Finite Group Databasebeta

← C127×#18666 #44582339

Subgroups — C127×#18666 #44582339

Order 32512.

936 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3507C211yesmaximal, characteristic, complemented
2C127×#3246C211yesmaximal, characteristic, complemented
3C127×#3169C211yesmaximal, characteristic
4C127×#3037C211yesmaximal, characteristic, complemented
5C127×#1648C212yesmaximal, complemented
6C127×#1640C212yesmaximal, complemented
7C127×#1626C212yesmaximal, complemented
8C127×#1620C212yesmaximal, complemented
9C127×#1547C211yesmaximal, characteristic, complemented
10C127×#1493C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8×C127C211yesmaximal, characteristic, complemented
12C2×#137×C127—21no—
13#559×C127C2212yes—
14#557×C127C2212yes—
15#556×C127—21no—
16#556×C127—22no—
17#554×C127—21no—
18#552×C127—22no—
19#539×C127—22no—
20#536×C127—21no—
21#533×C127C2211yescharacteristic
22#532×C127C2211yescharacteristic
23#532×C127C2211yescharacteristic
24#529×C127—22no—
25#520×C127—21no—
26C2×C4×Q8×C127C2211yescharacteristic
27#500×C127C2211yescharacteristic
28#493×C127C2211yescharacteristic
29#486×C127C2211yescharacteristic
30#486×C127C2211yescharacteristic
31#476×C127—21no—
32#475×C127C2212yes—
33#466×C127—22no—
34#465×C127—22no—
35#464×C127—22no—
36#462×C127C2212yes—
37#461×C127—21no—
38#460×C127C2212yes—
39#460×C127C2212yes—
40C4×D8×C127—22no—
41#423×C127C2211yescharacteristic
42C2×C4⋊C8×C127C2211yescharacteristic
43#417×C127—21no—
44C2×C4×C8×C127C2211yescharacteristic
45#332×C127C2212yes—
46#331×C127C2212yes—
47Q8⋊C8×C127C2212yes—
48Q8⋊C8×C127C2212yes—
49Q8⋊C8×C127C2212yes—
50Q8⋊C8×C127C2212yes—
51C2×#41×C127—21no—
52C2×#41×C127—21no—
53C22×Q8×C127D411yescharacteristic
54#137×C127—44no—
55C2×Q16×C127—22no—
56C2×SD16×C127—22no—
57C2×SD16×C127—22no—
58C2×D8×C127—22no—
59C2×M16×C127—21no—
60C22×C8×C127—21no—
61C22×C8×C127—21no—
62C4⋊Q8×C127—22no—
63C4⋊Q8×C127D411yescharacteristic
64C4⋊Q8×C127D411yescharacteristic
65C4⋊Q8×C127C2311yescharacteristic
66C4⋊Q8×C127C2311yescharacteristic
67#127×C127C2311yescharacteristic
68#127×C127C2311yescharacteristic
69#126×C127—24no—
70#126×C127—24no—
71#125×C127—22no—
72#125×C127—24no—
73#125×C127—21no—
74#124×C127—22no—
75#124×C127—22no—
76#124×C127—21no—
77#124×C127—21no—
78#123×C127—24no—
79#123×C127—24no—
80C22⋊Q8×C127—22no—
81C22⋊Q8×C127—22no—
82C22⋊Q8×C127—24no—
83C22⋊Q8×C127—21no—
84C22⋊Q8×C127D412yes—
85#121×C127—21no—
86#121×C127—21no—
87#121×C127—21no—
88#121×C127D412yes—
89C4×Q8×C127—22no—
90C4×Q8×C127—22no—
91C4×Q8×C127—21no—
92C4×Q8×C127—21no—
93C4×Q8×C127—21no—
94C4×Q8×C127D412yes—
95C4×Q8×C127C2312yes—
96C4×Q8×C127C2312yes—
97C4×D4×C127—24no—
98C4×D4×C127—22no—
99C4×D4×C127—22no—
100C4×D4×C127—21no—
101C4×D4×C127—21no—
102C4×D4×C127—21no—
103C4×D4×C127D412yes—
104#117×C127—22no—
105#117×C127—24no—
106#117×C127—21no—
107C2×C4⋊C4×C127—22no—
108C2×C4⋊C4×C127—21no—
109C2×C4⋊C4×C127D411yescharacteristic
110C2×C42×C127—21no—
111C2×C42×C127C2311yescharacteristic
112#107×C127—22no—
113#106×C127—22no—
114C4⋊C8×C127—21no—
115C4⋊C8×C127—21no—
116C4⋊C8×C127C2312yes—
117C4⋊C8×C127C2312yes—
118#103×C127—22no—
119#103×C127—22no—
