, complemented
30#380×C127C2211yescharacteristic, complemented
31C4×C22⋊C4×C127—21no—
32C4×C22⋊C4×C127C2211yescharacteristic, complemented
33#376×C127C2211yescharacteristic, complemented
34C2×#95×C127C2211yescharacteristic, complemented
35#327×C127C2214yescomplemented
36#327×C127C2214yescomplemented
37#327×C127C2214yescomplemented
38#327×C127C2214yescomplemented
39H(5,𝔽2)×C127—44no—
40H(5,𝔽2)×C127—44no—
41C2×#41×C127—24no—
42C2×#41×C127—22no—
43C22×D4×C127—22no—
44C22×D4×C127—22no—
45C22×D4×C127—24no—
46C22×D4×C127C2×C412yes—
47C22×D4×C127C2×C412yescomplemented
48C22×D4×C127D411yescharacteristic
49C23×C4×C127—21no—
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2311yescharacteristic
52C2×M16×C127—22no—
53C22×C8×C127—22no—
54C4⋊Q8×C127C2×C412yescomplemented
55#127×C127C2×C412yescomplemented
56#124×C127—22no—
57#123×C127—24no—
58#123×C127—24no—
59C22⋊Q8×C127—24no—
60#121×C127—22no—
61#121×C127—22no—
62#121×C127—22no—
63#121×C127—22no—
64#121×C127—22no—
65#121×C127—24no—
66#121×C127C2×C414yes—
67#121×C127C2×C414yescomplemented
68#121×C127C2314yes—
69#120×C127—24no—
70#120×C127—24no—
71C4×D4×C127—22no—
72C4×D4×C127C2×C414yes—
73C4×D4×C127C2×C414yescomplemented
74C4×D4×C127D414yes—
75C2×C4⋊C4×C127—21no—
76C2×C4⋊C4×C127C2×C412yes—
77C2×C4⋊C4×C127C2×C412yescomplemented
78C2×C4⋊C4×C127D411yescharacteristic
79C2×C22⋊C4×C127—22no—
80C2×C22⋊C4×C127—22no—
81C2×C22⋊C4×C127—21no—
82C2×C22⋊C4×C127—21no—
83C2×C22⋊C4×C127—21no—
84C2×C22⋊C4×C127D411yescharacteristic
85C2×C22⋊C4×C127D411yescharacteristic
86C2×C42×C127—21no—
87C2×C42×C127—21no—
88C2×C42×C127C2311yescharacteristic
89C4⋊C8×C127—22no—
90C22⋊C8×C127—22no—
91C22⋊C8×C127—22no—
92C22⋊C8×C127C2314yes—
93#97×C127—22no—
94#95×C127—21no—
95#95×C127—21no—
96#95×C127—21no—
97#95×C127—21no—
98#95×C127—21no—
99#95×C127C2311yescharacteristic
100#95×C127C2311yescharacteristic
101#95×C127C2311yescharacteristic
102#95×C127C2311yescharacteristic
103C24×C127—22no—
104C24×C127—22no—
105C24×C127C2×D411yescharacteristic
106C24×C127C2×D411yescharacteristic
107#41×C127—44no—
108#41×C127—44no—
109#41×C127—416no—
110C2×Q8×C127—22no—
111C2×D4×C127—22no—
112C2×D4×C127—22no—
113C2×D4×C127—24no—
114C2×D4×C127—24no—
115C2×D4×C127—24no—
116C2×D4×C127—22no—
117C2×D4×C127—22no—
118C2×D4×C127—22no—
119C2×D4×C127—22no—
120C2×D4×C127—416no—
121C2×D4×C127—44no—
122C2×D4×C127—44no—
123C2×D4×C127—44no—
124C2×D4×C127—44no—
125C2×D4×C127—82no—
126C2×D4×C127—82no—
127C2×D4×C127C22⋊C414yes—
128C2×D4×C127C22×C414yes—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—21no—
132C22×C4×C127—24no—
133C22×C4×C127—21no—
134C22×C4×C127—21no—
135C22×C4×C127—21no—
136C22×C4×C127—42no—
137C22×C4×C127—42no—
138C22×C4×C127—81no—
139C22×C4×C127—81no—
140C22×C4×C127—81no—
141C22×C4×C127C22×C412yes—
142C22×C4×C127C22×C412yes—
143C22×C4×C127C2×D411yescharacteristic
144C22×C4×C127C2×D411yescharacteristic
145C22×C4×C127C2×D411yescharacteristic
146C22×C4×C127C2×D411yescharacteristic
147C22×C4×C127C2×D412yes—
148C22×C4×C127C2×D412yes—
149C22×C4×C127C2411yescharacteristic
150M16×C127—82no—
151C2×C8×C127—24no—
152C2×C8×C127—82no—
