Finite Group Databasebeta

← C127×#12705 #44576378

Subgroups — C127×#12705 #44576378

Order 32512.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 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×#2121C214yesmaximal, complemented
3C127×#1783C211yesmaximal, characteristic, complemented
4C127×#1781C211yesmaximal, 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×C127C2212yes—
12#525×C127—21no—
13C2×#120×C127—21no—
14C2×#120×C127C412yescomplemented
15#518×C127C411yescharacteristic, complemented
16C2×#100×C127—21no—
17C2×#100×C127C2212yescomplemented
18C2×#99×C127—21no—
19C2×#99×C127C2212yescomplemented
20#407×C127—21no—
21#379×C127—21no—
22#352×C127—24no—
23AΣL(1,𝔽16)×C127—24no—
24C25×C127C2×C411yescharacteristic
25C25×C127D411yescharacteristic
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—22no—
33C22×D4×C127—22no—
34C22×D4×C127—24no—
35C22×D4×C127—42no—
36C22×D4×C127—42no—
37C22×D4×C127C2×C411yescharacteristic
38C22×D4×C127C2×C412yes—
39C22×D4×C127D411yescharacteristic
40C22×D4×C127D411yescharacteristic
41C22×D4×C127C2311yescharacteristic
42C23×C4×C127D411yescharacteristic
43C2×M16×C127—22no—
44#127×C127—22no—
45#124×C127—22no—
46#123×C127—22no—
47#123×C127—22no—
48#123×C127D414yes—
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—24no—
60#120×C127D414yes—
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—22no—
67C2×C22⋊C4×C127—22no—
68C2×C22⋊C4×C127—42no—
69C2×C22⋊C4×C127C2×C412yes—
70#100×C127—24no—
71#100×C127—42no—
72#99×C127—24no—
73#99×C127—42no—
74C22⋊C8×C127—42no—
75C24×C127—21no—
76C24×C127—22no—
77C24×C127—22no—
78C24×C127—22no—
79C24×C127—41no—
80C24×C127—41no—
81C24×C127—41no—
82C24×C127—82no—
83C24×C127—82no—
84C24×C127C22⋊C411yescharacteristic
85C24×C127C22⋊C411yescharacteristic
86C24×C127C22⋊C411yescharacteristic
87C24×C127C22⋊C412yescomplemented
88C24×C127C22×C411yescharacteristic
89C24×C127C2×D411yescharacteristic
90#41×C127—42no—
91#41×C127—48no—
92#41×C127—42no—
93C2×Q8×C127—21no—
94C2×D4×C127—22no—
95C2×D4×C127—22no—
96C2×D4×C127—24no—
97C2×D4×C127—22no—
98C2×D4×C127—21no—
99C2×D4×C127—22no—
100C2×D4×C127—22no—
101C2×D4×C127—22no—
102C2×D4×C127—22no—
103C2×D4×C127—24no—
104C2×D4×C127—24no—
105C2×D4×C127—22no—
106C2×D4×C127—22no—
107C2×D4×C127—22no—
108C2×D4×C127—41no—
109C2×D4×C127—41no—
110C2×D4×C127—41no—
111C2×D4×C127—41no—
112C2×D4×C127—416no—
113C2×D4×C127—48no—
114C2×D4×C127—42no—
115C2×D4×C127—42no—
116C2×D4×C127—42no—
117C2×D4×C127—42no—
118C2×D4×C127—84no—
119C2×D4×C127—84no—
120C2×D4×C127C2×D414yes—
121C22×C4×C127—21no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
127C22×C4×C127—41no—
128C22×C4×C127—42no—
129C22×C4×C127C22⋊C411yescharacteristic
130C22×C4×C127C22⋊C412yescomplemented
131C22×C4×C127C2×D411yescharacteristic
132M16×C127—42no—
133M16×C127—42no—
134C2×C8×C127—42no—
135C4⋊C4×C127—22no—
136C4⋊C4×C127—24no—
137C22⋊C4×C127—24no—
138C22⋊C4×C127—22no—
139C22⋊C4×C127—24no—
140C22⋊C4×C127—22no—
141C22⋊C4×C127—24no—
142C22⋊C4×C127—24no—
143C22⋊C4×C127—42no—
