Finite Group Databasebeta

← C127×#15015 #44578688

Subgroups — C127×#15015 #44578688

Order 32512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#380×C127C211yesmaximal, characteristic
2C22×#95×C127C211yesmaximal, characteristic
3C127×#1445C212yesmaximal, complemented
4C127×#1443C214yesmaximal, complemented
5C127×#1442C212yesmaximal, complemented
6C127×C22⋊#95C211yesmaximal, characteristic
7C127×C22⋊#95C212yesmaximal
8C127×C22⋊#95C212yesmaximal
9C24×C4×C127C2211yescharacteristic
10C24×C4×C127C2211yescharacteristic
11C24×C4×C127C2211yescharacteristic
12C22×C4⋊C4×C127C2211yescharacteristic
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C22⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16#380×C127—24no—
17C2×#95×C127—24no—
18C2×#95×C127C2214yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2214yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2214yes—
26C2×#95×C127C2214yes—
27C25×C127C2311yescharacteristic
28C23×C4×C127—22no—
29C23×C4×C127—22no—
30C23×C4×C127—24no—
31C23×C4×C127—22no—
32C23×C4×C127—24no—
33C23×C4×C127—24no—
34C23×C4×C127—24no—
35C23×C4×C127—42no—
36C23×C4×C127—42no—
37C23×C4×C127—42no—
38C23×C4×C127—42no—
39C23×C4×C127D414yes—
40C23×C4×C127Q814yes—
41C23×C4×C127C2312yes—
42C23×C4×C127C2312yes—
43C23×C4×C127C2314yes—
44C23×C4×C127C2311yescharacteristic
45C23×C4×C127C2312yes—
46C23×C4×C127C2312yes—
47C23×C4×C127C2311yescharacteristic
48C2×C4⋊C4×C127—22no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—24no—
51C2×C4⋊C4×C127—42no—
52C2×C22⋊C4×C127—24no—
53C2×C22⋊C4×C127—28no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127—24no—
57#95×C127—24no—
58#95×C127—24no—
59#95×C127—24no—
60#95×C127—28no—
61#95×C127—24no—
62#95×C127—42no—
63#95×C127—44no—
64#95×C127—44no—
65#95×C127—44no—
66C24×C127—22no—
67C24×C127—22no—
68C24×C127—22no—
69C24×C127—22no—
70C24×C127C2×D412yes—
71C24×C127C2×Q812yes—
72C24×C127#4112yes—
73C24×C127#4114yes—
74C24×C127#4114yes—
75C24×C127C2411yescharacteristic
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—28no—
79C22×C4×C127—28no—
80C22×C4×C127—28no—
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—24no—
88C22×C4×C127—22no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—28no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—24no—
96C22×C4×C127—24no—
97C22×C4×C127—24no—
98C22×C4×C127—24no—
99C22×C4×C127—24no—
100C22×C4×C127—24no—
101C22×C4×C127—24no—
102C22×C4×C127—24no—
103C22×C4×C127—24no—
104C22×C4×C127—24no—
105C22×C4×C127—24no—
106C22×C4×C127—24no—
107C22×C4×C127—24no—
108C22×C4×C127—44no—
109C22×C4×C127—48no—
110C22×C4×C127—416no—
111C22×C4×C127—416no—
112C22×C4×C127—44no—
113C22×C4×C127—44no—
114C22×C4×C127—44no—
115C22×C4×C127—48no—
116C22×C4×C127—44no—
117C22×C4×C127C2×D412yes—
118C22×C4×C127C2×D412yes—
119C22×C4×C127C2×Q814yes—
120C22×C4×C127#4114yes—
121C22×C4×C127#4114yes—
122C4⋊C4×C127—44no—
123C4⋊C4×C127—44no—
124C22⋊C4×C127—48no—
125C22⋊C4×C127—48no—
126C23×C127—22no—
127C23×C127—22no—
128C23×C127—24no—
129C23×C127—24no—
130C23×C127—22no—
131C23×C127—22no—
132C23×C127—24no—
133C23×C127—24no—
134C23×C127—28no—
135C23×C127—28no—
