Finite Group Databasebeta

← C127×#15761 #44579434

Subgroups — C127×#15761 #44579434

Order 32512.

2234 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
1C22×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
2C22×#124×C127C211yesmaximal, characteristic
3C2×#394×C127C212yesmaximal
4C2×#393×C127C212yesmaximal, complemented
5C127×#1446C211yesmaximal, characteristic, complemented
6C127×#1439C214yesmaximal, complemented
7C127×#1437C212yesmaximal, complemented
8C127×#1436C212yesmaximal, complemented
9C23×Q8×C127C2211yescharacteristic
10C23×Q8×C127C2211yescharacteristic
11C23×D4×C127C2211yescharacteristic
12C2×C4⋊Q8×C127—24no—
13C2×C4⋊Q8×C127—28no—
14C2×#124×C127—24no—
15C2×#124×C127—28no—
16C22×C4⋊C4×C127C2212yes—
17C22×C4⋊C4×C127C2212yes—
18C22×C22⋊C4×C127C2214yes—
19C22×C22⋊C4×C127C2212yes—
20C22×C22⋊C4×C127C2214yes—
21C22×C22⋊C4×C127C2212yes—
22C22×C42×C127C2211yescharacteristic
23C22×C42×C127C2211yescharacteristic
24#394×C127—28no—
25#393×C127—28no—
26C2×#95×C127C2214yes—
27C2×#95×C127C2212yes—
28C2×#95×C127C2212yes—
29C2×#95×C127C2214yes—
30C2×#95×C127C2212yes—
31C25×C127C2312yes—
32C22×Q8×C127—24no—
33C22×Q8×C127—24no—
34C22×Q8×C127—24no—
35C22×Q8×C127—22no—
36C22×Q8×C127—24no—
37C22×Q8×C127—42no—
38C22×Q8×C127—42no—
39C22×Q8×C127D414yes—
40C22×D4×C127—24no—
41C22×D4×C127—22no—
42C22×D4×C127—24no—
43C22×D4×C127—42no—
44C23×C4×C127C2314yes—
45C23×C4×C127C2312yes—
46C23×C4×C127C2312yes—
47C23×C4×C127C2311yescharacteristic
48C23×C4×C127C2311yescharacteristic
49C23×C4×C127C2311yescharacteristic
50C23×C4×C127C2312yes—
51C4⋊Q8×C127—232no—
52#124×C127—232no—
53C2×C4⋊C4×C127—24no—
54C2×C4⋊C4×C127—24no—
55C2×C4⋊C4×C127—24no—
56C2×C4⋊C4×C127—24no—
57C2×C4⋊C4×C127—24no—
58C2×C4⋊C4×C127D418yes—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—24no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—24no—
63C2×C22⋊C4×C127—24no—
64C2×C22⋊C4×C127—28no—
65C2×C22⋊C4×C127—28no—
66C2×C22⋊C4×C127—28no—
67C2×C22⋊C4×C127—28no—
68C2×C22⋊C4×C127—28no—
69C2×C22⋊C4×C127—44no—
70C2×C22⋊C4×C127D418yes—
71C2×C42×C127—24no—
72C2×C42×C127—22no—
73C2×C42×C127—22no—
74C2×C42×C127—24no—
75#95×C127—28no—
76#95×C127—24no—
77#95×C127—24no—
78#95×C127—28no—
79#95×C127—42no—
80C24×C127—22no—
81C24×C127—22no—
82C24×C127—24no—
83C24×C127—24no—
84C24×C127—24no—
85C24×C127—24no—
86C24×C127—24no—
87C24×C127—24no—
88C24×C127C2×D414yes—
89C24×C127C2411yescharacteristic
90C2×Q8×C127—28no—
91C2×Q8×C127—28no—
92C2×Q8×C127—28no—
93C2×Q8×C127—44no—
94C2×Q8×C127—416no—
95C2×Q8×C127—416no—
96C2×Q8×C127—44no—
97C2×Q8×C127—44no—
98C2×D4×C127—28no—
99C2×D4×C127—416no—
100C2×D4×C127—44no—
101C2×D4×C127—44no—
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—24no—
109C22×C4×C127—24no—
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—24no—
117C22×C4×C127—22no—
118C22×C4×C127—24no—
119C22×C4×C127—24no—
120C22×C4×C127—24no—
121C22×C4×C127—24no—
122C22×C4×C127—24no—
123C22×C4×C127—24no—
124C22×C4×C127—24no—
125C22×C4×C127—24no—
126C22×C4×C127—24no—
127C22×C4×C127C2×D414yes—
128C22×C4×C127C2×D412yes—
129C22×C4×C127C2×D414yes—
130C22×C4×C127C2×D414yes—
131C22×C4×C127C2×D414yes—
132C22×C4×C127C2×D412yes—
133C22×C4×C127C2×D412yes—
134C22×C4×C127C2×D414yes—
135C22×C4×C127#4118yes—
