Finite Group Databasebeta

← C127×#21581 #44585254

Subgroups — C127×#21581 #44585254

Order 32512.

964 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484×C127C211yesmaximal, characteristic
2C4×C22⋊Q8×C127C211yesmaximal, characteristic
3C2×C4×C4⋊C4×C127C211yesmaximal, characteristic
4C127×#1927C212yesmaximal, complemented
5C127×#1918C212yesmaximal, complemented
6C127×#1872C211yesmaximal, characteristic
7C127×#1853C211yesmaximal, characteristic
8C127×#1824C211yesmaximal, characteristic
9C127×#1776C212yesmaximal, complemented
10C127×#1763C212yesmaximal, complemented
11C127×#1749C211yesmaximal, characteristic
12C2×C22⋊Q8×C127C2211yescharacteristic
13C2×C4×Q8×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C42×C127—21no—
17C22×C42×C127C2211yescharacteristic
18#484×C127C418yescomplemented
19C2×#107×C127C2212yes—
20C2×#106×C127C2212yes—
21C2×#103×C127C2212yes—
22C2×#103×C127C2212yes—
23C2×C22⋊C8×C127C2211yescharacteristic
24C2×C22⋊C8×C127C2211yescharacteristic
25C2×#97×C127C2212yes—
26#391×C127C2211yescharacteristic
27#389×C127C2211yescharacteristic
28#387×C127—21no—
29#385×C127—21no—
30#380×C127C2211yescharacteristic
31C4×C4⋊C4×C127—21no—
32C4×C4⋊C4×C127—21no—
33C4×C4⋊C4×C127—21no—
34C4×C4⋊C4×C127C2212yes—
35C4×C4⋊C4×C127C2212yes—
36C4×C4⋊C4×C127C2211yescharacteristic
37C4×C22⋊C4×C127—21no—
38C2×#95×C127—21no—
39#340×C127C2212yes—
40#340×C127C2212yes—
41#340×C127C2212yes—
42#340×C127C2212yes—
43#336×C127C2212yes—
44C22×Q8×C127C2311yescharacteristic
45C23×C4×C127—21no—
46C23×C4×C127—21no—
47C23×C4×C127D411yescharacteristic
48C23×C4×C127D411yescharacteristic
49C23×C4×C127C2311yescharacteristic
50C22×C8×C127C2312yes—
51C22×C8×C127C2312yes—
52C22⋊Q8×C127—22no—
53C22⋊Q8×C127C2×C414yes—
54C4×Q8×C127—24no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127—21no—
57C2×C4⋊C4×C127—21no—
58C2×C4⋊C4×C127—22no—
59C2×C4⋊C4×C127—24no—
60C2×C4⋊C4×C127—21no—
61C2×C4⋊C4×C127—21no—
62C2×C4⋊C4×C127C2×C414yes—
63C2×C4⋊C4×C127C2311yescharacteristic
64C2×C4⋊C4×C127C2311yescharacteristic
65C2×C4⋊C4×C127C2312yes—
66C2×C4⋊C4×C127C2311yescharacteristic
67C2×C4⋊C4×C127C2312yes—
68C2×C22⋊C4×C127—21no—
69C2×C22⋊C4×C127—21no—
70C2×C42×C127—24no—
71C2×C42×C127—22no—
72C2×C42×C127—22no—
73C2×C42×C127—21no—
74C2×C42×C127—21no—
75C2×C42×C127—44no—
76C2×C42×C127D411yescharacteristic
77C2×C42×C127D411yescharacteristic
78C2×C42×C127C2311yescharacteristic
79C2×C42×C127C2311yescharacteristic
80#107×C127C2×C418yescomplemented
81#106×C127—24no—
82#103×C127—24no—
83#103×C127C2×C418yescomplemented
84C22⋊C8×C127—22no—
85C22⋊C8×C127C2×C414yes—
86#97×C127—24no—
87#95×C127—22no—
88#95×C127—22no—
89#95×C127—21no—
90#95×C127—21no—
91#95×C127—21no—
92#95×C127—21no—
93#95×C127—21no—
94C24×C127C2×D411yescharacteristic
95C2×Q8×C127—21no—
96C2×Q8×C127—22no—
97C2×Q8×C127—22no—
98C2×Q8×C127C22×C412yes—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—24no—
105C22×C4×C127—24no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—21no—
110C22×C4×C127—21no—
111C22×C4×C127—22no—
112C22×C4×C127—21no—
113C22×C4×C127—21no—
114C22×C4×C127—41no—
115C22×C4×C127—42no—
116C22×C4×C127—42no—
117C22×C4×C127—42no—
118C22×C4×C127—42no—
119C22×C4×C127C22×C412yes—
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123C22×C4×C127C2×D411yescharacteristic
124C22×C4×C127C2×D411yescharacteristic
125C22×C4×C127#4112yes—
126C22×C4×C127C2411yescharacteristic
127C2×C8×C127—22no—
128C2×C8×C127—22no—
129C2×C8×C127—22no—
130C2×C8×C127—22no—
131C2×C8×C127—22no—
132C2×C8×C127C22×C414yes—
133C4⋊C4×C127—24no—
134C4⋊C4×C127—24no—
135C4⋊C4×C127—24no—
136C4⋊C4×C127—22no—
137C4⋊C4×C127—22no—
138C4⋊C4×C127—21no—
139C4⋊C4×C127—22no—
140C4⋊C4×C127—22no—
141C4⋊C4×C127—42no—
