Finite Group Databasebeta

← C127×#21784 #44585457

Subgroups — C127×#21784 #44585457

Order 32512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3500C211yesmaximal, characteristic
2C127×#2888C211yesmaximal, characteristic, complemented
3C127×#2883C211yesmaximal, characteristic, complemented
4C127×#1911C212yesmaximal, complemented
5C127×#1909C212yesmaximal, complemented
6C127×#1885C212yesmaximal, complemented
7C127×#1884C212yesmaximal, complemented
8C127×#1827C212yesmaximal, complemented
9C127×#1796C212yesmaximal, complemented
10C2×2−1+4×C127C2211yescharacteristic
11C2×H(5,𝔽2)×C127C2211yescharacteristic
12#568×C127C2212yes—
13#548×C127—22no—
14#546×C127—22no—
15#546×C127—24no—
16#536×C127C412yescomplemented
17#536×C127C2211yescharacteristic
18#535×C127C412yescomplemented
19#535×C127C2211yescharacteristic
20#532×C127C412yescomplemented
21#532×C127C2211yescharacteristic
22C2×#124×C127C412yescomplemented
23C2×#124×C127C2211yescharacteristic
24#518×C127C2212yes—
25C4×#41×C127C2211yescharacteristic
26#421×C127C2214yescomplemented
27#419×C127C2214yescomplemented
28#413×C127—22no—
29#413×C127C2212yes—
30C2×#100×C127—22no—
31C2×#100×C127C2212yes—
32#410×C127C2212yes—
33#410×C127C2212yes—
34C2×#99×C127—24no—
35#339×C127C2214yescomplemented
36H(3,ℤ/4)×C127C2214yescomplemented
372−1+4×C127—44no—
38H(5,𝔽2)×C127—44no—
39C2×#41×C127—21no—
40C2×#41×C127—22no—
41C2×#41×C127C2×C412yes—
42C2×#41×C127C2×C412yescomplemented
43C2×#41×C127D412yes—
44C2×#41×C127C2311yescharacteristic
45C2×#41×C127C2311yescharacteristic
46C2×#41×C127C2311yescharacteristic
47C22×Q8×C127—21no—
48C22×Q8×C127C2×C412yescomplemented
49C22×Q8×C127C2311yescharacteristic
50C22×D4×C127C2×C412yes—
51C22×D4×C127D414yes—
52C22×D4×C127D412yes—
53C22×D4×C127C2311yescharacteristic
54#131×C127—44no—
55C2×M16×C127—24no—
56C2×M16×C127C2314yes—
57C4⋊Q8×C127C2×C412yescomplemented
58C4⋊Q8×C127C2×C411yescharacteristic, complemented
59#127×C127C2×C412yes—
60#127×C127C2×C411yescharacteristic, complemented
61#124×C127—24no—
62#124×C127—22no—
63#124×C127C2×C412yes—
64#124×C127C2×C412yes—
65#124×C127C2×C412yes—
66#124×C127C2×C411yescharacteristic, complemented
67#124×C127C2×C412yescomplemented
68#124×C127C2×C411yescharacteristic, complemented
69#123×C127—24no—
70#123×C127—22no—
71C22⋊Q8×C127—24no—
72C22⋊Q8×C127—22no—
73#121×C127—24no—
74#121×C127—24no—
75#121×C127—22no—
76#121×C127—22no—
77#120×C127—24no—
78#120×C127—22no—
79C4×Q8×C127—21no—
80C4×D4×C127—22no—
81C4×D4×C127—21no—
82C4×D4×C127—24no—
83C4×D4×C127—22no—
84C4×D4×C127—44no—
85#117×C127C2×C412yescomplemented
86#117×C127C2314yes—
87#117×C127C2311yescharacteristic
88C2×C22⋊C4×C127—24no—
89C2×C22⋊C4×C127—22no—
90C2×C22⋊C4×C127—24no—
91C2×C42×C127C2×C412yescomplemented
92C2×C42×C127C2311yescharacteristic
93#104×C127—44no—
94#103×C127—44no—
95#101×C127—22no—
96#100×C127—22no—
97#100×C127—24no—
98#99×C127—24no—
99#99×C127—24no—
100C24×C127C22⋊C414yes—
101C24×C127C2×D412yes—
102#41×C127—24no—
103#41×C127—24no—
104#41×C127—24no—
105#41×C127—44no—
106#41×C127—44no—
107#41×C127—44no—
108#41×C127—48no—
109#41×C127—44no—
110C2×Q8×C127—22no—
111C2×Q8×C127—21no—
112C2×Q8×C127—24no—
113C2×Q8×C127—44no—
114C2×Q8×C127—44no—
115C2×Q8×C127C22×C411yescharacteristic
116C2×Q8×C127C22×C412yes—
117C2×Q8×C127C22×C411yescharacteristic
