Finite Group Databasebeta

← C127×#21556 #44585229

Subgroups — C127×#21556 #44585229

Order 32512.

874 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#473×C127C211yesmaximal, characteristic
2C4×#137×C127C212yesmaximal
3C2×#441×C127C211yesmaximal, characteristic, complemented
4C127×#2875C211yesmaximal, characteristic
5C127×#1979C211yesmaximal, characteristic, complemented
6C127×#1951C211yesmaximal, characteristic, complemented
7C127×#1930C212yesmaximal, complemented
8C127×#1908C212yesmaximal
9C127×#1815C211yesmaximal, characteristic
10C127×#1778C211yesmaximal, characteristic, complemented
11C4×#101×C127C212yesmaximal, complemented
12C2×#137×C127C2212yes—
13C22×Q16×C127C2211yescharacteristic
14C2×#131×C127C2211yescharacteristic
15C2×#131×C127C2211yescharacteristic
16C4×#41×C127C2211yescharacteristic
17C2×C4×Q8×C127C2212yes—
18#473×C127C418yescomplemented
19#441×C127—22no—
20#441×C127—21no—
21#441×C127—21no—
22#441×C127—21no—
23#441×C127—21no—
24#440×C127—22no—
25C4×Q16×C127—22no—
26C4×SD16×C127—22no—
27C2×#108×C127C2211yescharacteristic
28C2×#108×C127C2211yescharacteristic
29#428×C127C2212yes—
30C2×#106×C127C2211yescharacteristic
31#419×C127C2212yes—
32#417×C127C2212yes—
33C2×#103×C127C2212yes—
34C2×#101×C127C2212yes—
35C2×#101×C127C2212yes—
36#405×C127—21no—
37#405×C127—21no—
38#405×C127C2211yescharacteristic
39#405×C127C2211yescharacteristic
40C4×M16×C127—21no—
41C4×M16×C127C2211yescharacteristic
42C4×M16×C127C2212yes—
43C2×#97×C127—21no—
44C2×#97×C127C2211yescharacteristic
45#343×C127C2212yes—
46#343×C127C2212yes—
47#341×C127C2212yes—
48C2×#41×C127C2311yescharacteristic
49C22×Q8×C127C2312yes—
50#137×C127—24no—
51#137×C127C2×C418yes—
52C2×Q16×C127—22no—
53C2×Q16×C127—21no—
54C2×Q16×C127—21no—
55C2×Q16×C127—22no—
56C2×Q16×C127C2×C414yescomplemented
57C2×SD16×C127—22no—
58#131×C127—22no—
59#131×C127—22no—
60#131×C127—22no—
61#131×C127C2×C414yes—
62C2×M16×C127—21no—
63C2×M16×C127—21no—
64C2×M16×C127C2312yes—
65C2×M16×C127C2311yescharacteristic
66C2×M16×C127C2312yes—
67C2×M16×C127C2311yescharacteristic
68C22×C8×C127—21no—
69C22×C8×C127—21no—
70C22×C8×C127C2311yescharacteristic
71C22×C8×C127C2311yescharacteristic
72C4×Q8×C127—22no—
73C4×Q8×C127—22no—
74C4×Q8×C127—22no—
75C4×Q8×C127—22no—
76C4×Q8×C127D412yes—
77C4×D4×C127—22no—
78C4×D4×C127D412yes—
79#117×C127—21no—
80#117×C127C2311yescharacteristic
81C2×C4⋊C4×C127—22no—
82C2×C4⋊C4×C127C2312yes—
83C2×C42×C127—21no—
84C2×C42×C127—22no—
85C2×C42×C127C2311yescharacteristic
86#108×C127—22no—
87#108×C127C2×C414yescomplemented
88#107×C127—22no—
89#106×C127—21no—
90#106×C127—21no—
91#106×C127—22no—
92#103×C127—22no—
93#103×C127—22no—
94#103×C127—22no—
95#103×C127—22no—
96#103×C127—22no—
97#102×C127—22no—
98#101×C127—24no—
99#101×C127C2×C418yescomplemented
100#97×C127—21no—
101#97×C127—21no—
102#97×C127—21no—
103#97×C127—22no—
104#97×C127D411yescharacteristic
105#97×C127D411yescharacteristic
106#97×C127D411yescharacteristic
