Finite Group Databasebeta

← C127×#16981 #44580654

Subgroups — C127×#16981 #44580654

Order 32512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389×C127C211yesmaximal, characteristic
2C2×#387×C127C211yesmaximal, characteristic
3C127×#1444C211yesmaximal, characteristic
4C127×#1444C212yesmaximal
5C127×#1444C211yesmaximal, characteristic
6C127×#1441C211yesmaximal, characteristic
7C127×#1441C211yesmaximal, characteristic
8C127×#1441C212yesmaximal
9C127×#1441C212yesmaximal
10C127×#1441C212yesmaximal
11C127×C4⋊#95C211yesmaximal, characteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2212yes—
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2212yes—
17C22×C4⋊C4×C127C2211yescharacteristic
18C22×C4⋊C4×C127C2212yes—
19C22×C4⋊C4×C127C2212yes—
20C22×C42×C127C2211yescharacteristic
21#389×C127—24no—
22#387×C127—24no—
23C2×#95×C127C2211yescharacteristic
24C2×#95×C127C2211yescharacteristic
25C2×#95×C127C2212yes—
26C2×#95×C127C2211yescharacteristic
27C2×#95×C127C2211yescharacteristic
28C2×#95×C127C2211yescharacteristic
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2212yes—
32C2×#95×C127C2212yes—
33C2×#95×C127C2212yes—
34C2×#95×C127C2212yes—
35C2×#95×C127C2212yes—
36C2×#95×C127C2212yes—
37C23×C4×C127C2311yescharacteristic
38C23×C4×C127C2312yes—
39C23×C4×C127C2311yescharacteristic
40C23×C4×C127C2311yescharacteristic
41C23×C4×C127C2311yescharacteristic
42C23×C4×C127C2312yes—
43C23×C4×C127C2312yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2311yescharacteristic
47C23×C4×C127C2311yescharacteristic
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—22no—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—24no—
56C2×C4⋊C4×C127—42no—
57C2×C4⋊C4×C127—42no—
58C2×C4⋊C4×C127—42no—
59C2×C4⋊C4×C127—42no—
60C2×C4⋊C4×C127—42no—
61C2×C4⋊C4×C127—42no—
62C2×C4⋊C4×C127—42no—
63C2×C4⋊C4×C127—42no—
64C2×C4⋊C4×C127—42no—
65C2×C4⋊C4×C127—42no—
66C2×C4⋊C4×C127—42no—
67C2×C4⋊C4×C127Q814yes—
68C2×C4⋊C4×C127Q818yes—
69C2×C42×C127—42no—
70C2×C42×C127D414yes—
71#95×C127—22no—
72#95×C127—22no—
73#95×C127—22no—
74#95×C127—22no—
75#95×C127—24no—
76#95×C127—24no—
77#95×C127—22no—
78#95×C127—22no—
79#95×C127—42no—
80#95×C127—42no—
81#95×C127—42no—
82#95×C127—42no—
83#95×C127—42no—
84#95×C127—42no—
85C24×C127C2411yescharacteristic
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—21no—
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—22no—
97C22×C4×C127—22no—
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—21no—
103C22×C4×C127—21no—
104C22×C4×C127—21no—
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—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—22no—
119C22×C4×C127—21no—
120C22×C4×C127—22no—
121C22×C4×C127—21no—
122C22×C4×C127—21no—
123C22×C4×C127—21no—
124C22×C4×C127—21no—
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—21no—
136C22×C4×C127—21no—
137C22×C4×C127—21no—
138C22×C4×C127—21no—
139C22×C4×C127—22no—
140C22×C4×C127C2×D412yes—
141C22×C4×C127C2×D412yes—
142C22×C4×C127C2×D412yes—
143C22×C4×C127C2×Q812yes—
144C22×C4×C127C2×Q812yes—
145C22×C4×C127C2×Q814yes—
146C22×C4×C127C2×Q814yes—
147C22×C4×C127C2×Q812yes—
148C22×C4×C127C2×Q814yes—
149C22×C4×C127#4114yes—
150C22×C4×C127#4112yes—
151C4⋊C4×C127—28no—
152C4⋊C4×C127—28no—
153C4⋊C4×C127—48no—
154C4⋊C4×C127—44no—
155C4⋊C4×C127—44no—
156C4⋊C4×C127—48no—
157C4⋊C4×C127—48no—
158C4⋊C4×C127—48no—
159C42×C127—44no—
160C23×C127C22×D411yescharacteristic
161C23×C127C22×Q811yescharacteristic
162C23×C127C22×Q812yes—
163C23×C127C2×#4111yescharacteristic
