Finite Group Databasebeta

← C127×#9975 #44573648

Subgroups — C127×#9975 #44573648

Order 32512.

742 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3187C211yesmaximal, characteristic, complemented
2C127×#3007C211yesmaximal, characteristic, complemented
3C127×#3006C211yesmaximal, characteristic
4C127×#1780C211yesmaximal, characteristic, complemented
5C127×#1472C211yesmaximal, characteristic, complemented
6C127×#1471C211yesmaximal, characteristic, complemented
7C127×#1293C211yesmaximal, characteristic, complemented
8C22×#41×C127C2211yescharacteristic
9C2×#137×C127—21no—
10C2×AGL(1,ℤ/8)×C127—21no—
11C2×C22⋊Q8×C127C2211yescharacteristic
12C2×#121×C127C2211yescharacteristic
13#483×C127—21no—
14#481×C127—21no—
15#465×C127—22no—
16#452×C127—22no—
17#451×C127—21no—
18#450×C127—21no—
19#450×C127—21no—
20#449×C127—22no—
21#448×C127—21no—
22#448×C127—21no—
23#447×C127—21no—
24#417×C127—21no—
25#417×C127—21no—
26#407×C127C2211yescharacteristic
27#407×C127C2211yescharacteristic
28C2×C22⋊C8×C127—22no—
29C2×C22⋊C8×C127C2211yescharacteristic
30C2×C22⋊C8×C127C2211yescharacteristic
31#324×C127—22no—
32#323×C127—22no—
33C2×#41×C127—21no—
34C2×#41×C127—21no—
35C2×#41×C127—21no—
36C2×#41×C127—21no—
37C2×#41×C127—44no—
38C22×Q8×C127D411yescharacteristic
39C22×Q8×C127D411yescharacteristic
40C22×D4×C127—21no—
41C22×D4×C127—21no—
42C22×D4×C127D411yescharacteristic
43C22×D4×C127D411yescharacteristic
44C23×C4×C127—21no—
45C23×C4×C127C2311yescharacteristic
46#137×C127—44no—
47AGL(1,ℤ/8)×C127—44no—
48C2×Q16×C127—22no—
49C2×SD16×C127—22no—
50C2×SD16×C127—22no—
51C2×SD16×C127—22no—
52C2×D8×C127—22no—
53C2×M16×C127—21no—
54C2×M16×C127—21no—
55C22×C8×C127—21no—
56C22×C8×C127D412yes—
57C22⋊Q8×C127—21no—
58C22⋊Q8×C127—21no—
59C22⋊Q8×C127—21no—
60C22⋊Q8×C127—21no—
61#121×C127—21no—
62#121×C127—21no—
63#121×C127—21no—
64#121×C127—21no—
65C2×C4⋊C4×C127—21no—
66C2×C4⋊C4×C127D411yescharacteristic
67C2×C4⋊C4×C127D411yescharacteristic
68C2×C22⋊C4×C127—21no—
69C2×C22⋊C4×C127—21no—
70#106×C127—22no—
71#103×C127—22no—
72#103×C127—22no—
73#103×C127—22no—
74#102×C127—22no—
75#102×C127—22no—
76#102×C127—22no—
77C22⋊C8×C127—21no—
78C22⋊C8×C127—21no—
79C22⋊C8×C127—21no—
80C22⋊C8×C127—21no—
81C22⋊C8×C127—21no—
82C22⋊C8×C127—21no—
83C22⋊C8×C127—44no—
84C22⋊C8×C127D412yes—
85C22⋊C8×C127D412yes—
86C24×C127—21no—
87C24×C127—21no—
88C24×C127#4111yescharacteristic
89#41×C127—24no—
90#41×C127—24no—
91#41×C127—42no—
92#41×C127—42no—
93#41×C127—42no—
94#41×C127—42no—
95#41×C127—42no—
96#41×C127—42no—
97C2×Q8×C127—21no—
98C2×Q8×C127—21no—
99C2×Q8×C127—21no—
100C2×Q8×C127—21no—
101C2×Q8×C127—42no—
102C2×Q8×C127—42no—
103C2×D4×C127—21no—
104C2×D4×C127—21no—
105C2×D4×C127—21no—
106C2×D4×C127—21no—
107C2×D4×C127—42no—
108C2×D4×C127—42no—
109C2×D4×C127—42no—
110C2×D4×C127—42no—
111C2×D4×C127—41no—
112C2×D4×C127—41no—
113C2×D4×C127—41no—
114C2×D4×C127—41no—
115C2×D4×C127—82no—
116C22×C4×C127—24no—
117C22×C4×C127—21no—
118C22×C4×C127—21no—
119C22×C4×C127—21no—
120C22×C4×C127—21no—
121C22×C4×C127—42no—
122C22×C4×C127—42no—
123C22×C4×C127—41no—
124C22×C4×C127—41no—
125C22×C4×C127C2×D411yescharacteristic
126C22×C4×C127C2×D411yescharacteristic
127C22×C4×C127C2×D411yescharacteristic
128C22×C4×C127C2×D411yescharacteristic
129C22×C4×C127C2×D411yescharacteristic
