Finite Group Databasebeta

← C2×C127×#1897 #44584651

Subgroups — C2×C127×#1897 #44584651

Order 32512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525×C127C211yesmaximal, characteristic
2C22×#117×C127C211yesmaximal, characteristic, complemented
3C22×#104×C127C212yesmaximal, complemented
4C2×#407×C127C211yesmaximal, characteristic, complemented
5C127×#1897C218yesmaximal, direct product factor
6C2×#339×C127C212yesmaximal, complemented
7C22×#41×C127C2211yescharacteristic
8C24×C4×C127C2211yescharacteristic
9C22×M16×C127C2212yes—
10#525×C127—22no—
11#525×C127—22no—
12#525×C127C414yescomplemented
13#525×C127C2214yes—
14C2×#123×C127—21no—
15C2×C22⋊Q8×C127C412yescomplemented
16C2×#121×C127C412yescomplemented
17C2×#120×C127—21no—
18C2×C4×D4×C127—22no—
19C2×#117×C127—22no—
20C2×#117×C127—28no—
21C2×#117×C127C2211yescharacteristic
22C2×#117×C127C2214yescomplemented
23C22×C4⋊C4×C127—21no—
24C22×C22⋊C4×C127—21no—
25C22×C42×C127C2212yes—
26C2×#104×C127—28no—
27C2×#104×C127C2218yescomplemented
28#407×C127—22no—
29#407×C127C2214yescomplemented
30C2×C22⋊C8×C127—22no—
31#339×C127C2218yescomplemented
32C25×C127D411yescharacteristic
33C2×#41×C127—21no—
34C2×#41×C127—21no—
35C2×#41×C127—24no—
36C2×#41×C127—24no—
37C2×#41×C127C2×C412yes—
38C2×#41×C127C2312yes—
39C22×Q8×C127C2×C411yescharacteristic
40C22×D4×C127—22no—
41C22×D4×C127—21no—
42C22×D4×C127C2×C411yescharacteristic
43C23×C4×C127—24no—
44C23×C4×C127—24no—
45C23×C4×C127—22no—
46C23×C4×C127—21no—
47C23×C4×C127C2×C412yes—
48C23×C4×C127C2×C412yes—
49C23×C4×C127D411yescharacteristic
50C23×C4×C127D412yes—
51C23×C4×C127D412yes—
52C23×C4×C127D412yes—
53C23×C4×C127C2312yes—
54C23×C4×C127C2311yescharacteristic
55C2×M16×C127—22no—
56C2×M16×C127—24no—
57C2×M16×C127—24no—
58C2×M16×C127C2314yes—
59C22×C8×C127—22no—
60#123×C127—24no—
61#123×C127—24no—
62C22⋊Q8×C127—24no—
63C22⋊Q8×C127C2×C418yescomplemented
64#121×C127—24no—
65#121×C127C2×C418yescomplemented
66#120×C127—24no—
67#120×C127—24no—
68C4×D4×C127—28no—
69C4×D4×C127—28no—
70#117×C127—216no—
71#117×C127—21no—
72#117×C127—21no—
73#117×C127—216no—
74#117×C127C2×C412yes—
75#117×C127C2312yes—
76C2×C4⋊C4×C127—28no—
77C2×C4⋊C4×C127—22no—
78C2×C4⋊C4×C127—24no—
79C2×C4⋊C4×C127C2×C412yes—
80C2×C22⋊C4×C127—28no—
81C2×C22⋊C4×C127—22no—
82C2×C22⋊C4×C127—24no—
83C2×C22⋊C4×C127—21no—
84C2×C22⋊C4×C127—22no—
85C2×C42×C127—22no—
86C2×C42×C127—24no—
87C2×C42×C127—21no—
88C2×C42×C127D418yescomplemented
89C2×C42×C127C2314yes—
90#104×C127—216no—
91C22⋊C8×C127—28no—
92C24×C127—22no—
93C24×C127—22no—
94C24×C127—21no—
95C24×C127—44no—
96C24×C127C22⋊C412yes—
97C24×C127C22⋊C412yes—
98C24×C127C2×D411yescharacteristic
99C24×C127C2×D412yes—
100#41×C127—28no—
101#41×C127—24no—
102#41×C127—24no—
103#41×C127—42no—
104#41×C127—44no—
105#41×C127—42no—
106C2×Q8×C127—21no—
107C2×Q8×C127—24no—
108C2×Q8×C127C22×C412yes—
109C2×D4×C127—28no—
110C2×D4×C127—22no—
111C2×D4×C127—22no—
112C2×D4×C127—21no—
113C2×D4×C127—24no—
114C2×D4×C127—48no—
115C2×D4×C127—42no—
116C2×D4×C127—42no—
117C2×D4×C127C22×C412yes—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—28no—
121C22×C4×C127—28no—
122C22×C4×C127—24no—
123C22×C4×C127—28no—
124C22×C4×C127—28no—
125C22×C4×C127—24no—
126C22×C4×C127—24no—
127C22×C4×C127—24no—
