Finite Group Databasebeta

← C127×#21034 #44584707

Subgroups — C127×#21034 #44584707

Order 32512.

974 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#428×C127C211yesmaximal, characteristic
2C2×#389×C127C211yesmaximal, characteristic
3C4×#117×C127C211yesmaximal, characteristic
4C127×#1921C212yesmaximal, complemented
5C127×#1920C212yesmaximal, complemented
6C127×#1919C212yesmaximal, complemented
7C127×#1918C212yesmaximal, complemented
8C127×#1736C211yesmaximal, characteristic
9C127×#1736C211yesmaximal, characteristic
10C127×#1736C211yesmaximal, characteristic
11C127×#1728C211yesmaximal, characteristic
12C22×M16×C127C2211yescharacteristic
13C22×M16×C127C2211yescharacteristic
14C2×#117×C127C2211yescharacteristic
15C2×#117×C127C2211yescharacteristic
16C22×C4⋊C4×C127—21no—
17C22×C4⋊C4×C127C2211yescharacteristic
18C22×C4⋊C4×C127C2211yescharacteristic
19C22×C42×C127C2211yescharacteristic
20#428×C127C418yescomplemented
21C2×#107×C127C2212yes—
22C2×#106×C127C2212yes—
23#389×C127—21no—
24#389×C127—21no—
25#389×C127C2212yes—
26#389×C127C2212yes—
27C4×C4⋊C4×C127C2211yescharacteristic
28C4×C4⋊C4×C127C2211yescharacteristic
29C4×C4⋊C4×C127C2211yescharacteristic
30C4×C4⋊C4×C127C2211yescharacteristic
31C4×C22⋊C4×C127—21no—
32C4×C22⋊C4×C127—21no—
33#376×C127—21no—
34C2×#95×C127—21no—
35C43×C127—21no—
36#340×C127C2212yes—
37#340×C127C2212yes—
38#340×C127C2212yes—
39#340×C127C2212yes—
40#340×C127C2212yes—
41#340×C127C2212yes—
42#336×C127C2212yes—
43#336×C127C2212yes—
44C23×C4×C127—21no—
45C23×C4×C127—21no—
46C23×C4×C127D411yescharacteristic
47C23×C4×C127D411yescharacteristic
48C23×C4×C127C2311yescharacteristic
49C2×M16×C127—22no—
50C2×M16×C127—22no—
51C2×M16×C127—22no—
52C2×M16×C127—22no—
53C2×M16×C127—22no—
54C2×M16×C127C2×C414yes—
55C22×C8×C127C2312yes—
56C22×C8×C127C2312yes—
57#117×C127—22no—
58#117×C127—22no—
59#117×C127C2×C414yes—
60#117×C127Q814yes—
61C2×C4⋊C4×C127—21no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—22no—
64C2×C4⋊C4×C127—22no—
65C2×C4⋊C4×C127—22no—
66C2×C4⋊C4×C127—22no—
67C2×C4⋊C4×C127—21no—
68C2×C4⋊C4×C127—44no—
69C2×C4⋊C4×C127C2×C414yes—
70C2×C4⋊C4×C127C2312yes—
71C2×C4⋊C4×C127C2312yes—
72C2×C4⋊C4×C127C2311yescharacteristic
73C2×C4⋊C4×C127C2311yescharacteristic
74C2×C4⋊C4×C127C2311yescharacteristic
75C2×C4⋊C4×C127C2311yescharacteristic
76C2×C22⋊C4×C127—21no—
77C2×C22⋊C4×C127—21no—
78C2×C42×C127—21no—
79C2×C42×C127—21no—
80C2×C42×C127—22no—
81C2×C42×C127—22no—
82C2×C42×C127—21no—
83C2×C42×C127—21no—
84C2×C42×C127—21no—
85C2×C42×C127—21no—
86C2×C42×C127D411yescharacteristic
87C2×C42×C127D411yescharacteristic
88C2×C42×C127C2311yescharacteristic
89C2×C42×C127C2311yescharacteristic
90#107×C127C2×C418yescomplemented
91#106×C127C2×C418yescomplemented
92#95×C127—22no—
93#95×C127—22no—
94#95×C127—21no—
95#95×C127—21no—
96#95×C127—21no—
97#95×C127—21no—
98C24×C127C2×D411yescharacteristic
99C22×C4×C127—24no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105C22×C4×C127—21no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—41no—
