Finite Group Databasebeta

← C127×#20616 #44584289

Subgroups — C127×#20616 #44584289

Order 32512.

3986 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×H(5,𝔽2)×C127C211yesmaximal, characteristic
2C2×#421×C127C212yesmaximal, complemented
3C127×#2896C212yesmaximal, complemented
4C22×#100×C127C211yesmaximal, characteristic, complemented
5C127×#2309C211yesmaximal, characteristic, complemented
6C127×#1792C218yesmaximal, complemented
7C2×H(5,𝔽2)×C127—28no—
8C2×H(5,𝔽2)×C127—24no—
9C2×H(5,𝔽2)×C127—24no—
10C2×H(5,𝔽2)×C127—22no—
11C2×H(5,𝔽2)×C127C418yescomplemented
12C2×H(5,𝔽2)×C127C2214yes—
13C22×#41×C127—22no—
14C22×#41×C127C2212yes—
15C23×D4×C127—22no—
16C23×D4×C127—22no—
17C23×D4×C127C2211yescharacteristic
18C22×M16×C127C2212yescomplemented
19C2×#117×C127C2211yescharacteristic, complemented
20C2×#117×C127C2211yescharacteristic, complemented
21#421×C127—28no—
22C2×#104×C127C2218yescomplemented
23#419×C127—22no—
24#418×C127—22no—
25#417×C127C2218yescomplemented
26#416×C127—24no—
27C2×#100×C127—24no—
28C2×#100×C127C2214yescomplemented
29#390×C127—22no—
30#390×C127C2214yescomplemented
31#379×C127—22no—
32C25×C127—22no—
33C25×C127D412yes—
34H(5,𝔽2)×C127—216no—
35H(5,𝔽2)×C127—232no—
36H(5,𝔽2)×C127—232no—
37H(5,𝔽2)×C127—216no—
38H(5,𝔽2)×C127—216no—
39H(5,𝔽2)×C127—28no—
40H(5,𝔽2)×C127C2×C4116yes—
41C2×#41×C127—28no—
42C2×#41×C127—216no—
43C2×#41×C127—216no—
44C2×#41×C127—28no—
45C2×#41×C127—24no—
46C2×#41×C127—22no—
47C2×#41×C127—24no—
48C2×#41×C127—22no—
49C2×#41×C127C2×C418yes—
50C2×#41×C127D418yes—
51C2×#41×C127D414yes—
52C2×#41×C127C2314yes—
53C22×Q8×C127D412yes—
54C22×D4×C127—24no—
55C22×D4×C127—28no—
56C22×D4×C127—216no—
57C22×D4×C127—28no—
58C22×D4×C127—216no—
59C22×D4×C127—216no—
60C22×D4×C127—22no—
61C22×D4×C127—28no—
62C22×D4×C127—216no—
63C22×D4×C127—28no—
64C22×D4×C127—22no—
65C22×D4×C127—22no—
66C22×D4×C127—24no—
67C22×D4×C127C2×C414yes—
68C22×D4×C127D412yes—
69C22×D4×C127D414yes—
70C22×D4×C127D412yes—
71C22×D4×C127C2312yes—
72C23×C4×C127—22no—
73C23×C4×C127—22no—
74C23×C4×C127C2311yescharacteristic
75C2×M16×C127—22no—
76C2×M16×C127—24no—
77C2×M16×C127—24no—
78C2×M16×C127C2314yescomplemented
79C22×C8×C127—22no—
80#117×C127—24no—
81#117×C127—22no—
82#117×C127—22no—
83C2×C4⋊C4×C127—21no—
84C2×C4⋊C4×C127C2312yescomplemented
85C2×C22⋊C4×C127—21no—
86C2×C22⋊C4×C127—21no—
87C2×C22⋊C4×C127—28no—
88C2×C42×C127—21no—
89C2×C42×C127C2312yescomplemented
90#104×C127—216no—
91#103×C127—28no—
92#102×C127—28no—
93#100×C127—28no—
94C24×C127—28no—
95C24×C127—216no—
96C24×C127—22no—
97C24×C127—22no—
98C24×C127—22no—
99C24×C127—24no—
100C24×C127—84no—
101C24×C127—88no—
102C24×C127C22⋊C418yes—
103C24×C127C2×D414yes—
104C24×C127C2×D411yescharacteristic
105#41×C127—232no—
106#41×C127—264no—
107#41×C127—232no—
108#41×C127—28no—
109#41×C127—28no—
110#41×C127—28no—
111#41×C127—28no—
112#41×C127—28no—
113#41×C127—24no—
114#41×C127—24no—
115#41×C127C22⋊C4116yescomplemented
116#41×C127C22×C418yes—
117#41×C127C2×D418yes—
118C2×Q8×C127—22no—
119C2×Q8×C127—24no—
120C2×Q8×C127C22⋊C418yescomplemented
121C2×Q8×C127C2×D414yes—
122C2×D4×C127—216no—
123C2×D4×C127—232no—
124C2×D4×C127—232no—
125C2×D4×C127—264no—
126C2×D4×C127—232no—
127C2×D4×C127—264no—
128C2×D4×C127—28no—
129C2×D4×C127—232no—
