Finite Group Databasebeta

← C2×C127×#1788 #44584269

Subgroups — C2×C127×#1788 #44584269

Order 32512.

4070 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#41×C127C211yesmaximal, characteristic
2C22×#103×C127C212yesmaximal, complemented
3C22×#102×C127C212yesmaximal, complemented
4C22×C22⋊C8×C127C211yesmaximal, characteristic, complemented
5C2×#380×C127C211yesmaximal, characteristic, complemented
6C127×#1788C218yesmaximal, direct product factor
7C22×#41×C127—28no—
8C22×#41×C127—28no—
9C22×#41×C127—24no—
10C22×#41×C127—22no—
11C22×#41×C127C418yescomplemented
12C22×#41×C127C2214yes—
13C23×Q8×C127C2211yescharacteristic
14C23×D4×C127—21no—
15C23×D4×C127C2211yescharacteristic
16C24×C4×C127—21no—
17C24×C4×C127C2211yescharacteristic
18C23×C8×C127C2212yes—
19C22×C4⋊C4×C127C2212yes—
20C22×C22⋊C4×C127—21no—
21C2×#103×C127—28no—
22C2×#103×C127C2218yescomplemented
23C2×#102×C127—28no—
24C2×#102×C127C2218yescomplemented
25C2×C22⋊C8×C127—24no—
26C2×C22⋊C8×C127C2214yescomplemented
27#380×C127—22no—
28#380×C127C2214yescomplemented
29C2×#95×C127—21no—
30C25×C127—21no—
31C25×C127D411yescharacteristic
32C2×#41×C127—232no—
33C2×#41×C127—264no—
34C2×#41×C127—232no—
35C2×#41×C127—216no—
36C2×#41×C127—232no—
37C2×#41×C127—216no—
38C2×#41×C127—216no—
39C2×#41×C127—28no—
40C2×#41×C127C2×C4116yescomplemented
41C22×Q8×C127—22no—
42C22×Q8×C127—24no—
43C22×Q8×C127—21no—
44C22×Q8×C127C2×C414yes—
45C22×Q8×C127D418yes—
46C22×Q8×C127C2312yes—
47C22×D4×C127—216no—
48C22×D4×C127—28no—
49C22×D4×C127—24no—
50C22×D4×C127—21no—
51C22×D4×C127—24no—
52C22×D4×C127—22no—
53C22×D4×C127C2×C414yes—
54C22×D4×C127D418yes—
55C22×D4×C127C2312yes—
56C23×C4×C127—216no—
57C23×C4×C127—21no—
58C23×C4×C127—28no—
59C23×C4×C127—24no—
60C23×C4×C127—24no—
61C23×C4×C127—24no—
62C23×C4×C127—22no—
63C23×C4×C127C2×C414yes—
64C23×C4×C127D412yes—
65C23×C4×C127D411yescharacteristic
66C23×C4×C127D414yes—
67C23×C4×C127C2312yes—
68C23×C4×C127C2311yescharacteristic
69C22×C8×C127—24no—
70C22×C8×C127—24no—
71C22×C8×C127—24no—
72C22×C8×C127C2314yes—
73C2×C4⋊C4×C127—24no—
74C2×C4⋊C4×C127—24no—
75C2×C4⋊C4×C127—22no—
76C2×C4⋊C4×C127C2314yes—
77C2×C22⋊C4×C127—24no—
78C2×C22⋊C4×C127—44no—
79#103×C127—216no—
80#102×C127—216no—
81C22⋊C8×C127—28no—
82#95×C127—24no—
83C24×C127—24no—
84C24×C127—24no—
85C24×C127—28no—
86C24×C127—24no—
87C24×C127—22no—
88C24×C127—22no—
89C24×C127—22no—
90C24×C127—44no—
91C24×C127—42no—
92C24×C127—42no—
93C24×C127C22⋊C414yes—
94C24×C127C2×D412yes—
95C24×C127C2×D411yescharacteristic
96#41×C127—264no—
97#41×C127—2128no—
98#41×C127—232no—
99#41×C127—232no—
100C2×Q8×C127—24no—
101C2×Q8×C127—28no—
102C2×Q8×C127—28no—
103C2×Q8×C127—216no—
104C2×Q8×C127—22no—
105C2×Q8×C127—24no—
106C2×Q8×C127C22⋊C4116yescomplemented
107C2×Q8×C127C22×C414yes—
108C2×Q8×C127C2×D418yes—
109C2×D4×C127—232no—
110C2×D4×C127—264no—
111C2×D4×C127—216no—
112C2×D4×C127—28no—
113C2×D4×C127—24no—
114C2×D4×C127—216no—
115C2×D4×C127—28no—
116C2×D4×C127—24no—
117C2×D4×C127—22no—
118C2×D4×C127C22⋊C4116yescomplemented
119C2×D4×C127C22×C414yes—
120C2×D4×C127C2×D418yes—
121C22×C4×C127—232no—
122C22×C4×C127—264no—
123C22×C4×C127—24no—
124C22×C4×C127—24no—
125C22×C4×C127—28no—
