Finite Group Databasebeta

← C127×#15019 #44578692

Subgroups — C127×#15019 #44578692

Order 32512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#381×C127C212yesmaximal
2C2×#380×C127C211yesmaximal, characteristic
3C22×#95×C127C211yesmaximal, characteristic
4C127×#1445C212yesmaximal, complemented
5C127×#1443C214yesmaximal, complemented
6C127×#1442C212yesmaximal, complemented
7C127×C22⋊#95C211yesmaximal, characteristic
8C127×C22⋊#95C212yesmaximal
9C24×C4×C127C2211yescharacteristic
10C24×C4×C127C2211yescharacteristic
11C24×C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16#381×C127—28no—
17#380×C127—24no—
18C2×#95×C127—24no—
19C2×#95×C127C2212yes—
20C2×#95×C127C2214yes—
21C2×#95×C127C2214yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2214yes—
26C2×#95×C127C2214yes—
27C2×#95×C127C2212yes—
28C25×C127C2311yescharacteristic
29C23×C4×C127—22no—
30C23×C4×C127—22no—
31C23×C4×C127—24no—
32C23×C4×C127—22no—
33C23×C4×C127—24no—
34C23×C4×C127—24no—
35C23×C4×C127—24no—
36C23×C4×C127—24no—
37C23×C4×C127—24no—
38C23×C4×C127—42no—
39C23×C4×C127—42no—
40C23×C4×C127D414yes—
41C23×C4×C127D414yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2314yes—
45C23×C4×C127C2312yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2312yes—
48C23×C4×C127C2311yescharacteristic
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—22no—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—42no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127—24no—
57C2×C22⋊C4×C127—24no—
58C2×C22⋊C4×C127—24no—
59#95×C127—24no—
60#95×C127—24no—
61#95×C127—24no—
62#95×C127—28no—
63#95×C127—28no—
64#95×C127—28no—
65#95×C127—24no—
66#95×C127—42no—
67#95×C127—44no—
68C24×C127—22no—
69C24×C127—22no—
70C24×C127—22no—
71C24×C127—22no—
72C24×C127C2×D412yes—
73C24×C127C2×D412yes—
74C24×C127#4112yes—
75C24×C127#4114yes—
76C24×C127#4114yes—
77C24×C127C2411yescharacteristic
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—28no—
81C22×C4×C127—28no—
82C22×C4×C127—28no—
83C22×C4×C127—216no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—24no—
88C22×C4×C127—24no—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—24no—
93C22×C4×C127—22no—
94C22×C4×C127—24no—
95C22×C4×C127—28no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—24no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—24no—
102C22×C4×C127—24no—
103C22×C4×C127—24no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—24no—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127—24no—
111C22×C4×C127—24no—
112C22×C4×C127—24no—
113C22×C4×C127—44no—
114C22×C4×C127—416no—
115C22×C4×C127—416no—
116C22×C4×C127—44no—
117C22×C4×C127—44no—
118C22×C4×C127—44no—
119C22×C4×C127—44no—
120C22×C4×C127—48no—
121C22×C4×C127C2×D414yes—
122C22×C4×C127C2×D412yes—
123C22×C4×C127C2×D412yes—
124C22×C4×C127#4114yes—
125C22×C4×C127#4114yes—
126C4⋊C4×C127—44no—
127C4⋊C4×C127—44no—
128C22⋊C4×C127—48no—
129C22⋊C4×C127—48no—
130C23×C127—22no—
131C23×C127—22no—
132C23×C127—24no—
133C23×C127—24no—
134C23×C127—22no—
135C23×C127—22no—
136C23×C127—24no—
137C23×C127—24no—
138C23×C127—28no—
139C23×C127—28no—
140C23×C127—28no—
141C23×C127—28no—
142C23×C127#12114yes—
143C23×C127#12318yes—
