Finite Group Databasebeta

← C127×#8512 #44572185

Subgroups — C127×#8512 #44572185

Order 32512.

652 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3563C211yesmaximal, characteristic
2C127×#1989C211yesmaximal, characteristic, complemented
3C127×#1986C211yesmaximal, characteristic, complemented
4C127×#1552C211yesmaximal, characteristic, complemented
5C127×#1528C211yesmaximal, characteristic, complemented
6C127×#1509C211yesmaximal, characteristic
7C127×#1483C211yesmaximal, characteristic
8#564×C127C412yescomplemented
9#563×C127C412yescomplemented
10#557×C127—21no—
11#554×C127—21no—
12#552×C127—21no—
13#552×C127—22no—
14#552×C127—21no—
15#544×C127—21no—
16#544×C127—21no—
17#541×C127—21no—
18#541×C127—22no—
19#539×C127—21no—
20#533×C127C2211yescharacteristic
21#529×C127C2211yescharacteristic
22C2×#125×C127C2211yescharacteristic
23#432×C127C2211yescharacteristic
24C2×C4⋊C8×C127C2211yescharacteristic
25C4×M16×C127C2211yescharacteristic
26C4×C4⋊C4×C127C2211yescharacteristic
27#340×C127—21no—
28#340×C127—21no—
29#339×C127—22no—
30#332×C127—21no—
31#332×C127—21no—
32#330×C127—22no—
33#328×C127—21no—
34#327×C127—21no—
35Q8⋊C8×C127—21no—
36D4⋊C8×C127—21no—
37C2×M16×C127—21no—
38C22×C8×C127—21no—
39C4⋊Q8×C127—22no—
40C4⋊Q8×C127—21no—
41C4⋊Q8×C127—22no—
42C4⋊Q8×C127D411yescharacteristic
43C4⋊Q8×C127D411yescharacteristic
44#126×C127—22no—
45#126×C127—22no—
46#126×C127—22no—
47#126×C127—22no—
48#126×C127—21no—
49#125×C127—22no—
50#125×C127—22no—
51#125×C127—21no—
52#125×C127—22no—
53#125×C127—22no—
54#125×C127—22no—
55#125×C127—22no—
56#125×C127—22no—
57#125×C127—22no—
58#125×C127—21no—
59#125×C127—21no—
60#125×C127C2×C412yes—
61#125×C127C2×C411yescharacteristic
62#125×C127C2×C412yes—
63#125×C127D412yes—
64#124×C127C2×C411yescharacteristic
65#123×C127—22no—
66#123×C127—22no—
67#123×C127—21no—
68C22⋊Q8×C127—22no—
69C22⋊Q8×C127—21no—
70C22⋊Q8×C127—21no—
71C22⋊Q8×C127—22no—
72C22⋊Q8×C127—21no—
73C22⋊Q8×C127—21no—
74C22⋊Q8×C127—22no—
75C22⋊Q8×C127—22no—
76C22⋊Q8×C127D411yescharacteristic
77#121×C127D411yescharacteristic
78C4×Q8×C127—21no—
79C4×Q8×C127—22no—
80C4×Q8×C127—21no—
81C4×Q8×C127—21no—
82C4×Q8×C127—21no—
83C4×Q8×C127D411yescharacteristic
84C4×D4×C127—22no—
85C4×D4×C127—21no—
86C4×D4×C127D411yescharacteristic
87#117×C127—22no—
88#117×C127—21no—
89#117×C127—22no—
90#117×C127—21no—
91#117×C127D412yes—
92C2×C4⋊C4×C127—21no—
93C2×C4⋊C4×C127—21no—
94C2×C4⋊C4×C127—21no—
95C2×C4⋊C4×C127—21no—
96C2×C4⋊C4×C127D411yescharacteristic
97C2×C4⋊C4×C127D411yescharacteristic
98C2×C42×C127—21no—
99C2×C42×C127C2311yescharacteristic
100C4⋊C8×C127—21no—
101C4⋊C8×C127—21no—
102C4⋊C8×C127—21no—
103C22⋊C8×C127—21no—
104#97×C127—21no—
105#97×C127—21no—
106C4×C8×C127—21no—
107#95×C127—21no—
108C2×Q8×C127—22no—
109C2×Q8×C127—21no—
110C2×Q8×C127—21no—
111C2×Q8×C127C22⋊C411yescharacteristic
112C2×D4×C127—21no—
113C2×D4×C127C22⋊C411yescharacteristic
114C22×C4×C127—21no—
115C22×C4×C127—22no—
116C22×C4×C127—21no—
117C22×C4×C127—21no—
118C22×C4×C127—41no—
119C22×C4×C127C2×D411yescharacteristic
120C22×C4×C127C2×D411yescharacteristic
121C22×C4×C127C2×D411yescharacteristic
122M16×C127—22no—
123M16×C127—41no—
124C2×C8×C127—41no—
125C2×C8×C127—42no—
126C2×C8×C127—41no—
127C2×C8×C127—41no—
128C4⋊C4×C127—22no—
129C4⋊C4×C127—22no—
130C4⋊C4×C127—22no—
131C4⋊C4×C127—22no—
132C4⋊C4×C127—22no—
133C4⋊C4×C127—22no—
134C4⋊C4×C127—22no—
135C4⋊C4×C127—22no—
136C4⋊C4×C127—22no—
137C4⋊C4×C127—22no—
138C4⋊C4×C127—21no—
139C4⋊C4×C127—21no—
140C4⋊C4×C127—22no—
141C4⋊C4×C127—22no—
142C4⋊C4×C127—21no—
143C4⋊C4×C127—21no—
144C4⋊C4×C127—22no—
