Finite Group Databasebeta

← C127×#12575 #44576248

Subgroups — C127×#12575 #44576248

Order 32512.

810 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#413×C127C211yesmaximal, characteristic
2C2×#410×C127C211yesmaximal, characteristic
3C127×#2311C211yesmaximal, characteristic
4C127×#1390C214yesmaximal, complemented
5C22×#41×C127C2211yescharacteristic
6C22×M16×C127—21no—
7C2×#117×C127—21no—
8C2×#117×C127C411yescharacteristic, complemented
9C22×C4⋊C4×C127C411yescharacteristic, complemented
10#413×C127—22no—
11#413×C127C414yescomplemented
12C2×#101×C127—21no—
13C2×#100×C127C2212yes—
14#410×C127—22no—
15#410×C127C414yescomplemented
16C2×#99×C127—21no—
17C2×#99×C127C2212yes—
18#391×C127—21no—
19#390×C127—21no—
20#386×C127C412yescomplemented
21#386×C127C2212yes—
22#380×C127—21no—
23#380×C127—21no—
24C2×#41×C127—41no—
25C2×#41×C127—41no—
26C2×#41×C127—42no—
27C2×#41×C127C2×C412yes—
28C2×#41×C127C2×C412yes—
29C2×#41×C127D412yes—
30C2×#41×C127Q812yes—
31C22×Q8×C127D411yescharacteristic
32C22×D4×C127C2×C411yescharacteristic
33C22×D4×C127D411yescharacteristic
34C22×D4×C127C2311yescharacteristic
35C23×C4×C127C2×C411yescharacteristic
36C23×C4×C127C2×C412yes—
37C23×C4×C127D411yescharacteristic
38C23×C4×C127D411yescharacteristic
39C2×M16×C127—24no—
40C2×M16×C127—22no—
41C2×M16×C127—41no—
42C2×M16×C127—42no—
43C22×C8×C127—41no—
44#117×C127—24no—
45#117×C127—42no—
46#117×C127—42no—
47C2×C4⋊C4×C127—21no—
48C2×C4⋊C4×C127—21no—
49C2×C4⋊C4×C127—21no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—41no—
52C2×C4⋊C4×C127—41no—
53C2×C22⋊C4×C127—22no—
54C2×C22⋊C4×C127—22no—
55C2×C22⋊C4×C127—22no—
56C2×C22⋊C4×C127C2×C412yes—
57C2×C42×C127—21no—
58C2×C42×C127—41no—
59#101×C127—24no—
60#100×C127—22no—
61#100×C127C2×C414yes—
62#99×C127—24no—
63#99×C127—22no—
64#99×C127C2×C414yes—
65#95×C127—22no—
66#95×C127—22no—
67C24×C127C22⋊C411yescharacteristic
68C24×C127C22×C411yescharacteristic
69C24×C127C2×D411yescharacteristic
70#41×C127—42no—
71#41×C127—42no—
72#41×C127—42no—
73#41×C127—42no—
74#41×C127—42no—
75#41×C127—42no—
76#41×C127—42no—
77#41×C127—42no—
78C2×Q8×C127—41no—
79C2×Q8×C127—41no—
80C2×Q8×C127C22⋊C412yes—
81C2×Q8×C127C4⋊C412yes—
82C2×D4×C127—41no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C2×D4×C127—41no—
87C2×D4×C127—41no—
88C2×D4×C127C4212yes—
89C2×D4×C127C22⋊C412yes—
90C2×D4×C127C4⋊C412yes—
91C2×D4×C127C4⋊C412yes—
92C2×D4×C127C22×C411yescharacteristic
93C2×D4×C127C22×C411yescharacteristic
94C2×D4×C127C2×D411yescharacteristic
95C2×D4×C127C2×Q811yescharacteristic
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—21no—
103C22×C4×C127—22no—
104C22×C4×C127—21no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—42no—
108C22×C4×C127—42no—
109C22×C4×C127—82no—
110C22×C4×C127C4212yes—
111C22×C4×C127C22⋊C411yescharacteristic
112C22×C4×C127C22⋊C411yescharacteristic
113C22×C4×C127C22⋊C412yes—
114C22×C4×C127C22⋊C412yes—
115C22×C4×C127C22⋊C411yescharacteristic
116C22×C4×C127C4⋊C412yes—
117C22×C4×C127C4⋊C412yes—
118C22×C4×C127C4⋊C412yes—
119C22×C4×C127C2×D411yescharacteristic
120M16×C127—24no—
121M16×C127—42no—
122M16×C127—44no—
123C2×C8×C127—42no—
124C2×C8×C127—44no—
125C4⋊C4×C127—42no—
126C4⋊C4×C127—44no—
127C4⋊C4×C127—41no—
128C4⋊C4×C127—41no—
129C4⋊C4×C127—42no—
130C22⋊C4×C127—24no—
131C22⋊C4×C127—42no—
132C22⋊C4×C127—42no—