153C4⋊C4×C127—22no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—22no—
156C4⋊C4×C127—81no—
157C4⋊C4×C127C22⋊C414yes—
158C4⋊C4×C127C22×C414yes—
159C22⋊C4×C127—24no—
160C22⋊C4×C127—24no—
161C22⋊C4×C127—24no—
162C22⋊C4×C127—24no—
163C22⋊C4×C127—42no—
164C22⋊C4×C127—42no—
165C22⋊C4×C127—81no—
166C22⋊C4×C127C22⋊C414yes—
167C22⋊C4×C127C22⋊C414yes—
168C42×C127—42no—
169C42×C127—81no—
170C42×C127C22×C412yes—
171C42×C127C2×D412yes—
172C23×C127—21no—
173C23×C127—21no—
174C23×C127—24no—
175C23×C127—41no—
176C23×C127—41no—
177C23×C127—41no—
178C23×C127—41no—
179C23×C127—44no—
180C23×C127—44no—
181C23×C127—82no—
182C23×C127—82no—
183C23×C127—82no—
184C23×C127—82no—
185C23×C127C2×C22⋊C412yes—
186C23×C127C2×C22⋊C412yes—
187C23×C127AGL(1,ℤ/8)12yes—
188C23×C127AGL(1,ℤ/8)12yes—
189C23×C127C22×D411yescharacteristic
190Q8×C127—44no—
191D4×C127—28no—
192D4×C127—44no—
193D4×C127—44no—
194D4×C127—48no—
195D4×C127—44no—
196D4×C127—44no—
197D4×C127—48no—
198D4×C127—82no—
199D4×C127—82no—
200D4×C127—82no—
201C2×C4×C127—21no—
202C2×C4×C127—21no—
203C2×C4×C127—22no—
204C2×C4×C127—24no—
205C2×C4×C127—41no—
206C2×C4×C127—41no—
207C2×C4×C127—41no—
208C2×C4×C127—42no—
209C2×C4×C127—42no—
210C2×C4×C127—41no—
211C2×C4×C127—42no—
212C2×C4×C127—42no—
213C2×C4×C127—48no—
214C2×C4×C127—41no—
215C2×C4×C127—44no—
216C2×C4×C127—41no—
217C2×C4×C127—44no—
218C2×C4×C127—41no—
219C2×C4×C127—41no—
220C2×C4×C127—41no—
221C2×C4×C127—41no—
222C2×C4×C127—41no—
223C2×C4×C127—41no—
224C2×C4×C127—41no—
225C2×C4×C127—41no—
226C2×C4×C127—41no—
227C2×C4×C127—84no—
228C2×C4×C127C2×C22⋊C412yes—
229C2×C4×C127C2×C22⋊C412yes—
230C2×C4×C127C2×C22⋊C412yes—
231C2×C4×C127C2×C22⋊C412yes—
232C2×C4×C127C23×C411yescharacteristic
233C2×C4×C127C22×D411yescharacteristic
234C8×C127—44no—
235C22×C127—21no—
236C22×C127—22no—
237C22×C127—21no—
238C22×C127—22no—
239C22×C127—22no—
240C22×C127—22no—
241C22×C127—22no—
242C22×C127—42no—
243C22×C127—42no—
244C22×C127—44no—
245C22×C127—44no—
246C22×C127—44no—
247C22×C127—81no—
248C22×C127—81no—
249C22×C127—164no—
250C22×C127—164no—
251C22×C127#41814yes—
252C22×C127C22×C22⋊C411yescharacteristic
253C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
255C4×C127—21no—
256C4×C127—21no—
257C4×C127—22no—
258C4×C127—22no—
259C4×C127—22no—
260C4×C127—22no—
261C4×C127—42no—
262C4×C127—44no—
263C4×C127—81no—
264C4×C127—81no—
265C4×C127—81no—
266#17620C12711yesmaximal, characteristic, direct product factor, Sylow
267C2×C127—21no—
268C2×C127—21no—
269C2×C127—24no—
270C2×C127—42no—
271C2×C127—84no—
272C2×C127#288311yescharacteristic
273C2×C127C2×#41811yescharacteristic
274C2×C127#290111yescharacteristic
275#3496C2×C12711yescharacteristic
276#2973C2×C12711yescharacteristic, complemented
277#2353C2×C12711yescharacteristic, complemented
278#1519C2×C12712yescomplemented
279#1514C2×C12712yescomplemented
280#1510C2×C12712yescomplemented
281#1507C2×C12712yescomplemented
282#1505C2×C12712yescomplemented
283C2×#327C2×C12712yescomplemented
284C127#1762011yescharacteristic, direct product factor, Sylow
285C2×H(5,𝔽2)—21no—
286#547C4×C12714yescomplemented
287#546—24no—
288D42C4×C12714yescomplemented
289#537—21no—