120#102×C127—22no—
121#102×C127—22no—
122#102×C127—22no—
123#102×C127—22no—
124C4×C8×C127—21no—
125C4×C8×C127C2311yescharacteristic
126C4×C8×C127C2311yescharacteristic
127#41×C127—42no—
128#41×C127—42no—
129#41×C127—42no—
130#41×C127—42no—
131C2×Q8×C127—21no—
132C2×Q8×C127—21no—
133C2×Q8×C127—21no—
134C2×Q8×C127—22no—
135C2×Q8×C127—22no—
136C2×Q8×C127—21no—
137C2×Q8×C127C2×D412yes—
138C2×Q8×C127C2×D412yes—
139C2×D4×C127—22no—
140C2×D4×C127—21no—
141C2×D4×C127—21no—
142C2×D4×C127—21no—
143C2×D4×C127—21no—
144C2×D4×C127—21no—
145C2×D4×C127—41no—
146C2×D4×C127C2×D412yes—
147C22×C4×C127—21no—
148C22×C4×C127—24no—
149C22×C4×C127—21no—
150C22×C4×C127—22no—
151C22×C4×C127—41no—
152C22×C4×C127C2×D411yescharacteristic
153C22×C4×C127C2×D411yescharacteristic
154C22×C4×C127C2×D411yescharacteristic
155Q16×C127—44no—
156SD16×C127—44no—
157SD16×C127—82no—
158D8×C127—44no—
159M16×C127—42no—
160C2×C8×C127—21no—
161C2×C8×C127—21no—
162C2×C8×C127—22no—
163C2×C8×C127—22no—
164C2×C8×C127—42no—
165C2×C8×C127—81no—
166C4⋊C4×C127—24no—
167C4⋊C4×C127—24no—
168C4⋊C4×C127—24no—
169C4⋊C4×C127—21no—
170C4⋊C4×C127—21no—
171C4⋊C4×C127—22no—
172C4⋊C4×C127—21no—
173C4⋊C4×C127—22no—
174C4⋊C4×C127—21no—
175C4⋊C4×C127—22no—
176C4⋊C4×C127—21no—
177C4⋊C4×C127—22no—
178C4⋊C4×C127—21no—
179C4⋊C4×C127—21no—
180C4⋊C4×C127C2×D412yes—
181C4⋊C4×C127C2×D412yes—
182C4⋊C4×C127C2×D412yes—
183C22⋊C4×C127—24no—
184C22⋊C4×C127—24no—
185C22⋊C4×C127—24no—
186C22⋊C4×C127—22no—
187C22⋊C4×C127—22no—
188C22⋊C4×C127—22no—
189C22⋊C4×C127—41no—
190C22⋊C4×C127—41no—
191C42×C127—24no—
192C42×C127—22no—
193C42×C127—22no—
194C42×C127—22no—
195C42×C127C2×D411yescharacteristic
196C42×C127C2×D411yescharacteristic
197C42×C127C2×D411yescharacteristic
198C42×C127C2411yescharacteristic
199C23×C127—22no—
200C23×C127—41no—
201C23×C127#12011yescharacteristic
202Q8×C127—22no—
203Q8×C127—22no—
204Q8×C127—22no—
205Q8×C127—22no—
206Q8×C127—41no—
207Q8×C127—41no—
208Q8×C127—42no—
209Q8×C127—41no—
210Q8×C127—41no—
211Q8×C127—81no—
212D4×C127—42no—
213D4×C127—42no—
214D4×C127—44no—
215D4×C127—41no—
216D4×C127—41no—
217D4×C127—41no—
218D4×C127—41no—
219D4×C127—41no—
220D4×C127—41no—
221D4×C127—41no—
222D4×C127—41no—
223D4×C127—82no—
224C2×C4×C127—21no—
225C2×C4×C127—21no—
226C2×C4×C127—22no—
227C2×C4×C127—22no—
228C2×C4×C127—22no—
229C2×C4×C127—22no—
230C2×C4×C127—22no—
231C2×C4×C127—41no—
232C2×C4×C127—42no—
233C2×C4×C127—44no—
234C2×C4×C127—41no—
235C2×C4×C127—81no—
236C2×C4×C127—84no—
237C2×C4×C127—82no—
238C2×C4×C127#12011yescharacteristic
239C2×C4×C127#12011yescharacteristic
240C2×C4×C127#12011yescharacteristic
241C2×C4×C127#13512yes—
242C2×C4×C127AGL(1,ℤ/8)12yes—
243C2×C4×C127C22×D411yescharacteristic
244C2×C4×C127C22×D411yescharacteristic
245C2×C4×C127C22×D411yescharacteristic
246C8×C127—22no—
247C8×C127—41no—
248C8×C127—42no—
249C8×C127—42no—
250C22×C127—21no—
251C22×C127—21no—
252C22×C127—42no—
253C22×C127—42no—
254C22×C127—42no—
255C22×C127—81no—
256C22×C127—81no—
257C22×C127—81no—
258C22×C127#44912yes—
259C22×C127C2×#12011yescharacteristic
260C4×C127—21no—
261C4×C127—21no—
262C4×C127—21no—
263C4×C127—42no—
264C4×C127—42no—
