Finite Group Databasebeta

← C127×#20606 #44584279

Subgroups — C127×#20606 #44584279

Order 32512.

3722 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585×C127C211yesmaximal, characteristic
2C2×#421×C127C212yesmaximal, complemented
3C22×#104×C127C212yesmaximal, complemented
4C127×#2879C211yesmaximal, characteristic, complemented
5C127×#2297C211yesmaximal, characteristic, complemented
6C127×#1789C218yesmaximal, complemented
7#585×C127—28no—
8#585×C127—28no—
9C2×2−1+4×C127—21no—
10C2×2−1+4×C127C414yescomplemented
11C2×H(5,𝔽2)×C127—22no—
12C2×H(5,𝔽2)×C127—21no—
13C2×H(5,𝔽2)×C127C414yescomplemented
14C22×#41×C127—24no—
15C22×#41×C127C2212yes—
16C22×#41×C127C2214yes—
17C22×#41×C127C2211yescharacteristic
18C22×M16×C127C2212yescomplemented
19C2×#117×C127C2211yescharacteristic, complemented
20C22×C42×C127C2211yescharacteristic, complemented
21#421×C127—28no—
22C2×#104×C127—28no—
23C2×#104×C127C2218yescomplemented
24C2×#104×C127C2218yescomplemented
25#408×C127—24no—
26#407×C127C2214yescomplemented
27#391×C127—21no—
28#390×C127—21no—
29#386×C127—22no—
30C4×C22⋊C4×C127C2214yescomplemented
312−1+4×C127—28no—
322−1+4×C127—28no—
332−1+4×C127—216no—
342−1+4×C127—216no—
35H(5,𝔽2)×C127—28no—
36H(5,𝔽2)×C127—28no—
37H(5,𝔽2)×C127—216no—
38H(5,𝔽2)×C127—216no—
39H(5,𝔽2)×C127—216no—
40H(5,𝔽2)×C127—216no—
41C2×#41×C127—232no—
42C2×#41×C127—232no—
43C2×#41×C127—216no—
44C2×#41×C127—216no—
45C2×#41×C127—24no—
46C2×#41×C127—28no—
47C2×#41×C127—24no—
48C2×#41×C127—24no—
49C2×#41×C127—22no—
50C2×#41×C127—22no—
51C2×#41×C127—24no—
52C2×#41×C127—24no—
53C2×#41×C127—24no—
54C2×#41×C127—24no—
55C2×#41×C127—24no—
56C2×#41×C127C2×C414yes—
57C2×#41×C127C2×C414yes—
58C2×#41×C127C2×C418yescomplemented
59C2×#41×C127D418yes—
60C2×#41×C127D418yes—
61C2×#41×C127D414yes—
62C2×#41×C127C2314yes—
63C22×Q8×C127—24no—
64C22×Q8×C127C2×C412yes—
65C22×Q8×C127C2×C414yescomplemented
66C22×Q8×C127D411yescharacteristic
67C22×D4×C127—22no—
68C22×D4×C127—24no—
69C22×D4×C127—24no—
70C22×D4×C127—28no—
71C22×D4×C127C2×C412yes—
72C22×D4×C127C2×C414yescomplemented
73C22×D4×C127D412yes—
74C22×D4×C127D411yescharacteristic
75C23×C4×C127—24no—
76C23×C4×C127—22no—
77C23×C4×C127—21no—
78C23×C4×C127C2312yes—
79C23×C4×C127C2311yescharacteristic
80C2×M16×C127—22no—
81C2×M16×C127—24no—
82C2×M16×C127—24no—
83C2×M16×C127C2314yes—
84C22×C8×C127—22no—
85#117×C127—24no—
86C2×C4⋊C4×C127—21no—
87C2×C22⋊C4×C127—21no—
88C2×C22⋊C4×C127—24no—
89C2×C42×C127—24no—
90C2×C42×C127—21no—
91C2×C42×C127C2312yes—
92C2×C42×C127C2312yes—
93#104×C127—216no—
94#104×C127—216no—
95C22⋊C8×C127—28no—
96#95×C127—24no—
97C24×C127—22no—
98C24×C127—24no—
99C24×C127C2×D412yes—
