Finite Group Databasebeta

← C127×#21814 #44585487

Subgroups — C127×#21814 #44585487

Order 32512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536×C127C211yesmaximal, characteristic
2C2×#433×C127C211yesmaximal, characteristic, complemented
3C127×#2314C211yesmaximal, characteristic, complemented
4C127×#1983C212yesmaximal, complemented
5C127×#1981C212yesmaximal, complemented
6C127×#1960C211yesmaximal, characteristic, complemented
7C127×#1959C211yesmaximal, characteristic, complemented
8C127×#1958C211yesmaximal, characteristic, complemented
9C127×#1956C211yesmaximal, characteristic, complemented
10C127×#1788C212yesmaximal
11C127×#1787C212yesmaximal, complemented
12C22×#41×C127C2211yescharacteristic
13C23×Q8×C127C2211yescharacteristic
14#536×C127—24no—
15#536×C127C418yescomplemented
16C2×C4⋊Q8×C127C2211yescharacteristic
17C2×C4⋊Q8×C127C2211yescharacteristic
18C2×#124×C127C2211yescharacteristic
19C2×#124×C127C2211yescharacteristic
20C2×#123×C127—22no—
21C2×C22⋊Q8×C127—22no—
22C2×#117×C127C2211yescharacteristic
23#433×C127—24no—
24C2×C4⋊C8×C127C2212yes—
25C2×#103×C127C2212yes—
26C2×#103×C127C2212yes—
27C2×#103×C127C2212yes—
28C2×#103×C127C2212yes—
29C2×#103×C127C2212yes—
30C2×#103×C127C2212yes—
31C2×#102×C127C2212yes—
32C2×#102×C127C2212yes—
33C2×C22⋊C8×C127C2212yes—
34C2×#97×C127C2211yescharacteristic
35C2×C4×C8×C127C2211yescharacteristic
36#388×C127—22no—
37#387×C127—22no—
38#384×C127C2211yescharacteristic
39#382×C127C2211yescharacteristic
40#380×C127C2212yes—
41C4×C4⋊C4×C127C2212yes—
42C2×#41×C127—22no—
43C2×#41×C127—42no—
44C2×#41×C127—42no—
45C2×#41×C127C2×C414yes—
46C22×Q8×C127—21no—
47C22×Q8×C127—22no—
48C22×Q8×C127—42no—
49C22×Q8×C127—42no—
50C22×Q8×C127C2×C414yes—
51C22×Q8×C127C2311yescharacteristic
52C22×Q8×C127C2311yescharacteristic
53C22×Q8×C127C2311yescharacteristic
54C22×D4×C127—21no—
55C22×D4×C127C2311yescharacteristic
56C23×C4×C127—21no—
57C23×C4×C127—21no—
58C23×C4×C127C2311yescharacteristic
59C22×C8×C127C2312yes—
60C22×C8×C127C2312yes—
61C4⋊Q8×C127—22no—
62C4⋊Q8×C127—22no—
63C4⋊Q8×C127C2×C414yes—
64C4⋊Q8×C127C2×C414yes—
65#124×C127—22no—
66#124×C127—22no—
67#124×C127C2×C414yes—
68#124×C127C2×C414yes—
69#123×C127—216no—
70C22⋊Q8×C127—216no—
71#117×C127C2×C414yes—
72#117×C127D414yes—
73C2×C4⋊C4×C127—22no—
74C2×C4⋊C4×C127—22no—
75C2×C4⋊C4×C127—22no—
76C2×C4⋊C4×C127D412yes—
77C2×C4⋊C4×C127C2312yes—
78C2×C4⋊C4×C127C2312yes—
79C2×C22⋊C4×C127—22no—
80C2×C22⋊C4×C127—22no—
81C2×C22⋊C4×C127—22no—
82C2×C22⋊C4×C127D412yes—
83C2×C42×C127—22no—
84C2×C42×C127C2311yescharacteristic
85C2×C42×C127C2311yescharacteristic
86C4⋊C8×C127—24no—
87#103×C127—42no—
88#103×C127—42no—
89#103×C127—42no—
90#103×C127—42no—
91#103×C127—42no—
92#103×C127—42no—
93#102×C127—42no—
94#102×C127—42no—
95C22⋊C8×C127—24no—
96#97×C127—22no—
97C4×C8×C127—22no—
98#95×C127—22no—
99#95×C127—22no—
100#95×C127—22no—
101C24×C127—21no—
102C24×C127C2×D411yescharacteristic
103#41×C127—48no—
104#41×C127—48no—