164C23×C127C2×#4112yes—
165C23×C127H(5,𝔽2)12yes—
166C23×C127H(5,𝔽2)12yes—
167C23×C127H(5,𝔽2)11yescharacteristic
168C23×C127H(5,𝔽2)11yescharacteristic
169C23×C1272−1+411yescharacteristic
170C23×C1272−1+411yescharacteristic
171C2×C4×C127—22no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—22no—
175C2×C4×C127—24no—
176C2×C4×C127—24no—
177C2×C4×C127—24no—
178C2×C4×C127—24no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—24no—
182C2×C4×C127—22no—
183C2×C4×C127—22no—
184C2×C4×C127—22no—
185C2×C4×C127—24no—
186C2×C4×C127—24no—
187C2×C4×C127—24no—
188C2×C4×C127—24no—
189C2×C4×C127—22no—
190C2×C4×C127—22no—
191C2×C4×C127—24no—
192C2×C4×C127—24no—
193C2×C4×C127—24no—
194C2×C4×C127—24no—
195C2×C4×C127—24no—
196C2×C4×C127—41no—
197C2×C4×C127—41no—
198C2×C4×C127—42no—
199C2×C4×C127—41no—
200C2×C4×C127—41no—
201C2×C4×C127—42no—
202C2×C4×C127—42no—
203C2×C4×C127—42no—
204C2×C4×C127—42no—
205C2×C4×C127—42no—
206C2×C4×C127—41no—
207C2×C4×C127—42no—
208C2×C4×C127—42no—
209C2×C4×C127—41no—
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—41no—
218C2×C4×C127—41no—
219C2×C4×C127—42no—
220C2×C4×C127—42no—
221C2×C4×C127—41no—
222C2×C4×C127—41no—
223C2×C4×C127—41no—
224C2×C4×C127—41no—
225C2×C4×C127—42no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—42no—
230C2×C4×C127—42no—
231C2×C4×C127—42no—
232C2×C4×C127—42no—
233C2×C4×C127—42no—
234C2×C4×C127C22⋊Q814yes—
235C2×C4×C127C4⋊Q814yes—
236C22×C127C2×C22⋊Q811yescharacteristic
237C22×C127#52511yescharacteristic
238C22×C127C2×C4⋊Q811yescharacteristic
239C22×C127#53411yescharacteristic
240C22×C127#53511yescharacteristic
241C22×C127#53711yescharacteristic
242C22×C127#53711yescharacteristic
243C22×C127#53812yes—
244C22×C127#53912yes—
245C22×C127#54011yescharacteristic
246C22×C127#54111yescharacteristic
247C22×C127#54312yes—
248C22×C127#54411yescharacteristic
249C22×C127#54411yescharacteristic
250C22×C127#54412yes—
251C22×C127#55312yes—
252C22×C127#55412yes—
253C22×C127#55412yes—
254C22×C127#55412yes—
255C22×C127#55712yes—
256C22×C127Q8212yes—
257C22×C127#56012yes—
258C22×C127#56212yes—
259C4×C127—24no—
260C4×C127—24no—
261C4×C127—42no—
262C4×C127—42no—
263C4×C127—44no—
264C4×C127—44no—
265C4×C127—44no—
266C4×C127—42no—
267C4×C127—81no—
268C4×C127—82no—
269C4×C127—81no—
270#16981C12711yesmaximal, characteristic, direct product factor, Sylow
271C2×C127#256711yescharacteristic
272C2×C127#257911yescharacteristic
273C2×C127#258411yescharacteristic
274C2×C127#262911yescharacteristic
275C2×C127#263611yescharacteristic
276C2×C127#265911yescharacteristic
277C2×C127#278012yes—
278C2×C127#279212yes—
279C2×C127#280212yes—
280C2×C127#280612yes—
281C2×C127#281311yescharacteristic
282C2×#389C2×C12711yescharacteristic
283C2×#387C2×C12711yescharacteristic
284#1444C2×C12711yescharacteristic
285#1444C2×C12712yes—
286#1444C2×C12711yescharacteristic
287#1441C2×C12711yescharacteristic
288#1441C2×C12711yescharacteristic
289#1441C2×C12712yes—
290#1441C2×C12712yes—
291#1441C2×C12712yes—
292C4⋊#95C2×C12711yescharacteristic
293C127#1698111yescharacteristic, direct product factor, Sylow
294C22×C4⋊C4C22×C12711yescharacteristic
295C22×C4⋊C4C22×C12711yescharacteristic
296C22×C4⋊C4C22×C12712yes—
297C22×C4⋊C4C22×C12711yescharacteristic
298C22×C4⋊C4C22×C12712yes—
299C22×C4⋊C4C22×C12711yescharacteristic
300C22×C4⋊C4C22×C12712yes—
301C22×C4⋊C4C22×C12712yes—
302C22×C42C22×C12711yescharacteristic
303#389—24no—
304#387—24no—
305C2×#95C22×C12711yescharacteristic
306C2×#95C22×C12711yescharacteristic
