Finite Group Databasebeta

← C127×#19554 #44583227

Subgroups — C127×#19554 #44583227

Order 32512.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3253C211yesmaximal, characteristic, complemented
2C127×#3205C211yesmaximal, characteristic, complemented
3C127×#3205C211yesmaximal, characteristic, complemented
4C127×#2979C211yesmaximal, characteristic, complemented
5C127×#1697C212yesmaximal, complemented
6C127×#1687C212yesmaximal, complemented
7C127×#1677C212yesmaximal, complemented
8C127×#1667C212yesmaximal, complemented
9C127×#1574C211yesmaximal, characteristic, complemented
10C127×#1545C211yesmaximal, characteristic, complemented
11C127×#1543C211yesmaximal, characteristic, complemented
12#532×C127C2211yescharacteristic
13#532×C127C2211yescharacteristic
14C2×#127×C127C2211yescharacteristic
15#493×C127—21no—
16#493×C127C2212yes—
17#486×C127—21no—
18#486×C127C2212yes—
19#476×C127—21no—
20#476×C127C2212yescomplemented
21#469×C127—22no—
22#459×C127—21no—
23#459×C127—21no—
24#459×C127—21no—
25#459×C127C2212yes—
26#459×C127C2212yes—
27#459×C127C2212yescomplemented
28#447×C127—22no—
29#447×C127—22no—
30#436×C127—21no—
31#436×C127C2212yescomplemented
32C8×D4×C127—21no—
33C8×D4×C127C2212yescomplemented
34#431×C127C2211yescharacteristic
35#431×C127C2211yescharacteristic
36#431×C127C2211yescharacteristic
37#423×C127C2211yescharacteristic
38#408×C127—22no—
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—
48C2×#41×C127—21no—
49C22×D4×C127—21no—
50C22×D4×C127—21no—
51C22×D4×C127—21no—
52C2×D8×C127—22no—
53C2×D8×C127—22no—
54C2×D8×C127—22no—
55C2×D8×C127—22no—
56C2×D8×C127—22no—
57C2×M16×C127—21no—
58C2×M16×C127—22no—
59C22×C8×C127—22no—
60C4⋊Q8×C127D411yescharacteristic
61C4⋊Q8×C127C2311yescharacteristic
62#127×C127—21no—
63#127×C127—21no—
64#127×C127—21no—
65#127×C127D411yescharacteristic
66#127×C127C2312yes—
67#127×C127C2311yescharacteristic
68#124×C127—21no—
69#124×C127—21no—
70#121×C127—21no—
71#121×C127—21no—
72#121×C127—21no—
73#121×C127D412yes—
74C4×D4×C127—21no—
75C4×D4×C127—21no—
76C4×D4×C127—21no—
77C4×D4×C127C2312yes—
78C2×C42×C127C2311yescharacteristic
79#106×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#102×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×C127—22no—
97C22⋊C8×C127—22no—
98C22⋊C8×C127D412yes—
99C4×C8×C127—21no—
100C4×C8×C127—21no—
101C4×C8×C127C2312yes—
102C24×C127—41no—
103#41×C127—81no—
104#41×C127—81no—
105#41×C127—82no—
106C2×Q8×C127—21no—
107C2×Q8×C127—21no—
108C2×D4×C127—22no—
109C2×D4×C127—21no—
110C2×D4×C127—21no—
111C2×D4×C127—21no—
112C2×D4×C127—22no—
113C2×D4×C127—22no—
114C2×D4×C127—21no—
115C2×D4×C127—21no—
116C2×D4×C127—22no—
117C2×D4×C127—41no—
118C2×D4×C127—41no—
119C2×D4×C127—41no—
120C2×D4×C127—41no—
121C2×D4×C127—81no—
122C2×D4×C127—81no—
123C2×D4×C127—81no—
124C2×D4×C127—81no—
125C2×D4×C127—82no—
126C2×D4×C127C2×D412yes—
127C22×C4×C127—22no—
128C22×C4×C127—41no—
129C22×C4×C127C2×D411yescharacteristic
130C22×C4×C127C2×D411yescharacteristic
131C22×C4×C127#4111yescharacteristic
132D8×C127—82no—
133D8×C127—82no—
134D8×C127—82no—
135D8×C127—82no—
136D8×C127—82no—
137M16×C127—82no—
138M16×C127—81no—
139C2×C8×C127—21no—
140C2×C8×C127—21no—
141C2×C8×C127—21no—
142C2×C8×C127—21no—
143C2×C8×C127—22no—
144C2×C8×C127—82no—
145C2×C8×C127C2×D412yes—
146C2×C8×C127C2×D412yes—
147C4⋊C4×C127—21no—
148C4⋊C4×C127—21no—
149C4⋊C4×C127—21no—
150C4⋊C4×C127C2×D412yes—
151C22⋊C4×C127—22no—
152C22⋊C4×C127—22no—
153C22⋊C4×C127—41no—
154C22⋊C4×C127—41no—
155C42×C127C2×D411yescharacteristic
156C42×C127C2×D411yescharacteristic
157C42×C127#4111yescharacteristic
158C42×C127C2411yescharacteristic
159C23×C127—22no—
160C23×C127—41no—
161C23×C127—42no—
162C23×C127—82no—
163C23×C127—82no—
164C23×C127—82no—
165C23×C127#12111yescharacteristic
166Q8×C127—81no—
167Q8×C127—81no—
168D4×C127—22no—
169D4×C127—22no—
