Finite Group Databasebeta

← C127×#21988 #44585661

Subgroups — C127×#21988 #44585661

Order 32512.

830 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#432×C127C211yesmaximal, characteristic
2C127×#2285C211yesmaximal, characteristic
3C127×#2285C211yesmaximal, characteristic
4C127×#1948C212yesmaximal, complemented
5C127×#1947C212yesmaximal, complemented
6C127×#1854C214yesmaximal
7C127×#1847C212yesmaximal, complemented
8C127×#1836C212yesmaximal, complemented
9C2×#117×C127—21no—
10C2×#117×C127C2212yes—
11C22×C42×C127C2211yescharacteristic
12#432×C127C418yescomplemented
13C2×#107×C127C2214yes—
14C2×#106×C127C2214yes—
15C2×C4⋊C8×C127C2212yes—
16C2×C22⋊C8×C127C2212yes—
17C2×#97×C127C2212yes—
18#384×C127C2211yescharacteristic
19#384×C127C2211yescharacteristic
20#384×C127C2211yescharacteristic
21#384×C127C2211yescharacteristic
22#382×C127—21no—
23#382×C127C2212yes—
24#380×C127—21no—
25#380×C127—21no—
26#380×C127—21no—
27#380×C127C2212yes—
28C4×C4⋊C4×C127—21no—
29C4×C4⋊C4×C127—21no—
30C4×C4⋊C4×C127—21no—
31C4×C4⋊C4×C127C2212yes—
32#340×C127C2214yes—
33#340×C127C2214yes—
34C23×C4×C127D411yescharacteristic
35C23×C4×C127D411yescharacteristic
36C23×C4×C127C2311yescharacteristic
37C22×C8×C127C2314yes—
38#117×C127—44no—
39#117×C127—44no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127—21no—
46C2×C4⋊C4×C127—21no—
47C2×C4⋊C4×C127—21no—
48C2×C4⋊C4×C127C2312yes—
49C2×C4⋊C4×C127C2312yes—
50C2×C4⋊C4×C127C2312yes—
51C2×C4⋊C4×C127C2312yes—
52C2×C22⋊C4×C127—22no—
53C2×C22⋊C4×C127—22no—
54C2×C42×C127—22no—
55C2×C42×C127—22no—
56C2×C42×C127C2×C414yes—
57C2×C42×C127D411yescharacteristic
58C2×C42×C127D411yescharacteristic
59C2×C42×C127Q814yes—
60C2×C42×C127C2311yescharacteristic
61C2×C42×C127C2311yescharacteristic
62#107×C127—44no—
63#106×C127—44no—
64C4⋊C8×C127C2×C418yes—
65C22⋊C8×C127C2×C418yes—
66#97×C127C2×C418yes—
67#95×C127—22no—
68#95×C127—22no—
69#95×C127—22no—
70#95×C127—22no—
71C24×C127C2×D411yescharacteristic
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—41no—
81C22×C4×C127—41no—
82C22×C4×C127C4⋊C414yes—
83C22×C4×C127C4⋊C414yes—
84C22×C4×C127C22×C412yes—
85C22×C4×C127C2×D411yescharacteristic
86C22×C4×C127C2×D411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D411yescharacteristic
90C22×C4×C127C2×Q812yes—
91C22×C4×C127C2411yescharacteristic
92C2×C8×C127—24no—
93C2×C8×C127—24no—
94C2×C8×C127C22×C418yes—
95C4⋊C4×C127—22no—
96C4⋊C4×C127—24no—
97C4⋊C4×C127—22no—
98C4⋊C4×C127—24no—
99C4⋊C4×C127—24no—
100C4⋊C4×C127—42no—
101C4⋊C4×C127—42no—
102C4⋊C4×C127—42no—
103C4⋊C4×C127—42no—
104C4⋊C4×C127—42no—
105C4⋊C4×C127—42no—
106C4⋊C4×C127—42no—
107C4⋊C4×C127—42no—
108C22⋊C4×C127—44no—
109C22⋊C4×C127—44no—
110C42×C127—44no—
111C42×C127—44no—
112C42×C127C4⋊C414yes—
113C42×C127C4⋊C414yes—
114C42×C127C22×C412yes—
115C42×C127C22×C412yes—
116C42×C127C2×Q812yes—
