Finite Group Databasebeta

← C127×#21613 #44585286

Subgroups — C127×#21613 #44585286

Order 32512.

988 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474×C127C211yesmaximal, characteristic
2C4×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
3C42×D4×C127C211yesmaximal, characteristic
4C127×#1988C211yesmaximal, characteristic, complemented
5C127×#1948C211yesmaximal, characteristic, complemented
6C127×#1921C212yesmaximal, complemented
7C127×#1864C212yesmaximal
8C127×#1802C211yesmaximal, characteristic
9C127×#1776C212yesmaximal, complemented
10C127×#1767C211yesmaximal, characteristic, complemented
11C127×#1762C212yesmaximal
12C2×C4⋊Q8×C127C2211yescharacteristic
13C2×C4×Q8×C127—21no—
14C2×C4×D4×C127C2211yescharacteristic
15C2×C4×D4×C127C2211yescharacteristic
16C2×C4×D4×C127C2211yescharacteristic
17C22×C42×C127—21no—
18#474×C127C418yescomplemented
19C2×#107×C127C2212yes—
20C2×#106×C127C2212yes—
21C2×C4⋊C8×C127C2211yescharacteristic
22C2×C4⋊C8×C127C2211yescharacteristic
23C2×#102×C127C2212yes—
24C2×#102×C127C2212yes—
25C2×#97×C127C2212yes—
26#391×C127—21no—
27#389×C127—21no—
28#389×C127C2212yes—
29#384×C127C2211yescharacteristic
30C4×C4⋊C4×C127—21no—
31C4×C4⋊C4×C127C2211yescharacteristic
32C4×C4⋊C4×C127C2211yescharacteristic
33C4×C4⋊C4×C127C2211yescharacteristic
34C4×C4⋊C4×C127C2212yes—
35C4×C22⋊C4×C127—21no—
36C4×C22⋊C4×C127—21no—
37C4×C22⋊C4×C127—21no—
38C43×C127C2211yescharacteristic
39#340×C127C2212yes—
40#340×C127C2212yes—
41#340×C127C2212yes—
42#340×C127C2212yes—
43#336×C127C2212yes—
44C22×Q8×C127—21no—
45C22×D4×C127C2311yescharacteristic
46C23×C4×C127—21no—
47C23×C4×C127—21no—
48C23×C4×C127—21no—
49C22×C8×C127C2312yes—
50C22×C8×C127C2312yes—
51C4⋊Q8×C127—22no—
52C4⋊Q8×C127C2×C414yescomplemented
53C4×Q8×C127—44no—
54C4×D4×C127—22no—
55C4×D4×C127—22no—
56C4×D4×C127—22no—
57C4×D4×C127—22no—
58C4×D4×C127C2×C414yes—
59C4×D4×C127Q814yescomplemented
60C2×C4⋊C4×C127—21no—
61C2×C4⋊C4×C127—21no—
62C2×C4⋊C4×C127—22no—
63C2×C4⋊C4×C127—21no—
64C2×C4⋊C4×C127C2312yes—
65C2×C4⋊C4×C127C2311yescharacteristic
66C2×C4⋊C4×C127C2312yes—
67C2×C4⋊C4×C127C2311yescharacteristic
68C2×C4⋊C4×C127C2311yescharacteristic
69C2×C22⋊C4×C127—21no—
70C2×C22⋊C4×C127—21no—
71C2×C22⋊C4×C127—21no—
72C2×C42×C127—21no—
73C2×C42×C127—21no—
74C2×C42×C127—21no—
75C2×C42×C127—22no—
76C2×C42×C127—44no—
77C2×C42×C127D411yescharacteristic
78C2×C42×C127D411yescharacteristic
79C2×C42×C127D411yescharacteristic
80C2×C42×C127D411yescharacteristic
81C2×C42×C127C2311yescharacteristic
82C2×C42×C127C2311yescharacteristic
83C2×C42×C127C2311yescharacteristic
84#107×C127C2×C418yescomplemented
85#106×C127—24no—
86C4⋊C8×C127—22no—
87C4⋊C8×C127C2×C414yescomplemented
88#102×C127—24no—
89#102×C127C2×C418yes—
90#97×C127—24no—
91#95×C127—21no—
92#95×C127—21no—
93#95×C127—21no—
94#95×C127—22no—
95#95×C127—22no—
96C24×C127—21no—
97C2×Q8×C127—24no—
98C2×Q8×C127—42no—
99C2×Q8×C127—42no—
100C2×D4×C127—21no—
101C2×D4×C127—21no—
102C2×D4×C127—21no—
103C2×D4×C127C22×C412yes—
104C2×D4×C127C2×Q812yes—
105C2×D4×C127#4112yes—
106C22×C4×C127—24no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—21no—
110C22×C4×C127—22no—
