Finite Group Databasebeta

← C127×#19566 #44583239

Subgroups — C127×#19566 #44583239

Order 32512.

648 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3254C211yesmaximal, characteristic, complemented
2C127×#3208C211yesmaximal, characteristic, complemented
3C127×#3205C211yesmaximal, characteristic
4C127×#2986C211yesmaximal, characteristic, complemented
5C127×#1698C212yesmaximal, complemented
6C127×#1688C212yesmaximal, complemented
7C127×#1678C212yesmaximal, complemented
8C127×#1668C212yesmaximal, complemented
9C127×#1574C211yesmaximal, characteristic, complemented
10C127×#1545C211yesmaximal, characteristic, complemented
11C127×#1543C211yesmaximal, characteristic, complemented
12#533×C127C2211yescharacteristic
13#532×C127C2211yescharacteristic
14C2×#127×C127C2211yescharacteristic
15#493×C127C2212yes—
16#492×C127—21no—
17#486×C127—21no—
18#486×C127C2212yes—
19#477×C127C2212yescomplemented
20#474×C127—21no—
21#468×C127—22no—
22#461×C127—21no—
23#460×C127C2212yes—
24#460×C127C2212yes—
25#460×C127C2212yescomplemented
26#459×C127—21no—
27#459×C127—21no—
28#450×C127—22no—
29#447×C127—22no—
30#446×C127—21no—
31#446×C127C2212yescomplemented
32C8×Q8×C127—21no—
33C8×Q8×C127C2212yescomplemented
34#433×C127—22no—
35#431×C127C2211yescharacteristic
36#431×C127C2211yescharacteristic
37#431×C127C2211yescharacteristic
38#423×C127C2211yescharacteristic
39#335×C127C2212yescomplemented
40#334×C127C2212yescomplemented
41#333×C127C2211yescharacteristic, complemented
42#333×C127C2211yescharacteristic, complemented
43#331×C127C2212yes—
44#331×C127C2212yes—
45#331×C127C2211yescharacteristic, complemented
46#331×C127C2211yescharacteristic, complemented
47C2×#41×C127—21no—
48C22×D4×C127—21no—
49C22×D4×C127—21no—
50C22×D4×C127—21no—
51C2×SD16×C127—22no—
52C2×SD16×C127—22no—
53C2×SD16×C127—22no—
54C2×D8×C127—22no—
55C2×D8×C127—22no—
56C2×M16×C127—21no—
57C4⋊Q8×C127D411yescharacteristic
58C4⋊Q8×C127D411yescharacteristic
59C4⋊Q8×C127C2311yescharacteristic
60#127×C127—21no—
61#127×C127—21no—
62#127×C127—21no—
63#127×C127C2312yes—
64#127×C127C2311yescharacteristic
65#125×C127—21no—
66#124×C127—21no—
67C22⋊Q8×C127—21no—
68C22⋊Q8×C127D412yes—
69#121×C127—21no—
70#121×C127—21no—
71C4×Q8×C127—21no—
72C4×Q8×C127C2312yes—
73C4×D4×C127—21no—
74C4×D4×C127—21no—
75#117×C127—21no—
76C2×C42×C127C2311yescharacteristic
77#107×C127—22no—
78C4⋊C8×C127—22no—
79C4⋊C8×C127—22no—
80C4⋊C8×C127—21no—
81C4⋊C8×C127D411yescharacteristic
82C4⋊C8×C127D411yescharacteristic
83C4⋊C8×C127C2311yescharacteristic
84C4⋊C8×C127C2311yescharacteristic
85C4⋊C8×C127C2312yes—
86C4⋊C8×C127C2311yescharacteristic
87C4⋊C8×C127C2311yescharacteristic
88#103×C127—22no—
89#102×C127—22no—
90#102×C127—22no—
91#102×C127—22no—
92#102×C127—22no—
93#102×C127—22no—
94C22⋊C8×C127—21no—
95C22⋊C8×C127—21no—
96C22⋊C8×C127D412yes—
97#97×C127—22no—
98C4×C8×C127—22no—
99C4×C8×C127—21no—
100C4×C8×C127—21no—
101C4×C8×C127C2312yes—
102C24×C127—41no—
103#41×C127—82no—
