Finite Group Databasebeta

← C127×#13344 #44577017

Subgroups — C127×#13344 #44577017

Order 32512.

892 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#458×C127C211yesmaximal, characteristic, complemented
2C2×B(4,𝔽2)×C127C211yesmaximal, characteristic
3C127×#2697C211yesmaximal, characteristic, complemented
4C2×#352×C127C211yesmaximal, characteristic, complemented
5C2×AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
6C127×#1792C211yesmaximal, characteristic, complemented
7C127×#1393C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C127C2211yescharacteristic
9C2×#123×C127—21no—
10C2×#123×C127C2211yescharacteristic
11C2×#120×C127—21no—
12C2×#120×C127C2211yescharacteristic
13#458×C127—22no—
14#458×C127—22no—
15B(4,𝔽2)×C127—22no—
16B(4,𝔽2)×C127—22no—
17C2×#104×C127—21no—
18#417×C127—21no—
19C2×#100×C127C2211yescharacteristic
20C2×#99×C127—21no—
21C2×#99×C127—21no—
22C2×#99×C127C2211yescharacteristic
23C2×#99×C127C2211yescharacteristic
24#397×C127—21no—
25#396×C127—21no—
26#394×C127—21no—
27#390×C127C2211yescharacteristic
28#388×C127—21no—
29#386×C127—21no—
30#379×C127—21no—
31#352×C127—22no—
32AΣL(1,𝔽16)×C127—22no—
33C25×C127D411yescharacteristic
34H(5,𝔽2)×C127—21no—
35H(5,𝔽2)×C127—21no—
36H(5,𝔽2)×C127—21no—
37H(5,𝔽2)×C127—42no—
38H(5,𝔽2)×C127D412yes—
39C2×#41×C127—21no—
40C2×#41×C127—41no—
41C22×D4×C127—21no—
42C22×D4×C127—21no—
43C22×D4×C127—21no—
44C22×D4×C127—41no—
45C22×D4×C127—41no—
46C22×D4×C127C2311yescharacteristic
47C23×C4×C127D411yescharacteristic, complemented
48C2×M16×C127—21no—
49#123×C127—24no—
50#123×C127—21no—
51#123×C127—21no—
52#123×C127—21no—
53#123×C127—21no—
54#123×C127—42no—
55#120×C127—24no—
56#120×C127—21no—
57#120×C127—21no—
58#120×C127—21no—
59#120×C127—21no—
60#120×C127—42no—
61C2×C4⋊C4×C127—21no—
62C2×C4⋊C4×C127—21no—
63C2×C4⋊C4×C127—41no—
64C2×C4⋊C4×C127D411yescharacteristic, complemented
65C2×C22⋊C4×C127—21no—
66C2×C22⋊C4×C127—21no—
67C2×C22⋊C4×C127—21no—
68C2×C22⋊C4×C127—21no—
69C2×C22⋊C4×C127—21no—
70C2×C22⋊C4×C127—21no—
71C2×C22⋊C4×C127—21no—
72C2×C22⋊C4×C127—21no—
73C2×C22⋊C4×C127—21no—
74C2×C22⋊C4×C127—21no—
75C2×C22⋊C4×C127—41no—
76C2×C22⋊C4×C127—41no—
77C2×C22⋊C4×C127D411yescharacteristic
78C2×C22⋊C4×C127D411yescharacteristic, complemented
79C2×C42×C127D411yescharacteristic, complemented
80#104×C127—42no—
81#103×C127—41no—
82#102×C127—41no—
83#100×C127—21no—
84#100×C127—21no—
85#99×C127—24no—
86#99×C127—24no—
87#99×C127—21no—
88#99×C127—21no—
89#99×C127D412yes—
90#99×C127D412yescomplemented
91#95×C127—21no—
92#95×C127—21no—
93#95×C127—21no—
94C24×C127—21no—
95C24×C127—21no—
96C24×C127—21no—
97C24×C127—21no—
98C24×C127—21no—
99C24×C127—41no—
100C24×C127—41no—
101C24×C127—42no—
102C24×C127—81no—
103C24×C127C2×D411yescharacteristic
104C24×C127C2×D411yescharacteristic
105#41×C127—22no—
106#41×C127—22no—
107#41×C127—42no—
108#41×C127—41no—
109#41×C127—42no—
110#41×C127—42no—
111#41×C127—42no—
112#41×C127—41no—
113C2×Q8×C127—21no—
114C2×D4×C127—21no—
115C2×D4×C127—22no—
116C2×D4×C127—22no—
117C2×D4×C127—22no—
118C2×D4×C127—22no—
119C2×D4×C127—41no—
120C2×D4×C127—41no—
121C2×D4×C127—42no—
122C2×D4×C127—42no—