136C4⋊C4×C127—216no—
137C4⋊C4×C127—216no—
138C22⋊C4×C127—216no—
139C22⋊C4×C127—216no—
140C22⋊C4×C127—416no—
141C22⋊C4×C127—416no—
142C42×C127—28no—
143C42×C127—28no—
144C23×C127—24no—
145C23×C127—24no—
146C23×C127—24no—
147C23×C127—28no—
148C23×C127—28no—
149C23×C127—42no—
150C23×C127—44no—
151C23×C127—42no—
152C23×C127—42no—
153C23×C127—44no—
154C23×C127—42no—
155C23×C127—44no—
156C23×C127—44no—
157C23×C127—44no—
158C23×C127—44no—
159C23×C127—44no—
160C23×C127—44no—
161C23×C127—44no—
162C23×C127—44no—
163C23×C127—44no—
164C23×C127—44no—
165C23×C127C22×D411yescharacteristic
166C23×C127C22×D412yes—
167C23×C127C22×D412yes—
168C23×C127C2×#4114yes—
169C23×C127H(5,𝔽2)12yes—
170C23×C1272−1+414yes—
171Q8×C127—416no—
172Q8×C127—416no—
173D4×C127—416no—
174C2×C4×C127—24no—
175C2×C4×C127—22no—
176C2×C4×C127—22no—
177C2×C4×C127—22no—
178C2×C4×C127—22no—
179C2×C4×C127—24no—
180C2×C4×C127—28no—
181C2×C4×C127—28no—
182C2×C4×C127—28no—
183C2×C4×C127—28no—
184C2×C4×C127—24no—
185C2×C4×C127—22no—
186C2×C4×C127—22no—
187C2×C4×C127—24no—
188C2×C4×C127—22no—
189C2×C4×C127—28no—
190C2×C4×C127—28no—
191C2×C4×C127—42no—
192C2×C4×C127—42no—
193C2×C4×C127—42no—
194C2×C4×C127—42no—
195C2×C4×C127—44no—
196C2×C4×C127—44no—
197C2×C4×C127—44no—
198C2×C4×C127—44no—
199C2×C4×C127—44no—
200C2×C4×C127—44no—
201C2×C4×C127—44no—
202C2×C4×C127—44no—
203C2×C4×C127—44no—
204C2×C4×C127—44no—
205C2×C4×C127#12014yes—
206C2×C4×C127#12014yes—
207C2×C4×C127#12418yescomplemented
208C22×C127—44no—
209C22×C127—48no—
210C22×C127—44no—
211C22×C127—44no—
212C22×C127—48no—
213C22×C127—84no—
214C22×C127—84no—
215C22×C127—84no—
216C22×C127—84no—
217C22×C127C2×#12011yescharacteristic
218C22×C127C2×#12011yescharacteristic
219C22×C127C2×#12412yes—
220C22×C127#53212yes—
221C22×C127#53411yescharacteristic
222C22×C127#53612yes—
223C22×C127#53612yes—
224C22×C127#54214yes—
225C22×C127D4214yes—
226C22×C127#54714yes—
227C22×C127#54814yes—
228C22×C127#54814yes—
229C22×C127#55512yes—
230C22×C127#56312yes—
231C4×C127—24no—
232C4×C127—24no—
233C4×C127—28no—
234C4×C127—42no—
235C4×C127—44no—
236C4×C127—44no—
237C4×C127—84no—
238#15761C12711yesmaximal, characteristic, direct product factor, Sylow
239C2×C127—84no—
240C2×C127#240311yescharacteristic
241C2×C127#243212yes—
242C2×C127#243612yes—
243C2×C127#249114yes—
244C2×C127#267414yes—
245C2×C127#273112yes—
246C22×C4⋊Q8C2×C12711yescharacteristic, complemented
247C22×#124C2×C12711yescharacteristic
248C2×#394C2×C12712yes—
249C2×#393C2×C12712yescomplemented
250#1446C2×C12711yescharacteristic, complemented
251#1439C2×C12714yescomplemented
252#1437C2×C12712yescomplemented
253#1436C2×C12712yescomplemented
254C127#1576111yescharacteristic, direct product factor, Sylow
255C23×Q8C22×C12711yescharacteristic
256C23×Q8C22×C12711yescharacteristic
257C23×D4C22×C12711yescharacteristic
258C2×C4⋊Q8—24no—
259C2×C4⋊Q8—28no—
260C2×#124—24no—
261C2×#124—28no—
262C22×C4⋊C4C22×C12712yes—
263C22×C4⋊C4C22×C12712yes—
264C22×C22⋊C4C22×C12714yes—
265C22×C22⋊C4C22×C12712yes—
266C22×C22⋊C4C22×C12714yes—
267C22×C22⋊C4C22×C12712yes—
268C22×C42C22×C12711yescharacteristic
269C22×C42C22×C12711yescharacteristic
270#394—28no—
271#393—28no—
272C2×#95C22×C12714yes—
273C2×#95C22×C12712yes—
