Finite Group Databasebeta

← C127×#19658 #44583331

Subgroups — C127×#19658 #44583331

Order 32512.

776 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3230C211yesmaximal, characteristic, complemented
2C127×#3208C211yesmaximal, characteristic, complemented
3C127×#3206C211yesmaximal, characteristic, complemented
4C127×#3205C211yesmaximal, characteristic
5C127×#3131C211yesmaximal, characteristic, complemented
6C2×#486×C127C211yesmaximal, characteristic, complemented
7C127×#1719C212yesmaximal, complemented
8C127×#1717C212yesmaximal, complemented
9C127×#1711C212yesmaximal, complemented
10C127×#1706C212yesmaximal, complemented
11C127×#1450C211yesmaximal, characteristic, complemented
12#533×C127C2211yescharacteristic
13#532×C127C2211yescharacteristic
14C2×C4⋊Q8×C127C2211yescharacteristic
15C2×#127×C127C2211yescharacteristic
16#500×C127C2212yescomplemented
17#493×C127C2212yes—
18#492×C127C2212yes—
19#492×C127C2212yescomplemented
20#488×C127—22no—
21#487×C127C2211yescharacteristic, complemented
22#487×C127C2211yescharacteristic, complemented
23#487×C127C2212yescomplemented
24#487×C127C2212yescomplemented
25#486×C127—22no—
26#486×C127C2212yes—
27#486×C127C2212yes—
28#486×C127C2212yes—
29#486×C127C2212yes—
30#486×C127C2211yescharacteristic, complemented
31#486×C127C2211yescharacteristic, complemented
32#484×C127—22no—
33#477×C127—22no—
34#461×C127—22no—
35#460×C127—22no—
36#459×C127—22no—
37#450×C127—22no—
38#448×C127—22no—
39#447×C127—22no—
40#431×C127C2211yescharacteristic
41#431×C127C2211yescharacteristic
42#416×C127—22no—
43C2×#102×C127—22no—
44C2×C4×C8×C127C2211yescharacteristic
45#322×C127C2212yescomplemented
46C82×C127C2212yescomplemented
47C2×#41×C127—21no—
48C22×Q8×C127—21no—
49C22×D4×C127—21no—
50C22×D4×C127—21no—
51C22×D4×C127—21no—
52C2×SD16×C127—24no—
53C2×SD16×C127—24no—
54C2×D8×C127—24no—
55C22×C8×C127D412yes—
56C4⋊Q8×C127—21no—
57C4⋊Q8×C127—21no—
58C4⋊Q8×C127—21no—
59C4⋊Q8×C127C2311yescharacteristic
60C4⋊Q8×C127C2312yes—
61C4⋊Q8×C127C2311yescharacteristic
62C4⋊Q8×C127C2311yescharacteristic
63#127×C127—21no—
64#127×C127—21no—
65#127×C127—21no—
66#127×C127C2312yes—
67#127×C127C2311yescharacteristic
68#125×C127—21no—
69#124×C127—21no—
70C22⋊Q8×C127—21no—
71C22⋊Q8×C127—21no—
72#121×C127—21no—
73#121×C127—21no—
74C4×Q8×C127—21no—
75C4×Q8×C127—21no—
76C4×D4×C127—21no—
77C4×D4×C127—21no—
78#117×C127—21no—
79C2×C4⋊C4×C127—21no—
80C2×C4⋊C4×C127—21no—
81C2×C42×C127C2311yescharacteristic
82#107×C127—24no—
83C4⋊C8×C127D412yes—
84C4⋊C8×C127D412yes—
85#103×C127—22no—
86#103×C127—22no—
87#103×C127—24no—
88#103×C127—24no—
89#102×C127—24no—
90#102×C127—24no—
91#102×C127—22no—
92#102×C127—22no—
93#102×C127—24no—
94#102×C127—24no—
95C22⋊C8×C127D412yes—
96C22⋊C8×C127D412yes—
97C4×C8×C127D412yes—
98C4×C8×C127C2312yes—
99C4×C8×C127C2311yescharacteristic
100C4×C8×C127C2312yes—
101C4×C8×C127C2311yescharacteristic
102C24×C127—41no—