123C2×D4×C127—41no—
124C2×D4×C127—41no—
125C2×D4×C127—41no—
126C2×D4×C127—41no—
127C2×D4×C127—42no—
128C2×D4×C127—42no—
129C2×D4×C127—41no—
130C2×D4×C127—41no—
131C2×D4×C127—41no—
132C2×D4×C127—44no—
133C2×D4×C127—42no—
134C2×D4×C127—41no—
135C2×D4×C127—41no—
136C2×D4×C127—81no—
137C2×D4×C127—82no—
138C2×D4×C127—82no—
139C2×D4×C127C2×D411yescharacteristic
140C2×D4×C127C2×D411yescharacteristic
141C2×D4×C127C2×D411yescharacteristic
142C2×D4×C127#4111yescharacteristic
143C22×C4×C127—21no—
144C22×C4×C127—21no—
145C22×C4×C127—21no—
146C22×C4×C127—21no—
147C22×C4×C127—21no—
148C22×C4×C127—21no—
149C22×C4×C127—21no—
150C22×C4×C127—21no—
151C22×C4×C127—21no—
152C22×C4×C127—41no—
153C22×C4×C127—41no—
154C22×C4×C127—41no—
155C22×C4×C127—41no—
156C22×C4×C127—41no—
157C22×C4×C127C2×D411yescharacteristic
158M16×C127—41no—
159M16×C127—41no—
160C2×C8×C127—41no—
161C4⋊C4×C127—41no—
162C4⋊C4×C127—44no—
163C4⋊C4×C127—41no—
164C4⋊C4×C127—41no—
165C4⋊C4×C127—81no—
166C22⋊C4×C127—21no—
167C22⋊C4×C127—21no—
168C22⋊C4×C127—22no—
169C22⋊C4×C127—22no—
170C22⋊C4×C127—21no—
171C22⋊C4×C127—21no—
172C22⋊C4×C127—44no—
173C22⋊C4×C127—42no—
174C22⋊C4×C127—41no—
175C22⋊C4×C127—41no—
176C22⋊C4×C127—44no—
177C22⋊C4×C127—41no—
178C22⋊C4×C127—41no—
179C22⋊C4×C127—41no—
180C22⋊C4×C127—41no—
181C22⋊C4×C127—81no—
182C22⋊C4×C127—81no—
183C22⋊C4×C127—81no—
184C22⋊C4×C127—81no—
185C22⋊C4×C127—81no—
186C42×C127—41no—
187C23×C127—21no—
188C23×C127—21no—
189C23×C127—21no—
190C23×C127—21no—
191C23×C127—22no—
192C23×C127—22no—
193C23×C127—22no—
194C23×C127—41no—
195C23×C127—41no—
196C23×C127—41no—
197C23×C127—41no—
198C23×C127—42no—
199C23×C127—41no—
200C23×C127—41no—
201C23×C127—41no—
202C23×C127—41no—
203C23×C127—41no—
204C23×C127—42no—
205C23×C127—42no—
206C23×C127—41no—
207C23×C127—41no—
208C23×C127—41no—
209C23×C127—41no—
210C23×C127—81no—
211C23×C127—82no—
212C23×C127—81no—
213C23×C127—82no—
214C23×C127—84no—
215C23×C127—81no—
216C23×C127—81no—
217C23×C127—81no—
218C23×C127—81no—
219C23×C127—81no—
220C23×C127#12011yescharacteristic
221C23×C127#12011yescharacteristic
222C23×C127#12011yescharacteristic
223C23×C127#12111yescharacteristic
224C23×C127#12111yescharacteristic
225Q8×C127—22no—
226Q8×C127—41no—
227D4×C127—22no—
228D4×C127—41no—
229D4×C127—41no—
230D4×C127—41no—
231D4×C127—41no—
232D4×C127—41no—
233D4×C127—42no—
234D4×C127—42no—
235D4×C127—42no—
236D4×C127—42no—
237D4×C127—42no—
238D4×C127—42no—
239D4×C127—82no—
240D4×C127—82no—
241D4×C127—88no—
242C2×C4×C127—21no—
243C2×C4×C127—21no—
244C2×C4×C127—22no—
245C2×C4×C127—22no—
246C2×C4×C127—42no—
247C2×C4×C127—42no—
248C2×C4×C127—41no—
249C2×C4×C127—41no—
250C2×C4×C127—41no—
251C2×C4×C127—42no—
252C2×C4×C127—42no—
253C2×C4×C127—42no—
254C2×C4×C127—41no—
255C2×C4×C127—41no—
256C2×C4×C127—41no—
257C2×C4×C127—41no—
258C2×C4×C127—41no—
259C2×C4×C127—41no—
260C2×C4×C127—41no—
261C2×C4×C127—41no—
262C2×C4×C127—41no—
263C2×C4×C127—41no—
264C2×C4×C127—41no—
265C2×C4×C127—41no—
266C2×C4×C127—41no—
267C2×C4×C127—41no—
268C2×C4×C127—41no—
269C2×C4×C127—81no—
270C2×C4×C127—81no—
271C2×C4×C127—81no—
272C2×C4×C127—81no—
273C2×C4×C127—81no—
