Finite Group Databasebeta

← C127×#17011 #44580684

Subgroups — C127×#17011 #44580684

Order 32512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394×C127C212yesmaximal
2C2×#390×C127C211yesmaximal, characteristic
3C2×#389×C127C211yesmaximal, characteristic, complemented
4C127×#1441C212yesmaximal, complemented
5C127×#1440C214yesmaximal, complemented
6C127×#1439C212yesmaximal, complemented
7C127×#1439C212yesmaximal, complemented
8C127×#1434C211yesmaximal, characteristic, complemented
9C23×D4×C127C2211yescharacteristic
10C22×C4⋊C4×C127C2212yes—
11C22×C4⋊C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2214yes—
16C22×C22⋊C4×C127C2214yes—
17C22×C22⋊C4×C127C2212yes—
18C22×C42×C127C2211yescharacteristic
19#394×C127—28no—
20#390×C127—24no—
21#389×C127—24no—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C2×#95×C127C2214yes—
27C2×#95×C127C2214yes—
28C25×C127C2312yes—
29C22×D4×C127—22no—
30C22×D4×C127—22no—
31C22×D4×C127—42no—
32C22×D4×C127—42no—
33C22×D4×C127D414yes—
34C23×C4×C127C2312yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2314yes—
37C23×C4×C127C2311yescharacteristic
38C23×C4×C127C2312yes—
39C23×C4×C127C2311yescharacteristic
40C23×C4×C127C2311yescharacteristic
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—24no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—22no—
46C2×C4⋊C4×C127—24no—
47C2×C4⋊C4×C127—24no—
48C2×C4⋊C4×C127—42no—
49C2×C4⋊C4×C127—42no—
50C2×C22⋊C4×C127—28no—
51C2×C22⋊C4×C127—28no—
52C2×C22⋊C4×C127—24no—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—28no—
55C2×C22⋊C4×C127—28no—
56C2×C22⋊C4×C127—44no—
57C2×C22⋊C4×C127—44no—
58C2×C22⋊C4×C127—44no—
59C2×C22⋊C4×C127—44no—
60C2×C42×C127—42no—
61C2×C42×C127D414yes—
62#95×C127—24no—
63#95×C127—24no—
64#95×C127—24no—
65#95×C127—44no—
66#95×C127—42no—
67#95×C127—44no—
68C24×C127—22no—
69C24×C127—24no—
70C24×C127—24no—
71C24×C127—22no—
72C24×C127—24no—
73C24×C127—24no—
74C24×C127—24no—
75C24×C127—24no—
76C24×C127C2×D414yes—
77C24×C127C2411yescharacteristic
78C2×D4×C127—28no—
79C2×D4×C127—48no—
80C2×D4×C127—44no—
81C2×D4×C127—44no—
82C2×D4×C127—48no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
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—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—24no—
97C22×C4×C127—22no—
98C22×C4×C127—22no—
99C22×C4×C127—22no—
100C22×C4×C127—24no—
101C22×C4×C127—24no—
102C22×C4×C127—24no—
103C22×C4×C127—22no—
104C22×C4×C127—24no—
105C22×C4×C127—24no—
106C22×C4×C127—24no—
107C22×C4×C127—24no—
108C22×C4×C127—24no—
109C22×C4×C127—24no—
110C22×C4×C127—24no—
111C22×C4×C127—24no—
112C22×C4×C127C2×D412yes—
113C22×C4×C127C2×D412yes—
114C22×C4×C127C2×D412yes—
115C22×C4×C127C2×D412yes—
116C22×C4×C127#4114yes—
117C22×C4×C127#4112yes—
118C22×C4×C127#4114yes—
119C4⋊C4×C127—28no—
120C4⋊C4×C127—44no—
121C4⋊C4×C127—44no—
122C4⋊C4×C127—48no—
123C22⋊C4×C127—416no—
124C22⋊C4×C127—48no—
125C22⋊C4×C127—416no—
126C22⋊C4×C127—48no—
127C42×C127—44no—
128C23×C127—24no—
129C23×C127—28no—
130C23×C127—28no—
131C23×C127—28no—
132C23×C127—42no—
133C23×C127—42no—
134C23×C127—42no—
135C23×C127—42no—
136C23×C127—42no—
137C23×C127—42no—
138C23×C127—44no—
139C23×C127—48no—
