Finite Group Databasebeta

← C127×#18442 #44582115

Subgroups — C127×#18442 #44582115

Order 32512.

848 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3563C211yesmaximal, characteristic
2C127×#3238C211yesmaximal, characteristic
3C127×#3229C211yesmaximal, characteristic, complemented
4C127×#2660C211yesmaximal, characteristic, complemented
5C127×#1611C214yesmaximal, complemented
6C127×#1609C214yesmaximal
7C127×#1506C211yesmaximal, characteristic, complemented
8C127×#1500C211yesmaximal, characteristic
9C127×#1470C211yesmaximal, characteristic, complemented
10#564×C127—21no—
11#563×C127—21no—
12#557×C127—21no—
13#554×C127—21no—
14#552×C127—21no—
15#552×C127—21no—
16#552×C127—22no—
17#544×C127—21no—
18#544×C127C2212yes—
19#541×C127—21no—
20#541×C127—22no—
21#539×C127C2212yes—
22#533×C127C2211yescharacteristic
23C2×C4⋊Q8×C127C2211yescharacteristic
24#529×C127C2211yescharacteristic
25C2×#125×C127C2211yescharacteristic
26#501×C127—21no—
27#498×C127—21no—
28#489×C127—21no—
29#487×C127—21no—
30#483×C127C2214yes—
31#483×C127C2212yes—
32#480×C127C2212yes—
33#477×C127—21no—
34#476×C127—21no—
35#475×C127—22no—
36#432×C127C2211yescharacteristic
37#431×C127C2211yescharacteristic
38#398×C127C2214yes—
39#395×C127—22no—
40#384×C127—21no—
41#382×C127—21no—
42#376×C127C2211yescharacteristic
43#328×C127C2214yes—
44#328×C127C2212yes—
45#327×C127C2212yes—
46#324×C127C2214yes—
47C22×Q8×C127—21no—
48C4⋊Q8×C127—21no—
49C4⋊Q8×C127—22no—
50C4⋊Q8×C127—22no—
51C4⋊Q8×C127—21no—
52C4⋊Q8×C127—21no—
53C4⋊Q8×C127—21no—
54C4⋊Q8×C127—21no—
55C4⋊Q8×C127D411yescharacteristic
56C4⋊Q8×C127D411yescharacteristic
57#126×C127—22no—
58#126×C127—22no—
59#126×C127—22no—
60#126×C127—22no—
61#126×C127—21no—
62#125×C127—22no—
63#125×C127—22no—
64#125×C127—22no—
65#125×C127—22no—
66#125×C127—21no—
67#125×C127—22no—
68#125×C127—22no—
69#125×C127—22no—
70#125×C127—22no—
71#125×C127—21no—
72#125×C127—21no—
73#125×C127—21no—
74#125×C127—21no—
75#125×C127—21no—
76#125×C127D411yescharacteristic
77#124×C127D411yescharacteristic
78#123×C127—22no—
79#123×C127—21no—
80#123×C127—22no—
81C22⋊Q8×C127—21no—
82C22⋊Q8×C127—21no—
83C22⋊Q8×C127—22no—
84C22⋊Q8×C127—22no—
85C22⋊Q8×C127—22no—
86C22⋊Q8×C127—21no—
87C22⋊Q8×C127—22no—
88C22⋊Q8×C127C2311yescharacteristic
89C22⋊Q8×C127C2312yes—
90#121×C127C2311yescharacteristic
91C4×Q8×C127—21no—
92C4×Q8×C127—22no—
93C4×Q8×C127—21no—
94C4×Q8×C127—21no—
95C4×Q8×C127D411yescharacteristic
96C4×Q8×C127D412yes—
97C4×D4×C127—21no—
98C4×D4×C127—22no—
99C4×D4×C127D411yescharacteristic
100#117×C127—22no—
101#117×C127—21no—
102#117×C127—21no—
103#117×C127—22no—
104#117×C127—21no—
105C2×C4⋊C4×C127—21no—
106C2×C4⋊C4×C127—21no—
107C2×C4⋊C4×C127—24no—
108C2×C4⋊C4×C127D411yescharacteristic
109C2×C4⋊C4×C127D412yes—
110C2×C4⋊C4×C127D411yescharacteristic
111C2×C4⋊C4×C127C2312yes—
