Finite Group Databasebeta

← C127×#17066 #44580739

Subgroups — C127×#17066 #44580739

Order 32512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#388×C127C211yesmaximal, characteristic
2C2×#384×C127C211yesmaximal, characteristic, complemented
3C127×#1442C212yesmaximal, complemented
4C127×#1442C212yesmaximal, complemented
5C127×#1441C212yesmaximal, complemented
6C127×#1439C212yesmaximal
7C127×#1439C212yesmaximal
8C127×#1439C212yesmaximal
9C127×#1434C211yesmaximal, characteristic, complemented
10C22×C4⋊C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C22⋊C4×C127C2211yescharacteristic
15C22×C22⋊C4×C127C2211yescharacteristic
16C22×C22⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2214yes—
18C22×C42×C127C2211yescharacteristic
19#388×C127—24no—
20#384×C127—24no—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2211yescharacteristic
25C2×#95×C127C2211yescharacteristic
26C2×#95×C127C2214yes—
27C2×#95×C127C2214yes—
28C2×#95×C127C2214yes—
29C25×C127C2311yescharacteristic
30C23×C4×C127C2314yes—
31C23×C4×C127C2311yescharacteristic
32C23×C4×C127C2311yescharacteristic
33C23×C4×C127C2314yes—
34C23×C4×C127C2311yescharacteristic
35C23×C4×C127C2311yescharacteristic
36C23×C4×C127C2311yescharacteristic
37C23×C4×C127C2311yescharacteristic
38C2×C4⋊C4×C127—24no—
39C2×C4⋊C4×C127—24no—
40C2×C4⋊C4×C127—24no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—44no—
44C2×C4⋊C4×C127—44no—
45C2×C4⋊C4×C127—44no—
46C2×C22⋊C4×C127—22no—
47C2×C22⋊C4×C127—22no—
48C2×C22⋊C4×C127—22no—
49C2×C22⋊C4×C127—22no—
50C2×C22⋊C4×C127—24no—
51C2×C22⋊C4×C127—24no—
52C2×C22⋊C4×C127—44no—
53C2×C22⋊C4×C127—44no—
54C2×C22⋊C4×C127—44no—
55C2×C22⋊C4×C127Q814yes—
56C2×C42×C127—24no—
57C2×C42×C127D414yes—
58#95×C127—22no—
59#95×C127—22no—
60#95×C127—24no—
61#95×C127—24no—
62#95×C127—24no—
63#95×C127—44no—
64#95×C127—44no—
65#95×C127—44no—
66C24×C127—21no—
67C24×C127—21no—
68C24×C127—22no—
69C24×C127—22no—
70C24×C127—22no—
71C24×C127—22no—
72C24×C127—22no—
73C24×C127—22no—
74C24×C127C2×Q812yes—
75C24×C127C2411yescharacteristic
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—21no—
83C22×C4×C127—21no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—24no—
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—24no—
105C22×C4×C127—24no—
106C22×C4×C127—22no—
107C22×C4×C127—24no—
108C22×C4×C127C2×D412yes—
109C22×C4×C127C2×D412yes—
110C22×C4×C127C2×D412yes—
111C22×C4×C127C2×Q812yes—
112C22×C4×C127C2×Q812yes—
113C22×C4×C127#4112yes—
114C22×C4×C127#4114yes—
115C22×C4×C127#4114yes—
116C4⋊C4×C127—48no—
117C4⋊C4×C127—48no—
118C4⋊C4×C127—48no—
119C4⋊C4×C127—44no—
120C22⋊C4×C127—28no—
121C22⋊C4×C127—416no—
122C22⋊C4×C127—44no—
123C22⋊C4×C127—44no—
124C42×C127—28no—
125C23×C127—22no—
126C23×C127—24no—
127C23×C127—24no—
128C23×C127—24no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127—41no—
134C23×C127—41no—
135C23×C127—44no—
136C23×C127—42no—
137C23×C127—44no—
138C23×C127—42no—
139C23×C127—44no—
140C23×C127—42no—
141C23×C127—44no—
142C23×C127C22×D411yescharacteristic
143C23×C127C22×Q811yescharacteristic
144C23×C127C2×#4111yescharacteristic