130C22×C4×C127C2×D411yescharacteristic
131Q16×C127—44no—
132SD16×C127—44no—
133SD16×C127—44no—
134SD16×C127—82no—
135D8×C127—44no—
136M16×C127—42no—
137M16×C127—42no—
138C2×C8×C127—22no—
139C2×C8×C127—22no—
140C2×C8×C127—22no—
141C2×C8×C127—42no—
142C2×C8×C127—42no—
143C2×C8×C127—41no—
144C4⋊C4×C127—21no—
145C4⋊C4×C127—21no—
146C4⋊C4×C127—21no—
147C4⋊C4×C127—21no—
148C4⋊C4×C127—41no—
149C4⋊C4×C127—41no—
150C22⋊C4×C127—41no—
151C22⋊C4×C127—41no—
152C22⋊C4×C127—41no—
153C22⋊C4×C127—41no—
154C23×C127—22no—
155C23×C127—22no—
156C23×C127—41no—
157C23×C127—41no—
158C23×C127—41no—
159C23×C127—42no—
160C23×C127—42no—
161C23×C127—41no—
162C23×C127—42no—
163C23×C127—41no—
164C23×C127—81no—
165C23×C127—81no—
166C23×C127#12111yescharacteristic
167C23×C127#12111yescharacteristic
168C23×C127#12111yescharacteristic
169Q8×C127—22no—
170Q8×C127—22no—
171Q8×C127—41no—
172Q8×C127—41no—
173Q8×C127—42no—
174Q8×C127—42no—
175D4×C127—22no—
176D4×C127—22no—
177D4×C127—41no—
178D4×C127—41no—
179D4×C127—42no—
180D4×C127—44no—
181D4×C127—44no—
182D4×C127—42no—
183D4×C127—82no—
184D4×C127—82no—
185C2×C4×C127—22no—
186C2×C4×C127—22no—
187C2×C4×C127—22no—
188C2×C4×C127—22no—
189C2×C4×C127—41no—
190C2×C4×C127—41no—
191C2×C4×C127—41no—
192C2×C4×C127—44no—
193C2×C4×C127—41no—
194C2×C4×C127—41no—
195C2×C4×C127—44no—
196C2×C4×C127—42no—
197C2×C4×C127—41no—
198C2×C4×C127—41no—
199C2×C4×C127—41no—
200C2×C4×C127—81no—
201C2×C4×C127—81no—
202C2×C4×C127#12011yescharacteristic
203C2×C4×C127#12011yescharacteristic
204C2×C4×C127#12011yescharacteristic
205C2×C4×C127#12711yescharacteristic
206C8×C127—42no—
207C8×C127—42no—
208C8×C127—42no—
209C8×C127—81no—
210C22×C127—21no—
211C22×C127—21no—
212C22×C127—21no—
213C22×C127—21no—
214C22×C127—21no—
215C22×C127—21no—
216C22×C127—21no—
217C22×C127—21no—
218C22×C127—44no—
219C22×C127—42no—
220C22×C127—82no—
221C22×C127—82no—
222C22×C127—81no—
223C22×C127—81no—
224C22×C127—81no—
225C22×C127—81no—
226C22×C127—81no—
227C22×C127—162no—
228C22×C127#39211yescharacteristic
229C22×C127#47112yes—
230C4×C127—21no—
231C4×C127—21no—
232C4×C127—21no—
233C4×C127—42no—
234C4×C127—81no—
235C4×C127—81no—
236C4×C127—81no—
237C4×C127B(4,𝔽2)12yes—
238#9975C12711yesmaximal, characteristic, direct product factor, Sylow
239C2×C127—22no—
240C2×C127—41no—
241C2×C127—41no—
242C2×C127—82no—
243C2×C127—161no—
244C2×C127#201711yescharacteristic
245C2×C127#201911yescharacteristic
246C2×C127#202111yescharacteristic
247#3187C2×C12711yescharacteristic, complemented
248#3007C2×C12711yescharacteristic, complemented
249#3006C2×C12711yescharacteristic
250#1780C2×C12711yescharacteristic, complemented
251#1472C2×C12711yescharacteristic, complemented
252#1471C2×C12711yescharacteristic, complemented
253#1293C2×C12711yescharacteristic, complemented
254C127#997511yescharacteristic, direct product factor, Sylow
255C22×#41C22×C12711yescharacteristic
256C2×#137—21no—
257C2×AGL(1,ℤ/8)—21no—
258C2×C22⋊Q8C22×C12711yescharacteristic
259C2×#121C22×C12711yescharacteristic
260#483—21no—
261#481—21no—
262#465—22no—
263#452—22no—
264#451—21no—
265#450—21no—
266#450—21no—
267#449—22no—
268#448—21no—
269#448—21no—
270#447—21no—
271#417—21no—
272#417—21no—
273#407C22×C12711yescharacteristic
274#407C22×C12711yescharacteristic
275C2×C22⋊C8—22no—