140C23×C127—48no—
141C23×C127—48no—
142C23×C127—44no—
143C23×C127—44no—
144C23×C127—48no—
145C23×C127C22×D411yescharacteristic
146C23×C127C22×D411yescharacteristic
147C23×C127C2×#4111yescharacteristic
148C23×C127C2×#4112yes—
149C23×C127C2×#4112yes—
150C23×C127H(5,𝔽2)12yes—
151C23×C127H(5,𝔽2)12yes—
152C23×C127H(5,𝔽2)12yes—
153C23×C1272−1+412yes—
154D4×C127—416no—
155C2×C4×C127—22no—
156C2×C4×C127—22no—
157C2×C4×C127—22no—
158C2×C4×C127—24no—
159C2×C4×C127—22no—
160C2×C4×C127—24no—
161C2×C4×C127—28no—
162C2×C4×C127—24no—
163C2×C4×C127—22no—
164C2×C4×C127—22no—
165C2×C4×C127—22no—
166C2×C4×C127—24no—
167C2×C4×C127—24no—
168C2×C4×C127—42no—
169C2×C4×C127—42no—
170C2×C4×C127—42no—
171C2×C4×C127—42no—
172C2×C4×C127—44no—
173C2×C4×C127—42no—
174C2×C4×C127—42no—
175C2×C4×C127—44no—
176C2×C4×C127—44no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—42no—
181C2×C4×C127—44no—
182C2×C4×C127—48no—
183C2×C4×C127—44no—
184C2×C4×C127—44no—
185C2×C4×C127—44no—
186C2×C4×C127—44no—
187C2×C4×C127—44no—
188C2×C4×C127—44no—
189C2×C4×C127#12114yes—
190C2×C4×C127#12414yes—
191C22×C127—44no—
192C22×C127—416no—
193C22×C127—44no—
194C22×C127—44no—
195C22×C127—82no—
196C22×C127—82no—
197C22×C127—82no—
198C22×C127—82no—
199C22×C127—88no—
200C22×C127C2×#12111yescharacteristic
201C22×C127#52511yescharacteristic
202C22×C127C2×#12411yescharacteristic
203C22×C127#53411yescharacteristic
204C22×C127#53412yes—
205C22×C127#53511yescharacteristic
206C22×C127#53511yescharacteristic
207C22×C127#53611yescharacteristic
208C22×C127#53812yes—
209C22×C127#54014yes—
210C22×C127#54214yes—
211C22×C127#54612yes—
212C22×C127#54712yes—
213C22×C127#55314yes—
214C22×C127#55312yes—
215C22×C127#55612yes—
216C22×C127#56214yes—
217C4×C127—24no—
218C4×C127—24no—
219C4×C127—42no—
220C4×C127—44no—
221C4×C127—82no—
222C4×C127—82no—
223C4×C127—84no—
224#17011C12711yesmaximal, characteristic, direct product factor, Sylow
225C2×C127—84no—
226C2×C127#253511yescharacteristic
227C2×C127#254711yescharacteristic
228C2×C127#259111yescharacteristic
229C2×C127#261711yescharacteristic
230C2×C127#263211yescharacteristic
231C2×C127#264111yescharacteristic
232C2×C127#270114yes—
233C2×C127#280014yes—
234C2×C127#281611yescharacteristic
235C2×#394C2×C12712yes—
236C2×#390C2×C12711yescharacteristic
237C2×#389C2×C12711yescharacteristic, complemented
238#1441C2×C12712yescomplemented
239#1440C2×C12714yescomplemented
240#1439C2×C12712yescomplemented
241#1439C2×C12712yescomplemented
242#1434C2×C12711yescharacteristic, complemented
243C127#1701111yescharacteristic, direct product factor, Sylow
244C23×D4C22×C12711yescharacteristic
245C22×C4⋊C4C22×C12712yes—
246C22×C4⋊C4C22×C12711yescharacteristic
247C22×C4⋊C4C22×C12711yescharacteristic
248C22×C4⋊C4C22×C12711yescharacteristic
249C22×C22⋊C4C22×C12712yes—
250C22×C22⋊C4C22×C12714yes—
251C22×C22⋊C4C22×C12714yes—
252C22×C22⋊C4C22×C12712yes—
253C22×C42C22×C12711yescharacteristic
254#394—28no—
255#390—24no—
256#389—24no—
257C2×#95C22×C12712yes—
258C2×#95C22×C12712yes—
259C2×#95C22×C12712yes—
260C2×#95C22×C12712yes—
261C2×#95C22×C12714yes—
262C2×#95C22×C12714yes—
263C25C23×C12712yes—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—42no—
