Finite Group Databasebeta

← C127×#17106 #44580779

Subgroups — C127×#17106 #44580779

Order 32512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#399×C127C211yesmaximal, characteristic
2C2×#397×C127C212yesmaximal
3C2×#395×C127C212yesmaximal, complemented
4C2×#393×C127C211yesmaximal, characteristic
5C127×#1444C212yesmaximal, complemented
6C127×#1443C212yesmaximal, complemented
7C127×#1442C212yesmaximal, complemented
8C127×#1440C211yesmaximal, characteristic
9C127×#1439C212yesmaximal
10C23×Q8×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2212yes—
13C22×C4⋊C4×C127C2212yes—
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2211yescharacteristic
17C22×C22⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2211yescharacteristic
19C22×C22⋊C4×C127C2211yescharacteristic
20#399×C127—24no—
21#397×C127—28no—
22#395×C127—28no—
23#393×C127—24no—
24C2×#95×C127C2211yescharacteristic
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2212yes—
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C25×C127C2311yescharacteristic
36C22×Q8×C127—22no—
37C22×Q8×C127—24no—
38C22×Q8×C127—41no—
39C22×Q8×C127—42no—
40C22×Q8×C127—41no—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2312yes—
47C23×C4×C127C2311yescharacteristic
48C23×C4×C127C2312yes—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—24no—
52C2×C4⋊C4×C127—24no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—42no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127—42no—
58C2×C4⋊C4×C127—42no—
59C2×C4⋊C4×C127—41no—
60C2×C4⋊C4×C127—41no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127—24no—
64C2×C22⋊C4×C127—24no—
65C2×C22⋊C4×C127—24no—
66C2×C22⋊C4×C127—24no—
67C2×C22⋊C4×C127—41no—
68C2×C22⋊C4×C127—41no—
69C2×C22⋊C4×C127—41no—
70C2×C22⋊C4×C127—41no—
71C2×C22⋊C4×C127—42no—
72C2×C22⋊C4×C127—42no—
73C2×C22⋊C4×C127D414yes—
74#95×C127—24no—
75#95×C127—22no—
76#95×C127—24no—
77#95×C127—24no—
78#95×C127—24no—
79#95×C127—24no—
80#95×C127—22no—
81#95×C127—24no—
82#95×C127—24no—
83#95×C127—42no—
84#95×C127D418yes—
85C24×C127—22no—
86C24×C127—22no—
87C24×C127—21no—
88C24×C127—21no—
89C24×C127—22no—
90C24×C127—22no—
91C24×C127—22no—
92C24×C127—22no—
93C24×C127C2×D412yes—
94C24×C127C2411yescharacteristic
95C2×Q8×C127—48no—
96C2×Q8×C127—44no—
97C2×Q8×C127—44no—
98C2×Q8×C127—48no—
99C2×Q8×C127—44no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—21no—
112C22×C4×C127—21no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—22no—
133C22×C4×C127—22no—
134C22×C4×C127—22no—
135C22×C4×C127—22no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—22no—
139C22×C4×C127—22no—
140C22×C4×C127—22no—
141C22×C4×C127—22no—
142C22×C4×C127—22no—
143C22×C4×C127—22no—
144C22×C4×C127—22no—
145C22×C4×C127—22no—
146C22×C4×C127C2×D412yes—
147C22×C4×C127C2×D414yes—
148C22×C4×C127C2×D414yes—
149C22×C4×C127C2×D412yes—
150C22×C4×C127C2×D414yes—
151C22×C4×C127#4114yes—
152C4⋊C4×C127—48no—
153C4⋊C4×C127—48no—
154C4⋊C4×C127—44no—
155C4⋊C4×C127—48no—
156C22⋊C4×C127—44no—
157C22⋊C4×C127—44no—
158C22⋊C4×C127—44no—
159C22⋊C4×C127—48no—
160C22⋊C4×C127—48no—
161C23×C127—22no—
162C23×C127—22no—
163C23×C127—22no—
164C23×C127—24no—
165C23×C127—24no—
166C23×C127—42no—
167C23×C127—42no—
168C23×C127—42no—
169C23×C127—42no—