130C2×D4×C127—24no—
131C2×D4×C127—28no—
132C2×D4×C127—24no—
133C2×D4×C127—232no—
134C2×D4×C127—28no—
135C2×D4×C127—28no—
136C2×D4×C127—28no—
137C2×D4×C127—24no—
138C2×D4×C127—24no—
139C2×D4×C127—22no—
140C2×D4×C127—28no—
141C2×D4×C127—24no—
142C2×D4×C127—24no—
143C2×D4×C127—24no—
144C2×D4×C127C22⋊C418yes—
145C2×D4×C127C22⋊C418yescomplemented
146C2×D4×C127C22×C414yes—
147C2×D4×C127C2×D414yes—
148C2×D4×C127C2×D414yes—
149C22×C4×C127—28no—
150C22×C4×C127—216no—
151C22×C4×C127—22no—
152C22×C4×C127—28no—
153C22×C4×C127—24no—
154C22×C4×C127—28no—
155C22×C4×C127—24no—
156C22×C4×C127—22no—
157C22×C4×C127—22no—
158C22×C4×C127C22×C412yes—
159C22×C4×C127C2×D411yescharacteristic
160C22×C4×C127C2×D412yes—
161C22×C4×C127C2×D412yes—
162C22×C4×C127C2×D414yes—
163C22×C4×C127C2×D412yes—
164C22×C4×C127C2411yescharacteristic
165M16×C127—24no—
166M16×C127—24no—
167M16×C127—44no—
168C2×C8×C127—24no—
169C2×C8×C127—44no—
170C4⋊C4×C127—24no—
171C4⋊C4×C127—42no—
172C22⋊C4×C127—42no—
173C22⋊C4×C127—41no—
174C22⋊C4×C127—41no—
175C22⋊C4×C127—88no—
176C42×C127—24no—
177C42×C127—42no—
178C23×C127—232no—
179C23×C127—24no—
180C23×C127—24no—
181C23×C127—28no—
182C23×C127—28no—
183C23×C127—24no—
184C23×C127—28no—
185C23×C127—42no—
186C23×C127—48no—
187C23×C127—48no—
188C23×C127—48no—
189C23×C127—84no—
190C23×C127—88no—
191C23×C127—88no—
192C23×C127—832no—
193C23×C127—816no—
194C23×C127C2×C22⋊C418yes—
195C23×C127C2×C22⋊C412yes—
196C23×C127#12011yescharacteristic
197C23×C127#12011yescharacteristic
198C23×C127#12012yes—
199C23×C127C22×D411yescharacteristic
200Q8×C127—24no—
201Q8×C127—28no—
202Q8×C127C2×C22⋊C418yes—
203D4×C127—264no—
204D4×C127—28no—
205D4×C127—28no—
206D4×C127—216no—
207D4×C127—28no—
208D4×C127—28no—
209D4×C127—24no—
210D4×C127—216no—
211D4×C127—28no—
212D4×C127C2×C22⋊C418yes—
213C2×C4×C127—232no—
214C2×C4×C127—24no—
215C2×C4×C127—28no—
216C2×C4×C127—28no—
217C2×C4×C127—28no—
218C2×C4×C127—216no—
219C2×C4×C127—44no—
220C2×C4×C127—44no—
221C2×C4×C127—42no—
222C2×C4×C127—42no—
223C2×C4×C127C2×C22⋊C414yes—
224C2×C4×C127C2×C22⋊C414yes—
225C2×C4×C127C2×C22⋊C412yes—
226C2×C4×C127#12014yes—
227C2×C4×C127#12014yes—
228C2×C4×C127#12012yes—
229C2×C4×C127#12012yes—
230C2×C4×C127C23×C411yescharacteristic
231C2×C4×C127C22×D412yes—
232C2×C4×C127C22×D412yes—
233C2×C4×C127C22×D411yescharacteristic
234C8×C127—44no—
235C22×C127—22no—
236C22×C127—22no—
237C22×C127—22no—
238C22×C127—28no—
239C22×C127—28no—
240C22×C127—42no—
241C22×C127—42no—
242C22×C127—48no—
243C22×C127—416no—
244C22×C127—416no—
245C22×C127—48no—
246C22×C127—48no—
247C22×C127—84no—
248C22×C127—88no—
249C22×C127—832no—
250C22×C127#37912yes—
251C22×C127#37912yes—
252C22×C127C22×C22⋊C411yescharacteristic
253C22×C127C2×#12011yescharacteristic
254C22×C127C2×#12011yescharacteristic
255C4×C127—28no—
256C4×C127—82no—
257C4×C127—82no—
258C4×C127#37914yes—
259C4×C127C22×C22⋊C412yes—
260C4×C127C2×#12012yes—
261#20616C12711yesmaximal, characteristic, direct product factor, Sylow
262C2×C127—21no—
263C2×C127—22no—
264C2×C127—21no—
265C2×C127—22no—
266C2×C127—48no—
267C2×C127—48no—
268C2×C127C2×#37911yescharacteristic
269C2×C127#301912yes—
270C22×H(5,𝔽2)C2×C12711yescharacteristic
271C2×#421C2×C12712yescomplemented
272#2896C2×C12712yescomplemented