145C23×C127C2×#4112yes—
146C23×C127C2×#4112yes—
147C23×C127H(5,𝔽2)12yes—
148C23×C127H(5,𝔽2)12yes—
149C23×C127H(5,𝔽2)12yes—
150C23×C1272−1+412yes—
151C2×C4×C127—22no—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—22no—
155C2×C4×C127—22no—
156C2×C4×C127—22no—
157C2×C4×C127—28no—
158C2×C4×C127—24no—
159C2×C4×C127—28no—
160C2×C4×C127—24no—
161C2×C4×C127—22no—
162C2×C4×C127—24no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—42no—
166C2×C4×C127—41no—
167C2×C4×C127—41no—
168C2×C4×C127—41no—
169C2×C4×C127—44no—
170C2×C4×C127—44no—
171C2×C4×C127—44no—
172C2×C4×C127—41no—
173C2×C4×C127—41no—
174C2×C4×C127—41no—
175C2×C4×C127—44no—
176C2×C4×C127—44no—
177C2×C4×C127—44no—
178C2×C4×C127—44no—
179C2×C4×C127—44no—
180C2×C4×C127—44no—
181C2×C4×C127—44no—
182C2×C4×C127—44no—
183C2×C4×C127—44no—
184C2×C4×C127—44no—
185C2×C4×C127—44no—
186C2×C4×C127—44no—
187C2×C4×C127—44no—
188C2×C4×C127C22⋊Q814yes—
189C2×C4×C127#12314yes—
190C2×C4×C127#12414yes—
191C22×C127—42no—
192C22×C127—42no—
193C22×C127—42no—
194C22×C127—48no—
195C22×C127—81no—
196C22×C127—81no—
197C22×C127—81no—
198C22×C127—81no—
199C22×C127—84no—
200C22×C127C2×C22⋊Q811yescharacteristic
201C22×C127C2×#12311yescharacteristic
202C22×C127C2×#12411yescharacteristic
203C22×C127#53511yescharacteristic
204C22×C127#53511yescharacteristic
205C22×C127#53511yescharacteristic
206C22×C127#53611yescharacteristic
207C22×C127#53914yes—
208C22×C127#54212yes—
209C22×C127#54214yes—
210C22×C127#54412yes—
211C22×C127#55312yes—
212C22×C127#55314yes—
213C22×C127#55412yes—
214C22×C127#55412yes—
215C22×C127#55512yes—
216C22×C127#56014yes—
217C4×C127—24no—
218C4×C127—24no—
219C4×C127—24no—
220C4×C127—42no—
221C4×C127—84no—
222C4×C127—81no—
223C4×C127—81no—
224C4×C127—84no—
225#17066C12711yesmaximal, characteristic, direct product factor, Sylow
226C2×C127—82no—
227C2×C127#248711yescharacteristic
228C2×C127#250411yescharacteristic
229C2×C127#256311yescharacteristic
230C2×C127#262311yescharacteristic
231C2×C127#262811yescharacteristic
232C2×C127#264011yescharacteristic
233C2×C127#279014yes—
234C2×C127#279514yes—
235C2×C127#281811yescharacteristic
236C2×#388C2×C12711yescharacteristic
237C2×#384C2×C12711yescharacteristic, complemented
238#1442C2×C12712yescomplemented
239#1442C2×C12712yescomplemented
240#1441C2×C12712yescomplemented
241#1439C2×C12712yes—
242#1439C2×C12712yes—
243#1439C2×C12712yes—
244#1434C2×C12711yescharacteristic, complemented
245C127#1706611yescharacteristic, direct product factor, Sylow
246C22×C4⋊C4C22×C12711yescharacteristic
247C22×C4⋊C4C22×C12712yes—
248C22×C4⋊C4C22×C12712yes—
249C22×C4⋊C4C22×C12712yes—
250C22×C22⋊C4C22×C12711yescharacteristic
251C22×C22⋊C4C22×C12711yescharacteristic
252C22×C22⋊C4C22×C12711yescharacteristic
253C22×C22⋊C4C22×C12714yes—
254C22×C42C22×C12711yescharacteristic
255#388—24no—
256#384—24no—
257C2×#95C22×C12712yes—
258C2×#95C22×C12712yes—
259C2×#95C22×C12712yes—
260C2×#95C22×C12711yescharacteristic
261C2×#95C22×C12711yescharacteristic
262C2×#95C22×C12714yes—
263C2×#95C22×C12714yes—
264C2×#95C22×C12714yes—
