Finite Group Databasebeta

← C127×#14271 #44577944

Subgroups — C127×#14271 #44577944

Order 32512.

2718 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#121×C127C211yesmaximal, characteristic
2C2×#386×C127C214yesmaximal, complemented
3C2×#380×C127C211yesmaximal, characteristic, complemented
4C22×#95×C127C211yesmaximal, characteristic, complemented
5C127×#1445C212yesmaximal, complemented
6C127×#1439C214yesmaximal, complemented
7C127×#1438C212yesmaximal, complemented
8C23×D4×C127C2212yes—
9C23×D4×C127C2211yescharacteristic
10C24×C4×C127C2212yes—
11C24×C4×C127C2211yescharacteristic
12C2×#121×C127—24no—
13C2×#121×C127—24no—
14C2×#121×C127C418yescomplemented
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2212yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2218yes—
20#386×C127—216no—
21#380×C127—24no—
22C2×#95×C127—24no—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2218yes—
27C25×C127C2312yes—
28C25×C127C2311yescharacteristic
29C22×D4×C127—24no—
30C22×D4×C127—24no—
31C22×D4×C127—28no—
32C22×D4×C127—28no—
33C22×D4×C127—22no—
34C22×D4×C127—22no—
35C22×D4×C127—44no—
36C22×D4×C127C2×C418yes—
37C22×D4×C127C2×C414yescomplemented
38C23×C4×C127—24no—
39C23×C4×C127—24no—
40C23×C4×C127—44no—
41C23×C4×C127—44no—
42C23×C4×C127—44no—
43C23×C4×C127C2×C414yes—
44C23×C4×C127D414yes—
45C23×C4×C127D414yes—
46C23×C4×C127D418yescomplemented
47C23×C4×C127C2314yes—
48C23×C4×C127C2312yes—
49C23×C4×C127C2314yes—
50C23×C4×C127C2311yescharacteristic
51C23×C4×C127C2311yescharacteristic
52#121×C127—232no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—24no—
56C2×C4⋊C4×C127—44no—
57C2×C4⋊C4×C127C2×C414yescomplemented
58C2×C22⋊C4×C127—24no—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—28no—
62C2×C22⋊C4×C127—216no—
63C2×C22⋊C4×C127—48no—
64C2×C22⋊C4×C127—48no—
65C2×C22⋊C4×C127C2×C418yescomplemented
66#95×C127—24no—
67#95×C127—24no—
68#95×C127—24no—
69#95×C127—216no—
70C24×C127—24no—
71C24×C127—22no—
72C24×C127—22no—
73C24×C127—24no—
74C24×C127—22no—
75C24×C127—22no—
76C24×C127—24no—
77C24×C127—24no—
78C24×C127—28no—
79C24×C127—24no—
80C24×C127C22×C414yes—
81C24×C127C22×C412yes—
82C24×C127C2×D412yes—
83C24×C127C2×D412yes—
84C24×C127C2×D414yes—
85C24×C127#4114yes—
86C24×C127C2411yescharacteristic
87C2×D4×C127—216no—
88C2×D4×C127—232no—
89C2×D4×C127—28no—
90C2×D4×C127—48no—
91C2×D4×C127—48no—
92C2×D4×C127—48no—
93C2×D4×C127—416no—
94C2×D4×C127—48no—
95C2×D4×C127—48no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—216no—
99C22×C4×C127—24no—
100C22×C4×C127—24no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—24no—
104C22×C4×C127—24no—
105C22×C4×C127—28no—
106C22×C4×C127—28no—
107C22×C4×C127—28no—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127—28no—
111C22×C4×C127—24no—
112C22×C4×C127—48no—
113C22×C4×C127—48no—
114C22×C4×C127—48no—
