Finite Group Databasebeta

← C127×#20617 #44584290

Subgroups — C127×#20617 #44584290

Order 32512.

3610 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×2−1+4×C127C211yesmaximal, characteristic
2C2×#421×C127C212yesmaximal, complemented
3C127×#2896C212yesmaximal, complemented
4C22×#101×C127C211yesmaximal, characteristic, complemented
5C127×#2310C211yesmaximal, characteristic, complemented
6C127×#1793C218yesmaximal, complemented
7C2×2−1+4×C127—28no—
8C2×2−1+4×C127—24no—
9C2×2−1+4×C127—24no—
10C2×2−1+4×C127—22no—
11C2×2−1+4×C127C418yescomplemented
12C2×2−1+4×C127C2214yes—
13C22×#41×C127—22no—
14C22×#41×C127—22no—
15C22×#41×C127C2212yes—
16C23×Q8×C127—22no—
17C23×Q8×C127C2211yescharacteristic
18C22×M16×C127C2212yes—
19C2×#117×C127C2211yescharacteristic
20C2×#117×C127C2211yescharacteristic
21#421×C127—28no—
22C2×#104×C127C2218yescomplemented
23#419×C127—22no—
24#418×C127—22no—
25#417×C127C2218yescomplemented
26#416×C127—24no—
27C2×#101×C127—24no—
28C2×#101×C127C2214yescomplemented
29#391×C127—22no—
30#391×C127C2214yescomplemented
31#381×C127—22no—
322−1+4×C127—232no—
332−1+4×C127—216no—
342−1+4×C127—232no—
352−1+4×C127—216no—
362−1+4×C127—216no—
372−1+4×C127—28no—
382−1+4×C127C2×C4116yes—
39C2×#41×C127—28no—
40C2×#41×C127—216no—
41C2×#41×C127—28no—
42C2×#41×C127—216no—
43C2×#41×C127—216no—
44C2×#41×C127—28no—
45C2×#41×C127—216no—
46C2×#41×C127—28no—
47C2×#41×C127—24no—
48C2×#41×C127—22no—
49C2×#41×C127—24no—
50C2×#41×C127—22no—
51C2×#41×C127C2×C418yes—
52C2×#41×C127D418yes—
53C2×#41×C127D414yes—
54C2×#41×C127C2314yes—
55C22×Q8×C127—22no—
56C22×Q8×C127—28no—
57C22×Q8×C127—216no—
58C22×Q8×C127—216no—
59C22×Q8×C127—28no—
60C22×Q8×C127—22no—
61C22×Q8×C127—22no—
62C22×Q8×C127—24no—
63C22×Q8×C127C2×C414yes—
64C22×Q8×C127D412yes—
65C22×Q8×C127D414yes—
66C22×Q8×C127D412yes—
67C22×Q8×C127C2312yes—
68C22×D4×C127—22no—
69C22×D4×C127—22no—
70C22×D4×C127D412yes—
71C23×C4×C127—24no—
72C23×C4×C127—22no—
73C23×C4×C127D412yes—
74C23×C4×C127C2311yescharacteristic
75C2×M16×C127—22no—
76C2×M16×C127—24no—
77C2×M16×C127—24no—
78C2×M16×C127C2314yes—
79C22×C8×C127—22no—
80#117×C127—24no—
81#117×C127—22no—
82#117×C127—22no—
83C2×C4⋊C4×C127—21no—
84C2×C4⋊C4×C127C2312yes—
85C2×C22⋊C4×C127—21no—
86C2×C22⋊C4×C127—21no—
87C2×C42×C127—21no—
88C2×C42×C127C2312yes—
89#104×C127—216no—
90#103×C127—28no—
91#102×C127—28no—
92#101×C127—28no—
93#95×C127—28no—
94C24×C127—22no—
95C24×C127C2×D411yescharacteristic
96#41×C127—232no—
97#41×C127—264no—
98#41×C127—232no—
99#41×C127—264no—
100#41×C127—232no—
101#41×C127—232no—
102#41×C127—28no—
103#41×C127—28no—
104#41×C127—28no—
105#41×C127—28no—
106#41×C127—28no—
107#41×C127—24no—
108#41×C127—24no—
109#41×C127C22⋊C4116yes—
110#41×C127C22×C418yes—
111#41×C127C2×D418yes—
112C2×Q8×C127—28no—
113C2×Q8×C127—216no—
114C2×Q8×C127—232no—
115C2×Q8×C127—264no—
116C2×Q8×C127—232no—
117C2×Q8×C127—28no—
118C2×Q8×C127—28no—
