Finite Group Databasebeta

← C127×#13081 #44576754

Subgroups — C127×#13081 #44576754

Order 32512.

722 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#413×C127C211yesmaximal, characteristic
2C127×#2406C211yesmaximal, characteristic
3C2×#352×C127C212yesmaximal, complemented
4C127×#1782C211yesmaximal, characteristic
5C127×#1390C212yesmaximal, complemented
6C22×#41×C127C2211yescharacteristic
7C22×M16×C127—21no—
8C2×#123×C127C411yescharacteristic, complemented
9C2×#123×C127C2211yescharacteristic
10C22×C4⋊C4×C127C411yescharacteristic, complemented
11#413×C127—24no—
12C2×#101×C127—21no—
13C2×#100×C127C2212yes—
14C2×#99×C127—21no—
15C2×#99×C127C2212yes—
16#395×C127—21no—
17#394×C127—21no—
18#386×C127C411yescharacteristic, complemented
19#386×C127C2211yescharacteristic
20#385×C127—21no—
21#385×C127—21no—
22#381×C127—21no—
23#381×C127C411yescharacteristic, complemented
24#352×C127—24no—
25C2×#41×C127—22no—
26C2×#41×C127—22no—
27C2×#41×C127—41no—
28C2×#41×C127—41no—
29C2×#41×C127—42no—
30C22×Q8×C127D411yescharacteristic
31C22×D4×C127C2×C411yescharacteristic
32C22×D4×C127D411yescharacteristic
33C22×D4×C127C2311yescharacteristic
34C23×C4×C127C2×C411yescharacteristic
35C23×C4×C127C2×C411yescharacteristic
36C23×C4×C127C2×C411yescharacteristic
37C23×C4×C127D411yescharacteristic
38C23×C4×C127D411yescharacteristic
39C2×M16×C127—24no—
40C2×M16×C127—22no—
41C2×M16×C127—42no—
42C2×M16×C127—41no—
43C22×C8×C127—41no—
44#123×C127—22no—
45#123×C127—22no—
46#123×C127—42no—
47C2×C4⋊C4×C127—21no—
48C2×C4⋊C4×C127—21no—
49C2×C4⋊C4×C127—21no—
50C2×C4⋊C4×C127—21no—
51C2×C4⋊C4×C127—41no—
52C2×C4⋊C4×C127—41no—
53C2×C22⋊C4×C127—21no—
54C2×C22⋊C4×C127—21no—
55C2×C22⋊C4×C127—21no—
56C2×C22⋊C4×C127—21no—
57C2×C22⋊C4×C127—22no—
58C2×C22⋊C4×C127C2×C411yescharacteristic
59C2×C22⋊C4×C127C2×C411yescharacteristic
60#101×C127—24no—
61#100×C127—22no—
62#100×C127D414yes—
63#99×C127—22no—
64#99×C127—22no—
65#99×C127—42no—
66#95×C127—21no—
67#95×C127—21no—
68#95×C127—21no—
69#95×C127—21no—
70#95×C127—41no—
71#95×C127—41no—
72C24×C127C22⋊C411yescharacteristic
73C24×C127C22×C411yescharacteristic
74C24×C127C2×D411yescharacteristic
75#41×C127—44no—
76#41×C127—44no—
77#41×C127—44no—
78#41×C127—44no—
79C2×Q8×C127—21no—
80C2×Q8×C127—21no—
81C2×Q8×C127—41no—
82C2×Q8×C127—41no—
83C2×D4×C127—22no—
84C2×D4×C127—22no—
85C2×D4×C127—22no—
86C2×D4×C127—22no—
87C2×D4×C127—41no—
88C2×D4×C127—41no—
89C2×D4×C127—41no—
90C2×D4×C127—41no—
91C2×D4×C127C2×D412yes—
92C2×D4×C127#4112yes—
93C22×C4×C127—22no—
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—21no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105C22×C4×C127—21no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—22no—
109C22×C4×C127—41no—
110C22×C4×C127—41no—
111C22×C4×C127—42no—
112C22×C4×C127—41no—
113C22×C4×C127—41no—
114C22×C4×C127—41no—
115C22×C4×C127—42no—
116C22×C4×C127C22⋊C411yescharacteristic
117C22×C4×C127C22⋊C411yescharacteristic
118C22×C4×C127C22⋊C411yescharacteristic