136C23×C127—28no—
137C23×C127—28no—
138C23×C127C22⋊Q814yes—
139C23×C127#12314yes—
140C23×C127#12414yes—
141C23×C127#12518yes—
142C23×C127#12618yes—
143C23×C127C22×D411yescharacteristic
144C23×C127C22×Q811yescharacteristic
145C23×C127C2×#4111yescharacteristic
146C23×C127C2×#4112yes—
147C23×C127C2×#4112yes—
148C23×C127C2×#4112yes—
149C23×C127C2×#4112yes—
150C23×C127H(5,𝔽2)12yes—
151C23×C127H(5,𝔽2)12yes—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—28no—
155C2×C4×C127—24no—
156C2×C4×C127—22no—
157C2×C4×C127—22no—
158C2×C4×C127—24no—
159C2×C4×C127—22no—
160C2×C4×C127—28no—
161C2×C4×C127—28no—
162C2×C4×C127—416no—
163C2×C4×C127—416no—
164C2×C4×C127—416no—
165C2×C4×C127—42no—
166C2×C4×C127—42no—
167C2×C4×C127—42no—
168C2×C4×C127—42no—
169C2×C4×C127—44no—
170C2×C4×C127—44no—
171C2×C4×C127—44no—
172C2×C4×C127—42no—
173C2×C4×C127—42no—
174C2×C4×C127—42no—
175C2×C4×C127—42no—
176C2×C4×C127—44no—
177C2×C4×C127—44no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—44no—
181C2×C4×C127—44no—
182C2×C4×C127—44no—
183C2×C4×C127—48no—
184C2×C4×C127—48no—
185C2×C4×C127—44no—
186C2×C4×C127—44no—
187C2×C4×C127#12314yes—
188C22×C127—216no—
189C22×C127—24no—
190C22×C127—24no—
191C22×C127—24no—
192C22×C127—24no—
193C22×C127—24no—
194C22×C127—24no—
195C22×C127—216no—
196C22×C127#40018yes—
197C22×C127C2×C22⋊Q811yescharacteristic
198C22×C127C2×#12311yescharacteristic
199C22×C127C2×#12311yescharacteristic
200C22×C127#52511yescharacteristic
201C22×C127C2×#12411yescharacteristic
202C22×C127C2×#12512yes—
203C22×C127C2×#12612yes—
204C22×C127#52912yes—
205C22×C127#53312yes—
206C22×C127#53411yescharacteristic
207C22×C127#53511yescharacteristic
208C22×C127#53812yes—
209C22×C127#54312yes—
210C22×C127#55114yes—
211C22×C127#55114yes—
212C22×C127#55314yes—
213C22×C127#55314yes—
214C4×C127—24no—
215C4×C127—44no—
216C4×C127—44no—
217C4×C127—42no—
218C4×C127—82no—
219C4×C127—82no—
220C4×C127—84no—
221#15015C12711yesmaximal, characteristic, direct product factor, Sylow
222C2×C127—28no—
223C2×C127C2×#40011yescharacteristic
224C2×C127#248211yescharacteristic
225C2×C127#248311yescharacteristic
226C2×C127#250411yescharacteristic
227C2×C127#251014yes—
228C2×C127#256914yes—
229C2×C127#259111yescharacteristic
230C2×C127#262611yescharacteristic
231C2×C127#285511yescharacteristic
232C2×#380C2×C12711yescharacteristic
233C22×#95C2×C12711yescharacteristic
234#1445C2×C12712yescomplemented
235#1443C2×C12714yescomplemented
236#1442C2×C12712yescomplemented
237C22⋊#95C2×C12711yescharacteristic
238C22⋊#95C2×C12712yes—
239C22⋊#95C2×C12712yes—
240C127#1501511yescharacteristic, direct product factor, Sylow
241C24×C4C22×C12711yescharacteristic
242C24×C4C22×C12711yescharacteristic
243C24×C4C22×C12711yescharacteristic
244C22×C4⋊C4C22×C12711yescharacteristic
245C22×C4⋊C4C22×C12711yescharacteristic
246C22×C22⋊C4C22×C12712yes—
247C22×C22⋊C4C22×C12712yes—
248#380—24no—
249C2×#95—24no—
250C2×#95C22×C12714yes—
251C2×#95C22×C12712yes—
252C2×#95C22×C12712yes—
253C2×#95C22×C12712yes—
254C2×#95C22×C12712yes—
255C2×#95C22×C12714yes—
256C2×#95C22×C12712yes—
257C2×#95C22×C12714yes—
258C2×#95C22×C12714yes—
