Finite Group Databasebeta

← C127×#8500 #44572173

Subgroups — C127×#8500 #44572173

Order 32512.

736 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3546C211yesmaximal, characteristic
2C127×#1553C211yesmaximal, characteristic, complemented
3C127×#1535C211yesmaximal, characteristic, complemented
4C127×#1521C211yesmaximal, characteristic, complemented
5C127×#1514C211yesmaximal, characteristic, complemented
6C127×#1514C211yesmaximal, characteristic, complemented
7C127×#1496C211yesmaximal, characteristic, complemented
8#560×C127—22no—
9#556×C127C411yescharacteristic, complemented
10#553×C127—21no—
11#550×C127—21no—
12#550×C127C411yescharacteristic, complemented
13#547×C127C411yescharacteristic, complemented
14#546×C127—21no—
15#546×C127—21no—
16D42×C127C411yescharacteristic, complemented
17#542×C127—21no—
18#538×C127—22no—
19#537×C127—21no—
20#535×C127—21no—
21#535×C127—21no—
22#532×C127C2211yescharacteristic
23#532×C127C2211yescharacteristic
24#529×C127C2211yescharacteristic
25#432×C127C2211yescharacteristic, complemented
26#432×C127C2211yescharacteristic, complemented
27#431×C127C2211yescharacteristic, complemented
28#383×C127C2211yescharacteristic, complemented
29#332×C127—21no—
30#331×C127—21no—
31#330×C127—21no—
32#329×C127—21no—
33#328×C127—21no—
34#327×C127—21no—
35#327×C127—21no—
36#327×C127—22no—
37#327×C127—21no—
38D4⋊C8×C127—22no—
39C2×#41×C127—21no—
40C2×#41×C127—21no—
41C2×#41×C127—21no—
42C2×#41×C127—21no—
43C22×D4×C127—21no—
44C22×D4×C127—21no—
45C4⋊Q8×C127C2×C411yescharacteristic
46C4⋊Q8×C127D411yescharacteristic
47#127×C127—22no—
48#127×C127—21no—
49#127×C127C2×C411yescharacteristic
50#127×C127D411yescharacteristic
51#126×C127—22no—
52#126×C127—22no—
53#126×C127—21no—
54#125×C127D411yescharacteristic
55#124×C127—22no—
56#124×C127—21no—
57#124×C127—21no—
58#124×C127—21no—
59#124×C127D411yescharacteristic
60#123×C127—22no—
61#123×C127—22no—
62#123×C127—21no—
63#123×C127—22no—
64C22⋊Q8×C127—22no—
65C22⋊Q8×C127—21no—
66C22⋊Q8×C127—21no—
67C22⋊Q8×C127D411yescharacteristic
68#121×C127—22no—
69#121×C127—22no—
70#121×C127—22no—
71#121×C127—22no—
72#121×C127—22no—
73#121×C127—21no—
74#121×C127—22no—
75#121×C127—21no—
76#121×C127—21no—
77#121×C127—21no—
78#121×C127—21no—
79#121×C127—21no—
80#121×C127D411yescharacteristic
81#121×C127D411yescharacteristic
82#121×C127D411yescharacteristic
83#121×C127D412yes—
84#120×C127—22no—
85#120×C127—22no—
86#120×C127—22no—
87#120×C127—22no—
88C4×Q8×C127—21no—
89C4×Q8×C127C2×C411yescharacteristic
90C4×D4×C127—22no—
91C4×D4×C127—21no—
92C4×D4×C127—21no—
93C4×D4×C127—21no—
94C4×D4×C127—21no—
95C4×D4×C127C2×C411yescharacteristic
96C4×D4×C127C2×C411yescharacteristic
97C4×D4×C127C2×C411yescharacteristic
98C4×D4×C127D412yes—
99#117×C127—21no—
100#117×C127—21no—
101C2×C4⋊C4×C127—21no—
102C2×C22⋊C4×C127—22no—
103C2×C42×C127C2311yescharacteristic
104C4⋊C8×C127—21no—
105C4⋊C8×C127—21no—
106C4⋊C8×C127—21no—
107C22⋊C8×C127—21no—
108C22⋊C8×C127—21no—
109C22⋊C8×C127—21no—
110#97×C127—21no—
111#97×C127—21no—
112C4×C8×C127—21no—
113#95×C127—21no—
114#95×C127—21no—
115#95×C127—21no—
