Finite Group Databasebeta

← C127×#12690 #44576363

Subgroups — C127×#12690 #44576363

Order 32512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#534×C127C211yesmaximal, characteristic
2C2×#352×C127C212yesmaximal, complemented
3C2×AΣL(1,𝔽16)×C127C212yesmaximal, complemented
4C127×#1783C211yesmaximal, characteristic, complemented
5C127×#1781C211yesmaximal, characteristic, complemented
6C23×D4×C127C411yescharacteristic, complemented
7C23×D4×C127C2211yescharacteristic
8#534×C127—24no—
9#534×C127—24no—
10C2×#123×C127—21no—
11C2×#123×C127C411yescharacteristic, complemented
12C2×#123×C127C2211yescharacteristic
13C2×#121×C127—21no—
14C2×#121×C127—21no—
15C2×#120×C127—21no—
16C2×#120×C127C411yescharacteristic, complemented
17C2×#120×C127C2211yescharacteristic
18C22×C22⋊C4×C127C411yescharacteristic, complemented
19C2×#100×C127—21no—
20C2×#100×C127C2212yes—
21C2×#99×C127—21no—
22C2×#99×C127C2212yes—
23#407×C127—21no—
24#379×C127—21no—
25#352×C127—24no—
26AΣL(1,𝔽16)×C127—24no—
27C25×C127C2×C411yescharacteristic
28C25×C127C2×C411yescharacteristic
29C25×C127D411yescharacteristic
30C22×D4×C127—21no—
31C22×D4×C127—21no—
32C22×D4×C127—22no—
33C22×D4×C127—21no—
34C22×D4×C127—21no—
35C22×D4×C127—22no—
36C22×D4×C127—22no—
37C22×D4×C127—22no—
38C22×D4×C127—42no—
39C22×D4×C127—42no—
40C22×D4×C127—42no—
41C22×D4×C127—42no—
42C22×D4×C127C2×C411yescharacteristic
43C22×D4×C127D411yescharacteristic
44C22×D4×C127D411yescharacteristic
45C22×D4×C127C2311yescharacteristic
46C23×C4×C127—21no—
47C23×C4×C127C2×C411yescharacteristic
48C23×C4×C127D411yescharacteristic
49C2×M16×C127—22no—
50#123×C127—28no—
51#123×C127—22no—
52#123×C127—22no—
53#123×C127—22no—
54#123×C127D414yes—
55#121×C127—28no—
56#121×C127—28no—
57#120×C127—28no—
58#120×C127—22no—
59#120×C127—22no—
60#120×C127—22no—
61#120×C127D414yes—
62C2×C4⋊C4×C127—21no—
63C2×C4⋊C4×C127—21no—
64C2×C22⋊C4×C127—21no—
65C2×C22⋊C4×C127—21no—
66C2×C22⋊C4×C127—21no—
67C2×C22⋊C4×C127—21no—
68C2×C22⋊C4×C127—21no—
69C2×C22⋊C4×C127—22no—
70C2×C22⋊C4×C127—22no—
71C2×C22⋊C4×C127—22no—
72C2×C22⋊C4×C127—42no—
73C2×C22⋊C4×C127C2×C411yescharacteristic
74C2×C22⋊C4×C127C2×C411yescharacteristic
75#100×C127—22no—
76#100×C127—22no—
77#100×C127—42no—
78#99×C127—22no—
79#99×C127—22no—
80#99×C127—42no—
81C22⋊C8×C127—42no—
82C24×C127—22no—
83C24×C127—22no—
84C24×C127—21no—
85C24×C127—21no—
86C24×C127—21no—
87C24×C127—22no—
88C24×C127—21no—
89C24×C127—22no—
90C24×C127—41no—
91C24×C127—41no—
92C24×C127—41no—
93C24×C127—82no—
94C24×C127—82no—
95C24×C127—82no—
96C24×C127C22⋊C411yescharacteristic
97C24×C127C22⋊C411yescharacteristic
98C24×C127C22⋊C411yescharacteristic
99C24×C127C22⋊C412yescomplemented
100C24×C127C22×C411yescharacteristic
101C24×C127C2×D411yescharacteristic
102C2×D4×C127—24no—
103C2×D4×C127—24no—
104C2×D4×C127—22no—
105C2×D4×C127—22no—
106C2×D4×C127—21no—
107C2×D4×C127—21no—
108C2×D4×C127—22no—
109C2×D4×C127—24no—
110C2×D4×C127—24no—
111C2×D4×C127—22no—
112C2×D4×C127—41no—
113C2×D4×C127—41no—
114C2×D4×C127—41no—
115C2×D4×C127—41no—
116C2×D4×C127—41no—
117C2×D4×C127—41no—
118C2×D4×C127—48no—
119C2×D4×C127—48no—
120C2×D4×C127—48no—
121C2×D4×C127—42no—
122C2×D4×C127—42no—
123C2×D4×C127—48no—
124C2×D4×C127—42no—
125C2×D4×C127—42no—
126C2×D4×C127—82no—
127C2×D4×C127—82no—
128C2×D4×C127C2×D412yes—
129C2×D4×C127C2×D412yes—
130C22×C4×C127—22no—
131C22×C4×C127—21no—
132C22×C4×C127—21no—
133C22×C4×C127—21no—
134C22×C4×C127—21no—
135C22×C4×C127—24no—
136C22×C4×C127—22no—
137C22×C4×C127—41no—
138C22×C4×C127—41no—
139C22×C4×C127—41no—
140C22×C4×C127—42no—
141C22×C4×C127—82no—
142C22×C4×C127C22⋊C411yescharacteristic