105C2×Q8×C127—24no—
106C2×Q8×C127—21no—
107C2×Q8×C127—21no—
108C2×Q8×C127—21no—
109C2×Q8×C127—22no—
110C2×Q8×C127—22no—
111C2×Q8×C127—22no—
112C2×Q8×C127—22no—
113C2×Q8×C127—22no—
114C2×Q8×C127—22no—
115C2×Q8×C127—48no—
116C2×Q8×C127—42no—
117C2×Q8×C127—42no—
118C2×Q8×C127—48no—
119C2×Q8×C127C22×C412yes—
120C2×Q8×C127C22×C412yes—
121C2×Q8×C127C22×C412yes—
122C2×D4×C127—24no—
123C2×D4×C127—21no—
124C2×D4×C127—22no—
125C2×D4×C127—22no—
126C2×D4×C127—42no—
127C2×D4×C127—42no—
128C2×D4×C127C22×C412yes—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127—24no—
132C22×C4×C127—21no—
133C22×C4×C127—21no—
134C22×C4×C127—21no—
135C22×C4×C127—24no—
136C22×C4×C127—22no—
137C22×C4×C127—22no—
138C22×C4×C127—42no—
139C22×C4×C127—42no—
140C22×C4×C127—42no—
141C22×C4×C127—42no—
142C22×C4×C127C22×C412yes—
143C22×C4×C127C2×D411yescharacteristic
144C22×C4×C127C2×D412yes—
145C22×C4×C127C2×D412yes—
146C22×C4×C127C2×D412yes—
147C22×C4×C127C2411yescharacteristic
148C2×C8×C127—22no—
149C2×C8×C127—22no—
150C2×C8×C127—22no—
151C2×C8×C127—22no—
152C2×C8×C127—22no—
153C2×C8×C127—22no—
154C4⋊C4×C127—28no—
155C4⋊C4×C127—28no—
156C4⋊C4×C127—22no—
157C4⋊C4×C127—22no—
158C4⋊C4×C127—22no—
159C4⋊C4×C127—22no—
160C4⋊C4×C127—82no—
161C4⋊C4×C127C22⋊C418yes—
162C22⋊C4×C127—28no—
163C22⋊C4×C127—28no—
164C22⋊C4×C127—82no—
165C22⋊C4×C127C22⋊C418yes—
166C42×C127—82no—
167C42×C127C22×C412yes—
168C42×C127C22×C412yes—
169C42×C127C2×D412yes—
170C42×C127C2×D412yes—
171C23×C127—21no—
172C23×C127—21no—
173C23×C127—22no—
174C23×C127—22no—
175C23×C127—22no—
176C23×C127—41no—
177C23×C127—41no—
178C23×C127—42no—
179C23×C127—42no—
180C23×C127C2×C22⋊C412yes—
181C23×C127C22×D411yescharacteristic
182Q8×C127—24no—
183Q8×C127—24no—
184Q8×C127—24no—
185Q8×C127—48no—
186Q8×C127—42no—
187Q8×C127—42no—
188Q8×C127—42no—
189D4×C127—24no—
190D4×C127—48no—
191D4×C127—42no—
192C2×C4×C127—21no—
193C2×C4×C127—21no—
194C2×C4×C127—22no—
195C2×C4×C127—22no—
196C2×C4×C127—22no—
197C2×C4×C127—22no—
198C2×C4×C127—24no—
199C2×C4×C127—24no—
200C2×C4×C127—22no—
201C2×C4×C127—22no—
202C2×C4×C127—22no—
203C2×C4×C127—48no—
204C2×C4×C127—48no—
205C2×C4×C127—41no—
206C2×C4×C127—41no—
207C2×C4×C127—41no—
208C2×C4×C127—41no—
209C2×C4×C127—41no—
210C2×C4×C127—41no—
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×C127C2×C22⋊C414yes—
218C2×C4×C127C2×C22⋊C414yes—
219C2×C4×C127C2×C22⋊C412yes—
220C2×C4×C127C23×C411yescharacteristic
221C2×C4×C127C22×D411yescharacteristic
222C2×C4×C127H(5,𝔽2)12yes—
223C2×C4×C127H(5,𝔽2)12yes—
224C8×C127—42no—
225C8×C127—42no—
226C22×C127—22no—
227C22×C127—24no—
228C22×C127—41no—
229C22×C127—41no—
230C22×C127—42no—
231C22×C127—42no—
232C22×C127—44no—
233C22×C127—81no—
234C22×C127—81no—
235C22×C127—82no—
236C22×C127C22×C22⋊C411yescharacteristic
237C22×C127#53511yescharacteristic
238C22×C127#53511yescharacteristic
239C22×C127#57712yes—
240C22×C127#57812yes—
241C4×C127—22no—
242C4×C127—24no—
243C4×C127—24no—
244C4×C127—42no—
