Finite Group Databasebeta

← C127×#8503 #44572176

Subgroups — C127×#8503 #44572176

Order 32512.

672 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3554C211yesmaximal, characteristic
2C127×#1554C211yesmaximal, characteristic, complemented
3C127×#1535C211yesmaximal, characteristic, complemented
4C127×#1521C211yesmaximal, characteristic, complemented
5C127×#1515C211yesmaximal, characteristic, complemented
6C127×#1514C211yesmaximal, characteristic, complemented
7C127×#1496C211yesmaximal, characteristic, complemented
8#562×C127—22no—
9#559×C127—21no—
10#559×C127C411yescharacteristic, complemented
11#557×C127C411yescharacteristic, complemented
12#554×C127—21no—
13#550×C127C411yescharacteristic, complemented
14D4×Q8×C127C411yescharacteristic, complemented
15#548×C127—21no—
16#547×C127—21no—
17#542×C127—21no—
18#542×C127—21no—
19#540×C127—21no—
20#539×C127—22no—
21#536×C127—21no—
22#533×C127C2211yescharacteristic
23#532×C127C2211yescharacteristic
24#529×C127C2211yescharacteristic
25#432×C127C2211yescharacteristic
26#432×C127C2211yescharacteristic
27#431×C127C2211yescharacteristic
28#383×C127C2211yescharacteristic
29#333×C127—21no—
30#332×C127—21no—
31#330×C127—21no—
32#329×C127—21no—
33#328×C127—21no—
34#328×C127—21no—
35#328×C127—21no—
36#327×C127—22no—
37#327×C127—21no—
38D4⋊C8×C127—22no—
39C2×#41×C127—21no—
40C2×#41×C127—21no—
41C22×Q8×C127—21no—
42C4⋊Q8×C127—21no—
43C4⋊Q8×C127—21no—
44C4⋊Q8×C127C2×C411yescharacteristic
45C4⋊Q8×C127D411yescharacteristic
46C4⋊Q8×C127D411yescharacteristic
47#127×C127—21no—
48#127×C127C2×C411yescharacteristic
49#126×C127—22no—
50#126×C127—21no—
51#125×C127—22no—
52#125×C127—21no—
53#125×C127—21no—
54#125×C127D411yescharacteristic
55#124×C127—22no—
56#124×C127—22no—
57#124×C127—21no—
58#124×C127—21no—
59#124×C127—21no—
60#124×C127D411yescharacteristic
61#123×C127—22no—
62#123×C127—22no—
63#123×C127—22no—
64#123×C127—22no—
65#123×C127—22no—
66#123×C127—22no—
67#123×C127—21no—
68C22⋊Q8×C127—22no—
69C22⋊Q8×C127—22no—
70C22⋊Q8×C127—22no—
71C22⋊Q8×C127—22no—
72C22⋊Q8×C127—21no—
73C22⋊Q8×C127—22no—
74C22⋊Q8×C127—21no—
75C22⋊Q8×C127D411yescharacteristic
76C22⋊Q8×C127D411yescharacteristic
77C22⋊Q8×C127D411yescharacteristic
78#121×C127—22no—
79#121×C127—22no—
80#121×C127—22no—
81#121×C127—21no—
82#121×C127—21no—
83#121×C127D411yescharacteristic
84#121×C127D412yes—
85C4×Q8×C127—21no—
86C4×Q8×C127—21no—
87C4×Q8×C127—21no—
88C4×Q8×C127C2×C411yescharacteristic
89C4×Q8×C127C2×C411yescharacteristic
90C4×Q8×C127C2×C411yescharacteristic
91C4×D4×C127—22no—
92C4×D4×C127—21no—
93C4×D4×C127—21no—
94C4×D4×C127—21no—
95C4×D4×C127—21no—
96C4×D4×C127—21no—
97C4×D4×C127—21no—
98C4×D4×C127C2×C411yescharacteristic
99C4×D4×C127D412yes—
100#117×C127—21no—
101#117×C127—21no—
102C2×C4⋊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—
116#41×C127—82no—
117#41×C127—82no—
118C2×Q8×C127—22no—
119C2×Q8×C127—21no—
120C2×Q8×C127—21no—
121C2×Q8×C127—82no—
122C2×Q8×C127C22⋊C411yescharacteristic
123C2×Q8×C127C22⋊C411yescharacteristic
124C2×Q8×C127C22⋊C411yescharacteristic
125C2×D4×C127—22no—
126C2×D4×C127—22no—
127C2×D4×C127—21no—
128C2×D4×C127—22no—
129C2×D4×C127—21no—
130C2×D4×C127—21no—
131C2×D4×C127C22⋊C411yescharacteristic
132C2×D4×C127C22⋊C412yes—
133C22×C4×C127—22no—
134C22×C4×C127—22no—
135C22×C4×C127—22no—
136C22×C4×C127—21no—
137C22×C4×C127—22no—
138C22×C4×C127—41no—
139C22×C4×C127C2×D411yescharacteristic
140C22×C4×C127C2×D411yescharacteristic
141C22×C4×C127C2×D411yescharacteristic
142C2×C8×C127—41no—
143C2×C8×C127—41no—
144C2×C8×C127—41no—
145C4⋊C4×C127—22no—
146C4⋊C4×C127—22no—
147C4⋊C4×C127—22no—
148C4⋊C4×C127—22no—
149C4⋊C4×C127—21no—
150C4⋊C4×C127—21no—
151C4⋊C4×C127—21no—
152C4⋊C4×C127—21no—
153C4⋊C4×C127—21no—
154C4⋊C4×C127—21no—
155C4⋊C4×C127—21no—
