Finite Group Databasebeta

← C2×C127×#1459 #44581019

Subgroups — C2×C127×#1459 #44581019

Order 32512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C127C211yesmaximal, characteristic
2C22×C22⋊C8×C127C212yesmaximal, complemented
3C127×#1459C218yesmaximal, direct product factor
4C2×#324×C127C212yesmaximal, complemented
5C2×#323×C127C212yesmaximal, complemented
6C23×D4×C127C411yescharacteristic
7C24×C4×C127C411yescharacteristic
8C24×C4×C127C2211yescharacteristic
9C23×C8×C127—22no—
10C2×C4×D4×C127—22no—
11C2×C4×D4×C127—24no—
12C2×C4×D4×C127—22no—
13C2×C4×D4×C127C414yes—
14C2×C4×D4×C127C2214yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C411yescharacteristic
17C22×C22⋊C4×C127C2211yescharacteristic
18C22×C42×C127C411yescharacteristic
19C2×C22⋊C8×C127—24no—
20C2×C22⋊C8×C127—22no—
21C2×C22⋊C8×C127C2214yes—
22C2×C22⋊C8×C127C2218yescomplemented
23#324×C127C2218yescomplemented
24#323×C127C2218yescomplemented
25C25×C127C2×C411yescharacteristic
26C25×C127C2×C411yescharacteristic
27C22×D4×C127—22no—
28C22×D4×C127—22no—
29C22×D4×C127—24no—
30C22×D4×C127C818yescomplemented
31C22×D4×C127C2×C414yes—
32C23×C4×C127—22no—
33C23×C4×C127—21no—
34C23×C4×C127—21no—
35C23×C4×C127—44no—
36C23×C4×C127—44no—
37C23×C4×C127C2×C411yescharacteristic
38C23×C4×C127C2×C411yescharacteristic
39C23×C4×C127C2×C411yescharacteristic
40C23×C4×C127C2×C411yescharacteristic
41C23×C4×C127C2×C414yes—
42C23×C4×C127C2×C412yes—
43C23×C4×C127D412yes—
44C23×C4×C127D414yes—
45C23×C4×C127D412yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2312yes—
48C22×C8×C127—24no—
49C22×C8×C127—28no—
50C22×C8×C127—48no—
51C4×D4×C127—28no—
52C4×D4×C127—28no—
53C4×D4×C127—28no—
54C4×D4×C127—28no—
55C2×C4⋊C4×C127—22no—
56C2×C4⋊C4×C127—22no—
57C2×C4⋊C4×C127C2×C412yes—
58C2×C4⋊C4×C127C2312yes—
59C2×C22⋊C4×C127—22no—
60C2×C22⋊C4×C127—22no—
61C2×C22⋊C4×C127—22no—
62C2×C22⋊C4×C127—22no—
63C2×C22⋊C4×C127C2×C414yes—
64C2×C22⋊C4×C127C2×C412yes—
65C2×C22⋊C4×C127C2312yes—
66C2×C42×C127—21no—
67C2×C42×C127—22no—
68C2×C42×C127—21no—
69C2×C42×C127C2×C414yes—
70C22⋊C8×C127—24no—
71C22⋊C8×C127—28no—
72C22⋊C8×C127C2318yes—
73C24×C127—21no—
74C24×C127—21no—
75C24×C127—22no—
76C24×C127—42no—
77C24×C127—42no—
78C24×C127—42no—
79C24×C127—42no—
80C24×C127C22⋊C412yes—
81C24×C127C22⋊C412yes—
82C24×C127C2×C814yes—
83C24×C127C2×C814yescomplemented
84C24×C127M1614yes—
85C24×C127C22×C411yescharacteristic
86C24×C127C22×C412yes—
87C24×C127C22×C412yes—
88C2×D4×C127—216no—
89C2×D4×C127—216no—
90C2×D4×C127—28no—
91C2×D4×C127—28no—
92C2×D4×C127C2×C8116yescomplemented
93C22×C4×C127—21no—
94C22×C4×C127—21no—
95C22×C4×C127—22no—
96C22×C4×C127—28no—
97C22×C4×C127—24no—
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127—21no—
101C22×C4×C127—21no—
102C22×C4×C127—24no—
103C22×C4×C127—24no—
104C22×C4×C127—44no—
105C22×C4×C127—48no—
106C22×C4×C127—44no—
107C22×C4×C127—44no—
108C22×C4×C127—42no—
109C22×C4×C127—44no—
110C22×C4×C127—48no—
111C22×C4×C127—48no—
112C22×C4×C127—44no—
113C22×C4×C127—42no—