133C22⋊C4×C127—82no—
134C22⋊C4×C127—82no—
135C42×C127—42no—
136C42×C127—44no—
137C23×C127—21no—
138C23×C127—21no—
139C23×C127—21no—
140C23×C127—21no—
141C23×C127—21no—
142C23×C127—21no—
143C23×C127—41no—
144C23×C127—41no—
145C23×C127—41no—
146C23×C127—41no—
147C23×C127—41no—
148C23×C127—41no—
149C23×C127#9512yes—
150C23×C127C2×C4211yescharacteristic
151C23×C127C2×C22⋊C411yescharacteristic
152C23×C127C2×C22⋊C411yescharacteristic
153C23×C127C2×C4⋊C411yescharacteristic
154C23×C127C2×C4⋊C411yescharacteristic
155Q8×C127—42no—
156Q8×C127—42no—
157D4×C127—42no—
158D4×C127—42no—
159D4×C127—42no—
160D4×C127—42no—
161D4×C127—41no—
162D4×C127—41no—
163D4×C127—41no—
164D4×C127—41no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—21no—
169C2×C4×C127—21no—
170C2×C4×C127—21no—
171C2×C4×C127—22no—
172C2×C4×C127—22no—
173C2×C4×C127—22no—
174C2×C4×C127—22no—
175C2×C4×C127—22no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—22no—
180C2×C4×C127—22no—
181C2×C4×C127—22no—
182C2×C4×C127—22no—
183C2×C4×C127—44no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—42no—
189C2×C4×C127—42no—
190C2×C4×C127—82no—
191C2×C4×C127—82no—
192C2×C4×C127—84no—
193C2×C4×C127#9512yes—
194C2×C4×C127#9512yes—
195C2×C4×C127#9512yes—
196C2×C4×C127#9912yes—
197C2×C4×C127#9911yescharacteristic
198C2×C4×C127#9911yescharacteristic
199C2×C4×C127C2×C22⋊C411yescharacteristic
200C2×C4×C127C2×C4⋊C411yescharacteristic
201C8×C127—44no—
202C22×C127—21no—
203C22×C127—21no—
204C22×C127—21no—
205C22×C127—22no—
206C22×C127—21no—
207C22×C127—21no—
208C22×C127—21no—
209C22×C127—42no—
210C22×C127—41no—
211C22×C127—41no—
212C22×C127—41no—
213C22×C127—41no—
214C22×C127—42no—
215C22×C127—41no—
216C22×C127—41no—
217C22×C127—41no—
218C22×C127—41no—
219C22×C127—41no—
220C22×C127—41no—
221C22×C127—81no—
222C22×C127—81no—
223C22×C127C2×#9511yescharacteristic
224C22×C127C2×#9911yescharacteristic
225C22×C127C2×#9911yescharacteristic
226C4×C127—21no—
227C4×C127—22no—
228C4×C127—22no—
229C4×C127—21no—
230C4×C127—82no—
231C4×C127—84no—
232C4×C127—82no—
233C4×C127#34212yes—
234C4×C127#34212yes—
235#12575C12711yesmaximal, characteristic, direct product factor, Sylow
236C2×C127—21no—
237C2×C127—21no—
238C2×C127—41no—
239C2×C127—41no—
240C2×C127—41no—
241C2×C127—42no—
242C2×C127—41no—
243C2×C127C2×#34211yescharacteristic
244C2×C127#212211yescharacteristic
245C2×C127#212311yescharacteristic
246C2×#413C2×C12711yescharacteristic
247C2×#410C2×C12711yescharacteristic
248#2311C2×C12711yescharacteristic
249#1390C2×C12714yescomplemented
250C127#1257511yescharacteristic, direct product factor, Sylow
251C22×#41C22×C12711yescharacteristic
252C22×M16—21no—
253C2×#117—21no—
254C2×#117C4×C12711yescharacteristic, complemented
255C22×C4⋊C4C4×C12711yescharacteristic, complemented
256#413—22no—
257#413C4×C12714yescomplemented
258C2×#101—21no—
259C2×#100C22×C12712yes—
260#410—22no—
261#410C4×C12714yescomplemented
262C2×#99—21no—
263C2×#99C22×C12712yes—
264#391—21no—
265#390—21no—
266#386C4×C12712yescomplemented
267#386C22×C12712yes—
268#380—21no—
269#380—21no—
270C2×#41—41no—
271C2×#41—41no—
272C2×#41—42no—
273C2×#41C2×C4×C12712yes—
274C2×#41C2×C4×C12712yes—
275C2×#41D4×C12712yes—
276C2×#41Q8×C12712yes—
277C22×Q8D4×C12711yescharacteristic
278C22×D4C2×C4×C12711yescharacteristic
279C22×D4D4×C12711yescharacteristic