156C4⋊C4×C127—21no—
157C4⋊C4×C127—21no—
158C4⋊C4×C127—21no—
159C4⋊C4×C127—21no—
160C4⋊C4×C127—21no—
161C4⋊C4×C127C22⋊C411yescharacteristic
162C4⋊C4×C127C22⋊C411yescharacteristic
163C4⋊C4×C127C22⋊C411yescharacteristic
164C4⋊C4×C127C22⋊C412yes—
165C4⋊C4×C127C22⋊C411yescharacteristic
166C22⋊C4×C127—22no—
167C22⋊C4×C127—22no—
168C22⋊C4×C127—22no—
169C22⋊C4×C127—22no—
170C22⋊C4×C127—22no—
171C22⋊C4×C127—21no—
172C22⋊C4×C127—21no—
173C22⋊C4×C127—21no—
174C22⋊C4×C127—21no—
175C22⋊C4×C127—21no—
176C22⋊C4×C127—21no—
177C22⋊C4×C127—21no—
178C22⋊C4×C127—22no—
179C22⋊C4×C127—22no—
180C42×C127—22no—
181C42×C127—21no—
182C42×C127—21no—
183C42×C127—21no—
184C42×C127C22×C411yescharacteristic
185C42×C127C2×D411yescharacteristic
186C42×C127C2×D411yescharacteristic
187C42×C127C2×D411yescharacteristic
188C23×C127—22no—
189C23×C127—21no—
190C23×C127#12011yescharacteristic
191Q8×C127—41no—
192Q8×C127—41no—
193Q8×C127—41no—
194Q8×C127—42no—
195Q8×C127—81no—
196Q8×C127—82no—
197D4×C127—42no—
198D4×C127—41no—
199D4×C127—42no—
200D4×C127—42no—
201D4×C127—81no—
202D4×C127—82no—
203D4×C127—82no—
204D4×C127—82no—
205C2×C4×C127—21no—
206C2×C4×C127—21no—
207C2×C4×C127—21no—
208C2×C4×C127—21no—
209C2×C4×C127—21no—
210C2×C4×C127—22no—
211C2×C4×C127—21no—
212C2×C4×C127—21no—
213C2×C4×C127—41no—
214C2×C4×C127—41no—
215C2×C4×C127—41no—
216C2×C4×C127—42no—
217C2×C4×C127—82no—
218C2×C4×C127—82no—
219C2×C4×C127—82no—
220C2×C4×C127—82no—
221C2×C4×C127—81no—
222C2×C4×C127—81no—
223C2×C4×C127—81no—
224C2×C4×C127C2×C22⋊C411yescharacteristic
225C2×C4×C127C2×C22⋊C411yescharacteristic
226C2×C4×C127C2×C22⋊C411yescharacteristic
227C2×C4×C127#12011yescharacteristic
228C2×C4×C127#12011yescharacteristic
229C2×C4×C127#12011yescharacteristic
230C8×C127—81no—
231C8×C127—81no—
232C8×C127—81no—
233C22×C127—21no—
234C22×C127—21no—
235C22×C127—21no—
236C22×C127—42no—
237C22×C127—41no—
238C22×C127—41no—
239C22×C127—42no—
240C22×C127—42no—
241C22×C127—41no—
242C22×C127#37911yescharacteristic
243C4×C127—21no—
244C4×C127—21no—
245C4×C127—21no—
246C4×C127—21no—
247C4×C127—21no—
248C4×C127—21no—
249C4×C127—41no—
250C4×C127—41no—
251C4×C127—41no—
252C4×C127—41no—
253C4×C127—41no—
254C4×C127—42no—
255C4×C127—41no—
256C4×C127—41no—
257C4×C127—161no—
258#8503C12711yesmaximal, characteristic, direct product factor, Sylow
259C2×C127—41no—
260C2×C127—82no—
261C2×C127—81no—
262C2×C127#179411yescharacteristic
263C2×C127#179511yescharacteristic
264C2×C127#179611yescharacteristic
265#3554C2×C12711yescharacteristic
266#1554C2×C12711yescharacteristic, complemented
267#1535C2×C12711yescharacteristic, complemented
268#1521C2×C12711yescharacteristic, complemented
269#1515C2×C12711yescharacteristic, complemented
270#1514C2×C12711yescharacteristic, complemented
271#1496C2×C12711yescharacteristic, complemented
272C127#850311yescharacteristic, direct product factor, Sylow
273#562—22no—
274#559—21no—
275#559C4×C12711yescharacteristic, complemented
276#557C4×C12711yescharacteristic, complemented
277#554—21no—
278#550C4×C12711yescharacteristic, complemented
279D4×Q8C4×C12711yescharacteristic, complemented
280#548—21no—
281#547—21no—
282#542—21no—
283#542—21no—
284#540—21no—
285#539—22no—
286#536—21no—
287#533C22×C12711yescharacteristic
288#532C22×C12711yescharacteristic
289#529C22×C12711yescharacteristic
290#432C22×C12711yescharacteristic
291#432C22×C12711yescharacteristic
292#431C22×C12711yescharacteristic
293#383C22×C12711yescharacteristic
294#333—21no—
295#332—21no—
296#330—21no—
297#329—21no—
298#328—21no—
299#328—21no—
300#328—21no—
301#327—22no—
302#327—21no—
303D4⋊C8—22no—
304C2×#41—21no—
305C2×#41—21no—
306C22×Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8C2×C4×C12711yescharacteristic
310C4⋊Q8D4×C12711yescharacteristic