274C2×C4×C127#12111yescharacteristic
275C2×C4×C127#12711yescharacteristic
276C8×C127—81no—
277C22×C127—21no—
278C22×C127—21no—
279C22×C127—21no—
280C22×C127—21no—
281C22×C127—21no—
282C22×C127—21no—
283C22×C127—41no—
284C22×C127—41no—
285C22×C127—41no—
286C22×C127—41no—
287C22×C127—42no—
288C22×C127—42no—
289C22×C127—41no—
290C22×C127—41no—
291C22×C127—41no—
292C22×C127—41no—
293C22×C127—44no—
294C22×C127—42no—
295C22×C127—42no—
296C22×C127—81no—
297C22×C127—81no—
298C22×C127—81no—
299C22×C127—81no—
300C22×C127—82no—
301C22×C127—82no—
302C22×C127—82no—
303C22×C127—82no—
304C22×C127—82no—
305C22×C127—81no—
306C22×C127—81no—
307C22×C127—81no—
308C22×C127—81no—
309C22×C127—161no—
310C22×C127—161no—
311C22×C127#39211yescharacteristic
312C22×C127B(4,𝔽2)11yescharacteristic
313C22×C127B(4,𝔽2)11yescharacteristic
314C4×C127—21no—
315C4×C127—21no—
316C4×C127—42no—
317C4×C127—81no—
318C4×C127—81no—
319C4×C127—81no—
320C4×C127—82no—
321C4×C127—82no—
322C4×C127—81no—
323C4×C127—81no—
324#13344C12711yesmaximal, characteristic, direct product factor, Sylow
325C2×C127—21no—
326C2×C127—21no—
327C2×C127—41no—
328C2×C127—41no—
329C2×C127—41no—
330C2×C127—41no—
331C2×C127—82no—
332C2×C127—82no—
333C2×C127#202111yescharacteristic
334C2×C127#219711yescharacteristic
335C2×C127#220011yescharacteristic
336C2×#458C2×C12711yescharacteristic, complemented
337C2×B(4,𝔽2)C2×C12711yescharacteristic
338#2697C2×C12711yescharacteristic, complemented
339C2×#352C2×C12711yescharacteristic, complemented
340C2×AΣL(1,𝔽16)C2×C12711yescharacteristic, complemented
341#1792C2×C12711yescharacteristic, complemented
342#1393C2×C12711yescharacteristic, complemented
343C127#1334411yescharacteristic, direct product factor, Sylow
344C2×H(5,𝔽2)C22×C12711yescharacteristic
345C2×#123—21no—
346C2×#123C22×C12711yescharacteristic
347C2×#120—21no—
348C2×#120C22×C12711yescharacteristic
349#458—22no—
350#458—22no—
351B(4,𝔽2)—22no—
352B(4,𝔽2)—22no—
353C2×#104—21no—
354#417—21no—
355C2×#100C22×C12711yescharacteristic
356C2×#99—21no—
357C2×#99—21no—
358C2×#99C22×C12711yescharacteristic
359C2×#99C22×C12711yescharacteristic
360#397—21no—
361#396—21no—
362#394—21no—
363#390C22×C12711yescharacteristic
364#388—21no—
365#386—21no—
366#379—21no—
367#352—22no—
368AΣL(1,𝔽16)—22no—
369C25D4×C12711yescharacteristic
370H(5,𝔽2)—21no—
371H(5,𝔽2)—21no—
372H(5,𝔽2)—21no—
373H(5,𝔽2)—42no—
374H(5,𝔽2)D4×C12712yes—
375C2×#41—21no—
376C2×#41—41no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—21no—
380C22×D4—41no—
381C22×D4—41no—
382C22×D4C23×C12711yescharacteristic
383C23×C4D4×C12711yescharacteristic, complemented
384C2×M16—21no—
385#123—24no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—21no—
390#123—42no—
391#120—24no—
392#120—21no—
393#120—21no—
394#120—21no—
395#120—21no—
396#120—42no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4D4×C12711yescharacteristic, complemented
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—41no—
412C2×C22⋊C4—41no—
413C2×C22⋊C4D4×C12711yescharacteristic
414C2×C22⋊C4D4×C12711yescharacteristic, complemented
415C2×C42D4×C12711yescharacteristic, complemented
416#104—42no—
417#103—41no—