112C2×C4⋊C4×C127C2312yes—
113C2×C42×C127—21no—
114C2×C42×C127C2311yescharacteristic
115#107×C127—22no—
116#106×C127—24no—
117#106×C127—22no—
118C4⋊C8×C127—21no—
119C4⋊C8×C127—21no—
120#103×C127—24no—
121#103×C127—22no—
122#102×C127—22no—
123C22⋊C8×C127C2312yes—
124C22⋊C8×C127C2312yes—
125#97×C127—21no—
126C4×C8×C127—21no—
127#95×C127—21no—
128#95×C127—24no—
129#95×C127C2312yes—
130C2×Q8×C127—22no—
131C2×Q8×C127—21no—
132C2×Q8×C127—41no—
133C2×Q8×C127—41no—
134C2×Q8×C127—82no—
135C2×Q8×C127C2×D412yes—
136C2×Q8×C127C2×D411yescharacteristic
137C2×D4×C127—21no—
138C2×D4×C127C2×D411yescharacteristic
139C22×C4×C127—22no—
140C22×C4×C127—22no—
141C22×C4×C127—21no—
142C22×C4×C127—22no—
143C22×C4×C127C2×D412yes—
144C22×C4×C127C2×D411yescharacteristic
145C22×C4×C127C2×D411yescharacteristic
146C22×C4×C127C2×D412yes—
147C22×C4×C127C2411yescharacteristic
148C2×C8×C127—22no—
149C2×C8×C127—22no—
150C4⋊C4×C127—22no—
151C4⋊C4×C127—22no—
152C4⋊C4×C127—22no—
153C4⋊C4×C127—22no—
154C4⋊C4×C127—22no—
155C4⋊C4×C127—22no—
156C4⋊C4×C127—22no—
157C4⋊C4×C127—22no—
158C4⋊C4×C127—21no—
159C4⋊C4×C127—21no—
160C4⋊C4×C127—21no—
161C4⋊C4×C127—22no—
162C4⋊C4×C127—21no—
163C4⋊C4×C127—22no—
164C4⋊C4×C127—21no—
165C4⋊C4×C127—21no—
166C4⋊C4×C127—21no—
167C4⋊C4×C127—21no—
168C4⋊C4×C127—21no—
169C4⋊C4×C127—21no—
170C4⋊C4×C127—22no—
171C4⋊C4×C127—22no—
172C4⋊C4×C127—22no—
173C4⋊C4×C127—21no—
174C4⋊C4×C127—22no—
175C4⋊C4×C127—22no—
176C4⋊C4×C127—22no—
177C4⋊C4×C127—41no—
178C4⋊C4×C127—41no—
179C4⋊C4×C127—84no—
180C4⋊C4×C127C2×D411yescharacteristic
181C4⋊C4×C127C2×D412yes—
182C4⋊C4×C127C2×D411yescharacteristic
183C22⋊C4×C127—22no—
184C22⋊C4×C127—22no—
185C22⋊C4×C127—22no—
186C22⋊C4×C127—22no—
187C22⋊C4×C127—21no—
188C22⋊C4×C127—21no—
189C22⋊C4×C127—21no—
190C22⋊C4×C127—21no—
191C22⋊C4×C127—21no—
192C42×C127—24no—
193C42×C127—22no—
194C42×C127—22no—
195C42×C127—22no—
196C42×C127—22no—
197C42×C127—21no—
198C42×C127C2×D411yescharacteristic
199C42×C127C2×D411yescharacteristic
200C42×C127C2×D411yescharacteristic
201C42×C127C2×D411yescharacteristic
202C23×C127—21no—
203C23×C127C22×D411yescharacteristic
204Q8×C127—41no—
205Q8×C127—42no—
206Q8×C127—42no—
207Q8×C127—82no—
208Q8×C127—82no—
209Q8×C127—82no—
210D4×C127—41no—
211D4×C127—42no—
212C2×C4×C127—22no—
213C2×C4×C127—22no—
214C2×C4×C127—24no—
215C2×C4×C127—22no—
216C2×C4×C127—21no—
217C2×C4×C127—21no—
218C2×C4×C127—22no—
219C2×C4×C127—21no—
220C2×C4×C127—22no—
221C2×C4×C127—22no—
222C2×C4×C127—24no—
223C2×C4×C127—41no—
224C2×C4×C127—42no—
225C2×C4×C127—42no—
226C2×C4×C127—42no—
227C2×C4×C127—42no—
228C2×C4×C127—42no—
229C2×C4×C127—44no—
230C2×C4×C127—82no—
231C2×C4×C127#12011yescharacteristic
232C2×C4×C127#12011yescharacteristic
233C2×C4×C127#12011yescharacteristic
234C2×C4×C127#12011yescharacteristic
235C2×C4×C127C22×D411yescharacteristic