126C22×C4×C127—22no—
127C22×C4×C127—216no—
128C22×C4×C127—28no—
129C22×C4×C127—28no—
130C22×C4×C127—28no—
131C22×C4×C127—216no—
132C22×C4×C127—24no—
133C22×C4×C127—24no—
134C22×C4×C127—24no—
135C22×C4×C127—44no—
136C22×C4×C127—44no—
137C22×C4×C127—44no—
138C22×C4×C127C22⋊C414yes—
139C22×C4×C127C22⋊C418yes—
140C22×C4×C127C22×C414yes—
141C22×C4×C127C22×C412yes—
142C22×C4×C127C2×D412yes—
143C22×C4×C127C2×D414yes—
144C22×C4×C127C2×D411yescharacteristic
145C22×C4×C127C2×D414yes—
146C22×C4×C127C2×D414yes—
147C22×C4×C127C2×D412yes—
148C22×C4×C127C2411yescharacteristic
149C2×C8×C127—24no—
150C2×C8×C127—24no—
151C2×C8×C127—24no—
152C2×C8×C127—48no—
153C4⋊C4×C127—24no—
154C4⋊C4×C127—24no—
155C4⋊C4×C127—44no—
156C22⋊C4×C127—48no—
157C23×C127—24no—
158C23×C127—24no—
159C23×C127—28no—
160C23×C127—24no—
161C23×C127—22no—
162C23×C127—22no—
163C23×C127—28no—
164C23×C127—216no—
165C23×C127—28no—
166C23×C127—28no—
167C23×C127—44no—
168C23×C127—42no—
169C23×C127—42no—
170C23×C127—48no—
171C23×C127—48no—
172C23×C127—48no—
173C23×C127—416no—
174C23×C127—48no—
175C23×C127—416no—
176C23×C127C2×C22⋊C414yes—
177C23×C127C2×C22⋊C412yes—
178C23×C127#12014yes—
179C23×C127#13514yes—
180C23×C127AGL(1,ℤ/8)12yes—
181C23×C127#13712yes—
182C23×C127C22×D411yescharacteristic
183Q8×C127—24no—
184Q8×C127—28no—
185Q8×C127—216no—
186Q8×C127C2×C22⋊C418yescomplemented
187D4×C127—264no—
188D4×C127—24no—
189D4×C127—28no—
190D4×C127—216no—
191D4×C127C2×C22⋊C418yescomplemented
192C2×C4×C127—264no—
193C2×C4×C127—24no—
194C2×C4×C127—24no—
195C2×C4×C127—28no—
196C2×C4×C127—28no—
197C2×C4×C127—216no—
198C2×C4×C127—24no—
199C2×C4×C127—24no—
200C2×C4×C127—28no—
201C2×C4×C127—216no—
202C2×C4×C127—44no—
203C2×C4×C127—44no—
204C2×C4×C127—44no—
205C2×C4×C127—88no—
206C2×C4×C127C2×C22⋊C414yes—
207C2×C4×C127C2×C22⋊C414yes—
208C2×C4×C127C2×C22⋊C414yes—
209C2×C4×C127C2×C22⋊C412yes—
210C2×C4×C127#12018yes—
211C2×C4×C127#12014yes—
212C2×C4×C127C23×C411yescharacteristic
213C2×C4×C127C22×D412yes—
214C2×C4×C127C22×D412yes—
215C2×C4×C127C22×D411yescharacteristic
216C8×C127—44no—
217C22×C127—24no—
218C22×C127—24no—
219C22×C127—24no—
220C22×C127—28no—
221C22×C127—44no—
222C22×C127—44no—
223C22×C127—44no—
224C22×C127—48no—
225C22×C127—44no—
226C22×C127—48no—
227C22×C127—416no—
228C22×C127—432no—
229C22×C127#37914yes—
230C22×C127#41614yes—
231C22×C127#41714yes—
232C22×C127#44918yes—
233C22×C127#45218yes—
234C22×C127C22×C22⋊C411yescharacteristic
235C22×C127C2×#12012yes—
236C22×C127C2×#13512yes—
237C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C127C2×#13711yescharacteristic
239C4×C127—24no—
240C4×C127—28no—
241C4×C127—84no—
242C4×C127#37914yes—
243C4×C127C22×C22⋊C411yescharacteristic
244C4×C127C22×C22⋊C411yescharacteristic
245C4×C127C2×#12012yes—
246C2×#1788C12711yesmaximal, characteristic, direct product factor, Sylow
247C2×C127—44no—
248C2×C127—48no—
249C2×C127#178818yesdirect product factor
250C2×C127C2×#37911yescharacteristic
251C2×C127C2×#41611yescharacteristic
252C2×C127C2×#41711yescharacteristic
253C2×C127C2×#44912yes—
254C2×C127C2×#45212yes—
255C23×#41C2×C12711yescharacteristic
256C22×#103C2×C12712yescomplemented
257C22×#102C2×C12712yescomplemented
