Finite Group Databasebeta

← C127×#10731 #44574404

Subgroups — C127×#10731 #44574404

Order 32512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C127C211yesmaximal, characteristic
2C22×C22⋊C8×C127C211yesmaximal, characteristic
3C2×#380×C127C211yesmaximal, characteristic
4C2×#340×C127C214yesmaximal, complemented
5C24×C4×C127C2211yescharacteristic
6C23×C8×C127C2212yes—
7C2×#117×C127—22no—
8C2×#117×C127—28no—
9C2×#117×C127C414yescomplemented
10C22×C4⋊C4×C127C2212yes—
11C22×C4⋊C4×C127C2212yes—
12C22×C22⋊C4×C127—21no—
13C22×C22⋊C4×C127—21no—
14C22×C42×C127—21no—
15C2×C22⋊C8×C127—22no—
16C2×C22⋊C8×C127—22no—
17C2×C22⋊C8×C127C414yescomplemented
18#380×C127—22no—
19#380×C127C414yescomplemented
20C2×#95×C127—21no—
21#340×C127—28no—
22C25×C127D411yescharacteristic
23C23×C4×C127—24no—
24C23×C4×C127—24no—
25C23×C4×C127—22no—
26C23×C4×C127—22no—
27C23×C4×C127C2×C412yes—
28C23×C4×C127C2×C412yes—
29C23×C4×C127C2×C412yes—
30C23×C4×C127D411yescharacteristic
31C23×C4×C127D412yes—
32C23×C4×C127D412yes—
33C23×C4×C127Q812yes—
34C23×C4×C127C2311yescharacteristic
35C22×C8×C127—24no—
36C22×C8×C127—22no—
37C22×C8×C127—22no—
38C22×C8×C127—24no—
39C22×C8×C127C2×C414yes—
40#117×C127—216no—
41#117×C127—216no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—24no—
46C2×C4⋊C4×C127—22no—
47C2×C4⋊C4×C127C2×C414yes—
48C2×C4⋊C4×C127C2×C414yes—
49C2×C4⋊C4×C127D418yes—
50C2×C22⋊C4×C127—28no—
51C2×C22⋊C4×C127—24no—
52C2×C22⋊C4×C127—24no—
53C2×C22⋊C4×C127—44no—
54C2×C42×C127—28no—
55C2×C42×C127—24no—
56C22⋊C8×C127—28no—
57#95×C127—24no—
58C24×C127—22no—
59C24×C127—22no—
60C24×C127—24no—
61C24×C127—22no—
62C24×C127—22no—
63C24×C127C22⋊C412yes—
64C24×C127C22⋊C412yes—
65C24×C127C4⋊C412yes—
66C24×C127C2×D411yescharacteristic
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—28no—
72C22×C4×C127—28no—
73C22×C4×C127—24no—
74C22×C4×C127—28no—
75C22×C4×C127—28no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—28no—
79C22×C4×C127—24no—
80C22×C4×C127—44no—
81C22×C4×C127—42no—
82C22×C4×C127—42no—
83C22×C4×C127—42no—
84C22×C4×C127—42no—
85C22×C4×C127—44no—
86C22×C4×C127—44no—
87C22×C4×C127C4214yes—
88C22×C4×C127C22⋊C412yes—
89C22×C4×C127C22⋊C412yes—
90C22×C4×C127C22⋊C414yes—
91C22×C4×C127C4⋊C412yes—
92C22×C4×C127C4⋊C414yes—
93C22×C4×C127C4⋊C414yes—
94C22×C4×C127C22×C411yescharacteristic
95C22×C4×C127C22×C411yescharacteristic
96C22×C4×C127C22×C411yescharacteristic
97C22×C4×C127C2×D411yescharacteristic
98C22×C4×C127C2×D411yescharacteristic
99C22×C4×C127C2×D414yes—
100C22×C4×C127C2×Q811yescharacteristic
101C22×C4×C127#4114yes—
102C2×C8×C127—24no—
103C2×C8×C127—28no—
104C2×C8×C127—42no—
105C2×C8×C127—44no—
106C2×C8×C127—42no—
107C4⋊C4×C127—24no—
108C4⋊C4×C127—28no—
109C4⋊C4×C127—28no—
110C4⋊C4×C127—44no—
111C4⋊C4×C127C22⋊C418yescomplemented
112C22⋊C4×C127—216no—
113C22⋊C4×C127—48no—
114C42×C127—216no—
115C23×C127—22no—
116C23×C127—22no—
117C23×C127—24no—
118C23×C127—24no—
119C23×C127—24no—
120C23×C127—24no—
121C23×C127—24no—
122C23×C127—22no—
123C23×C127—22no—
124C23×C127—24no—
125C23×C127—24no—
126C23×C127—44no—
127C23×C127—44no—
128C23×C127—44no—
129C23×C127—44no—
130C23×C127#9514yes—
131C23×C127C2×C22⋊C411yescharacteristic
132C23×C127C2×C22⋊C411yescharacteristic