276C2×C22⋊C8C22×C12711yescharacteristic
277C2×C22⋊C8C22×C12711yescharacteristic
278#324—22no—
279#323—22no—
280C2×#41—21no—
281C2×#41—21no—
282C2×#41—21no—
283C2×#41—21no—
284C2×#41—44no—
285C22×Q8D4×C12711yescharacteristic
286C22×Q8D4×C12711yescharacteristic
287C22×D4—21no—
288C22×D4—21no—
289C22×D4D4×C12711yescharacteristic
290C22×D4D4×C12711yescharacteristic
291C23×C4—21no—
292C23×C4C23×C12711yescharacteristic
293#137—44no—
294AGL(1,ℤ/8)—44no—
295C2×Q16—22no—
296C2×SD16—22no—
297C2×SD16—22no—
298C2×SD16—22no—
299C2×D8—22no—
300C2×M16—21no—
301C2×M16—21no—
302C22×C8—21no—
303C22×C8D4×C12712yes—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308#121—21no—
309#121—21no—
310#121—21no—
311#121—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4D4×C12711yescharacteristic
314C2×C4⋊C4D4×C12711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317#106—22no—
318#103—22no—
319#103—22no—
320#103—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C22⋊C8—44no—
331C22⋊C8D4×C12712yes—
332C22⋊C8D4×C12712yes—
333C24—21no—
334C24—21no—
335C24#41×C12711yescharacteristic
336#41—24no—
337#41—24no—
338#41—42no—
339#41—42no—
340#41—42no—
341#41—42no—
342#41—42no—
343#41—42no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—82no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4C2×D4×C12711yescharacteristic
373C22×C4C2×D4×C12711yescharacteristic
374C22×C4C2×D4×C12711yescharacteristic
375C22×C4C2×D4×C12711yescharacteristic
376C22×C4C2×D4×C12711yescharacteristic
377C22×C4C2×D4×C12711yescharacteristic
378Q16—44no—
379SD16—44no—
380SD16—44no—
381SD16—82no—
382D8—44no—
383M16—42no—
384M16—42no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—81no—
412C23—81no—
413C23#121×C12711yescharacteristic
414C23#121×C12711yescharacteristic
415C23#121×C12711yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—41no—
419Q8—41no—
420Q8—42no—
421Q8—42no—
422D4—22no—
423D4—22no—
424D4—41no—
425D4—41no—
426D4—42no—
427D4—44no—
428D4—44no—
429D4—42no—
430D4—82no—
431D4—82no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4#120×C12711yescharacteristic
450C2×C4#120×C12711yescharacteristic
451C2×C4#120×C12711yescharacteristic
452C2×C4#127×C12711yescharacteristic
453C8—42no—
454C8—42no—
455C8—42no—
456C8—81no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—44no—
466C22—42no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—162no—
475C22#392×C12711yescharacteristic
476C22#471×C12712yes—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4B(4,𝔽2)×C12712yes—
485C2—22no—
486C2—41no—
487C2—41no—
488C2—82no—
489C2—161no—
490C2C127×#201711yescharacteristic
491C2C127×#201911yescharacteristic
492C2C127×#202111yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3006C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3187C2semidirect
2. C127×#3007C2semidirect
4. C127×#1780C2semidirect
5. C127×#1472C2semidirect
6. C127×#1471C2semidirect
7. C127×#1293C2semidirect
238. #9975C127direct (N × H)
247. #3187C2×C127semidirect
248. #3007C2×C127semidirect
250. #1780C2×C127semidirect
251. #1472C2×C127semidirect
252. #1471C2×C127semidirect
253. #1293C2×C127semidirect
254. C127#9975direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31872
C127×#30072
C127×#30062
C127×#17802
C127×#14722
C127×#14712
C127×#12932
#9975127