Finite Group Databasebeta

← C127×#8534 #44572207

Subgroups — C127×#8534 #44572207

Order 32512.

1408 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C127×#3484C211yesmaximal, characteristic
2C127×#1898C214yesmaximal, complemented
3C127×#1781C211yesmaximal, characteristic, complemented
4C127×#1438C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C127C411yescharacteristic, complemented
6C23×D4×C127C2211yescharacteristic
7#546×C127—24no—
8D42×C127—24no—
9#535×C127—21no—
10#534×C127—21no—
11#534×C127C412yescomplemented
12#525×C127—21no—
13C2×#120×C127—21no—
14C2×#120×C127C2212yes—
15#518×C127C411yescharacteristic, complemented
16C22×C22⋊C4×C127—21no—
17C22×C22⋊C4×C127C2212yescomplemented
18C2×#99×C127—24no—
19C2×#99×C127—21no—
20C2×#99×C127—21no—
21#379×C127C2212yescomplemented
22C2×#95×C127—21no—
23#342×C127—24no—
24C25×C127D411yescharacteristic
25C25×C127C2311yescharacteristic
26H(5,𝔽2)×C127—42no—
27H(5,𝔽2)×C127—42no—
28C2×#41×C127—21no—
29C2×#41×C127—22no—
30C22×D4×C127—21no—
31C22×D4×C127—22no—
32C22×D4×C127—24no—
33C22×D4×C127—22no—
34C22×D4×C127—42no—
35C22×D4×C127—42no—
36C22×D4×C127C2×C411yescharacteristic
37C22×D4×C127C2×C411yescharacteristic
38C22×D4×C127C2×C412yes—
39C22×D4×C127D411yescharacteristic
40C22×D4×C127D411yescharacteristic
41C22×D4×C127D414yes—
42C23×C4×C127—22no—
43C23×C4×C127D411yescharacteristic
44#127×C127—22no—
45#124×C127—22no—
46#123×C127—22no—
47#123×C127—22no—
48#123×C127—22no—
49C22⋊Q8×C127—22no—
50#121×C127—22no—
51#121×C127—24no—
52#121×C127—24no—
53#121×C127—24no—
54#120×C127—24no—
55#120×C127—22no—
56#120×C127—22no—
57#120×C127—22no—
58#120×C127—22no—
59#120×C127—22no—
60#120×C127—24no—
61C4×D4×C127—24no—
62C4×D4×C127—24no—
63#117×C127—21no—
64C2×C22⋊C4×C127—22no—
65C2×C22⋊C4×C127—21no—
66C2×C22⋊C4×C127—24no—
67C2×C22⋊C4×C127—24no—
68C2×C22⋊C4×C127—22no—
69C2×C22⋊C4×C127—44no—
70C2×C22⋊C4×C127—44no—
71C2×C22⋊C4×C127—42no—
72C2×C22⋊C4×C127C2×C412yes—
73C2×C22⋊C4×C127D414yes—
74#99×C127—48no—
75#99×C127—42no—
76#99×C127—42no—
77#95×C127—42no—
78C24×C127—21no—
79C24×C127—22no—
80C24×C127—22no—
81C24×C127—42no—
82C24×C127—41no—
83C24×C127—41no—
84C24×C127—41no—
85C24×C127—42no—
86C24×C127—42no—
87C24×C127—42no—
88C24×C127C22⋊C411yescharacteristic
89C24×C127C22⋊C411yescharacteristic
90C24×C127C22⋊C412yescomplemented
91C24×C127C22×C411yescharacteristic
92C24×C127C2×D411yescharacteristic
93C24×C127C2×D414yes—
94C24×C127C2×D411yescharacteristic
95#41×C127—42no—
96#41×C127—48no—
97#41×C127—42no—
98C2×Q8×C127—21no—
99C2×D4×C127—22no—
100C2×D4×C127—22no—
101C2×D4×C127—22no—
102C2×D4×C127—21no—
103C2×D4×C127—22no—
104C2×D4×C127—22no—
105C2×D4×C127—22no—
106C2×D4×C127—22no—
107C2×D4×C127—24no—
108C2×D4×C127—22no—
109C2×D4×C127—22no—
110C2×D4×C127—24no—
111C2×D4×C127—24no—
112C2×D4×C127—22no—
113C2×D4×C127—22no—
114C2×D4×C127—41no—
115C2×D4×C127—41no—