104C2×Q8×C127—21no—
105C2×Q8×C127—21no—
106C2×Q8×C127C2×D412yes—
107C2×D4×C127—21no—
108C2×D4×C127—22no—
109C2×D4×C127—22no—
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127—22no—
113C2×D4×C127—41no—
114C2×D4×C127—41no—
115C2×D4×C127—41no—
116C2×D4×C127—41no—
117C2×D4×C127—81no—
118C2×D4×C127—81no—
119C2×D4×C127—81no—
120C2×D4×C127—81no—
121C2×D4×C127—82no—
122C22×C4×C127—41no—
123C22×C4×C127C2×D411yescharacteristic
124C22×C4×C127C2×D411yescharacteristic
125C22×C4×C127#4111yescharacteristic
126SD16×C127—82no—
127SD16×C127—82no—
128SD16×C127—82no—
129D8×C127—82no—
130D8×C127—82no—
131M16×C127—81no—
132C2×C8×C127—21no—
133C2×C8×C127—21no—
134C2×C8×C127—21no—
135C2×C8×C127—21no—
136C2×C8×C127—22no—
137C2×C8×C127C2×D412yes—
138C2×C8×C127C2×D412yes—
139C4⋊C4×C127—21no—
140C4⋊C4×C127—21no—
141C4⋊C4×C127—21no—
142C4⋊C4×C127—22no—
143C4⋊C4×C127—22no—
144C4⋊C4×C127—21no—
145C4⋊C4×C127—21no—
146C4⋊C4×C127C2×D412yes—
147C22⋊C4×C127—22no—
148C22⋊C4×C127—41no—
149C22⋊C4×C127—41no—
150C42×C127—22no—
151C42×C127C2×D411yescharacteristic
152C42×C127C2×D411yescharacteristic
153C42×C127#4111yescharacteristic
154C42×C127C2411yescharacteristic
155C23×C127—41no—
156C23×C127—42no—
157C23×C127—82no—
158C23×C127—82no—
159C23×C127—82no—
160C23×C127#12111yescharacteristic
161Q8×C127—22no—
162Q8×C127—22no—
163Q8×C127—42no—
164Q8×C127—81no—
165D4×C127—41no—
166D4×C127—42no—
167D4×C127—41no—
168D4×C127—41no—
169D4×C127—42no—
170D4×C127—42no—
171D4×C127—42no—
172D4×C127—41no—
173D4×C127—82no—
174D4×C127—81no—
175D4×C127—82no—
176D4×C127—82no—
177D4×C127—82no—
178D4×C127—82no—
179C2×C4×C127—21no—
180C2×C4×C127—21no—
181C2×C4×C127—21no—
182C2×C4×C127—21no—
183C2×C4×C127—22no—
184C2×C4×C127—22no—
185C2×C4×C127—22no—
186C2×C4×C127—41no—
187C2×C4×C127—82no—
188C2×C4×C127#12111yescharacteristic
189C2×C4×C127#12111yescharacteristic
190C2×C4×C127#12111yescharacteristic
191C2×C4×C127C22×D411yescharacteristic
192C2×C4×C127C22×D411yescharacteristic
193C2×C4×C127C2×#4111yescharacteristic
194C8×C127—22no—
195C8×C127—22no—
196C8×C127—42no—
197C8×C127—41no—
198C8×C127—41no—
199C8×C127—41no—
200C8×C127—41no—
201C22×C127—21no—
202C22×C127—21no—
203C22×C127—21no—
204C22×C127—82no—
205C22×C127—81no—
206C22×C127—82no—
207C22×C127—81no—
208C22×C127—81no—
209C22×C127—82no—
210C22×C127—164no—
211C22×C127C2×#12111yescharacteristic
212C4×C127—21no—
213C4×C127—21no—
214C4×C127—21no—
215C4×C127—42no—
216C4×C127—42no—
217C4×C127—81no—
218C4×C127#47212yes—
219C4×C127#57611yescharacteristic
220C4×C127#57711yescharacteristic
221C4×C127#57711yescharacteristic
222C4×C127#57711yescharacteristic
223#19566C12711yesmaximal, characteristic, direct product factor, Sylow
224C2×C127—41no—
225C2×C127—161no—
226C2×C127—162no—
227C2×C127#304311yescharacteristic
228C2×C127#306211yescharacteristic
229C2×C127C2×#47211yescharacteristic
230#3254C2×C12711yescharacteristic, complemented
231#3208C2×C12711yescharacteristic, complemented
232#3205C2×C12711yescharacteristic