107#97×C127D411yescharacteristic
108C4×C8×C127—21no—
109C4×C8×C127—21no—
110C4×C8×C127—21no—
111C4×C8×C127—22no—
112#41×C127—21no—
113#41×C127—21no—
114#41×C127C22×C412yes—
115#41×C127#4112yes—
116C2×Q8×C127—22no—
117C2×Q8×C127—22no—
118C2×Q8×C127—22no—
119C2×Q8×C127—22no—
120C2×Q8×C127—22no—
121C2×Q8×C127C22×C414yes—
122C2×Q8×C127C2×D411yescharacteristic
123C2×D4×C127—21no—
124C2×D4×C127C2×D411yescharacteristic
125C22×C4×C127—21no—
126C22×C4×C127—22no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127#4112yes—
130C22×C4×C127C2411yescharacteristic
131Q16×C127—28no—
132Q16×C127—24no—
133Q16×C127—24no—
134SD16×C127—28no—
135M16×C127—24no—
136M16×C127—21no—
137M16×C127—21no—
138M16×C127—22no—
139M16×C127—24no—
140M16×C127—22no—
141M16×C127—22no—
142M16×C127C22×C412yes—
143M16×C127C22×C414yes—
144M16×C127#4112yes—
145C2×C8×C127—24no—
146C2×C8×C127—21no—
147C2×C8×C127—21no—
148C2×C8×C127—22no—
149C2×C8×C127—21no—
150C2×C8×C127—21no—
151C2×C8×C127—24no—
152C2×C8×C127—22no—
153C2×C8×C127C22×C412yes—
154C2×C8×C127C2×D411yescharacteristic
155C2×C8×C127C2×D411yescharacteristic
156C2×C8×C127C2×D411yescharacteristic
157C2×C8×C127C2×D411yescharacteristic
158C2×C8×C127#4112yes—
159C4⋊C4×C127—22no—
160C4⋊C4×C127—22no—
161C4⋊C4×C127—22no—
162C4⋊C4×C127—42no—
163C4⋊C4×C127—42no—
164C4⋊C4×C127C2×D412yes—
165C22⋊C4×C127—21no—
166C22⋊C4×C127—22no—
167C42×C127—22no—
168C42×C127—21no—
169C42×C127—42no—
170C42×C127—42no—
171C42×C127C2×D412yes—
172C42×C127C2×D411yescharacteristic
173C42×C127C2×D411yescharacteristic
174C23×C127—21no—
175C23×C127C2×#4111yescharacteristic
176Q8×C127—21no—
177Q8×C127—24no—
178Q8×C127—24no—
179Q8×C127—42no—
180Q8×C127—42no—
181Q8×C127C4×D412yes—
182D4×C127—22no—
183D4×C127—22no—
184D4×C127—21no—
185D4×C127C4×D412yes—
186C2×C4×C127—21no—
187C2×C4×C127—22no—
188C2×C4×C127—22no—
189C2×C4×C127—24no—
190C2×C4×C127—22no—
191C2×C4×C127—42no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—82no—
196C2×C4×C127—82no—
197C2×C4×C127#12112yes—
198C2×C4×C127#12112yes—
199C2×C4×C127C23×C411yescharacteristic
200C2×C4×C127C22×D411yescharacteristic
201C2×C4×C127C22×D411yescharacteristic
202C2×C4×C127C2×#4111yescharacteristic
203C2×C4×C127C2×#4111yescharacteristic
204C2×C4×C127C2×#4111yescharacteristic
205C8×C127—22no—
206C8×C127—24no—
207C8×C127—21no—
208C8×C127—22no—
209C8×C127—21no—
210C8×C127—42no—
211C8×C127C4×D412yes—
212C8×C127C4×D412yes—
213C22×C127—22no—
214C22×C127—22no—
215C22×C127—42no—
216C22×C127C4×#4111yescharacteristic
217C22×C127C2×#12111yescharacteristic
218C22×C127#52911yescharacteristic
219C4×C127—22no—
220C4×C127—22no—
221C4×C127—22no—
222C4×C127—44no—
223C4×C127—42no—
224C4×C127—82no—
225C4×C127C2×C4×D411yescharacteristic
226C4×C127C2×C4×D411yescharacteristic
227C4×C127#52511yescharacteristic
228C4×C127#53211yescharacteristic
229#21556C12711yesmaximal, characteristic, direct product factor, Sylow
230C2×C127—21no—
231C2×C127—21no—
232C2×C127—42no—