116C2×D4×C127—41no—
117C2×D4×C127—41no—
118C2×D4×C127—416no—
119C2×D4×C127—48no—
120C2×D4×C127—42no—
121C2×D4×C127—42no—
122C2×D4×C127—42no—
123C2×D4×C127—42no—
124C2×D4×C127—44no—
125C2×D4×C127—44no—
126C2×D4×C127—82no—
127C2×D4×C127—82no—
128C22×C4×C127—21no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—22no—
132C22×C4×C127—22no—
133C22×C4×C127—24no—
134C22×C4×C127—41no—
135C22×C4×C127—42no—
136C22×C4×C127—42no—
137C22×C4×C127—42no—
138C22×C4×C127—42no—
139C22×C4×C127—42no—
140C22×C4×C127—42no—
141C22×C4×C127—42no—
142C22×C4×C127C22⋊C411yescharacteristic
143C22×C4×C127C22⋊C411yescharacteristic
144C22×C4×C127C22⋊C414yescomplemented
145C22×C4×C127C22⋊C412yescomplemented
146C4⋊C4×C127—22no—
147C4⋊C4×C127—24no—
148C22⋊C4×C127—22no—
149C22⋊C4×C127—24no—
150C22⋊C4×C127—24no—
151C22⋊C4×C127—22no—
152C22⋊C4×C127—22no—
153C22⋊C4×C127—24no—
154C22⋊C4×C127—24no—
155C22⋊C4×C127—28no—
156C22⋊C4×C127—44no—
157C22⋊C4×C127—48no—
158C22⋊C4×C127—42no—
159C22⋊C4×C127—42no—
160C22⋊C4×C127—44no—
161C22⋊C4×C127—44no—
162C42×C127—22no—
163C23×C127—21no—
164C23×C127—21no—
165C23×C127—22no—
166C23×C127—22no—
167C23×C127—21no—
168C23×C127—21no—
169C23×C127—21no—
170C23×C127—21no—
171C23×C127—21no—
172C23×C127—21no—
173C23×C127—24no—
174C23×C127—24no—
175C23×C127—21no—
176C23×C127—22no—
177C23×C127—21no—
178C23×C127—21no—
179C23×C127—44no—
180C23×C127—42no—
181C23×C127—44no—
182C23×C127—42no—
183C23×C127—44no—
184C23×C127—42no—
185C23×C127—42no—
186C23×C127—42no—
187C23×C127—44no—
188C23×C127—42no—
189C23×C127—42no—
190C23×C127—42no—
191C23×C127—44no—
192C23×C127—42no—
193C23×C127—42no—
194C23×C127—82no—
195C23×C127—81no—
196C23×C127—82no—
197C23×C127—82no—
198C23×C127—81no—
199C23×C127—81no—
200C23×C127—82no—
201C23×C127—81no—
202C23×C127—81no—
203C23×C127—81no—
204C23×C127—82no—
205C23×C127#9912yescomplemented
206C23×C127C2×C22⋊C412yes—
207C23×C127C2×C22⋊C411yescharacteristic
208C23×C127#12012yes—
209C23×C127#12012yes—
210Q8×C127—42no—
211D4×C127—28no—
212D4×C127—44no—
213D4×C127—42no—
214D4×C127—44no—
215D4×C127—44no—
216D4×C127—44no—
217D4×C127—44no—
218D4×C127—42no—
219D4×C127—42no—
220D4×C127—44no—
221D4×C127—44no—
222D4×C127—84no—
223D4×C127—84no—
224C2×C4×C127—21no—
225C2×C4×C127—21no—
226C2×C4×C127—24no—
227C2×C4×C127—22no—
228C2×C4×C127—21no—
229C2×C4×C127—21no—
230C2×C4×C127—21no—
231C2×C4×C127—44no—
232C2×C4×C127—44no—
233C2×C4×C127—42no—
234C2×C4×C127—44no—
235C2×C4×C127—42no—
236C2×C4×C127—44no—
237C2×C4×C127—44no—
238C2×C4×C127—44no—
239C2×C4×C127—82no—
240C2×C4×C127—82no—
241C2×C4×C127—82no—
242C2×C4×C127—82no—
243C2×C4×C127—82no—
244C2×C4×C127—82no—
245C2×C4×C127—84no—
246C2×C4×C127#9912yescomplemented
247C22×C127—21no—
248C22×C127—22no—
249C22×C127—21no—
250C22×C127—22no—
251C22×C127—21no—
252C22×C127—21no—
253C22×C127—22no—
254C22×C127—22no—
255C22×C127—22no—
256C22×C127—22no—
