Finite Group Databasebeta

← C2×C127×#1912 #44584684

Subgroups — C2×C127×#1912 #44584684

Order 32512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C127C211yesmaximal, characteristic
2C2×#519×C127C211yesmaximal, characteristic, complemented
3C2×#421×C127C212yesmaximal
4C22×#101×C127C211yesmaximal, characteristic, complemented
5C127×#1912C218yesmaximal, direct product factor
6C2×H(3,ℤ/4)×C127C212yesmaximal, complemented
7C22×#41×C127C2211yescharacteristic
8C23×Q8×C127C2211yescharacteristic
9C22×M16×C127C2212yes—
10#536×C127—22no—
11#536×C127—22no—
12#536×C127C414yescomplemented
13#536×C127C2214yes—
14C2×C4⋊Q8×C127C412yescomplemented
15C2×#124×C127C412yescomplemented
16C2×#123×C127—22no—
17C2×C22⋊Q8×C127—22no—
18#519×C127—22no—
19#519×C127—24no—
20#519×C127C2214yescomplemented
21C2×C4×Q8×C127—24no—
22C2×#117×C127—22no—
23C2×#117×C127C2211yescharacteristic
24C2×#117×C127C2212yes—
25#421×C127—24no—
26#421×C127C2218yes—
27C2×#104×C127—24no—
28C2×#101×C127—22no—
29C2×#101×C127—22no—
30C2×#101×C127C2214yescomplemented
31H(3,ℤ/4)×C127C2218yescomplemented
32C2×#41×C127—21no—
33C2×#41×C127—21no—
34C2×#41×C127—22no—
35C2×#41×C127—22no—
36C2×#41×C127—22no—
37C2×#41×C127—22no—
38C2×#41×C127C2×C412yes—
39C2×#41×C127C2312yes—
40C22×Q8×C127—22no—
41C22×Q8×C127—22no—
42C22×Q8×C127—22no—
43C22×Q8×C127—22no—
44C22×Q8×C127C2×C411yescharacteristic
45C22×Q8×C127C2×C412yes—
46C22×Q8×C127C2×C412yes—
47C22×Q8×C127D412yes—
48C22×Q8×C127D412yes—
49C22×Q8×C127D412yes—
50C22×Q8×C127C2312yes—
51C22×D4×C127—21no—
52C22×D4×C127C2×C411yescharacteristic
53C23×C4×C127—21no—
54C23×C4×C127D412yes—
55C23×C4×C127C2311yescharacteristic
56C2×M16×C127—22no—
57C2×M16×C127—22no—
58C2×M16×C127—22no—
59C2×M16×C127—22no—
60C2×M16×C127—22no—
61C2×M16×C127C2314yes—
62C22×C8×C127—22no—
63C4⋊Q8×C127—24no—
64C4⋊Q8×C127C2×C418yescomplemented
65#124×C127—24no—
66#124×C127C2×C418yescomplemented
67#123×C127—28no—
68#123×C127—28no—
69C22⋊Q8×C127—28no—
70C22⋊Q8×C127—28no—
71C4×Q8×C127—216no—
72C4×Q8×C127—216no—
73#117×C127—28no—
74#117×C127—28no—
75#117×C127—22no—
76#117×C127—22no—
77#117×C127—21no—
78#117×C127—21no—
79#117×C127C2×C412yes—
80#117×C127D414yes—
81#117×C127C2312yes—
82#117×C127C2314yes—
83C2×C4⋊C4×C127—24no—
84C2×C4⋊C4×C127—24no—
85C2×C4⋊C4×C127—22no—
86C2×C4⋊C4×C127—21no—
87C2×C22⋊C4×C127—22no—
88C2×C22⋊C4×C127—21no—
89C2×C22⋊C4×C127—24no—
90C2×C42×C127—24no—
91C2×C42×C127C2×C412yes—
92C2×C42×C127D414yes—
93#104×C127—216no—
94#101×C127—28no—
95C24×C127—21no—
96C24×C127C2×D411yescharacteristic
97#41×C127—24no—
98#41×C127—24no—
99#41×C127—24no—
100#41×C127—24no—
101#41×C127—42no—
102#41×C127—42no—
103#41×C127—42no—
104#41×C127—42no—
105C2×Q8×C127—21no—
106C2×Q8×C127—24no—
107C2×Q8×C127—24no—
108C2×Q8×C127—24no—
109C2×Q8×C127—24no—
110C2×Q8×C127—24no—
111C2×Q8×C127—24no—
112C2×Q8×C127—24no—
113C2×Q8×C127—24no—
114C2×Q8×C127—24no—
115C2×Q8×C127—24no—
116C2×Q8×C127—24no—
117C2×Q8×C127—24no—
118C2×Q8×C127—22no—
119C2×Q8×C127—22no—
120C2×Q8×C127C22⋊C414yes—
121C2×Q8×C127C22⋊C414yes—
122C2×Q8×C127C22×C412yes—
123C2×Q8×C127C22×C414yes—
124C2×Q8×C127C2×D414yes—
125C2×D4×C127—22no—