103#41×C127—44no—
104C2×Q8×C127—21no—
105C2×Q8×C127—21no—
106C2×Q8×C127—22no—
107C2×Q8×C127—21no—
108C2×Q8×C127—41no—
109C2×Q8×C127—82no—
110C2×D4×C127—21no—
111C2×D4×C127—22no—
112C2×D4×C127—22no—
113C2×D4×C127—22no—
114C2×D4×C127—21no—
115C2×D4×C127—21no—
116C2×D4×C127—41no—
117C2×D4×C127—41no—
118C2×D4×C127—41no—
119C2×D4×C127—41no—
120C2×D4×C127—44no—
121C2×D4×C127—82no—
122C2×D4×C127—82no—
123C22×C4×C127—41no—
124C22×C4×C127—41no—
125C22×C4×C127C2×D411yescharacteristic
126C22×C4×C127C2×D411yescharacteristic
127C22×C4×C127C2×D411yescharacteristic
128SD16×C127—84no—
129SD16×C127—84no—
130D8×C127—84no—
131C2×C8×C127—24no—
132C2×C8×C127C2×D412yes—
133C2×C8×C127C2×D412yes—
134C2×C8×C127C2×D414yes—
135C2×C8×C127C2×D414yes—
136C4⋊C4×C127—22no—
137C4⋊C4×C127—21no—
138C4⋊C4×C127—21no—
139C4⋊C4×C127—21no—
140C4⋊C4×C127—21no—
141C4⋊C4×C127—21no—
142C4⋊C4×C127—22no—
143C4⋊C4×C127—21no—
144C4⋊C4×C127—41no—
145C4⋊C4×C127—41no—
146C4⋊C4×C127—41no—
147C4⋊C4×C127—41no—
148C22⋊C4×C127—41no—
149C22⋊C4×C127—41no—
150C22⋊C4×C127—41no—
151C42×C127—41no—
152C42×C127C2×D411yescharacteristic
153C42×C127C2×D411yescharacteristic
154C42×C127C2×D411yescharacteristic
155C42×C127C2411yescharacteristic
156C23×C127—41no—
157C23×C127—42no—
158C23×C127—82no—
159C23×C127—82no—
160C23×C127—82no—
161C23×C127#12711yescharacteristic
162Q8×C127—42no—
163Q8×C127—42no—
164Q8×C127—44no—
165Q8×C127—42no—
166Q8×C127—82no—
167D4×C127—44no—
168D4×C127—42no—
169D4×C127—42no—
170D4×C127—44no—
171D4×C127—42no—
172D4×C127—44no—
173D4×C127—82no—
174D4×C127—82no—
175D4×C127—82no—
176D4×C127—82no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—41no—
180C2×C4×C127—41no—
181C2×C4×C127—41no—
182C2×C4×C127—42no—
183C2×C4×C127—82no—
184C2×C4×C127—82no—
185C2×C4×C127#12711yescharacteristic
186C2×C4×C127#12711yescharacteristic
187C2×C4×C127#12711yescharacteristic
188C2×C4×C127#13514yes—
189C2×C4×C127C22×D411yescharacteristic
190C2×C4×C127C22×D411yescharacteristic
191C2×C4×C127C22×D411yescharacteristic
192C8×C127—22no—
193C8×C127—22no—
194C8×C127—24no—
195C8×C127—24no—
196C22×C127—21no—
197C22×C127—21no—
198C22×C127—81no—
199C22×C127—82no—
200C22×C127—82no—
201C22×C127—81no—
202C22×C127—82no—
203C22×C127—81no—
204C22×C127—164no—
205C22×C127#49512yes—
206C22×C127C2×#12711yescharacteristic
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—81no—
210C4×C127—81no—
211C4×C127—82no—
212C4×C127—81no—
213C4×C127#49512yes—
214C4×C127C2×#13512yes—
215C4×C127#57612yes—
216C4×C127#57712yes—
217#19658C12711yesmaximal, characteristic, direct product factor, Sylow
218C2×C127—22no—
219C2×C127—161no—
220C2×C127—162no—
221C2×C127C2×#49511yescharacteristic
222C2×C127#315411yescharacteristic
223C2×C127#315511yescharacteristic
224#3230C2×C12711yescharacteristic, complemented
225#3208C2×C12711yescharacteristic, complemented
226#3206C2×C12711yescharacteristic, complemented
