Finite Group Databasebeta

← C127×#8372 #44572045

Subgroups — C127×#8372 #44572045

Order 32512.

788 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554×C127C211yesmaximal, characteristic
2C4×#101×C127C211yesmaximal, characteristic, complemented
3C127×#1528C212yesmaximal, complemented
4C127×#1490C212yesmaximal
5C127×#1467C211yesmaximal, characteristic, complemented
6#554×C127—24no—
7#554×C127—24no—
8C2×C4⋊Q8×C127C411yescharacteristic
9C2×#125×C127C411yescharacteristic
10C2×#125×C127C2211yescharacteristic
11C2×C22⋊Q8×C127—21no—
12C2×C22⋊Q8×C127—21no—
13C2×C22⋊Q8×C127—21no—
14C2×C4×Q8×C127C2211yescharacteristic
15C2×C4×D4×C127C411yescharacteristic
16C2×C4×D4×C127C411yescharacteristic
17C2×C4×D4×C127C2211yescharacteristic
18C22×C4⋊C4×C127—21no—
19#432×C127C2212yes—
20C2×#101×C127—21no—
21C4×M16×C127C2212yes—
22#343×C127—21no—
23#330×C127—24no—
24D4⋊C8×C127—24no—
25#324×C127—21no—
26#324×C127—21no—
27C22×Q8×C127—21no—
28C22×Q8×C127D411yescharacteristic
29C22×D4×C127C2×C411yescharacteristic
30C23×C4×C127—21no—
31C23×C4×C127—21no—
32C23×C4×C127—21no—
33C2×M16×C127—22no—
34C4⋊Q8×C127—22no—
35C4⋊Q8×C127—22no—
36#125×C127—22no—
37#125×C127—22no—
38#125×C127—22no—
39#125×C127D414yes—
40C22⋊Q8×C127—28no—
41C22⋊Q8×C127—28no—
42C22⋊Q8×C127—28no—
43C4×Q8×C127—22no—
44C4×Q8×C127—22no—
45C4×D4×C127—22no—
46C4×D4×C127—22no—
47C4×D4×C127—22no—
48C4×D4×C127—22no—
49C4×D4×C127—22no—
50C4×D4×C127D414yes—
51C2×C4⋊C4×C127—28no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127—21no—
54C2×C4⋊C4×C127—21no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127—21no—
57C2×C4⋊C4×C127C2×C411yescharacteristic
58C2×C4⋊C4×C127C2×C411yescharacteristic
59C2×C4⋊C4×C127C2×C411yescharacteristic
60C2×C4⋊C4×C127D411yescharacteristic
61C2×C4⋊C4×C127D411yescharacteristic
62C2×C4⋊C4×C127D411yescharacteristic
63C2×C22⋊C4×C127—21no—
64C2×C22⋊C4×C127—21no—
65C2×C22⋊C4×C127—21no—
66C2×C42×C127C2×C411yescharacteristic
67C2×C42×C127C2×C411yescharacteristic
68C2×C42×C127C2311yescharacteristic
69C4⋊C8×C127—22no—
70#101×C127—24no—
71C22⋊C8×C127—22no—
72#97×C127—22no—
73#97×C127—22no—
74C4×C8×C127—22no—
75C24×C127—21no—
76C2×Q8×C127—24no—
77C2×Q8×C127—22no—
78C2×Q8×C127—22no—
79C2×Q8×C127—41no—
80C2×Q8×C127—82no—
81C2×D4×C127—24no—
82C2×D4×C127—21no—
83C2×D4×C127C22⋊C412yes—
84C22×C4×C127—24no—
85C22×C4×C127—21no—
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—42no—
91C22×C4×C127—42no—
92C22×C4×C127—82no—
93C22×C4×C127—82no—
94C22×C4×C127C22⋊C411yescharacteristic
95C22×C4×C127C22⋊C411yescharacteristic
96C22×C4×C127C22⋊C411yescharacteristic
97C22×C4×C127C22⋊C411yescharacteristic
98C22×C4×C127C22×C411yescharacteristic
99C22×C4×C127C2×D411yescharacteristic
100C22×C4×C127C2×D411yescharacteristic
101M16×C127—24no—
102M16×C127—42no—
103C2×C8×C127—42no—
104C2×C8×C127—42no—
105C4⋊C4×C127—24no—
106C4⋊C4×C127—24no—
107C4⋊C4×C127—24no—
108C4⋊C4×C127—21no—
109C4⋊C4×C127—21no—
110C4⋊C4×C127—21no—
111C4⋊C4×C127—22no—
112C4⋊C4×C127—24no—
113C4⋊C4×C127—24no—
114C4⋊C4×C127—22no—
115C4⋊C4×C127—22no—
116C4⋊C4×C127C22⋊C412yes—
117C4⋊C4×C127C22⋊C412yes—
118C4⋊C4×C127C22⋊C412yes—
119C22⋊C4×C127—24no—
120C22⋊C4×C127—24no—
121C22⋊C4×C127—24no—
122C42×C127—21no—
123C42×C127—21no—
124C42×C127—21no—
125C42×C127—21no—
126C42×C127C2×D412yes—
127C42×C127C2×D412yes—
128C23×C127—24no—
129C23×C127—22no—
130C23×C127—42no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127C2×C22⋊C411yescharacteristic
134Q8×C127—44no—
135Q8×C127—84no—
136D4×C127—24no—
137D4×C127—22no—
138D4×C127#10114yescomplemented
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—22no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—21no—
147C2×C4×C127—22no—
148C2×C4×C127—22no—
149C2×C4×C127—22no—
150C2×C4×C127—21no—
151C2×C4×C127—21no—
152C2×C4×C127—41no—
153C2×C4×C127—48no—
154C2×C4×C127—42no—
155C2×C4×C127—42no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—41no—
160C2×C4×C127—84no—
161C2×C4×C127—84no—
162C2×C4×C127#10412yes—
163C2×C4×C127C2×C22⋊C411yescharacteristic
164C2×C4×C127C2×C22⋊C411yescharacteristic
165C2×C4×C127#12012yes—
166C2×C4×C127#12012yes—
167C8×C127—44no—
168C8×C127—82no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—24no—
