Finite Group Databasebeta

← C127×#14667 #44578340

Subgroups — C127×#14667 #44578340

Order 32512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#390×C127C211yesmaximal, characteristic
2C127×#1441C212yesmaximal, complemented
3C127×#1440C214yesmaximal, complemented
4C127×#1439C214yesmaximal, complemented
5C127×#1438C212yesmaximal
6C127×#1434C212yesmaximal, complemented
7C23×D4×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2212yes—
9C22×C4⋊C4×C127C2211yescharacteristic
10C22×C22⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2214yes—
12C22×C22⋊C4×C127C2212yes—
13C22×C22⋊C4×C127C2214yes—
14C22×C42×C127C2211yescharacteristic
15#390×C127C418yescomplemented
16C2×#95×C127C2214yes—
17C2×#95×C127C2212yes—
18C2×#95×C127C2214yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2214yes—
21C2×#95×C127C2212yes—
22C25×C127C2312yes—
23C22×D4×C127—24no—
24C22×D4×C127—42no—
25C22×D4×C127—41no—
26C22×D4×C127—41no—
27C22×D4×C127C2×C414yes—
28C23×C4×C127C2314yes—
29C23×C4×C127C2314yes—
30C23×C4×C127C2312yes—
31C23×C4×C127C2312yes—
32C23×C4×C127C2311yescharacteristic
33C2×C4⋊C4×C127—41no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127—44no—
36C2×C4⋊C4×C127—42no—
37C2×C4⋊C4×C127C2×C414yes—
38C2×C22⋊C4×C127—44no—
39C2×C22⋊C4×C127—44no—
40C2×C22⋊C4×C127—44no—
41C2×C22⋊C4×C127—42no—
42C2×C22⋊C4×C127—44no—
43C2×C22⋊C4×C127—44no—
44C2×C22⋊C4×C127—44no—
45C2×C22⋊C4×C127—44no—
46C2×C22⋊C4×C127—42no—
47C2×C22⋊C4×C127C2×C4116yes—
48C2×C42×C127—42no—
49C2×C42×C127C2×C414yes—
50#95×C127—28no—
51#95×C127—42no—
52#95×C127—42no—
53#95×C127—44no—
54#95×C127—44no—
55#95×C127—42no—
56C24×C127—24no—
57C24×C127—24no—
58C24×C127—24no—
59C24×C127—24no—
60C24×C127—24no—
61C24×C127—22no—
62C24×C127—22no—
63C24×C127—24no—
64C24×C127C22×C414yes—
65C24×C127C2411yescharacteristic
66C2×D4×C127—28no—
67C2×D4×C127—44no—
68C2×D4×C127—44no—
69C2×D4×C127—48no—
70C2×D4×C127—48no—
71C22×C4×C127—22no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—22no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—22no—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
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—24no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127C22×C414yes—
97C22×C4×C127C22×C414yes—
98C22×C4×C127C22×C412yes—
99C22×C4×C127#4114yes—
100C4⋊C4×C127—44no—
101C4⋊C4×C127—48no—
102C22⋊C4×C127—416no—
103C22⋊C4×C127—416no—
104C22⋊C4×C127—48no—
105C22⋊C4×C127—48no—
106C42×C127—44no—
107C23×C127—24no—
108C23×C127—28no—
109C23×C127—24no—
110C23×C127—24no—
111C23×C127—28no—
112C23×C127—44no—
113C23×C127—44no—
114C23×C127—44no—
115C23×C127—44no—
116C23×C127—44no—
117C23×C127—44no—
118C23×C127—44no—
119C23×C127—42no—
120C23×C127—44no—
121C23×C127—44no—
122C23×C127—42no—
123C23×C127—42no—
124C23×C127—44no—
125C23×C127—44no—
126C23×C127—44no—
127C23×C127—42no—
128C23×C127C23×C411yescharacteristic
129C23×C127C2×#4112yes—
130C23×C127H(5,𝔽2)11yescharacteristic
131C23×C127H(5,𝔽2)12yes—
132C23×C127H(5,𝔽2)11yescharacteristic
133C23×C127H(5,𝔽2)14yes—
134C23×C127H(5,𝔽2)12yes—
135C23×C1272−1+412yes—
136D4×C127—416no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—24no—
140C2×C4×C127—24no—
141C2×C4×C127—24no—
142C2×C4×C127—24no—
143C2×C4×C127—28no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—42no—
147C2×C4×C127—44no—
148C2×C4×C127—42no—
149C2×C4×C127—44no—
150C2×C4×C127—44no—
151C2×C4×C127—44no—
152C2×C4×C127—44no—
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#11714yes—
166C22×C127—44no—
167C22×C127—48no—
168C22×C127—48no—
169C22×C127—44no—
170C22×C127—44no—
171C22×C127—84no—
172C22×C127—84no—
173C22×C127—84no—
174C22×C127—84no—
175C22×C127C2×#11711yescharacteristic
176C22×C127#51811yescharacteristic
177C22×C127#51811yescharacteristic
178C22×C127#51812yes—
179C22×C127#52012yes—
180C22×C127#53812yes—
181C22×C127#53812yes—
182C22×C127#54014yes—
183C22×C127#54214yes—
184C22×C127#56014yes—
185C22×C127#56012yes—
186C22×C127#56012yes—
187C22×C127H(3,𝔽4)11yescharacteristic
188C22×C127#56212yes—
189C22×C127#56214yes—
190C22×C127#56311yescharacteristic
191C4×C127—24no—
192C4×C127—44no—
193C4×C127—44no—