145C4⋊C4×C127—22no—
146C4⋊C4×C127—21no—
147C4⋊C4×C127—21no—
148C4⋊C4×C127—21no—
149C4⋊C4×C127—21no—
150C4⋊C4×C127—22no—
151C4⋊C4×C127—41no—
152C4⋊C4×C127C22⋊C412yes—
153C4⋊C4×C127C22⋊C412yes—
154C4⋊C4×C127C22⋊C412yes—
155C4⋊C4×C127C22⋊C412yes—
156C4⋊C4×C127C22⋊C411yescharacteristic
157C4⋊C4×C127C22⋊C411yescharacteristic
158C22⋊C4×C127—22no—
159C22⋊C4×C127—22no—
160C22⋊C4×C127—22no—
161C22⋊C4×C127—21no—
162C22⋊C4×C127—22no—
163C22⋊C4×C127—21no—
164C22⋊C4×C127—21no—
165C22⋊C4×C127—21no—
166C22⋊C4×C127—21no—
167C42×C127—22no—
168C42×C127—22no—
169C42×C127—22no—
170C42×C127—22no—
171C42×C127—21no—
172C42×C127—42no—
173C42×C127C22×C411yescharacteristic
174C42×C127C2×D411yescharacteristic
175C42×C127C2×D411yescharacteristic
176C42×C127C2×D411yescharacteristic
177C23×C127—21no—
178C23×C127#12011yescharacteristic
179Q8×C127—41no—
180Q8×C127—41no—
181Q8×C127—41no—
182Q8×C127—81no—
183Q8×C127—82no—
184D4×C127—41no—
185D4×C127—41no—
186D4×C127—41no—
187C2×C4×C127—21no—
188C2×C4×C127—21no—
189C2×C4×C127—21no—
190C2×C4×C127—21no—
191C2×C4×C127—22no—
192C2×C4×C127—21no—
193C2×C4×C127—21no—
194C2×C4×C127—21no—
195C2×C4×C127—22no—
196C2×C4×C127—22no—
197C2×C4×C127—22no—
198C2×C4×C127—41no—
199C2×C4×C127—41no—
200C2×C4×C127—41no—
201C2×C4×C127—41no—
202C2×C4×C127—41no—
203C2×C4×C127—42no—
204C2×C4×C127—81no—
205C2×C4×C127—82no—
206C2×C4×C127—81no—
207C2×C4×C127—81no—
208C2×C4×C127C2×C22⋊C411yescharacteristic
209C2×C4×C127C2×C22⋊C411yescharacteristic
210C2×C4×C127C2×C22⋊C411yescharacteristic
211C2×C4×C127#12011yescharacteristic
212C2×C4×C127#12011yescharacteristic
213C2×C4×C127#12011yescharacteristic
214C8×C127—42no—
215C8×C127—81no—
216C8×C127—81no—
217C22×C127—21no—
218C22×C127—21no—
219C22×C127—41no—
220C22×C127—41no—
221C22×C127—41no—
222C22×C127#37911yescharacteristic
223C22×C127#45411yescharacteristic
224C22×C127#45611yescharacteristic
225C4×C127—21no—
226C4×C127—21no—
227C4×C127—21no—
228C4×C127—21no—
229C4×C127—21no—
230C4×C127—42no—
231C4×C127—41no—
232C4×C127—41no—
233C4×C127—42no—
234C4×C127—42no—
235C4×C127—42no—
236C4×C127—41no—
237C4×C127—82no—
238C4×C127#45411yescharacteristic
239C4×C127#45611yescharacteristic
240#8512C12711yesmaximal, characteristic, direct product factor, Sylow
241C2×C127—21no—
242C2×C127—21no—
243C2×C127—81no—
244C2×C127#179311yescharacteristic
245C2×C127#179311yescharacteristic
246C2×C127#179411yescharacteristic
247#3563C2×C12711yescharacteristic
248#1989C2×C12711yescharacteristic, complemented
249#1986C2×C12711yescharacteristic, complemented
250#1552C2×C12711yescharacteristic, complemented
251#1528C2×C12711yescharacteristic, complemented
252#1509C2×C12711yescharacteristic
253#1483C2×C12711yescharacteristic
254C127#851211yescharacteristic, direct product factor, Sylow
255#564C4×C12712yescomplemented
256#563C4×C12712yescomplemented
257#557—21no—
258#554—21no—
259#552—21no—
260#552—22no—
261#552—21no—
262#544—21no—
263#544—21no—
264#541—21no—
265#541—22no—
266#539—21no—
267#533C22×C12711yescharacteristic
268#529C22×C12711yescharacteristic
269C2×#125C22×C12711yescharacteristic
270#432C22×C12711yescharacteristic
271C2×C4⋊C8C22×C12711yescharacteristic
272C4×M16C22×C12711yescharacteristic
273C4×C4⋊C4C22×C12711yescharacteristic
274#340—21no—
275#340—21no—
276#339—22no—
277#332—21no—
278#332—21no—
279#330—22no—
280#328—21no—
281#327—21no—
282Q8⋊C8—21no—
283D4⋊C8—21no—
284C2×M16—21no—
285C22×C8—21no—
286C4⋊Q8—22no—
287C4⋊Q8—21no—
288C4⋊Q8—22no—
289C4⋊Q8D4×C12711yescharacteristic
290C4⋊Q8D4×C12711yescharacteristic
291#126—22no—
292#126—22no—
293#126—22no—
294#126—22no—
295#126—21no—