119C22×C4×C127C2×D411yescharacteristic
120M16×C127—24no—
121M16×C127—44no—
122M16×C127—42no—
123C2×C8×C127—44no—
124C2×C8×C127—42no—
125C4⋊C4×C127—42no—
126C4⋊C4×C127—42no—
127C4⋊C4×C127—41no—
128C4⋊C4×C127—41no—
129C4⋊C4×C127—41no—
130C4⋊C4×C127—41no—
131C22⋊C4×C127—22no—
132C22⋊C4×C127—42no—
133C22⋊C4×C127—42no—
134C22⋊C4×C127—42no—
135C22⋊C4×C127—42no—
136C22⋊C4×C127—41no—
137C22⋊C4×C127—42no—
138C22⋊C4×C127—42no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—41no—
146C23×C127—41no—
147C23×C127—41no—
148C23×C127—41no—
149C23×C127—41no—
150C23×C127—41no—
151C23×C127#9912yes—
152C23×C127C2×C22⋊C411yescharacteristic
153C23×C127C4×D412yes—
154C23×C127#12112yes—
155Q8×C127—42no—
156Q8×C127—42no—
157D4×C127—42no—
158D4×C127—42no—
159D4×C127—44no—
160D4×C127—44no—
161C2×C4×C127—22no—
162C2×C4×C127—22no—
163C2×C4×C127—22no—
164C2×C4×C127—22no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—22no—
170C2×C4×C127—21no—
171C2×C4×C127—21no—
172C2×C4×C127—21no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—21no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—41no—
181C2×C4×C127—42no—
182C2×C4×C127—41no—
183C2×C4×C127—42no—
184C2×C4×C127—41no—
185C2×C4×C127—41no—
186C2×C4×C127—41no—
187C2×C4×C127—41no—
188C2×C4×C127—41no—
189C2×C4×C127—41no—
190C2×C4×C127—41no—
191C2×C4×C127—41no—
192C2×C4×C127—41no—
193C2×C4×C127—82no—
194C2×C4×C127—82no—
195C2×C4×C127—82no—
196C2×C4×C127—82no—
197C2×C4×C127#9911yescharacteristic
198C2×C4×C127#9911yescharacteristic
199C2×C4×C127#12011yescharacteristic
200C2×C4×C127#12311yescharacteristic
201C8×C127—44no—
202C22×C127—21no—
203C22×C127—21no—
204C22×C127—22no—
205C22×C127—22no—
206C22×C127—22no—
207C22×C127—42no—
208C22×C127—42no—
209C22×C127—42no—
210C22×C127—42no—
211C22×C127—41no—
212C22×C127—41no—
213C22×C127—41no—
214C22×C127—41no—
215C22×C127—41no—
216C22×C127—41no—
217C22×C127—81no—
218C22×C127—81no—
219C22×C127#38611yescharacteristic
220C22×C127C2×#9911yescharacteristic
221C22×C127C2×#9911yescharacteristic
222C4×C127—21no—
223C4×C127—21no—
224C4×C127—21no—
225C4×C127—21no—
226C4×C127—42no—
227C4×C127—41no—
228C4×C127—41no—
229C4×C127—81no—
230C4×C127—81no—
231C4×C127—81no—
232C4×C127—162no—
233#13081C12711yesmaximal, characteristic, direct product factor, Sylow
234C2×C127—21no—
235C2×C127—21no—
236C2×C127—42no—
237C2×C127—42no—
238C2×C127—42no—
239C2×C127#189811yescharacteristic
240C2×C127#212011yescharacteristic
241C2×C127#212311yescharacteristic
242C2×#413C2×C12711yescharacteristic
243#2406C2×C12711yescharacteristic
244C2×#352C2×C12712yescomplemented
245#1782C2×C12711yescharacteristic
246#1390C2×C12712yescomplemented
247C127#1308111yescharacteristic, direct product factor, Sylow
248C22×#41C22×C12711yescharacteristic
249C22×M16—21no—
250C2×#123C4×C12711yescharacteristic, complemented
251C2×#123C22×C12711yescharacteristic
252C22×C4⋊C4C4×C12711yescharacteristic, complemented
253#413—24no—
254C2×#101—21no—
255C2×#100C22×C12712yes—
