Finite Group Databasebeta

← C127×#17144 #44580817

Subgroups — C127×#17144 #44580817

Order 32512.

1074 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×#400×C127C211yesmaximal, characteristic
2C2×#395×C127C211yesmaximal, characteristic
3C127×#1446C212yesmaximal
4C127×#1445C211yesmaximal, characteristic
5C127×#1443C212yesmaximal
6C127×#1443C212yesmaximal
7C127×#1443C212yesmaximal
8C127×#1442C212yesmaximal
9C127×#1441C212yesmaximal
10C23×Q8×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2212yes—
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14#400×C127—24no—
15#395×C127—24no—
16C2×#95×C127C2211yescharacteristic
17C2×#95×C127C2212yes—
18C2×#95×C127C2211yescharacteristic
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2212yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2212yes—
30C2×#95×C127C2211yescharacteristic
31C2×#95×C127C2211yescharacteristic
32C2×#95×C127C2212yes—
33C22×Q8×C127—22no—
34C22×Q8×C127—24no—
35C22×Q8×C127—42no—
36C22×Q8×C127—41no—
37C22×Q8×C127—41no—
38C23×C4×C127C2312yes—
39C23×C4×C127C2312yes—
40C23×C4×C127C2312yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2311yescharacteristic
43C23×C4×C127C2311yescharacteristic
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2312yes—
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—41no—
51C2×C4⋊C4×C127—41no—
52C2×C4⋊C4×C127—42no—
53C2×C4⋊C4×C127—41no—
54C2×C4⋊C4×C127—41no—
55C2×C4⋊C4×C127D414yes—
56#95×C127—22no—
57#95×C127—24no—
58#95×C127—22no—
59#95×C127—24no—
60#95×C127—24no—
61#95×C127—24no—
62#95×C127—24no—
63#95×C127—24no—
64#95×C127—24no—
65#95×C127—42no—
66#95×C127—42no—
67#95×C127—41no—
68#95×C127—41no—
69#95×C127—42no—
70#95×C127—42no—
71#95×C127—42no—
72#95×C127—42no—
73C24×C127C2411yescharacteristic
74C2×Q8×C127—48no—
75C2×Q8×C127—44no—
76C2×Q8×C127—44no—
77C2×Q8×C127—48no—
78C2×Q8×C127—44no—
79C22×C4×C127—22no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—21no—
86C22×C4×C127—21no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—22no—
93C22×C4×C127—21no—
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—22no—
102C22×C4×C127—22no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—22no—
108C22×C4×C127—22no—
109C22×C4×C127—22no—
110C22×C4×C127—22no—
111C22×C4×C127—22no—
112C22×C4×C127—22no—
113C22×C4×C127—22no—
114C22×C4×C127—22no—
115C22×C4×C127—22no—
116C22×C4×C127—22no—
117C22×C4×C127—22no—
118C22×C4×C127—22no—
119C22×C4×C127—22no—
120C22×C4×C127—22no—
121C22×C4×C127—22no—
122C22×C4×C127—22no—
123C22×C4×C127—22no—
124C22×C4×C127—22no—
125C22×C4×C127—22no—
126C22×C4×C127—22no—
127C22×C4×C127—22no—
128C22×C4×C127—22no—
129C22×C4×C127—22no—
130C22×C4×C127—22no—
131C22×C4×C127C2×D412yes—
132C22×C4×C127C2×D412yes—
133C22×C4×C127C2×D412yes—
134C22×C4×C127#4114yes—
135C22×C4×C127#4114yes—
136C4⋊C4×C127—44no—
137C4⋊C4×C127—48no—
138C4⋊C4×C127—44no—
139C23×C127C22×D411yescharacteristic
140C23×C127C2×#4112yes—
141C23×C127C2×#4112yes—
142C23×C127H(5,𝔽2)11yescharacteristic
143C23×C127H(5,𝔽2)11yescharacteristic
144C23×C127H(5,𝔽2)12yes—
145C23×C1272−1+412yes—
146C23×C1272−1+412yes—
147C23×C1272−1+412yes—
148Q8×C127—416no—
149C2×C4×C127—22no—
150C2×C4×C127—22no—
151C2×C4×C127—22no—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—24no—
155C2×C4×C127—24no—
156C2×C4×C127—24no—
157C2×C4×C127—24no—
158C2×C4×C127—24no—
159C2×C4×C127—24no—
160C2×C4×C127—24no—
161C2×C4×C127—24no—
162C2×C4×C127—42no—
163C2×C4×C127—42no—
164C2×C4×C127—42no—
165C2×C4×C127—42no—
166C2×C4×C127—42no—
167C2×C4×C127—42no—
168C2×C4×C127—42no—
169C2×C4×C127—42no—
170C2×C4×C127—41no—
171C2×C4×C127—41no—
172C2×C4×C127—41no—
173C2×C4×C127—41no—
174C2×C4×C127—42no—
175C2×C4×C127—42no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—42no—
182C2×C4×C127—42no—
183C2×C4×C127—42no—
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—42no—
191C2×C4×C127—42no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—42no—
196C2×C4×C127—42no—
197C2×C4×C127—42no—
198C2×C4×C127—42no—
199C2×C4×C127—42no—
200C2×C4×C127—42no—
201C2×C4×C127—42no—
202C2×C4×C127—42no—
203C2×C4×C127—42no—
204C2×C4×C127—42no—
205C2×C4×C127—42no—
206C22×C127#53411yescharacteristic
207C22×C127#53611yescharacteristic
208C22×C127#53611yescharacteristic
209C22×C127#53812yes—
210C22×C127#53912yes—
211C22×C127#54112yes—
