Finite Group Databasebeta

← C127×#15239 #44578912

Subgroups — C127×#15239 #44578912

Order 32512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389×C127C211yesmaximal, characteristic
2C2×#382×C127C211yesmaximal, characteristic
3C127×#1443C214yesmaximal
4C127×#1442C214yesmaximal
5C127×C4⋊#95C211yesmaximal, characteristic
6C127×C4⋊#95C211yesmaximal, characteristic
7C127×C4⋊#95C211yesmaximal, characteristic
8C127×C4⋊#95C211yesmaximal, characteristic
9C4×#95×C127C211yesmaximal, characteristic
10C22×C4⋊C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2212yes—
15C22×C42×C127C2211yescharacteristic
16C22×C42×C127C2211yescharacteristic
17C22×C42×C127C2211yescharacteristic
18#389×C127—24no—
19#382×C127—24no—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C2×#95×C127C2214yes—
23C2×#95×C127C2214yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2214yes—
28C2×#95×C127C2214yes—
29C23×C4×C127C2312yes—
30C23×C4×C127C2312yes—
31C23×C4×C127C2312yes—
32C23×C4×C127C2311yescharacteristic
33C23×C4×C127C2311yescharacteristic
34C23×C4×C127C2312yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2312yes—
37C23×C4×C127C2311yescharacteristic
38C2×C4⋊C4×C127—22no—
39C2×C4⋊C4×C127—22no—
40C2×C4⋊C4×C127—22no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—22no—
45C2×C4⋊C4×C127—22no—
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127—22no—
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—44no—
51C2×C4⋊C4×C127Q814yes—
52C2×C42×C127—22no—
53C2×C42×C127—22no—
54C2×C42×C127—22no—
55C2×C42×C127—22no—
56C2×C42×C127—22no—
57C2×C42×C127—22no—
58C2×C42×C127—22no—
59C2×C42×C127—22no—
60C2×C42×C127D414yes—
61#95×C127—24no—
62#95×C127—24no—
63#95×C127—24no—
64#95×C127—24no—
65#95×C127—24no—
66#95×C127—28no—
67#95×C127—28no—
68#95×C127—44no—
69#95×C127—44no—
70C24×C127C2411yescharacteristic
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—24no—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—24no—
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—24no—
99C22×C4×C127—24no—
100C22×C4×C127—24no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127C2×D412yes—
111C22×C4×C127C2×D412yes—
112C22×C4×C127C2×D412yes—
113C22×C4×C127C2×Q814yes—
114C22×C4×C127C2×Q812yes—
115C22×C4×C127#4114yes—
116C22×C4×C127#4114yes—
117C22×C4×C127#4112yes—
118C22×C4×C127#4112yes—
119C22×C4×C127#4112yes—
120C22×C4×C127#4112yes—
121C22×C4×C127#4112yes—
122C4⋊C4×C127—28no—
123C4⋊C4×C127—48no—
124C4⋊C4×C127—44no—
125C4⋊C4×C127—44no—
126C4⋊C4×C127—44no—
127C42×C127—28no—
128C42×C127—44no—
129C42×C127—44no—
130C23×C127C22×D411yescharacteristic
131C23×C127C22×Q811yescharacteristic
132C23×C127C2×#4111yescharacteristic
133C23×C127C2×#4111yescharacteristic
134C23×C127C2×#4111yescharacteristic
135C23×C127C2×#4111yescharacteristic
136C23×C127C2×#4111yescharacteristic
137C23×C127C2×#4112yes—
138C23×C127C2×#4112yes—
139C23×C127H(5,𝔽2)12yes—
140C23×C1272−1+412yes—
141C2×C4×C127—24no—
142C2×C4×C127—22no—