118C2×Q8×C127C2×D411yescharacteristic
119C2×D4×C127—22no—
120C2×D4×C127—21no—
121C2×D4×C127—24no—
122C2×D4×C127—22no—
123C2×D4×C127—24no—
124C2×D4×C127—44no—
125C2×D4×C127—44no—
126C2×D4×C127—48no—
127C2×D4×C127C22⋊C414yes—
128C2×D4×C127C22⋊C414yes—
129C2×D4×C127C22×C411yescharacteristic
130C2×D4×C127C22×C412yes—
131C2×D4×C127C22×C411yescharacteristic
132C2×D4×C127C2×D412yes—
133C2×D4×C127C2×D412yes—
134C2×D4×C127C2×D411yescharacteristic
135C2×D4×C127C2×D412yes—
136C22×C4×C127—22no—
137C22×C4×C127—21no—
138C22×C4×C127—22no—
139C22×C4×C127—21no—
140C22×C4×C127—22no—
141C22×C4×C127—21no—
142C22×C4×C127—44no—
143C22×C4×C127C22⋊C414yes—
144C22×C4×C127C22×C412yes—
145C22×C4×C127C2×D412yes—
146C22×C4×C127C2411yescharacteristic
147M16×C127—24no—
148M16×C127—24no—
149M16×C127—44no—
150C2×C8×C127—24no—
151C2×C8×C127—44no—
152C4⋊C4×C127—22no—
153C4⋊C4×C127—24no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—21no—
156C4⋊C4×C127—24no—
157C22⋊C4×C127—24no—
158C22⋊C4×C127—22no—
159C22⋊C4×C127—24no—
160C22⋊C4×C127—24no—
161C22⋊C4×C127—24no—
162C22⋊C4×C127—22no—
163C22⋊C4×C127—21no—
164C22⋊C4×C127—22no—
165C22⋊C4×C127—24no—
166C22⋊C4×C127—44no—
167C42×C127—22no—
168C42×C127—21no—
169C42×C127—24no—
170C42×C127C22×C411yescharacteristic
171C42×C127C22×C412yes—
172C42×C127C22×C411yescharacteristic
173C23×C127—24no—
174C23×C127—21no—
175C23×C127—44no—
176C23×C127—82no—
177C23×C127—84no—
178C23×C127C2×C22⋊C414yes—
179C23×C127C2×C22⋊C412yes—
180C23×C127C2×C22⋊C412yes—
181C23×C127C22×D411yescharacteristic
182Q8×C127—22no—
183Q8×C127—22no—
184Q8×C127—24no—
185Q8×C127—22no—
186Q8×C127—44no—
187Q8×C127—42no—
188D4×C127—24no—
189D4×C127—24no—
190D4×C127—24no—
191D4×C127—22no—
192D4×C127—22no—
193D4×C127—24no—
194D4×C127—22no—
195D4×C127—44no—
196D4×C127—44no—
197D4×C127—44no—
198D4×C127—42no—
199D4×C127—84no—
200D4×C127—82no—
201C2×C4×C127—24no—
202C2×C4×C127—24no—
203C2×C4×C127—21no—
204C2×C4×C127—22no—
205C2×C4×C127—21no—
206C2×C4×C127—22no—
207C2×C4×C127—21no—
208C2×C4×C127—44no—
209C2×C4×C127—44no—
210C2×C4×C127—44no—
211C2×C4×C127—42no—
212C2×C4×C127—42no—
213C2×C4×C127—44no—
214C2×C4×C127—81no—
215C2×C4×C127—82no—
216C2×C4×C127—164no—
217C2×C4×C127C2×C22⋊C412yes—
218C2×C4×C127C2×C22⋊C412yes—
219C2×C4×C127C23×C411yescharacteristic
220C2×C4×C127C22×D411yescharacteristic
221C2×C4×C127H(5,𝔽2)14yes—
222C8×C127—44no—
223C22×C127—22no—
224C22×C127—22no—
225C22×C127—24no—
226C22×C127—41no—
227C22×C127—42no—
228C22×C127—44no—
229C22×C127—42no—
230C22×C127—42no—
231C22×C127—41no—
232C22×C127—84no—
233C22×C127—164no—
234C22×C127C22×C22⋊C411yescharacteristic
235C22×C127#53412yes—
236C4×C127—22no—
237C4×C127—22no—
238C4×C127—41no—
239C4×C127—42no—
240C4×C127—41no—
241C4×C127—42no—
242C4×C127—41no—
243C4×C127—84no—
244C4×C127#51812yes—
245C4×C127#53412yes—
246#21784C12711yesmaximal, characteristic, direct product factor, Sylow
247C2×C127—21no—
248C2×C127—22no—
249C2×C127—44no—
250C2×C127—42no—
251C2×C127—42no—
252C2×C127—81no—
253C2×C127#230811yescharacteristic
254#3500C2×C12711yescharacteristic
255#2888C2×C12711yescharacteristic, complemented
256#2883C2×C12711yescharacteristic, complemented
257#1911C2×C12712yescomplemented
258#1909C2×C12712yescomplemented
259#1885C2×C12712yescomplemented
260#1884C2×C12712yescomplemented