273C22×#100C2×C12711yescharacteristic, complemented
274#2309C2×C12711yescharacteristic, complemented
275#1792C2×C12718yescomplemented
276C127#2061611yescharacteristic, direct product factor, Sylow
277C2×H(5,𝔽2)—28no—
278C2×H(5,𝔽2)—24no—
279C2×H(5,𝔽2)—24no—
280C2×H(5,𝔽2)—22no—
281C2×H(5,𝔽2)C4×C12718yescomplemented
282C2×H(5,𝔽2)C22×C12714yes—
283C22×#41—22no—
284C22×#41C22×C12712yes—
285C23×D4—22no—
286C23×D4—22no—
287C23×D4C22×C12711yescharacteristic
288C22×M16C22×C12712yescomplemented
289C2×#117C22×C12711yescharacteristic, complemented
290C2×#117C22×C12711yescharacteristic, complemented
291#421—28no—
292C2×#104C22×C12718yescomplemented
293#419—22no—
294#418—22no—
295#417C22×C12718yescomplemented
296#416—24no—
297C2×#100—24no—
298C2×#100C22×C12714yescomplemented
299#390—22no—
300#390C22×C12714yescomplemented
301#379—22no—
302C25—22no—
303C25D4×C12712yes—
304H(5,𝔽2)—216no—
305H(5,𝔽2)—232no—
306H(5,𝔽2)—232no—
307H(5,𝔽2)—216no—
308H(5,𝔽2)—216no—
309H(5,𝔽2)—28no—
310H(5,𝔽2)C2×C4×C127116yes—
311C2×#41—28no—
312C2×#41—216no—
313C2×#41—216no—
314C2×#41—28no—
315C2×#41—24no—
316C2×#41—22no—
317C2×#41—24no—
318C2×#41—22no—
319C2×#41C2×C4×C12718yes—
320C2×#41D4×C12718yes—
321C2×#41D4×C12714yes—
322C2×#41C23×C12714yes—
323C22×Q8D4×C12712yes—
324C22×D4—24no—
325C22×D4—28no—
326C22×D4—216no—
327C22×D4—28no—
328C22×D4—216no—
329C22×D4—216no—
330C22×D4—22no—
331C22×D4—28no—
332C22×D4—216no—
333C22×D4—28no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—24no—
337C22×D4C2×C4×C12714yes—
338C22×D4D4×C12712yes—
339C22×D4D4×C12714yes—
340C22×D4D4×C12712yes—
341C22×D4C23×C12712yes—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4C23×C12711yescharacteristic
345C2×M16—22no—
346C2×M16—24no—
347C2×M16—24no—
348C2×M16C23×C12714yescomplemented
349C22×C8—22no—
350#117—24no—
351#117—22no—
352#117—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4C23×C12712yescomplemented
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—28no—
358C2×C42—21no—
359C2×C42C23×C12712yescomplemented
360#104—216no—
361#103—28no—
362#102—28no—
363#100—28no—
364C24—28no—
365C24—216no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—84no—
371C24—88no—
372C24C22⋊C4×C12718yes—
373C24C2×D4×C12714yes—
374C24C2×D4×C12711yescharacteristic
375#41—232no—
376#41—264no—
377#41—232no—
378#41—28no—
379#41—28no—
380#41—28no—
381#41—28no—
382#41—28no—
383#41—24no—
384#41—24no—
385#41C22⋊C4×C127116yescomplemented
386#41C22×C4×C12718yes—
387#41C2×D4×C12718yes—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8C22⋊C4×C12718yescomplemented
391C2×Q8C2×D4×C12714yes—
392C2×D4—216no—
393C2×D4—232no—
394C2×D4—232no—
395C2×D4—264no—
396C2×D4—232no—
397C2×D4—264no—
398C2×D4—28no—
399C2×D4—232no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—232no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4C22⋊C4×C12718yes—
415C2×D4C22⋊C4×C12718yescomplemented
416C2×D4C22×C4×C12714yes—
417C2×D4C2×D4×C12714yes—
418C2×D4C2×D4×C12714yes—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4C22×C4×C12712yes—
429C22×C4C2×D4×C12711yescharacteristic
430C22×C4C2×D4×C12712yes—
431C22×C4C2×D4×C12712yes—
432C22×C4C2×D4×C12714yes—
433C22×C4C2×D4×C12712yes—
434C22×C4C24×C12711yescharacteristic
435M16—24no—