194C4×C127—84no—
195C4×C127—84no—
196#14667C12711yesmaximal, characteristic, direct product factor, Sylow
197C2×C127—84no—
198C2×C127#231211yescharacteristic
199C2×C127#231912yes—
200C2×C127#237712yes—
201C2×C127#237911yescharacteristic
202C2×C127#238011yescharacteristic
203C2×C127#282812yes—
204C2×C127#282914yes—
205C2×C127#283012yes—
206C2×#390C2×C12711yescharacteristic
207#1441C2×C12712yescomplemented
208#1440C2×C12714yescomplemented
209#1439C2×C12714yescomplemented
210#1438C2×C12712yes—
211#1434C2×C12712yescomplemented
212C127#1466711yescharacteristic, direct product factor, Sylow
213C23×D4C22×C12711yescharacteristic
214C22×C4⋊C4C22×C12712yes—
215C22×C4⋊C4C22×C12711yescharacteristic
216C22×C22⋊C4C22×C12712yes—
217C22×C22⋊C4C22×C12714yes—
218C22×C22⋊C4C22×C12712yes—
219C22×C22⋊C4C22×C12714yes—
220C22×C42C22×C12711yescharacteristic
221#390C4×C12718yescomplemented
222C2×#95C22×C12714yes—
223C2×#95C22×C12712yes—
224C2×#95C22×C12714yes—
225C2×#95C22×C12712yes—
226C2×#95C22×C12714yes—
227C2×#95C22×C12712yes—
228C25C23×C12712yes—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4—41no—
232C22×D4—41no—
233C22×D4C2×C4×C12714yes—
234C23×C4C23×C12714yes—
235C23×C4C23×C12714yes—
236C23×C4C23×C12712yes—
237C23×C4C23×C12712yes—
238C23×C4C23×C12711yescharacteristic
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C2×C4×C12714yes—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4C2×C4×C127116yes—
254C2×C42—42no—
255C2×C42C2×C4×C12714yes—
256#95—28no—
257#95—42no—
258#95—42no—
259#95—44no—
260#95—44no—
261#95—42no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—22no—
268C24—22no—
269C24—24no—
270C24C22×C4×C12714yes—
271C24C24×C12711yescharacteristic
272C2×D4—28no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—48no—
276C2×D4—48no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
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—24no—
301C22×C4—24no—
302C22×C4C22×C4×C12714yes—
303C22×C4C22×C4×C12714yes—
304C22×C4C22×C4×C12712yes—
305C22×C4#41×C12714yes—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C22⋊C4—416no—
309C22⋊C4—416no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C42—44no—
313C23—24no—
314C23—28no—
315C23—24no—
316C23—24no—
317C23—28no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—42no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—42no—
334C23C23×C4×C12711yescharacteristic
335C23C2×#41×C12712yes—
336C23H(5,𝔽2)×C12711yescharacteristic
337C23H(5,𝔽2)×C12712yes—
338C23H(5,𝔽2)×C12711yescharacteristic
339C23H(5,𝔽2)×C12714yes—
340C23H(5,𝔽2)×C12712yes—
341C232−1+4×C12712yes—
342D4—416no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—28no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
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#117×C12714yes—
372C22—44no—
373C22—48no—
374C22—48no—
375C22—44no—
376C22—44no—
377C22—84no—
378C22—84no—
379C22—84no—
380C22—84no—
381C22C2×#117×C12711yescharacteristic
382C22#518×C12711yescharacteristic
383C22#518×C12711yescharacteristic
384C22#518×C12712yes—
385C22#520×C12712yes—
386C22#538×C12712yes—
387C22#538×C12712yes—
388C22#540×C12714yes—
389C22#542×C12714yes—
390C22#560×C12714yes—
391C22#560×C12712yes—
392C22#560×C12712yes—
393C22H(3,𝔽4)×C12711yescharacteristic
394C22#562×C12712yes—
395C22#562×C12714yes—
396C22#563×C12711yescharacteristic
397C4—24no—
398C4—44no—
399C4—44no—
400C4—84no—
401C4—84no—
402C2—84no—
403C2C127×#231211yescharacteristic
404C2C127×#231912yes—
405C2C127×#237712yes—
406C2C127×#237911yescharacteristic
407C2C127×#238011yescharacteristic
408C2C127×#282812yes—
409C2C127×#282914yes—
410C2C127×#283012yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1441C2semidirect
3. C127×#1440C2semidirect
4. C127×#1439C2semidirect
6. C127×#1434C2semidirect
15. #390×C127C4semidirect
196. #14667C127direct (N × H)
207. #1441C2×C127semidirect
208. #1440C2×C127semidirect
209. #1439C2×C127semidirect
211. #1434C2×C127semidirect
212. C127#14667direct (N × H)
221. #390C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#390×C1272
C127×#14412
C127×#14402
C127×#14392
C127×#14382
C127×#14342
#14667127