128C22×C4×C127—22no—
129C22×C4×C127—21no—
130C22×C4×C127—28no—
131C22×C4×C127—24no—
132C22×C4×C127—22no—
133C22×C4×C127—22no—
134C22×C4×C127—48no—
135C22×C4×C127—44no—
136C22×C4×C127C22⋊C412yes—
137C22×C4×C127C22⋊C414yes—
138C22×C4×C127C22⋊C414yes—
139C22×C4×C127C22⋊C412yes—
140C22×C4×C127C22×C414yes—
141C22×C4×C127C22×C411yescharacteristic
142C22×C4×C127C22×C411yescharacteristic
143C22×C4×C127C2×D412yes—
144C22×C4×C127C2×D414yes—
145C22×C4×C127C2×D411yescharacteristic
146C22×C4×C127C2×D411yescharacteristic
147C22×C4×C127C2×D414yes—
148C22×C4×C127C2×D411yescharacteristic
149C22×C4×C127#4114yes—
150C22×C4×C127C2411yescharacteristic
151M16×C127—24no—
152M16×C127—24no—
153M16×C127—44no—
154C2×C8×C127—24no—
155C2×C8×C127—44no—
156C4⋊C4×C127—216no—
157C4⋊C4×C127—28no—
158C4⋊C4×C127—22no—
159C4⋊C4×C127C22×C414yes—
160C22⋊C4×C127—216no—
161C22⋊C4×C127—22no—
162C22⋊C4×C127—28no—
163C22⋊C4×C127—22no—
164C22⋊C4×C127—28no—
165C42×C127—22no—
166C42×C127—28no—
167C42×C127—24no—
168C42×C127—22no—
169C42×C127C2×D418yescomplemented
170C23×C127—24no—
171C23×C127—22no—
172C23×C127—22no—
173C23×C127—24no—
174C23×C127—24no—
175C23×C127—24no—
176C23×C127—24no—
177C23×C127—22no—
178C23×C127—44no—
179C23×C127—44no—
180C23×C127—48no—
181C23×C127—48no—
182C23×C127—42no—
183C23×C127—44no—
184C23×C127C2×C22⋊C414yes—
185C23×C127C2×C22⋊C411yescharacteristic
186C23×C127C2×C22⋊C411yescharacteristic
187C23×C127#12012yes—
188C23×C127#12312yes—
189C23×C127C22×D411yescharacteristic
190Q8×C127—24no—
191Q8×C127—42no—
192D4×C127—24no—
193D4×C127—416no—
194D4×C127—42no—
195D4×C127—42no—
196D4×C127—44no—
197D4×C127—42no—
198C2×C4×C127—22no—
199C2×C4×C127—22no—
200C2×C4×C127—24no—
201C2×C4×C127—24no—
202C2×C4×C127—28no—
203C2×C4×C127—28no—
204C2×C4×C127—28no—
205C2×C4×C127—28no—
206C2×C4×C127—24no—
207C2×C4×C127—22no—
208C2×C4×C127—28no—
209C2×C4×C127—48no—
210C2×C4×C127—416no—
211C2×C4×C127—42no—
212C2×C4×C127—44no—
213C2×C4×C127—44no—
214C2×C4×C127—44no—
215C2×C4×C127C2×C22⋊C414yes—
216C2×C4×C127C2×C22⋊C411yescharacteristic
217C2×C4×C127C2×C22⋊C411yescharacteristic
218C2×C4×C127C4×D414yes—
219C2×C4×C127C4×D414yes—
220C2×C4×C127#12012yes—
221C2×C4×C127#12114yes—
222C2×C4×C127#12114yes—
223C2×C4×C127#12312yes—
224C2×C4×C127C23×C411yescharacteristic
225C2×C4×C127C22×D412yes—
226C2×C4×C127C22×D411yescharacteristic
227C2×C4×C127C2×#4112yes—
228C8×C127—44no—
229C22×C127—22no—
230C22×C127—22no—
231C22×C127—24no—
232C22×C127—24no—
233C22×C127—24no—
234C22×C127—24no—
235C22×C127—24no—
236C22×C127—44no—
237C22×C127—44no—
238C22×C127—48no—
239C22×C127—48no—
240C22×C127—44no—
241C22×C127—42no—
242C22×C127—84no—
243C22×C127#38612yes—
244C22×C127#38612yes—
245C22×C127#42114yes—
246C22×C127C22×C22⋊C411yescharacteristic
247C22×C127C2×#12011yescharacteristic
248C22×C127C2×#12311yescharacteristic
249C4×C127—24no—
250C4×C127—48no—
251C4×C127—44no—
252C4×C127—42no—
253C4×C127#38612yes—
254C4×C127#38612yes—
255C4×C127C2×C4×D412yes—
256C4×C127C2×#12112yes—
257C2×#1897C12711yesmaximal, characteristic, direct product factor, Sylow
258C2×C127—21no—
259C2×C127—21no—
260C2×C127—22no—
261C2×C127—44no—
262C2×C127—82no—
263C2×C127#189714yesdirect product factor
264C2×C127C2×#38611yescharacteristic
265C2×C127C2×#42112yes—