114C22×C4×C127—42no—
115C22×C4×C127C22⋊C411yescharacteristic
116C22×C4×C127C22⋊C412yes—
117C22×C4×C127C22⋊C414yes—
118C22×C4×C127C22⋊C414yes—
119C22×C4×C127C22⋊C411yescharacteristic
120C22×C4×C127C2×C814yescomplemented
121C22×C4×C127C22×C412yes—
122C22×C4×C127C22×C412yes—
123C22×C4×C127C22×C412yes—
124C22×C4×C127C22×C412yes—
125C22×C4×C127C22×C411yescharacteristic
126C22×C4×C127C2×D412yes—
127C22×C4×C127C2×D414yes—
128C22×C4×C127C2×D412yes—
129C22×C4×C127C2×D414yes—
130C22×C4×C127C2411yescharacteristic
131C2×C8×C127—28no—
132C2×C8×C127—48no—
133C2×C8×C127—416no—
134C4⋊C4×C127—24no—
135C4⋊C4×C127—24no—
136C22⋊C4×C127—28no—
137C22⋊C4×C127—24no—
138C22⋊C4×C127—24no—
139C42×C127—24no—
140C42×C127—24no—
141C23×C127—22no—
142C23×C127—24no—
143C23×C127—22no—
144C23×C127—24no—
145C23×C127—22no—
146C23×C127—22no—
147C23×C127—24no—
148C23×C127—24no—
149C23×C127—42no—
150C23×C127—42no—
151C23×C127—42no—
152C23×C127—48no—
153C23×C127—48no—
154C23×C127—42no—
155C23×C127—42no—
156C23×C127—48no—
157C23×C127—42no—
158C23×C127—42no—
159C23×C127—42no—
160C23×C127—48no—
161C23×C127—42no—
162C23×C127—42no—
163C23×C127C22⋊C818yes—
164C23×C127C22⋊C818yes—
165C23×C127C2×C22⋊C414yes—
166C23×C127C2×C22⋊C411yescharacteristic
167C23×C127C2×C22⋊C411yescharacteristic
168C23×C127C22×C812yes—
169C23×C127C22×C814yes—
170C23×C127C22×C814yescomplemented
171C23×C127C2×M1612yes—
172C23×C127C2×M1614yes—
173C23×C127C23×C411yescharacteristic
174D4×C127—232no—
175C2×C4×C127—22no—
176C2×C4×C127—22no—
177C2×C4×C127—24no—
178C2×C4×C127—24no—
179C2×C4×C127—24no—
180C2×C4×C127—22no—
181C2×C4×C127—28no—
182C2×C4×C127—24no—
183C2×C4×C127—24no—
184C2×C4×C127—28no—
185C2×C4×C127—22no—
186C2×C4×C127—22no—
187C2×C4×C127—44no—
188C2×C4×C127—41no—
189C2×C4×C127—44no—
190C2×C4×C127—42no—
191C2×C4×C127—41no—
192C2×C4×C127—48no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—42no—
196C2×C4×C127—42no—
197C2×C4×C127—48no—
198C2×C4×C127—42no—
199C2×C4×C127—42no—
200C2×C4×C127#9914yes—
201C2×C4×C127C2×C22⋊C412yes—
202C2×C4×C127C2×C22⋊C412yes—
203C2×C4×C127C2×C22⋊C412yes—
204C2×C4×C127C22×C814yescomplemented
205C2×C4×C127C22×D412yes—
206C8×C127—48no—
207C22×C127—21no—
208C22×C127—21no—
209C22×C127—24no—
210C22×C127—21no—
211C22×C127—21no—
212C22×C127—24no—
213C22×C127—24no—
214C22×C127—24no—
215C22×C127—216no—
216C22×C127—24no—
217C22×C127—48no—
218C22×C127—44no—
219C22×C127—44no—
220C22×C127—44no—
221C22×C127—44no—
222C22×C127—48no—
223C22×C127—42no—
224C22×C127—42no—
225C22×C127—48no—
226C22×C127C2×C22⋊C812yes—
227C22×C127C2×C22⋊C818yes—
228C22×C127C2×C22⋊C812yes—
229C22×C127C2×#9912yes—
230C22×C127#41312yes—
231C22×C127C22×C22⋊C411yescharacteristic
232C22×C127C23×C811yescharacteristic
233C22×C127C22×M1611yescharacteristic
234C4×C127—24no—
235C4×C127—22no—
236C4×C127—22no—
237C4×C127—22no—
238C4×C127—42no—
239C4×C127—42no—
240C4×C127C2×#9914yes—
241C2×#1459C12711yesmaximal, characteristic, direct product factor, Sylow
242C2×C127—21no—
243C2×C127—21no—