170C23×C127—42no—
171C23×C127—42no—
172C23×C127—41no—
173C23×C127—42no—
174C23×C127—42no—
175C23×C127—41no—
176C23×C127—41no—
177C23×C127—42no—
178C23×C127—42no—
179C23×C127—42no—
180C23×C127—42no—
181C23×C127—41no—
182C23×C127C22×D411yescharacteristic
183C23×C127C22×D412yes—
184C23×C127C2×#4112yes—
185C23×C127H(5,𝔽2)11yescharacteristic
186C23×C127H(5,𝔽2)11yescharacteristic
187C23×C127H(5,𝔽2)12yes—
188C23×C1272−1+412yes—
189C23×C1272−1+412yes—
190C23×C1272−1+412yes—
191Q8×C127—416no—
192C2×C4×C127—22no—
193C2×C4×C127—22no—
194C2×C4×C127—24no—
195C2×C4×C127—24no—
196C2×C4×C127—24no—
197C2×C4×C127—24no—
198C2×C4×C127—24no—
199C2×C4×C127—24no—
200C2×C4×C127—24no—
201C2×C4×C127—24no—
202C2×C4×C127—24no—
203C2×C4×C127—24no—
204C2×C4×C127—24no—
205C2×C4×C127—24no—
206C2×C4×C127—24no—
207C2×C4×C127—24no—
208C2×C4×C127—42no—
209C2×C4×C127—42no—
210C2×C4×C127—42no—
211C2×C4×C127—42no—
212C2×C4×C127—42no—
213C2×C4×C127—42no—
214C2×C4×C127—42no—
215C2×C4×C127—42no—
216C2×C4×C127—42no—
217C2×C4×C127—42no—
218C2×C4×C127—41no—
219C2×C4×C127—42no—
220C2×C4×C127—42no—
221C2×C4×C127—42no—
222C2×C4×C127—42no—
223C2×C4×C127—41no—
224C2×C4×C127—41no—
225C2×C4×C127—42no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—41no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127—42no—
235C2×C4×C127—42no—
236C2×C4×C127—42no—
237C2×C4×C127—42no—
238C2×C4×C127—42no—
239C2×C4×C127—42no—
240C2×C4×C127—42no—
241C2×C4×C127—42no—
242C2×C4×C127—42no—
243C2×C4×C127—42no—
244C22×C127—42no—
245C22×C127—42no—
246C22×C127—44no—
247C22×C127—44no—
248C22×C127—42no—
249C22×C127—82no—
250C22×C127—82no—
251C22×C127—82no—
252C22×C127—82no—
253C22×C127#53411yescharacteristic
254C22×C127#53512yes—
255C22×C127#53611yescharacteristic
256C22×C127#53611yescharacteristic
257C22×C127#53612yes—
258C22×C127#53612yes—
259C22×C127#53912yes—
260C22×C127#54212yes—
261C22×C127#54212yes—
262C22×C127D4211yescharacteristic
263C22×C127D4212yes—
264C22×C127#54712yes—
265C22×C127#54712yes—
266C22×C127#54812yes—
267C22×C127#55111yescharacteristic
268C22×C127#56011yescharacteristic
269C22×C127#56312yes—
270C22×C127#56312yes—
271C22×C127#56311yescharacteristic
272C22×C127#56312yes—
273C22×C127#56312yes—
274C4×C127—42no—
275C4×C127—44no—
276C4×C127—44no—
277C4×C127—44no—
278C4×C127—42no—
279C4×C127—44no—
280C4×C127—82no—
281C4×C127—82no—
282#17106C12711yesmaximal, characteristic, direct product factor, Sylow
283C2×C127—82no—
284C2×C127#266911yescharacteristic
285C2×C127#267412yes—
286C2×C127#267412yes—
287C2×C127#268012yes—
288C2×C127#271711yescharacteristic
289C2×C127#272312yes—
290C2×C127#281711yescharacteristic
291C2×C127#282112yes—
292C2×C127#283812yes—
293C2×#399C2×C12711yescharacteristic
294C2×#397C2×C12712yes—
295C2×#395C2×C12712yescomplemented
296C2×#393C2×C12711yescharacteristic
297#1444C2×C12712yescomplemented
298#1443C2×C12712yescomplemented
299#1442C2×C12712yescomplemented
300#1440C2×C12711yescharacteristic
301#1439C2×C12712yes—
302C127#1710611yescharacteristic, direct product factor, Sylow
303C23×Q8C22×C12711yescharacteristic
304C22×C4⋊C4C22×C12712yes—
305C22×C4⋊C4C22×C12712yes—
306C22×C4⋊C4C22×C12712yes—
307C22×C4⋊C4C22×C12711yescharacteristic
308C22×C22⋊C4C22×C12712yes—
309C22×C22⋊C4C22×C12711yescharacteristic