280C22×D4C23×C12711yescharacteristic
281C23×C4C2×C4×C12711yescharacteristic
282C23×C4C2×C4×C12712yes—
283C23×C4D4×C12711yescharacteristic
284C23×C4D4×C12711yescharacteristic
285C2×M16—24no—
286C2×M16—22no—
287C2×M16—41no—
288C2×M16—42no—
289C22×C8—41no—
290#117—24no—
291#117—42no—
292#117—42no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—41no—
298C2×C4⋊C4—41no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4C2×C4×C12712yes—
303C2×C42—21no—
304C2×C42—41no—
305#101—24no—
306#100—22no—
307#100C2×C4×C12714yes—
308#99—24no—
309#99—22no—
310#99C2×C4×C12714yes—
311#95—22no—
312#95—22no—
313C24C22⋊C4×C12711yescharacteristic
314C24C22×C4×C12711yescharacteristic
315C24C2×D4×C12711yescharacteristic
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8C22⋊C4×C12712yes—
327C2×Q8C4⋊C4×C12712yes—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C42×C12712yes—
335C2×D4C22⋊C4×C12712yes—
336C2×D4C4⋊C4×C12712yes—
337C2×D4C4⋊C4×C12712yes—
338C2×D4C22×C4×C12711yescharacteristic
339C2×D4C22×C4×C12711yescharacteristic
340C2×D4C2×D4×C12711yescharacteristic
341C2×D4C2×Q8×C12711yescharacteristic
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—82no—
356C22×C4C42×C12712yes—
357C22×C4C22⋊C4×C12711yescharacteristic
358C22×C4C22⋊C4×C12711yescharacteristic
359C22×C4C22⋊C4×C12712yes—
360C22×C4C22⋊C4×C12712yes—
361C22×C4C22⋊C4×C12711yescharacteristic
362C22×C4C4⋊C4×C12712yes—
363C22×C4C4⋊C4×C12712yes—
364C22×C4C4⋊C4×C12712yes—
365C22×C4C2×D4×C12711yescharacteristic
366M16—24no—
367M16—42no—
368M16—44no—
369C2×C8—42no—
370C2×C8—44no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C42—42no—
382C42—44no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23#95×C12712yes—
396C23C2×C42×C12711yescharacteristic
397C23C2×C22⋊C4×C12711yescharacteristic
398C23C2×C22⋊C4×C12711yescharacteristic
399C23C2×C4⋊C4×C12711yescharacteristic
400C23C2×C4⋊C4×C12711yescharacteristic
401Q8—42no—
402Q8—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—84no—
439C2×C4#95×C12712yes—
440C2×C4#95×C12712yes—
441C2×C4#95×C12712yes—
442C2×C4#99×C12712yes—
443C2×C4#99×C12711yescharacteristic
444C2×C4#99×C12711yescharacteristic
445C2×C4C2×C22⋊C4×C12711yescharacteristic
446C2×C4C2×C4⋊C4×C12711yescharacteristic
447C8—44no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—81no—
468C22—81no—
469C22C2×#95×C12711yescharacteristic
470C22C2×#99×C12711yescharacteristic
471C22C2×#99×C12711yescharacteristic
472C4—21no—
473C4—22no—
474C4—22no—
475C4—21no—
476C4—82no—
477C4—84no—
478C4—82no—
479C4#342×C12712yes—
480C4#342×C12712yes—
481C2—21no—
482C2—21no—
483C2—41no—
484C2—41no—
485C2—41no—
486C2—42no—
487C2—41no—
488C2C2×#342×C12711yescharacteristic
489C2C127×#212211yescharacteristic
490C2C127×#212311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC23C2×C42×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1390C2semidirect
8. C2×#117×C127C4semidirect
9. C22×C4⋊C4×C127C4semidirect
11. #413×C127C4semidirect
15. #410×C127C4semidirect
20. #386×C127C4semidirect
235. #12575C127direct (N × H)
249. #1390C2×C127semidirect
250. C127#12575direct (N × H)
254. C2×#117C4×C127semidirect
255. C22×C4⋊C4C4×C127semidirect
257. #413C4×C127semidirect
261. #410C4×C127semidirect
266. #386C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#413×C1272
C2×#410×C1272
C127×#23112
C127×#13902
#12575127