143C22×C4×C127C22⋊C412yescomplemented
144C22×C4×C127C2×D411yescharacteristic
145M16×C127—42no—
146M16×C127—42no—
147C2×C8×C127—42no—
148C4⋊C4×C127—24no—
149C4⋊C4×C127—24no—
150C22⋊C4×C127—24no—
151C22⋊C4×C127—24no—
152C22⋊C4×C127—24no—
153C22⋊C4×C127—21no—
154C22⋊C4×C127—21no—
155C22⋊C4×C127—24no—
156C22⋊C4×C127—24no—
157C22⋊C4×C127—42no—
158C22⋊C4×C127—42no—
159C22⋊C4×C127—84no—
160C22⋊C4×C127C22⋊C412yescomplemented
161C22⋊C4×C127C22⋊C412yescomplemented
162C23×C127—24no—
163C23×C127—24no—
164C23×C127—21no—
165C23×C127—21no—
166C23×C127—21no—
167C23×C127—21no—
168C23×C127—21no—
169C23×C127—21no—
170C23×C127—21no—
171C23×C127—21no—
172C23×C127—21no—
173C23×C127—22no—
174C23×C127—21no—
175C23×C127—21no—
176C23×C127—42no—
177C23×C127—44no—
178C23×C127—42no—
179C23×C127—42no—
180C23×C127—42no—
181C23×C127—42no—
182C23×C127—44no—
183C23×C127—42no—
184C23×C127—42no—
185C23×C127—42no—
186C23×C127—42no—
187C23×C127—41no—
188C23×C127—41no—
189C23×C127—82no—
190C23×C127—82no—
191C23×C127—82no—
192C23×C127—82no—
193C23×C127—84no—
194C23×C127—82no—
195C23×C127—82no—
196C23×C127—82no—
197C23×C127—84no—
198C23×C127—84no—
199C23×C127—84no—
200C23×C127—84no—
201C23×C127#9912yes—
202C23×C127#9912yes—
203C23×C127C2×C22⋊C411yescharacteristic
204C23×C127C2×C22⋊C411yescharacteristic
205C23×C127C2×C22⋊C411yescharacteristic
206C23×C127#12012yes—
207D4×C127—42no—
208D4×C127—42no—
209D4×C127—44no—
210D4×C127—44no—
211D4×C127—42no—
212D4×C127—42no—
213D4×C127—48no—
214D4×C127—48no—
215D4×C127—84no—
216D4×C127—84no—
217C2×C4×C127—21no—
218C2×C4×C127—21no—
219C2×C4×C127—22no—
220C2×C4×C127—22no—
221C2×C4×C127—21no—
222C2×C4×C127—22no—
223C2×C4×C127—21no—
224C2×C4×C127—21no—
225C2×C4×C127—44no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—41no—
230C2×C4×C127—41no—
231C2×C4×C127—42no—
232C2×C4×C127—84no—
233C2×C4×C127—82no—
234C2×C4×C127—84no—
235C2×C4×C127#12012yes—
236C8×C127—82no—
237C22×C127—22no—
238C22×C127—22no—
239C22×C127—22no—
240C22×C127—22no—
241C22×C127—21no—
242C22×C127—21no—
243C22×C127—22no—
244C22×C127—22no—
245C22×C127—42no—
246C22×C127—41no—
247C22×C127—41no—
248C22×C127—42no—
249C22×C127—42no—
250C22×C127—42no—
251C22×C127—42no—
252C22×C127—42no—
253C22×C127—42no—
254C22×C127—42no—
255C22×C127—42no—
256C22×C127—42no—
257C22×C127—42no—
258C22×C127—42no—
259C22×C127—42no—
260C22×C127—41no—
261C22×C127—42no—
262C22×C127—81no—
263C22×C127—81no—
264C22×C127—82no—
265C22×C127—84no—
266C22×C127—84no—
267C22×C127—88no—
268C22×C127—84no—
269C22×C127—84no—
270C22×C127—82no—
271C22×C127#37911yescharacteristic
272C22×C127C2×#9911yescharacteristic
273C22×C127C2×#9911yescharacteristic
274C4×C127—22no—
275C4×C127—42no—
276C4×C127—42no—
277C4×C127—41no—
278C4×C127—42no—
279C4×C127—162no—
280#12690C12711yesmaximal, characteristic, direct product factor, Sylow
281C2×C127—21no—
282C2×C127—21no—
283C2×C127—42no—
284C2×C127—42no—
285C2×C127—42no—
286C2×C127—41no—
287C2×C127—41no—
288C2×C127—81no—
289C2×C127—82no—
290C2×C127#178111yescharacteristic
291C2×C127#212112yes—
292C2×#534C2×C12711yescharacteristic
293C2×#352C2×C12712yescomplemented
294C2×AΣL(1,𝔽16)C2×C12712yescomplemented
295#1783C2×C12711yescharacteristic, complemented
296#1781C2×C12711yescharacteristic, complemented
297C127#1269011yescharacteristic, direct product factor, Sylow
298C23×D4C4×C12711yescharacteristic, complemented
299C23×D4C22×C12711yescharacteristic
300#534—24no—
301#534—24no—
302C2×#123—21no—
303C2×#123C4×C12711yescharacteristic, complemented
304C2×#123C22×C12711yescharacteristic
305C2×#121—21no—
306C2×#121—21no—
307C2×#120—21no—
308C2×#120C4×C12711yescharacteristic, complemented