265C4×C127—42no—
266C4×C127—42no—
267C4×C127—42no—
268C4×C127—81no—
269C4×C127#44912yes—
270C4×C127C2×#13511yescharacteristic
271C4×C127C2×AGL(1,ℤ/8)11yescharacteristic
272C4×C127#57711yescharacteristic
273C4×C127#57811yescharacteristic
274#18666C12711yesmaximal, characteristic, direct product factor, Sylow
275C2×C127—22no—
276C2×C127—82no—
277C2×C127—161no—
278C2×C127C2×#44911yescharacteristic
279C2×C127#301311yescharacteristic
280C2×C127#301811yescharacteristic
281#3507C2×C12711yescharacteristic, complemented
282#3246C2×C12711yescharacteristic, complemented
283#3169C2×C12711yescharacteristic
284#3037C2×C12711yescharacteristic, complemented
285#1648C2×C12712yescomplemented
286#1640C2×C12712yescomplemented
287#1626C2×C12712yescomplemented
288#1620C2×C12712yescomplemented
289#1547C2×C12711yescharacteristic, complemented
290#1493C2×C12711yescharacteristic, complemented
291C2×Q8⋊C8C2×C12711yescharacteristic, complemented
292C127#1866611yescharacteristic, direct product factor, Sylow
293C2×#137—21no—
294#559C22×C12712yes—
295#557C22×C12712yes—
296#556—21no—
297#556—22no—
298#554—21no—
299#552—22no—
300#539—22no—
301#536—21no—
302#533C22×C12711yescharacteristic
303#532C22×C12711yescharacteristic
304#532C22×C12711yescharacteristic
305#529—22no—
306#520—21no—
307C2×C4×Q8C22×C12711yescharacteristic
308#500C22×C12711yescharacteristic
309#493C22×C12711yescharacteristic
310#486C22×C12711yescharacteristic
311#486C22×C12711yescharacteristic
312#476—21no—
313#475C22×C12712yes—
314#466—22no—
315#465—22no—
316#464—22no—
317#462C22×C12712yes—
318#461—21no—
319#460C22×C12712yes—
320#460C22×C12712yes—
321C4×D8—22no—
322#423C22×C12711yescharacteristic
323C2×C4⋊C8C22×C12711yescharacteristic
324#417—21no—
325C2×C4×C8C22×C12711yescharacteristic
326#332C22×C12712yes—
327#331C22×C12712yes—
328Q8⋊C8C22×C12712yes—
329Q8⋊C8C22×C12712yes—
330Q8⋊C8C22×C12712yes—
331Q8⋊C8C22×C12712yes—
332C2×#41—21no—
333C2×#41—21no—
334C22×Q8D4×C12711yescharacteristic
335#137—44no—
336C2×Q16—22no—
337C2×SD16—22no—
338C2×SD16—22no—
339C2×D8—22no—
340C2×M16—21no—
341C22×C8—21no—
342C22×C8—21no—
343C4⋊Q8—22no—
344C4⋊Q8D4×C12711yescharacteristic
345C4⋊Q8D4×C12711yescharacteristic
346C4⋊Q8C23×C12711yescharacteristic
347C4⋊Q8C23×C12711yescharacteristic
348#127C23×C12711yescharacteristic
349#127C23×C12711yescharacteristic
350#126—24no—
351#126—24no—
352#125—22no—
353#125—24no—
354#125—21no—
355#124—22no—
356#124—22no—
357#124—21no—
358#124—21no—
359#123—24no—
360#123—24no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—24no—
364C22⋊Q8—21no—
365C22⋊Q8D4×C12712yes—
366#121—21no—
367#121—21no—
368#121—21no—
369#121D4×C12712yes—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—21no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×Q8D4×C12712yes—
376C4×Q8C23×C12712yes—
377C4×Q8C23×C12712yes—
378C4×D4—24no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4D4×C12712yes—
385#117—22no—
386#117—24no—
387#117—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4D4×C12711yescharacteristic
391C2×C42—21no—
392C2×C42C23×C12711yescharacteristic
393#107—22no—
394#106—22no—
395C4⋊C8—21no—
396C4⋊C8—21no—