233#2986C2×C12711yescharacteristic, complemented
234#1698C2×C12712yescomplemented
235#1688C2×C12712yescomplemented
236#1678C2×C12712yescomplemented
237#1668C2×C12712yescomplemented
238#1574C2×C12711yescharacteristic, complemented
239#1545C2×C12711yescharacteristic, complemented
240#1543C2×C12711yescharacteristic, complemented
241C127#1956611yescharacteristic, direct product factor, Sylow
242#533C22×C12711yescharacteristic
243#532C22×C12711yescharacteristic
244C2×#127C22×C12711yescharacteristic
245#493C22×C12712yes—
246#492—21no—
247#486—21no—
248#486C22×C12712yes—
249#477C22×C12712yescomplemented
250#474—21no—
251#468—22no—
252#461—21no—
253#460C22×C12712yes—
254#460C22×C12712yes—
255#460C22×C12712yescomplemented
256#459—21no—
257#459—21no—
258#450—22no—
259#447—22no—
260#446—21no—
261#446C22×C12712yescomplemented
262C8×Q8—21no—
263C8×Q8C22×C12712yescomplemented
264#433—22no—
265#431C22×C12711yescharacteristic
266#431C22×C12711yescharacteristic
267#431C22×C12711yescharacteristic
268#423C22×C12711yescharacteristic
269#335C22×C12712yescomplemented
270#334C22×C12712yescomplemented
271#333C22×C12711yescharacteristic, complemented
272#333C22×C12711yescharacteristic, complemented
273#331C22×C12712yes—
274#331C22×C12712yes—
275#331C22×C12711yescharacteristic, complemented
276#331C22×C12711yescharacteristic, complemented
277C2×#41—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4—21no—
281C2×SD16—22no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×D8—22no—
285C2×D8—22no—
286C2×M16—21no—
287C4⋊Q8D4×C12711yescharacteristic
288C4⋊Q8D4×C12711yescharacteristic
289C4⋊Q8C23×C12711yescharacteristic
290#127—21no—
291#127—21no—
292#127—21no—
293#127C23×C12712yes—
294#127C23×C12711yescharacteristic
295#125—21no—
296#124—21no—
297C22⋊Q8—21no—
298C22⋊Q8D4×C12712yes—
299#121—21no—
300#121—21no—
301C4×Q8—21no—
302C4×Q8C23×C12712yes—
303C4×D4—21no—
304C4×D4—21no—
305#117—21no—
306C2×C42C23×C12711yescharacteristic
307#107—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8D4×C12711yescharacteristic
312C4⋊C8D4×C12711yescharacteristic
313C4⋊C8C23×C12711yescharacteristic
314C4⋊C8C23×C12711yescharacteristic
315C4⋊C8C23×C12712yes—
316C4⋊C8C23×C12711yescharacteristic
317C4⋊C8C23×C12711yescharacteristic
318#103—22no—
319#102—22no—
320#102—22no—
321#102—22no—
322#102—22no—
323#102—22no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8D4×C12712yes—
327#97—22no—
328C4×C8—22no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8C23×C12712yes—
332C24—41no—
333#41—82no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C2×D4×C12712yes—
337C2×D4—21no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—81no—
350C2×D4—81no—
351C2×D4—82no—
352C22×C4—41no—
353C22×C4C2×D4×C12711yescharacteristic
354C22×C4C2×D4×C12711yescharacteristic
355C22×C4#41×C12711yescharacteristic
356SD16—82no—
357SD16—82no—
358SD16—82no—
359D8—82no—
360D8—82no—
361M16—81no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—22no—