143C2×C4×C127—22no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—22no—
147C2×C4×C127—22no—
148C2×C4×C127—22no—
149C2×C4×C127—22no—
150C2×C4×C127—24no—
151C2×C4×C127—24no—
152C2×C4×C127—24no—
153C2×C4×C127—24no—
154C2×C4×C127—28no—
155C2×C4×C127—42no—
156C2×C4×C127—42no—
157C2×C4×C127—42no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—42no—
161C2×C4×C127—42no—
162C2×C4×C127—42no—
163C2×C4×C127—44no—
164C2×C4×C127—42no—
165C2×C4×C127—42no—
166C2×C4×C127—42no—
167C2×C4×C127—42no—
168C2×C4×C127—42no—
169C2×C4×C127—42no—
170C2×C4×C127—42no—
171C2×C4×C127—42no—
172C2×C4×C127—42no—
173C2×C4×C127—44no—
174C2×C4×C127—44no—
175C2×C4×C127—42no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—44no—
182C2×C4×C127—44no—
183C2×C4×C127—44no—
184C2×C4×C127—44no—
185C2×C4×C127C22⋊Q814yes—
186C2×C4×C127#12314yes—
187C2×C4×C127#12414yes—
188C2×C4×C127#12514yes—
189C2×C4×C127#12514yes—
190C2×C4×C127#12614yes—
191C2×C4×C127#12614yes—
192C22×C127C2×C22⋊Q811yescharacteristic
193C22×C127C2×#12311yescharacteristic
194C22×C127C2×#12411yescharacteristic
195C22×C127C2×#12511yescharacteristic
196C22×C127C2×#12511yescharacteristic
197C22×C127C2×#12611yescharacteristic
198C22×C127C2×#12611yescharacteristic
199C22×C127#52912yes—
200C22×C127#52912yes—
201C22×C127#53211yescharacteristic
202C22×C127#53211yescharacteristic
203C22×C127#53511yescharacteristic
204C22×C127#53611yescharacteristic
205C22×C127#54212yes—
206C22×C127#54212yes—
207C22×C127#55212yes—
208C22×C127#55212yes—
209C22×C127#55312yes—
210C22×C127#55312yes—
211C22×C127#55412yes—
212C22×C127#55512yes—
213C22×C127#55612yes—
214C22×C127#55712yes—
215C4×C127—44no—
216C4×C127—44no—
217C4×C127—44no—
218C4×C127—42no—
219C4×C127—42no—
220C4×C127—82no—
221C4×C127—82no—
222C4×C127—82no—
223C4×C127#40018yes—
224#15239C12711yesmaximal, characteristic, direct product factor, Sylow
225C2×C127C2×#40011yescharacteristic
226C2×C127#249412yes—
227C2×C127#249512yes—
228C2×C127#249911yescharacteristic
229C2×C127#250011yescharacteristic
230C2×C127#250812yes—
231C2×C127#250912yes—
232C2×C127#256511yescharacteristic
233C2×C127#256611yescharacteristic
234C2×C127#264011yescharacteristic
235C2×C127#286111yescharacteristic
236C2×#389C2×C12711yescharacteristic
237C2×#382C2×C12711yescharacteristic
238#1443C2×C12714yes—
239#1442C2×C12714yes—
240C4⋊#95C2×C12711yescharacteristic
241C4⋊#95C2×C12711yescharacteristic
242C4⋊#95C2×C12711yescharacteristic
243C4⋊#95C2×C12711yescharacteristic
244C4×#95C2×C12711yescharacteristic
245C127#1523911yescharacteristic, direct product factor, Sylow
246C22×C4⋊C4C22×C12711yescharacteristic
247C22×C4⋊C4C22×C12711yescharacteristic
248C22×C4⋊C4C22×C12711yescharacteristic
249C22×C4⋊C4C22×C12711yescharacteristic
250C22×C4⋊C4C22×C12712yes—
251C22×C42C22×C12711yescharacteristic
252C22×C42C22×C12711yescharacteristic
253C22×C42C22×C12711yescharacteristic
254#389—24no—
255#382—24no—
256C2×#95C22×C12712yes—
257C2×#95C22×C12712yes—
258C2×#95C22×C12714yes—
259C2×#95C22×C12714yes—
260C2×#95C22×C12712yes—