172C22×C127—44no—
173C22×C127—44no—
174C22×C127—42no—
175C22×C127—42no—
176C22×C127#37911yescharacteristic
177C22×C127C2×#10411yescharacteristic
178C22×C127#42111yescharacteristic
179C4×C127—22no—
180C4×C127—21no—
181C4×C127—22no—
182C4×C127—22no—
183C4×C127—42no—
184C4×C127—42no—
185C4×C127—41no—
186C4×C127—41no—
187C4×C127—41no—
188C4×C127—42no—
189C4×C127C2×#10112yes—
190#8372C12711yesmaximal, characteristic, direct product factor, Sylow
191C2×C127—21no—
192C2×C127—21no—
193C2×C127—44no—
194C2×C127#178411yescharacteristic
195C2×C127#178911yescharacteristic
196C2×C127#179511yescharacteristic
197C2×#554C2×C12711yescharacteristic
198C4×#101C2×C12711yescharacteristic, complemented
199#1528C2×C12712yescomplemented
200#1490C2×C12712yes—
201#1467C2×C12711yescharacteristic, complemented
202C127#837211yescharacteristic, direct product factor, Sylow
203#554—24no—
204#554—24no—
205C2×C4⋊Q8C4×C12711yescharacteristic
206C2×#125C4×C12711yescharacteristic
207C2×#125C22×C12711yescharacteristic
208C2×C22⋊Q8—21no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8—21no—
211C2×C4×Q8C22×C12711yescharacteristic
212C2×C4×D4C4×C12711yescharacteristic
213C2×C4×D4C4×C12711yescharacteristic
214C2×C4×D4C22×C12711yescharacteristic
215C22×C4⋊C4—21no—
216#432C22×C12712yes—
217C2×#101—21no—
218C4×M16C22×C12712yes—
219#343—21no—
220#330—24no—
221D4⋊C8—24no—
222#324—21no—
223#324—21no—
224C22×Q8—21no—
225C22×Q8D4×C12711yescharacteristic
226C22×D4C2×C4×C12711yescharacteristic
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C2×M16—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233#125—22no—
234#125—22no—
235#125—22no—
236#125D4×C12714yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4D4×C12714yes—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C2×C4×C12711yescharacteristic
255C2×C4⋊C4C2×C4×C12711yescharacteristic
256C2×C4⋊C4C2×C4×C12711yescharacteristic
257C2×C4⋊C4D4×C12711yescharacteristic
258C2×C4⋊C4D4×C12711yescharacteristic
259C2×C4⋊C4D4×C12711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42C2×C4×C12711yescharacteristic
264C2×C42C2×C4×C12711yescharacteristic
265C2×C42C23×C12711yescharacteristic
266C4⋊C8—22no—
267#101—24no—
268C22⋊C8—22no—
269#97—22no—
270#97—22no—
271C4×C8—22no—
272C24—21no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—82no—
278C2×D4—24no—
279C2×D4—21no—
280C2×D4C22⋊C4×C12712yes—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C22⋊C4×C12711yescharacteristic
292C22×C4C22⋊C4×C12711yescharacteristic
293C22×C4C22⋊C4×C12711yescharacteristic
294C22×C4C22⋊C4×C12711yescharacteristic
295C22×C4C22×C4×C12711yescharacteristic
296C22×C4C2×D4×C12711yescharacteristic
297C22×C4C2×D4×C12711yescharacteristic
298M16—24no—
299M16—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4C22⋊C4×C12712yes—
314C4⋊C4C22⋊C4×C12712yes—
315C4⋊C4C22⋊C4×C12712yes—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42C2×D4×C12712yes—
324C42C2×D4×C12712yes—
325C23—24no—
326C23—22no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23C2×C22⋊C4×C12711yescharacteristic
331Q8—44no—
332Q8—84no—
333D4—24no—
334D4—22no—
335D4#101×C12714yescomplemented
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#104×C12712yes—
360C2×C4C2×C22⋊C4×C12711yescharacteristic
361C2×C4C2×C22⋊C4×C12711yescharacteristic
362C2×C4#120×C12712yes—
363C2×C4#120×C12712yes—
364C8—44no—
365C8—82no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22#379×C12711yescharacteristic
374C22C2×#104×C12711yescharacteristic
375C22#421×C12711yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4C2×#101×C12712yes—
387C2—21no—
388C2—21no—
389C2—44no—
390C2C127×#178411yescharacteristic
391C2C127×#178911yescharacteristic
392C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#101×C127C2semidirect
3. C127×#1528C2semidirect
5. C127×#1467C2semidirect
138. D4×C127#101semidirect
190. #8372C127direct (N × H)
198. C4×#101C2×C127semidirect
199. #1528C2×C127semidirect
201. #1467C2×C127semidirect
202. C127#8372direct (N × H)
335. D4#101×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#554×C1272
C4×#101×C1272
C127×#15282
C127×#14902
C127×#14672
#8372127