144C23×C127#12314yes—
145C23×C127#12414yes—
146C23×C127#12618yes—
147C23×C127C22×D411yescharacteristic
148C23×C127C22×D411yescharacteristic
149C23×C127C2×#4111yescharacteristic
150C23×C127C2×#4112yes—
151C23×C127C2×#4112yes—
152C23×C127C2×#4112yes—
153C23×C127C2×#4112yes—
154C23×C127H(5,𝔽2)12yes—
155C23×C1272−1+412yes—
156C2×C4×C127—22no—
157C2×C4×C127—22no—
158C2×C4×C127—28no—
159C2×C4×C127—24no—
160C2×C4×C127—28no—
161C2×C4×C127—22no—
162C2×C4×C127—22no—
163C2×C4×C127—24no—
164C2×C4×C127—22no—
165C2×C4×C127—28no—
166C2×C4×C127—416no—
167C2×C4×C127—416no—
168C2×C4×C127—42no—
169C2×C4×C127—42no—
170C2×C4×C127—42no—
171C2×C4×C127—44no—
172C2×C4×C127—42no—
173C2×C4×C127—44no—
174C2×C4×C127—44no—
175C2×C4×C127—416no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—44no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—44no—
182C2×C4×C127—44no—
183C2×C4×C127—44no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—44no—
187C2×C4×C127—44no—
188C2×C4×C127—44no—
189C2×C4×C127—44no—
190C2×C4×C127—44no—
191C2×C4×C127—44no—
192C2×C4×C127—44no—
193C2×C4×C127#12314yes—
194C22×C127—24no—
195C22×C127—24no—
196C22×C127—24no—
197C22×C127—24no—
198C22×C127—24no—
199C22×C127—216no—
200C22×C127—216no—
201C22×C127—24no—
202C22×C127#39718yes—
203C22×C127C2×#12111yescharacteristic
204C22×C127C2×#12311yescharacteristic
205C22×C127C2×#12311yescharacteristic
206C22×C127C2×#12312yes—
207C22×C127#52511yescharacteristic
208C22×C127C2×#12411yescharacteristic
209C22×C127C2×#12612yes—
210C22×C127#52912yes—
211C22×C127#53212yes—
212C22×C127#53411yescharacteristic
213C22×C127#53611yescharacteristic
214C22×C127#53712yes—
215C22×C127#53912yes—
216C22×C127#55114yes—
217C22×C127#55114yes—
218C22×C127#55214yes—
219C22×C127#55214yes—
220C4×C127—24no—
221C4×C127—44no—
222C4×C127—44no—
223C4×C127—42no—
224C4×C127—82no—
225C4×C127—82no—
226C4×C127—84no—
227#15019C12711yesmaximal, characteristic, direct product factor, Sylow
228C2×C127—28no—
229C2×C127C2×#39711yescharacteristic
230C2×C127#248311yescharacteristic
231C2×C127#248411yescharacteristic
232C2×C127#249314yes—
233C2×C127#249814yes—
234C2×C127#250311yescharacteristic
235C2×C127#256111yescharacteristic
236C2×C127#261811yescharacteristic
237C2×C127#285411yescharacteristic
238C2×#381C2×C12712yes—
239C2×#380C2×C12711yescharacteristic
240C22×#95C2×C12711yescharacteristic
241#1445C2×C12712yescomplemented
242#1443C2×C12714yescomplemented
243#1442C2×C12712yescomplemented
244C22⋊#95C2×C12711yescharacteristic
245C22⋊#95C2×C12712yes—
246C127#1501911yescharacteristic, direct product factor, Sylow
247C24×C4C22×C12711yescharacteristic
248C24×C4C22×C12711yescharacteristic
249C24×C4C22×C12711yescharacteristic
250C22×C4⋊C4C22×C12711yescharacteristic
251C22×C4⋊C4C22×C12711yescharacteristic
252C22×C22⋊C4C22×C12712yes—
253C22×C22⋊C4C22×C12712yes—
254#381—28no—
255#380—24no—
256C2×#95—24no—
257C2×#95C22×C12712yes—
258C2×#95C22×C12714yes—
259C2×#95C22×C12714yes—
260C2×#95C22×C12712yes—
261C2×#95C22×C12712yes—
262C2×#95C22×C12712yes—
263C2×#95C22×C12714yes—
264C2×#95C22×C12714yes—
265C2×#95C22×C12712yes—
266C25C23×C12711yescharacteristic
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—24no—