119C2×Q8×C127—22no—
120C2×Q8×C127—24no—
121C2×Q8×C127—24no—
122C2×Q8×C127—28no—
123C2×Q8×C127—24no—
124C2×Q8×C127—28no—
125C2×Q8×C127—24no—
126C2×Q8×C127—24no—
127C2×Q8×C127C22⋊C418yes—
128C2×Q8×C127C22⋊C418yes—
129C2×Q8×C127C22×C414yes—
130C2×Q8×C127C2×D414yes—
131C2×Q8×C127C2×D414yes—
132C2×D4×C127—28no—
133C2×D4×C127—216no—
134C2×D4×C127—28no—
135C2×D4×C127—24no—
136C2×D4×C127—28no—
137C2×D4×C127—24no—
138C2×D4×C127—24no—
139C2×D4×C127—22no—
140C2×D4×C127C22⋊C418yes—
141C2×D4×C127C2×D414yes—
142C22×C4×C127—216no—
143C22×C4×C127—232no—
144C22×C4×C127—22no—
145C22×C4×C127—22no—
146C22×C4×C127—24no—
147C22×C4×C127—22no—
148C22×C4×C127—28no—
149C22×C4×C127—24no—
150C22×C4×C127—28no—
151C22×C4×C127—24no—
152C22×C4×C127—22no—
153C22×C4×C127—22no—
154C22×C4×C127C22⋊C418yes—
155C22×C4×C127C22×C412yes—
156C22×C4×C127C2×D414yes—
157C22×C4×C127C2×D411yescharacteristic
158C22×C4×C127C2×D412yes—
159C22×C4×C127C2×D412yes—
160C22×C4×C127C2×D414yes—
161C22×C4×C127C2×D412yes—
162C22×C4×C127C2411yescharacteristic
163M16×C127—24no—
164M16×C127—24no—
165M16×C127—44no—
166C2×C8×C127—24no—
167C2×C8×C127—44no—
168C4⋊C4×C127—24no—
169C4⋊C4×C127—42no—
170C22⋊C4×C127—42no—
171C22⋊C4×C127—41no—
172C22⋊C4×C127—41no—
173C42×C127—24no—
174C42×C127—42no—
175C23×C127—24no—
176C23×C127—28no—
177C23×C127—42no—
178C23×C127—48no—
179C23×C127C2×C22⋊C412yes—
180C23×C127#12011yescharacteristic
181C23×C127#12011yescharacteristic
182C23×C127#12012yes—
183C23×C127C22×D411yescharacteristic
184Q8×C127—232no—
185Q8×C127—28no—
186Q8×C127—24no—
187Q8×C127—28no—
188Q8×C127—28no—
189Q8×C127—216no—
190Q8×C127C2×C22⋊C418yes—
191D4×C127—232no—
192D4×C127—28no—
193D4×C127—28no—
194D4×C127—216no—
195D4×C127—24no—
196D4×C127—28no—
197D4×C127C2×C22⋊C418yes—
198C2×C4×C127—264no—
199C2×C4×C127—24no—
200C2×C4×C127—24no—
201C2×C4×C127—24no—
202C2×C4×C127—28no—
203C2×C4×C127—28no—
204C2×C4×C127—28no—
205C2×C4×C127—28no—
206C2×C4×C127—28no—
207C2×C4×C127—216no—
208C2×C4×C127—44no—
209C2×C4×C127—44no—
210C2×C4×C127—42no—
211C2×C4×C127—42no—
212C2×C4×C127C2×C22⋊C418yes—
213C2×C4×C127C2×C22⋊C414yes—
214C2×C4×C127C2×C22⋊C414yes—
215C2×C4×C127C2×C22⋊C412yes—
216C2×C4×C127#12014yes—
217C2×C4×C127#12014yes—
218C2×C4×C127#12012yes—
219C2×C4×C127#12012yes—
220C2×C4×C127C23×C411yescharacteristic
221C2×C4×C127C22×D412yes—
222C2×C4×C127C22×D412yes—
223C2×C4×C127C22×D411yescharacteristic
224C8×C127—44no—
225C22×C127—22no—
226C22×C127—22no—
227C22×C127—22no—
228C22×C127—28no—
229C22×C127—42no—
230C22×C127—42no—
231C22×C127—416no—
232C22×C127—48no—
233C22×C127#37912yes—
234C22×C127#37912yes—
235C22×C127C22×C22⋊C411yescharacteristic
236C22×C127C2×#12011yescharacteristic
237C22×C127C2×#12011yescharacteristic
238C4×C127—28no—
239C4×C127—28no—
240C4×C127—82no—
241C4×C127—82no—
242C4×C127#37914yes—
243C4×C127C22×C22⋊C412yes—
244C4×C127C2×#12012yes—
245#20617C12711yesmaximal, characteristic, direct product factor, Sylow