227#3205C2×C12711yescharacteristic
228#3131C2×C12711yescharacteristic, complemented
229C2×#486C2×C12711yescharacteristic, complemented
230#1719C2×C12712yescomplemented
231#1717C2×C12712yescomplemented
232#1711C2×C12712yescomplemented
233#1706C2×C12712yescomplemented
234#1450C2×C12711yescharacteristic, complemented
235C127#1965811yescharacteristic, direct product factor, Sylow
236#533C22×C12711yescharacteristic
237#532C22×C12711yescharacteristic
238C2×C4⋊Q8C22×C12711yescharacteristic
239C2×#127C22×C12711yescharacteristic
240#500C22×C12712yescomplemented
241#493C22×C12712yes—
242#492C22×C12712yes—
243#492C22×C12712yescomplemented
244#488—22no—
245#487C22×C12711yescharacteristic, complemented
246#487C22×C12711yescharacteristic, complemented
247#487C22×C12712yescomplemented
248#487C22×C12712yescomplemented
249#486—22no—
250#486C22×C12712yes—
251#486C22×C12712yes—
252#486C22×C12712yes—
253#486C22×C12712yes—
254#486C22×C12711yescharacteristic, complemented
255#486C22×C12711yescharacteristic, complemented
256#484—22no—
257#477—22no—
258#461—22no—
259#460—22no—
260#459—22no—
261#450—22no—
262#448—22no—
263#447—22no—
264#431C22×C12711yescharacteristic
265#431C22×C12711yescharacteristic
266#416—22no—
267C2×#102—22no—
268C2×C4×C8C22×C12711yescharacteristic
269#322C22×C12712yescomplemented
270C82C22×C12712yescomplemented
271C2×#41—21no—
272C22×Q8—21no—
273C22×D4—21no—
274C22×D4—21no—
275C22×D4—21no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×D8—24no—
279C22×C8D4×C12712yes—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8C23×C12712yes—
284C4⋊Q8C23×C12711yescharacteristic
285C4⋊Q8C23×C12711yescharacteristic
286C4⋊Q8C23×C12711yescharacteristic
287#127—21no—
288#127—21no—
289#127—21no—
290#127C23×C12712yes—
291#127C23×C12711yescharacteristic
292#125—21no—
293#124—21no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296#121—21no—
297#121—21no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×D4—21no—
301C4×D4—21no—
302#117—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C42C23×C12711yescharacteristic
306#107—24no—
307C4⋊C8D4×C12712yes—
308C4⋊C8D4×C12712yes—
309#103—22no—
310#103—22no—
311#103—24no—
312#103—24no—
313#102—24no—
314#102—24no—
315#102—22no—
316#102—22no—
317#102—24no—
318#102—24no—
319C22⋊C8D4×C12712yes—
320C22⋊C8D4×C12712yes—
321C4×C8D4×C12712yes—
322C4×C8C23×C12711yescharacteristic
323C4×C8C23×C12712yes—
324C4×C8C23×C12712yes—
325C4×C8C23×C12711yescharacteristic
326C24—41no—
327#41—44no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8—41no—
333C2×Q8—82no—
334C2×D4—21no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—44no—
345C2×D4—82no—
346C2×D4—82no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4C2×D4×C12711yescharacteristic
350C22×C4C2×D4×C12711yescharacteristic
351C22×C4C2×D4×C12711yescharacteristic
352SD16—84no—
353SD16—84no—
354D8—84no—
355C2×C8—24no—
356C2×C8C2×D4×C12712yes—