309C2×#120C22×C12711yescharacteristic
310C22×C22⋊C4C4×C12711yescharacteristic, complemented
311C2×#100—21no—
312C2×#100C22×C12712yes—
313C2×#99—21no—
314C2×#99C22×C12712yes—
315#407—21no—
316#379—21no—
317#352—24no—
318AΣL(1,𝔽16)—24no—
319C25C2×C4×C12711yescharacteristic
320C25C2×C4×C12711yescharacteristic
321C25D4×C12711yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4—42no—
334C22×D4C2×C4×C12711yescharacteristic
335C22×D4D4×C12711yescharacteristic
336C22×D4D4×C12711yescharacteristic
337C22×D4C23×C12711yescharacteristic
338C23×C4—21no—
339C23×C4C2×C4×C12711yescharacteristic
340C23×C4D4×C12711yescharacteristic
341C2×M16—22no—
342#123—28no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123D4×C12714yes—
347#121—28no—
348#121—28no—
349#120—28no—
350#120—22no—
351#120—22no—
352#120—22no—
353#120D4×C12714yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4C2×C4×C12711yescharacteristic
366C2×C22⋊C4C2×C4×C12711yescharacteristic
367#100—22no—
368#100—22no—
369#100—42no—
370#99—22no—
371#99—22no—
372#99—42no—
373C22⋊C8—42no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24C22⋊C4×C12711yescharacteristic
389C24C22⋊C4×C12711yescharacteristic
390C24C22⋊C4×C12711yescharacteristic
391C24C22⋊C4×C12712yescomplemented
392C24C22×C4×C12711yescharacteristic
393C24C2×D4×C12711yescharacteristic
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×D4×C12712yes—
421C2×D4C2×D4×C12712yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4C22⋊C4×C12711yescharacteristic
435C22×C4C22⋊C4×C12712yescomplemented
436C22×C4C2×D4×C12711yescharacteristic
437M16—42no—
438M16—42no—
439C2×C8—42no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—84no—
452C22⋊C4C22⋊C4×C12712yescomplemented
453C22⋊C4C22⋊C4×C12712yescomplemented
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23#99×C12712yes—
494C23#99×C12712yes—
495C23C2×C22⋊C4×C12711yescharacteristic
496C23C2×C22⋊C4×C12711yescharacteristic
497C23C2×C22⋊C4×C12711yescharacteristic
498C23#120×C12712yes—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—48no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4#120×C12712yes—
528C8—82no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—88no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22#379×C12711yescharacteristic
564C22C2×#99×C12711yescharacteristic
565C22C2×#99×C12711yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—162no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—41no—
577C2—41no—
578C2—42no—
579C2—81no—
580C2—82no—
581C2C127×#178111yescharacteristic
582C2C127×#212112yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC2×#534×C127C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C23×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#352×C127C2semidirect
3. C2×AΣL(1,𝔽16)×C127C2semidirect
4. C127×#1783C2semidirect
5. C127×#1781C2semidirect
6. C23×D4×C127C4semidirect
11. C2×#123×C127C4semidirect
16. C2×#120×C127C4semidirect
18. C22×C22⋊C4×C127C4semidirect
99. C24×C127C22⋊C4semidirect
143. C22×C4×C127C22⋊C4semidirect
160. C22⋊C4×C127C22⋊C4semidirect
161. C22⋊C4×C127C22⋊C4semidirect
280. #12690C127direct (N × H)
293. C2×#352C2×C127semidirect
294. C2×AΣL(1,𝔽16)C2×C127semidirect
295. #1783C2×C127semidirect
296. #1781C2×C127semidirect
297. C127#12690direct (N × H)
298. C23×D4C4×C127semidirect
303. C2×#123C4×C127semidirect
308. C2×#120C4×C127semidirect
310. C22×C22⋊C4C4×C127semidirect
391. C24C22⋊C4×C127semidirect
435. C22×C4C22⋊C4×C127semidirect
452. C22⋊C4C22⋊C4×C127semidirect
453. C22⋊C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#534×C1272
C2×#352×C1272
C2×AΣL(1,𝔽16)×C1272
C127×#17832
C127×#17812
#12690127