257C22×C127—22no—
258C22×C127—22no—
259C22×C127—24no—
260C22×C127—24no—
261C22×C127—42no—
262C22×C127—42no—
263C22×C127—42no—
264C22×C127—42no—
265C22×C127—41no—
266C22×C127—44no—
267C22×C127—42no—
268C22×C127—84no—
269C22×C127—81no—
270C22×C127—81no—
271C22×C127—81no—
272C22×C127—81no—
273C22×C127—81no—
274C22×C127—84no—
275C22×C127—84no—
276C22×C127—84no—
277C22×C127—81no—
278C22×C127—162no—
279C22×C127—162no—
280C22×C127#37911yescharacteristic
281C22×C127C2×#9911yescharacteristic
282C22×C127C2×#9911yescharacteristic
283C22×C127B(4,𝔽2)12yes—
284C22×C127#45812yes—
285C4×C127—22no—
286C4×C127—41no—
287C4×C127—42no—
288C4×C127—44no—
289C4×C127—84no—
290C4×C127—162no—
291#8534C12711yesmaximal, characteristic, direct product factor, Sylow
292C2×C127—22no—
293C2×C127—21no—
294C2×C127—21no—
295C2×C127—22no—
296C2×C127—42no—
297C2×C127—42no—
298C2×C127—42no—
299C2×C127—81no—
300C2×C127—82no—
301C2×C127#178111yescharacteristic
302C2×C127#179012yes—
303#3484C2×C12711yescharacteristic
304#1898C2×C12714yescomplemented
305#1781C2×C12711yescharacteristic, complemented
306#1438C2×C12711yescharacteristic, complemented
307C127#853411yescharacteristic, direct product factor, Sylow
308C2×H(5,𝔽2)C4×C12711yescharacteristic, complemented
309C23×D4C22×C12711yescharacteristic
310#546—24no—
311D42—24no—
312#535—21no—
313#534—21no—
314#534C4×C12712yescomplemented
315#525—21no—
316C2×#120—21no—
317C2×#120C22×C12712yes—
318#518C4×C12711yescharacteristic, complemented
319C22×C22⋊C4—21no—
320C22×C22⋊C4C22×C12712yescomplemented
321C2×#99—24no—
322C2×#99—21no—
323C2×#99—21no—
324#379C22×C12712yescomplemented
325C2×#95—21no—
326#342—24no—
327C25D4×C12711yescharacteristic
328C25C23×C12711yescharacteristic
329H(5,𝔽2)—42no—
330H(5,𝔽2)—42no—
331C2×#41—21no—
332C2×#41—22no—
333C22×D4—21no—
334C22×D4—22no—
335C22×D4—24no—
336C22×D4—22no—
337C22×D4—42no—
338C22×D4—42no—
339C22×D4C2×C4×C12711yescharacteristic
340C22×D4C2×C4×C12711yescharacteristic
341C22×D4C2×C4×C12712yes—
342C22×D4D4×C12711yescharacteristic
343C22×D4D4×C12711yescharacteristic
344C22×D4D4×C12714yes—
345C23×C4—22no—
346C23×C4D4×C12711yescharacteristic
347#127—22no—
348#124—22no—
349#123—22no—
350#123—22no—
351#123—22no—
352C22⋊Q8—22no—
353#121—22no—
354#121—24no—
355#121—24no—
356#121—24no—
357#120—24no—
358#120—22no—
359#120—22no—
360#120—22no—
361#120—22no—
362#120—22no—
363#120—24no—
364C4×D4—24no—
365C4×D4—24no—
366#117—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4—44no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4C2×C4×C12712yes—
376C2×C22⋊C4D4×C12714yes—
377#99—48no—
378#99—42no—
379#99—42no—
380#95—42no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—42no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22⋊C4×C12711yescharacteristic
392C24C22⋊C4×C12711yescharacteristic
393C24C22⋊C4×C12712yescomplemented
