Finite Group Databasebeta

← C127×#14252 #44577925

Subgroups — C127×#14252 #44577925

Order 32512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400×C127C211yesmaximal, characteristic
2C127×#1445C211yesmaximal, characteristic
3C127×#1443C212yesmaximal
4C127×#1436C211yesmaximal, characteristic
5C127×#1436C212yesmaximal
6C127×#1434C212yesmaximal
7C127×#1434C212yesmaximal
8C127×#1434C212yesmaximal
9C4×#95×C127C212yesmaximal
10C22×C4⋊C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C42×C127C2212yes—
13C22×C42×C127C2211yescharacteristic
14C22×C42×C127C2212yes—
15#400×C127C418yescomplemented
16C2×#95×C127C2212yes—
17C2×#95×C127C2212yes—
18C2×#95×C127C2212yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2211yescharacteristic
27C2×#95×C127C2212yes—
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2212yes—
30C2×#95×C127C2212yes—
31C2×#95×C127C2211yescharacteristic
32C23×C4×C127C2312yes—
33C23×C4×C127C2311yescharacteristic
34C23×C4×C127C2312yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2311yescharacteristic
37C23×C4×C127C2311yescharacteristic
38C23×C4×C127C2312yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C2×C4⋊C4×C127—41no—
42C2×C4⋊C4×C127—41no—
43C2×C4⋊C4×C127—41no—
44C2×C4⋊C4×C127—41no—
45C2×C4⋊C4×C127C2×C414yes—
46C2×C4⋊C4×C127C2×C414yes—
47C2×C42×C127—24no—
48C2×C42×C127—24no—
49C2×C42×C127—24no—
50C2×C42×C127—24no—
51C2×C42×C127—24no—
52C2×C42×C127—24no—
53C2×C42×C127—41no—
54C2×C42×C127—42no—
55#95×C127—24no—
56#95×C127—24no—
57#95×C127—24no—
58#95×C127—24no—
59#95×C127—24no—
60#95×C127—24no—
61#95×C127—24no—
62#95×C127—24no—
63#95×C127—24no—
64#95×C127—41no—
65#95×C127—42no—
66#95×C127—41no—
67#95×C127—41no—
68#95×C127C2×C414yes—
69#95×C127C2×C418yes—
70#95×C127C2×C418yes—
71C24×C127C2411yescharacteristic
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—22no—
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—22no—
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—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—21no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—22no—
119C22×C4×C127C22×C414yes—
120C22×C4×C127C22×C414yes—
121C22×C4×C127C22×C412yes—
122C22×C4×C127C22×C412yes—
123C22×C4×C127C22×C412yes—
124C22×C4×C127#4114yes—
125C22×C4×C127#4114yes—
126C22×C4×C127#4114yes—
127C22×C4×C127#4114yes—
128C22×C4×C127#4114yes—
129C4⋊C4×C127—44no—
130C4⋊C4×C127—44no—
131C42×C127—44no—
132C42×C127—48no—
133C42×C127—48no—
134C23×C127C23×C411yescharacteristic
135C23×C127C2×#4112yes—
136C23×C127C2×#4112yes—
137C23×C127C2×#4112yes—
138C23×C127C2×#4112yes—
139C23×C127C2×#4112yes—
140C23×C127H(5,𝔽2)11yescharacteristic
141C23×C127H(5,𝔽2)11yescharacteristic
142C23×C1272−1+412yes—
143C2×C4×C127—24no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—24no—
147C2×C4×C127—24no—
148C2×C4×C127—24no—
149C2×C4×C127—24no—
150C2×C4×C127—24no—
151C2×C4×C127—24no—
152C2×C4×C127—22no—
153C2×C4×C127—24no—
154C2×C4×C127—24no—
155C2×C4×C127—24no—
156C2×C4×C127—24no—
157C2×C4×C127—22no—
158C2×C4×C127—22no—
159C2×C4×C127—24no—
160C2×C4×C127—22no—
161C2×C4×C127—22no—
162C2×C4×C127—24no—
163C2×C4×C127—24no—
164C2×C4×C127—24no—
165C2×C4×C127—24no—
166C2×C4×C127—24no—
167C2×C4×C127—24no—
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—42no—
175C2×C4×C127—42no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—41no—
180C2×C4×C127—42no—
181C2×C4×C127—41no—
182C2×C4×C127—42no—
183C2×C4×C127—42no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—42no—
189C2×C4×C127—42no—
190C2×C4×C127—41no—
191C2×C4×C127—41no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—42no—
196C2×C4×C127—42no—
197C2×C4×C127—42no—
198C2×C4×C127—42no—
199C2×C4×C127#12618yes—
200C22×C127C4×#4111yescharacteristic
201C22×C127C4×#4112yes—
202C22×C127C4×#4112yes—
203C22×C127#51811yescharacteristic
204C22×C127#51911yescharacteristic
205C22×C127C2×#12612yes—
206C22×C127#52912yes—
207C22×C127#52912yes—
208C22×C127#52912yes—
209C22×C127#53912yes—
210C22×C127#54212yes—
211C22×C127#55112yes—
212C22×C127#55111yescharacteristic
213C22×C127#55212yes—
214C22×C127#55212yes—
215C22×C127#55212yes—
216C22×C127#55212yes—
217C22×C127#55312yes—
218C22×C127#55511yescharacteristic
219C22×C127#55511yescharacteristic
220C22×C127#55911yescharacteristic
221C4×C127—28no—
222C4×C127—44no—