311C4⋊Q8D4×C12711yescharacteristic
312#127—21no—
313#127C2×C4×C12711yescharacteristic
314#126—22no—
315#126—21no—
316#125—22no—
317#125—21no—
318#125—21no—
319#125D4×C12711yescharacteristic
320#124—22no—
321#124—22no—
322#124—21no—
323#124—21no—
324#124—21no—
325#124D4×C12711yescharacteristic
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332#123—21no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8D4×C12711yescharacteristic
341C22⋊Q8D4×C12711yescharacteristic
342C22⋊Q8D4×C12711yescharacteristic
343#121—22no—
344#121—22no—
345#121—22no—
346#121—21no—
347#121—21no—
348#121D4×C12711yescharacteristic
349#121D4×C12712yes—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8C2×C4×C12711yescharacteristic
354C4×Q8C2×C4×C12711yescharacteristic
355C4×Q8C2×C4×C12711yescharacteristic
356C4×D4—22no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×C4×C12711yescharacteristic
364C4×D4D4×C12712yes—
365#117—21no—
366#117—21no—
367C2×C4⋊C4—22no—
368C2×C42C23×C12711yescharacteristic
369C4⋊C8—21no—
370C4⋊C8—21no—
371C4⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375#97—21no—
376#97—21no—
377C4×C8—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#41—82no—
382#41—82no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—82no—
387C2×Q8C22⋊C4×C12711yescharacteristic
388C2×Q8C22⋊C4×C12711yescharacteristic
389C2×Q8C22⋊C4×C12711yescharacteristic
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4C22⋊C4×C12711yescharacteristic
397C2×D4C22⋊C4×C12712yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4C2×D4×C12711yescharacteristic
405C22×C4C2×D4×C12711yescharacteristic
406C22×C4C2×D4×C12711yescharacteristic
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4C22⋊C4×C12711yescharacteristic
427C4⋊C4C22⋊C4×C12711yescharacteristic
428C4⋊C4C22⋊C4×C12711yescharacteristic
429C4⋊C4C22⋊C4×C12712yes—
430C4⋊C4C22⋊C4×C12711yescharacteristic
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42C22×C4×C12711yescharacteristic
450C42C2×D4×C12711yescharacteristic
451C42C2×D4×C12711yescharacteristic
452C42C2×D4×C12711yescharacteristic
453C23—22no—
454C23—21no—
455C23#120×C12711yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—81no—
461Q8—82no—
462D4—42no—
463D4—41no—
464D4—42no—
465D4—42no—
466D4—82no—
467D4—81no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4C2×C22⋊C4×C12711yescharacteristic
490C2×C4C2×C22⋊C4×C12711yescharacteristic
491C2×C4C2×C22⋊C4×C12711yescharacteristic
492C2×C4#120×C12711yescharacteristic
493C2×C4#120×C12711yescharacteristic
494C2×C4#120×C12711yescharacteristic
495C8—81no—
496C8—81no—
497C8—81no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—42no—
502C22—41no—
503C22—41no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22#379×C12711yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—161no—
523C2—41no—
524C2—82no—
525C2—81no—
526C2C127×#179411yescharacteristic
527C2C127×#179511yescharacteristic
528C2C127×#179611yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3554C2
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×#1554C2semidirect
3. C127×#1535C2semidirect
4. C127×#1521C2semidirect
5. C127×#1515C2semidirect
6. C127×#1514C2semidirect
7. C127×#1496C2semidirect
10. #559×C127C4semidirect
11. #557×C127C4semidirect
13. #550×C127C4semidirect
14. D4×Q8×C127C4semidirect
258. #8503C127direct (N × H)
266. #1554C2×C127semidirect
267. #1535C2×C127semidirect
268. #1521C2×C127semidirect
269. #1515C2×C127semidirect
270. #1514C2×C127semidirect
271. #1496C2×C127semidirect
272. C127#8503direct (N × H)
275. #559C4×C127semidirect
276. #557C4×C127semidirect
278. #550C4×C127semidirect
279. D4×Q8C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35542
C127×#15542
C127×#15352
C127×#15212
C127×#15152
C127×#15142
C127×#14962
#8503127