274C2×#95C22×C12712yes—
275C2×#95C22×C12714yes—
276C2×#95C22×C12712yes—
277C25C23×C12712yes—
278C22×Q8—24no—
279C22×Q8—24no—
280C22×Q8—24no—
281C22×Q8—22no—
282C22×Q8—24no—
283C22×Q8—42no—
284C22×Q8—42no—
285C22×Q8D4×C12714yes—
286C22×D4—24no—
287C22×D4—22no—
288C22×D4—24no—
289C22×D4—42no—
290C23×C4C23×C12714yes—
291C23×C4C23×C12712yes—
292C23×C4C23×C12711yescharacteristic
293C23×C4C23×C12712yes—
294C23×C4C23×C12711yescharacteristic
295C23×C4C23×C12711yescharacteristic
296C23×C4C23×C12712yes—
297C4⋊Q8—232no—
298#124—232no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4D4×C12718yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4D4×C12718yes—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—42no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24C2×D4×C12714yes—
335C24C24×C12711yescharacteristic
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—44no—
340C2×Q8—416no—
341C2×Q8—416no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×D4—28no—
345C2×D4—416no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4C2×D4×C12714yes—
374C22×C4C2×D4×C12712yes—
375C22×C4C2×D4×C12714yes—
376C22×C4C2×D4×C12714yes—
377C22×C4C2×D4×C12714yes—
378C22×C4C2×D4×C12712yes—
379C22×C4C2×D4×C12712yes—
380C22×C4C2×D4×C12714yes—
381C22×C4#41×C12718yes—
382C4⋊C4—216no—
383C4⋊C4—216no—
384C22⋊C4—216no—
385C22⋊C4—216no—
386C22⋊C4—416no—
387C22⋊C4—416no—
388C42—28no—
389C42—28no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—28no—
394C23—28no—
395C23—42no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23C22×D4×C12711yescharacteristic
412C23C22×D4×C12712yes—
413C23C22×D4×C12712yes—
414C23C2×#41×C12714yes—
415C23H(5,𝔽2)×C12712yes—
416C232−1+4×C12714yes—
417Q8—416no—
418Q8—416no—
419D4—416no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4#120×C12714yes—
452C2×C4#120×C12714yes—
453C2×C4#124×C12718yescomplemented
454C22—44no—
455C22—48no—
456C22—44no—
457C22—44no—
458C22—48no—
459C22—84no—
460C22—84no—
461C22—84no—
462C22—84no—
463C22C2×#120×C12711yescharacteristic
464C22C2×#120×C12711yescharacteristic
465C22C2×#124×C12712yes—
466C22#532×C12712yes—
467C22#534×C12711yescharacteristic
468C22#536×C12712yes—
469C22#536×C12712yes—
470C22#542×C12714yes—
471C22D42×C12714yes—
472C22#547×C12714yes—
473C22#548×C12714yes—
474C22#548×C12714yes—
475C22#555×C12712yes—
476C22#563×C12712yes—
477C4—24no—
478C4—24no—
479C4—28no—
480C4—42no—
481C4—44no—
482C4—44no—
483C4—84no—
484C2—84no—
485C2C127×#240311yescharacteristic
486C2C127×#243212yes—
487C2C127×#243612yes—
488C2C127×#249114yes—
489C2C127×#267414yes—
490C2C127×#273112yes—
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Q8×C127C2semidirect
4. C2×#393×C127C2semidirect
5. C127×#1446C2semidirect
6. C127×#1439C2semidirect
7. C127×#1437C2semidirect
8. C127×#1436C2semidirect
207. C2×C4×C127#124semidirect
238. #15761C127direct (N × H)
246. C22×C4⋊Q8C2×C127semidirect
249. C2×#393C2×C127semidirect
250. #1446C2×C127semidirect
251. #1439C2×C127semidirect
252. #1437C2×C127semidirect
253. #1436C2×C127semidirect
254. C127#15761direct (N × H)
453. C2×C4#124×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q8×C1272
C22×#124×C1272
C2×#394×C1272
C2×#393×C1272
C127×#14462
C127×#14392
C127×#14372
C127×#14362
#15761127