Finite Group Databasebeta

← C127×#14287 #44577960

Subgroups — C127×#14287 #44577960

Order 32512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C127C211yesmaximal, characteristic
2C2×#380×C127C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
4C127×#1441C212yesmaximal, complemented
5C127×#1440C212yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1438C212yesmaximal
8C127×#1437C212yesmaximal, complemented
9C127×#1436C212yesmaximal, complemented
10C23×D4×C127C2211yescharacteristic
11C24×C4×C127C2211yescharacteristic
12C2×#123×C127—24no—
13C2×#123×C127—24no—
14C2×#123×C127C418yescomplemented
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2211yescharacteristic
17C22×C4⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2212yes—
19C22×C22⋊C4×C127C2211yescharacteristic
20C22×C22⋊C4×C127C2211yescharacteristic
21C22×C22⋊C4×C127C2211yescharacteristic
22C22×C22⋊C4×C127C2214yes—
23C22×C22⋊C4×C127C2212yes—
24C22×C42×C127C2212yes—
25#380×C127—24no—
26C4×C22⋊C4×C127—24no—
27C2×#95×C127C2212yes—
28C2×#95×C127C2214yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2214yes—
31C2×#95×C127C2214yes—
32C25×C127C2311yescharacteristic
33C25×C127C2311yescharacteristic
34C22×D4×C127—22no—
35C22×D4×C127—22no—
36C22×D4×C127—44no—
37C22×D4×C127C2×C414yes—
38C23×C4×C127—44no—
39C23×C4×C127C2×C414yes—
40C23×C4×C127D414yes—
41C23×C4×C127D414yes—
42C23×C4×C127C2314yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2311yescharacteristic
46C23×C4×C127C2314yes—
47C23×C4×C127C2311yescharacteristic
48C23×C4×C127C2311yescharacteristic
49#123×C127—232no—
50C2×C4⋊C4×C127—22no—
51C2×C4⋊C4×C127—22no—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127—24no—
55C2×C4⋊C4×C127—44no—
56C2×C4⋊C4×C127C2×C414yes—
57C2×C4⋊C4×C127C2×C414yes—
58C2×C22⋊C4×C127—22no—
59C2×C22⋊C4×C127—22no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—22no—
62C2×C22⋊C4×C127—22no—
63C2×C22⋊C4×C127—22no—
64C2×C22⋊C4×C127—22no—
65C2×C22⋊C4×C127—28no—
66C2×C22⋊C4×C127—24no—
67C2×C22⋊C4×C127—28no—
68C2×C22⋊C4×C127—44no—
69C2×C22⋊C4×C127—44no—
70C2×C22⋊C4×C127—44no—
71C2×C22⋊C4×C127C2×C414yes—
72C2×C22⋊C4×C127C2×C414yes—
73C2×C22⋊C4×C127C2×C414yes—
74C2×C42×C127—24no—
75C2×C42×C127—44no—
76#95×C127—24no—
77#95×C127—24no—
78#95×C127—28no—
79#95×C127—44no—
80#95×C127—44no—
81C24×C127—24no—
82C24×C127—22no—
83C24×C127—24no—
84C24×C127—22no—
85C24×C127—21no—
86C24×C127—21no—
87C24×C127—24no—
88C24×C127—22no—
89C24×C127—24no—
90C24×C127—22no—
91C24×C127C22×C412yes—
92C24×C127C22×C412yes—
93C24×C127C2×D412yes—
94C24×C127C2×D412yes—
95C24×C127C2411yescharacteristic
96C2×D4×C127—28no—
97C2×D4×C127—48no—
98C2×D4×C127—48no—
99C2×D4×C127—48no—
100C22×C4×C127—24no—
101C22×C4×C127—24no—
102C22×C4×C127—24no—
103C22×C4×C127—21no—
104C22×C4×C127—21no—