244C2×C127—22no—
245C2×C127—28no—
246C2×C127—44no—
247C2×C127#145914yesdirect product factor
248C2×C127C22×C22⋊C811yescharacteristic
249C2×C127C22×#9911yescharacteristic
250C2×C127C2×#41311yescharacteristic
251C22×C4×D4C2×C12711yescharacteristic
252C22×C22⋊C8C2×C12712yescomplemented
253#1459C2×C12718yesdirect product factor
254C2×#324C2×C12712yescomplemented
255C2×#323C2×C12712yescomplemented
256C127C2×#145911yescharacteristic, direct product factor, Sylow
257C23×D4C4×C12711yescharacteristic
258C24×C4C4×C12711yescharacteristic
259C24×C4C22×C12711yescharacteristic
260C23×C8—22no—
261C2×C4×D4—22no—
262C2×C4×D4—22no—
263C2×C4×D4—24no—
264C2×C4×D4C4×C12714yes—
265C2×C4×D4C22×C12714yes—
266C22×C4⋊C4C22×C12711yescharacteristic
267C22×C22⋊C4C4×C12711yescharacteristic
268C22×C22⋊C4C22×C12711yescharacteristic
269C22×C42C4×C12711yescharacteristic
270C2×C22⋊C8—24no—
271C2×C22⋊C8—22no—
272C2×C22⋊C8C22×C12714yes—
273C2×C22⋊C8C22×C12718yescomplemented
274#324C22×C12718yescomplemented
275#323C22×C12718yescomplemented
276C25C2×C4×C12711yescharacteristic
277C25C2×C4×C12711yescharacteristic
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—24no—
281C22×D4C8×C12718yescomplemented
282C22×D4C2×C4×C12714yes—
283C23×C4—22no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4—44no—
287C23×C4—44no—
288C23×C4C2×C4×C12711yescharacteristic
289C23×C4C2×C4×C12711yescharacteristic
290C23×C4C2×C4×C12711yescharacteristic
291C23×C4C2×C4×C12711yescharacteristic
292C23×C4C2×C4×C12714yes—
293C23×C4C2×C4×C12712yes—
294C23×C4D4×C12712yes—
295C23×C4D4×C12714yes—
296C23×C4D4×C12712yes—
297C23×C4C23×C12711yescharacteristic
298C23×C4C23×C12712yes—
299C22×C8—24no—
300C22×C8—28no—
301C22×C8—48no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—28no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×C4×C12712yes—
309C2×C4⋊C4C23×C12712yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×C4×C12714yes—
315C2×C22⋊C4C2×C4×C12712yes—
316C2×C22⋊C4C23×C12712yes—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42C2×C4×C12714yes—
321C22⋊C8—24no—
322C22⋊C8—28no—
323C22⋊C8C23×C12718yes—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24C22⋊C4×C12712yes—
332C24C22⋊C4×C12712yes—
333C24C2×C8×C12714yes—
334C24C2×C8×C12714yescomplemented
335C24M16×C12714yes—
336C24C22×C4×C12711yescharacteristic
337C24C22×C4×C12712yes—
338C24C22×C4×C12712yes—
339C2×D4—216no—
340C2×D4—216no—
341C2×D4—28no—
342C2×D4—28no—
343C2×D4C2×C8×C127116yescomplemented
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4C22⋊C4×C12711yescharacteristic
367C22×C4C22⋊C4×C12712yes—
368C22×C4C22⋊C4×C12714yes—
369C22×C4C22⋊C4×C12714yes—
370C22×C4C22⋊C4×C12711yescharacteristic
371C22×C4C2×C8×C12714yescomplemented
372C22×C4C22×C4×C12712yes—
373C22×C4C22×C4×C12712yes—
374C22×C4C22×C4×C12712yes—
375C22×C4C22×C4×C12712yes—
376C22×C4C22×C4×C12711yescharacteristic
377C22×C4C2×D4×C12712yes—
378C22×C4C2×D4×C12714yes—
379C22×C4C2×D4×C12712yes—
380C22×C4C2×D4×C12714yes—