267C22×D4—42no—
268C22×D4D4×C12714yes—
269C23×C4C23×C12712yes—
270C23×C4C23×C12712yes—
271C23×C4C23×C12714yes—
272C23×C4C23×C12711yescharacteristic
273C23×C4C23×C12712yes—
274C23×C4C23×C12711yescharacteristic
275C23×C4C23×C12711yescharacteristic
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C42—42no—
296C2×C42D4×C12714yes—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—44no—
301#95—42no—
302#95—44no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24C2×D4×C12714yes—
312C24C24×C12711yescharacteristic
313C2×D4—28no—
314C2×D4—48no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—48no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4C2×D4×C12712yes—
348C22×C4C2×D4×C12712yes—
349C22×C4C2×D4×C12712yes—
350C22×C4C2×D4×C12712yes—
351C22×C4#41×C12714yes—
352C22×C4#41×C12712yes—
353C22×C4#41×C12714yes—
354C4⋊C4—28no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—48no—
358C22⋊C4—416no—
359C22⋊C4—48no—
360C22⋊C4—416no—
361C22⋊C4—48no—
362C42—44no—
363C23—24no—
364C23—28no—
365C23—28no—
366C23—28no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—48no—
375C23—48no—
376C23—48no—
377C23—44no—
378C23—44no—
379C23—48no—
380C23C22×D4×C12711yescharacteristic
381C23C22×D4×C12711yescharacteristic
382C23C2×#41×C12711yescharacteristic
383C23C2×#41×C12712yes—
384C23C2×#41×C12712yes—
385C23H(5,𝔽2)×C12712yes—
386C23H(5,𝔽2)×C12712yes—
387C23H(5,𝔽2)×C12712yes—
388C232−1+4×C12712yes—
389D4—416no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—28no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4#121×C12714yes—
425C2×C4#124×C12714yes—
426C22—44no—
427C22—416no—
428C22—44no—
429C22—44no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—88no—
434C22—82no—
435C22C2×#121×C12711yescharacteristic
436C22#525×C12711yescharacteristic
437C22C2×#124×C12711yescharacteristic
438C22#534×C12711yescharacteristic
439C22#534×C12712yes—
440C22#535×C12711yescharacteristic
441C22#535×C12711yescharacteristic
442C22#536×C12711yescharacteristic
443C22#538×C12712yes—
444C22#540×C12714yes—
445C22#542×C12714yes—
446C22#546×C12712yes—
447C22#547×C12712yes—
448C22#553×C12714yes—
449C22#553×C12712yes—
450C22#556×C12712yes—
451C22#562×C12714yes—
452C4—24no—
453C4—24no—
454C4—44no—
455C4—42no—
456C4—82no—
457C4—82no—
458C4—84no—
459C2—84no—
460C2C127×#253511yescharacteristic
461C2C127×#254711yescharacteristic
462C2C127×#259111yescharacteristic
463C2C127×#261711yescharacteristic
464C2C127×#263211yescharacteristic
465C2C127×#264111yescharacteristic
466C2C127×#270114yes—
467C2C127×#280014yes—
468C2C127×#281611yescharacteristic
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
3. C2×#389×C127C2semidirect
4. C127×#1441C2semidirect
5. C127×#1440C2semidirect
6. C127×#1439C2semidirect
7. C127×#1439C2semidirect
8. C127×#1434C2semidirect
224. #17011C127direct (N × H)
237. C2×#389C2×C127semidirect
238. #1441C2×C127semidirect
239. #1440C2×C127semidirect
240. #1439C2×C127semidirect
241. #1439C2×C127semidirect
242. #1434C2×C127semidirect
243. C127#17011direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#394×C1272
C2×#390×C1272
C2×#389×C1272
C127×#14412
C127×#14402
C127×#14392
C127×#14342
#17011127