307C2×#95C22×C12712yes—
308C2×#95C22×C12711yescharacteristic
309C2×#95C22×C12711yescharacteristic
310C2×#95C22×C12711yescharacteristic
311C2×#95C22×C12712yes—
312C2×#95C22×C12711yescharacteristic
313C2×#95C22×C12712yes—
314C2×#95C22×C12712yes—
315C2×#95C22×C12712yes—
316C2×#95C22×C12712yes—
317C2×#95C22×C12712yes—
318C2×#95C22×C12712yes—
319C23×C4C23×C12711yescharacteristic
320C23×C4C23×C12712yes—
321C23×C4C23×C12711yescharacteristic
322C23×C4C23×C12711yescharacteristic
323C23×C4C23×C12711yescharacteristic
324C23×C4C23×C12712yes—
325C23×C4C23×C12712yes—
326C23×C4C23×C12711yescharacteristic
327C23×C4C23×C12712yes—
328C23×C4C23×C12711yescharacteristic
329C23×C4C23×C12711yescharacteristic
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4—42no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—42no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—42no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4Q8×C12714yes—
350C2×C4⋊C4Q8×C12718yes—
351C2×C42—42no—
352C2×C42D4×C12714yes—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—42no—
362#95—42no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—42no—
367C24C24×C12711yescharacteristic
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
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—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4C2×D4×C12712yes—
423C22×C4C2×D4×C12712yes—
424C22×C4C2×D4×C12712yes—
425C22×C4C2×Q8×C12712yes—
426C22×C4C2×Q8×C12712yes—
427C22×C4C2×Q8×C12714yes—
428C22×C4C2×Q8×C12714yes—
429C22×C4C2×Q8×C12712yes—
430C22×C4C2×Q8×C12714yes—
431C22×C4#41×C12714yes—
432C22×C4#41×C12712yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C42—44no—
442C23C22×D4×C12711yescharacteristic
443C23C22×Q8×C12711yescharacteristic
444C23C22×Q8×C12712yes—
445C23C2×#41×C12711yescharacteristic
446C23C2×#41×C12712yes—
447C23H(5,𝔽2)×C12712yes—
448C23H(5,𝔽2)×C12712yes—
449C23H(5,𝔽2)×C12711yescharacteristic
450C23H(5,𝔽2)×C12711yescharacteristic
451C232−1+4×C12711yescharacteristic
452C232−1+4×C12711yescharacteristic
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4C22⋊Q8×C12714yes—
517C2×C4C4⋊Q8×C12714yes—
518C22C2×C22⋊Q8×C12711yescharacteristic
519C22#525×C12711yescharacteristic
520C22C2×C4⋊Q8×C12711yescharacteristic
521C22#534×C12711yescharacteristic
522C22#535×C12711yescharacteristic
523C22#537×C12711yescharacteristic
524C22#537×C12711yescharacteristic
525C22#538×C12712yes—
526C22#539×C12712yes—
527C22#540×C12711yescharacteristic
528C22#541×C12711yescharacteristic
529C22#543×C12712yes—
530C22#544×C12711yescharacteristic
531C22#544×C12711yescharacteristic
532C22#544×C12712yes—
533C22#553×C12712yes—
534C22#554×C12712yes—
535C22#554×C12712yes—
536C22#554×C12712yes—
537C22#557×C12712yes—
538C22Q82×C12712yes—
539C22#560×C12712yes—
540C22#562×C12712yes—
541C4—24no—
542C4—24no—
543C4—42no—
544C4—42no—
545C4—44no—
546C4—44no—
547C4—44no—
548C4—42no—
549C4—81no—
550C4—82no—
551C4—81no—
552C2C127×#256711yescharacteristic
553C2C127×#257911yescharacteristic
554C2C127×#258411yescharacteristic
555C2C127×#262911yescharacteristic
556C2C127×#263611yescharacteristic
557C2C127×#265911yescharacteristic
558C2C127×#278012yes—
559C2C127×#279212yes—
560C2C127×#280212yes—
561C2C127×#280612yes—
562C2C127×#281311yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
270. #16981C127direct (N × H)
293. C127#16981direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#389×C1272
C2×#387×C1272
C127×#14442
C127×#14412
C127×C4⋊#952
#16981127