100C24×C127C2×D411yescharacteristic
101#41×C127—232no—
102#41×C127—216no—
103#41×C127—216no—
104#41×C127—264no—
105#41×C127—232no—
106#41×C127—232no—
107#41×C127—216no—
108#41×C127—216no—
109#41×C127—28no—
110#41×C127—28no—
111#41×C127—28no—
112#41×C127—28no—
113#41×C127—28no—
114#41×C127—28no—
115#41×C127—28no—
116#41×C127—28no—
117#41×C127—28no—
118#41×C127—24no—
119#41×C127—24no—
120#41×C127C22⋊C4116yescomplemented
121#41×C127C2×D418yes—
122#41×C127C2×D418yes—
123C2×Q8×C127—232no—
124C2×Q8×C127—216no—
125C2×Q8×C127—22no—
126C2×Q8×C127—24no—
127C2×Q8×C127—24no—
128C2×Q8×C127—22no—
129C2×Q8×C127—22no—
130C2×Q8×C127C22⋊C414yes—
131C2×Q8×C127C22⋊C418yescomplemented
132C2×Q8×C127C22×C414yes—
133C2×D4×C127—216no—
134C2×D4×C127—216no—
135C2×D4×C127—264no—
136C2×D4×C127—232no—
137C2×D4×C127—24no—
138C2×D4×C127—28no—
139C2×D4×C127—28no—
140C2×D4×C127—28no—
141C2×D4×C127—24no—
142C2×D4×C127—24no—
143C2×D4×C127—24no—
144C2×D4×C127—216no—
145C2×D4×C127—22no—
146C2×D4×C127—24no—
147C2×D4×C127—24no—
148C2×D4×C127—22no—
149C2×D4×C127—22no—
150C2×D4×C127C22⋊C418yes—
151C2×D4×C127C22⋊C414yes—
152C2×D4×C127C22⋊C418yescomplemented
153C2×D4×C127C22×C414yes—
154C22×C4×C127—232no—
155C22×C4×C127—22no—
156C22×C4×C127—24no—
157C22×C4×C127—24no—
158C22×C4×C127—28no—
159C22×C4×C127—28no—
160C22×C4×C127—28no—
161C22×C4×C127—24no—
162C22×C4×C127—24no—
163C22×C4×C127—24no—
164C22×C4×C127—22no—
165C22×C4×C127—22no—
166C22×C4×C127—44no—
167C22×C4×C127—42no—
168C22×C4×C127C22×C414yes—
169C22×C4×C127C22×C412yes—
170C22×C4×C127C2×D412yes—
171C22×C4×C127C2×D414yes—
172C22×C4×C127C2×D411yescharacteristic
173C22×C4×C127C2×D414yes—
174C22×C4×C127C2×D414yes—
175C22×C4×C127C2×D412yes—
176C22×C4×C127C2411yescharacteristic
177M16×C127—24no—
178M16×C127—24no—
179M16×C127—44no—
180C2×C8×C127—24no—
181C2×C8×C127—44no—
182C4⋊C4×C127—42no—
183C22⋊C4×C127—48no—
184C22⋊C4×C127—42no—
185C42×C127—24no—
186C42×C127—24no—
187C42×C127—42no—
188C23×C127—216no—
189C23×C127—24no—
190C23×C127—22no—
191C23×C127—22no—
192C23×C127—28no—
193C23×C127—42no—
194C23×C127—44no—
195C23×C127—84no—
196C23×C127—88no—
197C23×C127C2×C22⋊C414yes—
198C23×C127C2×C22⋊C412yes—
199C23×C127#12012yes—
200C23×C127#12012yes—
201C23×C127C22×D411yescharacteristic
202Q8×C127—216no—
203Q8×C127—24no—
204Q8×C127—28no—
205Q8×C127—28no—
206Q8×C127C2×C22⋊C418yes—
207D4×C127—232no—
208D4×C127—216no—
209D4×C127—28no—
210D4×C127—28no—
211D4×C127—216no—
212D4×C127—216no—
213D4×C127—24no—
214D4×C127—28no—
215D4×C127—28no—
216D4×C127C2×C22⋊C418yes—
217C2×C4×C127—216no—
218C2×C4×C127—232no—
219C2×C4×C127—24no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—28no—
223C2×C4×C127—28no—
224C2×C4×C127—28no—