367C2×C8C2×D4×C12712yes—
368C2×C8C2×D4×C12712yes—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4C2×D4×C12712yes—
377C22⋊C4—22no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42—22no—
381C42C2×D4×C12711yescharacteristic
382C42C2×D4×C12711yescharacteristic
383C42#41×C12711yescharacteristic
384C42C24×C12711yescharacteristic
385C23—41no—
386C23—42no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23#121×C12711yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—42no—
394Q8—81no—
395D4—41no—
396D4—42no—
397D4—41no—
398D4—41no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—41no—
403D4—82no—
404D4—81no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4#121×C12711yescharacteristic
419C2×C4#121×C12711yescharacteristic
420C2×C4#121×C12711yescharacteristic
421C2×C4C22×D4×C12711yescharacteristic
422C2×C4C22×D4×C12711yescharacteristic
423C2×C4C2×#41×C12711yescharacteristic
424C8—22no—
425C8—22no—
426C8—42no—
427C8—41no—
428C8—41no—
429C8—41no—
430C8—41no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—82no—
435C22—81no—
436C22—82no—
437C22—81no—
438C22—81no—
439C22—82no—
440C22—164no—
441C22C2×#121×C12711yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—42no—
447C4—81no—
448C4#472×C12712yes—
449C4#576×C12711yescharacteristic
450C4#577×C12711yescharacteristic
451C4#577×C12711yescharacteristic
452C4#577×C12711yescharacteristic
453C2—41no—
454C2—162no—
455C2—161no—
456C2C127×#304311yescharacteristic
457C2C127×#306211yescharacteristic
458C2C2×#472×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#121
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×#3254C2semidirect
2. C127×#3208C2semidirect
4. C127×#2986C2semidirect
5. C127×#1698C2semidirect
6. C127×#1688C2semidirect
7. C127×#1678C2semidirect
8. C127×#1668C2semidirect
9. C127×#1574C2semidirect
10. C127×#1545C2semidirect
11. C127×#1543C2semidirect
19. #477×C127C22semidirect
25. #460×C127C22semidirect
31. #446×C127C22semidirect
33. C8×Q8×C127C22semidirect
39. #335×C127C22semidirect
40. #334×C127C22semidirect
41. #333×C127C22semidirect
42. #333×C127C22semidirect
45. #331×C127C22semidirect
46. #331×C127C22semidirect
223. #19566C127direct (N × H)
230. #3254C2×C127semidirect
231. #3208C2×C127semidirect
233. #2986C2×C127semidirect
234. #1698C2×C127semidirect
235. #1688C2×C127semidirect
236. #1678C2×C127semidirect
237. #1668C2×C127semidirect
238. #1574C2×C127semidirect
239. #1545C2×C127semidirect
240. #1543C2×C127semidirect
241. C127#19566direct (N × H)
249. #477C22×C127semidirect
255. #460C22×C127semidirect
261. #446C22×C127semidirect
263. C8×Q8C22×C127semidirect
269. #335C22×C127semidirect
270. #334C22×C127semidirect
271. #333C22×C127semidirect
272. #333C22×C127semidirect
275. #331C22×C127semidirect
276. #331C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32542
C127×#32082
C127×#32052
C127×#29862
C127×#16982
C127×#16882
C127×#16782
C127×#16682
C127×#15742
C127×#15452
C127×#15432
#19566127