Finite Group Databasebeta

← C127×#20597 #44584270

Subgroups — C127×#20597 #44584270

Order 32512.

3998 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C127C211yesmaximal, characteristic
2C2×#416×C127C214yesmaximal, complemented
3C22×C22⋊C8×C127C211yesmaximal, characteristic, complemented
4C127×#2280C211yesmaximal, characteristic, complemented
5C127×#1788C214yesmaximal, complemented
6C127×#1787C212yesmaximal, complemented
7C127×#1786C212yesmaximal, complemented
8C22×#41×C127—216no—
9C22×#41×C127—24no—
10C22×#41×C127—21no—
11C22×#41×C127—21no—
12C22×#41×C127C418yescomplemented
13C22×#41×C127C2212yes—
14C22×#41×C127C2212yes—
15C23×Q8×C127C2211yescharacteristic
16C23×D4×C127—21no—
17C23×D4×C127C2211yescharacteristic
18C24×C4×C127—21no—
19C24×C4×C127C2211yescharacteristic
20C23×C8×C127C2212yescomplemented
21C2×#117×C127C2212yescomplemented
22#416×C127—216no—
23C2×#103×C127C2218yescomplemented
24C2×#102×C127C2218yescomplemented
25C2×C22⋊C8×C127—24no—
26C2×C22⋊C8×C127C2212yes—
27C2×C22⋊C8×C127C2212yescomplemented
28#381×C127—21no—
29#380×C127C2212yes—
30#380×C127C2212yescomplemented
31#379×C127—21no—
32C4×C22⋊C4×C127—21no—
33C4×C22⋊C4×C127—21no—
34C25×C127—21no—
35C25×C127D411yescharacteristic
36C2×#41×C127—216no—
37C2×#41×C127—264no—
38C2×#41×C127—216no—
39C2×#41×C127—264no—
40C2×#41×C127—28no—
41C2×#41×C127—28no—
42C2×#41×C127—28no—
43C2×#41×C127—28no—
44C2×#41×C127—24no—
45C2×#41×C127—24no—
46C2×#41×C127—28no—
47C2×#41×C127C2×C418yes—
48C2×#41×C127C2×C418yes—
49C2×#41×C127D4116yes—
50C22×Q8×C127—28no—
51C22×Q8×C127—22no—
52C22×Q8×C127—21no—
53C22×Q8×C127C2×C414yes—
54C22×Q8×C127C2312yes—
55C22×D4×C127—216no—
56C22×D4×C127—28no—
57C22×D4×C127—22no—
58C22×D4×C127—22no—
59C22×D4×C127—28no—
60C22×D4×C127—21no—
61C22×D4×C127—22no—
62C22×D4×C127C2×C414yes—
63C22×D4×C127C2312yes—
64C23×C4×C127—216no—
65C23×C4×C127—21no—
66C23×C4×C127—28no—
67C23×C4×C127—22no—
68C23×C4×C127—28no—
69C23×C4×C127—22no—
70C23×C4×C127C2×C414yes—
71C23×C4×C127D411yescharacteristic
72C23×C4×C127D411yescharacteristic
73C23×C4×C127D411yescharacteristic
74C23×C4×C127D414yes—
75C23×C4×C127C2311yescharacteristic
76C23×C4×C127C2311yescharacteristic
77C23×C4×C127C2311yescharacteristic
78C22×C8×C127—24no—
79C22×C8×C127—28no—
80C22×C8×C127C2314yes—
81#117×C127—28no—
82C2×C4⋊C4×C127C2314yes—
83C2×C22⋊C4×C127—22no—
84C2×C22⋊C4×C127—22no—
85C2×C22⋊C4×C127—22no—
86C2×C42×C127—22no—
87#103×C127—216no—
88#102×C127—216no—
89C22⋊C8×C127—24no—
90C22⋊C8×C127—24no—
91#95×C127—22no—
92#95×C127—22no—
93C24×C127—28no—
94C24×C127—24no—
95C24×C127—24no—
96C24×C127—24no—
97C24×C127—22no—
98C24×C127—42no—
99C24×C127—44no—
100C24×C127—44no—
101C24×C127C22⋊C414yes—
102C24×C127C2×D411yescharacteristic
103C24×C127C2×D411yescharacteristic
104C24×C127C2×D411yescharacteristic
105#41×C127—264no—
106#41×C127—232no—
107#41×C127—264no—
108#41×C127—216no—
109#41×C127—216no—
110#41×C127—216no—
111#41×C127—232no—
112#41×C127C22⋊C4132yescomplemented
113C2×Q8×C127—216no—
114C2×Q8×C127—28no—
115C2×Q8×C127—216no—
116C2×Q8×C127—24no—
117C2×Q8×C127—22no—
118C2×Q8×C127—24no—
119C2×Q8×C127C22×C414yes—
120C2×Q8×C127C2×D418yes—
121C2×D4×C127—264no—
122C2×D4×C127—216no—
123C2×D4×C127—28no—
124C2×D4×C127—216no—
125C2×D4×C127—28no—
126C2×D4×C127—28no—
127C2×D4×C127—216no—