133C23×C127C2×C4⋊C411yescharacteristic
134C23×C127#12012yes—
135C23×C127#12312yes—
136C23×C127#13514yes—
137C23×C127AGL(1,ℤ/8)12yes—
138C23×C127#13712yes—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—24no—
142C2×C4×C127—24no—
143C2×C4×C127—28no—
144C2×C4×C127—28no—
145C2×C4×C127—28no—
146C2×C4×C127—28no—
147C2×C4×C127—28no—
148C2×C4×C127—44no—
149C2×C4×C127—44no—
150C2×C4×C127—48no—
151C2×C4×C127—48no—
152C2×C4×C127—44no—
153C2×C4×C127—44no—
154C2×C4×C127—44no—
155C2×C4×C127—84no—
156C2×C4×C127—84no—
157C2×C4×C127#9514yes—
158C2×C4×C127C2×C4211yescharacteristic
159C2×C4×C127C2×C22⋊C412yes—
160C2×C4×C127C2×C22⋊C411yescharacteristic
161C2×C4×C127C2×C4⋊C411yescharacteristic
162C2×C4×C127C2×C4⋊C411yescharacteristic
163C2×C4×C127#11712yes—
164C2×C4×C127C4×D414yes—
165C2×C4×C127C4×D414yes—
166C2×C4×C127#12012yes—
167C2×C4×C127#12114yes—
168C2×C4×C127C22⋊Q814yes—
169C2×C4×C127#12312yes—
170C8×C127—44no—
171C22×C127—24no—
172C22×C127—24no—
173C22×C127—24no—
174C22×C127—44no—
175C22×C127—48no—
176C22×C127—44no—
177C22×C127—44no—
178C22×C127—44no—
179C22×C127C2×#9511yescharacteristic
180C22×C127#37911yescharacteristic
181C22×C127#38111yescharacteristic
182C22×C127#38512yes—
183C22×C127#38612yes—
184C22×C127#41612yes—
185C22×C127#41612yes—
186C22×C127#41712yes—
187C22×C127#41811yescharacteristic
188C22×C127#41911yescharacteristic
189C22×C127#42712yes—
190C22×C127#42812yes—
191C22×C127#44914yes—
192C22×C127#45214yes—
193C22×C127#48214yes—
194C22×C127#48514yes—
195C4×C127—24no—
196C4×C127—48no—
197C4×C127—84no—
198C4×C127C4×C22⋊C412yes—
199C4×C127#38012yes—
200C4×C127#38512yes—
201C4×C127#38612yes—
202#10731C12711yesmaximal, characteristic, direct product factor, Sylow
203C2×C127—44no—
204C2×C127C22⋊#9511yescharacteristic
205C2×C127#173511yescharacteristic
206C2×C127#173611yescharacteristic
207C2×C127#178812yes—
208C2×C127#182512yes—
209C2×C127#185414yes—
210C2×C127#187312yes—
211C2×C127#187512yes—
212C22×#117C2×C12711yescharacteristic
213C22×C22⋊C8C2×C12711yescharacteristic
214C2×#380C2×C12711yescharacteristic
215C2×#340C2×C12714yescomplemented
216C127#1073111yescharacteristic, direct product factor, Sylow
217C24×C4C22×C12711yescharacteristic
218C23×C8C22×C12712yes—
219C2×#117—22no—
220C2×#117—28no—
221C2×#117C4×C12714yescomplemented
222C22×C4⋊C4C22×C12712yes—
223C22×C4⋊C4C22×C12712yes—
224C22×C22⋊C4—21no—
225C22×C22⋊C4—21no—
226C22×C42—21no—
227C2×C22⋊C8—22no—
228C2×C22⋊C8—22no—
229C2×C22⋊C8C4×C12714yescomplemented
230#380—22no—
231#380C4×C12714yescomplemented
232C2×#95—21no—
233#340—28no—
234C25D4×C12711yescharacteristic
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4C2×C4×C12712yes—
240C23×C4C2×C4×C12712yes—
241C23×C4C2×C4×C12712yes—
242C23×C4D4×C12711yescharacteristic
243C23×C4D4×C12712yes—
244C23×C4D4×C12712yes—
245C23×C4Q8×C12712yes—
246C23×C4C23×C12711yescharacteristic
247C22×C8—24no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—24no—
251C22×C8C2×C4×C12714yes—
252#117—216no—
253#117—216no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4C2×C4×C12714yes—
260C2×C4⋊C4C2×C4×C12714yes—
261C2×C4⋊C4D4×C12718yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—44no—
266C2×C42—28no—
267C2×C42—24no—
268C22⋊C8—28no—