117C42×C127C2×Q812yes—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—21no—
124C23×C127C2×C4⋊C412yes—
125C23×C127C22×D411yescharacteristic
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—24no—
135C2×C4×C127—24no—
136C2×C4×C127—28no—
137C2×C4×C127—44no—
138C2×C4×C127—42no—
139C2×C4×C127—42no—
140C2×C4×C127—42no—
141C2×C4×C127—42no—
142C2×C4×C127—42no—
143C2×C4×C127—42no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127C2×C4⋊C412yes—
151C2×C4×C127C2×C4⋊C412yes—
152C2×C4×C127C2×C4⋊C412yes—
153C2×C4×C127C2×C4⋊C412yes—
154C2×C4×C127C2×C4⋊C412yes—
155C2×C4×C127AGL(1,ℤ/8)12yes—
156C2×C4×C127#13712yes—
157C2×C4×C127C23×C411yescharacteristic
158C2×C4×C127C22×Q811yescharacteristic
159C2×C4×C127H(5,𝔽2)12yes—
160C2×C4×C1272−1+412yes—
161C8×C127—28no—
162C22×C127—22no—
163C22×C127—22no—
164C22×C127—22no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127C22×C4⋊C411yescharacteristic
170C22×C127#53511yescharacteristic
171C22×C127#53611yescharacteristic
172C22×C127C2×AGL(1,ℤ/8)11yescharacteristic
173C22×C127C2×#13711yescharacteristic
174C22×C127#57511yescharacteristic
175C22×C127#57511yescharacteristic
176C4×C127—22no—
177C4×C127—22no—
178C4×C127—22no—
179C4×C127—22no—
180C4×C127—82no—
181C4×C127—82no—
182C4×C127—82no—
183C4×C127—82no—
184C4×C127#42814yes—
185C4×C127#52012yes—
186C4×C127#54412yes—
187#21988C12711yesmaximal, characteristic, direct product factor, Sylow
188C2×C127—42no—
189C2×C127#231711yescharacteristic
190C2×C127C2×#42811yescharacteristic
191C2×C127#291111yescharacteristic
192C2×C127#321411yescharacteristic
193C2×C127#321611yescharacteristic
194C2×C127#323511yescharacteristic
195C2×C127#323711yescharacteristic
196C2×#432C2×C12711yescharacteristic
197#2285C2×C12711yescharacteristic
198#2285C2×C12711yescharacteristic
199#1948C2×C12712yescomplemented
200#1947C2×C12712yescomplemented
201#1854C2×C12714yes—
202#1847C2×C12712yescomplemented
203#1836C2×C12712yescomplemented
204C127#2198811yescharacteristic, direct product factor, Sylow
205C2×#117—21no—
206C2×#117C22×C12712yes—
207C22×C42C22×C12711yescharacteristic
208#432C4×C12718yescomplemented
209C2×#107C22×C12714yes—
210C2×#106C22×C12714yes—
211C2×C4⋊C8C22×C12712yes—
212C2×C22⋊C8C22×C12712yes—
213C2×#97C22×C12712yes—
214#384C22×C12711yescharacteristic
215#384C22×C12711yescharacteristic
216#384C22×C12711yescharacteristic
217#384C22×C12711yescharacteristic
218#382—21no—
219#382C22×C12712yes—
220#380—21no—
221#380—21no—
222#380—21no—
223#380C22×C12712yes—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4C22×C12712yes—
228#340C22×C12714yes—
229#340C22×C12714yes—
230C23×C4D4×C12711yescharacteristic
231C23×C4D4×C12711yescharacteristic
232C23×C4C23×C12711yescharacteristic
233C22×C8C23×C12714yes—
234#117—44no—
235#117—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C23×C12712yes—
245C2×C4⋊C4C23×C12712yes—
246C2×C4⋊C4C23×C12712yes—