115C22×C4×C127—416no—
116C22×C4×C127—48no—
117C22×C4×C127—48no—
118C22×C4×C127—416no—
119C22×C4×C127C22⋊C4116yescomplemented
120C22×C4×C127C22×C414yes—
121C22×C4×C127C22×C412yes—
122C22×C4×C127C22×C412yes—
123C22×C4×C127C2×D418yes—
124C22×C4×C127C2×D412yes—
125C22×C4×C127C2×D412yes—
126C22×C4×C127C2×D412yes—
127C22×C4×C127C2×D412yes—
128C4⋊C4×C127—28no—
129C4⋊C4×C127—48no—
130C22⋊C4×C127—216no—
131C22⋊C4×C127—432no—
132C22⋊C4×C127—48no—
133C23×C127—24no—
134C23×C127—24no—
135C23×C127—24no—
136C23×C127—24no—
137C23×C127—28no—
138C23×C127—22no—
139C23×C127—22no—
140C23×C127—24no—
141C23×C127—28no—
142C23×C127—28no—
143C23×C127—28no—
144C23×C127—24no—
145C23×C127—28no—
146C23×C127—28no—
147C23×C127—28no—
148C23×C127—44no—
149C23×C127—42no—
150C23×C127—42no—
151C23×C127—48no—
152C23×C127—44no—
153C23×C127—44no—
154C23×C127—42no—
155C23×C127—42no—
156C23×C127—44no—
157C23×C127—48no—
158C23×C127—48no—
159C23×C127—48no—
160C23×C127C2×C22⋊C414yes—
161C23×C127C4×D418yes—
162C23×C127#12118yes—
163C23×C127#12414yes—
164C23×C127#12714yes—
165C23×C127C23×C411yescharacteristic
166C23×C127C22×D411yescharacteristic
167C23×C127C22×D411yescharacteristic
168C23×C127C22×D412yes—
169C23×C127C2×#4112yes—
170C23×C127H(5,𝔽2)14yes—
171C23×C127H(5,𝔽2)12yes—
172C23×C1272−1+412yes—
173D4×C127—416no—
174D4×C127—432no—
175C2×C4×C127—24no—
176C2×C4×C127—24no—
177C2×C4×C127—24no—
178C2×C4×C127—24no—
179C2×C4×C127—24no—
180C2×C4×C127—24no—
181C2×C4×C127—24no—
182C2×C4×C127—28no—
183C2×C4×C127—28no—
184C2×C4×C127—216no—
185C2×C4×C127—416no—
186C2×C4×C127—432no—
187C2×C4×C127—44no—
188C2×C4×C127—42no—
189C2×C4×C127—42no—
190C2×C4×C127—44no—
191C2×C4×C127—44no—
192C2×C4×C127—48no—
193C2×C4×C127—44no—
194C2×C4×C127—48no—
195C2×C4×C127—44no—
196C2×C4×C127—44no—
197C2×C4×C127—48no—
198C2×C4×C127C2×C22⋊C414yes—
199C2×C4×C127C2×C22⋊C414yes—
200C22×C127—24no—
201C22×C127—24no—
202C22×C127—216no—
203C22×C127—28no—
204C22×C127—28no—
205C22×C127—28no—
206C22×C127—28no—
207C22×C127—44no—
208C22×C127—44no—
209C22×C127—416no—
210C22×C127—44no—
211C22×C127—84no—
212C22×C127—84no—
213C22×C127—84no—
214C22×C127—84no—
215C22×C127#39018yes—
216C22×C127C22×C22⋊C411yescharacteristic
217C22×C127C2×C4×D412yes—
218C22×C127#51812yes—
219C22×C127#52012yes—
220C22×C127C2×#12112yes—
221C22×C127C2×#12411yescharacteristic
222C22×C127C2×#12711yescharacteristic
223C22×C127#53412yes—
224C22×C127#53412yes—
225C22×C127#53414yes—
226C22×C127#53511yescharacteristic
227C22×C127#53611yescharacteristic
228C22×C127#53712yes—
229C22×C127#53714yes—
230C22×C127#53814yes—
231C22×C127#53914yes—
232C4×C127—24no—
233C4×C127—24no—
234C4×C127—48no—
235C4×C127—44no—
236C4×C127—84no—
237#14271C12711yesmaximal, characteristic, direct product factor, Sylow
238C2×C127—28no—
239C2×C127—84no—
240C2×C127C2×#39011yescharacteristic
241C2×C127#230811yescharacteristic