126C2×D4×C127—22no—
127C2×D4×C127—21no—
128C2×D4×C127—22no—
129C2×D4×C127—22no—
130C2×D4×C127—41no—
131C2×D4×C127—41no—
132C2×D4×C127—41no—
133C2×D4×C127—41no—
134C2×D4×C127C22×C412yes—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—22no—
140C22×C4×C127—21no—
141C22×C4×C127—24no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—42no—
145C22×C4×C127—42no—
146C22×C4×C127C22⋊C414yes—
147C22×C4×C127C22⋊C414yes—
148C22×C4×C127C22×C411yescharacteristic
149C22×C4×C127C22×C411yescharacteristic
150C22×C4×C127C2×D414yes—
151C22×C4×C127C2×D411yescharacteristic
152C22×C4×C127C2×D411yescharacteristic
153C22×C4×C127C2×D411yescharacteristic
154C22×C4×C127C2×D412yes—
155C22×C4×C127C2×D412yes—
156C22×C4×C127#4112yes—
157C22×C4×C127#4112yes—
158C22×C4×C127C2411yescharacteristic
159M16×C127—24no—
160M16×C127—24no—
161M16×C127—42no—
162M16×C127—42no—
163C2×C8×C127—24no—
164C2×C8×C127—42no—
165C2×C8×C127—42no—
166C4⋊C4×C127—216no—
167C4⋊C4×C127—22no—
168C4⋊C4×C127—216no—
169C4⋊C4×C127—24no—
170C4⋊C4×C127—24no—
171C4⋊C4×C127—22no—
172C22⋊C4×C127—22no—
173C22⋊C4×C127—24no—
174C22⋊C4×C127—24no—
175C22⋊C4×C127—22no—
176C22⋊C4×C127—216no—
177C42×C127—216no—
178C42×C127—22no—
179C42×C127—24no—
180C42×C127C22×C414yes—
181C42×C127C2×D418yes—
182C23×C127—22no—
183C23×C127—42no—
184C23×C127—44no—
185C23×C127—41no—
186C23×C127—41no—
187C23×C127C2×C22⋊C411yescharacteristic
188C23×C127C2×C22⋊C411yescharacteristic
189C23×C127#12011yescharacteristic
190C23×C127#12011yescharacteristic
191C23×C127#12311yescharacteristic
192C23×C127#12311yescharacteristic
193C23×C127C22×D411yescharacteristic
194Q8×C127—28no—
195Q8×C127—28no—
196Q8×C127—28no—
197Q8×C127—28no—
198Q8×C127—24no—
199Q8×C127—42no—
200D4×C127—24no—
201D4×C127—42no—
202D4×C127—42no—
203D4×C127—42no—
204D4×C127—42no—
205D4×C127—42no—
206C2×C4×C127—24no—
207C2×C4×C127—24no—
208C2×C4×C127—24no—
209C2×C4×C127—24no—
210C2×C4×C127—24no—
211C2×C4×C127—24no—
212C2×C4×C127—24no—
213C2×C4×C127—22no—
214C2×C4×C127—28no—
215C2×C4×C127—48no—
216C2×C4×C127—44no—
217C2×C4×C127—41no—
218C2×C4×C127—41no—
219C2×C4×C127—44no—
220C2×C4×C127—44no—
221C2×C4×C127C2×C22⋊C418yes—
222C2×C4×C127C2×C22⋊C411yescharacteristic
223C2×C4×C127C2×C22⋊C411yescharacteristic
224C2×C4×C127C4×D412yes—
225C2×C4×C127C4×D412yes—
226C2×C4×C127C4×D412yes—
227C2×C4×C127C4×D412yes—
228C2×C4×C127#12012yes—
229C2×C4×C127#12112yes—
230C2×C4×C127#12112yes—
231C2×C4×C127#12112yes—
232C2×C4×C127#12112yes—
233C2×C4×C127#12312yes—
234C2×C4×C127C23×C411yescharacteristic
235C2×C4×C127C22×D412yes—
236C2×C4×C127C22×D411yescharacteristic
237C2×C4×C127C2×#4112yes—
238C8×C127—44no—
239C22×C127—22no—
240C22×C127—22no—
241C22×C127—22no—
242C22×C127—41no—
243C22×C127—41no—
244C22×C127—41no—
245C22×C127—41no—
246C22×C127—44no—
247C22×C127—42no—
248C22×C127—82no—
249C22×C127—82no—
250C22×C127#38611yescharacteristic
251C22×C127#38611yescharacteristic
252C22×C127#38611yescharacteristic
253C22×C127#38611yescharacteristic
254C22×C127C22×C22⋊C411yescharacteristic
255C22×C127C2×#12011yescharacteristic
256C22×C127C2×#12311yescharacteristic
257C4×C127—28no—
258C4×C127—48no—
259C4×C127—44no—
