Finite Group Databasebeta

← C127×#12718 #44576391

Subgroups — C127×#12718 #44576391

Order 32512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3498C211yesmaximal, characteristic
2C127×#2132C211yesmaximal, characteristic, complemented
3C127×#2126C211yesmaximal, characteristic, complemented
4C127×#2122C212yesmaximal, complemented
5C127×#1795C212yesmaximal, complemented
6C2×H(5,𝔽2)×C127C411yescharacteristic, complemented
7C2×H(5,𝔽2)×C127C2211yescharacteristic
8#550×C127—22no—
9D42×C127—22no—
10D42×C127—24no—
11#535×C127—21no—
12#535×C127—21no—
13#535×C127C411yescharacteristic, complemented
14#535×C127C2211yescharacteristic
15#532×C127—21no—
16#532×C127C2211yescharacteristic
17C2×#127×C127—21no—
18C2×#127×C127C411yescharacteristic, complemented
19C4×#41×C127C411yescharacteristic, complemented
20#421×C127—22no—
21#418×C127—22no—
22#413×C127C2212yescomplemented
23#410×C127C2212yescomplemented
24#356×C127—21no—
25#355×C127—21no—
26#354×C127—21no—
27#353×C127—21no—
28AΣL(1,𝔽16)×C127—24no—
29H(5,𝔽2)×C127—42no—
30H(5,𝔽2)×C127—42no—
31H(5,𝔽2)×C127—42no—
32H(5,𝔽2)×C127—42no—
33C2×#41×C127—21no—
34C2×#41×C127—21no—
35C2×#41×C127—21no—
36C2×#41×C127C2×C411yescharacteristic
37C2×#41×C127C2×C411yescharacteristic
38C2×#41×C127D412yes—
39C2×#41×C127C2311yescharacteristic
40C22×D4×C127—22no—
41C22×D4×C127—24no—
42C22×D4×C127—22no—
43C22×D4×C127—21no—
44C22×D4×C127—21no—
45C22×D4×C127—21no—
46C22×D4×C127—22no—
47C22×D4×C127C2×C411yescharacteristic
48C22×D4×C127C2×C411yescharacteristic
49C22×D4×C127D412yes—
50C2×M16×C127—22no—
51C4⋊Q8×C127D411yescharacteristic
52#127×C127—24no—
53#127×C127—21no—
54#127×C127—22no—
55#127×C127—21no—
56#127×C127—21no—
57#127×C127—21no—
58#127×C127D411yescharacteristic
59#124×C127—21no—
60#124×C127—21no—
61#124×C127D411yescharacteristic
62#124×C127D411yescharacteristic
63#121×C127—24no—
64#121×C127—24no—
65#121×C127—24no—
66#121×C127—22no—
67#121×C127—22no—
68#121×C127—22no—
69#120×C127—24no—
70#120×C127—24no—
71#120×C127—22no—
72#120×C127D414yes—
73C4×Q8×C127—21no—
74C4×D4×C127—24no—
75C4×D4×C127—21no—
76C4×D4×C127—22no—
77C4×D4×C127—22no—
78#117×C127—21no—
79#117×C127—22no—
80#117×C127C2×C411yescharacteristic
81C2×C42×C127—21no—
82C2×C42×C127C2×C411yescharacteristic
83#104×C127—44no—
84#102×C127—44no—
85#101×C127—22no—
86#100×C127—22no—
87#99×C127—22no—
88#99×C127—22no—
89C24×C127—22no—
90C24×C127—21no—
91C24×C127—22no—
92C24×C127C22⋊C412yes—
93#41×C127—22no—
94#41×C127—22no—
95#41×C127—24no—
96#41×C127—41no—
97#41×C127—41no—
98#41×C127—42no—
99#41×C127—44no—
100#41×C127—44no—
101#41×C127—41no—
102#41×C127—41no—
103#41×C127—42no—
104#41×C127—42no—
105C2×Q8×C127C22⋊C411yescharacteristic
106C2×Q8×C127C22⋊C411yescharacteristic
107C2×Q8×C127C2×D411yescharacteristic
108C2×D4×C127—21no—
109C2×D4×C127—24no—
110C2×D4×C127—24no—
