Finite Group Databasebeta

← C127×#20716 #44584389

Subgroups — C127×#20716 #44584389

Order 32512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#117×C127C211yesmaximal, characteristic
2C2×#433×C127C212yesmaximal, complemented
3C127×#1818C212yesmaximal, complemented
4C127×#1798C212yesmaximal, complemented
5C127×#1797C212yesmaximal, complemented
6C127×#1779C212yesmaximal
7C127×#1750C212yesmaximal, complemented
8C127×#1749C211yesmaximal, characteristic, complemented
9C4×C22⋊C8×C127C211yesmaximal, characteristic, complemented
10C24×C4×C127C2211yescharacteristic
11C2×#117×C127—28no—
12C2×#117×C127C412yes—
13C2×#117×C127C412yes—
14C2×#117×C127C2211yescharacteristic
15C2×#117×C127C2211yescharacteristic
16C2×#117×C127C2211yescharacteristic
17C2×#117×C127C2211yescharacteristic
18C22×C4⋊C4×C127C412yes—
19C22×C22⋊C4×C127C412yes—
20C22×C42×C127C2211yescharacteristic
21C22×C42×C127C2211yescharacteristic
22#433×C127—28no—
23C2×C4⋊C8×C127C2214yescomplemented
24C2×C22⋊C8×C127C2214yes—
25C2×C22⋊C8×C127C2212yes—
26C2×C22⋊C8×C127C2212yes—
27C2×C22⋊C8×C127C2214yes—
28C2×#97×C127C2212yescomplemented
29C2×C4×C8×C127C2212yescomplemented
30#336×C127C2212yescomplemented
31#336×C127C2212yescomplemented
32#336×C127C2214yescomplemented
33C25×C127C2×C411yescharacteristic
34C23×C4×C127—24no—
35C23×C4×C127—24no—
36C23×C4×C127—24no—
37C23×C4×C127C2×C411yescharacteristic
38C23×C4×C127C2×C411yescharacteristic
39C23×C4×C127C2×C411yescharacteristic
40C23×C4×C127C2×C412yes—
41C23×C4×C127C2×C412yes—
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C22×C8×C127C2314yes—
46C22×C8×C127C2314yes—
47#117×C127—24no—
48#117×C127—28no—
49#117×C127—28no—
50#117×C127—24no—
51#117×C127—24no—
52#117×C127D418yes—
53C2×C4⋊C4×C127—28no—
54C2×C4⋊C4×C127C2×C412yes—
55C2×C4⋊C4×C127C2×C412yes—
56C2×C4⋊C4×C127C2×C412yes—
57C2×C4⋊C4×C127C2×C412yes—
58C2×C22⋊C4×C127—28no—
59C2×C22⋊C4×C127C2×C412yes—
60C2×C22⋊C4×C127C2×C412yes—
61C2×C22⋊C4×C127C2×C412yes—
62C2×C22⋊C4×C127C2×C412yes—
63C2×C42×C127—24no—
64C2×C42×C127—24no—
65C2×C42×C127C2×C412yes—
66C2×C42×C127C2×C412yes—
67C2×C42×C127C2311yescharacteristic
68C2×C42×C127C2311yescharacteristic
69C2×C42×C127C2311yescharacteristic
70C2×C42×C127C2311yescharacteristic
71C4⋊C8×C127—28no—
72C22⋊C8×C127—28no—
73C22⋊C8×C127—24no—
74C22⋊C8×C127—24no—
75C22⋊C8×C127—28no—
76#97×C127—24no—
77C4×C8×C127—24no—
78C24×C127—22no—
79C24×C127—22no—
80C24×C127—22no—
81C24×C127—24no—
82C24×C127—42no—
83C24×C127C22×C411yescharacteristic
84C24×C127C22×C411yescharacteristic
85C24×C127C22×C411yescharacteristic
86C22×C4×C127—232no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—24no—
103C22×C4×C127—24no—
104C22×C4×C127—24no—
105C22×C4×C127—24no—
106C22×C4×C127—42no—
107C22×C4×C127—42no—
108C22×C4×C127C22×C412yes—
109C22×C4×C127C22×C412yes—
110C22×C4×C127C22×C412yes—
111C22×C4×C127C22×C412yes—
112C22×C4×C127C22×C411yescharacteristic
113C22×C4×C127C22×C411yescharacteristic
114C22×C4×C127C22×C411yescharacteristic
115C22×C4×C127C2×D414yes—
116C22×C4×C127#4114yes—
117C22×C4×C127#4114yes—
118C22×C4×C127C2411yescharacteristic
119C2×C8×C127—24no—
120C2×C8×C127—24no—
121C2×C8×C127—24no—
122C2×C8×C127—24no—
123C2×C8×C127—24no—
124C2×C8×C127—24no—
125C4⋊C4×C127—24no—
126C4⋊C4×C127—24no—
127C4⋊C4×C127—24no—
128C4⋊C4×C127C22⋊C418yes—
129C22⋊C4×C127—24no—
130C22⋊C4×C127—24no—
131C22⋊C4×C127—24no—
132C22⋊C4×C127C22⋊C418yes—
133C42×C127—24no—
134C42×C127—24no—
135C42×C127—22no—
136C42×C127—22no—
137C42×C127C2×D414yes—
138C42×C127C2×D414yes—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—22no—
142C23×C127—28no—
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—21no—
146C23×C127—21no—
