Finite Group Databasebeta

← C127×#15234 #44578907

Subgroups — C127×#15234 #44578907

Order 32512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389×C127C211yesmaximal, characteristic
2C2×#389×C127C211yesmaximal, characteristic
3C2×#382×C127C211yesmaximal, characteristic
4C2×C4×C4⋊C4×C127C211yesmaximal, characteristic
5C127×#1442C214yesmaximal
6C127×#1441C214yesmaximal
7C127×C4⋊#95C211yesmaximal, characteristic
8C127×C4⋊#95C211yesmaximal, characteristic
9C127×C4⋊#95C211yesmaximal, characteristic
10C22×C4⋊C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2212yes—
17C22×C42×C127C2211yescharacteristic
18C22×C42×C127C2211yescharacteristic
19C22×C42×C127C2211yescharacteristic
20#389×C127—24no—
21#389×C127—24no—
22#382×C127—24no—
23C4×C4⋊C4×C127—24no—
24C2×#95×C127C2214yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2214yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2214yes—
30C2×#95×C127C2214yes—
31C23×C4×C127C2312yes—
32C23×C4×C127C2312yes—
33C23×C4×C127C2311yescharacteristic
34C23×C4×C127C2312yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2312yes—
37C23×C4×C127C2312yes—
38C23×C4×C127C2311yescharacteristic
39C23×C4×C127C2311yescharacteristic
40C2×C4⋊C4×C127—24no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—22no—
43C2×C4⋊C4×C127—24no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—22no—
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127—22no—
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—22no—
51C2×C4⋊C4×C127—22no—
52C2×C4⋊C4×C127—24no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—28no—
56C2×C4⋊C4×C127—44no—
57C2×C4⋊C4×C127Q814yes—
58C2×C4⋊C4×C127Q814yes—
59C2×C42×C127—22no—
60C2×C42×C127—22no—
61C2×C42×C127—22no—
62C2×C42×C127—22no—
63C2×C42×C127—22no—
64C2×C42×C127—22no—
65C2×C42×C127—22no—
66C2×C42×C127D414yes—
67C2×C42×C127D414yes—
68#95×C127—24no—
69#95×C127—28no—
70#95×C127—24no—
71#95×C127—24no—
72#95×C127—28no—
73#95×C127—44no—
74#95×C127—44no—
75C24×C127C2411yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—22no—
79C22×C4×C127—21no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—24no—
84C22×C4×C127—21no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—24no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—24no—
98C22×C4×C127—24no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
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—22no—
109C22×C4×C127—22no—
110C22×C4×C127—24no—
111C22×C4×C127—24no—
112C22×C4×C127—24no—
113C22×C4×C127C2×D414yes—
114C22×C4×C127C2×D412yes—
115C22×C4×C127C2×D412yes—
116C22×C4×C127C2×D412yes—
117C22×C4×C127C2×D412yes—
118C22×C4×C127C2×Q814yes—
119C22×C4×C127C2×Q814yes—
120C22×C4×C127C2×Q812yes—
121C22×C4×C127C2×Q812yes—
122C22×C4×C127#4114yes—
123C22×C4×C127#4114yes—
124C22×C4×C127#4112yes—
125C22×C4×C127#4112yes—
126C22×C4×C127#4112yes—
127C4⋊C4×C127—216no—
128C4⋊C4×C127—28no—
129C4⋊C4×C127—28no—
130C4⋊C4×C127—48no—
131C4⋊C4×C127—44no—
132C4⋊C4×C127—44no—
133C4⋊C4×C127—48no—
134C42×C127—28no—
135C42×C127—28no—
136C42×C127—44no—
137C23×C127C22×D411yescharacteristic
138C23×C127C22×D411yescharacteristic
139C23×C127C22×Q811yescharacteristic
140C23×C127C22×Q811yescharacteristic
141C23×C127C2×#4111yescharacteristic