246C2×C127—21no—
247C2×C127—22no—
248C2×C127—21no—
249C2×C127—22no—
250C2×C127—48no—
251C2×C127C2×#37911yescharacteristic
252C2×C127#302012yes—
253C22×2−1+4C2×C12711yescharacteristic
254C2×#421C2×C12712yescomplemented
255#2896C2×C12712yescomplemented
256C22×#101C2×C12711yescharacteristic, complemented
257#2310C2×C12711yescharacteristic, complemented
258#1793C2×C12718yescomplemented
259C127#2061711yescharacteristic, direct product factor, Sylow
260C2×2−1+4—28no—
261C2×2−1+4—24no—
262C2×2−1+4—24no—
263C2×2−1+4—22no—
264C2×2−1+4C4×C12718yescomplemented
265C2×2−1+4C22×C12714yes—
266C22×#41—22no—
267C22×#41—22no—
268C22×#41C22×C12712yes—
269C23×Q8—22no—
270C23×Q8C22×C12711yescharacteristic
271C22×M16C22×C12712yes—
272C2×#117C22×C12711yescharacteristic
273C2×#117C22×C12711yescharacteristic
274#421—28no—
275C2×#104C22×C12718yescomplemented
276#419—22no—
277#418—22no—
278#417C22×C12718yescomplemented
279#416—24no—
280C2×#101—24no—
281C2×#101C22×C12714yescomplemented
282#391—22no—
283#391C22×C12714yescomplemented
284#381—22no—
2852−1+4—232no—
2862−1+4—216no—
2872−1+4—232no—
2882−1+4—216no—
2892−1+4—216no—
2902−1+4—28no—
2912−1+4C2×C4×C127116yes—
292C2×#41—28no—
293C2×#41—216no—
294C2×#41—28no—
295C2×#41—216no—
296C2×#41—216no—
297C2×#41—28no—
298C2×#41—216no—
299C2×#41—28no—
300C2×#41—24no—
301C2×#41—22no—
302C2×#41—24no—
303C2×#41—22no—
304C2×#41C2×C4×C12718yes—
305C2×#41D4×C12718yes—
306C2×#41D4×C12714yes—
307C2×#41C23×C12714yes—
308C22×Q8—22no—
309C22×Q8—28no—
310C22×Q8—216no—
311C22×Q8—216no—
312C22×Q8—28no—
313C22×Q8—22no—
314C22×Q8—22no—
315C22×Q8—24no—
316C22×Q8C2×C4×C12714yes—
317C22×Q8D4×C12712yes—
318C22×Q8D4×C12714yes—
319C22×Q8D4×C12712yes—
320C22×Q8C23×C12712yes—
321C22×D4—22no—
322C22×D4—22no—
323C22×D4D4×C12712yes—
324C23×C4—24no—
325C23×C4—22no—
326C23×C4D4×C12712yes—
327C23×C4C23×C12711yescharacteristic
328C2×M16—22no—
329C2×M16—24no—
330C2×M16—24no—
331C2×M16C23×C12714yes—
332C22×C8—22no—
333#117—24no—
334#117—22no—
335#117—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4C23×C12712yes—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C42—21no—
341C2×C42C23×C12712yes—
342#104—216no—
343#103—28no—
344#102—28no—
345#101—28no—
346#95—28no—
347C24—22no—
348C24C2×D4×C12711yescharacteristic
349#41—232no—
350#41—264no—
351#41—232no—
352#41—264no—
353#41—232no—
354#41—232no—
355#41—28no—
356#41—28no—
357#41—28no—
358#41—28no—
359#41—28no—
360#41—24no—
361#41—24no—
362#41C22⋊C4×C127116yes—
363#41C22×C4×C12718yes—
364#41C2×D4×C12718yes—
365C2×Q8—28no—
366C2×Q8—216no—
367C2×Q8—232no—
368C2×Q8—264no—
369C2×Q8—232no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—28no—
376C2×Q8—24no—
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8C22⋊C4×C12718yes—
381C2×Q8C22⋊C4×C12718yes—
382C2×Q8C22×C4×C12714yes—
383C2×Q8C2×D4×C12714yes—
384C2×Q8C2×D4×C12714yes—
385C2×D4—28no—
386C2×D4—216no—
387C2×D4—28no—