266C2×#525C2×C12711yescharacteristic
267C22×#117C2×C12711yescharacteristic, complemented
268C22×#104C2×C12712yescomplemented
269C2×#407C2×C12711yescharacteristic, complemented
270#1897C2×C12718yesdirect product factor
271C2×#339C2×C12712yescomplemented
272C127C2×#189711yescharacteristic, direct product factor, Sylow
273C22×#41C22×C12711yescharacteristic
274C24×C4C22×C12711yescharacteristic
275C22×M16C22×C12712yes—
276#525—22no—
277#525—22no—
278#525C4×C12714yescomplemented
279#525C22×C12714yes—
280C2×#123—21no—
281C2×C22⋊Q8C4×C12712yescomplemented
282C2×#121C4×C12712yescomplemented
283C2×#120—21no—
284C2×C4×D4—22no—
285C2×#117—22no—
286C2×#117—28no—
287C2×#117C22×C12711yescharacteristic
288C2×#117C22×C12714yescomplemented
289C22×C4⋊C4—21no—
290C22×C22⋊C4—21no—
291C22×C42C22×C12712yes—
292C2×#104—28no—
293C2×#104C22×C12718yescomplemented
294#407—22no—
295#407C22×C12714yescomplemented
296C2×C22⋊C8—22no—
297#339C22×C12718yescomplemented
298C25D4×C12711yescharacteristic
299C2×#41—21no—
300C2×#41—21no—
301C2×#41—24no—
302C2×#41—24no—
303C2×#41C2×C4×C12712yes—
304C2×#41C23×C12712yes—
305C22×Q8C2×C4×C12711yescharacteristic
306C22×D4—22no—
307C22×D4—21no—
308C22×D4C2×C4×C12711yescharacteristic
309C23×C4—24no—
310C23×C4—24no—
311C23×C4—22no—
312C23×C4—21no—
313C23×C4C2×C4×C12712yes—
314C23×C4C2×C4×C12712yes—
315C23×C4D4×C12711yescharacteristic
316C23×C4D4×C12712yes—
317C23×C4D4×C12712yes—
318C23×C4D4×C12712yes—
319C23×C4C23×C12712yes—
320C23×C4C23×C12711yescharacteristic
321C2×M16—22no—
322C2×M16—24no—
323C2×M16—24no—
324C2×M16C23×C12714yes—
325C22×C8—22no—
326#123—24no—
327#123—24no—
328C22⋊Q8—24no—
329C22⋊Q8C2×C4×C12718yescomplemented
330#121—24no—
331#121C2×C4×C12718yescomplemented
332#120—24no—
333#120—24no—
334C4×D4—28no—
335C4×D4—28no—
336#117—216no—
337#117—21no—
338#117—21no—
339#117—216no—
340#117C2×C4×C12712yes—
341#117C23×C12712yes—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4C2×C4×C12712yes—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C42—22no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42D4×C12718yescomplemented
355C2×C42C23×C12714yes—
356#104—216no—
357C22⋊C8—28no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—44no—
362C24C22⋊C4×C12712yes—
363C24C22⋊C4×C12712yes—
364C24C2×D4×C12711yescharacteristic
365C24C2×D4×C12712yes—
366#41—28no—
367#41—24no—
368#41—24no—
369#41—42no—
370#41—44no—
371#41—42no—
372C2×Q8—21no—
373C2×Q8—24no—
374C2×Q8C22×C4×C12712yes—
375C2×D4—28no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—24no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4C22×C4×C12712yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—48no—
401C22×C4—44no—
402C22×C4C22⋊C4×C12712yes—
403C22×C4C22⋊C4×C12714yes—
404C22×C4C22⋊C4×C12714yes—
405C22×C4C22⋊C4×C12712yes—
406C22×C4C22×C4×C12714yes—
407C22×C4C22×C4×C12711yescharacteristic
408C22×C4C22×C4×C12711yescharacteristic
409C22×C4C2×D4×C12712yes—
410C22×C4C2×D4×C12714yes—
411C22×C4C2×D4×C12711yescharacteristic
412C22×C4C2×D4×C12711yescharacteristic
413C22×C4C2×D4×C12711yescharacteristic
414C22×C4C2×D4×C12714yes—
415C22×C4#41×C12714yes—
416C22×C4C24×C12711yescharacteristic
417M16—24no—
418M16—24no—
419M16—44no—
420C2×C8—24no—
421C2×C8—44no—