170D4×C127—41no—
171D4×C127—41no—
172D4×C127—41no—
173D4×C127—42no—
174D4×C127—42no—
175D4×C127—42no—
176D4×C127—42no—
177D4×C127—41no—
178D4×C127—42no—
179D4×C127—81no—
180D4×C127—82no—
181D4×C127—81no—
182D4×C127—82no—
183D4×C127—82no—
184D4×C127—82no—
185D4×C127—82no—
186D4×C127—82no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—22no—
192C2×C4×C127—22no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—41no—
196C2×C4×C127—82no—
197C2×C4×C127#12111yescharacteristic
198C2×C4×C127#12111yescharacteristic
199C2×C4×C127#12111yescharacteristic
200C2×C4×C127C22×D411yescharacteristic
201C2×C4×C127C22×D411yescharacteristic
202C2×C4×C127C2×#4111yescharacteristic
203C8×C127—22no—
204C8×C127—22no—
205C8×C127—41no—
206C8×C127—41no—
207C8×C127—42no—
208C8×C127—41no—
209C8×C127—41no—
210C22×C127—21no—
211C22×C127—21no—
212C22×C127—21no—
213C22×C127—42no—
214C22×C127—42no—
215C22×C127—42no—
216C22×C127—82no—
217C22×C127—81no—
218C22×C127—82no—
219C22×C127—81no—
220C22×C127—81no—
221C22×C127—82no—
222C22×C127—164no—
223C22×C127C2×#12111yescharacteristic
224C4×C127—21no—
225C4×C127—21no—
226C4×C127—21no—
227C4×C127—42no—
228C4×C127—81no—
229C4×C127#47212yes—
230C4×C127#57611yescharacteristic
231C4×C127#57611yescharacteristic
232C4×C127#57611yescharacteristic
233C4×C127#57711yescharacteristic
234#19554C12711yesmaximal, characteristic, direct product factor, Sylow
235C2×C127—41no—
236C2×C127—82no—
237C2×C127—161no—
238C2×C127—162no—
239C2×C127#304111yescharacteristic
240C2×C127#306211yescharacteristic
241C2×C127C2×#47211yescharacteristic
242#3253C2×C12711yescharacteristic, complemented
243#3205C2×C12711yescharacteristic, complemented
244#3205C2×C12711yescharacteristic, complemented
245#2979C2×C12711yescharacteristic, complemented
246#1697C2×C12712yescomplemented
247#1687C2×C12712yescomplemented
248#1677C2×C12712yescomplemented
249#1667C2×C12712yescomplemented
250#1574C2×C12711yescharacteristic, complemented
251#1545C2×C12711yescharacteristic, complemented
252#1543C2×C12711yescharacteristic, complemented
253C127#1955411yescharacteristic, direct product factor, Sylow
254#532C22×C12711yescharacteristic
255#532C22×C12711yescharacteristic
256C2×#127C22×C12711yescharacteristic
257#493—21no—
258#493C22×C12712yes—
259#486—21no—
260#486C22×C12712yes—
261#476—21no—
262#476C22×C12712yescomplemented
263#469—22no—
264#459—21no—
265#459—21no—
266#459—21no—
267#459C22×C12712yes—
268#459C22×C12712yes—
269#459C22×C12712yescomplemented
270#447—22no—
271#447—22no—
272#436—21no—
273#436C22×C12712yescomplemented
274C8×D4—21no—
275C8×D4C22×C12712yescomplemented
276#431C22×C12711yescharacteristic
277#431C22×C12711yescharacteristic
278#431C22×C12711yescharacteristic
279#423C22×C12711yescharacteristic
280#408—22no—
281#335C22×C12712yescomplemented
282#334C22×C12712yescomplemented
283#333C22×C12711yescharacteristic, complemented
284#333C22×C12711yescharacteristic, complemented
285#331C22×C12712yes—
286#331C22×C12712yes—
287#331C22×C12711yescharacteristic, complemented
288#331C22×C12711yescharacteristic, complemented
289C2×#41—21no—
290C2×#41—21no—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C2×D8—22no—
295C2×D8—22no—
296C2×D8—22no—
297C2×D8—22no—
298C2×D8—22no—
299C2×M16—21no—
300C2×M16—22no—
301C22×C8—22no—
302C4⋊Q8D4×C12711yescharacteristic
303C4⋊Q8C23×C12711yescharacteristic
304#127—21no—
305#127—21no—
306#127—21no—
307#127D4×C12711yescharacteristic
308#127C23×C12712yes—
309#127C23×C12711yescharacteristic
310#124—21no—
311#124—21no—
312#121—21no—
313#121—21no—
314#121—21no—
315#121D4×C12712yes—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4C23×C12712yes—
320C2×C42C23×C12711yescharacteristic
321#106—22no—
322C4⋊C8—21no—
323C4⋊C8D4×C12711yescharacteristic
324C4⋊C8D4×C12711yescharacteristic
325C4⋊C8C23×C12711yescharacteristic
326C4⋊C8C23×C12711yescharacteristic