142C23×C127C2×#4111yescharacteristic
143C23×C127C2×#4111yescharacteristic
144C23×C127C2×#4112yes—
145C23×C127C2×#4112yes—
146C23×C127H(5,𝔽2)12yes—
147C23×C1272−1+412yes—
148C2×C4×C127—22no—
149C2×C4×C127—24no—
150C2×C4×C127—24no—
151C2×C4×C127—22no—
152C2×C4×C127—22no—
153C2×C4×C127—24no—
154C2×C4×C127—24no—
155C2×C4×C127—24no—
156C2×C4×C127—24no—
157C2×C4×C127—24no—
158C2×C4×C127—22no—
159C2×C4×C127—22no—
160C2×C4×C127—24no—
161C2×C4×C127—24no—
162C2×C4×C127—24no—
163C2×C4×C127—28no—
164C2×C4×C127—28no—
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—42no—
174C2×C4×C127—44no—
175C2×C4×C127—42no—
176C2×C4×C127—44no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—42no—
182C2×C4×C127—42no—
183C2×C4×C127—44no—
184C2×C4×C127—42no—
185C2×C4×C127—44no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—42no—
189C2×C4×C127—44no—
190C2×C4×C127—44no—
191C2×C4×C127#12114yes—
192C2×C4×C127C22⋊Q814yes—
193C2×C4×C127C22⋊Q814yes—
194C2×C4×C127#12314yes—
195C2×C4×C127#12414yes—
196C2×C4×C127#12514yes—
197C2×C4×C127#12514yes—
198C2×C4×C127C4⋊Q814yes—
199C22×C127C2×#12111yescharacteristic
200C22×C127C2×C22⋊Q811yescharacteristic
201C22×C127C2×C22⋊Q811yescharacteristic
202C22×C127C2×#12311yescharacteristic
203C22×C127#52512yes—
204C22×C127C2×#12411yescharacteristic
205C22×C127C2×#12511yescharacteristic
206C22×C127C2×#12511yescharacteristic
207C22×C127#52912yes—
208C22×C127C2×C4⋊Q811yescharacteristic
209C22×C127#53211yescharacteristic
210C22×C127#53511yescharacteristic
211C22×C127#53611yescharacteristic
212C22×C127#53712yes—
213C22×C127#54212yes—
214C22×C127#55212yes—
215C22×C127#55312yes—
216C22×C127#55412yes—
217C22×C127#55412yes—
218C22×C127#55512yes—
219C22×C127#55612yes—
220C22×C127#55712yes—
221C22×C127#55712yes—
222C4×C127—24no—
223C4×C127—44no—
224C4×C127—44no—
225C4×C127—42no—
226C4×C127—44no—
227C4×C127—44no—
228C4×C127—82no—
229C4×C127—82no—
230C4×C127#39818yes—
231#15234C12711yesmaximal, characteristic, direct product factor, Sylow
232C2×C127C2×#39811yescharacteristic
233C2×C127#248512yes—
234C2×C127#250812yes—
235C2×C127#250912yes—
236C2×C127#256311yescharacteristic
237C2×C127#256511yescharacteristic
238C2×C127#256611yescharacteristic
239C2×C127#256811yescharacteristic
240C2×C127#258412yes—
241C2×C127#262011yescharacteristic
242C2×C127#285011yescharacteristic
243C2×#389C2×C12711yescharacteristic
244C2×#389C2×C12711yescharacteristic
245C2×#382C2×C12711yescharacteristic
246C2×C4×C4⋊C4C2×C12711yescharacteristic
247#1442C2×C12714yes—
248#1441C2×C12714yes—
249C4⋊#95C2×C12711yescharacteristic
250C4⋊#95C2×C12711yescharacteristic
251C4⋊#95C2×C12711yescharacteristic
252C127#1523411yescharacteristic, direct product factor, Sylow
253C22×C4⋊C4C22×C12711yescharacteristic
254C22×C4⋊C4C22×C12711yescharacteristic
255C22×C4⋊C4C22×C12712yes—
256C22×C4⋊C4C22×C12712yes—
257C22×C4⋊C4C22×C12711yescharacteristic
258C22×C4⋊C4C22×C12711yescharacteristic
259C22×C4⋊C4C22×C12712yes—
260C22×C42C22×C12711yescharacteristic
261C22×C42C22×C12711yescharacteristic
262C22×C42C22×C12711yescharacteristic
263#389—24no—
264#389—24no—
265#382—24no—
266C4×C4⋊C4—24no—
267C2×#95C22×C12714yes—
268C2×#95C22×C12712yes—
269C2×#95C22×C12714yes—
270C2×#95C22×C12712yes—