290#534—21no—
291#534—21no—
292#532C22×C12712yes—
293C2×#121C22×C12712yes—
294C2×#121C22×C12712yes—
295C2×C4×D4C22×C12711yescharacteristic
296#435—24no—
297#432C22×C12712yescomplemented
298#407C22×C12712yescomplemented
299C2×C22⋊C8C22×C12712yescomplemented
300#388—21no—
301#387—21no—
302#386—21no—
303#383C22×C12711yescharacteristic, complemented
304#381C22×C12711yescharacteristic, complemented
305#380C22×C12711yescharacteristic, complemented
306C4×C22⋊C4—21no—
307C4×C22⋊C4C22×C12711yescharacteristic, complemented
308#376C22×C12711yescharacteristic, complemented
309C2×#95C22×C12711yescharacteristic, complemented
310#327C22×C12714yescomplemented
311#327C22×C12714yescomplemented
312#327C22×C12714yescomplemented
313#327C22×C12714yescomplemented
314H(5,𝔽2)—44no—
315H(5,𝔽2)—44no—
316C2×#41—24no—
317C2×#41—22no—
318C22×D4—22no—
319C22×D4—22no—
320C22×D4—24no—
321C22×D4C2×C4×C12712yes—
322C22×D4C2×C4×C12712yescomplemented
323C22×D4D4×C12711yescharacteristic
324C23×C4—21no—
325C23×C4C23×C12711yescharacteristic
326C23×C4C23×C12711yescharacteristic
327C2×M16—22no—
328C22×C8—22no—
329C4⋊Q8C2×C4×C12712yescomplemented
330#127C2×C4×C12712yescomplemented
331#124—22no—
332#123—24no—
333#123—24no—
334C22⋊Q8—24no—
335#121—22no—
336#121—22no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121—24no—
341#121C2×C4×C12714yes—
342#121C2×C4×C12714yescomplemented
343#121C23×C12714yes—
344#120—24no—
345#120—24no—
346C4×D4—22no—
347C4×D4C2×C4×C12714yes—
348C4×D4C2×C4×C12714yescomplemented
349C4×D4D4×C12714yes—
350C2×C4⋊C4—21no—
351C2×C4⋊C4C2×C4×C12712yes—
352C2×C4⋊C4C2×C4×C12712yescomplemented
353C2×C4⋊C4D4×C12711yescharacteristic
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4D4×C12711yescharacteristic
360C2×C22⋊C4D4×C12711yescharacteristic
361C2×C42—21no—
362C2×C42—21no—
363C2×C42C23×C12711yescharacteristic
364C4⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8C23×C12714yes—
368#97—22no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95C23×C12711yescharacteristic
375#95C23×C12711yescharacteristic
376#95C23×C12711yescharacteristic
377#95C23×C12711yescharacteristic
378C24—22no—
379C24—22no—
380C24C2×D4×C12711yescharacteristic
381C24C2×D4×C12711yescharacteristic
382#41—44no—
383#41—44no—
384#41—416no—
385C2×Q8—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—82no—
401C2×D4—82no—
402C2×D4C22⋊C4×C12714yes—
403C2×D4C22×C4×C12714yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4C22×C4×C12712yes—
417C22×C4C22×C4×C12712yes—
418C22×C4C2×D4×C12711yescharacteristic
419C22×C4C2×D4×C12711yescharacteristic
420C22×C4C2×D4×C12711yescharacteristic
421C22×C4C2×D4×C12711yescharacteristic
422C22×C4C2×D4×C12712yes—
423C22×C4C2×D4×C12712yes—
424C22×C4C24×C12711yescharacteristic
425M16—82no—
426C2×C8—24no—
427C2×C8—82no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—81no—
432C4⋊C4C22⋊C4×C12714yes—
433C4⋊C4C22×C4×C12714yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—81no—
441C22⋊C4C22⋊C4×C12714yes—
442C22⋊C4C22⋊C4×C12714yes—
443C42—42no—
444C42—81no—
445C42C22×C4×C12712yes—
446C42C2×D4×C12712yes—
447C23—21no—
448C23—21no—