111C22×C4×C127—41no—
112C22×C4×C127—42no—
113C22×C4×C127—42no—
114C22×C4×C127—42no—
115C22×C4×C127—42no—
116C22×C4×C127—42no—
117C22×C4×C127C22×C412yes—
118C22×C4×C127C2×D411yescharacteristic
119C22×C4×C127C2×D411yescharacteristic
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123C22×C4×C127C2×Q812yes—
124C22×C4×C127#4112yes—
125C22×C4×C127#4112yes—
126C22×C4×C127#4112yes—
127C22×C4×C127#4112yes—
128C22×C4×C127C2411yescharacteristic
129M16×C127—28no—
130M16×C127—44no—
131C2×C8×C127—22no—
132C2×C8×C127—22no—
133C2×C8×C127—22no—
134C2×C8×C127—22no—
135C2×C8×C127—22no—
136C2×C8×C127C22×C414yes—
137C4⋊C4×C127—24no—
138C4⋊C4×C127—22no—
139C4⋊C4×C127—21no—
140C4⋊C4×C127—21no—
141C4⋊C4×C127—21no—
142C4⋊C4×C127—21no—
143C4⋊C4×C127—22no—
144C4⋊C4×C127—22no—
145C4⋊C4×C127—44no—
146C4⋊C4×C127—42no—
147C4⋊C4×C127—44no—
148C4⋊C4×C127C22×C412yes—
149C4⋊C4×C127C22×C414yes—
150C4⋊C4×C127C22×C412yes—
151C4⋊C4×C127C2×Q812yes—
152C4⋊C4×C127C2×Q812yes—
153C22⋊C4×C127—24no—
154C22⋊C4×C127—24no—
155C42×C127—24no—
156C42×C127—24no—
157C42×C127—22no—
158C42×C127—22no—
159C42×C127—21no—
160C42×C127—21no—
161C42×C127—21no—
162C42×C127—21no—
163C42×C127—42no—
164C42×C127—42no—
165C42×C127—42no—
166C42×C127—42no—
167C23×C127—21no—
168C23×C127—21no—
169C23×C127—21no—
170C23×C127—21no—
171C23×C127C4×D412yes—
172C23×C127C22⋊Q812yes—
173C23×C127#12312yes—
174C23×C127C22×D411yescharacteristic
175C2×C4×C127—21no—
176C2×C4×C127—21no—
177C2×C4×C127—21no—
178C2×C4×C127—21no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—24no—
182C2×C4×C127—22no—
183C2×C4×C127—22no—
184C2×C4×C127—21no—
185C2×C4×C127—21no—
186C2×C4×C127—21no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—44no—
191C2×C4×C127—44no—
192C2×C4×C127—48no—
193C2×C4×C127—41no—
194C2×C4×C127—41no—
195C2×C4×C127—42no—
196C2×C4×C127—41no—
197C2×C4×C127—41no—
198C2×C4×C127—41no—
199C2×C4×C127—41no—
200C2×C4×C127—41no—
201C2×C4×C127—41no—
202C2×C4×C127—42no—
203C2×C4×C127—42no—
204C2×C4×C127—42no—
205C2×C4×C127—42no—
206C2×C4×C127—84no—
207C2×C4×C127#11714yes—
208C2×C4×C127C4×D412yes—
209C2×C4×C127C4×Q814yes—
210C2×C4×C127C22⋊Q812yes—
211C2×C4×C127#12312yes—
212C2×C4×C127#12514yes—
213C2×C4×C127#12614yes—
214C2×C4×C127AGL(1,ℤ/8)12yes—
215C2×C4×C127#13712yes—
216C2×C4×C127C23×C411yescharacteristic
217C2×C4×C127C22×Q811yescharacteristic
218C2×C4×C127C2×#4111yescharacteristic
219C2×C4×C127C2×#4111yescharacteristic
220C2×C4×C127C2×#4111yescharacteristic
221C2×C4×C127C2×#4111yescharacteristic
222C8×C127—24no—
223C8×C127—42no—
224C22×C127—22no—
225C22×C127—22no—
226C22×C127—22no—
227C22×C127—22no—
228C22×C127—22no—
229C22×C127—22no—
230C22×C127#38714yes—
231C22×C127C2×C4×D411yescharacteristic
232C22×C127C2×C22⋊Q811yescharacteristic
233C22×C127C2×#12311yescharacteristic
234C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
235C22×C127C2×#13711yescharacteristic
236C22×C127#57511yescharacteristic
237C22×C127#57511yescharacteristic
238C4×C127—22no—