225C2×C4×C127—28no—
226C2×C4×C127—28no—
227C2×C4×C127—216no—
228C2×C4×C127—24no—
229C2×C4×C127—42no—
230C2×C4×C127—48no—
231C2×C4×C127—42no—
232C2×C4×C127—44no—
233C2×C4×C127—42no—
234C2×C4×C127C2×C22⋊C414yes—
235C2×C4×C127C2×C22⋊C414yes—
236C2×C4×C127C2×C22⋊C414yes—
237C2×C4×C127C2×C22⋊C412yes—
238C2×C4×C127#12014yes—
239C2×C4×C127#12014yes—
240C2×C4×C127#12012yes—
241C2×C4×C127#12012yes—
242C2×C4×C127C23×C411yescharacteristic
243C2×C4×C127C22×D412yes—
244C2×C4×C127C22×D412yes—
245C2×C4×C127C22×D411yescharacteristic
246C8×C127—44no—
247C22×C127—22no—
248C22×C127—24no—
249C22×C127—28no—
250C22×C127—24no—
251C22×C127—28no—
252C22×C127—42no—
253C22×C127—42no—
254C22×C127—48no—
255C22×C127—82no—
256C22×C127—82no—
257C22×C127—88no—
258C22×C127—816no—
259C22×C127#37914yes—
260C22×C127C22×C22⋊C411yescharacteristic
261C22×C127C2×#12011yescharacteristic
262C22×C127C2×#12011yescharacteristic
263C4×C127—28no—
264C4×C127—24no—
265C4×C127—44no—
266C4×C127—82no—
267C4×C127#37914yes—
268C4×C127C22×C22⋊C412yes—
269C4×C127C2×#12012yes—
270#20606C12711yesmaximal, characteristic, direct product factor, Sylow
271C2×C127—21no—
272C2×C127—21no—
273C2×C127—24no—
274C2×C127—44no—
275C2×C127—48no—
276C2×C127C2×#37911yescharacteristic
277C2×C127#290112yes—
278C2×#585C2×C12711yescharacteristic
279C2×#421C2×C12712yescomplemented
280C22×#104C2×C12712yescomplemented
281#2879C2×C12711yescharacteristic, complemented
282#2297C2×C12711yescharacteristic, complemented
283#1789C2×C12718yescomplemented
284C127#2060611yescharacteristic, direct product factor, Sylow
285#585—28no—
286#585—28no—
287C2×2−1+4—21no—
288C2×2−1+4C4×C12714yescomplemented
289C2×H(5,𝔽2)—22no—
290C2×H(5,𝔽2)—21no—
291C2×H(5,𝔽2)C4×C12714yescomplemented
292C22×#41—24no—
293C22×#41C22×C12712yes—
294C22×#41C22×C12714yes—
295C22×#41C22×C12711yescharacteristic
296C22×M16C22×C12712yescomplemented
297C2×#117C22×C12711yescharacteristic, complemented
298C22×C42C22×C12711yescharacteristic, complemented
299#421—28no—
300C2×#104—28no—
301C2×#104C22×C12718yescomplemented
302C2×#104C22×C12718yescomplemented
303#408—24no—
304#407C22×C12714yescomplemented
305#391—21no—
306#390—21no—
307#386—22no—
308C4×C22⋊C4C22×C12714yescomplemented
3092−1+4—28no—
3102−1+4—28no—
3112−1+4—216no—
3122−1+4—216no—
313H(5,𝔽2)—28no—
314H(5,𝔽2)—28no—
315H(5,𝔽2)—216no—
316H(5,𝔽2)—216no—
317H(5,𝔽2)—216no—
318H(5,𝔽2)—216no—
319C2×#41—232no—
320C2×#41—232no—
321C2×#41—216no—
322C2×#41—216no—
323C2×#41—24no—
324C2×#41—28no—
325C2×#41—24no—
326C2×#41—24no—
327C2×#41—22no—
328C2×#41—22no—
329C2×#41—24no—
330C2×#41—24no—
331C2×#41—24no—
332C2×#41—24no—
333C2×#41—24no—
334C2×#41C2×C4×C12714yes—