259C25C23×C12711yescharacteristic
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4—24no—
265C23×C4—24no—
266C23×C4—24no—
267C23×C4—42no—
268C23×C4—42no—
269C23×C4—42no—
270C23×C4—42no—
271C23×C4D4×C12714yes—
272C23×C4Q8×C12714yes—
273C23×C4C23×C12712yes—
274C23×C4C23×C12712yes—
275C23×C4C23×C12714yes—
276C23×C4C23×C12711yescharacteristic
277C23×C4C23×C12712yes—
278C23×C4C23×C12712yes—
279C23×C4C23×C12711yescharacteristic
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24C2×D4×C12712yes—
303C24C2×Q8×C12712yes—
304C24#41×C12712yes—
305C24#41×C12714yes—
306C24#41×C12714yes—
307C24C24×C12711yescharacteristic
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—28no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—28no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—44no—
341C22×C4—48no—
342C22×C4—416no—
343C22×C4—416no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4C2×D4×C12712yes—
350C22×C4C2×D4×C12712yes—
351C22×C4C2×Q8×C12714yes—
352C22×C4#41×C12714yes—
353C22×C4#41×C12714yes—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—28no—
367C23—28no—
368C23—28no—
369C23—28no—
370C23C22⋊Q8×C12714yes—
371C23#123×C12714yes—
372C23#124×C12714yes—
373C23#125×C12718yes—
374C23#126×C12718yes—
375C23C22×D4×C12711yescharacteristic
376C23C22×Q8×C12711yescharacteristic
377C23C2×#41×C12711yescharacteristic
378C23C2×#41×C12712yes—
379C23C2×#41×C12712yes—
380C23C2×#41×C12712yes—
381C23C2×#41×C12712yes—
382C23H(5,𝔽2)×C12712yes—
383C23H(5,𝔽2)×C12712yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—28no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—416no—
395C2×C4—416no—
396C2×C4—416no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4#123×C12714yes—
420C22—216no—
421C22—24no—
422C22—24no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—216no—
428C22#400×C12718yes—
429C22C2×C22⋊Q8×C12711yescharacteristic
430C22C2×#123×C12711yescharacteristic
431C22C2×#123×C12711yescharacteristic
432C22#525×C12711yescharacteristic
433C22C2×#124×C12711yescharacteristic
434C22C2×#125×C12712yes—
435C22C2×#126×C12712yes—
436C22#529×C12712yes—
437C22#533×C12712yes—
438C22#534×C12711yescharacteristic
439C22#535×C12711yescharacteristic
440C22#538×C12712yes—
441C22#543×C12712yes—
442C22#551×C12714yes—
443C22#551×C12714yes—
444C22#553×C12714yes—
445C22#553×C12714yes—
446C4—24no—
447C4—44no—
448C4—44no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4—84no—
453C2—28no—
454C2C2×#400×C12711yescharacteristic
455C2C127×#248211yescharacteristic
456C2C127×#248311yescharacteristic
457C2C127×#250411yescharacteristic
458C2C127×#251014yes—
459C2C127×#256914yes—
460C2C127×#259111yescharacteristic
461C2C127×#262611yescharacteristic
462C2C127×#285511yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1445C2semidirect
4. C127×#1443C2semidirect
5. C127×#1442C2semidirect
221. #15015C127direct (N × H)
234. #1445C2×C127semidirect
235. #1443C2×C127semidirect
236. #1442C2×C127semidirect
240. C127#15015direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#380×C1272
C22×#95×C1272
C127×#14452
C127×#14432
C127×#14422
C127×C22⋊#952
#15015127