Finite Group Databasebeta

← C127×#14946 #44578619

Subgroups — C127×#14946 #44578619

Order 32512.

2162 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#386×C127C211yesmaximal, characteristic
2C2×#386×C127C212yesmaximal
3C2×#385×C127C212yesmaximal, complemented
4C22×#95×C127C211yesmaximal, characteristic, complemented
5C127×#1445C214yesmaximal, complemented
6C127×#1439C214yesmaximal, complemented
7C127×C22⋊#95C211yesmaximal, characteristic, complemented
8C23×D4×C127C2211yescharacteristic
9C24×C4×C127C2212yes—
10C24×C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2214yes—
12C22×C22⋊C4×C127C2211yescharacteristic
13C22×C22⋊C4×C127C2214yes—
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16#386×C127—24no—
17#386×C127—28no—
18#385×C127—28no—
19C2×#95×C127—24no—
20C2×#95×C127C2214yes—
21C2×#95×C127C2214yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2214yes—
24C2×#95×C127C2214yes—
25C25×C127C2311yescharacteristic
26C25×C127C2311yescharacteristic
27C22×D4×C127—24no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—24no—
31C22×D4×C127Q814yes—
32C23×C4×C127—24no—
33C23×C4×C127—24no—
34C23×C4×C127—28no—
35C23×C4×C127—24no—
36C23×C4×C127—24no—
37C23×C4×C127—44no—
38C23×C4×C127—44no—
39C23×C4×C127D414yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2314yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2314yes—
45C2×C4⋊C4×C127—28no—
46C2×C4⋊C4×C127—28no—
47C2×C4⋊C4×C127—44no—
48C2×C22⋊C4×C127—24no—
49C2×C22⋊C4×C127—24no—
50C2×C22⋊C4×C127—24no—
51C2×C22⋊C4×C127—22no—
52C2×C22⋊C4×C127—28no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—28no—
55C2×C22⋊C4×C127—44no—
56C2×C22⋊C4×C127—44no—
57C2×C22⋊C4×C127D418yes—
58#95×C127—28no—
59#95×C127—28no—
60#95×C127—28no—
61#95×C127—24no—
62#95×C127—44no—
63C24×C127—24no—
64C24×C127—24no—
65C24×C127—21no—
66C24×C127—22no—
67C24×C127—24no—
68C24×C127—22no—
69C24×C127—21no—
70C24×C127—24no—
71C24×C127C2×D414yes—
72C24×C127C2×D412yes—
73C24×C127C2×Q812yes—
74C24×C127C2×Q812yes—
75C24×C127#4112yes—
76C24×C127#4114yes—
77C24×C127C2411yescharacteristic
78C2×D4×C127—28no—
79C2×D4×C127—216no—
80C2×D4×C127—48no—
81C2×D4×C127—44no—
82C2×D4×C127—44no—
83C22×C4×C127—216no—
84C22×C4×C127—28no—
85C22×C4×C127—24no—
86C22×C4×C127—21no—
87C22×C4×C127—22no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—24no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127—22no—
97C22×C4×C127—21no—
98C22×C4×C127—24no—
99C22×C4×C127—24no—
100C22×C4×C127—24no—
101C22×C4×C127—24no—
102C22×C4×C127—24no—
103C22×C4×C127—24no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—48no—
110C22×C4×C127—416no—
111C22×C4×C127—416no—
112C22×C4×C127—48no—
113C22×C4×C127—416no—
114C22×C4×C127—48no—
115C22×C4×C127C2×D414yes—
116C22×C4×C127C2×D418yes—
117C22×C4×C127C2×Q812yes—
118C22×C4×C127#4118yes—
119C4⋊C4×C127—416no—
120C22⋊C4×C127—216no—
121C22⋊C4×C127—416no—
122C22⋊C4×C127—48no—
123C22⋊C4×C127—44no—
124C23×C127—24no—
125C23×C127—28no—
126C23×C127—24no—
127C23×C127—28no—
128C23×C127—28no—
129C23×C127—28no—
130C23×C127—28no—
131C23×C127—28no—
132C23×C127—22no—
133C23×C127—24no—
134C23×C127—28no—