142C4⋊C4×C127—42no—
143C4⋊C4×C127C22×C414yes—
144C4⋊C4×C127C22×C412yes—
145C4⋊C4×C127#4114yes—
146C22⋊C4×C127—24no—
147C22⋊C4×C127—42no—
148C42×C127—22no—
149C42×C127—22no—
150C42×C127—22no—
151C42×C127—42no—
152C42×C127—44no—
153C42×C127—42no—
154C42×C127—44no—
155C42×C127#4114yes—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—21no—
159C23×C127—21no—
160C23×C127C4×D412yes—
161C23×C127#13712yes—
162C23×C127#13712yes—
163C23×C127C22×D411yescharacteristic
164Q8×C127—24no—
165Q8×C127—42no—
166C2×C4×C127—28no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—24no—
176C2×C4×C127—24no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—24no—
180C2×C4×C127—21no—
181C2×C4×C127—24no—
182C2×C4×C127—21no—
183C2×C4×C127—21no—
184C2×C4×C127—44no—
185C2×C4×C127—48no—
186C2×C4×C127—48no—
187C2×C4×C127—41no—
188C2×C4×C127—41no—
189C2×C4×C127—42no—
190C2×C4×C127—42no—
191C2×C4×C127—41no—
192C2×C4×C127—42no—
193C2×C4×C127—41no—
194C2×C4×C127—42no—
195C2×C4×C127—41no—
196C2×C4×C127—41no—
197C2×C4×C127—41no—
198C2×C4×C127—41no—
199C2×C4×C127—41no—
200C2×C4×C127—41no—
201C2×C4×C127C4×D412yes—
202C2×C4×C127#12312yes—
203C2×C4×C127#12312yes—
204C2×C4×C127AGL(1,ℤ/8)12yes—
205C2×C4×C127C23×C411yescharacteristic
206C2×C4×C127C2×#4112yes—
207C2×C4×C127C2×#4112yes—
208C2×C4×C127C2×#4111yescharacteristic
209C8×C127—24no—
210C8×C127—42no—
211C22×C127—22no—
212C22×C127—22no—
213C22×C127—22no—
214C22×C127—22no—
215C22×C127—22no—
216C22×C127—22no—
217C22×C127#44114yes—
218C22×C127C2×C4×D411yescharacteristic
219C22×C127C2×#12311yescharacteristic
220C22×C127#52511yescharacteristic
221C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C127C2×#13711yescharacteristic
223C22×C127C2×#13711yescharacteristic
224C22×C127#57511yescharacteristic
225C4×C127—22no—
226C4×C127—22no—
227C4×C127—22no—
228C4×C127—22no—
229C4×C127—41no—
230C4×C127—44no—
231C4×C127—42no—
232C4×C127—44no—
233C4×C127—42no—
234C4×C127—81no—
235C4×C127—81no—
236C4×C127C4×#4112yes—
237C4×C127#52912yes—
238#21581C12711yesmaximal, characteristic, direct product factor, Sylow
239C2×C127—24no—
240C2×C127C4×#12311yescharacteristic
241C2×C127C2×#44111yescharacteristic
242C2×C127C4×AGL(1,ℤ/8)11yescharacteristic
243C2×C127#309211yescharacteristic
244C2×C127#309511yescharacteristic
245C2×C127#311011yescharacteristic
246C2×C127#311711yescharacteristic
247C2×#484C2×C12711yescharacteristic
248C4×C22⋊Q8C2×C12711yescharacteristic
249C2×C4×C4⋊C4C2×C12711yescharacteristic
250#1927C2×C12712yescomplemented
251#1918C2×C12712yescomplemented
252#1872C2×C12711yescharacteristic
253#1853C2×C12711yescharacteristic
254#1824C2×C12711yescharacteristic
255#1776C2×C12712yescomplemented
256#1763C2×C12712yescomplemented
257#1749C2×C12711yescharacteristic
258C127#2158111yescharacteristic, direct product factor, Sylow
259C2×C22⋊Q8C22×C12711yescharacteristic
260C2×C4×Q8C22×C12711yescharacteristic
261C22×C4⋊C4C22×C12711yescharacteristic
262C22×C4⋊C4C22×C12711yescharacteristic
263C22×C42—21no—
264C22×C42C22×C12711yescharacteristic
265#484C4×C12718yescomplemented
266C2×#107C22×C12712yes—
267C2×#106C22×C12712yes—
268C2×#103C22×C12712yes—
269C2×#103C22×C12712yes—
270C2×C22⋊C8C22×C12711yescharacteristic
271C2×C22⋊C8C22×C12711yescharacteristic
272C2×#97C22×C12712yes—
273#391C22×C12711yescharacteristic
274#389C22×C12711yescharacteristic
275#387—21no—
276#385—21no—
277#380C22×C12711yescharacteristic
278C4×C4⋊C4—21no—
279C4×C4⋊C4—21no—
280C4×C4⋊C4—21no—
281C4×C4⋊C4C22×C12712yes—
282C4×C4⋊C4C22×C12712yes—
283C4×C4⋊C4C22×C12711yescharacteristic
284C4×C22⋊C4—21no—
285C2×#95—21no—
286#340C22×C12712yes—
287#340C22×C12712yes—
288#340C22×C12712yes—
289#340C22×C12712yes—
290#336C22×C12712yes—
291C22×Q8C23×C12711yescharacteristic