258C22×C22⋊C8C2×C12711yescharacteristic, complemented
259C2×#380C2×C12711yescharacteristic, complemented
260#1788C2×C12718yesdirect product factor
261C127C2×#178811yescharacteristic, direct product factor, Sylow
262C22×#41—28no—
263C22×#41—28no—
264C22×#41—24no—
265C22×#41—22no—
266C22×#41C4×C12718yescomplemented
267C22×#41C22×C12714yes—
268C23×Q8C22×C12711yescharacteristic
269C23×D4—21no—
270C23×D4C22×C12711yescharacteristic
271C24×C4—21no—
272C24×C4C22×C12711yescharacteristic
273C23×C8C22×C12712yes—
274C22×C4⋊C4C22×C12712yes—
275C22×C22⋊C4—21no—
276C2×#103—28no—
277C2×#103C22×C12718yescomplemented
278C2×#102—28no—
279C2×#102C22×C12718yescomplemented
280C2×C22⋊C8—24no—
281C2×C22⋊C8C22×C12714yescomplemented
282#380—22no—
283#380C22×C12714yescomplemented
284C2×#95—21no—
285C25—21no—
286C25D4×C12711yescharacteristic
287C2×#41—232no—
288C2×#41—264no—
289C2×#41—232no—
290C2×#41—216no—
291C2×#41—232no—
292C2×#41—216no—
293C2×#41—216no—
294C2×#41—28no—
295C2×#41C2×C4×C127116yescomplemented
296C22×Q8—22no—
297C22×Q8—24no—
298C22×Q8—21no—
299C22×Q8C2×C4×C12714yes—
300C22×Q8D4×C12718yes—
301C22×Q8C23×C12712yes—
302C22×D4—216no—
303C22×D4—28no—
304C22×D4—24no—
305C22×D4—21no—
306C22×D4—24no—
307C22×D4—22no—
308C22×D4C2×C4×C12714yes—
309C22×D4D4×C12718yes—
310C22×D4C23×C12712yes—
311C23×C4—216no—
312C23×C4—21no—
313C23×C4—28no—
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—24no—
317C23×C4—22no—
318C23×C4C2×C4×C12714yes—
319C23×C4D4×C12711yescharacteristic
320C23×C4D4×C12712yes—
321C23×C4D4×C12714yes—
322C23×C4C23×C12712yes—
323C23×C4C23×C12711yescharacteristic
324C22×C8—24no—
325C22×C8—24no—
326C22×C8—24no—
327C22×C8C23×C12714yes—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4C23×C12714yes—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—44no—
334#103—216no—
335#102—216no—
336C22⋊C8—28no—
337#95—24no—
338C24—24no—
339C24—24no—
340C24—28no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—44no—
346C24—42no—
347C24—42no—
348C24C22⋊C4×C12714yes—
349C24C2×D4×C12712yes—
350C24C2×D4×C12711yescharacteristic
351#41—264no—
352#41—2128no—
353#41—232no—
354#41—232no—
355C2×Q8—24no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—216no—
359C2×Q8—22no—
360C2×Q8—24no—
361C2×Q8C22⋊C4×C127116yescomplemented
362C2×Q8C22×C4×C12714yes—
363C2×Q8C2×D4×C12718yes—
364C2×D4—232no—
365C2×D4—264no—
366C2×D4—216no—
367C2×D4—28no—
368C2×D4—24no—
369C2×D4—216no—
370C2×D4—28no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4C22⋊C4×C127116yescomplemented
374C2×D4C22×C4×C12714yes—
375C2×D4C2×D4×C12718yes—
376C22×C4—232no—
377C22×C4—264no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4C22⋊C4×C12714yes—
394C22×C4C22⋊C4×C12718yes—
395C22×C4C22×C4×C12714yes—
396C22×C4C22×C4×C12712yes—
397C22×C4C2×D4×C12712yes—
398C22×C4C2×D4×C12714yes—
399C22×C4C2×D4×C12711yescharacteristic
400C22×C4C2×D4×C12714yes—
401C22×C4C2×D4×C12714yes—
402C22×C4C2×D4×C12712yes—
403C22×C4C24×C12711yescharacteristic
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—48no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—44no—
411C22⋊C4—48no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—28no—
419C23—216no—
420C23—28no—