260C4×C127—42no—
261C4×C127#38612yes—
262C4×C127#38612yes—
263C4×C127C2×C4×D412yes—
264C4×C127C2×#12112yes—
265C2×#1912C12711yesmaximal, characteristic, direct product factor, Sylow
266C2×C127—21no—
267C2×C127—21no—
268C2×C127—21no—
269C2×C127—21no—
270C2×C127—21no—
271C2×C127—21no—
272C2×C127—82no—
273C2×C127#191212yesdirect product factor
274C2×C127C2×#38611yescharacteristic
275C2×#536C2×C12711yescharacteristic
276C2×#519C2×C12711yescharacteristic, complemented
277C2×#421C2×C12712yes—
278C22×#101C2×C12711yescharacteristic, complemented
279#1912C2×C12718yesdirect product factor
280C2×H(3,ℤ/4)C2×C12712yescomplemented
281C127C2×#191211yescharacteristic, direct product factor, Sylow
282C22×#41C22×C12711yescharacteristic
283C23×Q8C22×C12711yescharacteristic
284C22×M16C22×C12712yes—
285#536—22no—
286#536—22no—
287#536C4×C12714yescomplemented
288#536C22×C12714yes—
289C2×C4⋊Q8C4×C12712yescomplemented
290C2×#124C4×C12712yescomplemented
291C2×#123—22no—
292C2×C22⋊Q8—22no—
293#519—22no—
294#519—24no—
295#519C22×C12714yescomplemented
296C2×C4×Q8—24no—
297C2×#117—22no—
298C2×#117C22×C12711yescharacteristic
299C2×#117C22×C12712yes—
300#421—24no—
301#421C22×C12718yes—
302C2×#104—24no—
303C2×#101—22no—
304C2×#101—22no—
305C2×#101C22×C12714yescomplemented
306H(3,ℤ/4)C22×C12718yescomplemented
307C2×#41—21no—
308C2×#41—21no—
309C2×#41—22no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41C2×C4×C12712yes—
314C2×#41C23×C12712yes—
315C22×Q8—22no—
316C22×Q8—22no—
317C22×Q8—22no—
318C22×Q8—22no—
319C22×Q8C2×C4×C12711yescharacteristic
320C22×Q8C2×C4×C12712yes—
321C22×Q8C2×C4×C12712yes—
322C22×Q8D4×C12712yes—
323C22×Q8D4×C12712yes—
324C22×Q8D4×C12712yes—
325C22×Q8C23×C12712yes—
326C22×D4—21no—
327C22×D4C2×C4×C12711yescharacteristic
328C23×C4—21no—
329C23×C4D4×C12712yes—
330C23×C4C23×C12711yescharacteristic
331C2×M16—22no—
332C2×M16—22no—
333C2×M16—22no—
334C2×M16—22no—
335C2×M16—22no—
336C2×M16C23×C12714yes—
337C22×C8—22no—
338C4⋊Q8—24no—
339C4⋊Q8C2×C4×C12718yescomplemented
340#124—24no—
341#124C2×C4×C12718yescomplemented
342#123—28no—
343#123—28no—
344C22⋊Q8—28no—
345C22⋊Q8—28no—
346C4×Q8—216no—
347C4×Q8—216no—
348#117—28no—
349#117—28no—
350#117—22no—
351#117—22no—
352#117—21no—
353#117—21no—
354#117C2×C4×C12712yes—
355#117D4×C12714yes—
356#117C23×C12712yes—
357#117C23×C12714yes—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—24no—
365C2×C42—24no—
366C2×C42C2×C4×C12712yes—
367C2×C42D4×C12714yes—
368#104—216no—
369#101—28no—
370C24—21no—
371C24C2×D4×C12711yescharacteristic
372#41—24no—
373#41—24no—
374#41—24no—
375#41—24no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22⋊C4×C12714yes—
396C2×Q8C22⋊C4×C12714yes—
397C2×Q8C22×C4×C12712yes—
398C2×Q8C22×C4×C12714yes—
399C2×Q8C2×D4×C12714yes—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4C22×C4×C12712yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C12714yes—
422C22×C4C22⋊C4×C12714yes—
423C22×C4C22×C4×C12711yescharacteristic
424C22×C4C22×C4×C12711yescharacteristic
425C22×C4C2×D4×C12714yes—
426C22×C4C2×D4×C12711yescharacteristic
427C22×C4C2×D4×C12711yescharacteristic
428C22×C4C2×D4×C12711yescharacteristic