111C2×D4×C127—22no—
112C2×D4×C127—22no—
113C2×D4×C127—21no—
114C2×D4×C127—22no—
115C2×D4×C127—22no—
116C2×D4×C127—24no—
117C2×D4×C127—24no—
118C2×D4×C127—24no—
119C2×D4×C127—21no—
120C2×D4×C127—21no—
121C2×D4×C127—21no—
122C2×D4×C127—22no—
123C2×D4×C127—41no—
124C2×D4×C127—41no—
125C2×D4×C127—42no—
126C2×D4×C127—44no—
127C2×D4×C127—44no—
128C2×D4×C127—41no—
129C2×D4×C127—41no—
130C2×D4×C127—48no—
131C2×D4×C127—48no—
132C2×D4×C127—42no—
133C2×D4×C127—42no—
134C2×D4×C127—84no—
135C2×D4×C127—88no—
136C2×D4×C127C22⋊C411yescharacteristic
137C2×D4×C127C22⋊C412yes—
138C2×D4×C127C22⋊C411yescharacteristic
139C2×D4×C127C2×D411yescharacteristic
140C2×D4×C127C2×D412yes—
141C22×C4×C127—21no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—21no—
145C22×C4×C127—21no—
146C22×C4×C127—22no—
147C22×C4×C127C22×C411yescharacteristic
148C22×C4×C127C2×D412yes—
149M16×C127—42no—
150M16×C127—42no—
151C2×C8×C127—42no—
152C4⋊C4×C127—22no—
153C4⋊C4×C127—21no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—42no—
156C22⋊C4×C127—22no—
157C22⋊C4×C127—24no—
158C22⋊C4×C127—22no—
159C22⋊C4×C127—22no—
160C22⋊C4×C127—24no—
161C22⋊C4×C127—42no—
162C22⋊C4×C127C22⋊C412yes—
163C42×C127—22no—
164C42×C127—21no—
165C42×C127—21no—
166C42×C127—42no—
167C42×C127C22⋊C411yescharacteristic
168C42×C127C22⋊C411yescharacteristic
169C23×C127—21no—
170C23×C127—22no—
171C23×C127—22no—
172C23×C127—21no—
173C23×C127—44no—
174C23×C127—42no—
175C23×C127—44no—
176C23×C127—42no—
177C23×C127—82no—
178C23×C127—82no—
179C23×C127—84no—
180C23×C127—84no—
181C23×C127C2×C22⋊C411yescharacteristic
182C23×C127#12012yes—
183Q8×C127—21no—
184Q8×C127—41no—
185Q8×C127—41no—
186Q8×C127#9912yes—
187D4×C127—24no—
188D4×C127—21no—
189D4×C127—22no—
190D4×C127—22no—
191D4×C127—42no—
192D4×C127—41no—
193D4×C127—44no—
194D4×C127—44no—
195D4×C127—41no—
196D4×C127—42no—
197D4×C127—42no—
198D4×C127—44no—
199D4×C127—44no—
200D4×C127—48no—
201D4×C127—42no—
202D4×C127—44no—
203D4×C127—41no—
204D4×C127—41no—
205D4×C127—84no—
206D4×C127—84no—
207D4×C127—84no—
208D4×C127#9912yes—
209C2×C4×C127—24no—
210C2×C4×C127—21no—
211C2×C4×C127—21no—
212C2×C4×C127—22no—
213C2×C4×C127—22no—
214C2×C4×C127—21no—
215C2×C4×C127—21no—
216C2×C4×C127—22no—
217C2×C4×C127—21no—
218C2×C4×C127—21no—
219C2×C4×C127—44no—
220C2×C4×C127—44no—
221C2×C4×C127—42no—
222C2×C4×C127—44no—
223C2×C4×C127—41no—
224C2×C4×C127—41no—
225C2×C4×C127—82no—
226C2×C4×C127C2×C22⋊C411yescharacteristic
227C2×C4×C127C2×C22⋊C411yescharacteristic
228C2×C4×C127#12012yes—
229C8×C127—82no—
230C22×C127—22no—
231C22×C127—21no—
232C22×C127—22no—
233C22×C127—41no—
234C22×C127—42no—
235C22×C127—42no—
236C22×C127—44no—
237C22×C127—44no—
238C22×C127—42no—
239C22×C127—42no—
240C22×C127—42no—
241C22×C127—41no—
242C22×C127—164no—
243C22×C127—162no—
244C22×C127—164no—
245C22×C127—162no—
