Finite Group Databasebeta

← C127×#17082 #44580755

Subgroups — C127×#17082 #44580755

Order 32512.

2346 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394×C127C212yesmaximal, complemented
2C2×#392×C127C214yesmaximal, complemented
3C127×#1445C211yesmaximal, characteristic, complemented
4C127×#1440C214yesmaximal, complemented
5C127×#1438C212yesmaximal, complemented
6C127×#1438C212yesmaximal, complemented
7C23×D4×C127C2212yes—
8C23×D4×C127C2212yes—
9C23×D4×C127C2212yes—
10C22×C4⋊C4×C127C2211yescharacteristic
11C22×C22⋊C4×C127C2214yes—
12C22×C22⋊C4×C127C2214yes—
13C22×C22⋊C4×C127C2214yes—
14C22×C22⋊C4×C127C2214yes—
15#394×C127—28no—
16#392×C127—216no—
17C2×#95×C127C2214yes—
18C2×#95×C127C2212yes—
19C2×#95×C127C2211yescharacteristic
20C2×#95×C127C2211yescharacteristic
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C25×C127C2314yes—
24C22×D4×C127—28no—
25C22×D4×C127—28no—
26C22×D4×C127—24no—
27C22×D4×C127—24no—
28C22×D4×C127—42no—
29C22×D4×C127—42no—
30C22×D4×C127—42no—
31C22×D4×C127—42no—
32C22×D4×C127—44no—
33C22×D4×C127—44no—
34C22×D4×C127—44no—
35C22×D4×C127—44no—
36C22×D4×C127—42no—
37C22×D4×C127—42no—
38C22×D4×C127D418yescomplemented
39C23×C4×C127C2314yes—
40C23×C4×C127C2311yescharacteristic
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2312yes—
44C2×C4⋊C4×C127—41no—
45C2×C4⋊C4×C127—41no—
46C2×C4⋊C4×C127D414yescomplemented
47C2×C22⋊C4×C127—28no—
48C2×C22⋊C4×C127—28no—
49C2×C22⋊C4×C127—28no—
50C2×C22⋊C4×C127—28no—
51C2×C22⋊C4×C127—28no—
52C2×C22⋊C4×C127—28no—
53C2×C22⋊C4×C127—44no—
54C2×C22⋊C4×C127—44no—
55C2×C22⋊C4×C127—44no—
56C2×C22⋊C4×C127—44no—
57C2×C22⋊C4×C127—44no—
58C2×C22⋊C4×C127—44no—
59#95×C127—28no—
60#95×C127—24no—
61#95×C127—41no—
62#95×C127—41no—
63#95×C127—42no—
64#95×C127—42no—
65C24×C127—24no—
66C24×C127—24no—
67C24×C127—24no—
68C24×C127—24no—
69C24×C127—24no—
70C24×C127—24no—
71C24×C127—24no—
72C24×C127—24no—
73C24×C127—24no—
74C24×C127—24no—
75C24×C127—24no—
76C24×C127—24no—
77C24×C127—24no—
78C24×C127—24no—
79C24×C127C2×D418yes—
80C24×C127C2411yescharacteristic
81C2×D4×C127—416no—
82C2×D4×C127—48no—
83C2×D4×C127—416no—
84C2×D4×C127—416no—
85C2×D4×C127—48no—
86C2×D4×C127—48no—
87C2×D4×C127—48no—
88C2×D4×C127—48no—
89C2×D4×C127—416no—
90C2×D4×C127—416no—
91C2×D4×C127—48no—
92C2×D4×C127—416no—
93C2×D4×C127—48no—
94C2×D4×C127—48no—
95C2×D4×C127—48no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—24no—
105C22×C4×C127—24no—
106C22×C4×C127—24no—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—24no—
112C22×C4×C127—24no—
113C22×C4×C127—24no—
114C22×C4×C127—24no—
115C22×C4×C127—24no—
116C22×C4×C127—24no—
117C22×C4×C127—24no—
118C22×C4×C127C2×D414yes—
119C22×C4×C127C2×D414yes—
120C22×C4×C127C2×D412yes—
121C22×C4×C127#4114yes—
122C4⋊C4×C127—44no—
123C22⋊C4×C127—416no—
124C22⋊C4×C127—416no—
125C22⋊C4×C127—416no—
126C22⋊C4×C127—416no—
127C23×C127—28no—
128C23×C127—28no—
129C23×C127—28no—
130C23×C127—28no—
131C23×C127—28no—
132C23×C127—28no—
133C23×C127—28no—
134C23×C127—44no—
135C23×C127—44no—
136C23×C127—44no—
137C23×C127—44no—
138C23×C127—44no—