357C2×C8C2×D4×C12712yes—
358C2×C8C2×D4×C12714yes—
359C2×C8C2×D4×C12714yes—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C42—41no—
376C42C2×D4×C12711yescharacteristic
377C42C2×D4×C12711yescharacteristic
378C42C2×D4×C12711yescharacteristic
379C42C24×C12711yescharacteristic
380C23—41no—
381C23—42no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23#127×C12711yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—44no—
389Q8—42no—
390Q8—82no—
391D4—44no—
392D4—42no—
393D4—42no—
394D4—44no—
395D4—42no—
396D4—44no—
397D4—82no—
398D4—82no—
399D4—82no—
400D4—82no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#127×C12711yescharacteristic
410C2×C4#127×C12711yescharacteristic
411C2×C4#127×C12711yescharacteristic
412C2×C4#135×C12714yes—
413C2×C4C22×D4×C12711yescharacteristic
414C2×C4C22×D4×C12711yescharacteristic
415C2×C4C22×D4×C12711yescharacteristic
416C8—22no—
417C8—22no—
418C8—24no—
419C8—24no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—82no—
424C22—82no—
425C22—81no—
426C22—82no—
427C22—81no—
428C22—164no—
429C22#495×C12712yes—
430C22C2×#127×C12711yescharacteristic
431C4—21no—
432C4—21no—
433C4—81no—
434C4—81no—
435C4—82no—
436C4—81no—
437C4#495×C12712yes—
438C4C2×#135×C12712yes—
439C4#576×C12712yes—
440C4#577×C12712yes—
441C2—22no—
442C2—162no—
443C2—161no—
444C2C2×#495×C12711yescharacteristic
445C2C127×#315411yescharacteristic
446C2C127×#315511yescharacteristic
subgroupG/H
center, socleC22×C127C2×#127
commutator, Frattini, cosocleC42C24×C127
prime radicalC127×#3205C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3230C2semidirect
2. C127×#3208C2semidirect
3. C127×#3206C2semidirect
5. C127×#3131C2semidirect
6. C2×#486×C127C2semidirect
7. C127×#1719C2semidirect
8. C127×#1717C2semidirect
9. C127×#1711C2semidirect
10. C127×#1706C2semidirect
11. C127×#1450C2semidirect
16. #500×C127C22semidirect
19. #492×C127C22semidirect
21. #487×C127C22semidirect
22. #487×C127C22semidirect
23. #487×C127C22semidirect
24. #487×C127C22semidirect
30. #486×C127C22semidirect
31. #486×C127C22semidirect
45. #322×C127C22semidirect
46. C82×C127C22semidirect
217. #19658C127direct (N × H)
224. #3230C2×C127semidirect
225. #3208C2×C127semidirect
226. #3206C2×C127semidirect
228. #3131C2×C127semidirect
229. C2×#486C2×C127semidirect
230. #1719C2×C127semidirect
231. #1717C2×C127semidirect
232. #1711C2×C127semidirect
233. #1706C2×C127semidirect
234. #1450C2×C127semidirect
235. C127#19658direct (N × H)
240. #500C22×C127semidirect
243. #492C22×C127semidirect
245. #487C22×C127semidirect
246. #487C22×C127semidirect
247. #487C22×C127semidirect
248. #487C22×C127semidirect
254. #486C22×C127semidirect
255. #486C22×C127semidirect
269. #322C22×C127semidirect
270. C82C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32302
C127×#32082
C127×#32062
C127×#32052
C127×#31312
C2×#486×C1272
C127×#17192
C127×#17172
C127×#17112
C127×#17062
C127×#14502
#19658127