261#1827C2×C12712yescomplemented
262#1796C2×C12712yescomplemented
263C127#2178411yescharacteristic, direct product factor, Sylow
264C2×2−1+4C22×C12711yescharacteristic
265C2×H(5,𝔽2)C22×C12711yescharacteristic
266#568C22×C12712yes—
267#548—22no—
268#546—22no—
269#546—24no—
270#536C4×C12712yescomplemented
271#536C22×C12711yescharacteristic
272#535C4×C12712yescomplemented
273#535C22×C12711yescharacteristic
274#532C4×C12712yescomplemented
275#532C22×C12711yescharacteristic
276C2×#124C4×C12712yescomplemented
277C2×#124C22×C12711yescharacteristic
278#518C22×C12712yes—
279C4×#41C22×C12711yescharacteristic
280#421C22×C12714yescomplemented
281#419C22×C12714yescomplemented
282#413—22no—
283#413C22×C12712yes—
284C2×#100—22no—
285C2×#100C22×C12712yes—
286#410C22×C12712yes—
287#410C22×C12712yes—
288C2×#99—24no—
289#339C22×C12714yescomplemented
290H(3,ℤ/4)C22×C12714yescomplemented
2912−1+4—44no—
292H(5,𝔽2)—44no—
293C2×#41—21no—
294C2×#41—22no—
295C2×#41C2×C4×C12712yes—
296C2×#41C2×C4×C12712yescomplemented
297C2×#41D4×C12712yes—
298C2×#41C23×C12711yescharacteristic
299C2×#41C23×C12711yescharacteristic
300C2×#41C23×C12711yescharacteristic
301C22×Q8—21no—
302C22×Q8C2×C4×C12712yescomplemented
303C22×Q8C23×C12711yescharacteristic
304C22×D4C2×C4×C12712yes—
305C22×D4D4×C12714yes—
306C22×D4D4×C12712yes—
307C22×D4C23×C12711yescharacteristic
308#131—44no—
309C2×M16—24no—
310C2×M16C23×C12714yes—
311C4⋊Q8C2×C4×C12712yescomplemented
312C4⋊Q8C2×C4×C12711yescharacteristic, complemented
313#127C2×C4×C12712yes—
314#127C2×C4×C12711yescharacteristic, complemented
315#124—24no—
316#124—22no—
317#124C2×C4×C12712yes—
318#124C2×C4×C12712yes—
319#124C2×C4×C12712yes—
320#124C2×C4×C12711yescharacteristic, complemented
321#124C2×C4×C12712yescomplemented
322#124C2×C4×C12711yescharacteristic, complemented
323#123—24no—
324#123—22no—
325C22⋊Q8—24no—
326C22⋊Q8—22no—
327#121—24no—
328#121—24no—
329#121—22no—
330#121—22no—
331#120—24no—
332#120—22no—
333C4×Q8—21no—
334C4×D4—22no—
335C4×D4—21no—
336C4×D4—24no—
337C4×D4—22no—
338C4×D4—44no—
339#117C2×C4×C12712yescomplemented
340#117C23×C12714yes—
341#117C23×C12711yescharacteristic
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C42C2×C4×C12712yescomplemented
346C2×C42C23×C12711yescharacteristic
347#104—44no—
348#103—44no—
349#101—22no—
350#100—22no—
351#100—24no—
352#99—24no—
353#99—24no—
354C24C22⋊C4×C12714yes—
355C24C2×D4×C12712yes—
356#41—24no—
357#41—24no—
358#41—24no—
359#41—44no—
360#41—44no—
361#41—44no—
362#41—48no—
363#41—44no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—24no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8C22×C4×C12711yescharacteristic
370C2×Q8C22×C4×C12712yes—
371C2×Q8C22×C4×C12711yescharacteristic
372C2×Q8C2×D4×C12711yescharacteristic
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4C22⋊C4×C12714yes—
382C2×D4C22⋊C4×C12714yes—
383C2×D4C22×C4×C12711yescharacteristic
384C2×D4C22×C4×C12712yes—
385C2×D4C22×C4×C12711yescharacteristic
386C2×D4C2×D4×C12712yes—
387C2×D4C2×D4×C12712yes—
388C2×D4C2×D4×C12711yescharacteristic
389C2×D4C2×D4×C12712yes—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—44no—
397C22×C4C22⋊C4×C12714yes—
398C22×C4C22×C4×C12712yes—
399C22×C4C2×D4×C12712yes—
400C22×C4C24×C12711yescharacteristic
401M16—24no—
402M16—24no—