236C2×C4×C127C22×D411yescharacteristic
237C8×C127—42no—
238C8×C127—42no—
239C22×C127—21no—
240C22×C127—21no—
241C22×C127—41no—
242C22×C127—41no—
243C22×C127—41no—
244C22×C127C2×#12011yescharacteristic
245C22×C127#57712yes—
246C4×C127—21no—
247C4×C127—21no—
248C4×C127—21no—
249C4×C127—21no—
250C4×C127—21no—
251C4×C127—44no—
252C4×C127—42no—
253C4×C127—41no—
254C4×C127—41no—
255C4×C127—42no—
256C4×C127—42no—
257C4×C127—42no—
258C4×C127—41no—
259C4×C127—82no—
260C4×C127#45612yes—
261#18442C12711yesmaximal, characteristic, direct product factor, Sylow
262C2×C127—22no—
263C2×C127—81no—
264C2×C127#301011yescharacteristic
265C2×C127C2×#45611yescharacteristic
266C2×C127#302811yescharacteristic
267#3563C2×C12711yescharacteristic
268#3238C2×C12711yescharacteristic
269#3229C2×C12711yescharacteristic, complemented
270#2660C2×C12711yescharacteristic, complemented
271#1611C2×C12714yescomplemented
272#1609C2×C12714yes—
273#1506C2×C12711yescharacteristic, complemented
274#1500C2×C12711yescharacteristic
275#1470C2×C12711yescharacteristic, complemented
276C127#1844211yescharacteristic, direct product factor, Sylow
277#564—21no—
278#563—21no—
279#557—21no—
280#554—21no—
281#552—21no—
282#552—21no—
283#552—22no—
284#544—21no—
285#544C22×C12712yes—
286#541—21no—
287#541—22no—
288#539C22×C12712yes—
289#533C22×C12711yescharacteristic
290C2×C4⋊Q8C22×C12711yescharacteristic
291#529C22×C12711yescharacteristic
292C2×#125C22×C12711yescharacteristic
293#501—21no—
294#498—21no—
295#489—21no—
296#487—21no—
297#483C22×C12714yes—
298#483C22×C12712yes—
299#480C22×C12712yes—
300#477—21no—
301#476—21no—
302#475—22no—
303#432C22×C12711yescharacteristic
304#431C22×C12711yescharacteristic
305#398C22×C12714yes—
306#395—22no—
307#384—21no—
308#382—21no—
309#376C22×C12711yescharacteristic
310#328C22×C12714yes—
311#328C22×C12712yes—
312#327C22×C12712yes—
313#324C22×C12714yes—
314C22×Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—22no—
317C4⋊Q8—22no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8D4×C12711yescharacteristic
323C4⋊Q8D4×C12711yescharacteristic
324#126—22no—
325#126—22no—
326#126—22no—
327#126—22no—
328#126—21no—
329#125—22no—
330#125—22no—
331#125—22no—
332#125—22no—
333#125—21no—
334#125—22no—
335#125—22no—
336#125—22no—
337#125—22no—
338#125—21no—
339#125—21no—
340#125—21no—
341#125—21no—
342#125—21no—
343#125D4×C12711yescharacteristic
344#124D4×C12711yescharacteristic
345#123—22no—
346#123—21no—
347#123—22no—
348C22⋊Q8—21no—
349C22⋊Q8—21no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8C23×C12711yescharacteristic
356C22⋊Q8C23×C12712yes—
357#121C23×C12711yescharacteristic
358C4×Q8—21no—
359C4×Q8—22no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8D4×C12711yescharacteristic
363C4×Q8D4×C12712yes—
364C4×D4—21no—
365C4×D4—22no—