246C22×C127#37911yescharacteristic
247C4×C127—22no—
248C4×C127—22no—
249C4×C127—21no—
250C4×C127—22no—
251C4×C127—42no—
252C4×C127—41no—
253C4×C127—41no—
254C4×C127—162no—
255C4×C127C2×#9911yescharacteristic
256C4×C127C2×#9911yescharacteristic
257#12718C12711yesmaximal, characteristic, direct product factor, Sylow
258C2×C127—21no—
259C2×C127—42no—
260C2×C127—42no—
261C2×C127—41no—
262C2×C127—82no—
263C2×C127—82no—
264C2×C127—81no—
265C2×C127#178111yescharacteristic
266#3498C2×C12711yescharacteristic
267#2132C2×C12711yescharacteristic, complemented
268#2126C2×C12711yescharacteristic, complemented
269#2122C2×C12712yescomplemented
270#1795C2×C12712yescomplemented
271C127#1271811yescharacteristic, direct product factor, Sylow
272C2×H(5,𝔽2)C4×C12711yescharacteristic, complemented
273C2×H(5,𝔽2)C22×C12711yescharacteristic
274#550—22no—
275D42—22no—
276D42—24no—
277#535—21no—
278#535—21no—
279#535C4×C12711yescharacteristic, complemented
280#535C22×C12711yescharacteristic
281#532—21no—
282#532C22×C12711yescharacteristic
283C2×#127—21no—
284C2×#127C4×C12711yescharacteristic, complemented
285C4×#41C4×C12711yescharacteristic, complemented
286#421—22no—
287#418—22no—
288#413C22×C12712yescomplemented
289#410C22×C12712yescomplemented
290#356—21no—
291#355—21no—
292#354—21no—
293#353—21no—
294AΣL(1,𝔽16)—24no—
295H(5,𝔽2)—42no—
296H(5,𝔽2)—42no—
297H(5,𝔽2)—42no—
298H(5,𝔽2)—42no—
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—21no—
302C2×#41C2×C4×C12711yescharacteristic
303C2×#41C2×C4×C12711yescharacteristic
304C2×#41D4×C12712yes—
305C2×#41C23×C12711yescharacteristic
306C22×D4—22no—
307C22×D4—24no—
308C22×D4—22no—
309C22×D4—21no—
310C22×D4—21no—
311C22×D4—21no—
312C22×D4—22no—
313C22×D4C2×C4×C12711yescharacteristic
314C22×D4C2×C4×C12711yescharacteristic
315C22×D4D4×C12712yes—
316C2×M16—22no—
317C4⋊Q8D4×C12711yescharacteristic
318#127—24no—
319#127—21no—
320#127—22no—
321#127—21no—
322#127—21no—
323#127—21no—
324#127D4×C12711yescharacteristic
325#124—21no—
326#124—21no—
327#124D4×C12711yescharacteristic
328#124D4×C12711yescharacteristic
329#121—24no—
330#121—24no—
331#121—24no—
332#121—22no—
333#121—22no—
334#121—22no—
335#120—24no—
336#120—24no—
337#120—22no—
338#120D4×C12714yes—
339C4×Q8—21no—
340C4×D4—24no—
341C4×D4—21no—
342C4×D4—22no—
343C4×D4—22no—
344#117—21no—
345#117—22no—
346#117C2×C4×C12711yescharacteristic
347C2×C42—21no—
348C2×C42C2×C4×C12711yescharacteristic
349#104—44no—
350#102—44no—
351#101—22no—
352#100—22no—
353#99—22no—
354#99—22no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24C22⋊C4×C12712yes—
359#41—22no—
360#41—22no—
361#41—24no—
362#41—41no—
363#41—41no—
364#41—42no—
365#41—44no—
366#41—44no—
367#41—41no—
368#41—41no—
369#41—42no—
370#41—42no—
371C2×Q8C22⋊C4×C12711yescharacteristic
372C2×Q8C22⋊C4×C12711yescharacteristic