335C2×#41C2×C4×C12714yes—
336C2×#41C2×C4×C12718yescomplemented
337C2×#41D4×C12718yes—
338C2×#41D4×C12718yes—
339C2×#41D4×C12714yes—
340C2×#41C23×C12714yes—
341C22×Q8—24no—
342C22×Q8C2×C4×C12712yes—
343C22×Q8C2×C4×C12714yescomplemented
344C22×Q8D4×C12711yescharacteristic
345C22×D4—22no—
346C22×D4—24no—
347C22×D4—24no—
348C22×D4—28no—
349C22×D4C2×C4×C12712yes—
350C22×D4C2×C4×C12714yescomplemented
351C22×D4D4×C12712yes—
352C22×D4D4×C12711yescharacteristic
353C23×C4—24no—
354C23×C4—22no—
355C23×C4—21no—
356C23×C4C23×C12712yes—
357C23×C4C23×C12711yescharacteristic
358C2×M16—22no—
359C2×M16—24no—
360C2×M16—24no—
361C2×M16C23×C12714yes—
362C22×C8—22no—
363#117—24no—
364C2×C4⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—24no—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42C23×C12712yes—
370C2×C42C23×C12712yes—
371#104—216no—
372#104—216no—
373C22⋊C8—28no—
374#95—24no—
375C24—22no—
376C24—24no—
377C24C2×D4×C12712yes—
378C24C2×D4×C12711yescharacteristic
379#41—232no—
380#41—216no—
381#41—216no—
382#41—264no—
383#41—232no—
384#41—232no—
385#41—216no—
386#41—216no—
387#41—28no—
388#41—28no—
389#41—28no—
390#41—28no—
391#41—28no—
392#41—28no—
393#41—28no—
394#41—28no—
395#41—28no—
396#41—24no—
397#41—24no—
398#41C22⋊C4×C127116yescomplemented
399#41C2×D4×C12718yes—
400#41C2×D4×C12718yes—
401C2×Q8—232no—
402C2×Q8—216no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C22⋊C4×C12714yes—
409C2×Q8C22⋊C4×C12718yescomplemented
410C2×Q8C22×C4×C12714yes—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—264no—
414C2×D4—232no—
415C2×D4—24no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—216no—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4C22⋊C4×C12718yes—
429C2×D4C22⋊C4×C12714yes—
430C2×D4C22⋊C4×C12718yescomplemented
431C2×D4C22×C4×C12714yes—
432C22×C4—232no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4C22×C4×C12714yes—
447C22×C4C22×C4×C12712yes—
448C22×C4C2×D4×C12712yes—
449C22×C4C2×D4×C12714yes—
450C22×C4C2×D4×C12711yescharacteristic
451C22×C4C2×D4×C12714yes—
452C22×C4C2×D4×C12714yes—
453C22×C4C2×D4×C12712yes—
454C22×C4C24×C12711yescharacteristic
455M16—24no—
456M16—24no—
457M16—44no—
458C2×C8—24no—
459C2×C8—44no—
460C4⋊C4—42no—
461C22⋊C4—48no—
462C22⋊C4—42no—
463C42—24no—
464C42—24no—
465C42—42no—
466C23—216no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—42no—
472C23—44no—
473C23—84no—
474C23—88no—
475C23C2×C22⋊C4×C12714yes—
476C23C2×C22⋊C4×C12712yes—
477C23#120×C12712yes—
478C23#120×C12712yes—
479C23C22×D4×C12711yescharacteristic
480Q8—216no—
481Q8—24no—
482Q8—28no—
483Q8—28no—
484Q8C2×C22⋊C4×C12718yes—
485D4—232no—
486D4—216no—
487D4—28no—
488D4—28no—
489D4—216no—