388C2×D4—24no—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4C22⋊C4×C12718yes—
394C2×D4C2×D4×C12714yes—
395C22×C4—216no—
396C22×C4—232no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4C22⋊C4×C12718yes—
408C22×C4C22×C4×C12712yes—
409C22×C4C2×D4×C12714yes—
410C22×C4C2×D4×C12711yescharacteristic
411C22×C4C2×D4×C12712yes—
412C22×C4C2×D4×C12712yes—
413C22×C4C2×D4×C12714yes—
414C22×C4C2×D4×C12712yes—
415C22×C4C24×C12711yescharacteristic
416M16—24no—
417M16—24no—
418M16—44no—
419C2×C8—24no—
420C2×C8—44no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C42—24no—
427C42—42no—
428C23—24no—
429C23—28no—
430C23—42no—
431C23—48no—
432C23C2×C22⋊C4×C12712yes—
433C23#120×C12711yescharacteristic
434C23#120×C12711yescharacteristic
435C23#120×C12712yes—
436C23C22×D4×C12711yescharacteristic
437Q8—232no—
438Q8—28no—
439Q8—24no—
440Q8—28no—
441Q8—28no—
442Q8—216no—
443Q8C2×C22⋊C4×C12718yes—
444D4—232no—
445D4—28no—
446D4—28no—
447D4—216no—
448D4—24no—
449D4—28no—
450D4C2×C22⋊C4×C12718yes—
451C2×C4—264no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—216no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4C2×C22⋊C4×C12718yes—
466C2×C4C2×C22⋊C4×C12714yes—
467C2×C4C2×C22⋊C4×C12714yes—
468C2×C4C2×C22⋊C4×C12712yes—
469C2×C4#120×C12714yes—
470C2×C4#120×C12714yes—
471C2×C4#120×C12712yes—
472C2×C4#120×C12712yes—
473C2×C4C23×C4×C12711yescharacteristic
474C2×C4C22×D4×C12712yes—
475C2×C4C22×D4×C12712yes—
476C2×C4C22×D4×C12711yescharacteristic
477C8—44no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—28no—
482C22—42no—
483C22—42no—
484C22—416no—
485C22—48no—
486C22#379×C12712yes—
487C22#379×C12712yes—
488C22C22×C22⋊C4×C12711yescharacteristic
489C22C2×#120×C12711yescharacteristic
490C22C2×#120×C12711yescharacteristic
491C4—28no—
492C4—28no—
493C4—82no—
494C4—82no—
495C4#379×C12714yes—
496C4C22×C22⋊C4×C12712yes—
497C4C2×#120×C12712yes—
498C2—21no—
499C2—22no—
500C2—21no—
501C2—22no—
502C2—48no—
503C2C2×#379×C12711yescharacteristic
504C2C127×#302012yes—
subgroupG/H
center, socleC22×C127C2×#120
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×2−1+4×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×#101×C127C2semidirect
5. C127×#2310C2semidirect
6. C127×#1793C2semidirect
11. C2×2−1+4×C127C4semidirect
22. C2×#104×C127C22semidirect
25. #417×C127C22semidirect
28. C2×#101×C127C22semidirect
30. #391×C127C22semidirect
245. #20617C127direct (N × H)
254. C2×#421C2×C127semidirect
255. #2896C2×C127semidirect
256. C22×#101C2×C127semidirect
257. #2310C2×C127semidirect
258. #1793C2×C127semidirect
259. C127#20617direct (N × H)
264. C2×2−1+4C4×C127semidirect
275. C2×#104C22×C127semidirect
278. #417C22×C127semidirect
281. C2×#101C22×C127semidirect
283. #391C22×C127semidirect

Maximal subgroups

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

subgroupindex
C22×2−1+4×C1272
C2×#421×C1272
C127×#28962
C22×#101×C1272
C127×#23102
C127×#17932
#20617127