105C22×C4×C127—21no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—24no—
110C22×C4×C127—24no—
111C22×C4×C127—22no—
112C22×C4×C127—24no—
113C22×C4×C127—24no—
114C22×C4×C127—24no—
115C22×C4×C127—24no—
116C22×C4×C127—22no—
117C22×C4×C127—24no—
118C22×C4×C127—24no—
119C22×C4×C127—24no—
120C22×C4×C127—24no—
121C22×C4×C127—24no—
122C22×C4×C127—24no—
123C22×C4×C127—24no—
124C22×C4×C127—24no—
125C22×C4×C127—48no—
126C22×C4×C127—48no—
127C22×C4×C127—48no—
128C22×C4×C127C22⋊C4116yescomplemented
129C22×C4×C127C22×C412yes—
130C22×C4×C127C22×C412yes—
131C22×C4×C127C22×C412yes—
132C22×C4×C127C22×C412yes—
133C22×C4×C127C22×C412yes—
134C22×C4×C127C2×D412yes—
135C22×C4×C127C2×D412yes—
136C22×C4×C127C2×D412yes—
137C22×C4×C127C2×D412yes—
138C22×C4×C127#4118yes—
139C4⋊C4×C127—28no—
140C4⋊C4×C127—28no—
141C4⋊C4×C127—48no—
142C22⋊C4×C127—216no—
143C22⋊C4×C127—28no—
144C22⋊C4×C127—28no—
145C22⋊C4×C127—28no—
146C22⋊C4×C127—416no—
147C22⋊C4×C127—48no—
148C42×C127—48no—
149C23×C127—24no—
150C23×C127—24no—
151C23×C127—24no—
152C23×C127—24no—
153C23×C127—24no—
154C23×C127—24no—
155C23×C127—24no—
156C23×C127—22no—
157C23×C127—22no—
158C23×C127—24no—
159C23×C127—22no—
160C23×C127—24no—
161C23×C127—24no—
162C23×C127—24no—
163C23×C127—44no—
164C23×C127—44no—
165C23×C127—44no—
166C23×C127—44no—
167C23×C127—42no—
168C23×C127—42no—
169C23×C127—42no—
170C23×C127—42no—
171C23×C127—41no—
172C23×C127—41no—
173C23×C127—42no—
174C23×C127—44no—
175C23×C127—44no—
176C23×C127—44no—
177C23×C127—44no—
178C23×C127C2×C22⋊C414yes—
179C23×C127C23×C411yescharacteristic
180C23×C127C22×D411yescharacteristic
181C23×C127C22×D411yescharacteristic
182C23×C127C2×#4114yes—
183C23×C127H(5,𝔽2)14yes—
184C23×C127H(5,𝔽2)12yes—
185C23×C1272−1+412yes—
186D4×C127—416no—
187C2×C4×C127—24no—
188C2×C4×C127—24no—
189C2×C4×C127—24no—
190C2×C4×C127—24no—
191C2×C4×C127—24no—
192C2×C4×C127—24no—
193C2×C4×C127—22no—
194C2×C4×C127—22no—
195C2×C4×C127—22no—
196C2×C4×C127—24no—
197C2×C4×C127—24no—
198C2×C4×C127—24no—
199C2×C4×C127—28no—
200C2×C4×C127—28no—
201C2×C4×C127—28no—
202C2×C4×C127—416no—
203C2×C4×C127—42no—
204C2×C4×C127—42no—
205C2×C4×C127—44no—
206C2×C4×C127—42no—
207C2×C4×C127—42no—
208C2×C4×C127—41no—
209C2×C4×C127—41no—
210C2×C4×C127—42no—
211C2×C4×C127—44no—
212C2×C4×C127—44no—
213C2×C4×C127—44no—
214C2×C4×C127—44no—
215C2×C4×C127—44no—
216C2×C4×C127—44no—
217C2×C4×C127—44no—
218C2×C4×C127—44no—
219C2×C4×C127—44no—
220C2×C4×C127—44no—
221C2×C4×C127C2×C22⋊C414yes—
222C2×C4×C127C2×C22⋊C414yes—
223C22×C127—24no—
224C22×C127—24no—
225C22×C127—24no—
226C22×C127—48no—
227C22×C127—48no—
228C22×C127—44no—
229C22×C127—44no—
230C22×C127—42no—
231C22×C127—42no—
232C22×C127—48no—
233C22×C127—42no—
234C22×C127—84no—
235C22×C127—82no—
236C22×C127—82no—