135C23×C127—41no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127—44no—
139C23×C127—41no—
140C23×C127—42no—
141C23×C127—44no—
142C23×C127—44no—
143C23×C127—44no—
144C23×C127—44no—
145C23×C127C22⋊Q814yes—
146C23×C127C22⋊Q818yes—
147C23×C127#12318yes—
148C23×C127#12314yes—
149C23×C127#12714yes—
150C23×C127C22×D411yescharacteristic
151C23×C127C22×D412yes—
152C23×C127C22×Q811yescharacteristic
153C23×C127C2×#4112yes—
154C23×C127C2×#4111yescharacteristic
155C23×C127C2×#4114yes—
156C23×C127H(5,𝔽2)14yes—
157D4×C127—416no—
158C2×C4×C127—22no—
159C2×C4×C127—24no—
160C2×C4×C127—28no—
161C2×C4×C127—24no—
162C2×C4×C127—28no—
163C2×C4×C127—24no—
164C2×C4×C127—24no—
165C2×C4×C127—28no—
166C2×C4×C127—28no—
167C2×C4×C127—28no—
168C2×C4×C127—28no—
169C2×C4×C127—416no—
170C2×C4×C127—432no—
171C2×C4×C127—41no—
172C2×C4×C127—42no—
173C2×C4×C127—44no—
174C2×C4×C127—44no—
175C2×C4×C127—41no—
176C2×C4×C127—44no—
177C2×C4×C127—44no—
178C2×C4×C127—48no—
179C2×C4×C127—44no—
180C2×C4×C127—44no—
181C2×C4×C127—44no—
182C2×C4×C127—44no—
183C2×C4×C127—42no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—44no—
187C2×C4×C127—44no—
188C22×C127—28no—
189C22×C127—24no—
190C22×C127—24no—
191C22×C127—216no—
192C22×C127—216no—
193C22×C127—28no—
194C22×C127—44no—
195C22×C127—42no—
196C22×C127—48no—
197C22×C127—81no—
198C22×C127—81no—
199C22×C127—84no—
200C22×C127—82no—
201C22×C127#39918yes—
202C22×C127C2×C22⋊Q811yescharacteristic
203C22×C127C2×C22⋊Q812yes—
204C22×C127C2×#12311yescharacteristic
205C22×C127C2×#12312yes—
206C22×C127#52512yes—
207C22×C127#52914yes—
208C22×C127C2×#12711yescharacteristic
209C22×C127#53412yes—
210C22×C127#53814yes—
211C22×C127#54614yes—
212C22×C127#54614yes—
213C22×C127#55314yes—
214C22×C127#55414yes—
215C4×C127—44no—
216C4×C127—42no—
217C4×C127—48no—
218C4×C127—48no—
219C4×C127—82no—
220#14946C12711yesmaximal, characteristic, direct product factor, Sylow
221C2×C127—28no—
222C2×C127—82no—
223C2×C127C2×#39911yescharacteristic
224C2×C127#244414yes—
225C2×C127#245014yes—
226C2×C127#253912yes—
227C2×C127#257912yes—
228C2×C127#262211yescharacteristic
229C2×C127#285311yescharacteristic
230C2×#386C2×C12711yescharacteristic
231C2×#386C2×C12712yes—
232C2×#385C2×C12712yescomplemented
233C22×#95C2×C12711yescharacteristic, complemented
234#1445C2×C12714yescomplemented
235#1439C2×C12714yescomplemented
236C22⋊#95C2×C12711yescharacteristic, complemented
237C127#1494611yescharacteristic, direct product factor, Sylow
238C23×D4C22×C12711yescharacteristic
239C24×C4C22×C12712yes—
240C24×C4C22×C12711yescharacteristic
241C22×C4⋊C4C22×C12714yes—
242C22×C22⋊C4C22×C12711yescharacteristic
243C22×C22⋊C4C22×C12714yes—
244C22×C22⋊C4C22×C12712yes—
245C22×C22⋊C4C22×C12712yes—
246#386—24no—
247#386—28no—
248#385—28no—
249C2×#95—24no—
250C2×#95C22×C12714yes—
251C2×#95C22×C12714yes—
252C2×#95C22×C12712yes—
253C2×#95C22×C12714yes—
254C2×#95C22×C12714yes—
255C25C23×C12711yescharacteristic
256C25C23×C12711yescharacteristic
257C22×D4—24no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—24no—
261C22×D4Q8×C12714yes—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—28no—