147C23×C127—22no—
148C23×C127—22no—
149C23×C127—22no—
150C23×C127—22no—
151C23×C127—48no—
152C23×C127—48no—
153C23×C127—44no—
154C23×C127—44no—
155C23×C127C2×C22⋊C412yes—
156C23×C127#11712yes—
157C23×C127#11712yes—
158C23×C127#13114yes—
159C23×C127C23×C411yescharacteristic
160C2×C4×C127—28no—
161C2×C4×C127—28no—
162C2×C4×C127—28no—
163C2×C4×C127—21no—
164C2×C4×C127—21no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—22no—
170C2×C4×C127—22no—
171C2×C4×C127—22no—
172C2×C4×C127—22no—
173C2×C4×C127—28no—
174C2×C4×C127—28no—
175C2×C4×C127—28no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—22no—
182C2×C4×C127—48no—
183C2×C4×C127—48no—
184C2×C4×C127C2×C22⋊C414yes—
185C2×C4×C127C2×C22⋊C414yes—
186C2×C4×C127C2×C22⋊C412yes—
187C2×C4×C127#11712yes—
188C2×C4×C127#11712yes—
189C2×C4×C127C22×D412yes—
190C2×C4×C127C2×#4112yes—
191C2×C4×C127C2×#4112yes—
192C8×C127—44no—
193C8×C127—44no—
194C22×C127—24no—
195C22×C127—22no—
196C22×C127—24no—
197C22×C127—22no—
198C22×C127—24no—
199C22×C127—24no—
200C22×C127—22no—
201C22×C127—22no—
202C22×C127—416no—
203C22×C127—42no—
204C22×C127—41no—
205C22×C127—41no—
206C22×C127—41no—
207C22×C127—41no—
208C22×C127—42no—
209C22×C127#43314yescomplemented
210C22×C127C22×C22⋊C411yescharacteristic
211C22×C127C2×#11711yescharacteristic
212C22×C127C2×#11711yescharacteristic
213C22×C127C2×#13112yes—
214C22×C127#56812yes—
215C4×C127—24no—
216C4×C127—24no—
217C4×C127—22no—
218C4×C127—22no—
219C4×C127—22no—
220C4×C127—42no—
221C4×C127—42no—
222C4×C127#52514yes—
223#20716C12711yesmaximal, characteristic, direct product factor, Sylow
224C2×C127—24no—
225C2×C127—42no—
226C2×C127—42no—
227C2×C127#228011yescharacteristic
228C2×C127#288012yes—
229C2×C127C2×#43311yescharacteristic
230C2×C127#292111yescharacteristic
231C2×C127#292412yes—
232C22×#117C2×C12711yescharacteristic
233C2×#433C2×C12712yescomplemented
234#1818C2×C12712yescomplemented
235#1798C2×C12712yescomplemented
236#1797C2×C12712yescomplemented
237#1779C2×C12712yes—
238#1750C2×C12712yescomplemented
239#1749C2×C12711yescharacteristic, complemented
240C4×C22⋊C8C2×C12711yescharacteristic, complemented
241C127#2071611yescharacteristic, direct product factor, Sylow
242C24×C4C22×C12711yescharacteristic
243C2×#117—28no—
244C2×#117C4×C12712yes—
245C2×#117C4×C12712yes—
246C2×#117C22×C12711yescharacteristic
247C2×#117C22×C12711yescharacteristic
248C2×#117C22×C12711yescharacteristic
249C2×#117C22×C12711yescharacteristic
250C22×C4⋊C4C4×C12712yes—
251C22×C22⋊C4C4×C12712yes—
252C22×C42C22×C12711yescharacteristic
253C22×C42C22×C12711yescharacteristic
254#433—28no—
255C2×C4⋊C8C22×C12714yescomplemented
256C2×C22⋊C8C22×C12712yes—
257C2×C22⋊C8C22×C12712yes—
258C2×C22⋊C8C22×C12714yes—
259C2×C22⋊C8C22×C12714yes—
260C2×#97C22×C12712yescomplemented
261C2×C4×C8C22×C12712yescomplemented
262#336C22×C12714yescomplemented
263#336C22×C12712yescomplemented
264#336C22×C12712yescomplemented
265C25C2×C4×C12711yescharacteristic
266C23×C4—24no—
267C23×C4—24no—
268C23×C4—24no—
269C23×C4C2×C4×C12711yescharacteristic
270C23×C4C2×C4×C12711yescharacteristic
271C23×C4C2×C4×C12711yescharacteristic
272C23×C4C2×C4×C12712yes—
273C23×C4C2×C4×C12712yes—
274C23×C4C23×C12711yescharacteristic
275C23×C4C23×C12711yescharacteristic
276C23×C4C23×C12711yescharacteristic
277C22×C8C23×C12714yes—
278C22×C8C23×C12714yes—
279#117—24no—
280#117—28no—
281#117—28no—
282#117—24no—
283#117—24no—
284#117D4×C12718yes—
285C2×C4⋊C4—28no—
286C2×C4⋊C4C2×C4×C12712yes—
287C2×C4⋊C4C2×C4×C12712yes—
288C2×C4⋊C4C2×C4×C12712yes—
289C2×C4⋊C4C2×C4×C12712yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4C2×C4×C12712yes—
292C2×C22⋊C4C2×C4×C12712yes—