128C2×D4×C127—216no—
129C2×D4×C127—24no—
130C2×D4×C127—24no—
131C2×D4×C127—22no—
132C2×D4×C127C22×C414yes—
133C2×D4×C127C2×D418yes—
134C22×C4×C127—28no—
135C22×C4×C127—264no—
136C22×C4×C127—216no—
137C22×C4×C127—24no—
138C22×C4×C127—24no—
139C22×C4×C127—22no—
140C22×C4×C127—28no—
141C22×C4×C127—28no—
142C22×C4×C127—216no—
143C22×C4×C127—24no—
144C22×C4×C127—216no—
145C22×C4×C127—24no—
146C22×C4×C127—216no—
147C22×C4×C127—24no—
148C22×C4×C127—24no—
149C22×C4×C127—24no—
150C22×C4×C127C22⋊C414yes—
151C22×C4×C127C22⋊C414yes—
152C22×C4×C127C22⋊C414yes—
153C22×C4×C127C22×C412yes—
154C22×C4×C127C22×C412yes—
155C22×C4×C127C22×C412yes—
156C22×C4×C127C2×D411yescharacteristic
157C22×C4×C127C2×D411yescharacteristic
158C22×C4×C127C2×D412yes—
159C22×C4×C127C2×D412yes—
160C22×C4×C127C2×D418yes—
161C22×C4×C127C2×D412yes—
162C22×C4×C127C2×D411yescharacteristic
163C22×C4×C127C2411yescharacteristic
164C2×C8×C127—24no—
165C2×C8×C127—28no—
166C2×C8×C127—48no—
167C4⋊C4×C127—28no—
168C22⋊C4×C127—44no—
169C22⋊C4×C127—44no—
170C22⋊C4×C127—44no—
171C42×C127—44no—
172C23×C127—22no—
173C23×C127—22no—
174C23×C127—22no—
175C23×C127—22no—
176C23×C127—216no—
177C23×C127—24no—
178C23×C127—22no—
179C23×C127—28no—
180C23×C127—22no—
181C23×C127—22no—
182C23×C127—28no—
183C23×C127—28no—
184C23×C127—44no—
185C23×C127—416no—
186C23×C127—48no—
187C23×C127—432no—
188C23×C127—44no—
189C23×C127—44no—
190C23×C127—48no—
191C23×C127C2×C22⋊C412yes—
192C23×C127C2×C22⋊C412yes—
193C23×C127C2×C22⋊C412yes—
194C23×C127#12014yes—
195C23×C127#13514yes—
196C23×C127C22×D411yescharacteristic
197Q8×C127—216no—
198Q8×C127—24no—
199Q8×C127—28no—
200Q8×C127C2×C22⋊C418yes—
201D4×C127—232no—
202D4×C127—216no—
203D4×C127—232no—
204D4×C127—24no—
205D4×C127—28no—
206D4×C127C2×C22⋊C418yes—
207C2×C4×C127—216no—
208C2×C4×C127—232no—
209C2×C4×C127—24no—
210C2×C4×C127—22no—
211C2×C4×C127—28no—
212C2×C4×C127—22no—
213C2×C4×C127—28no—
214C2×C4×C127—24no—
215C2×C4×C127—24no—
216C2×C4×C127—24no—
217C2×C4×C127—24no—
218C2×C4×C127—28no—
219C2×C4×C127—232no—
220C2×C4×C127—28no—
221C2×C4×C127—44no—
222C2×C4×C127—44no—
223C2×C4×C127—44no—
224C2×C4×C127C2×C22⋊C412yes—
225C2×C4×C127C2×C22⋊C412yes—
226C2×C4×C127C2×C22⋊C418yes—
227C2×C4×C127C2×C22⋊C412yes—
228C2×C4×C127#12018yes—
229C2×C4×C127#12014yes—
230C2×C4×C127C23×C411yescharacteristic
231C2×C4×C127C22×D414yes—
232C2×C4×C127C22×D411yescharacteristic
233C8×C127—44no—
234C22×C127—22no—
235C22×C127—22no—
236C22×C127—22no—
237C22×C127—28no—
238C22×C127—28no—
239C22×C127—22no—
240C22×C127—22no—
241C22×C127—22no—
242C22×C127—22no—
243C22×C127—28no—
244C22×C127—416no—
245C22×C127—432no—
246C22×C127—44no—
247C22×C127—44no—
248C22×C127—44no—
249C22×C127—416no—
250C22×C127#37914yes—
251C22×C127#41614yescomplemented
252C22×C127C22×C22⋊C411yescharacteristic
253C22×C127C2×#12012yes—
254C22×C127C2×#13512yes—
255C22×C127#57512yes—
256C4×C127—22no—
257C4×C127—28no—
258C4×C127—22no—
259C4×C127—84no—
260C4×C127#37914yes—
261C4×C127C22×C22⋊C412yes—
262C4×C127C2×#12012yes—
263#20597C12711yesmaximal, characteristic, direct product factor, Sylow
264C2×C127—24no—
265C2×C127—24no—
266C2×C127—42no—
267C2×C127—48no—
268C2×C127C2×#37911yescharacteristic
269C2×C127C2×#41611yescharacteristic
270C2×C127#289611yescharacteristic
271C2×C127#300214yes—