237C22×C127C22×C22⋊C411yescharacteristic
238C22×C127C4×#4112yes—
239C22×C127#51812yes—
240C22×C127#52012yes—
241C22×C127#52512yes—
242C22×C127#53212yes—
243C22×C127#53412yes—
244C22×C127#53412yes—
245C22×C127#53511yescharacteristic
246C22×C127#53611yescharacteristic
247C22×C127#53712yes—
248C22×C127#53814yes—
249C22×C127#53914yes—
250C22×C127#54014yes—
251C22×C127#54214yes—
252C4×C127—24no—
253C4×C127—24no—
254C4×C127—42no—
255C4×C127—42no—
256C4×C127—48no—
257C4×C127—42no—
258C4×C127—84no—
259#14287C12711yesmaximal, characteristic, direct product factor, Sylow
260C2×C127—44no—
261C2×C127—82no—
262C2×C127#229711yescharacteristic
263C2×C127#230811yescharacteristic
264C2×C127#231111yescharacteristic
265C2×C127#233514yes—
266C2×C127#263014yes—
267C2×C127#266112yes—
268C2×C127#266312yes—
269C22×#123C2×C12711yescharacteristic
270C2×#380C2×C12711yescharacteristic, complemented
271C2×C4×C22⋊C4C2×C12711yescharacteristic, complemented
272#1441C2×C12712yescomplemented
273#1440C2×C12712yescomplemented
274#1439C2×C12712yescomplemented
275#1438C2×C12712yes—
276#1437C2×C12712yescomplemented
277#1436C2×C12712yescomplemented
278C127#1428711yescharacteristic, direct product factor, Sylow
279C23×D4C22×C12711yescharacteristic
280C24×C4C22×C12711yescharacteristic
281C2×#123—24no—
282C2×#123—24no—
283C2×#123C4×C12718yescomplemented
284C22×C4⋊C4C22×C12711yescharacteristic
285C22×C4⋊C4C22×C12711yescharacteristic
286C22×C4⋊C4C22×C12712yes—
287C22×C22⋊C4C22×C12712yes—
288C22×C22⋊C4C22×C12711yescharacteristic
289C22×C22⋊C4C22×C12711yescharacteristic
290C22×C22⋊C4C22×C12711yescharacteristic
291C22×C22⋊C4C22×C12714yes—
292C22×C22⋊C4C22×C12712yes—
293C22×C42C22×C12712yes—
294#380—24no—
295C4×C22⋊C4—24no—
296C2×#95C22×C12712yes—
297C2×#95C22×C12714yes—
298C2×#95C22×C12712yes—
299C2×#95C22×C12714yes—
300C2×#95C22×C12714yes—
301C25C23×C12711yescharacteristic
302C25C23×C12711yescharacteristic
303C22×D4—22no—
304C22×D4—22no—
305C22×D4—44no—
306C22×D4C2×C4×C12714yes—
307C23×C4—44no—
308C23×C4C2×C4×C12714yes—
309C23×C4D4×C12714yes—
310C23×C4D4×C12714yes—
311C23×C4C23×C12714yes—
312C23×C4C23×C12711yescharacteristic
313C23×C4C23×C12711yescharacteristic
314C23×C4C23×C12711yescharacteristic
315C23×C4C23×C12714yes—
316C23×C4C23×C12711yescharacteristic
317C23×C4C23×C12711yescharacteristic
318#123—232no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×C4×C12714yes—
326C2×C4⋊C4C2×C4×C12714yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4C2×C4×C12714yes—
341C2×C22⋊C4C2×C4×C12714yes—
342C2×C22⋊C4C2×C4×C12714yes—
343C2×C42—24no—
344C2×C42—44no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—44no—
349#95—44no—
350C24—24no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24C22×C4×C12712yes—
361C24C22×C4×C12712yes—
362C24C2×D4×C12712yes—
