Finite Group Databasebeta

← C127×#14345 #44578018

Subgroups — C127×#14345 #44578018

Order 32512.

2150 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C127C211yesmaximal, characteristic
2C2×#383×C127C211yesmaximal, characteristic, complemented
3C2×#382×C127C211yesmaximal, characteristic, complemented
4C127×#1445C212yesmaximal, complemented
5C127×#1439C214yesmaximal, complemented
6C127×#1438C212yesmaximal
7C127×C22⋊#95C214yesmaximal, complemented
8C23×D4×C127C2211yescharacteristic
9C24×C4×C127C2212yes—
10C2×C4×D4×C127—24no—
11C2×C4×D4×C127—24no—
12C2×C4×D4×C127C418yescomplemented
13C22×C4⋊C4×C127C2211yescharacteristic
14C22×C4⋊C4×C127C2212yes—
15C22×C22⋊C4×C127C2212yes—
16C22×C22⋊C4×C127C2218yes—
17C22×C42×C127C2211yescharacteristic
18#383×C127—24no—
19#382×C127—24no—
20C2×#95×C127C2212yes—
21C2×#95×C127C2212yes—
22C2×#95×C127C2212yes—
23C2×#95×C127C2212yes—
24C2×#95×C127C2212yes—
25C2×#95×C127C2218yes—
26C25×C127C2312yes—
27C22×D4×C127—28no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127C2×C414yes—
31C23×C4×C127—24no—
32C23×C4×C127—24no—
33C23×C4×C127—48no—
34C23×C4×C127C2×C418yes—
35C23×C4×C127C2314yes—
36C23×C4×C127C2314yes—
37C23×C4×C127C2312yes—
38C23×C4×C127C2311yescharacteristic
39C23×C4×C127C2311yescharacteristic
40C23×C4×C127C2311yescharacteristic
41C4×D4×C127—232no—
42C2×C4⋊C4×C127—22no—
43C2×C4⋊C4×C127—22no—
44C2×C4⋊C4×C127—24no—
45C2×C4⋊C4×C127—44no—
46C2×C4⋊C4×C127C2×C414yes—
47C2×C22⋊C4×C127—24no—
48C2×C22⋊C4×C127—24no—
49C2×C22⋊C4×C127—216no—
50C2×C22⋊C4×C127—48no—
51C2×C22⋊C4×C127—48no—
52C2×C22⋊C4×C127C2×C418yes—
53C2×C42×C127C2×C414yes—
54C2×C42×C127D414yes—
55C2×C42×C127D414yes—
56#95×C127—24no—
57#95×C127—24no—
58#95×C127—24no—
59#95×C127—24no—
60#95×C127—24no—
61#95×C127—48no—
62C24×C127—22no—
63C24×C127—22no—
64C24×C127—24no—
65C24×C127—24no—
66C24×C127—28no—
67C24×C127—24no—
68C24×C127C22×C414yes—
69C24×C127#4118yes—
70C24×C127C2411yescharacteristic
71C2×D4×C127—216no—
72C2×D4×C127—216no—
73C2×D4×C127—216no—
74C2×D4×C127—28no—
75C22×C4×C127—24no—
76C22×C4×C127—22no—
77C22×C4×C127—24no—
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—22no—
81C22×C4×C127—28no—
82C22×C4×C127—216no—
83C22×C4×C127—24no—
84C22×C4×C127—24no—
85C22×C4×C127—28no—
86C22×C4×C127—28no—
87C22×C4×C127—24no—
88C22×C4×C127—28no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—28no—
92C22×C4×C127—24no—
93C22×C4×C127—416no—
94C22×C4×C127—416no—
95C22×C4×C127—416no—
96C22×C4×C127C22×C414yes—
97C22×C4×C127C22×C412yes—
98C22×C4×C127C22×C412yes—
99C22×C4×C127C22×C412yes—
100C22×C4×C127C2×D412yes—
101C22×C4×C127C2×D412yes—
102C22×C4×C127C2×D412yes—
103C22×C4×C127C2×D412yes—
104C22×C4×C127C2×D412yes—
105C22×C4×C127C2×D412yes—
106C4⋊C4×C127—28no—
107C4⋊C4×C127—48no—
108C22⋊C4×C127—216no—
109C22⋊C4×C127—432no—
110C42×C127C22⋊C4116yescomplemented
111C23×C127—24no—
112C23×C127—24no—
113C23×C127—28no—
114C23×C127—28no—
115C23×C127—28no—
116C23×C127—28no—
117C23×C127—216no—
118C23×C127—216no—
119C23×C127—44no—
120C23×C127—44no—
