Finite Group Databasebeta

← C127×#14295 #44577968

Subgroups — C127×#14295 #44577968

Order 32512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#381×C127C211yesmaximal, characteristic
2C2×#380×C127C211yesmaximal, characteristic
3C2×C4×C22⋊C4×C127C211yesmaximal, characteristic
4C127×#1444C212yesmaximal, complemented
5C127×#1442C212yesmaximal, complemented
6C127×#1439C214yesmaximal
7C127×#1436C214yesmaximal, complemented
8C24×C4×C127C2211yescharacteristic
9C22×C4⋊C4×C127C2214yes—
10C22×C4⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2212yes—
12C22×C22⋊C4×C127C2212yes—
13C22×C22⋊C4×C127C2212yes—
14C22×C42×C127C2212yes—
15#381×C127C418yescomplemented
16#380×C127—24no—
17C4×C22⋊C4×C127—24no—
18C2×#95×C127C2212yes—
19C2×#95×C127C2214yes—
20C2×#95×C127C2211yescharacteristic
21C2×#95×C127C2211yescharacteristic
22C2×#95×C127C2214yes—
23C2×#95×C127C2214yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2212yes—
26C25×C127C2311yescharacteristic
27C23×C4×C127—44no—
28C23×C4×C127C2×C414yes—
29C23×C4×C127D414yes—
30C23×C4×C127Q814yes—
31C23×C4×C127C2314yes—
32C23×C4×C127C2314yes—
33C23×C4×C127C2312yes—
34C23×C4×C127C2312yes—
35C23×C4×C127C2311yescharacteristic
36C23×C4×C127C2311yescharacteristic
37C2×C4⋊C4×C127—24no—
38C2×C4⋊C4×C127—28no—
39C2×C4⋊C4×C127—44no—
40C2×C4⋊C4×C127—44no—
41C2×C4⋊C4×C127—44no—
42C2×C22⋊C4×C127—24no—
43C2×C22⋊C4×C127—24no—
44C2×C22⋊C4×C127—24no—
45C2×C22⋊C4×C127—24no—
46C2×C22⋊C4×C127—44no—
47C2×C22⋊C4×C127—42no—
48C2×C22⋊C4×C127—42no—
49C2×C22⋊C4×C127C2×C418yes—
50C2×C42×C127—24no—
51C2×C42×C127—44no—
52#95×C127—22no—
53#95×C127—22no—
54#95×C127—24no—
55#95×C127—28no—
56#95×C127—44no—
57#95×C127—44no—
58#95×C127C2×C418yes—
59#95×C127C2×C418yes—
60C24×C127—22no—
61C24×C127—22no—
62C24×C127—22no—
63C24×C127—24no—
64C24×C127—22no—
65C24×C127C22×C412yes—
66C24×C127C2×D412yes—
67C24×C127C2×Q812yes—
68C24×C127C2411yescharacteristic
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—24no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—24no—
83C22×C4×C127—24no—
84C22×C4×C127—24no—
85C22×C4×C127—24no—
86C22×C4×C127—24no—
87C22×C4×C127—24no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—24no—
92C22×C4×C127—48no—
93C22×C4×C127—48no—
94C22×C4×C127—48no—
95C22×C4×C127C4⋊C4116yescomplemented
96C22×C4×C127C22×C418yes—
97C22×C4×C127C22×C412yes—
98C22×C4×C127C22×C412yes—
99C22×C4×C127C2×D412yes—
100C22×C4×C127C2×D412yes—
101C22×C4×C127C2×Q812yes—
102C22×C4×C127C2×Q812yes—
103C22×C4×C127#4114yes—
104C22×C4×C127#4114yes—
105C4⋊C4×C127—48no—
106C4⋊C4×C127—416no—
107C22⋊C4×C127—216no—
108C22⋊C4×C127—48no—
109C22⋊C4×C127—48no—
110C42×C127—48no—
111C23×C127—24no—
112C23×C127—24no—
113C23×C127—24no—
114C23×C127—24no—
115C23×C127—22no—
116C23×C127—22no—
117C23×C127—24no—
118C23×C127—22no—
119C23×C127—22no—
120C23×C127—24no—
121C23×C127—24no—
122C23×C127—44no—
123C23×C127—44no—
124C23×C127—44no—