422C4⋊C4—216no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4C22×C4×C12714yes—
426C22⋊C4—216no—
427C22⋊C4—22no—
428C22⋊C4—28no—
429C22⋊C4—22no—
430C22⋊C4—28no—
431C42—22no—
432C42—28no—
433C42—24no—
434C42—22no—
435C42C2×D4×C12718yescomplemented
436C23—24no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—22no—
444C23—44no—
445C23—44no—
446C23—48no—
447C23—48no—
448C23—42no—
449C23—44no—
450C23C2×C22⋊C4×C12714yes—
451C23C2×C22⋊C4×C12711yescharacteristic
452C23C2×C22⋊C4×C12711yescharacteristic
453C23#120×C12712yes—
454C23#123×C12712yes—
455C23C22×D4×C12711yescharacteristic
456Q8—24no—
457Q8—42no—
458D4—24no—
459D4—416no—
460D4—42no—
461D4—42no—
462D4—44no—
463D4—42no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—28no—
475C2×C4—48no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4C2×C22⋊C4×C12714yes—
482C2×C4C2×C22⋊C4×C12711yescharacteristic
483C2×C4C2×C22⋊C4×C12711yescharacteristic
484C2×C4C4×D4×C12714yes—
485C2×C4C4×D4×C12714yes—
486C2×C4#120×C12712yes—
487C2×C4#121×C12714yes—
488C2×C4#121×C12714yes—
489C2×C4#123×C12712yes—
490C2×C4C23×C4×C12711yescharacteristic
491C2×C4C22×D4×C12712yes—
492C2×C4C22×D4×C12711yescharacteristic
493C2×C4C2×#41×C12712yes—
494C8—44no—
495C22—22no—
496C22—22no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—44no—
503C22—44no—
504C22—48no—
505C22—48no—
506C22—44no—
507C22—42no—
508C22—84no—
509C22#386×C12712yes—
510C22#386×C12712yes—
511C22#421×C12714yes—
512C22C22×C22⋊C4×C12711yescharacteristic
513C22C2×#120×C12711yescharacteristic
514C22C2×#123×C12711yescharacteristic
515C4—24no—
516C4—48no—
517C4—44no—
518C4—42no—
519C4#386×C12712yes—
520C4#386×C12712yes—
521C4C2×C4×D4×C12712yes—
522C4C2×#121×C12712yes—
523C2—21no—
524C2—21no—
525C2—22no—
526C2—44no—
527C2—82no—
528C2C127×#189714yesdirect product factor
529C2C2×#386×C12711yescharacteristic
530C2C2×#421×C12712yes—
subgroupG/H
centerC22×C4×C127C2×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C2×C22⋊C4
prime radicalC2×#525×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#117×C127C2semidirect
3. C22×#104×C127C2semidirect
4. C2×#407×C127C2semidirect
5. C127×#1897C2direct (N × H)
6. C2×#339×C127C2semidirect
12. #525×C127C4semidirect
15. C2×C22⋊Q8×C127C4semidirect
16. C2×#121×C127C4semidirect
22. C2×#117×C127C22semidirect
27. C2×#104×C127C22semidirect
29. #407×C127C22semidirect
31. #339×C127C22semidirect
63. C22⋊Q8×C127C2×C4semidirect
65. #121×C127C2×C4semidirect
88. C2×C42×C127D4semidirect
169. C42×C127C2×D4semidirect
257. C2×#1897C127direct (N × H)
263. C2×C127#1897direct (N × H)
267. C22×#117C2×C127semidirect
268. C22×#104C2×C127semidirect
269. C2×#407C2×C127semidirect
270. #1897C2×C127direct (N × H)
271. C2×#339C2×C127semidirect
272. C127C2×#1897direct (N × H)
278. #525C4×C127semidirect
281. C2×C22⋊Q8C4×C127semidirect
282. C2×#121C4×C127semidirect
288. C2×#117C22×C127semidirect
293. C2×#104C22×C127semidirect
295. #407C22×C127semidirect
297. #339C22×C127semidirect
329. C22⋊Q8C2×C4×C127semidirect
331. #121C2×C4×C127semidirect
354. C2×C42D4×C127semidirect
435. C42C2×D4×C127semidirect
528. C2C127×#1897direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#525×C1272
C22×#117×C1272
C22×#104×C1272
C2×#407×C1272
C127×#18972
C2×#339×C1272
C2×#1897127