111C22×C4×C127—24no—
112C22×C4×C127—24no—
113C22×C4×C127—22no—
114C22×C4×C127—42no—
115C22×C4×C127—42no—
116C22×C4×C127—42no—
117C22×C4×C127—42no—
118C22×C4×C127—42no—
119C22×C4×C127—42no—
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122C22×C4×C127C2×D411yescharacteristic
123C22×C4×C127C2×D411yescharacteristic
124C22×C4×C127C2×D411yescharacteristic
125C22×C4×C127C2×D411yescharacteristic
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—21no—
135C4⋊C4×C127—21no—
136C4⋊C4×C127—21no—
137C4⋊C4×C127—22no—
138C4⋊C4×C127—22no—
139C4⋊C4×C127—22no—
140C4⋊C4×C127—42no—
141C4⋊C4×C127—44no—
142C4⋊C4×C127—42no—
143C4⋊C4×C127C22×C414yes—
144C4⋊C4×C127C22×C412yes—
145C4⋊C4×C127C2×Q812yes—
146C4⋊C4×C127#4112yes—
147C22⋊C4×C127—24no—
148C22⋊C4×C127—24no—
149C22⋊C4×C127—24no—
150C42×C127—22no—
151C42×C127—22no—
152C42×C127—22no—
153C42×C127—21no—
154C42×C127—21no—
155C42×C127—41no—
156C42×C127—42no—
157C42×C127—42no—
158C42×C127—44no—
159C42×C127—42no—
160C42×C127C22×C412yes—
161C42×C127C2×Q812yes—
162C42×C127#4112yes—
163C42×C127#4112yes—
164C23×C127—22no—
165C23×C127—22no—
166C23×C127—22no—
167C23×C127—41no—
168C23×C127—41no—
169C23×C127—41no—
170C23×C127—41no—
171C23×C127C22×D411yescharacteristic
172Q8×C127—48no—
173D4×C127—22no—
174D4×C127—22no—
175D4×C127—22no—
176D4×C127C4×Q814yescomplemented
177C2×C4×C127—21no—
178C2×C4×C127—21no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127—21no—
184C2×C4×C127—21no—
185C2×C4×C127—22no—
186C2×C4×C127—24no—
187C2×C4×C127—22no—
188C2×C4×C127—21no—
189C2×C4×C127—22no—
190C2×C4×C127—21no—
191C2×C4×C127—21no—
192C2×C4×C127—48no—
193C2×C4×C127—48no—
194C2×C4×C127—41no—
195C2×C4×C127—41no—
196C2×C4×C127—41no—
197C2×C4×C127—42no—
198C2×C4×C127—41no—
199C2×C4×C127—42no—
200C2×C4×C127—41no—
201C2×C4×C127—41no—
202C2×C4×C127—48no—
203C2×C4×C127—42no—
204C2×C4×C127—42no—
205C2×C4×C127—42no—
206C2×C4×C127C4×D412yes—
207C2×C4×C127C4×D412yes—
208C2×C4×C127C22⋊Q812yes—
209C2×C4×C127C22⋊Q812yes—
210C2×C4×C127AGL(1,ℤ/8)12yes—
211C2×C4×C127AGL(1,ℤ/8)12yes—
212C2×C4×C127#13712yes—
213C2×C4×C127C23×C411yescharacteristic
214C2×C4×C127C22×Q811yescharacteristic
215C2×C4×C127C2×#4111yescharacteristic
216C2×C4×C127C2×#4111yescharacteristic
217C2×C4×C127C2×#4111yescharacteristic
218C2×C4×C127C2×#4111yescharacteristic
219C8×C127—24no—
220C8×C127—42no—
221C22×C127—24no—
222C22×C127—42no—
223C22×C127—42no—
224C22×C127—42no—
225C22×C127—42no—
226C22×C127—42no—
227C22×C127—42no—
228C22×C127C2×C4×D411yescharacteristic
229C22×C127C2×C22⋊Q811yescharacteristic
230C22×C127#52511yescharacteristic
231C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
233C22×C127C2×#13711yescharacteristic
234C22×C127#57511yescharacteristic
235C4×C127—22no—
236C4×C127—22no—
237C4×C127—22no—
238C4×C127—22no—
239C4×C127—42no—
240C4×C127—42no—
241C4×C127—41no—
242C4×C127—42no—
243C4×C127—44no—
244C4×C127—82no—
245C4×C127#44214yes—
246C4×C127C2×C4×Q811yescharacteristic
247C4×C127C4×#4111yescharacteristic
248C4×C127#52911yescharacteristic
249C4×C127#53311yescharacteristic
250#21613C12711yesmaximal, characteristic, direct product factor, Sylow
251C2×C127—44no—