381C22×C4C24×C12711yescharacteristic
382C2×C8—28no—
383C2×C8—48no—
384C2×C8—416no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C42—24no—
391C42—24no—
392C23—22no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—48no—
404C23—48no—
405C23—42no—
406C23—42no—
407C23—48no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—48no—
412C23—42no—
413C23—42no—
414C23C22⋊C8×C12718yes—
415C23C22⋊C8×C12718yes—
416C23C2×C22⋊C4×C12714yes—
417C23C2×C22⋊C4×C12711yescharacteristic
418C23C2×C22⋊C4×C12711yescharacteristic
419C23C22×C8×C12712yes—
420C23C22×C8×C12714yes—
421C23C22×C8×C12714yescomplemented
422C23C2×M16×C12712yes—
423C23C2×M16×C12714yes—
424C23C23×C4×C12711yescharacteristic
425D4—232no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4#99×C12714yes—
452C2×C4C2×C22⋊C4×C12712yes—
453C2×C4C2×C22⋊C4×C12712yes—
454C2×C4C2×C22⋊C4×C12712yes—
455C2×C4C22×C8×C12714yescomplemented
456C2×C4C22×D4×C12712yes—
457C8—48no—
458C22—21no—
459C22—21no—
460C22—24no—
461C22—21no—
462C22—21no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—216no—
467C22—24no—
468C22—48no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—48no—
474C22—42no—
475C22—42no—
476C22—48no—
477C22C2×C22⋊C8×C12712yes—
478C22C2×C22⋊C8×C12718yes—
479C22C2×C22⋊C8×C12712yes—
480C22C2×#99×C12712yes—
481C22#413×C12712yes—
482C22C22×C22⋊C4×C12711yescharacteristic
483C22C23×C8×C12711yescharacteristic
484C22C22×M16×C12711yescharacteristic
485C4—24no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—42no—
490C4—42no—
491C4C2×#99×C12714yes—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—28no—
496C2—44no—
497C2C127×#145914yesdirect product factor
498C2C22×C22⋊C8×C12711yescharacteristic
499C2C22×#99×C12711yescharacteristic
500C2C2×#413×C12711yescharacteristic
subgroupG/H
centerC22×C4×C127C22⋊C4
commutatorC22C23×C8×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C2×C22⋊C4
prime radicalC23×D4×C127C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊C8×C127C2semidirect
3. C127×#1459C2direct (N × H)
4. C2×#324×C127C2semidirect
5. C2×#323×C127C2semidirect
22. C2×C22⋊C8×C127C22semidirect
23. #324×C127C22semidirect
24. #323×C127C22semidirect
30. C22×D4×C127C8semidirect
83. C24×C127C2×C8semidirect
92. C2×D4×C127C2×C8semidirect
120. C22×C4×C127C2×C8semidirect
170. C23×C127C22×C8semidirect
204. C2×C4×C127C22×C8semidirect
241. C2×#1459C127direct (N × H)
247. C2×C127#1459direct (N × H)
252. C22×C22⋊C8C2×C127semidirect
253. #1459C2×C127direct (N × H)
254. C2×#324C2×C127semidirect
255. C2×#323C2×C127semidirect
256. C127C2×#1459direct (N × H)
273. C2×C22⋊C8C22×C127semidirect
274. #324C22×C127semidirect
275. #323C22×C127semidirect
281. C22×D4C8×C127semidirect
334. C24C2×C8×C127semidirect
343. C2×D4C2×C8×C127semidirect
371. C22×C4C2×C8×C127semidirect
421. C23C22×C8×C127semidirect
455. C2×C4C22×C8×C127semidirect
497. C2C127×#1459direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D4×C1272
C22×C22⋊C8×C1272
C127×#14592
C2×#324×C1272
C2×#323×C1272
C2×#1459127