223C4×C127—44no—
224C4×C127—42no—
225C4×C127—44no—
226C4×C127—42no—
227C4×C127—42no—
228C4×C127—44no—
229C4×C127—82no—
230#14252C12711yesmaximal, characteristic, direct product factor, Sylow
231C2×C127C4×#12612yes—
232C2×C127#233211yescharacteristic
233C2×C127#237112yes—
234C2×C127#237211yescharacteristic
235C2×C127#237411yescharacteristic
236C2×C127#249312yes—
237C2×C127#249512yes—
238C2×C127#252912yes—
239C2×C127#257512yes—
240C2×#400C2×C12711yescharacteristic
241#1445C2×C12711yescharacteristic
242#1443C2×C12712yes—
243#1436C2×C12712yes—
244#1436C2×C12711yescharacteristic
245#1434C2×C12712yes—
246#1434C2×C12712yes—
247#1434C2×C12712yes—
248C4×#95C2×C12712yes—
249C127#1425211yescharacteristic, direct product factor, Sylow
250C22×C4⋊C4C22×C12711yescharacteristic
251C22×C4⋊C4C22×C12711yescharacteristic
252C22×C42C22×C12712yes—
253C22×C42C22×C12711yescharacteristic
254C22×C42C22×C12712yes—
255#400C4×C12718yescomplemented
256C2×#95C22×C12712yes—
257C2×#95C22×C12712yes—
258C2×#95C22×C12712yes—
259C2×#95C22×C12712yes—
260C2×#95C22×C12712yes—
261C2×#95C22×C12712yes—
262C2×#95C22×C12712yes—
263C2×#95C22×C12712yes—
264C2×#95C22×C12712yes—
265C2×#95C22×C12711yescharacteristic
266C2×#95C22×C12711yescharacteristic
267C2×#95C22×C12712yes—
268C2×#95C22×C12711yescharacteristic
269C2×#95C22×C12712yes—
270C2×#95C22×C12711yescharacteristic
271C2×#95C22×C12712yes—
272C23×C4C23×C12712yes—
273C23×C4C23×C12711yescharacteristic
274C23×C4C23×C12711yescharacteristic
275C23×C4C23×C12712yes—
276C23×C4C23×C12711yescharacteristic
277C23×C4C23×C12712yes—
278C23×C4C23×C12712yes—
279C23×C4C23×C12712yes—
280C23×C4C23×C12712yes—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4C2×C4×C12714yes—
286C2×C4⋊C4C2×C4×C12714yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—41no—
294C2×C42—42no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—41no—
305#95—42no—
306#95—41no—
307#95—41no—
308#95C2×C4×C12714yes—
309#95C2×C4×C12718yes—
310#95C2×C4×C12718yes—
311C24C24×C12711yescharacteristic
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
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—22no—
327C22×C4—22no—
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—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4C22×C4×C12714yes—
360C22×C4C22×C4×C12714yes—
361C22×C4C22×C4×C12712yes—
362C22×C4C22×C4×C12712yes—
363C22×C4C22×C4×C12712yes—
364C22×C4#41×C12714yes—
365C22×C4#41×C12714yes—
366C22×C4#41×C12714yes—
367C22×C4#41×C12714yes—
368C22×C4#41×C12714yes—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C42—44no—
372C42—48no—
373C42—48no—
374C23C23×C4×C12711yescharacteristic
375C23C2×#41×C12712yes—
376C23C2×#41×C12712yes—
377C23C2×#41×C12712yes—
378C23C2×#41×C12712yes—
379C23C2×#41×C12712yes—
380C23H(5,𝔽2)×C12711yescharacteristic
381C23H(5,𝔽2)×C12711yescharacteristic
382C232−1+4×C12712yes—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
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—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4#126×C12718yes—
440C22C4×#41×C12711yescharacteristic
441C22C4×#41×C12712yes—
442C22C4×#41×C12712yes—
443C22#518×C12711yescharacteristic
444C22#519×C12711yescharacteristic
445C22C2×#126×C12712yes—
446C22#529×C12712yes—
447C22#529×C12712yes—
448C22#529×C12712yes—
449C22#539×C12712yes—
450C22#542×C12712yes—
451C22#551×C12712yes—
452C22#551×C12711yescharacteristic
453C22#552×C12712yes—
454C22#552×C12712yes—
455C22#552×C12712yes—
456C22#552×C12712yes—
457C22#553×C12712yes—
458C22#555×C12711yescharacteristic
459C22#555×C12711yescharacteristic
460C22#559×C12711yescharacteristic
461C4—28no—
462C4—44no—
463C4—42no—
464C4—44no—
465C4—42no—
466C4—42no—
467C4—44no—
468C4—44no—
469C4—82no—
470C2C4×#126×C12712yes—
471C2C127×#233211yescharacteristic
472C2C127×#237112yes—
473C2C127×#237211yescharacteristic
474C2C127×#237411yescharacteristic
475C2C127×#249312yes—
476C2C127×#249512yes—
477C2C127×#252912yes—
478C2C127×#257512yes—
subgroupG/H
center, socle, prime radicalC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
15. #400×C127C4semidirect
230. #14252C127direct (N × H)
249. C127#14252direct (N × H)
255. #400C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#400×C1272
C127×#14452
C127×#14432
C127×#14362
C127×#14342
C4×#95×C1272
#14252127