490D4—216no—
491D4—24no—
492D4—28no—
493D4—28no—
494D4C2×C22⋊C4×C12718yes—
495C2×C4—216no—
496C2×C4—232no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—216no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4C2×C22⋊C4×C12714yes—
513C2×C4C2×C22⋊C4×C12714yes—
514C2×C4C2×C22⋊C4×C12714yes—
515C2×C4C2×C22⋊C4×C12712yes—
516C2×C4#120×C12714yes—
517C2×C4#120×C12714yes—
518C2×C4#120×C12712yes—
519C2×C4#120×C12712yes—
520C2×C4C23×C4×C12711yescharacteristic
521C2×C4C22×D4×C12712yes—
522C2×C4C22×D4×C12712yes—
523C2×C4C22×D4×C12711yescharacteristic
524C8—44no—
525C22—22no—
526C22—24no—
527C22—28no—
528C22—24no—
529C22—28no—
530C22—42no—
531C22—42no—
532C22—48no—
533C22—82no—
534C22—82no—
535C22—88no—
536C22—816no—
537C22#379×C12714yes—
538C22C22×C22⋊C4×C12711yescharacteristic
539C22C2×#120×C12711yescharacteristic
540C22C2×#120×C12711yescharacteristic
541C4—28no—
542C4—24no—
543C4—44no—
544C4—82no—
545C4#379×C12714yes—
546C4C22×C22⋊C4×C12712yes—
547C4C2×#120×C12712yes—
548C2—21no—
549C2—21no—
550C2—24no—
551C2—44no—
552C2—48no—
553C2C2×#379×C12711yescharacteristic
554C2C127×#290112yes—
subgroupG/H
centerC2×C4×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC22×C127C22×C22⋊C4
prime radicalC2×#585×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#421×C127C2semidirect
3. C22×#104×C127C2semidirect
4. C127×#2879C2semidirect
5. C127×#2297C2semidirect
6. C127×#1789C2semidirect
10. C2×2−1+4×C127C4semidirect
13. C2×H(5,𝔽2)×C127C4semidirect
18. C22×M16×C127C22semidirect
19. C2×#117×C127C22semidirect
20. C22×C42×C127C22semidirect
23. C2×#104×C127C22semidirect
24. C2×#104×C127C22semidirect
26. #407×C127C22semidirect
30. C4×C22⋊C4×C127C22semidirect
58. C2×#41×C127C2×C4semidirect
65. C22×Q8×C127C2×C4semidirect
72. C22×D4×C127C2×C4semidirect
120. #41×C127C22⋊C4semidirect
131. C2×Q8×C127C22⋊C4semidirect
152. C2×D4×C127C22⋊C4semidirect
270. #20606C127direct (N × H)
279. C2×#421C2×C127semidirect
280. C22×#104C2×C127semidirect
281. #2879C2×C127semidirect
282. #2297C2×C127semidirect
283. #1789C2×C127semidirect
284. C127#20606direct (N × H)
288. C2×2−1+4C4×C127semidirect
291. C2×H(5,𝔽2)C4×C127semidirect
296. C22×M16C22×C127semidirect
297. C2×#117C22×C127semidirect
298. C22×C42C22×C127semidirect
301. C2×#104C22×C127semidirect
302. C2×#104C22×C127semidirect
304. #407C22×C127semidirect
308. C4×C22⋊C4C22×C127semidirect
336. C2×#41C2×C4×C127semidirect
343. C22×Q8C2×C4×C127semidirect
350. C22×D4C2×C4×C127semidirect
398. #41C22⋊C4×C127semidirect
409. C2×Q8C22⋊C4×C127semidirect
430. C2×D4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#585×C1272
C2×#421×C1272
C22×#104×C1272
C127×#28792
C127×#22972
C127×#17892
#20606127