116C24×C127—22no—
117#41×C127—82no—
118#41×C127—82no—
119#41×C127—82no—
120#41×C127—82no—
121C2×Q8×C127—21no—
122C2×Q8×C127—21no—
123C2×Q8×C127C22⋊C411yescharacteristic
124C2×D4×C127—22no—
125C2×D4×C127—22no—
126C2×D4×C127—22no—
127C2×D4×C127—22no—
128C2×D4×C127—22no—
129C2×D4×C127—21no—
130C2×D4×C127—21no—
131C2×D4×C127—22no—
132C2×D4×C127—21no—
133C2×D4×C127—21no—
134C2×D4×C127—21no—
135C2×D4×C127—21no—
136C2×D4×C127—21no—
137C2×D4×C127—82no—
138C2×D4×C127—82no—
139C2×D4×C127C22⋊C411yescharacteristic
140C2×D4×C127C22⋊C411yescharacteristic
141C2×D4×C127C22⋊C411yescharacteristic
142C2×D4×C127C22⋊C412yes—
143C22×C4×C127—22no—
144C22×C4×C127—22no—
145C22×C4×C127—21no—
146C22×C4×C127—21no—
147C22×C4×C127—21no—
148C22×C4×C127—22no—
149C22×C4×C127—41no—
150C22×C4×C127C2×D411yescharacteristic
151C22×C4×C127C2×D411yescharacteristic
152C22×C4×C127C2×D411yescharacteristic
153C2×C8×C127—41no—
154C2×C8×C127—41no—
155C2×C8×C127—41no—
156C4⋊C4×C127—22no—
157C4⋊C4×C127—22no—
158C4⋊C4×C127—21no—
159C4⋊C4×C127—21no—
160C4⋊C4×C127—21no—
161C4⋊C4×C127—21no—
162C4⋊C4×C127—21no—
163C4⋊C4×C127—21no—
164C4⋊C4×C127C22⋊C411yescharacteristic
165C4⋊C4×C127C22⋊C411yescharacteristic
166C4⋊C4×C127C22⋊C411yescharacteristic
167C4⋊C4×C127C22⋊C411yescharacteristic
168C4⋊C4×C127C22⋊C412yes—
169C22⋊C4×C127—22no—
170C22⋊C4×C127—22no—
171C22⋊C4×C127—22no—
172C22⋊C4×C127—22no—
173C22⋊C4×C127—22no—
174C22⋊C4×C127—21no—
175C22⋊C4×C127—21no—
176C22⋊C4×C127—21no—
177C22⋊C4×C127—21no—
178C22⋊C4×C127—21no—
179C22⋊C4×C127—21no—
180C22⋊C4×C127—21no—
181C22⋊C4×C127—21no—
182C22⋊C4×C127—21no—
183C22⋊C4×C127—22no—
184C22⋊C4×C127—22no—
185C42×C127—22no—
186C42×C127—21no—
187C42×C127C22×C411yescharacteristic
188C42×C127C2×D411yescharacteristic
189C42×C127C2×D411yescharacteristic
190C42×C127C2×D411yescharacteristic
191C23×C127—21no—
192C23×C127—21no—
193C23×C127—22no—
194C23×C127—21no—
195C23×C127—82no—
196C23×C127—82no—
197C23×C127—82no—
198C23×C127#12011yescharacteristic
199Q8×C127—41no—
200Q8×C127—42no—
201Q8×C127—81no—
202D4×C127—41no—
203D4×C127—41no—
204D4×C127—42no—
205D4×C127—41no—
206D4×C127—42no—
207D4×C127—42no—
208D4×C127—81no—
209D4×C127—81no—
210D4×C127—81no—
211D4×C127—81no—
212D4×C127—81no—
213D4×C127—82no—
214D4×C127—82no—
215D4×C127—82no—
216D4×C127—82no—
217D4×C127—82no—
218D4×C127—82no—
219C2×C4×C127—21no—
220C2×C4×C127—21no—
221C2×C4×C127—21no—
222C2×C4×C127—22no—
223C2×C4×C127—21no—
224C2×C4×C127—21no—
225C2×C4×C127—41no—
226C2×C4×C127—41no—
227C2×C4×C127—41no—
228C2×C4×C127—42no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—82no—
232C2×C4×C127—82no—
233C2×C4×C127—82no—
234C2×C4×C127—81no—
235C2×C4×C127—81no—
236C2×C4×C127—81no—
237C2×C4×C127C2×C22⋊C411yescharacteristic
238C2×C4×C127C2×C22⋊C411yescharacteristic
239C2×C4×C127C2×C22⋊C411yescharacteristic
240C2×C4×C127#12011yescharacteristic
241C2×C4×C127#12011yescharacteristic
242C2×C4×C127#12011yescharacteristic
243C8×C127—81no—
244C8×C127—81no—
245C8×C127—81no—
246C22×C127—21no—
247C22×C127—21no—
248C22×C127—21no—
249C22×C127—42no—
250C22×C127—41no—
251C22×C127—41no—