245C4×C127—42no—
246C4×C127—42no—
247C4×C127—82no—
248C4×C127—82no—
249C4×C127#51812yes—
250C4×C127#53512yes—
251#21814C12711yesmaximal, characteristic, direct product factor, Sylow
252C2×C127—42no—
253C2×C127—82no—
254C2×C127#230911yescharacteristic
255C2×C127#289812yes—
256C2×C127#322111yescharacteristic
257C2×C127#322311yescharacteristic
258C2×C127#322512yes—
259C2×#536C2×C12711yescharacteristic
260C2×#433C2×C12711yescharacteristic, complemented
261#2314C2×C12711yescharacteristic, complemented
262#1983C2×C12712yescomplemented
263#1981C2×C12712yescomplemented
264#1960C2×C12711yescharacteristic, complemented
265#1959C2×C12711yescharacteristic, complemented
266#1958C2×C12711yescharacteristic, complemented
267#1956C2×C12711yescharacteristic, complemented
268#1788C2×C12712yes—
269#1787C2×C12712yescomplemented
270C127#2181411yescharacteristic, direct product factor, Sylow
271C22×#41C22×C12711yescharacteristic
272C23×Q8C22×C12711yescharacteristic
273#536—24no—
274#536C4×C12718yescomplemented
275C2×C4⋊Q8C22×C12711yescharacteristic
276C2×C4⋊Q8C22×C12711yescharacteristic
277C2×#124C22×C12711yescharacteristic
278C2×#124C22×C12711yescharacteristic
279C2×#123—22no—
280C2×C22⋊Q8—22no—
281C2×#117C22×C12711yescharacteristic
282#433—24no—
283C2×C4⋊C8C22×C12712yes—
284C2×#103C22×C12712yes—
285C2×#103C22×C12712yes—
286C2×#103C22×C12712yes—
287C2×#103C22×C12712yes—
288C2×#103C22×C12712yes—
289C2×#103C22×C12712yes—
290C2×#102C22×C12712yes—
291C2×#102C22×C12712yes—
292C2×C22⋊C8C22×C12712yes—
293C2×#97C22×C12711yescharacteristic
294C2×C4×C8C22×C12711yescharacteristic
295#388—22no—
296#387—22no—
297#384C22×C12711yescharacteristic
298#382C22×C12711yescharacteristic
299#380C22×C12712yes—
300C4×C4⋊C4C22×C12712yes—
301C2×#41—22no—
302C2×#41—42no—
303C2×#41—42no—
304C2×#41C2×C4×C12714yes—
305C22×Q8—21no—
306C22×Q8—22no—
307C22×Q8—42no—
308C22×Q8—42no—
309C22×Q8C2×C4×C12714yes—
310C22×Q8C23×C12711yescharacteristic
311C22×Q8C23×C12711yescharacteristic
312C22×Q8C23×C12711yescharacteristic
313C22×D4—21no—
314C22×D4C23×C12711yescharacteristic
315C23×C4—21no—
316C23×C4—21no—
317C23×C4C23×C12711yescharacteristic
318C22×C8C23×C12712yes—
319C22×C8C23×C12712yes—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8C2×C4×C12714yes—
323C4⋊Q8C2×C4×C12714yes—
324#124—22no—
325#124—22no—
326#124C2×C4×C12714yes—
327#124C2×C4×C12714yes—
328#123—216no—
329C22⋊Q8—216no—
330#117C2×C4×C12714yes—
331#117D4×C12714yes—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4D4×C12712yes—
336C2×C4⋊C4C23×C12712yes—
337C2×C4⋊C4C23×C12712yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4D4×C12712yes—
342C2×C42—22no—
343C2×C42C23×C12711yescharacteristic
344C2×C42C23×C12711yescharacteristic
345C4⋊C8—24no—
346#103—42no—
347#103—42no—
348#103—42no—
349#103—42no—
350#103—42no—
351#103—42no—
352#102—42no—
353#102—42no—
354C22⋊C8—24no—
355#97—22no—
356C4×C8—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360C24—21no—