429C22×C4C2×D4×C12712yes—
430C22×C4C2×D4×C12712yes—
431C22×C4#41×C12712yes—
432C22×C4#41×C12712yes—
433C22×C4C24×C12711yescharacteristic
434M16—24no—
435M16—24no—
436M16—42no—
437M16—42no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C4⋊C4—216no—
442C4⋊C4—22no—
443C4⋊C4—216no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—216no—
452C42—216no—
453C42—22no—
454C42—24no—
455C42C22×C4×C12714yes—
456C42C2×D4×C12718yes—
457C23—22no—
458C23—42no—
459C23—44no—
460C23—41no—
461C23—41no—
462C23C2×C22⋊C4×C12711yescharacteristic
463C23C2×C22⋊C4×C12711yescharacteristic
464C23#120×C12711yescharacteristic
465C23#120×C12711yescharacteristic
466C23#123×C12711yescharacteristic
467C23#123×C12711yescharacteristic
468C23C22×D4×C12711yescharacteristic
469Q8—28no—
470Q8—28no—
471Q8—28no—
472Q8—28no—
473Q8—24no—
474Q8—42no—
475D4—24no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4C2×C22⋊C4×C12718yes—
497C2×C4C2×C22⋊C4×C12711yescharacteristic
498C2×C4C2×C22⋊C4×C12711yescharacteristic
499C2×C4C4×D4×C12712yes—
500C2×C4C4×D4×C12712yes—
501C2×C4C4×D4×C12712yes—
502C2×C4C4×D4×C12712yes—
503C2×C4#120×C12712yes—
504C2×C4#121×C12712yes—
505C2×C4#121×C12712yes—
506C2×C4#121×C12712yes—
507C2×C4#121×C12712yes—
508C2×C4#123×C12712yes—
509C2×C4C23×C4×C12711yescharacteristic
510C2×C4C22×D4×C12712yes—
511C2×C4C22×D4×C12711yescharacteristic
512C2×C4C2×#41×C12712yes—
513C8—44no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—44no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22#386×C12711yescharacteristic
526C22#386×C12711yescharacteristic
527C22#386×C12711yescharacteristic
528C22#386×C12711yescharacteristic
529C22C22×C22⋊C4×C12711yescharacteristic
530C22C2×#120×C12711yescharacteristic
531C22C2×#123×C12711yescharacteristic
532C4—28no—
533C4—48no—
534C4—44no—
535C4—42no—
536C4#386×C12712yes—
537C4#386×C12712yes—
538C4C2×C4×D4×C12712yes—
539C4C2×#121×C12712yes—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—82no—
547C2C127×#191212yesdirect product factor
548C2C2×#386×C12711yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×#41×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#519×C127C2semidirect
4. C22×#101×C127C2semidirect
5. C127×#1912C2direct (N × H)
6. C2×H(3,ℤ/4)×C127C2semidirect
12. #536×C127C4semidirect
14. C2×C4⋊Q8×C127C4semidirect
15. C2×#124×C127C4semidirect
20. #519×C127C22semidirect
30. C2×#101×C127C22semidirect
31. H(3,ℤ/4)×C127C22semidirect
64. C4⋊Q8×C127C2×C4semidirect
66. #124×C127C2×C4semidirect
265. C2×#1912C127direct (N × H)
273. C2×C127#1912direct (N × H)
276. C2×#519C2×C127semidirect
278. C22×#101C2×C127semidirect
279. #1912C2×C127direct (N × H)
280. C2×H(3,ℤ/4)C2×C127semidirect
281. C127C2×#1912direct (N × H)
287. #536C4×C127semidirect
289. C2×C4⋊Q8C4×C127semidirect
290. C2×#124C4×C127semidirect
295. #519C22×C127semidirect
305. C2×#101C22×C127semidirect
306. H(3,ℤ/4)C22×C127semidirect
339. C4⋊Q8C2×C4×C127semidirect
341. #124C2×C4×C127semidirect
547. C2C127×#1912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#536×C1272
C2×#519×C1272
C2×#421×C1272
C22×#101×C1272
C127×#19122
C2×H(3,ℤ/4)×C1272
C2×#1912127