269#95—24no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—22no—
274C24—22no—
275C24C22⋊C4×C12712yes—
276C24C22⋊C4×C12712yes—
277C24C4⋊C4×C12712yes—
278C24C2×D4×C12711yescharacteristic
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4C42×C12714yes—
300C22×C4C22⋊C4×C12712yes—
301C22×C4C22⋊C4×C12712yes—
302C22×C4C22⋊C4×C12714yes—
303C22×C4C4⋊C4×C12712yes—
304C22×C4C4⋊C4×C12714yes—
305C22×C4C4⋊C4×C12714yes—
306C22×C4C22×C4×C12711yescharacteristic
307C22×C4C22×C4×C12711yescharacteristic
308C22×C4C22×C4×C12711yescharacteristic
309C22×C4C2×D4×C12711yescharacteristic
310C22×C4C2×D4×C12711yescharacteristic
311C22×C4C2×D4×C12714yes—
312C22×C4C2×Q8×C12711yescharacteristic
313C22×C4#41×C12714yes—
314C2×C8—24no—
315C2×C8—28no—
316C2×C8—42no—
317C2×C8—44no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—44no—
323C4⋊C4C22⋊C4×C12718yescomplemented
324C22⋊C4—216no—
325C22⋊C4—48no—
326C42—216no—
327C23—22no—
328C23—22no—
329C23—24no—
330C23—24no—
331C23—24no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23#95×C12714yes—
343C23C2×C22⋊C4×C12711yescharacteristic
344C23C2×C22⋊C4×C12711yescharacteristic
345C23C2×C4⋊C4×C12711yescharacteristic
346C23#120×C12712yes—
347C23#123×C12712yes—
348C23#135×C12714yes—
349C23AGL(1,ℤ/8)×C12712yes—
350C23#137×C12712yes—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4#95×C12714yes—
370C2×C4C2×C42×C12711yescharacteristic
371C2×C4C2×C22⋊C4×C12712yes—
372C2×C4C2×C22⋊C4×C12711yescharacteristic
373C2×C4C2×C4⋊C4×C12711yescharacteristic
374C2×C4C2×C4⋊C4×C12711yescharacteristic
375C2×C4#117×C12712yes—
376C2×C4C4×D4×C12714yes—
377C2×C4C4×D4×C12714yes—
378C2×C4#120×C12712yes—
379C2×C4#121×C12714yes—
380C2×C4C22⋊Q8×C12714yes—
381C2×C4#123×C12712yes—
382C8—44no—
383C22—24no—
384C22—24no—
385C22—24no—
386C22—44no—
387C22—48no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22C2×#95×C12711yescharacteristic
392C22#379×C12711yescharacteristic
393C22#381×C12711yescharacteristic
394C22#385×C12712yes—
395C22#386×C12712yes—
396C22#416×C12712yes—
397C22#416×C12712yes—
398C22#417×C12712yes—
399C22#418×C12711yescharacteristic
400C22#419×C12711yescharacteristic
401C22#427×C12712yes—
402C22#428×C12712yes—
403C22#449×C12714yes—
404C22#452×C12714yes—
405C22#482×C12714yes—
406C22#485×C12714yes—
407C4—24no—
408C4—48no—
409C4—84no—
410C4C4×C22⋊C4×C12712yes—
411C4#380×C12712yes—
412C4#385×C12712yes—
413C4#386×C12712yes—
414C2—44no—
415C2C127×C22⋊#9511yescharacteristic
416C2C127×#173511yescharacteristic
417C2C127×#173611yescharacteristic
418C2C127×#178812yes—
419C2C127×#182512yes—
420C2C127×#185414yes—
421C2C127×#187312yes—
422C2C127×#187512yes—
subgroupG/H
center, socleC24×C127C2×D4
commutatorC2×C4C2×C42×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC25×C127D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×#340×C127C2semidirect
9. C2×#117×C127C4semidirect
17. C2×C22⋊C8×C127C4semidirect
19. #380×C127C4semidirect
111. C4⋊C4×C127C22⋊C4semidirect
202. #10731C127direct (N × H)
215. C2×#340C2×C127semidirect
216. C127#10731direct (N × H)
221. C2×#117C4×C127semidirect
229. C2×C22⋊C8C4×C127semidirect
231. #380C4×C127semidirect
323. C4⋊C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C1272
C22×C22⋊C8×C1272
C2×#380×C1272
C2×#340×C1272
#10731127