363C24C2×D4×C12712yes—
364C24C24×C12711yescharacteristic
365C2×D4—28no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4C22⋊C4×C127116yescomplemented
398C22×C4C22×C4×C12712yes—
399C22×C4C22×C4×C12712yes—
400C22×C4C22×C4×C12712yes—
401C22×C4C22×C4×C12712yes—
402C22×C4C22×C4×C12712yes—
403C22×C4C2×D4×C12712yes—
404C22×C4C2×D4×C12712yes—
405C22×C4C2×D4×C12712yes—
406C22×C4C2×D4×C12712yes—
407C22×C4#41×C12718yes—
408C4⋊C4—28no—
409C4⋊C4—28no—
410C4⋊C4—48no—
411C22⋊C4—216no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—416no—
416C22⋊C4—48no—
417C42—48no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—22no—
426C23—22no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23C2×C22⋊C4×C12714yes—
448C23C23×C4×C12711yescharacteristic
449C23C22×D4×C12711yescharacteristic
450C23C22×D4×C12711yescharacteristic
451C23C2×#41×C12714yes—
452C23H(5,𝔽2)×C12714yes—
453C23H(5,𝔽2)×C12712yes—
454C232−1+4×C12712yes—
455D4—416no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—416no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4C2×C22⋊C4×C12714yes—
491C2×C4C2×C22⋊C4×C12714yes—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—48no—
496C22—48no—
497C22—44no—
498C22—44no—
499C22—48no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—84no—
504C22—82no—
505C22—82no—
506C22C22×C22⋊C4×C12711yescharacteristic
507C22C4×#41×C12712yes—
508C22#518×C12712yes—
509C22#520×C12712yes—
510C22#525×C12712yes—
511C22#532×C12712yes—
512C22#534×C12712yes—
513C22#534×C12712yes—
514C22#535×C12711yescharacteristic
515C22#536×C12711yescharacteristic
516C22#537×C12712yes—
517C22#538×C12714yes—
518C22#539×C12714yes—
519C22#540×C12714yes—
520C22#542×C12714yes—
521C4—24no—
522C4—24no—
523C4—48no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—84no—
528C2—44no—
529C2—82no—
530C2C127×#229711yescharacteristic
531C2C127×#230811yescharacteristic
532C2C127×#231111yescharacteristic
533C2C127×#233514yes—
534C2C127×#263014yes—
535C2C127×#266112yes—
536C2C127×#266312yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#380×C127C2semidirect
3. C2×C4×C22⋊C4×C127C2semidirect
4. C127×#1441C2semidirect
5. C127×#1440C2semidirect
6. C127×#1439C2semidirect
8. C127×#1437C2semidirect
9. C127×#1436C2semidirect
14. C2×#123×C127C4semidirect
128. C22×C4×C127C22⋊C4semidirect
259. #14287C127direct (N × H)
270. C2×#380C2×C127semidirect
271. C2×C4×C22⋊C4C2×C127semidirect
272. #1441C2×C127semidirect
273. #1440C2×C127semidirect
274. #1439C2×C127semidirect
276. #1437C2×C127semidirect
277. #1436C2×C127semidirect
278. C127#14287direct (N × H)
283. C2×#123C4×C127semidirect
397. 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×#123×C1272
C2×#380×C1272
C2×C4×C22⋊C4×C1272
C127×#14412
C127×#14402
C127×#14392
C127×#14382
C127×#14372
C127×#14362
#14287127