121C23×C127—42no—
122C23×C127—42no—
123C23×C127—44no—
124C23×C127—48no—
125C23×C127—48no—
126C23×C127#11718yes—
127C23×C127C23×C411yescharacteristic
128C23×C127C22×D411yescharacteristic
129C23×C127C22×D411yescharacteristic
130C23×C127C2×#4114yes—
131C23×C127H(5,𝔽2)14yes—
132C23×C127H(5,𝔽2)12yes—
133C23×C1272−1+412yes—
134D4×C127—232no—
135C2×C4×C127—24no—
136C2×C4×C127—24no—
137C2×C4×C127—24no—
138C2×C4×C127—24no—
139C2×C4×C127—24no—
140C2×C4×C127—22no—
141C2×C4×C127—24no—
142C2×C4×C127—24no—
143C2×C4×C127—22no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—28no—
147C2×C4×C127—432no—
148C2×C4×C127—44no—
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—48no—
157C2×C4×C127—48no—
158C2×C4×C127—48no—
159C2×C4×C127—48no—
160C2×C4×C127C2×C22⋊C414yes—
161C2×C4×C127C2×C22⋊C414yes—
162C2×C4×C127C2×C22⋊C414yes—
163C22×C127—216no—
164C22×C127—28no—
165C22×C127—44no—
166C22×C127—44no—
167C22×C127—48no—
168C22×C127—48no—
169C22×C127—48no—
170C22×C127—416no—
171C22×C127C22×C22⋊C411yescharacteristic
172C22×C127C2×#11712yes—
173C22×C127#51812yes—
174C22×C127#52012yes—
175C22×C127#52512yes—
176C22×C127#52512yes—
177C22×C127#53512yes—
178C22×C127#53512yes—
179C22×C127#53511yescharacteristic
180C22×C127#53611yescharacteristic
181C22×C127#53712yes—
182C22×C127#53818yes—
183C22×C127#53918yes—
184C4×C127—24no—
185C4×C127—24no—
186C4×C127—24no—
187C4×C127—44no—
188C4×C127—84no—
189C4×C127—84no—
190#14345C12711yesmaximal, characteristic, direct product factor, Sylow
191C2×C127—48no—
192C2×C127#228011yescharacteristic
193C2×C127#230911yescharacteristic
194C2×C127#231111yescharacteristic
195C2×C127#231814yes—
196C2×C127#263112yes—
197C2×C127#263312yes—
198C2×C127#263614yes—
199C22×C4×D4C2×C12711yescharacteristic
200C2×#383C2×C12711yescharacteristic, complemented
201C2×#382C2×C12711yescharacteristic, complemented
202#1445C2×C12712yescomplemented
203#1439C2×C12714yescomplemented
204#1438C2×C12712yes—
205C22⋊#95C2×C12714yescomplemented
206C127#1434511yescharacteristic, direct product factor, Sylow
207C23×D4C22×C12711yescharacteristic
208C24×C4C22×C12712yes—
209C2×C4×D4—24no—
210C2×C4×D4—24no—
211C2×C4×D4C4×C12718yescomplemented
212C22×C4⋊C4C22×C12711yescharacteristic
213C22×C4⋊C4C22×C12712yes—
214C22×C22⋊C4C22×C12712yes—
215C22×C22⋊C4C22×C12718yes—
216C22×C42C22×C12711yescharacteristic
217#383—24no—
218#382—24no—
219C2×#95C22×C12712yes—
220C2×#95C22×C12712yes—
221C2×#95C22×C12712yes—
222C2×#95C22×C12712yes—
223C2×#95C22×C12712yes—
224C2×#95C22×C12718yes—
225C25C23×C12712yes—
226C22×D4—28no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4C2×C4×C12714yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—48no—
233C23×C4C2×C4×C12718yes—
234C23×C4C23×C12714yes—
235C23×C4C23×C12714yes—
236C23×C4C23×C12712yes—
237C23×C4C23×C12711yescharacteristic
238C23×C4C23×C12711yescharacteristic
239C23×C4C23×C12711yescharacteristic
240C4×D4—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×C4×C12714yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—216no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—48no—