125C23×C127—44no—
126C23×C127C2×C4⋊C414yes—
127C23×C127C23×C411yescharacteristic
128C23×C127C22×D411yescharacteristic
129C23×C127C22×Q811yescharacteristic
130C23×C127C2×#4112yes—
131C23×C127C2×#4112yes—
132C23×C127H(5,𝔽2)14yes—
133C23×C127H(5,𝔽2)12yes—
134C23×C1272−1+412yes—
135C2×C4×C127—24no—
136C2×C4×C127—24no—
137C2×C4×C127—24no—
138C2×C4×C127—24no—
139C2×C4×C127—24no—
140C2×C4×C127—24no—
141C2×C4×C127—28no—
142C2×C4×C127—28no—
143C2×C4×C127—28no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—28no—
147C2×C4×C127—28no—
148C2×C4×C127—416no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—44no—
154C2×C4×C127—44no—
155C2×C4×C127—44no—
156C2×C4×C127—44no—
157C2×C4×C127—44no—
158C2×C4×C127—44no—
159C2×C4×C127—44no—
160C2×C4×C127—44no—
161C2×C4×C127—44no—
162C2×C4×C127—44no—
163C2×C4×C127—44no—
164C2×C4×C127—44no—
165C2×C4×C127—44no—
166C2×C4×C127C2×C4⋊C414yes—
167C2×C4×C127C2×C4⋊C414yes—
168C22×C127—24no—
169C22×C127—24no—
170C22×C127—24no—
171C22×C127—48no—
172C22×C127—44no—
173C22×C127—44no—
174C22×C127—44no—
175C22×C127—44no—
176C22×C127C22×C4⋊C411yescharacteristic
177C22×C127C4×#4112yes—
178C22×C127#51812yes—
179C22×C127#52012yes—
180C22×C127#52512yes—
181C22×C127#53311yescharacteristic
182C22×C127#53311yescharacteristic
183C22×C127#53412yes—
184C22×C127#53511yescharacteristic
185C22×C127#53611yescharacteristic
186C22×C127#53814yes—
187C22×C127#53914yes—
188C22×C127#54014yes—
189C22×C127#54214yes—
190C22×C127#54312yes—
191C22×C127#54412yes—
192C4×C127—24no—
193C4×C127—24no—
194C4×C127—48no—
195C4×C127—44no—
196C4×C127—44no—
197C4×C127—84no—
198#14295C12711yesmaximal, characteristic, direct product factor, Sylow
199C2×C127—44no—
200C2×C127#229611yescharacteristic
201C2×C127#231511yescharacteristic
202C2×C127#231711yescharacteristic
203C2×C127#233514yes—
204C2×C127#263512yes—
205C2×C127#263812yes—
206C2×C127#266512yes—
207C2×C127#266712yes—
208C2×#381C2×C12711yescharacteristic
209C2×#380C2×C12711yescharacteristic
210C2×C4×C22⋊C4C2×C12711yescharacteristic
211#1444C2×C12712yescomplemented
212#1442C2×C12712yescomplemented
213#1439C2×C12714yes—
214#1436C2×C12714yescomplemented
215C127#1429511yescharacteristic, direct product factor, Sylow
216C24×C4C22×C12711yescharacteristic
217C22×C4⋊C4C22×C12714yes—
218C22×C4⋊C4C22×C12712yes—
219C22×C22⋊C4C22×C12712yes—
220C22×C22⋊C4C22×C12712yes—
221C22×C22⋊C4C22×C12712yes—
222C22×C42C22×C12712yes—
223#381C4×C12718yescomplemented
224#380—24no—
225C4×C22⋊C4—24no—
226C2×#95C22×C12712yes—
227C2×#95C22×C12714yes—
228C2×#95C22×C12711yescharacteristic
229C2×#95C22×C12711yescharacteristic
230C2×#95C22×C12714yes—
231C2×#95C22×C12714yes—
232C2×#95C22×C12712yes—
233C2×#95C22×C12712yes—
234C25C23×C12711yescharacteristic
235C23×C4—44no—
236C23×C4C2×C4×C12714yes—
237C23×C4D4×C12714yes—
238C23×C4Q8×C12714yes—
239C23×C4C23×C12714yes—
240C23×C4C23×C12714yes—
241C23×C4C23×C12712yes—
242C23×C4C23×C12712yes—