436M16—24no—
437M16—44no—
438C2×C8—24no—
439C2×C8—44no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—88no—
446C42—24no—
447C42—42no—
448C23—232no—
449C23—24no—
450C23—24no—
451C23—28no—
452C23—28no—
453C23—24no—
454C23—28no—
455C23—42no—
456C23—48no—
457C23—48no—
458C23—48no—
459C23—84no—
460C23—88no—
461C23—88no—
462C23—832no—
463C23—816no—
464C23C2×C22⋊C4×C12718yes—
465C23C2×C22⋊C4×C12712yes—
466C23#120×C12711yescharacteristic
467C23#120×C12711yescharacteristic
468C23#120×C12712yes—
469C23C22×D4×C12711yescharacteristic
470Q8—24no—
471Q8—28no—
472Q8C2×C22⋊C4×C12718yes—
473D4—264no—
474D4—28no—
475D4—28no—
476D4—216no—
477D4—28no—
478D4—28no—
479D4—24no—
480D4—216no—
481D4—28no—
482D4C2×C22⋊C4×C12718yes—
483C2×C4—232no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—216no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4C2×C22⋊C4×C12714yes—
494C2×C4C2×C22⋊C4×C12714yes—
495C2×C4C2×C22⋊C4×C12712yes—
496C2×C4#120×C12714yes—
497C2×C4#120×C12714yes—
498C2×C4#120×C12712yes—
499C2×C4#120×C12712yes—
500C2×C4C23×C4×C12711yescharacteristic
501C2×C4C22×D4×C12712yes—
502C2×C4C22×D4×C12712yes—
503C2×C4C22×D4×C12711yescharacteristic
504C8—44no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—28no—
509C22—28no—
510C22—42no—
511C22—42no—
512C22—48no—
513C22—416no—
514C22—416no—
515C22—48no—
516C22—48no—
517C22—84no—
518C22—88no—
519C22—832no—
520C22#379×C12712yes—
521C22#379×C12712yes—
522C22C22×C22⋊C4×C12711yescharacteristic
523C22C2×#120×C12711yescharacteristic
524C22C2×#120×C12711yescharacteristic
525C4—28no—
526C4—82no—
527C4—82no—
528C4#379×C12714yes—
529C4C22×C22⋊C4×C12712yes—
530C4C2×#120×C12712yes—
531C2—21no—
532C2—22no—
533C2—21no—
534C2—22no—
535C2—48no—
536C2—48no—
537C2C2×#379×C12711yescharacteristic
538C2C127×#301912yes—
subgroupG/H
center, socleC22×C127C2×#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×H(5,𝔽2)×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#421×C127C2semidirect
3. C127×#2896C2semidirect
4. C22×#100×C127C2semidirect
5. C127×#2309C2semidirect
6. C127×#1792C2semidirect
11. C2×H(5,𝔽2)×C127C4semidirect
18. C22×M16×C127C22semidirect
19. C2×#117×C127C22semidirect
20. C2×#117×C127C22semidirect
22. C2×#104×C127C22semidirect
25. #417×C127C22semidirect
28. C2×#100×C127C22semidirect
30. #390×C127C22semidirect
78. C2×M16×C127C23semidirect
84. C2×C4⋊C4×C127C23semidirect
89. C2×C42×C127C23semidirect
115. #41×C127C22⋊C4semidirect
120. C2×Q8×C127C22⋊C4semidirect
145. C2×D4×C127C22⋊C4semidirect
261. #20616C127direct (N × H)
271. C2×#421C2×C127semidirect
272. #2896C2×C127semidirect
273. C22×#100C2×C127semidirect
274. #2309C2×C127semidirect
275. #1792C2×C127semidirect
276. C127#20616direct (N × H)
281. C2×H(5,𝔽2)C4×C127semidirect
288. C22×M16C22×C127semidirect
289. C2×#117C22×C127semidirect
290. C2×#117C22×C127semidirect
292. C2×#104C22×C127semidirect
295. #417C22×C127semidirect
298. C2×#100C22×C127semidirect
300. #390C22×C127semidirect
348. C2×M16C23×C127semidirect
354. C2×C4⋊C4C23×C127semidirect
359. C2×C42C23×C127semidirect
385. #41C22⋊C4×C127semidirect
390. C2×Q8C22⋊C4×C127semidirect
415. C2×D4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C1272
C2×#421×C1272
C127×#28962
C22×#100×C1272
C127×#23092
C127×#17922
#20616127