233C2×C127C4×#12111yescharacteristic
234C2×C127#306912yes—
235C2×#473C2×C12711yescharacteristic
236C4×#137C2×C12712yes—
237C2×#441C2×C12711yescharacteristic, complemented
238#2875C2×C12711yescharacteristic
239#1979C2×C12711yescharacteristic, complemented
240#1951C2×C12711yescharacteristic, complemented
241#1930C2×C12712yescomplemented
242#1908C2×C12712yes—
243#1815C2×C12711yescharacteristic
244#1778C2×C12711yescharacteristic, complemented
245C4×#101C2×C12712yescomplemented
246C127#2155611yescharacteristic, direct product factor, Sylow
247C2×#137C22×C12712yes—
248C22×Q16C22×C12711yescharacteristic
249C2×#131C22×C12711yescharacteristic
250C2×#131C22×C12711yescharacteristic
251C4×#41C22×C12711yescharacteristic
252C2×C4×Q8C22×C12712yes—
253#473C4×C12718yescomplemented
254#441—22no—
255#441—21no—
256#441—21no—
257#441—21no—
258#441—21no—
259#440—22no—
260C4×Q16—22no—
261C4×SD16—22no—
262C2×#108C22×C12711yescharacteristic
263C2×#108C22×C12711yescharacteristic
264#428C22×C12712yes—
265C2×#106C22×C12711yescharacteristic
266#419C22×C12712yes—
267#417C22×C12712yes—
268C2×#103C22×C12712yes—
269C2×#101C22×C12712yes—
270C2×#101C22×C12712yes—
271#405—21no—
272#405—21no—
273#405C22×C12711yescharacteristic
274#405C22×C12711yescharacteristic
275C4×M16—21no—
276C4×M16C22×C12711yescharacteristic
277C4×M16C22×C12712yes—
278C2×#97—21no—
279C2×#97C22×C12711yescharacteristic
280#343C22×C12712yes—
281#343C22×C12712yes—
282#341C22×C12712yes—
283C2×#41C23×C12711yescharacteristic
284C22×Q8C23×C12712yes—
285#137—24no—
286#137C2×C4×C12718yes—
287C2×Q16—22no—
288C2×Q16—21no—
289C2×Q16—21no—
290C2×Q16—22no—
291C2×Q16C2×C4×C12714yescomplemented
292C2×SD16—22no—
293#131—22no—
294#131—22no—
295#131—22no—
296#131C2×C4×C12714yes—
297C2×M16—21no—
298C2×M16—21no—
299C2×M16C23×C12712yes—
300C2×M16C23×C12711yescharacteristic
301C2×M16C23×C12712yes—
302C2×M16C23×C12711yescharacteristic
303C22×C8—21no—
304C22×C8—21no—
305C22×C8C23×C12711yescharacteristic
306C22×C8C23×C12711yescharacteristic
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8D4×C12712yes—
312C4×D4—22no—
313C4×D4D4×C12712yes—
314#117—21no—
315#117C23×C12711yescharacteristic
316C2×C4⋊C4—22no—
317C2×C4⋊C4C23×C12712yes—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42C23×C12711yescharacteristic
321#108—22no—
322#108C2×C4×C12714yescomplemented
323#107—22no—
324#106—21no—
325#106—21no—
326#106—22no—
327#103—22no—
328#103—22no—
329#103—22no—
330#103—22no—
331#103—22no—
332#102—22no—
333#101—24no—
334#101C2×C4×C12718yescomplemented
335#97—21no—
336#97—21no—
337#97—21no—
338#97—22no—
339#97D4×C12711yescharacteristic
340#97D4×C12711yescharacteristic
341#97D4×C12711yescharacteristic
342#97D4×C12711yescharacteristic
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—22no—
347#41—21no—
348#41—21no—
349#41C22×C4×C12712yes—
350#41#41×C12712yes—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8C22×C4×C12714yes—
357C2×Q8C2×D4×C12711yescharacteristic