272C23×#41C2×C12711yescharacteristic
273C2×#416C2×C12714yescomplemented
274C22×C22⋊C8C2×C12711yescharacteristic, complemented
275#2280C2×C12711yescharacteristic, complemented
276#1788C2×C12714yescomplemented
277#1787C2×C12712yescomplemented
278#1786C2×C12712yescomplemented
279C127#2059711yescharacteristic, direct product factor, Sylow
280C22×#41—216no—
281C22×#41—24no—
282C22×#41—21no—
283C22×#41—21no—
284C22×#41C4×C12718yescomplemented
285C22×#41C22×C12712yes—
286C22×#41C22×C12712yes—
287C23×Q8C22×C12711yescharacteristic
288C23×D4—21no—
289C23×D4C22×C12711yescharacteristic
290C24×C4—21no—
291C24×C4C22×C12711yescharacteristic
292C23×C8C22×C12712yescomplemented
293C2×#117C22×C12712yescomplemented
294#416—216no—
295C2×#103C22×C12718yescomplemented
296C2×#102C22×C12718yescomplemented
297C2×C22⋊C8—24no—
298C2×C22⋊C8C22×C12712yes—
299C2×C22⋊C8C22×C12712yescomplemented
300#381—21no—
301#380C22×C12712yes—
302#380C22×C12712yescomplemented
303#379—21no—
304C4×C22⋊C4—21no—
305C4×C22⋊C4—21no—
306C25—21no—
307C25D4×C12711yescharacteristic
308C2×#41—216no—
309C2×#41—264no—
310C2×#41—216no—
311C2×#41—264no—
312C2×#41—28no—
313C2×#41—28no—
314C2×#41—28no—
315C2×#41—28no—
316C2×#41—24no—
317C2×#41—24no—
318C2×#41—28no—
319C2×#41C2×C4×C12718yes—
320C2×#41C2×C4×C12718yes—
321C2×#41D4×C127116yes—
322C22×Q8—28no—
323C22×Q8—22no—
324C22×Q8—21no—
325C22×Q8C2×C4×C12714yes—
326C22×Q8C23×C12712yes—
327C22×D4—216no—
328C22×D4—28no—
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—28no—
332C22×D4—21no—
333C22×D4—22no—
334C22×D4C2×C4×C12714yes—
335C22×D4C23×C12712yes—
336C23×C4—216no—
337C23×C4—21no—
338C23×C4—28no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—28no—
342C23×C4C2×C4×C12714yes—
343C23×C4D4×C12711yescharacteristic
344C23×C4D4×C12711yescharacteristic
345C23×C4D4×C12711yescharacteristic
346C23×C4D4×C12714yes—
347C23×C4C23×C12711yescharacteristic
348C23×C4C23×C12711yescharacteristic
349C23×C4C23×C12711yescharacteristic
350C22×C8—24no—
351C22×C8—28no—
352C22×C8C23×C12714yes—
353#117—28no—
354C2×C4⋊C4C23×C12714yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C42—22no—
359#103—216no—
360#102—216no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363#95—22no—
364#95—22no—
365C24—28no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—42no—
371C24—44no—
372C24—44no—
373C24C22⋊C4×C12714yes—
374C24C2×D4×C12711yescharacteristic
375C24C2×D4×C12711yescharacteristic
376C24C2×D4×C12711yescharacteristic
377#41—264no—
378#41—232no—
379#41—264no—
380#41—216no—
381#41—216no—
382#41—216no—
383#41—232no—
384#41C22⋊C4×C127132yescomplemented
385C2×Q8—216no—
386C2×Q8—28no—
387C2×Q8—216no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8C22×C4×C12714yes—
392C2×Q8C2×D4×C12718yes—
393C2×D4—264no—
394C2×D4—216no—
395C2×D4—28no—
396C2×D4—216no—
397C2×D4—28no—
398C2×D4—28no—
399C2×D4—216no—
400C2×D4—216no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4C22×C4×C12714yes—
405C2×D4C2×D4×C12718yes—
406C22×C4—28no—
407C22×C4—264no—
408C22×C4—216no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—216no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4C22⋊C4×C12714yes—
423C22×C4C22⋊C4×C12714yes—
424C22×C4C22⋊C4×C12714yes—
425C22×C4C22×C4×C12712yes—
426C22×C4C22×C4×C12712yes—
427C22×C4C22×C4×C12712yes—
428C22×C4C2×D4×C12711yescharacteristic