296#125—22no—
297#125—22no—
298#125—21no—
299#125—22no—
300#125—22no—
301#125—22no—
302#125—22no—
303#125—22no—
304#125—22no—
305#125—21no—
306#125—21no—
307#125C2×C4×C12712yes—
308#125C2×C4×C12711yescharacteristic
309#125C2×C4×C12712yes—
310#125D4×C12712yes—
311#124C2×C4×C12711yescharacteristic
312#123—22no—
313#123—22no—
314#123—21no—
315C22⋊Q8—22no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—22no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8D4×C12711yescharacteristic
324#121D4×C12711yescharacteristic
325C4×Q8—21no—
326C4×Q8—22no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8D4×C12711yescharacteristic
331C4×D4—22no—
332C4×D4—21no—
333C4×D4D4×C12711yescharacteristic
334#117—22no—
335#117—21no—
336#117—22no—
337#117—21no—
338#117D4×C12712yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4D4×C12711yescharacteristic
344C2×C4⋊C4D4×C12711yescharacteristic
345C2×C42—21no—
346C2×C42C23×C12711yescharacteristic
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C22⋊C8—21no—
351#97—21no—
352#97—21no—
353C4×C8—21no—
354#95—21no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8C22⋊C4×C12711yescharacteristic
359C2×D4—21no—
360C2×D4C22⋊C4×C12711yescharacteristic
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4C2×D4×C12711yescharacteristic
367C22×C4C2×D4×C12711yescharacteristic
368C22×C4C2×D4×C12711yescharacteristic
369M16—22no—
370M16—41no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—41no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4C22⋊C4×C12712yes—
400C4⋊C4C22⋊C4×C12712yes—
401C4⋊C4C22⋊C4×C12712yes—
402C4⋊C4C22⋊C4×C12712yes—
403C4⋊C4C22⋊C4×C12711yescharacteristic
404C4⋊C4C22⋊C4×C12711yescharacteristic
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—21no—
409C22⋊C4—22no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—21no—
419C42—42no—
420C42C22×C4×C12711yescharacteristic
421C42C2×D4×C12711yescharacteristic
422C42C2×D4×C12711yescharacteristic
423C42C2×D4×C12711yescharacteristic
424C23—21no—
425C23#120×C12711yescharacteristic
426Q8—41no—
427Q8—41no—
428Q8—41no—
429Q8—81no—
430Q8—82no—
431D4—41no—
432D4—41no—
433D4—41no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4C2×C22⋊C4×C12711yescharacteristic
456C2×C4C2×C22⋊C4×C12711yescharacteristic
457C2×C4C2×C22⋊C4×C12711yescharacteristic
458C2×C4#120×C12711yescharacteristic
459C2×C4#120×C12711yescharacteristic
460C2×C4#120×C12711yescharacteristic
461C8—42no—
462C8—81no—
463C8—81no—
464C22—21no—
465C22—21no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22#379×C12711yescharacteristic
470C22#454×C12711yescharacteristic
471C22#456×C12711yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—41no—
479C4—41no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—41no—
484C4—82no—
485C4#454×C12711yescharacteristic
486C4#456×C12711yescharacteristic
487C2—21no—
488C2—21no—
489C2—81no—
490C2C127×#179311yescharacteristic
491C2C127×#179311yescharacteristic
492C2C127×#179411yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radical#121×C127D4
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
2. C127×#1989C2semidirect
3. C127×#1986C2semidirect
4. C127×#1552C2semidirect
5. C127×#1528C2semidirect
8. #564×C127C4semidirect
9. #563×C127C4semidirect
240. #8512C127direct (N × H)
248. #1989C2×C127semidirect
249. #1986C2×C127semidirect
250. #1552C2×C127semidirect
251. #1528C2×C127semidirect
254. C127#8512direct (N × H)
255. #564C4×C127semidirect
256. #563C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35632
C127×#19892
C127×#19862
C127×#15522
C127×#15282
C127×#15092
C127×#14832
#8512127