247C2×C4⋊C4C23×C12712yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42C2×C4×C12714yes—
253C2×C42D4×C12711yescharacteristic
254C2×C42D4×C12711yescharacteristic
255C2×C42Q8×C12714yes—
256C2×C42C23×C12711yescharacteristic
257C2×C42C23×C12711yescharacteristic
258#107—44no—
259#106—44no—
260C4⋊C8C2×C4×C12718yes—
261C22⋊C8C2×C4×C12718yes—
262#97C2×C4×C12718yes—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24C2×D4×C12711yescharacteristic
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C4⋊C4×C12714yes—
279C22×C4C4⋊C4×C12714yes—
280C22×C4C22×C4×C12712yes—
281C22×C4C2×D4×C12711yescharacteristic
282C22×C4C2×D4×C12711yescharacteristic
283C22×C4C2×D4×C12711yescharacteristic
284C22×C4C2×D4×C12711yescharacteristic
285C22×C4C2×D4×C12711yescharacteristic
286C22×C4C2×Q8×C12712yes—
287C22×C4C24×C12711yescharacteristic
288C2×C8—24no—
289C2×C8—24no—
290C2×C8C22×C4×C12718yes—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C42—44no—
307C42—44no—
308C42C4⋊C4×C12714yes—
309C42C4⋊C4×C12714yes—
310C42C22×C4×C12712yes—
311C42C22×C4×C12712yes—
312C42C2×Q8×C12712yes—
313C42C2×Q8×C12712yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×C4⋊C4×C12712yes—
321C23C22×D4×C12711yescharacteristic
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4C2×C4⋊C4×C12712yes—
347C2×C4C2×C4⋊C4×C12712yes—
348C2×C4C2×C4⋊C4×C12712yes—
349C2×C4C2×C4⋊C4×C12712yes—
350C2×C4C2×C4⋊C4×C12712yes—
351C2×C4AGL(1,ℤ/8)×C12712yes—
352C2×C4#137×C12712yes—
353C2×C4C23×C4×C12711yescharacteristic
354C2×C4C22×Q8×C12711yescharacteristic
355C2×C4H(5,𝔽2)×C12712yes—
356C2×C42−1+4×C12712yes—
357C8—28no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22C22×C4⋊C4×C12711yescharacteristic
366C22#535×C12711yescharacteristic
367C22#536×C12711yescharacteristic
368C22C2×AGL(1,ℤ/8)×C12711yescharacteristic
369C22C2×#137×C12711yescharacteristic
370C22#575×C12711yescharacteristic
371C22#575×C12711yescharacteristic
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—82no—
377C4—82no—
378C4—82no—
379C4—82no—
380C4#428×C12714yes—
381C4#520×C12712yes—
382C4#544×C12712yes—
383C2—42no—
384C2C127×#231711yescharacteristic
385C2C2×#428×C12711yescharacteristic
386C2C127×#291111yescharacteristic
387C2C127×#321411yescharacteristic
388C2C127×#321611yescharacteristic
389C2C127×#323511yescharacteristic
390C2C127×#323711yescharacteristic
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1948C2semidirect
5. C127×#1947C2semidirect
7. C127×#1847C2semidirect
8. C127×#1836C2semidirect
12. #432×C127C4semidirect
187. #21988C127direct (N × H)
199. #1948C2×C127semidirect
200. #1947C2×C127semidirect
202. #1847C2×C127semidirect
203. #1836C2×C127semidirect
204. C127#21988direct (N × H)
208. #432C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#432×C1272
C127×#22852
C127×#19482
C127×#19472
C127×#18542
C127×#18472
C127×#18362
#21988127