144C22⋊C4×C127—42no—
145C22⋊C4×C127—84no—
146C22⋊C4×C127C22⋊C414yescomplemented
147C42×C127—22no—
148C23×C127—21no—
149C23×C127—21no—
150C23×C127—21no—
151C23×C127—21no—
152C23×C127—21no—
153C23×C127—21no—
154C23×C127—21no—
155C23×C127—21no—
156C23×C127—21no—
157C23×C127—21no—
158C23×C127—22no—
159C23×C127—21no—
160C23×C127—21no—
161C23×C127—42no—
162C23×C127—44no—
163C23×C127—44no—
164C23×C127—44no—
165C23×C127—44no—
166C23×C127—44no—
167C23×C127—42no—
168C23×C127—42no—
169C23×C127—82no—
170C23×C127—82no—
171C23×C127—82no—
172C23×C127—81no—
173C23×C127—81no—
174C23×C127—81no—
175C23×C127—81no—
176C23×C127—82no—
177C23×C127—82no—
178C23×C127—82no—
179C23×C127—82no—
180C23×C127—82no—
181C23×C127—82no—
182C23×C127—82no—
183C23×C127—82no—
184C23×C127#9912yes—
185C23×C127C2×C22⋊C412yes—
186C23×C127C2×C22⋊C411yescharacteristic
187C23×C127#12012yes—
188Q8×C127—42no—
189D4×C127—28no—
190D4×C127—44no—
191D4×C127—44no—
192D4×C127—44no—
193D4×C127—42no—
194D4×C127—44no—
195D4×C127—44no—
196D4×C127—48no—
197D4×C127—42no—
198D4×C127—42no—
199D4×C127—84no—
200D4×C127—84no—
201C2×C4×C127—21no—
202C2×C4×C127—21no—
203C2×C4×C127—21no—
204C2×C4×C127—21no—
205C2×C4×C127—24no—
206C2×C4×C127—22no—
207C2×C4×C127—44no—
208C2×C4×C127—44no—
209C2×C4×C127—42no—
210C2×C4×C127—42no—
211C2×C4×C127—82no—
212C2×C4×C127—82no—
213C2×C4×C127—82no—
214C2×C4×C127—84no—
215C2×C4×C127#9912yes—
216C2×C4×C127#12012yes—
217C8×C127—82no—
218C22×C127—22no—
219C22×C127—22no—
220C22×C127—22no—
221C22×C127—22no—
222C22×C127—22no—
223C22×C127—22no—
224C22×C127—41no—
225C22×C127—42no—
226C22×C127—42no—
227C22×C127—42no—
228C22×C127—42no—
229C22×C127—42no—
230C22×C127—41no—
231C22×C127—41no—
232C22×C127—41no—
233C22×C127—44no—
234C22×C127—42no—
235C22×C127—41no—
236C22×C127—42no—
237C22×C127—41no—
238C22×C127—81no—
239C22×C127—81no—
240C22×C127—81no—
241C22×C127—81no—
242C22×C127—81no—
243C22×C127—81no—
244C22×C127—88no—
245C22×C127—88no—
246C22×C127—162no—
247C22×C127—162no—
248C22×C127#37911yescharacteristic
249C22×C127C2×#9911yescharacteristic
250C22×C127C2×#9911yescharacteristic
251C4×C127—22no—
252C4×C127—22no—
253C4×C127—44no—
254C4×C127—42no—
255C4×C127—162no—
256#12705C12711yesmaximal, characteristic, direct product factor, Sylow
257C2×C127—21no—
258C2×C127—21no—
259C2×C127—21no—
260C2×C127—42no—
261C2×C127—42no—
262C2×C127—42no—
263C2×C127—42no—
264C2×C127—81no—
265C2×C127—82no—
266C2×C127#178111yescharacteristic
267#3484C2×C12711yescharacteristic
268#2121C2×C12714yescomplemented
269#1783C2×C12711yescharacteristic, complemented
270#1781C2×C12711yescharacteristic, complemented
271C127#1270511yescharacteristic, direct product factor, Sylow
272C2×H(5,𝔽2)C4×C12711yescharacteristic, complemented
273C23×D4C22×C12711yescharacteristic
274#546—24no—
275D42—24no—
276#535—21no—
277#534—21no—
278#534C22×C12712yes—
279#525—21no—
280C2×#120—21no—
281C2×#120C4×C12712yescomplemented
282#518C4×C12711yescharacteristic, complemented
283C2×#100—21no—
284C2×#100C22×C12712yescomplemented
285C2×#99—21no—