270C23×C4—22no—
271C23×C4—24no—
272C23×C4—24no—
273C23×C4—24no—
274C23×C4—24no—
275C23×C4—24no—
276C23×C4—42no—
277C23×C4—42no—
278C23×C4D4×C12714yes—
279C23×C4D4×C12714yes—
280C23×C4C23×C12712yes—
281C23×C4C23×C12712yes—
282C23×C4C23×C12714yes—
283C23×C4C23×C12712yes—
284C23×C4C23×C12711yescharacteristic
285C23×C4C23×C12712yes—
286C23×C4C23×C12711yescharacteristic
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—42no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—42no—
305#95—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24C2×D4×C12712yes—
311C24C2×D4×C12712yes—
312C24#41×C12712yes—
313C24#41×C12714yes—
314C24#41×C12714yes—
315C24C24×C12711yescharacteristic
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—216no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—44no—
352C22×C4—416no—
353C22×C4—416no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4C2×D4×C12714yes—
360C22×C4C2×D4×C12712yes—
361C22×C4C2×D4×C12712yes—
362C22×C4#41×C12714yes—
363C22×C4#41×C12714yes—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—24no—
376C23—28no—
377C23—28no—
378C23—28no—
379C23—28no—
380C23#121×C12714yes—
381C23#123×C12718yes—
382C23#123×C12714yes—
383C23#124×C12714yes—
384C23#126×C12718yes—
385C23C22×D4×C12711yescharacteristic
386C23C22×D4×C12711yescharacteristic
387C23C2×#41×C12711yescharacteristic
388C23C2×#41×C12712yes—
389C23C2×#41×C12712yes—
390C23C2×#41×C12712yes—
391C23C2×#41×C12712yes—
392C23H(5,𝔽2)×C12712yes—
393C232−1+4×C12712yes—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—28no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—416no—
405C2×C4—416no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—416no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4#123×C12714yes—
432C22—24no—
433C22—24no—
434C22—24no—
435C22—24no—
436C22—216no—
437C22—216no—
438C22—24no—
439C22—24no—
440C22#397×C12718yes—
441C22C2×#121×C12711yescharacteristic
442C22C2×#123×C12711yescharacteristic
443C22C2×#123×C12711yescharacteristic
444C22C2×#123×C12712yes—
445C22#525×C12711yescharacteristic
446C22C2×#124×C12711yescharacteristic
447C22C2×#126×C12712yes—
448C22#529×C12712yes—
449C22#532×C12712yes—
450C22#534×C12711yescharacteristic
451C22#536×C12711yescharacteristic
452C22#537×C12712yes—
453C22#539×C12712yes—
454C22#551×C12714yes—
455C22#551×C12714yes—
456C22#552×C12714yes—
457C22#552×C12714yes—
458C4—24no—
459C4—44no—
460C4—42no—
461C4—44no—
462C4—82no—
463C4—82no—
464C4—84no—
465C2—28no—
466C2C2×#397×C12711yescharacteristic
467C2C127×#248311yescharacteristic
468C2C127×#248411yescharacteristic
469C2C127×#249314yes—
470C2C127×#249814yes—
471C2C127×#250311yescharacteristic
472C2C127×#256111yescharacteristic
473C2C127×#261811yescharacteristic
474C2C127×#285411yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1445C2semidirect
5. C127×#1443C2semidirect
6. C127×#1442C2semidirect
227. #15019C127direct (N × H)
241. #1445C2×C127semidirect
242. #1443C2×C127semidirect
243. #1442C2×C127semidirect
246. C127#15019direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#381×C1272
C2×#380×C1272
C22×#95×C1272
C127×#14452
C127×#14432
C127×#14422
C127×C22⋊#952
#15019127