421C23—28no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—48no—
426C23—48no—
427C23—48no—
428C23—416no—
429C23—48no—
430C23—416no—
431C23C2×C22⋊C4×C12714yes—
432C23C2×C22⋊C4×C12712yes—
433C23#120×C12714yes—
434C23#135×C12714yes—
435C23AGL(1,ℤ/8)×C12712yes—
436C23#137×C12712yes—
437C23C22×D4×C12711yescharacteristic
438Q8—24no—
439Q8—28no—
440Q8—216no—
441Q8C2×C22⋊C4×C12718yescomplemented
442D4—264no—
443D4—24no—
444D4—28no—
445D4—216no—
446D4C2×C22⋊C4×C12718yescomplemented
447C2×C4—264no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—216no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—216no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—88no—
461C2×C4C2×C22⋊C4×C12714yes—
462C2×C4C2×C22⋊C4×C12714yes—
463C2×C4C2×C22⋊C4×C12714yes—
464C2×C4C2×C22⋊C4×C12712yes—
465C2×C4#120×C12718yes—
466C2×C4#120×C12714yes—
467C2×C4C23×C4×C12711yescharacteristic
468C2×C4C22×D4×C12712yes—
469C2×C4C22×D4×C12712yes—
470C2×C4C22×D4×C12711yescharacteristic
471C8—44no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—28no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—48no—
480C22—44no—
481C22—48no—
482C22—416no—
483C22—432no—
484C22#379×C12714yes—
485C22#416×C12714yes—
486C22#417×C12714yes—
487C22#449×C12718yes—
488C22#452×C12718yes—
489C22C22×C22⋊C4×C12711yescharacteristic
490C22C2×#120×C12712yes—
491C22C2×#135×C12712yes—
492C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
493C22C2×#137×C12711yescharacteristic
494C4—24no—
495C4—28no—
496C4—84no—
497C4#379×C12714yes—
498C4C22×C22⋊C4×C12711yescharacteristic
499C4C22×C22⋊C4×C12711yescharacteristic
500C4C2×#120×C12712yes—
501C2—44no—
502C2—48no—
503C2C127×#178818yesdirect product factor
504C2C2×#379×C12711yescharacteristic
505C2C2×#416×C12711yescharacteristic
506C2C2×#417×C12711yescharacteristic
507C2C2×#449×C12712yes—
508C2C2×#452×C12712yes—
subgroupG/H
center, socleC24×C127C2×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC23×#41×C127C2
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
2. C22×#103×C127C2semidirect
3. C22×#102×C127C2semidirect
4. C22×C22⋊C8×C127C2semidirect
5. C2×#380×C127C2semidirect
6. C127×#1788C2direct (N × H)
11. C22×#41×C127C4semidirect
22. C2×#103×C127C22semidirect
24. C2×#102×C127C22semidirect
26. C2×C22⋊C8×C127C22semidirect
28. #380×C127C22semidirect
40. C2×#41×C127C2×C4semidirect
106. C2×Q8×C127C22⋊C4semidirect
118. C2×D4×C127C22⋊C4semidirect
186. Q8×C127C2×C22⋊C4semidirect
191. D4×C127C2×C22⋊C4semidirect
246. C2×#1788C127direct (N × H)
249. C2×C127#1788direct (N × H)
256. C22×#103C2×C127semidirect
257. C22×#102C2×C127semidirect
258. C22×C22⋊C8C2×C127semidirect
259. C2×#380C2×C127semidirect
260. #1788C2×C127direct (N × H)
261. C127C2×#1788direct (N × H)
266. C22×#41C4×C127semidirect
277. C2×#103C22×C127semidirect
279. C2×#102C22×C127semidirect
281. C2×C22⋊C8C22×C127semidirect
283. #380C22×C127semidirect
295. C2×#41C2×C4×C127semidirect
361. C2×Q8C22⋊C4×C127semidirect
373. C2×D4C22⋊C4×C127semidirect
441. Q8C2×C22⋊C4×C127semidirect
446. D4C2×C22⋊C4×C127semidirect
503. C2C127×#1788direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C22×#103×C1272
C22×#102×C1272
C22×C22⋊C8×C1272
C2×#380×C1272
C127×#17882
C2×#1788127