394C24C22×C4×C12711yescharacteristic
395C24C2×D4×C12711yescharacteristic
396C24C2×D4×C12714yes—
397C24C2×D4×C12711yescharacteristic
398#41—42no—
399#41—48no—
400#41—42no—
401C2×Q8—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—82no—
430C2×D4—82no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C22⋊C4×C12711yescharacteristic
446C22×C4C22⋊C4×C12711yescharacteristic
447C22×C4C22⋊C4×C12714yescomplemented
448C22×C4C22⋊C4×C12712yescomplemented
449C4⋊C4—22no—
450C4⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C42—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—82no—
498C23—81no—
499C23—82no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—82no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—82no—
508C23#99×C12712yescomplemented
509C23C2×C22⋊C4×C12712yes—
510C23C2×C22⋊C4×C12711yescharacteristic
511C23#120×C12712yes—
512C23#120×C12712yes—
513Q8—42no—
514D4—28no—
515D4—44no—
516D4—42no—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—42no—
522D4—42no—
523D4—44no—
524D4—44no—
525D4—84no—
526D4—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4#99×C12712yescomplemented
550C22—21no—
551C22—22no—
552C22—21no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—41no—
569C22—44no—
570C22—42no—
571C22—84no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—81no—
581C22—162no—
582C22—162no—
583C22#379×C12711yescharacteristic
584C22C2×#99×C12711yescharacteristic
585C22C2×#99×C12711yescharacteristic
586C22B(4,𝔽2)×C12712yes—
587C22#458×C12712yes—
588C4—22no—
589C4—41no—
590C4—42no—
591C4—44no—
592C4—84no—
593C4—162no—
594C2—22no—
595C2—21no—
596C2—21no—
597C2—22no—
598C2—42no—
599C2—42no—
600C2—42no—
601C2—81no—
602C2—82no—
603C2C127×#178111yescharacteristic
604C2C127×#179012yes—
subgroupG/H
center, socleC22×C127#379
commutatorC24C22×C4×C127
Frattini, cosocleC25C23×C127
prime radicalC127×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C25×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1898C2semidirect
3. C127×#1781C2semidirect
4. C127×#1438C2semidirect
5. C2×H(5,𝔽2)×C127C4semidirect
11. #534×C127C4semidirect
15. #518×C127C4semidirect
17. C22×C22⋊C4×C127C22semidirect
21. #379×C127C22semidirect
90. C24×C127C22⋊C4semidirect
144. C22×C4×C127C22⋊C4semidirect
145. C22×C4×C127C22⋊C4semidirect
205. C23×C127#99semidirect
246. C2×C4×C127#99semidirect
291. #8534C127direct (N × H)
304. #1898C2×C127semidirect
305. #1781C2×C127semidirect
306. #1438C2×C127semidirect
307. C127#8534direct (N × H)
308. C2×H(5,𝔽2)C4×C127semidirect
314. #534C4×C127semidirect
318. #518C4×C127semidirect
320. C22×C22⋊C4C22×C127semidirect
324. #379C22×C127semidirect
393. C24C22⋊C4×C127semidirect
447. C22×C4C22⋊C4×C127semidirect
448. C22×C4C22⋊C4×C127semidirect
508. C23#99×C127semidirect
549. C2×C4#99×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34842
C127×#18982
C127×#17812
C127×#14382
#8534127