242C2×C127#231111yescharacteristic
243C2×C127#232514yes—
244C2×C127#261712yes—
245C2×C127#261812yes—
246C2×C127#263912yes—
247C2×C127#265612yes—
248C22×#121C2×C12711yescharacteristic
249C2×#386C2×C12714yescomplemented
250C2×#380C2×C12711yescharacteristic, complemented
251C22×#95C2×C12711yescharacteristic, complemented
252#1445C2×C12712yescomplemented
253#1439C2×C12714yescomplemented
254#1438C2×C12712yescomplemented
255C127#1427111yescharacteristic, direct product factor, Sylow
256C23×D4C22×C12712yes—
257C23×D4C22×C12711yescharacteristic
258C24×C4C22×C12712yes—
259C24×C4C22×C12711yescharacteristic
260C2×#121—24no—
261C2×#121—24no—
262C2×#121C4×C12718yescomplemented
263C22×C4⋊C4C22×C12711yescharacteristic
264C22×C4⋊C4C22×C12712yes—
265C22×C22⋊C4C22×C12712yes—
266C22×C22⋊C4C22×C12712yes—
267C22×C22⋊C4C22×C12718yes—
268#386—216no—
269#380—24no—
270C2×#95—24no—
271C2×#95C22×C12712yes—
272C2×#95C22×C12712yes—
273C2×#95C22×C12712yes—
274C2×#95C22×C12718yes—
275C25C23×C12712yes—
276C25C23×C12711yescharacteristic
277C22×D4—24no—
278C22×D4—24no—
279C22×D4—28no—
280C22×D4—28no—
281C22×D4—22no—
282C22×D4—22no—
283C22×D4—44no—
284C22×D4C2×C4×C12718yes—
285C22×D4C2×C4×C12714yescomplemented
286C23×C4—24no—
287C23×C4—24no—
288C23×C4—44no—
289C23×C4—44no—
290C23×C4—44no—
291C23×C4C2×C4×C12714yes—
292C23×C4D4×C12714yes—
293C23×C4D4×C12714yes—
294C23×C4D4×C12718yescomplemented
295C23×C4C23×C12714yes—
296C23×C4C23×C12712yes—
297C23×C4C23×C12714yes—
298C23×C4C23×C12711yescharacteristic
299C23×C4C23×C12711yescharacteristic
300#121—232no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×C4×C12714yescomplemented
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4C2×C4×C12718yescomplemented
314#95—24no—
315#95—24no—
316#95—24no—
317#95—216no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—28no—
327C24—24no—
328C24C22×C4×C12714yes—
329C24C22×C4×C12712yes—
330C24C2×D4×C12712yes—
331C24C2×D4×C12712yes—
332C24C2×D4×C12714yes—
333C24#41×C12714yes—
334C24C24×C12711yescharacteristic
335C2×D4—216no—
336C2×D4—232no—
337C2×D4—28no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—416no—
342C2×D4—48no—
343C2×D4—48no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—216no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—416no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—416no—
367C22×C4C22⋊C4×C127116yescomplemented
368C22×C4C22×C4×C12714yes—
369C22×C4C22×C4×C12712yes—
370C22×C4C22×C4×C12712yes—
371C22×C4C2×D4×C12718yes—
372C22×C4C2×D4×C12712yes—
373C22×C4C2×D4×C12712yes—
374C22×C4C2×D4×C12712yes—
375C22×C4C2×D4×C12712yes—
376C4⋊C4—28no—
377C4⋊C4—48no—
378C22⋊C4—216no—
379C22⋊C4—432no—
380C22⋊C4—48no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—28no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—28no—
389C23—24no—
390C23—28no—
391C23—28no—