252C2×C127C4×C22⋊Q811yescharacteristic
253C2×C127C2×#44211yescharacteristic
254C2×C127C4×#13711yescharacteristic
255C2×C127#307711yescharacteristic
256C2×C127#307911yescharacteristic
257C2×C127#311011yescharacteristic
258C2×C127#311511yescharacteristic
259C2×#474C2×C12711yescharacteristic
260C4×C4⋊Q8C2×C12711yescharacteristic, complemented
261C42×D4C2×C12711yescharacteristic
262#1988C2×C12711yescharacteristic, complemented
263#1948C2×C12711yescharacteristic, complemented
264#1921C2×C12712yescomplemented
265#1864C2×C12712yes—
266#1802C2×C12711yescharacteristic
267#1776C2×C12712yescomplemented
268#1767C2×C12711yescharacteristic, complemented
269#1762C2×C12712yes—
270C127#2161311yescharacteristic, direct product factor, Sylow
271C2×C4⋊Q8C22×C12711yescharacteristic
272C2×C4×Q8—21no—
273C2×C4×D4C22×C12711yescharacteristic
274C2×C4×D4C22×C12711yescharacteristic
275C2×C4×D4C22×C12711yescharacteristic
276C22×C42—21no—
277#474C4×C12718yescomplemented
278C2×#107C22×C12712yes—
279C2×#106C22×C12712yes—
280C2×C4⋊C8C22×C12711yescharacteristic
281C2×C4⋊C8C22×C12711yescharacteristic
282C2×#102C22×C12712yes—
283C2×#102C22×C12712yes—
284C2×#97C22×C12712yes—
285#391—21no—
286#389—21no—
287#389C22×C12712yes—
288#384C22×C12711yescharacteristic
289C4×C4⋊C4—21no—
290C4×C4⋊C4C22×C12712yes—
291C4×C4⋊C4C22×C12711yescharacteristic
292C4×C4⋊C4C22×C12711yescharacteristic
293C4×C4⋊C4C22×C12711yescharacteristic
294C4×C22⋊C4—21no—
295C4×C22⋊C4—21no—
296C4×C22⋊C4—21no—
297C43C22×C12711yescharacteristic
298#340C22×C12712yes—
299#340C22×C12712yes—
300#340C22×C12712yes—
301#340C22×C12712yes—
302#336C22×C12712yes—
303C22×Q8—21no—
304C22×D4C23×C12711yescharacteristic
305C23×C4—21no—
306C23×C4—21no—
307C23×C4—21no—
308C22×C8C23×C12712yes—
309C22×C8C23×C12712yes—
310C4⋊Q8—22no—
311C4⋊Q8C2×C4×C12714yescomplemented
312C4×Q8—44no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4C2×C4×C12714yes—
318C4×D4Q8×C12714yescomplemented
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4C23×C12712yes—
324C2×C4⋊C4C23×C12711yescharacteristic
325C2×C4⋊C4C23×C12712yes—
326C2×C4⋊C4C23×C12711yescharacteristic
327C2×C4⋊C4C23×C12711yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—44no—
336C2×C42D4×C12711yescharacteristic
337C2×C42D4×C12711yescharacteristic
338C2×C42D4×C12711yescharacteristic
339C2×C42D4×C12711yescharacteristic
340C2×C42C23×C12711yescharacteristic
341C2×C42C23×C12711yescharacteristic
342C2×C42C23×C12711yescharacteristic
343#107C2×C4×C12718yescomplemented
344#106—24no—
345C4⋊C8—22no—
346C4⋊C8C2×C4×C12714yescomplemented
347#102—24no—
348#102C2×C4×C12718yes—
349#97—24no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95—22no—
355C24—21no—
356C2×Q8—24no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4C22×C4×C12712yes—
363C2×D4C2×Q8×C12712yes—
364C2×D4#41×C12712yes—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4C2×D4×C12711yescharacteristic
380C22×C4C2×D4×C12711yescharacteristic
381C22×C4C2×D4×C12711yescharacteristic
382C22×C4C2×D4×C12711yescharacteristic
383C22×C4C2×D4×C12711yescharacteristic
384C22×C4C2×D4×C12711yescharacteristic
385C22×C4C24×C12711yescharacteristic