139C23×C127—44no—
140C23×C127—44no—
141C23×C127—44no—
142C23×C127—44no—
143C23×C127—44no—
144C23×C127—44no—
145C23×C127—44no—
146C23×C127—44no—
147C23×C127—44no—
148C23×C127—44no—
149C23×C127—44no—
150C23×C127—44no—
151C23×C127—44no—
152C23×C127—44no—
153C23×C127—44no—
154C23×C127—44no—
155C23×C127—44no—
156C23×C127—44no—
157C23×C127—44no—
158C23×C127—44no—
159C23×C127—44no—
160C23×C127—44no—
161C23×C127—44no—
162C23×C127C22×D412yes—
163C23×C127C22×D411yescharacteristic
164C23×C127C2×#4112yes—
165C23×C127H(5,𝔽2)11yescharacteristic
166C23×C127H(5,𝔽2)12yes—
167C23×C127H(5,𝔽2)14yes—
168C23×C127H(5,𝔽2)11yescharacteristic
169C23×C127H(5,𝔽2)12yes—
170D4×C127—432no—
171D4×C127—432no—
172D4×C127—432no—
173C2×C4×C127—28no—
174C2×C4×C127—22no—
175C2×C4×C127—24no—
176C2×C4×C127—24no—
177C2×C4×C127—24no—
178C2×C4×C127—24no—
179C2×C4×C127—28no—
180C2×C4×C127—28no—
181C2×C4×C127—44no—
182C2×C4×C127—44no—
183C2×C4×C127—42no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—44no—
189C2×C4×C127—42no—
190C2×C4×C127—42no—
191C2×C4×C127—42no—
192C2×C4×C127—44no—
193C2×C4×C127—44no—
194C2×C4×C127—44no—
195C2×C4×C127—44no—
196C2×C4×C127—44no—
197C2×C4×C127—44no—
198C2×C4×C127—44no—
199C22×C127—48no—
200C22×C127—48no—
201C22×C127—48no—
202C22×C127—48no—
203C22×C127—48no—
204C22×C127—48no—
205C22×C127—48no—
206C22×C127—84no—
207C22×C127—84no—
208C22×C127—84no—
209C22×C127—84no—
210C22×C127—84no—
211C22×C127—84no—
212C22×C127—84no—
213C22×C127—84no—
214C22×C127#53414yes—
215C22×C127#53412yes—
216C22×C127#53411yescharacteristic
217C22×C127#53512yes—
218C22×C127#53812yes—
219C22×C127#54014yes—
220C22×C127D4211yescharacteristic
221C22×C127D4212yes—
222C22×C127#54614yes—
223C22×C127#55012yes—
224C22×C127#55511yescharacteristic
225C22×C127#56012yes—
226C22×C127#56014yes—
227C22×C127#56012yes—
228C22×C127H(3,𝔽4)11yescharacteristic
229C22×C127H(3,𝔽4)11yescharacteristic
230C4×C127—44no—
231C4×C127—44no—
232C4×C127—44no—
233C4×C127—42no—
234C4×C127—84no—
235#17082C12711yesmaximal, characteristic, direct product factor, Sylow
236C2×C127—88no—
237C2×C127#266812yes—
238C2×C127#267314yes—
239C2×C127#267911yescharacteristic
240C2×C127#269912yes—
241C2×C127#273311yescharacteristic
242C2×C127#281212yes—
243C2×C127#281511yescharacteristic
244C2×C127#282812yes—
245C2×#394C2×C12712yescomplemented
246C2×#392C2×C12714yescomplemented
247#1445C2×C12711yescharacteristic, complemented
248#1440C2×C12714yescomplemented
249#1438C2×C12712yescomplemented
250#1438C2×C12712yescomplemented
251C127#1708211yescharacteristic, direct product factor, Sylow
252C23×D4C22×C12712yes—
253C23×D4C22×C12712yes—
254C23×D4C22×C12712yes—
255C22×C4⋊C4C22×C12711yescharacteristic
256C22×C22⋊C4C22×C12714yes—
257C22×C22⋊C4C22×C12714yes—
258C22×C22⋊C4C22×C12714yes—
259C22×C22⋊C4C22×C12714yes—
260#394—28no—
261#392—216no—
262C2×#95C22×C12714yes—
263C2×#95C22×C12712yes—
264C2×#95C22×C12711yescharacteristic
265C2×#95C22×C12711yescharacteristic
266C2×#95C22×C12712yes—
267C2×#95C22×C12712yes—
268C25C23×C12714yes—
269C22×D4—28no—
270C22×D4—28no—
271C22×D4—24no—