358C2×D4—21no—
359C2×D4C2×D4×C12711yescharacteristic
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4#41×C12712yes—
365C22×C4C24×C12711yescharacteristic
366Q16—28no—
367Q16—24no—
368Q16—24no—
369SD16—28no—
370M16—24no—
371M16—21no—
372M16—21no—
373M16—22no—
374M16—24no—
375M16—22no—
376M16—22no—
377M16C22×C4×C12712yes—
378M16C22×C4×C12714yes—
379M16#41×C12712yes—
380C2×C8—24no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8C22×C4×C12712yes—
389C2×C8C2×D4×C12711yescharacteristic
390C2×C8C2×D4×C12711yescharacteristic
391C2×C8C2×D4×C12711yescharacteristic
392C2×C8C2×D4×C12711yescharacteristic
393C2×C8#41×C12712yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4C2×D4×C12712yes—
400C22⋊C4—21no—
401C22⋊C4—22no—
402C42—22no—
403C42—21no—
404C42—42no—
405C42—42no—
406C42C2×D4×C12712yes—
407C42C2×D4×C12711yescharacteristic
408C42C2×D4×C12711yescharacteristic
409C23—21no—
410C23C2×#41×C12711yescharacteristic
411Q8—21no—
412Q8—24no—
413Q8—24no—
414Q8—42no—
415Q8—42no—
416Q8C4×D4×C12712yes—
417D4—22no—
418D4—22no—
419D4—21no—
420D4C4×D4×C12712yes—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#121×C12712yes—
433C2×C4#121×C12712yes—
434C2×C4C23×C4×C12711yescharacteristic
435C2×C4C22×D4×C12711yescharacteristic
436C2×C4C22×D4×C12711yescharacteristic
437C2×C4C2×#41×C12711yescharacteristic
438C2×C4C2×#41×C12711yescharacteristic
439C2×C4C2×#41×C12711yescharacteristic
440C8—22no—
441C8—24no—
442C8—21no—
443C8—22no—
444C8—21no—
445C8—42no—
446C8C4×D4×C12712yes—
447C8C4×D4×C12712yes—
448C22—22no—
449C22—22no—
450C22—42no—
451C22C4×#41×C12711yescharacteristic
452C22C2×#121×C12711yescharacteristic
453C22#529×C12711yescharacteristic
454C4—22no—
455C4—22no—
456C4—22no—
457C4—44no—
458C4—42no—
459C4—82no—
460C4C2×C4×D4×C12711yescharacteristic
461C4C2×C4×D4×C12711yescharacteristic
462C4#525×C12711yescharacteristic
463C4#532×C12711yescharacteristic
464C2—21no—
465C2—21no—
466C2—42no—
467C2C4×#121×C12711yescharacteristic
468C2C127×#306912yes—
subgroupG/H
center, socleC22×C127C2×#121
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC2×#41×C127C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#441×C127C2semidirect
5. C127×#1979C2semidirect
6. C127×#1951C2semidirect
7. C127×#1930C2semidirect
10. C127×#1778C2semidirect
11. C4×#101×C127C2semidirect
18. #473×C127C4semidirect
56. C2×Q16×C127C2×C4semidirect
87. #108×C127C2×C4semidirect
99. #101×C127C2×C4semidirect
229. #21556C127direct (N × H)
237. C2×#441C2×C127semidirect
239. #1979C2×C127semidirect
240. #1951C2×C127semidirect
241. #1930C2×C127semidirect
244. #1778C2×C127semidirect
245. C4×#101C2×C127semidirect
246. C127#21556direct (N × H)
253. #473C4×C127semidirect
291. C2×Q16C2×C4×C127semidirect
322. #108C2×C4×C127semidirect
334. #101C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#473×C1272
C4×#137×C1272
C2×#441×C1272
C127×#28752
C127×#19792
C127×#19512
C127×#19302
C127×#19082
C127×#18152
C127×#17782
C4×#101×C1272
#21556127