265C23×C4—24no—
266C23×C4—24no—
267C23×C4—44no—
268C23×C4—44no—
269C23×C4D4×C12714yes—
270C23×C4C23×C12712yes—
271C23×C4C23×C12714yes—
272C23×C4C23×C12712yes—
273C23×C4C23×C12711yescharacteristic
274C23×C4C23×C12714yes—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4D4×C12718yes—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—44no—
293C24—24no—
294C24—24no—
295C24—21no—
296C24—22no—
297C24—24no—
298C24—22no—
299C24—21no—
300C24—24no—
301C24C2×D4×C12714yes—
302C24C2×D4×C12712yes—
303C24C2×Q8×C12712yes—
304C24C2×Q8×C12712yes—
305C24#41×C12712yes—
306C24#41×C12714yes—
307C24C24×C12711yescharacteristic
308C2×D4—28no—
309C2×D4—216no—
310C2×D4—48no—
311C2×D4—44no—
312C2×D4—44no—
313C22×C4—216no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—48no—
340C22×C4—416no—
341C22×C4—416no—
342C22×C4—48no—
343C22×C4—416no—
344C22×C4—48no—
345C22×C4C2×D4×C12714yes—
346C22×C4C2×D4×C12718yes—
347C22×C4C2×Q8×C12712yes—
348C22×C4#41×C12718yes—
349C4⋊C4—416no—
350C22⋊C4—216no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C22⋊C4—44no—
354C23—24no—
355C23—28no—
356C23—24no—
357C23—28no—
358C23—28no—
359C23—28no—
360C23—28no—
361C23—28no—
362C23—22no—
363C23—24no—
364C23—28no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—41no—
370C23—42no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23C22⋊Q8×C12714yes—
376C23C22⋊Q8×C12718yes—
377C23#123×C12718yes—
378C23#123×C12714yes—
379C23#127×C12714yes—
380C23C22×D4×C12711yescharacteristic
381C23C22×D4×C12712yes—
382C23C22×Q8×C12711yescharacteristic
383C23C2×#41×C12712yes—
384C23C2×#41×C12711yescharacteristic
385C23C2×#41×C12714yes—
386C23H(5,𝔽2)×C12714yes—
387D4—416no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—416no—
400C2×C4—432no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C22—28no—
419C22—24no—
420C22—24no—
421C22—216no—
422C22—216no—
423C22—28no—
424C22—44no—
425C22—42no—
426C22—48no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—82no—
431C22#399×C12718yes—
432C22C2×C22⋊Q8×C12711yescharacteristic
433C22C2×C22⋊Q8×C12712yes—
434C22C2×#123×C12711yescharacteristic
435C22C2×#123×C12712yes—
436C22#525×C12712yes—
437C22#529×C12714yes—
438C22C2×#127×C12711yescharacteristic
439C22#534×C12712yes—
440C22#538×C12714yes—
441C22#546×C12714yes—
442C22#546×C12714yes—
443C22#553×C12714yes—
444C22#554×C12714yes—
445C4—44no—
446C4—42no—
447C4—48no—
448C4—48no—
449C4—82no—
450C2—28no—
451C2—82no—
452C2C2×#399×C12711yescharacteristic
453C2C127×#244414yes—
454C2C127×#245014yes—
455C2C127×#253912yes—
456C2C127×#257912yes—
457C2C127×#262211yescharacteristic
458C2C127×#285311yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#385×C127C2semidirect
4. C22×#95×C127C2semidirect
5. C127×#1445C2semidirect
6. C127×#1439C2semidirect
7. C127×C22⋊#95C2semidirect
220. #14946C127direct (N × H)
232. C2×#385C2×C127semidirect
233. C22×#95C2×C127semidirect
234. #1445C2×C127semidirect
235. #1439C2×C127semidirect
236. C22⋊#95C2×C127semidirect
237. C127#14946direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#386×C1272
C2×#385×C1272
C22×#95×C1272
C127×#14452
C127×#14392
C127×C22⋊#952
#14946127