256C2×#99—21no—
257C2×#99C22×C12712yes—
258#395—21no—
259#394—21no—
260#386C4×C12711yescharacteristic, complemented
261#386C22×C12711yescharacteristic
262#385—21no—
263#385—21no—
264#381—21no—
265#381C4×C12711yescharacteristic, complemented
266#352—24no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—41no—
270C2×#41—41no—
271C2×#41—42no—
272C22×Q8D4×C12711yescharacteristic
273C22×D4C2×C4×C12711yescharacteristic
274C22×D4D4×C12711yescharacteristic
275C22×D4C23×C12711yescharacteristic
276C23×C4C2×C4×C12711yescharacteristic
277C23×C4C2×C4×C12711yescharacteristic
278C23×C4C2×C4×C12711yescharacteristic
279C23×C4D4×C12711yescharacteristic
280C23×C4D4×C12711yescharacteristic
281C2×M16—24no—
282C2×M16—22no—
283C2×M16—42no—
284C2×M16—41no—
285C22×C8—41no—
286#123—22no—
287#123—22no—
288#123—42no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4C2×C4×C12711yescharacteristic
301C2×C22⋊C4C2×C4×C12711yescharacteristic
302#101—24no—
303#100—22no—
304#100D4×C12714yes—
305#99—22no—
306#99—22no—
307#99—42no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—41no—
313#95—41no—
314C24C22⋊C4×C12711yescharacteristic
315C24C22×C4×C12711yescharacteristic
316C24C2×D4×C12711yescharacteristic
317#41—44no—
318#41—44no—
319#41—44no—
320#41—44no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4C2×D4×C12712yes—
334C2×D4#41×C12712yes—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4C22⋊C4×C12711yescharacteristic
359C22×C4C22⋊C4×C12711yescharacteristic
360C22×C4C22⋊C4×C12711yescharacteristic
361C22×C4C2×D4×C12711yescharacteristic
362M16—24no—
363M16—44no—
364M16—42no—
365C2×C8—44no—
366C2×C8—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#99×C12712yes—
394C23C2×C22⋊C4×C12711yescharacteristic
395C23C4×D4×C12712yes—
396C23#121×C12712yes—
397Q8—42no—
398Q8—42no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—44no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#99×C12711yescharacteristic
440C2×C4#99×C12711yescharacteristic
441C2×C4#120×C12711yescharacteristic
442C2×C4#123×C12711yescharacteristic
443C8—44no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22#386×C12711yescharacteristic
462C22C2×#99×C12711yescharacteristic
463C22C2×#99×C12711yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—162no—
475C2—21no—
476C2—21no—
477C2—42no—
478C2—42no—
479C2—42no—
480C2C127×#189811yescharacteristic
481C2C127×#212011yescharacteristic
482C2C127×#212311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#352×C127C2semidirect
5. C127×#1390C2semidirect
8. C2×#123×C127C4semidirect
10. C22×C4⋊C4×C127C4semidirect
18. #386×C127C4semidirect
23. #381×C127C4semidirect
233. #13081C127direct (N × H)
244. C2×#352C2×C127semidirect
246. #1390C2×C127semidirect
247. C127#13081direct (N × H)
250. C2×#123C4×C127semidirect
252. C22×C4⋊C4C4×C127semidirect
260. #386C4×C127semidirect
265. #381C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#413×C1272
C127×#24062
C2×#352×C1272
C127×#17822
C127×#13902
#13081127