449C23—24no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—44no—
455C23—44no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23C2×C22⋊C4×C12712yes—
461C23C2×C22⋊C4×C12712yes—
462C23AGL(1,ℤ/8)×C12712yes—
463C23AGL(1,ℤ/8)×C12712yes—
464C23C22×D4×C12711yescharacteristic
465Q8—44no—
466D4—28no—
467D4—44no—
468D4—44no—
469D4—48no—
470D4—44no—
471D4—44no—
472D4—48no—
473D4—82no—
474D4—82no—
475D4—82no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—84no—
503C2×C4C2×C22⋊C4×C12712yes—
504C2×C4C2×C22⋊C4×C12712yes—
505C2×C4C2×C22⋊C4×C12712yes—
506C2×C4C2×C22⋊C4×C12712yes—
507C2×C4C23×C4×C12711yescharacteristic
508C2×C4C22×D4×C12711yescharacteristic
509C8—44no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—81no—
523C22—81no—
524C22—164no—
525C22—164no—
526C22#418×C12714yes—
527C22C22×C22⋊C4×C12711yescharacteristic
528C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
529C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—44no—
538C4—81no—
539C4—81no—
540C4—81no—
541C2—21no—
542C2—21no—
543C2—24no—
544C2—42no—
545C2—84no—
546C2C127×#288311yescharacteristic
547C2C2×#418×C12711yescharacteristic
548C2C127×#290111yescharacteristic
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3496C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2973C2semidirect
3. C127×#2353C2semidirect
4. C127×#1519C2semidirect
5. C127×#1514C2semidirect
6. C127×#1510C2semidirect
7. C127×#1507C2semidirect
8. C127×#1505C2semidirect
9. C2×#327×C127C2semidirect
11. #547×C127C4semidirect
13. D42×C127C4semidirect
22. #432×C127C22semidirect
23. #407×C127C22semidirect
24. C2×C22⋊C8×C127C22semidirect
28. #383×C127C22semidirect
29. #381×C127C22semidirect
30. #380×C127C22semidirect
32. C4×C22⋊C4×C127C22semidirect
33. #376×C127C22semidirect
34. C2×#95×C127C22semidirect
35. #327×C127C22semidirect
36. #327×C127C22semidirect
37. #327×C127C22semidirect
38. #327×C127C22semidirect
47. C22×D4×C127C2×C4semidirect
54. C4⋊Q8×C127C2×C4semidirect
55. #127×C127C2×C4semidirect
67. #121×C127C2×C4semidirect
73. C4×D4×C127C2×C4semidirect
77. C2×C4⋊C4×C127C2×C4semidirect
266. #17620C127direct (N × H)
276. #2973C2×C127semidirect
277. #2353C2×C127semidirect
278. #1519C2×C127semidirect
279. #1514C2×C127semidirect
280. #1510C2×C127semidirect
281. #1507C2×C127semidirect
282. #1505C2×C127semidirect
283. C2×#327C2×C127semidirect
284. C127#17620direct (N × H)
286. #547C4×C127semidirect
288. D42C4×C127semidirect
297. #432C22×C127semidirect
298. #407C22×C127semidirect
299. C2×C22⋊C8C22×C127semidirect
303. #383C22×C127semidirect
304. #381C22×C127semidirect
305. #380C22×C127semidirect
307. C4×C22⋊C4C22×C127semidirect
308. #376C22×C127semidirect
309. C2×#95C22×C127semidirect
310. #327C22×C127semidirect
311. #327C22×C127semidirect
312. #327C22×C127semidirect
313. #327C22×C127semidirect
322. C22×D4C2×C4×C127semidirect
329. C4⋊Q8C2×C4×C127semidirect
330. #127C2×C4×C127semidirect
342. #121C2×C4×C127semidirect
348. C4×D4C2×C4×C127semidirect
352. C2×C4⋊C4C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34962
C127×#29732
C127×#23532
C127×#15192
C127×#15142
C127×#15102
C127×#15072
C127×#15052
C2×#327×C1272
#17620127