239C4×C127—22no—
240C4×C127—42no—
241C4×C127—42no—
242C4×C127—41no—
243C4×C127—41no—
244C4×C127—42no—
245C4×C127—44no—
246C4×C127—42no—
247C4×C127—82no—
248C4×C127#38714yes—
249C4×C127C2×#11711yescharacteristic
250C4×C127C2×C4×Q811yescharacteristic
251C4×C127C2×#12511yescharacteristic
252C4×C127C2×#12611yescharacteristic
253#21034C12711yesmaximal, characteristic, direct product factor, Sylow
254C2×C127—24no—
255C2×C127C2×#38711yescharacteristic
256C2×C127C4×AGL(1,ℤ/8)11yescharacteristic
257C2×C127C4×#13711yescharacteristic
258C2×C127#307911yescharacteristic
259C2×C127#308011yescharacteristic
260C2×C127#309411yescharacteristic
261C2×C127#309511yescharacteristic
262C2×#428C2×C12711yescharacteristic
263C2×#389C2×C12711yescharacteristic
264C4×#117C2×C12711yescharacteristic
265#1921C2×C12712yescomplemented
266#1920C2×C12712yescomplemented
267#1919C2×C12712yescomplemented
268#1918C2×C12712yescomplemented
269#1736C2×C12711yescharacteristic
270#1736C2×C12711yescharacteristic
271#1736C2×C12711yescharacteristic
272#1728C2×C12711yescharacteristic
273C127#2103411yescharacteristic, direct product factor, Sylow
274C22×M16C22×C12711yescharacteristic
275C22×M16C22×C12711yescharacteristic
276C2×#117C22×C12711yescharacteristic
277C2×#117C22×C12711yescharacteristic
278C22×C4⋊C4—21no—
279C22×C4⋊C4C22×C12711yescharacteristic
280C22×C4⋊C4C22×C12711yescharacteristic
281C22×C42C22×C12711yescharacteristic
282#428C4×C12718yescomplemented
283C2×#107C22×C12712yes—
284C2×#106C22×C12712yes—
285#389—21no—
286#389—21no—
287#389C22×C12712yes—
288#389C22×C12712yes—
289C4×C4⋊C4C22×C12711yescharacteristic
290C4×C4⋊C4C22×C12711yescharacteristic
291C4×C4⋊C4C22×C12711yescharacteristic
292C4×C4⋊C4C22×C12711yescharacteristic
293C4×C22⋊C4—21no—
294C4×C22⋊C4—21no—
295#376—21no—
296C2×#95—21no—
297C43—21no—
298#340C22×C12712yes—
299#340C22×C12712yes—
300#340C22×C12712yes—
301#340C22×C12712yes—
302#340C22×C12712yes—
303#340C22×C12712yes—
304#336C22×C12712yes—
305#336C22×C12712yes—
306C23×C4—21no—
307C23×C4—21no—
308C23×C4D4×C12711yescharacteristic
309C23×C4D4×C12711yescharacteristic
310C23×C4C23×C12711yescharacteristic
311C2×M16—22no—
312C2×M16—22no—
313C2×M16—22no—
314C2×M16—22no—
315C2×M16—22no—
316C2×M16C2×C4×C12714yes—
317C22×C8C23×C12712yes—
318C22×C8C23×C12712yes—
319#117—22no—
320#117—22no—
321#117C2×C4×C12714yes—
322#117Q8×C12714yes—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2×C4×C12714yes—
332C2×C4⋊C4C23×C12712yes—
333C2×C4⋊C4C23×C12712yes—
334C2×C4⋊C4C23×C12711yescharacteristic
335C2×C4⋊C4C23×C12711yescharacteristic
336C2×C4⋊C4C23×C12711yescharacteristic
337C2×C4⋊C4C23×C12711yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42D4×C12711yescharacteristic
349C2×C42D4×C12711yescharacteristic
350C2×C42C23×C12711yescharacteristic
351C2×C42C23×C12711yescharacteristic
352#107C2×C4×C12718yescomplemented
353#106C2×C4×C12718yescomplemented
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360C24C2×D4×C12711yescharacteristic