272C22×D4—24no—
273C22×D4—42no—
274C22×D4—42no—
275C22×D4—42no—
276C22×D4—42no—
277C22×D4—44no—
278C22×D4—44no—
279C22×D4—44no—
280C22×D4—44no—
281C22×D4—42no—
282C22×D4—42no—
283C22×D4D4×C12718yescomplemented
284C23×C4C23×C12714yes—
285C23×C4C23×C12711yescharacteristic
286C23×C4C23×C12712yes—
287C23×C4C23×C12712yes—
288C23×C4C23×C12712yes—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4D4×C12714yescomplemented
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304#95—28no—
305#95—24no—
306#95—41no—
307#95—41no—
308#95—42no—
309#95—42no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24C2×D4×C12718yes—
325C24C24×C12711yescharacteristic
326C2×D4—416no—
327C2×D4—48no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—48no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4C2×D4×C12714yes—
364C22×C4C2×D4×C12714yes—
365C22×C4C2×D4×C12712yes—
366C22×C4#41×C12714yes—
367C4⋊C4—44no—
368C22⋊C4—416no—
369C22⋊C4—416no—
370C22⋊C4—416no—
371C22⋊C4—416no—
372C23—28no—
373C23—28no—
374C23—28no—
375C23—28no—
376C23—28no—
377C23—28no—
378C23—28no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23C22×D4×C12712yes—
408C23C22×D4×C12711yescharacteristic
409C23C2×#41×C12712yes—
410C23H(5,𝔽2)×C12711yescharacteristic
411C23H(5,𝔽2)×C12712yes—
412C23H(5,𝔽2)×C12714yes—
413C23H(5,𝔽2)×C12711yescharacteristic
414C23H(5,𝔽2)×C12712yes—
415D4—432no—
416D4—432no—
417D4—432no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C22—48no—
445C22—48no—
446C22—48no—
447C22—48no—
448C22—48no—
449C22—48no—
450C22—48no—
451C22—84no—
452C22—84no—
453C22—84no—
454C22—84no—
455C22—84no—
456C22—84no—
457C22—84no—
458C22—84no—
459C22#534×C12714yes—
460C22#534×C12712yes—
461C22#534×C12711yescharacteristic
462C22#535×C12712yes—
463C22#538×C12712yes—
464C22#540×C12714yes—
465C22D42×C12711yescharacteristic
466C22D42×C12712yes—
467C22#546×C12714yes—
468C22#550×C12712yes—
469C22#555×C12711yescharacteristic
470C22#560×C12712yes—
471C22#560×C12714yes—
472C22#560×C12712yes—
473C22H(3,𝔽4)×C12711yescharacteristic
474C22H(3,𝔽4)×C12711yescharacteristic
475C4—44no—
476C4—44no—
477C4—44no—
478C4—42no—
479C4—84no—
480C2—88no—
481C2C127×#266812yes—
482C2C127×#267314yes—
483C2C127×#267911yescharacteristic
484C2C127×#269912yes—
485C2C127×#273311yescharacteristic
486C2C127×#281212yes—
487C2C127×#281511yescharacteristic
488C2C127×#282812yes—
subgroupG/H
center, socleC24×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
1. C2×#394×C127C2semidirect
2. C2×#392×C127C2semidirect
3. C127×#1445C2semidirect
4. C127×#1440C2semidirect
5. C127×#1438C2semidirect
6. C127×#1438C2semidirect
38. C22×D4×C127D4semidirect
46. C2×C4⋊C4×C127D4semidirect
235. #17082C127direct (N × H)
245. C2×#394C2×C127semidirect
246. C2×#392C2×C127semidirect
247. #1445C2×C127semidirect
248. #1440C2×C127semidirect
249. #1438C2×C127semidirect
250. #1438C2×C127semidirect
251. C127#17082direct (N × H)
283. C22×D4D4×C127semidirect
291. C2×C4⋊C4D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#394×C1272
C2×#392×C1272
C127×#14452
C127×#14402
C127×#14382
#17082127