212C22×C127#54112yes—
213C22×C127#54212yes—
214C22×C127#54212yes—
215C22×C127#54612yes—
216C22×C127#54812yes—
217C22×C127#55111yescharacteristic
218C22×C127#55212yes—
219C22×C127#55511yescharacteristic
220C22×C127#55612yes—
221C22×C127#56011yescharacteristic
222C22×C127#56312yes—
223C22×C127#56312yes—
224C22×C127#56311yescharacteristic
225C22×C127#56312yes—
226C22×C127#56312yes—
227C4×C127—42no—
228C4×C127—44no—
229C4×C127—42no—
230C4×C127—42no—
231C4×C127—44no—
232C4×C127—82no—
233C4×C127—82no—
234C4×C127—82no—
235C4×C127—82no—
236#17144C12711yesmaximal, characteristic, direct product factor, Sylow
237C2×C127#269012yes—
238C2×C127#269711yescharacteristic
239C2×C127#271612yes—
240C2×C127#273111yescharacteristic
241C2×C127#275812yes—
242C2×C127#277512yes—
243C2×C127#281711yescharacteristic
244C2×C127#283312yes—
245C2×C127#283812yes—
246C2×#400C2×C12711yescharacteristic
247C2×#395C2×C12711yescharacteristic
248#1446C2×C12712yes—
249#1445C2×C12711yescharacteristic
250#1443C2×C12712yes—
251#1443C2×C12712yes—
252#1443C2×C12712yes—
253#1442C2×C12712yes—
254#1441C2×C12712yes—
255C127#1714411yescharacteristic, direct product factor, Sylow
256C23×Q8C22×C12711yescharacteristic
257C22×C4⋊C4C22×C12712yes—
258C22×C4⋊C4C22×C12711yescharacteristic
259C22×C4⋊C4C22×C12711yescharacteristic
260#400—24no—
261#395—24no—
262C2×#95C22×C12711yescharacteristic
263C2×#95C22×C12712yes—
264C2×#95C22×C12711yescharacteristic
265C2×#95C22×C12712yes—
266C2×#95C22×C12712yes—
267C2×#95C22×C12712yes—
268C2×#95C22×C12712yes—
269C2×#95C22×C12712yes—
270C2×#95C22×C12712yes—
271C2×#95C22×C12712yes—
272C2×#95C22×C12712yes—
273C2×#95C22×C12712yes—
274C2×#95C22×C12712yes—
275C2×#95C22×C12712yes—
276C2×#95C22×C12711yescharacteristic
277C2×#95C22×C12711yescharacteristic
278C2×#95C22×C12712yes—
279C22×Q8—22no—
280C22×Q8—24no—
281C22×Q8—42no—
282C22×Q8—41no—
283C22×Q8—41no—
284C23×C4C23×C12712yes—
285C23×C4C23×C12712yes—
286C23×C4C23×C12712yes—
287C23×C4C23×C12712yes—
288C23×C4C23×C12711yescharacteristic
289C23×C4C23×C12711yescharacteristic
290C23×C4C23×C12711yescharacteristic
291C23×C4C23×C12712yes—
292C23×C4C23×C12712yes—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—41no—
297C2×C4⋊C4—41no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—41no—
301C2×C4⋊C4D4×C12714yes—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—42no—
312#95—42no—
313#95—41no—
314#95—41no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—42no—
319C24C24×C12711yescharacteristic
320C2×Q8—48no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—44no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4C2×D4×C12712yes—
378C22×C4C2×D4×C12712yes—
379C22×C4C2×D4×C12712yes—
380C22×C4#41×C12714yes—
381C22×C4#41×C12714yes—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C23C22×D4×C12711yescharacteristic
386C23C2×#41×C12712yes—
387C23C2×#41×C12712yes—
388C23H(5,𝔽2)×C12711yescharacteristic
389C23H(5,𝔽2)×C12711yescharacteristic
390C23H(5,𝔽2)×C12712yes—
391C232−1+4×C12712yes—
392C232−1+4×C12712yes—
393C232−1+4×C12712yes—
394Q8—416no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C22#534×C12711yescharacteristic
453C22#536×C12711yescharacteristic
454C22#536×C12711yescharacteristic
455C22#538×C12712yes—
456C22#539×C12712yes—
457C22#541×C12712yes—
458C22#541×C12712yes—
459C22#542×C12712yes—
460C22#542×C12712yes—
461C22#546×C12712yes—
462C22#548×C12712yes—
463C22#551×C12711yescharacteristic
464C22#552×C12712yes—
465C22#555×C12711yescharacteristic
466C22#556×C12712yes—
467C22#560×C12711yescharacteristic
468C22#563×C12711yescharacteristic
469C22#563×C12712yes—
470C22#563×C12712yes—
471C22#563×C12712yes—
472C22#563×C12712yes—
473C4—44no—
474C4—42no—
475C4—44no—
476C4—42no—
477C4—42no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4—82no—
482C2C127×#269012yes—
483C2C127×#269711yescharacteristic
484C2C127×#271612yes—
485C2C127×#273111yescharacteristic
486C2C127×#275812yes—
487C2C127×#277512yes—
488C2C127×#281711yescharacteristic
489C2C127×#283312yes—
490C2C127×#283812yes—
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
236. #17144C127direct (N × H)
255. C127#17144direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#400×C1272
C2×#395×C1272
C127×#14462
C127×#14452
C127×#14432
C127×#14422
C127×#14412
#17144127