366C4×D4D4×C12711yescharacteristic
367#117—22no—
368#117—21no—
369#117—21no—
370#117—22no—
371#117—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4D4×C12711yescharacteristic
376C2×C4⋊C4D4×C12712yes—
377C2×C4⋊C4D4×C12711yescharacteristic
378C2×C4⋊C4C23×C12712yes—
379C2×C4⋊C4C23×C12712yes—
380C2×C42—21no—
381C2×C42C23×C12711yescharacteristic
382#107—22no—
383#106—24no—
384#106—22no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387#103—24no—
388#103—22no—
389#102—22no—
390C22⋊C8C23×C12712yes—
391C22⋊C8C23×C12712yes—
392#97—21no—
393C4×C8—21no—
394#95—21no—
395#95—24no—
396#95C23×C12712yes—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—82no—
402C2×Q8C2×D4×C12712yes—
403C2×Q8C2×D4×C12711yescharacteristic
404C2×D4—21no—
405C2×D4C2×D4×C12711yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4C2×D4×C12712yes—
411C22×C4C2×D4×C12711yescharacteristic
412C22×C4C2×D4×C12711yescharacteristic
413C22×C4C2×D4×C12712yes—
414C22×C4C24×C12711yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—84no—
447C4⋊C4C2×D4×C12711yescharacteristic
448C4⋊C4C2×D4×C12712yes—
449C4⋊C4C2×D4×C12711yescharacteristic
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C42—24no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42C2×D4×C12711yescharacteristic
466C42C2×D4×C12711yescharacteristic
467C42C2×D4×C12711yescharacteristic
468C42C2×D4×C12711yescharacteristic
469C23—21no—
470C23C22×D4×C12711yescharacteristic
471Q8—41no—
472Q8—42no—
473Q8—42no—
474Q8—82no—
475Q8—82no—
476Q8—82no—
477D4—41no—
478D4—42no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—82no—
498C2×C4#120×C12711yescharacteristic
499C2×C4#120×C12711yescharacteristic
500C2×C4#120×C12711yescharacteristic
501C2×C4#120×C12711yescharacteristic
502C2×C4C22×D4×C12711yescharacteristic
503C2×C4C22×D4×C12711yescharacteristic
504C8—42no—
505C8—42no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22C2×#120×C12711yescharacteristic
512C22#577×C12712yes—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—44no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—82no—
527C4#456×C12712yes—
528C2—22no—
529C2—81no—
530C2C127×#301011yescharacteristic
531C2C2×#456×C12711yescharacteristic
532C2C127×#302811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC22×C4C24×C127
prime radical#121×C127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#3229C2semidirect
4. C127×#2660C2semidirect
5. C127×#1611C2semidirect
7. C127×#1506C2semidirect
9. C127×#1470C2semidirect
261. #18442C127direct (N × H)
269. #3229C2×C127semidirect
270. #2660C2×C127semidirect
271. #1611C2×C127semidirect
273. #1506C2×C127semidirect
275. #1470C2×C127semidirect
276. C127#18442direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35632
C127×#32382
C127×#32292
C127×#26602
C127×#16112
C127×#16092
C127×#15062
C127×#15002
C127×#14702
#18442127