392C23—24no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—48no—
400C23—44no—
401C23—44no—
402C23—42no—
403C23—42no—
404C23—48no—
405C23—44no—
406C23—48no—
407C23—48no—
408C23C2×C22⋊C4×C12714yes—
409C23C4×D4×C12718yes—
410C23#121×C12718yes—
411C23#124×C12714yes—
412C23#127×C12714yes—
413C23C23×C4×C12711yescharacteristic
414C23C22×D4×C12711yescharacteristic
415C23C22×D4×C12711yescharacteristic
416C23C22×D4×C12712yes—
417C23C2×#41×C12712yes—
418C23H(5,𝔽2)×C12714yes—
419C23H(5,𝔽2)×C12712yes—
420C232−1+4×C12712yes—
421D4—416no—
422D4—432no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—216no—
433C2×C4—416no—
434C2×C4—432no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—48no—
441C2×C4—44no—
442C2×C4—48no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4C2×C22⋊C4×C12714yes—
447C2×C4C2×C22⋊C4×C12714yes—
448C22—216no—
449C22—28no—
450C22—28no—
451C22—28no—
452C22—28no—
453C22—24no—
454C22—24no—
455C22—416no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—84no—
460C22—84no—
461C22—84no—
462C22—84no—
463C22#390×C12718yes—
464C22C22×C22⋊C4×C12711yescharacteristic
465C22C2×C4×D4×C12712yes—
466C22#518×C12712yes—
467C22#520×C12712yes—
468C22C2×#121×C12712yes—
469C22C2×#124×C12711yescharacteristic
470C22C2×#127×C12711yescharacteristic
471C22#534×C12714yes—
472C22#534×C12712yes—
473C22#534×C12712yes—
474C22#535×C12711yescharacteristic
475C22#536×C12711yescharacteristic
476C22#537×C12714yes—
477C22#537×C12712yes—
478C22#538×C12714yes—
479C22#539×C12714yes—
480C4—24no—
481C4—24no—
482C4—48no—
483C4—44no—
484C4—84no—
485C2—28no—
486C2—84no—
487C2C2×#390×C12711yescharacteristic
488C2C127×#230811yescharacteristic
489C2C127×#231111yescharacteristic
490C2C127×#232514yes—
491C2C127×#261712yes—
492C2C127×#261812yes—
493C2C127×#263912yes—
494C2C127×#265612yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC22×#121×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#386×C127C2semidirect
3. C2×#380×C127C2semidirect
4. C22×#95×C127C2semidirect
5. C127×#1445C2semidirect
6. C127×#1439C2semidirect
7. C127×#1438C2semidirect
14. C2×#121×C127C4semidirect
37. C22×D4×C127C2×C4semidirect
46. C23×C4×C127D4semidirect
57. C2×C4⋊C4×C127C2×C4semidirect
65. C2×C22⋊C4×C127C2×C4semidirect
119. C22×C4×C127C22⋊C4semidirect
237. #14271C127direct (N × H)
249. C2×#386C2×C127semidirect
250. C2×#380C2×C127semidirect
251. C22×#95C2×C127semidirect
252. #1445C2×C127semidirect
253. #1439C2×C127semidirect
254. #1438C2×C127semidirect
255. C127#14271direct (N × H)
262. C2×#121C4×C127semidirect
285. C22×D4C2×C4×C127semidirect
294. C23×C4D4×C127semidirect
305. C2×C4⋊C4C2×C4×C127semidirect
313. C2×C22⋊C4C2×C4×C127semidirect
367. C22×C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#121×C1272
C2×#386×C1272
C2×#380×C1272
C22×#95×C1272
C127×#14452
C127×#14392
C127×#14382
#14271127