418#102—41no—
419#100—21no—
420#100—21no—
421#99—24no—
422#99—24no—
423#99—21no—
424#99—21no—
425#99D4×C12712yes—
426#99D4×C12712yescomplemented
427#95—21no—
428#95—21no—
429#95—21no—
430C24—21no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—41no—
436C24—41no—
437C24—42no—
438C24—81no—
439C24C2×D4×C12711yescharacteristic
440C24C2×D4×C12711yescharacteristic
441#41—22no—
442#41—22no—
443#41—42no—
444#41—41no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—41no—
449C2×Q8—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—81no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4C2×D4×C12711yescharacteristic
476C2×D4C2×D4×C12711yescharacteristic
477C2×D4C2×D4×C12711yescharacteristic
478C2×D4#41×C12711yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×D4×C12711yescharacteristic
494M16—41no—
495M16—41no—
496C2×C8—41no—
497C4⋊C4—41no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—81no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—44no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C42—41no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—81no—
547C23—82no—
548C23—81no—
549C23—82no—
550C23—84no—
551C23—81no—
552C23—81no—
553C23—81no—
554C23—81no—
555C23—81no—
556C23#120×C12711yescharacteristic
557C23#120×C12711yescharacteristic
558C23#120×C12711yescharacteristic
559C23#121×C12711yescharacteristic
560C23#121×C12711yescharacteristic
561Q8—22no—
562Q8—41no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—82no—
576D4—82no—
577D4—88no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4#121×C12711yescharacteristic
611C2×C4#127×C12711yescharacteristic
612C8—81no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—161no—
646C22—161no—
647C22#392×C12711yescharacteristic
648C22B(4,𝔽2)×C12711yescharacteristic
649C22B(4,𝔽2)×C12711yescharacteristic
650C4—21no—
651C4—21no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—81no—
660C2—21no—
661C2—21no—
662C2—41no—
663C2—41no—
664C2—41no—
665C2—41no—
666C2—82no—
667C2—82no—
668C2C127×#202111yescharacteristic
669C2C127×#219711yescharacteristic
670C2C127×#220011yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C127
prime radicalC2×B(4,𝔽2)×C127C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C22×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#458×C127C2semidirect
3. C127×#2697C2semidirect
4. C2×#352×C127C2semidirect
5. C2×AΣL(1,𝔽16)×C127C2semidirect
6. C127×#1792C2semidirect
7. C127×#1393C2semidirect
47. C23×C4×C127D4semidirect
64. C2×C4⋊C4×C127D4semidirect
78. C2×C22⋊C4×C127D4semidirect
79. C2×C42×C127D4semidirect
90. #99×C127D4semidirect
324. #13344C127direct (N × H)
336. C2×#458C2×C127semidirect
338. #2697C2×C127semidirect
339. C2×#352C2×C127semidirect
340. C2×AΣL(1,𝔽16)C2×C127semidirect
341. #1792C2×C127semidirect
342. #1393C2×C127semidirect
343. C127#13344direct (N × H)
383. C23×C4D4×C127semidirect
400. C2×C4⋊C4D4×C127semidirect
414. C2×C22⋊C4D4×C127semidirect
415. C2×C42D4×C127semidirect
426. #99D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#458×C1272
C2×B(4,𝔽2)×C1272
C127×#26972
C2×#352×C1272
C2×AΣL(1,𝔽16)×C1272
C127×#17922
C127×#13932
#13344127