361C24C2×D4×C12711yescharacteristic
362#41—48no—
363#41—48no—
364C2×Q8—24no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—48no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—48no—
378C2×Q8C22×C4×C12712yes—
379C2×Q8C22×C4×C12712yes—
380C2×Q8C22×C4×C12712yes—
381C2×D4—24no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4C22×C4×C12712yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4C22×C4×C12712yes—
402C22×C4C2×D4×C12711yescharacteristic
403C22×C4C2×D4×C12712yes—
404C22×C4C2×D4×C12712yes—
405C22×C4C2×D4×C12712yes—
406C22×C4C24×C12711yescharacteristic
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—82no—
420C4⋊C4C22⋊C4×C12718yes—
421C22⋊C4—28no—
422C22⋊C4—28no—
423C22⋊C4—82no—
424C22⋊C4C22⋊C4×C12718yes—
425C42—82no—
426C42C22×C4×C12712yes—
427C42C22×C4×C12712yes—
428C42C2×D4×C12712yes—
429C42C2×D4×C12712yes—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23C2×C22⋊C4×C12712yes—
440C23C22×D4×C12711yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—24no—
444Q8—48no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448D4—24no—
449D4—48no—
450D4—42no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4C2×C22⋊C4×C12714yes—
477C2×C4C2×C22⋊C4×C12714yes—
478C2×C4C2×C22⋊C4×C12712yes—
479C2×C4C23×C4×C12711yescharacteristic
480C2×C4C22×D4×C12711yescharacteristic
481C2×C4H(5,𝔽2)×C12712yes—
482C2×C4H(5,𝔽2)×C12712yes—
483C8—42no—
484C8—42no—
485C22—24no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—42no—
490C22—44no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—82no—
495C22C22×C22⋊C4×C12711yescharacteristic
496C22#535×C12711yescharacteristic
497C22#535×C12711yescharacteristic
498C22#577×C12712yes—
499C22#578×C12712yes—
500C4—24no—
501C4—24no—
502C4—22no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—82no—
508C4#518×C12712yes—
509C4#535×C12712yes—
510C2—42no—
511C2—82no—
512C2C127×#230911yescharacteristic
513C2C127×#289812yes—
514C2C127×#322111yescharacteristic
515C2C127×#322311yescharacteristic
516C2C127×#322512yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC22×#41×C127C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×#117×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#433×C127C2semidirect
3. C127×#2314C2semidirect
4. C127×#1983C2semidirect
5. C127×#1981C2semidirect
6. C127×#1960C2semidirect
7. C127×#1959C2semidirect
8. C127×#1958C2semidirect
9. C127×#1956C2semidirect
11. C127×#1787C2semidirect
15. #536×C127C4semidirect
251. #21814C127direct (N × H)
260. C2×#433C2×C127semidirect
261. #2314C2×C127semidirect
262. #1983C2×C127semidirect
263. #1981C2×C127semidirect
264. #1960C2×C127semidirect
265. #1959C2×C127semidirect
266. #1958C2×C127semidirect
267. #1956C2×C127semidirect
269. #1787C2×C127semidirect
270. C127#21814direct (N × H)
274. #536C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1272
C2×#433×C1272
C127×#23142
C127×#19832
C127×#19812
C127×#19602
C127×#19592
C127×#19582
C127×#19562
C127×#17882
C127×#17872
#21814127