261C2×#95C22×C12712yes—
262C2×#95C22×C12712yes—
263C2×#95C22×C12714yes—
264C2×#95C22×C12714yes—
265C23×C4C23×C12712yes—
266C23×C4C23×C12712yes—
267C23×C4C23×C12712yes—
268C23×C4C23×C12711yescharacteristic
269C23×C4C23×C12711yescharacteristic
270C23×C4C23×C12712yes—
271C23×C4C23×C12712yes—
272C23×C4C23×C12712yes—
273C23×C4C23×C12711yescharacteristic
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4Q8×C12714yes—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42D4×C12714yes—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306C24C24×C12711yescharacteristic
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4C2×D4×C12712yes—
347C22×C4C2×D4×C12712yes—
348C22×C4C2×D4×C12712yes—
349C22×C4C2×Q8×C12714yes—
350C22×C4C2×Q8×C12712yes—
351C22×C4#41×C12714yes—
352C22×C4#41×C12714yes—
353C22×C4#41×C12712yes—
354C22×C4#41×C12712yes—
355C22×C4#41×C12712yes—
356C22×C4#41×C12712yes—
357C22×C4#41×C12712yes—
358C4⋊C4—28no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C42—28no—
364C42—44no—
365C42—44no—
366C23C22×D4×C12711yescharacteristic
367C23C22×Q8×C12711yescharacteristic
368C23C2×#41×C12711yescharacteristic
369C23C2×#41×C12711yescharacteristic
370C23C2×#41×C12711yescharacteristic
371C23C2×#41×C12711yescharacteristic
372C23C2×#41×C12711yescharacteristic
373C23C2×#41×C12712yes—
374C23C2×#41×C12712yes—
375C23H(5,𝔽2)×C12712yes—
376C232−1+4×C12712yes—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4C22⋊Q8×C12714yes—
422C2×C4#123×C12714yes—
423C2×C4#124×C12714yes—
424C2×C4#125×C12714yes—
425C2×C4#125×C12714yes—
426C2×C4#126×C12714yes—
427C2×C4#126×C12714yes—
428C22C2×C22⋊Q8×C12711yescharacteristic
429C22C2×#123×C12711yescharacteristic
430C22C2×#124×C12711yescharacteristic
431C22C2×#125×C12711yescharacteristic
432C22C2×#125×C12711yescharacteristic
433C22C2×#126×C12711yescharacteristic
434C22C2×#126×C12711yescharacteristic
435C22#529×C12712yes—
436C22#529×C12712yes—
437C22#532×C12711yescharacteristic
438C22#532×C12711yescharacteristic
439C22#535×C12711yescharacteristic
440C22#536×C12711yescharacteristic
441C22#542×C12712yes—
442C22#542×C12712yes—
443C22#552×C12712yes—
444C22#552×C12712yes—
445C22#553×C12712yes—
446C22#553×C12712yes—
447C22#554×C12712yes—
448C22#555×C12712yes—
449C22#556×C12712yes—
450C22#557×C12712yes—
451C4—44no—
452C4—44no—
453C4—42no—
454C4—42no—
455C4—44no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4#400×C12718yes—
460C2C2×#400×C12711yescharacteristic
461C2C127×#249412yes—
462C2C127×#249512yes—
463C2C127×#249911yescharacteristic
464C2C127×#250011yescharacteristic
465C2C127×#250812yes—
466C2C127×#250912yes—
467C2C127×#256511yescharacteristic
468C2C127×#256611yescharacteristic
469C2C127×#264011yescharacteristic
470C2C127×#286111yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
224. #15239C127direct (N × H)
245. C127#15239direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#389×C1272
C2×#382×C1272
C127×#14432
C127×#14422
C127×C4⋊#952
C4×#95×C1272
#15239127