271C2×#95C22×C12712yes—
272C2×#95C22×C12714yes—
273C2×#95C22×C12714yes—
274C23×C4C23×C12712yes—
275C23×C4C23×C12712yes—
276C23×C4C23×C12711yescharacteristic
277C23×C4C23×C12712yes—
278C23×C4C23×C12712yes—
279C23×C4C23×C12712yes—
280C23×C4C23×C12712yes—
281C23×C4C23×C12711yescharacteristic
282C23×C4C23×C12711yescharacteristic
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—28no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4Q8×C12714yes—
301C2×C4⋊C4Q8×C12714yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42D4×C12714yes—
310C2×C42D4×C12714yes—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—44no—
317#95—44no—
318C24C24×C12711yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4C2×D4×C12714yes—
357C22×C4C2×D4×C12712yes—
358C22×C4C2×D4×C12712yes—
359C22×C4C2×D4×C12712yes—
360C22×C4C2×D4×C12712yes—
361C22×C4C2×Q8×C12714yes—
362C22×C4C2×Q8×C12714yes—
363C22×C4C2×Q8×C12712yes—
364C22×C4C2×Q8×C12712yes—
365C22×C4#41×C12714yes—
366C22×C4#41×C12714yes—
367C22×C4#41×C12712yes—
368C22×C4#41×C12712yes—
369C22×C4#41×C12712yes—
370C4⋊C4—216no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—48no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C42—28no—
378C42—28no—
379C42—44no—
380C23C22×D4×C12711yescharacteristic
381C23C22×D4×C12711yescharacteristic
382C23C22×Q8×C12711yescharacteristic
383C23C22×Q8×C12711yescharacteristic
384C23C2×#41×C12711yescharacteristic
385C23C2×#41×C12711yescharacteristic
386C23C2×#41×C12711yescharacteristic
387C23C2×#41×C12712yes—
388C23C2×#41×C12712yes—
389C23H(5,𝔽2)×C12712yes—
390C232−1+4×C12712yes—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4#121×C12714yes—
435C2×C4C22⋊Q8×C12714yes—
436C2×C4C22⋊Q8×C12714yes—
437C2×C4#123×C12714yes—
438C2×C4#124×C12714yes—
439C2×C4#125×C12714yes—
440C2×C4#125×C12714yes—
441C2×C4C4⋊Q8×C12714yes—
442C22C2×#121×C12711yescharacteristic
443C22C2×C22⋊Q8×C12711yescharacteristic
444C22C2×C22⋊Q8×C12711yescharacteristic
445C22C2×#123×C12711yescharacteristic
446C22#525×C12712yes—
447C22C2×#124×C12711yescharacteristic
448C22C2×#125×C12711yescharacteristic
449C22C2×#125×C12711yescharacteristic
450C22#529×C12712yes—
451C22C2×C4⋊Q8×C12711yescharacteristic
452C22#532×C12711yescharacteristic
453C22#535×C12711yescharacteristic
454C22#536×C12711yescharacteristic
455C22#537×C12712yes—
456C22#542×C12712yes—
457C22#552×C12712yes—
458C22#553×C12712yes—
459C22#554×C12712yes—
460C22#554×C12712yes—
461C22#555×C12712yes—
462C22#556×C12712yes—
463C22#557×C12712yes—
464C22#557×C12712yes—
465C4—24no—
466C4—42no—
467C4—44no—
468C4—44no—
469C4—44no—
470C4—44no—
471C4—82no—
472C4—82no—
473C4#398×C12718yes—
474C2C2×#398×C12711yescharacteristic
475C2C127×#248512yes—
476C2C127×#250812yes—
477C2C127×#250912yes—
478C2C127×#256311yescharacteristic
479C2C127×#256511yescharacteristic
480C2C127×#256611yescharacteristic
481C2C127×#256811yescharacteristic
482C2C127×#258412yes—
483C2C127×#262011yescharacteristic
484C2C127×#285011yescharacteristic
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
231. #15234C127direct (N × H)
252. C127#15234direct (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
C2×C4×C4⋊C4×C1272
C127×#14422
C127×#14412
C127×C4⋊#952
#15234127