327C4⋊C8C23×C12712yes—
328C4⋊C8C23×C12711yescharacteristic
329C4⋊C8C23×C12711yescharacteristic
330#102—22no—
331#102—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336C22⋊C8—21no—
337C22⋊C8—21no—
338C22⋊C8—22no—
339C22⋊C8—22no—
340C22⋊C8D4×C12712yes—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8C23×C12712yes—
344C24—41no—
345#41—81no—
346#41—81no—
347#41—82no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4—82no—
368C2×D4C2×D4×C12712yes—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4C2×D4×C12711yescharacteristic
372C22×C4C2×D4×C12711yescharacteristic
373C22×C4#41×C12711yescharacteristic
374D8—82no—
375D8—82no—
376D8—82no—
377D8—82no—
378D8—82no—
379M16—82no—
380M16—81no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—82no—
387C2×C8C2×D4×C12712yes—
388C2×C8C2×D4×C12712yes—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4C2×D4×C12712yes—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C42C2×D4×C12711yescharacteristic
398C42C2×D4×C12711yescharacteristic
399C42#41×C12711yescharacteristic
400C42C24×C12711yescharacteristic
401C23—22no—
402C23—41no—
403C23—42no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23#121×C12711yescharacteristic
408Q8—81no—
409Q8—81no—
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—42no—
419D4—41no—
420D4—42no—
421D4—81no—
422D4—82no—
423D4—81no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—82no—
428D4—82no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—82no—
439C2×C4#121×C12711yescharacteristic
440C2×C4#121×C12711yescharacteristic
441C2×C4#121×C12711yescharacteristic
442C2×C4C22×D4×C12711yescharacteristic
443C2×C4C22×D4×C12711yescharacteristic
444C2×C4C2×#41×C12711yescharacteristic
445C8—22no—
446C8—22no—
447C8—41no—
448C8—41no—
449C8—42no—
450C8—41no—
451C8—41no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22—81no—
460C22—82no—
461C22—81no—
462C22—81no—
463C22—82no—
464C22—164no—
465C22C2×#121×C12711yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—42no—
470C4—81no—
471C4#472×C12712yes—
472C4#576×C12711yescharacteristic
473C4#576×C12711yescharacteristic
474C4#576×C12711yescharacteristic
475C4#577×C12711yescharacteristic
476C2—41no—
477C2—82no—
478C2—162no—
479C2—161no—
480C2C127×#304111yescharacteristic
481C2C127×#306211yescharacteristic
482C2C2×#472×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#121
commutator, Frattini, cosocleC42C24×C127
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×#3253C2semidirect
2. C127×#3205C2semidirect
3. C127×#3205C2semidirect
4. C127×#2979C2semidirect
5. C127×#1697C2semidirect
6. C127×#1687C2semidirect
7. C127×#1677C2semidirect
8. C127×#1667C2semidirect
9. C127×#1574C2semidirect
10. C127×#1545C2semidirect
11. C127×#1543C2semidirect
20. #476×C127C22semidirect
27. #459×C127C22semidirect
31. #436×C127C22semidirect
33. C8×D4×C127C22semidirect
39. #335×C127C22semidirect
40. #334×C127C22semidirect
41. #333×C127C22semidirect
42. #333×C127C22semidirect
45. #331×C127C22semidirect
46. #331×C127C22semidirect
234. #19554C127direct (N × H)
242. #3253C2×C127semidirect
243. #3205C2×C127semidirect
244. #3205C2×C127semidirect
245. #2979C2×C127semidirect
246. #1697C2×C127semidirect
247. #1687C2×C127semidirect
248. #1677C2×C127semidirect
249. #1667C2×C127semidirect
250. #1574C2×C127semidirect
251. #1545C2×C127semidirect
252. #1543C2×C127semidirect
253. C127#19554direct (N × H)
262. #476C22×C127semidirect
269. #459C22×C127semidirect
273. #436C22×C127semidirect
275. C8×D4C22×C127semidirect
281. #335C22×C127semidirect
282. #334C22×C127semidirect
283. #333C22×C127semidirect
284. #333C22×C127semidirect
287. #331C22×C127semidirect
288. #331C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32532
C127×#32052
C127×#29792
C127×#16972
C127×#16872
C127×#16772
C127×#16672
C127×#15742
C127×#15452
C127×#15432
#19554127