252C22×C127—41no—
253C22×C127—41no—
254C22×C127—41no—
255C22×C127—41no—
256C22×C127—41no—
257C22×C127—42no—
258C22×C127—42no—
259C22×C127—41no—
260C22×C127—41no—
261C22×C127—164no—
262C22×C127#37911yescharacteristic
263C4×C127—21no—
264C4×C127—21no—
265C4×C127—21no—
266C4×C127—21no—
267C4×C127—21no—
268C4×C127—21no—
269C4×C127—41no—
270C4×C127—41no—
271C4×C127—41no—
272C4×C127—41no—
273C4×C127—41no—
274C4×C127—42no—
275C4×C127—161no—
276#8500C12711yesmaximal, characteristic, direct product factor, Sylow
277C2×C127—41no—
278C2×C127—81no—
279C2×C127—82no—
280C2×C127—81no—
281C2×C127—81no—
282C2×C127#179411yescharacteristic
283C2×C127#179511yescharacteristic
284C2×C127#179511yescharacteristic
285#3546C2×C12711yescharacteristic
286#1553C2×C12711yescharacteristic, complemented
287#1535C2×C12711yescharacteristic, complemented
288#1521C2×C12711yescharacteristic, complemented
289#1514C2×C12711yescharacteristic, complemented
290#1514C2×C12711yescharacteristic, complemented
291#1496C2×C12711yescharacteristic, complemented
292C127#850011yescharacteristic, direct product factor, Sylow
293#560—22no—
294#556C4×C12711yescharacteristic, complemented
295#553—21no—
296#550—21no—
297#550C4×C12711yescharacteristic, complemented
298#547C4×C12711yescharacteristic, complemented
299#546—21no—
300#546—21no—
301D42C4×C12711yescharacteristic, complemented
302#542—21no—
303#538—22no—
304#537—21no—
305#535—21no—
306#535—21no—
307#532C22×C12711yescharacteristic
308#532C22×C12711yescharacteristic
309#529C22×C12711yescharacteristic
310#432C22×C12711yescharacteristic, complemented
311#432C22×C12711yescharacteristic, complemented
312#431C22×C12711yescharacteristic, complemented
313#383C22×C12711yescharacteristic, complemented
314#332—21no—
315#331—21no—
316#330—21no—
317#329—21no—
318#328—21no—
319#327—21no—
320#327—21no—
321#327—22no—
322#327—21no—
323D4⋊C8—22no—
324C2×#41—21no—
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—21no—
328C22×D4—21no—
329C22×D4—21no—
330C4⋊Q8C2×C4×C12711yescharacteristic
331C4⋊Q8D4×C12711yescharacteristic
332#127—22no—
333#127—21no—
334#127C2×C4×C12711yescharacteristic
335#127D4×C12711yescharacteristic
336#126—22no—
337#126—22no—
338#126—21no—
339#125D4×C12711yescharacteristic
340#124—22no—
341#124—21no—
342#124—21no—
343#124—21no—
344#124D4×C12711yescharacteristic
345#123—22no—
346#123—22no—
347#123—21no—
348#123—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8D4×C12711yescharacteristic
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121—22no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121—21no—
365#121D4×C12711yescharacteristic
366#121D4×C12711yescharacteristic
367#121D4×C12711yescharacteristic
368#121D4×C12712yes—
369#120—22no—
370#120—22no—
371#120—22no—
372#120—22no—
373C4×Q8—21no—
374C4×Q8C2×C4×C12711yescharacteristic
375C4×D4—22no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C2×C4×C12711yescharacteristic
381C4×D4C2×C4×C12711yescharacteristic
382C4×D4C2×C4×C12711yescharacteristic
383C4×D4D4×C12712yes—
384#117—21no—
385#117—21no—