243C23×C4C23×C12711yescharacteristic
244C23×C4C23×C12711yescharacteristic
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4C2×C4×C12718yes—
258C2×C42—24no—
259C2×C42—44no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—28no—
264#95—44no—
265#95—44no—
266#95C2×C4×C12718yes—
267#95C2×C4×C12718yes—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—24no—
272C24—22no—
273C24C22×C4×C12712yes—
274C24C2×D4×C12712yes—
275C24C2×Q8×C12712yes—
276C24C24×C12711yescharacteristic
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4C4⋊C4×C127116yescomplemented
304C22×C4C22×C4×C12718yes—
305C22×C4C22×C4×C12712yes—
306C22×C4C22×C4×C12712yes—
307C22×C4C2×D4×C12712yes—
308C22×C4C2×D4×C12712yes—
309C22×C4C2×Q8×C12712yes—
310C22×C4C2×Q8×C12712yes—
311C22×C4#41×C12714yes—
312C22×C4#41×C12714yes—
313C4⋊C4—48no—
314C4⋊C4—416no—
315C22⋊C4—216no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C42—48no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—22no—
324C23—22no—
325C23—24no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—24no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23C2×C4⋊C4×C12714yes—
335C23C23×C4×C12711yescharacteristic
336C23C22×D4×C12711yescharacteristic
337C23C22×Q8×C12711yescharacteristic
338C23C2×#41×C12712yes—
339C23C2×#41×C12712yes—
340C23H(5,𝔽2)×C12714yes—
341C23H(5,𝔽2)×C12712yes—
342C232−1+4×C12712yes—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—416no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4C2×C4⋊C4×C12714yes—
375C2×C4C2×C4⋊C4×C12714yes—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—48no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—44no—
384C22C22×C4⋊C4×C12711yescharacteristic
385C22C4×#41×C12712yes—
386C22#518×C12712yes—
387C22#520×C12712yes—
388C22#525×C12712yes—
389C22#533×C12711yescharacteristic
390C22#533×C12711yescharacteristic
391C22#534×C12712yes—
392C22#535×C12711yescharacteristic
393C22#536×C12711yescharacteristic
394C22#538×C12714yes—
395C22#539×C12714yes—
396C22#540×C12714yes—
397C22#542×C12714yes—
398C22#543×C12712yes—
399C22#544×C12712yes—
400C4—24no—
401C4—24no—
402C4—48no—
403C4—44no—
404C4—44no—
405C4—84no—
406C2—44no—
407C2C127×#229611yescharacteristic
408C2C127×#231511yescharacteristic
409C2C127×#231711yescharacteristic
410C2C127×#233514yes—
411C2C127×#263512yes—
412C2C127×#263812yes—
413C2C127×#266512yes—
414C2C127×#266712yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1444C2semidirect
5. C127×#1442C2semidirect
7. C127×#1436C2semidirect
15. #381×C127C4semidirect
95. C22×C4×C127C4⋊C4semidirect
198. #14295C127direct (N × H)
211. #1444C2×C127semidirect
212. #1442C2×C127semidirect
214. #1436C2×C127semidirect
215. C127#14295direct (N × H)
223. #381C4×C127semidirect
303. C22×C4C4⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#381×C1272
C2×#380×C1272
C2×C4×C22⋊C4×C1272
C127×#14442
C127×#14422
C127×#14392
C127×#14362
#14295127