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8C22×C4×C12714yes—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4C22×C4×C12714yes—
403C4⋊C4C22×C4×C12712yes—
404C4⋊C4C2×Q8×C12712yes—
405C4⋊C4#41×C12712yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—21no—
413C42—21no—
414C42—41no—
415C42—42no—
416C42—42no—
417C42—44no—
418C42—42no—
419C42C22×C4×C12712yes—
420C42C2×Q8×C12712yes—
421C42#41×C12712yes—
422C42#41×C12712yes—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23C22×D4×C12711yescharacteristic
431Q8—48no—
432D4—22no—
433D4—22no—
434D4—22no—
435D4C4×Q8×C12714yescomplemented
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—48no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4C4×D4×C12712yes—
466C2×C4C4×D4×C12712yes—
467C2×C4C22⋊Q8×C12712yes—
468C2×C4C22⋊Q8×C12712yes—
469C2×C4AGL(1,ℤ/8)×C12712yes—
470C2×C4AGL(1,ℤ/8)×C12712yes—
471C2×C4#137×C12712yes—
472C2×C4C23×C4×C12711yescharacteristic
473C2×C4C22×Q8×C12711yescharacteristic
474C2×C4C2×#41×C12711yescharacteristic
475C2×C4C2×#41×C12711yescharacteristic
476C2×C4C2×#41×C12711yescharacteristic
477C2×C4C2×#41×C12711yescharacteristic
478C8—24no—
479C8—42no—
480C22—24no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22C2×C4×D4×C12711yescharacteristic
488C22C2×C22⋊Q8×C12711yescharacteristic
489C22#525×C12711yescharacteristic
490C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
491C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
492C22C2×#137×C12711yescharacteristic
493C22#575×C12711yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—41no—
500C4—42no—
501C4—44no—
502C4—42no—
503C4—82no—
504C4#442×C12714yes—
505C4C2×C4×Q8×C12711yescharacteristic
506C4C4×#41×C12711yescharacteristic
507C4#529×C12711yescharacteristic
508C4#533×C12711yescharacteristic
509C2—44no—
510C2C4×C22⋊Q8×C12711yescharacteristic
511C2C2×#442×C12711yescharacteristic
512C2C4×#137×C12711yescharacteristic
513C2C127×#307711yescharacteristic
514C2C127×#307911yescharacteristic
515C2C127×#311011yescharacteristic
516C2C127×#311511yescharacteristic
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×D4×C127C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C43×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊Q8×C127C2semidirect
4. C127×#1988C2semidirect
5. C127×#1948C2semidirect
6. C127×#1921C2semidirect
9. C127×#1776C2semidirect
10. C127×#1767C2semidirect
18. #474×C127C4semidirect
52. C4⋊Q8×C127C2×C4semidirect
59. C4×D4×C127Q8semidirect
84. #107×C127C2×C4semidirect
87. C4⋊C8×C127C2×C4semidirect
176. D4×C127C4×Q8semidirect
250. #21613C127direct (N × H)
260. C4×C4⋊Q8C2×C127semidirect
262. #1988C2×C127semidirect
263. #1948C2×C127semidirect
264. #1921C2×C127semidirect
267. #1776C2×C127semidirect
268. #1767C2×C127semidirect
270. C127#21613direct (N × H)
277. #474C4×C127semidirect
311. C4⋊Q8C2×C4×C127semidirect
318. C4×D4Q8×C127semidirect
343. #107C2×C4×C127semidirect
346. C4⋊C8C2×C4×C127semidirect
435. D4C4×Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#474×C1272
C4×C4⋊Q8×C1272
C42×D4×C1272
C127×#19882
C127×#19482
C127×#19212
C127×#18642
C127×#18022
C127×#17762
C127×#17672
C127×#17622
#21613127