265C25C23×C12711yescharacteristic
266C23×C4C23×C12714yes—
267C23×C4C23×C12711yescharacteristic
268C23×C4C23×C12711yescharacteristic
269C23×C4C23×C12714yes—
270C23×C4C23×C12711yescharacteristic
271C23×C4C23×C12711yescharacteristic
272C23×C4C23×C12711yescharacteristic
273C23×C4C23×C12711yescharacteristic
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4Q8×C12714yes—
292C2×C42—24no—
293C2×C42D4×C12714yes—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—44no—
300#95—44no—
301#95—44no—
302C24—21no—
303C24—21no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24C2×Q8×C12712yes—
311C24C24×C12711yescharacteristic
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4C2×D4×C12712yes—
345C22×C4C2×D4×C12712yes—
346C22×C4C2×D4×C12712yes—
347C22×C4C2×Q8×C12712yes—
348C22×C4C2×Q8×C12712yes—
349C22×C4#41×C12712yes—
350C22×C4#41×C12714yes—
351C22×C4#41×C12714yes—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—44no—
356C22⋊C4—28no—
357C22⋊C4—416no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C42—28no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—44no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—44no—
376C23—42no—
377C23—44no—
378C23C22×D4×C12711yescharacteristic
379C23C22×Q8×C12711yescharacteristic
380C23C2×#41×C12711yescharacteristic
381C23C2×#41×C12712yes—
382C23C2×#41×C12712yes—
383C23H(5,𝔽2)×C12712yes—
384C23H(5,𝔽2)×C12712yes—
385C23H(5,𝔽2)×C12712yes—
386C232−1+4×C12712yes—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4C22⋊Q8×C12714yes—
425C2×C4#123×C12714yes—
426C2×C4#124×C12714yes—
427C22—42no—
428C22—42no—
429C22—48no—
430C22—42no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—84no—
435C22—81no—
436C22C2×C22⋊Q8×C12711yescharacteristic
437C22C2×#123×C12711yescharacteristic
438C22C2×#124×C12711yescharacteristic
439C22#535×C12711yescharacteristic
440C22#535×C12711yescharacteristic
441C22#535×C12711yescharacteristic
442C22#536×C12711yescharacteristic
443C22#539×C12714yes—
444C22#542×C12712yes—
445C22#542×C12714yes—
446C22#544×C12712yes—
447C22#553×C12712yes—
448C22#553×C12714yes—
449C22#554×C12712yes—
450C22#554×C12712yes—
451C22#555×C12712yes—
452C22#560×C12714yes—
453C4—24no—
454C4—24no—
455C4—24no—
456C4—42no—
457C4—81no—
458C4—81no—
459C4—84no—
460C4—84no—
461C2—82no—
462C2C127×#248711yescharacteristic
463C2C127×#250411yescharacteristic
464C2C127×#256311yescharacteristic
465C2C127×#262311yescharacteristic
466C2C127×#262811yescharacteristic
467C2C127×#264011yescharacteristic
468C2C127×#279014yes—
469C2C127×#279514yes—
470C2C127×#281811yescharacteristic
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
2. C2×#384×C127C2semidirect
3. C127×#1442C2semidirect
4. C127×#1442C2semidirect
5. C127×#1441C2semidirect
9. C127×#1434C2semidirect
225. #17066C127direct (N × H)
237. C2×#384C2×C127semidirect
238. #1442C2×C127semidirect
239. #1442C2×C127semidirect
240. #1441C2×C127semidirect
244. #1434C2×C127semidirect
245. C127#17066direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#388×C1272
C2×#384×C1272
C127×#14422
C127×#14412
C127×#14392
C127×#14342
#17066127