386C2×C4⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C42C23×C12711yescharacteristic
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8—21no—
394C22⋊C8—21no—
395#97—21no—
396#97—21no—
397C4×C8—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401C24—22no—
402#41—82no—
403#41—82no—
404#41—82no—
405#41—82no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C22⋊C4×C12711yescharacteristic
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4C22⋊C4×C12711yescharacteristic
425C2×D4C22⋊C4×C12711yescharacteristic
426C2×D4C22⋊C4×C12711yescharacteristic
427C2×D4C22⋊C4×C12712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4C2×D4×C12711yescharacteristic
436C22×C4C2×D4×C12711yescharacteristic
437C22×C4C2×D4×C12711yescharacteristic
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4C22⋊C4×C12711yescharacteristic
450C4⋊C4C22⋊C4×C12711yescharacteristic
451C4⋊C4C22⋊C4×C12711yescharacteristic
452C4⋊C4C22⋊C4×C12711yescharacteristic
453C4⋊C4C22⋊C4×C12712yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C42—22no—
471C42—21no—
472C42C22×C4×C12711yescharacteristic
473C42C2×D4×C12711yescharacteristic
474C42C2×D4×C12711yescharacteristic
475C42C2×D4×C12711yescharacteristic
476C23—21no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23#120×C12711yescharacteristic
484Q8—41no—
485Q8—42no—
486Q8—81no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—82no—
498D4—81no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—82no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4C2×C22⋊C4×C12711yescharacteristic
523C2×C4C2×C22⋊C4×C12711yescharacteristic
524C2×C4C2×C22⋊C4×C12711yescharacteristic
525C2×C4#120×C12711yescharacteristic
526C2×C4#120×C12711yescharacteristic
527C2×C4#120×C12711yescharacteristic
528C8—81no—
529C8—81no—
530C8—81no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—41no—
545C22—41no—
546C22—164no—
547C22#379×C12711yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—41no—
560C4—161no—
561C2—41no—
562C2—81no—
563C2—81no—
564C2—82no—
565C2—81no—
566C2C127×#179411yescharacteristic
567C2C127×#179511yescharacteristic
568C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3546C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1553C2semidirect
3. C127×#1535C2semidirect
4. C127×#1521C2semidirect
5. C127×#1514C2semidirect
6. C127×#1514C2semidirect
7. C127×#1496C2semidirect
9. #556×C127C4semidirect
12. #550×C127C4semidirect
13. #547×C127C4semidirect
16. D42×C127C4semidirect
25. #432×C127C22semidirect
26. #432×C127C22semidirect
27. #431×C127C22semidirect
28. #383×C127C22semidirect
276. #8500C127direct (N × H)
286. #1553C2×C127semidirect
287. #1535C2×C127semidirect
288. #1521C2×C127semidirect
289. #1514C2×C127semidirect
290. #1514C2×C127semidirect
291. #1496C2×C127semidirect
292. C127#8500direct (N × H)
294. #556C4×C127semidirect
297. #550C4×C127semidirect
298. #547C4×C127semidirect
301. D42C4×C127semidirect
310. #432C22×C127semidirect
311. #432C22×C127semidirect
312. #431C22×C127semidirect
313. #383C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35462
C127×#15532
C127×#15352
C127×#15212
C127×#15142
C127×#14962
#8500127