373C2×Q8C2×D4×C12711yescharacteristic
374C2×D4—21no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—84no—
401C2×D4—88no—
402C2×D4C22⋊C4×C12712yes—
403C2×D4C22⋊C4×C12711yescharacteristic
404C2×D4C22⋊C4×C12711yescharacteristic
405C2×D4C2×D4×C12711yescharacteristic
406C2×D4C2×D4×C12712yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4C22×C4×C12711yescharacteristic
414C22×C4C2×D4×C12712yes—
415M16—42no—
416M16—42no—
417C2×C8—42no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4C22⋊C4×C12712yes—
429C42—22no—
430C42—21no—
431C42—21no—
432C42—42no—
433C42C22⋊C4×C12711yescharacteristic
434C42C22⋊C4×C12711yescharacteristic
435C23—21no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—44no—
440C23—42no—
441C23—44no—
442C23—42no—
443C23—82no—
444C23—82no—
445C23—84no—
446C23—84no—
447C23C2×C22⋊C4×C12711yescharacteristic
448C23#120×C12712yes—
449Q8—21no—
450Q8—41no—
451Q8—41no—
452Q8#99×C12712yes—
453D4—24no—
454D4—21no—
455D4—22no—
456D4—22no—
457D4—42no—
458D4—41no—
459D4—44no—
460D4—44no—
461D4—41no—
462D4—42no—
463D4—42no—
464D4—44no—
465D4—44no—
466D4—48no—
467D4—42no—
468D4—44no—
469D4—41no—
470D4—41no—
471D4—84no—
472D4—84no—
473D4—84no—
474D4#99×C12712yes—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4C2×C22⋊C4×C12711yescharacteristic
493C2×C4C2×C22⋊C4×C12711yescharacteristic
494C2×C4#120×C12712yes—
495C8—82no—
496C22—22no—
497C22—21no—
498C22—22no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—41no—
508C22—164no—
509C22—162no—
510C22—164no—
511C22—162no—
512C22#379×C12711yescharacteristic
513C4—22no—
514C4—22no—
515C4—21no—
516C4—22no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—162no—
521C4C2×#99×C12711yescharacteristic
522C4C2×#99×C12711yescharacteristic
523C2—21no—
524C2—42no—
525C2—42no—
526C2—41no—
527C2—82no—
528C2—81no—
529C2—82no—
530C2C127×#178111yescharacteristic
subgroupG/H
center, socleC2×C127#1781
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×#41C23×C127
prime radicalC127×#3498C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×H(5,𝔽2)×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2132C2semidirect
3. C127×#2126C2semidirect
4. C127×#2122C2semidirect
5. C127×#1795C2semidirect
6. C2×H(5,𝔽2)×C127C4semidirect
13. #535×C127C4semidirect
18. C2×#127×C127C4semidirect
19. C4×#41×C127C4semidirect
22. #413×C127C22semidirect
23. #410×C127C22semidirect
257. #12718C127direct (N × H)
267. #2132C2×C127semidirect
268. #2126C2×C127semidirect
269. #2122C2×C127semidirect
270. #1795C2×C127semidirect
271. C127#12718direct (N × H)
272. C2×H(5,𝔽2)C4×C127semidirect
279. #535C4×C127semidirect
284. C2×#127C4×C127semidirect
285. C4×#41C4×C127semidirect
288. #413C22×C127semidirect
289. #410C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34982
C127×#21322
C127×#21262
C127×#21222
C127×#17952
#12718127