286C2×#99C22×C12712yescomplemented
287#407—21no—
288#379—21no—
289#352—24no—
290AΣL(1,𝔽16)—24no—
291C25C2×C4×C12711yescharacteristic
292C25D4×C12711yescharacteristic
293H(5,𝔽2)—42no—
294H(5,𝔽2)—42no—
295C2×#41—21no—
296C2×#41—22no—
297C22×D4—21no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—24no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4C2×C4×C12711yescharacteristic
305C22×D4C2×C4×C12712yes—
306C22×D4D4×C12711yescharacteristic
307C22×D4D4×C12711yescharacteristic
308C22×D4C23×C12711yescharacteristic
309C23×C4D4×C12711yescharacteristic
310C2×M16—22no—
311#127—22no—
312#124—22no—
313#123—22no—
314#123—22no—
315#123D4×C12714yes—
316C22⋊Q8—22no—
317#121—22no—
318#121—24no—
319#121—24no—
320#121—24no—
321#120—24no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—22no—
326#120—24no—
327#120D4×C12714yes—
328C4×D4—24no—
329C4×D4—24no—
330#117—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4C2×C4×C12712yes—
337#100—24no—
338#100—42no—
339#99—24no—
340#99—42no—
341C22⋊C8—42no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C12711yescharacteristic
352C24C22⋊C4×C12711yescharacteristic
353C24C22⋊C4×C12711yescharacteristic
354C24C22⋊C4×C12712yescomplemented
355C24C22×C4×C12711yescharacteristic
356C24C2×D4×C12711yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C12714yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C12711yescharacteristic
397C22×C4C22⋊C4×C12712yescomplemented
398C22×C4C2×D4×C12711yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C12714yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23#99×C12712yes—
452C23C2×C22⋊C4×C12712yes—
453C23C2×C22⋊C4×C12711yescharacteristic
454C23#120×C12712yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4#99×C12712yes—
483C2×C4#120×C12712yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22#379×C12711yescharacteristic
516C22C2×#99×C12711yescharacteristic
517C22C2×#99×C12711yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C127×#178111yescharacteristic
subgroupG/H
center, socleC2×C127#1781
commutatorC24C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC127×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C23×C127 ◁ C23×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2121C2semidirect
3. C127×#1783C2semidirect
4. C127×#1781C2semidirect
5. C2×H(5,𝔽2)×C127C4semidirect
14. C2×#120×C127C4semidirect
15. #518×C127C4semidirect
17. C2×#100×C127C22semidirect
19. C2×#99×C127C22semidirect
87. C24×C127C22⋊C4semidirect
130. C22×C4×C127C22⋊C4semidirect
146. C22⋊C4×C127C22⋊C4semidirect
256. #12705C127direct (N × H)
268. #2121C2×C127semidirect
269. #1783C2×C127semidirect
270. #1781C2×C127semidirect
271. C127#12705direct (N × H)
272. C2×H(5,𝔽2)C4×C127semidirect
281. C2×#120C4×C127semidirect
282. #518C4×C127semidirect
284. C2×#100C22×C127semidirect
286. C2×#99C22×C127semidirect
354. C24C22⋊C4×C127semidirect
397. C22×C4C22⋊C4×C127semidirect
413. C22⋊C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34842
C127×#21212
C127×#17832
C127×#17812
#12705127