251C2×C22⋊C4C2×C4×C12718yes—
252C2×C42C2×C4×C12714yes—
253C2×C42D4×C12714yes—
254C2×C42D4×C12714yes—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—48no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—28no—
266C24—24no—
267C24C22×C4×C12714yes—
268C24#41×C12718yes—
269C24C24×C12711yescharacteristic
270C2×D4—216no—
271C2×D4—216no—
272C2×D4—216no—
273C2×D4—28no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—28no—
281C22×C4—216no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—416no—
293C22×C4—416no—
294C22×C4—416no—
295C22×C4C22×C4×C12714yes—
296C22×C4C22×C4×C12712yes—
297C22×C4C22×C4×C12712yes—
298C22×C4C22×C4×C12712yes—
299C22×C4C2×D4×C12712yes—
300C22×C4C2×D4×C12712yes—
301C22×C4C2×D4×C12712yes—
302C22×C4C2×D4×C12712yes—
303C22×C4C2×D4×C12712yes—
304C22×C4C2×D4×C12712yes—
305C4⋊C4—28no—
306C4⋊C4—48no—
307C22⋊C4—216no—
308C22⋊C4—432no—
309C42C22⋊C4×C127116yescomplemented
310C23—24no—
311C23—24no—
312C23—28no—
313C23—28no—
314C23—28no—
315C23—28no—
316C23—216no—
317C23—216no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23—44no—
323C23—48no—
324C23—48no—
325C23#117×C12718yes—
326C23C23×C4×C12711yescharacteristic
327C23C22×D4×C12711yescharacteristic
328C23C22×D4×C12711yescharacteristic
329C23C2×#41×C12714yes—
330C23H(5,𝔽2)×C12714yes—
331C23H(5,𝔽2)×C12712yes—
332C232−1+4×C12712yes—
333D4—232no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—28no—
346C2×C4—432no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4C2×C22⋊C4×C12714yes—
360C2×C4C2×C22⋊C4×C12714yes—
361C2×C4C2×C22⋊C4×C12714yes—
362C22—216no—
363C22—28no—
364C22—48no—
365C22—48no—
366C22—48no—
367C22—416no—
368C22—44no—
369C22—44no—
370C22C22×C22⋊C4×C12711yescharacteristic
371C22C2×#117×C12712yes—
372C22#518×C12712yes—
373C22#520×C12712yes—
374C22#525×C12712yes—
375C22#525×C12712yes—
376C22#535×C12711yescharacteristic
377C22#535×C12712yes—
378C22#535×C12712yes—
379C22#536×C12711yescharacteristic
380C22#537×C12712yes—
381C22#538×C12718yes—
382C22#539×C12718yes—
383C4—24no—
384C4—24no—
385C4—24no—
386C4—44no—
387C4—84no—
388C4—84no—
389C2—48no—
390C2C127×#228011yescharacteristic
391C2C127×#230911yescharacteristic
392C2C127×#231111yescharacteristic
393C2C127×#231814yes—
394C2C127×#263112yes—
395C2C127×#263312yes—
396C2C127×#263614yes—
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. C2×#383×C127C2semidirect
3. C2×#382×C127C2semidirect
4. C127×#1445C2semidirect
5. C127×#1439C2semidirect
7. C127×C22⋊#95C2semidirect
12. C2×C4×D4×C127C4semidirect
110. C42×C127C22⋊C4semidirect
190. #14345C127direct (N × H)
200. C2×#383C2×C127semidirect
201. C2×#382C2×C127semidirect
202. #1445C2×C127semidirect
203. #1439C2×C127semidirect
205. C22⋊#95C2×C127semidirect
206. C127#14345direct (N × H)
211. C2×C4×D4C4×C127semidirect
309. C42C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D4×C1272
C2×#383×C1272
C2×#382×C1272
C127×#14452
C127×#14392
C127×#14382
C127×C22⋊#952
#14345127