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C22×C4×C12712yes—
380C22×C4C2×D4×C12711yescharacteristic
381C22×C4C2×D4×C12711yescharacteristic
382C22×C4C2×D4×C12711yescharacteristic
383C22×C4C2×D4×C12711yescharacteristic
384C22×C4C2×D4×C12711yescharacteristic
385C22×C4C2×Q8×C12712yes—
386C22×C4#41×C12712yes—
387C22×C4#41×C12712yes—
388C22×C4#41×C12712yes—
389C22×C4#41×C12712yes—
390C22×C4C24×C12711yescharacteristic
391M16—28no—
392M16—44no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8C22×C4×C12714yes—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4C22×C4×C12712yes—
411C4⋊C4C22×C4×C12714yes—
412C4⋊C4C22×C4×C12712yes—
413C4⋊C4C2×Q8×C12712yes—
414C4⋊C4C2×Q8×C12712yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C42—24no—
418C42—24no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C4×D4×C12712yes—
434C23C22⋊Q8×C12712yes—
435C23#123×C12712yes—
436C23C22×D4×C12711yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4#117×C12714yes—
470C2×C4C4×D4×C12712yes—
471C2×C4C4×Q8×C12714yes—
472C2×C4C22⋊Q8×C12712yes—
473C2×C4#123×C12712yes—
474C2×C4#125×C12714yes—
475C2×C4#126×C12714yes—
476C2×C4AGL(1,ℤ/8)×C12712yes—
477C2×C4#137×C12712yes—
478C2×C4C23×C4×C12711yescharacteristic
479C2×C4C22×Q8×C12711yescharacteristic
480C2×C4C2×#41×C12711yescharacteristic
481C2×C4C2×#41×C12711yescharacteristic
482C2×C4C2×#41×C12711yescharacteristic
483C2×C4C2×#41×C12711yescharacteristic
484C8—24no—
485C8—42no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22#387×C12714yes—
493C22C2×C4×D4×C12711yescharacteristic
494C22C2×C22⋊Q8×C12711yescharacteristic
495C22C2×#123×C12711yescharacteristic
496C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
497C22C2×#137×C12711yescharacteristic
498C22#575×C12711yescharacteristic
499C22#575×C12711yescharacteristic
500C4—22no—
501C4—22no—
502C4—42no—
503C4—41no—
504C4—41no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—82no—
510C4#387×C12714yes—
511C4C2×#117×C12711yescharacteristic
512C4C2×C4×Q8×C12711yescharacteristic
513C4C2×#125×C12711yescharacteristic
514C4C2×#126×C12711yescharacteristic
515C2—24no—
516C2C2×#387×C12711yescharacteristic
517C2C4×AGL(1,ℤ/8)×C12711yescharacteristic
518C2C4×#137×C12711yescharacteristic
519C2C127×#307911yescharacteristic
520C2C127×#308011yescharacteristic
521C2C127×#309411yescharacteristic
522C2C127×#309511yescharacteristic
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×#1921C2semidirect
5. C127×#1920C2semidirect
6. C127×#1919C2semidirect
7. C127×#1918C2semidirect
20. #428×C127C4semidirect
90. #107×C127C2×C4semidirect
91. #106×C127C2×C4semidirect
253. #21034C127direct (N × H)
265. #1921C2×C127semidirect
266. #1920C2×C127semidirect
267. #1919C2×C127semidirect
268. #1918C2×C127semidirect
273. C127#21034direct (N × H)
282. #428C4×C127semidirect
352. #107C2×C4×C127semidirect
353. #106C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#428×C1272
C2×#389×C1272
C4×#117×C1272
C127×#19212
C127×#19202
C127×#19192
C127×#19182
C127×#17362
C127×#17282
#21034127