397C4⋊C8C23×C12712yes—
398C4⋊C8C23×C12712yes—
399#103—22no—
400#103—22no—
401#102—22no—
402#102—22no—
403#102—22no—
404#102—22no—
405C4×C8—21no—
406C4×C8C23×C12711yescharacteristic
407C4×C8C23×C12711yescharacteristic
408#41—42no—
409#41—42no—
410#41—42no—
411#41—42no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—21no—
418C2×Q8C2×D4×C12712yes—
419C2×Q8C2×D4×C12712yes—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—41no—
427C2×D4C2×D4×C12712yes—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—41no—
433C22×C4C2×D4×C12711yescharacteristic
434C22×C4C2×D4×C12711yescharacteristic
435C22×C4C2×D4×C12711yescharacteristic
436Q16—44no—
437SD16—44no—
438SD16—82no—
439D8—44no—
440M16—42no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—42no—
446C2×C8—81no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4C2×D4×C12712yes—
462C4⋊C4C2×D4×C12712yes—
463C4⋊C4C2×D4×C12712yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42C2×D4×C12711yescharacteristic
477C42C2×D4×C12711yescharacteristic
478C42C2×D4×C12711yescharacteristic
479C42C24×C12711yescharacteristic
480C23—22no—
481C23—41no—
482C23#120×C12711yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—22no—
487Q8—41no—
488Q8—41no—
489Q8—42no—
490Q8—41no—
491Q8—41no—
492Q8—81no—
493D4—42no—
494D4—42no—
495D4—44no—
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—82no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4#120×C12711yescharacteristic
520C2×C4#120×C12711yescharacteristic
521C2×C4#120×C12711yescharacteristic
522C2×C4#135×C12712yes—
523C2×C4AGL(1,ℤ/8)×C12712yes—
524C2×C4C22×D4×C12711yescharacteristic
525C2×C4C22×D4×C12711yescharacteristic
526C2×C4C22×D4×C12711yescharacteristic
527C8—22no—
528C8—41no—
529C8—42no—
530C8—42no—
531C22—21no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22#449×C12712yes—
540C22C2×#120×C12711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—81no—
550C4#449×C12712yes—
551C4C2×#135×C12711yescharacteristic
552C4C2×AGL(1,ℤ/8)×C12711yescharacteristic
553C4#577×C12711yescharacteristic
554C4#578×C12711yescharacteristic
555C2—22no—
556C2—82no—
557C2—161no—
558C2C2×#449×C12711yescharacteristic
559C2C127×#301311yescharacteristic
560C2C127×#301811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC42C24×C127
prime radicalC127×#3169C2
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
1. C127×#3507C2semidirect
2. C127×#3246C2semidirect
4. C127×#3037C2semidirect
5. C127×#1648C2semidirect
6. C127×#1640C2semidirect
7. C127×#1626C2semidirect
8. C127×#1620C2semidirect
9. C127×#1547C2semidirect
10. C127×#1493C2semidirect
11. C2×Q8⋊C8×C127C2semidirect
274. #18666C127direct (N × H)
281. #3507C2×C127semidirect
282. #3246C2×C127semidirect
284. #3037C2×C127semidirect
285. #1648C2×C127semidirect
286. #1640C2×C127semidirect
287. #1626C2×C127semidirect
288. #1620C2×C127semidirect
289. #1547C2×C127semidirect
290. #1493C2×C127semidirect
291. C2×Q8⋊C8C2×C127semidirect
292. C127#18666direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35072
C127×#32462
C127×#31692
C127×#30372
C127×#16482
C127×#16402
C127×#16262
C127×#16202
C127×#15472
C127×#14932
C2×Q8⋊C8×C1272
#18666127