293C2×C22⋊C4C2×C4×C12712yes—
294C2×C22⋊C4C2×C4×C12712yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42C2×C4×C12712yes—
298C2×C42C2×C4×C12712yes—
299C2×C42C23×C12711yescharacteristic
300C2×C42C23×C12711yescharacteristic
301C2×C42C23×C12711yescharacteristic
302C2×C42C23×C12711yescharacteristic
303C4⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307C22⋊C8—28no—
308#97—24no—
309C4×C8—24no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—42no—
315C24C22×C4×C12711yescharacteristic
316C24C22×C4×C12711yescharacteristic
317C24C22×C4×C12711yescharacteristic
318C22×C4—232no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4C22×C4×C12712yes—
341C22×C4C22×C4×C12712yes—
342C22×C4C22×C4×C12712yes—
343C22×C4C22×C4×C12712yes—
344C22×C4C22×C4×C12711yescharacteristic
345C22×C4C22×C4×C12711yescharacteristic
346C22×C4C22×C4×C12711yescharacteristic
347C22×C4C2×D4×C12714yes—
348C22×C4#41×C12714yes—
349C22×C4#41×C12714yes—
350C22×C4C24×C12711yescharacteristic
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4C22⋊C4×C12718yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4C22⋊C4×C12718yes—
365C42—24no—
366C42—24no—
367C42—22no—
368C42—22no—
369C42C2×D4×C12714yes—
370C42C2×D4×C12714yes—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—28no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—48no—
384C23—48no—
385C23—44no—
386C23—44no—
387C23C2×C22⋊C4×C12712yes—
388C23#117×C12712yes—
389C23#117×C12712yes—
390C23#131×C12714yes—
391C23C23×C4×C12711yescharacteristic
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4C2×C22⋊C4×C12714yes—
417C2×C4C2×C22⋊C4×C12714yes—
418C2×C4C2×C22⋊C4×C12712yes—
419C2×C4#117×C12712yes—
420C2×C4#117×C12712yes—
421C2×C4C22×D4×C12712yes—
422C2×C4C2×#41×C12712yes—
423C2×C4C2×#41×C12712yes—
424C8—44no—
425C8—44no—
426C22—24no—
427C22—22no—
428C22—24no—
429C22—22no—
430C22—24no—
431C22—24no—
432C22—22no—
433C22—22no—
434C22—416no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—42no—
441C22#433×C12714yescomplemented
442C22C22×C22⋊C4×C12711yescharacteristic
443C22C2×#117×C12711yescharacteristic
444C22C2×#117×C12711yescharacteristic
445C22C2×#131×C12712yes—
446C22#568×C12712yes—
447C4—24no—
448C4—24no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—42no—
454C4#525×C12714yes—
455C2—24no—
456C2—42no—
457C2—42no—
458C2C127×#228011yescharacteristic
459C2C127×#288012yes—
460C2C2×#433×C12711yescharacteristic
461C2C127×#292111yescharacteristic
462C2C127×#292412yes—
subgroupG/H
centerC22×C4×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C23×C4
prime radicalC25×C127C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#433×C127C2semidirect
3. C127×#1818C2semidirect
4. C127×#1798C2semidirect
5. C127×#1797C2semidirect
7. C127×#1750C2semidirect
8. C127×#1749C2semidirect
9. C4×C22⋊C8×C127C2semidirect
23. C2×C4⋊C8×C127C22semidirect
28. C2×#97×C127C22semidirect
29. C2×C4×C8×C127C22semidirect
30. #336×C127C22semidirect
31. #336×C127C22semidirect
32. #336×C127C22semidirect
209. C22×C127#433semidirect
223. #20716C127direct (N × H)
233. C2×#433C2×C127semidirect
234. #1818C2×C127semidirect
235. #1798C2×C127semidirect
236. #1797C2×C127semidirect
238. #1750C2×C127semidirect
239. #1749C2×C127semidirect
240. C4×C22⋊C8C2×C127semidirect
241. C127#20716direct (N × H)
255. C2×C4⋊C8C22×C127semidirect
260. C2×#97C22×C127semidirect
261. C2×C4×C8C22×C127semidirect
262. #336C22×C127semidirect
263. #336C22×C127semidirect
264. #336C22×C127semidirect
441. C22#433×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#117×C1272
C2×#433×C1272
C127×#18182
C127×#17982
C127×#17972
C127×#17792
C127×#17502
C127×#17492
C4×C22⋊C8×C1272
#20716127