310C22×C22⋊C4C22×C12712yes—
311C22×C22⋊C4C22×C12711yescharacteristic
312C22×C22⋊C4C22×C12711yescharacteristic
313#399—24no—
314#397—28no—
315#395—28no—
316#393—24no—
317C2×#95C22×C12711yescharacteristic
318C2×#95C22×C12712yes—
319C2×#95C22×C12712yes—
320C2×#95C22×C12712yes—
321C2×#95C22×C12712yes—
322C2×#95C22×C12712yes—
323C2×#95C22×C12711yescharacteristic
324C2×#95C22×C12712yes—
325C2×#95C22×C12712yes—
326C2×#95C22×C12712yes—
327C2×#95C22×C12712yes—
328C25C23×C12711yescharacteristic
329C22×Q8—22no—
330C22×Q8—24no—
331C22×Q8—41no—
332C22×Q8—42no—
333C22×Q8—41no—
334C23×C4C23×C12712yes—
335C23×C4C23×C12712yes—
336C23×C4C23×C12712yes—
337C23×C4C23×C12711yescharacteristic
338C23×C4C23×C12712yes—
339C23×C4C23×C12712yes—
340C23×C4C23×C12711yescharacteristic
341C23×C4C23×C12712yes—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—41no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—41no—
362C2×C22⋊C4—41no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4D4×C12714yes—
367#95—24no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—42no—
377#95D4×C12718yes—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24C2×D4×C12712yes—
387C24C24×C12711yescharacteristic
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×Q8—44no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4C2×D4×C12712yes—
440C22×C4C2×D4×C12714yes—
441C22×C4C2×D4×C12714yes—
442C22×C4C2×D4×C12712yes—
443C22×C4C2×D4×C12714yes—
444C22×C4#41×C12714yes—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23C22×D4×C12711yescharacteristic
476C23C22×D4×C12712yes—
477C23C2×#41×C12712yes—
478C23H(5,𝔽2)×C12711yescharacteristic
479C23H(5,𝔽2)×C12711yescharacteristic
480C23H(5,𝔽2)×C12712yes—
481C232−1+4×C12712yes—
482C232−1+4×C12712yes—
483C232−1+4×C12712yes—
484Q8—416no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22#534×C12711yescharacteristic
547C22#535×C12712yes—
548C22#536×C12711yescharacteristic
549C22#536×C12712yes—
550C22#536×C12712yes—
551C22#536×C12711yescharacteristic
552C22#539×C12712yes—
553C22#542×C12712yes—
554C22#542×C12712yes—
555C22D42×C12711yescharacteristic
556C22D42×C12712yes—
557C22#547×C12712yes—
558C22#547×C12712yes—
559C22#548×C12712yes—
560C22#551×C12711yescharacteristic
561C22#560×C12711yescharacteristic
562C22#563×C12712yes—
563C22#563×C12712yes—
564C22#563×C12712yes—
565C22#563×C12711yescharacteristic
566C22#563×C12712yes—
567C4—42no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—42no—
572C4—44no—
573C4—82no—
574C4—82no—
575C2—82no—
576C2C127×#266911yescharacteristic
577C2C127×#267412yes—
578C2C127×#267412yes—
579C2C127×#268012yes—
580C2C127×#271711yescharacteristic
581C2C127×#272312yes—
582C2C127×#281711yescharacteristic
583C2C127×#282112yes—
584C2C127×#283812yes—
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#395×C127C2semidirect
5. C127×#1444C2semidirect
6. C127×#1443C2semidirect
7. C127×#1442C2semidirect
282. #17106C127direct (N × H)
295. C2×#395C2×C127semidirect
297. #1444C2×C127semidirect
298. #1443C2×C127semidirect
299. #1442C2×C127semidirect
302. C127#17106direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#399×C1272
C2×#397×C1272
C2×#395×C1272
C2×#393×C1272
C127×#14442
C127×#14432
C127×#14422
C127×#14402
C127×#14392
#17106127