292C23×C4—21no—
293C23×C4—21no—
294C23×C4D4×C12711yescharacteristic
295C23×C4D4×C12711yescharacteristic
296C23×C4C23×C12711yescharacteristic
297C22×C8C23×C12712yes—
298C22×C8C23×C12712yes—
299C22⋊Q8—22no—
300C22⋊Q8C2×C4×C12714yes—
301C4×Q8—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×C4×C12714yes—
310C2×C4⋊C4C23×C12711yescharacteristic
311C2×C4⋊C4C23×C12711yescharacteristic
312C2×C4⋊C4C23×C12712yes—
313C2×C4⋊C4C23×C12711yescharacteristic
314C2×C4⋊C4C23×C12712yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C42—24no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—44no—
323C2×C42D4×C12711yescharacteristic
324C2×C42D4×C12711yescharacteristic
325C2×C42C23×C12711yescharacteristic
326C2×C42C23×C12711yescharacteristic
327#107C2×C4×C12718yescomplemented
328#106—24no—
329#103—24no—
330#103C2×C4×C12718yescomplemented
331C22⋊C8—22no—
332C22⋊C8C2×C4×C12714yes—
333#97—24no—
334#95—22no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341C24C2×D4×C12711yescharacteristic
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8C22×C4×C12712yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4C22×C4×C12712yes—
367C22×C4C2×D4×C12711yescharacteristic
368C22×C4C2×D4×C12711yescharacteristic
369C22×C4C2×D4×C12711yescharacteristic
370C22×C4C2×D4×C12711yescharacteristic
371C22×C4C2×D4×C12711yescharacteristic
372C22×C4#41×C12712yes—
373C22×C4C24×C12711yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8C22×C4×C12714yes—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4C22×C4×C12714yes—
391C4⋊C4C22×C4×C12712yes—
392C4⋊C4#41×C12714yes—
393C22⋊C4—24no—
394C22⋊C4—42no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—42no—
399C42—44no—
400C42—42no—
401C42—44no—
402C42#41×C12714yes—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23C4×D4×C12712yes—
408C23#137×C12712yes—
409C23#137×C12712yes—
410C23C22×D4×C12711yescharacteristic
411Q8—24no—
412Q8—42no—
413C2×C4—28no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4C4×D4×C12712yes—
449C2×C4#123×C12712yes—
450C2×C4#123×C12712yes—
451C2×C4AGL(1,ℤ/8)×C12712yes—
452C2×C4C23×C4×C12711yescharacteristic
453C2×C4C2×#41×C12712yes—
454C2×C4C2×#41×C12712yes—
455C2×C4C2×#41×C12711yescharacteristic
456C8—24no—
457C8—42no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22#441×C12714yes—
465C22C2×C4×D4×C12711yescharacteristic
466C22C2×#123×C12711yescharacteristic
467C22#525×C12711yescharacteristic
468C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
469C22C2×#137×C12711yescharacteristic
470C22C2×#137×C12711yescharacteristic
471C22#575×C12711yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—41no—
477C4—44no—
478C4—44no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4C4×#41×C12712yes—
484C4#529×C12712yes—
485C2—24no—
486C2C4×#123×C12711yescharacteristic
487C2C2×#441×C12711yescharacteristic
488C2C4×AGL(1,ℤ/8)×C12711yescharacteristic
489C2C127×#309211yescharacteristic
490C2C127×#309511yescharacteristic
491C2C127×#311011yescharacteristic
492C2C127×#311711yescharacteristic
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1927C2semidirect
5. C127×#1918C2semidirect
9. C127×#1776C2semidirect
10. C127×#1763C2semidirect
18. #484×C127C4semidirect
80. #107×C127C2×C4semidirect
83. #103×C127C2×C4semidirect
238. #21581C127direct (N × H)
250. #1927C2×C127semidirect
251. #1918C2×C127semidirect
255. #1776C2×C127semidirect
256. #1763C2×C127semidirect
258. C127#21581direct (N × H)
265. #484C4×C127semidirect
327. #107C2×C4×C127semidirect
330. #103C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#484×C1272
C4×C22⋊Q8×C1272
C2×C4×C4⋊C4×C1272
C127×#19272
C127×#19182
C127×#18722
C127×#18532
C127×#18242
C127×#17762
C127×#17632
C127×#17492
#21581127