429C22×C4C2×D4×C12711yescharacteristic
430C22×C4C2×D4×C12712yes—
431C22×C4C2×D4×C12712yes—
432C22×C4C2×D4×C12711yescharacteristic
433C22×C4C2×D4×C12718yes—
434C22×C4C2×D4×C12712yes—
435C22×C4C24×C12711yescharacteristic
436C2×C8—24no—
437C2×C8—28no—
438C2×C8—48no—
439C4⋊C4—28no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C42—44no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—216no—
449C23—24no—
450C23—22no—
451C23—28no—
452C23—22no—
453C23—22no—
454C23—28no—
455C23—28no—
456C23—44no—
457C23—416no—
458C23—48no—
459C23—432no—
460C23—44no—
461C23—44no—
462C23—48no—
463C23C2×C22⋊C4×C12712yes—
464C23C2×C22⋊C4×C12712yes—
465C23C2×C22⋊C4×C12712yes—
466C23#120×C12714yes—
467C23#135×C12714yes—
468C23C22×D4×C12711yescharacteristic
469Q8—216no—
470Q8—24no—
471Q8—28no—
472Q8C2×C22⋊C4×C12718yes—
473D4—232no—
474D4—216no—
475D4—232no—
476D4—24no—
477D4—28no—
478D4C2×C22⋊C4×C12718yes—
479C2×C4—216no—
480C2×C4—232no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—232no—
492C2×C4—28no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4C2×C22⋊C4×C12712yes—
497C2×C4C2×C22⋊C4×C12712yes—
498C2×C4C2×C22⋊C4×C12718yes—
499C2×C4C2×C22⋊C4×C12712yes—
500C2×C4#120×C12718yes—
501C2×C4#120×C12714yes—
502C2×C4C23×C4×C12711yescharacteristic
503C2×C4C22×D4×C12714yes—
504C2×C4C22×D4×C12711yescharacteristic
505C8—44no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—28no—
510C22—28no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—28no—
516C22—416no—
517C22—432no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—416no—
522C22#379×C12714yes—
523C22#416×C12714yescomplemented
524C22C22×C22⋊C4×C12711yescharacteristic
525C22C2×#120×C12712yes—
526C22C2×#135×C12712yes—
527C22#575×C12712yes—
528C4—22no—
529C4—28no—
530C4—22no—
531C4—84no—
532C4#379×C12714yes—
533C4C22×C22⋊C4×C12712yes—
534C4C2×#120×C12712yes—
535C2—24no—
536C2—24no—
537C2—42no—
538C2—48no—
539C2C2×#379×C12711yescharacteristic
540C2C2×#416×C12711yescharacteristic
541C2C127×#289611yescharacteristic
542C2C127×#300214yes—
subgroupG/H
centerC22×C4×C127C2×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C22×D4
prime radicalC23×#41×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#416×C127C2semidirect
3. C22×C22⋊C8×C127C2semidirect
4. C127×#2280C2semidirect
5. C127×#1788C2semidirect
6. C127×#1787C2semidirect
7. C127×#1786C2semidirect
12. C22×#41×C127C4semidirect
20. C23×C8×C127C22semidirect
21. C2×#117×C127C22semidirect
23. C2×#103×C127C22semidirect
24. C2×#102×C127C22semidirect
27. C2×C22⋊C8×C127C22semidirect
30. #380×C127C22semidirect
112. #41×C127C22⋊C4semidirect
251. C22×C127#416semidirect
263. #20597C127direct (N × H)
273. C2×#416C2×C127semidirect
274. C22×C22⋊C8C2×C127semidirect
275. #2280C2×C127semidirect
276. #1788C2×C127semidirect
277. #1787C2×C127semidirect
278. #1786C2×C127semidirect
279. C127#20597direct (N × H)
284. C22×#41C4×C127semidirect
292. C23×C8C22×C127semidirect
293. C2×#117C22×C127semidirect
295. C2×#103C22×C127semidirect
296. C2×#102C22×C127semidirect
299. C2×C22⋊C8C22×C127semidirect
302. #380C22×C127semidirect
384. #41C22⋊C4×C127semidirect
523. C22#416×C127semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C2×#416×C1272
C22×C22⋊C8×C1272
C127×#22802
C127×#17882
C127×#17872
C127×#17862
#20597127