403M16—44no—
404C2×C8—24no—
405C2×C8—44no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—22no—
417C22⋊C4—21no—
418C22⋊C4—22no—
419C22⋊C4—24no—
420C22⋊C4—44no—
421C42—22no—
422C42—21no—
423C42—24no—
424C42C22×C4×C12711yescharacteristic
425C42C22×C4×C12712yes—
426C42C22×C4×C12711yescharacteristic
427C23—24no—
428C23—21no—
429C23—44no—
430C23—82no—
431C23—84no—
432C23C2×C22⋊C4×C12714yes—
433C23C2×C22⋊C4×C12712yes—
434C23C2×C22⋊C4×C12712yes—
435C23C22×D4×C12711yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—24no—
439Q8—22no—
440Q8—44no—
441Q8—42no—
442D4—24no—
443D4—24no—
444D4—24no—
445D4—22no—
446D4—22no—
447D4—24no—
448D4—22no—
449D4—44no—
450D4—44no—
451D4—44no—
452D4—42no—
453D4—84no—
454D4—82no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—164no—
471C2×C4C2×C22⋊C4×C12712yes—
472C2×C4C2×C22⋊C4×C12712yes—
473C2×C4C23×C4×C12711yescharacteristic
474C2×C4C22×D4×C12711yescharacteristic
475C2×C4H(5,𝔽2)×C12714yes—
476C8—44no—
477C22—22no—
478C22—22no—
479C22—24no—
480C22—41no—
481C22—42no—
482C22—44no—
483C22—42no—
484C22—42no—
485C22—41no—
486C22—84no—
487C22—164no—
488C22C22×C22⋊C4×C12711yescharacteristic
489C22#534×C12712yes—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—42no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—84no—
498C4#518×C12712yes—
499C4#534×C12712yes—
500C2—21no—
501C2—22no—
502C2—44no—
503C2—42no—
504C2—42no—
505C2—81no—
506C2C127×#230811yescharacteristic
subgroupG/H
center, socleC2×C127#2308
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3500C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2888C2semidirect
3. C127×#2883C2semidirect
4. C127×#1911C2semidirect
5. C127×#1909C2semidirect
6. C127×#1885C2semidirect
7. C127×#1884C2semidirect
8. C127×#1827C2semidirect
9. C127×#1796C2semidirect
16. #536×C127C4semidirect
18. #535×C127C4semidirect
20. #532×C127C4semidirect
22. C2×#124×C127C4semidirect
26. #421×C127C22semidirect
27. #419×C127C22semidirect
35. #339×C127C22semidirect
36. H(3,ℤ/4)×C127C22semidirect
42. C2×#41×C127C2×C4semidirect
48. C22×Q8×C127C2×C4semidirect
57. C4⋊Q8×C127C2×C4semidirect
58. C4⋊Q8×C127C2×C4semidirect
60. #127×C127C2×C4semidirect
66. #124×C127C2×C4semidirect
67. #124×C127C2×C4semidirect
68. #124×C127C2×C4semidirect
85. #117×C127C2×C4semidirect
91. C2×C42×C127C2×C4semidirect
246. #21784C127direct (N × H)
255. #2888C2×C127semidirect
256. #2883C2×C127semidirect
257. #1911C2×C127semidirect
258. #1909C2×C127semidirect
259. #1885C2×C127semidirect
260. #1884C2×C127semidirect
261. #1827C2×C127semidirect
262. #1796C2×C127semidirect
263. C127#21784direct (N × H)
270. #536C4×C127semidirect
272. #535C4×C127semidirect
274. #532C4×C127semidirect
276. C2×#124C4×C127semidirect
280. #421C22×C127semidirect
281. #419C22×C127semidirect
289. #339C22×C127semidirect
290. H(3,ℤ/4)C22×C127semidirect
296. C2×#41C2×C4×C127semidirect
302. C22×Q8C2×C4×C127semidirect
311. C4⋊Q8C2×C4×C127semidirect
312. C4⋊Q8C2×C4×C127semidirect
314. #127C2×C4×C127semidirect
320. #124C2×C4×C127semidirect
321. #124C2×C4×C127semidirect
322. #124C2×C4×C127semidirect
339. #117C2×C4×C127semidirect
345. C2×C42C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35002
C127×#28882
C127×#28832
C127×#19112
C127×#19092
C127×#18852
C127×#18842
C127×#18272
C127×#17962
#21784127