Finite Group Databasebeta

← C127×#12735 #44576408

Subgroups — C127×#12735 #44576408

Order 32512.

948 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#135×C127C211yesmaximal, characteristic
2C2×#360×C127C214yesmaximal, complemented
3C127×#2112C211yesmaximal, characteristic, complemented
4C127×#1807C211yesmaximal, characteristic, complemented
5C22×#41×C127—21no—
6C2×#135×C127—28no—
7C2×#135×C127—21no—
8C2×#135×C127—21no—
9C2×#135×C127C412yescomplemented
10C2×#135×C127C2212yes—
11C22×Q16×C127C411yescharacteristic, complemented
12C22×SD16×C127—21no—
13C22×D8×C127C411yescharacteristic, complemented
14C23×C8×C127C2211yescharacteristic
15C2×M32×C127C2212yescomplemented
16C2×#106×C127C2212yescomplemented
17#380×C127—21no—
18#360×C127—28no—
19C22⋊C16×C127—21no—
20C22⋊C16×C127—21no—
21#340×C127—21no—
22C2×#41×C127—28no—
23C2×#41×C127—22no—
24C2×#41×C127—22no—
25C2×#41×C127—44no—
26C2×#41×C127—41no—
27C2×#41×C127—41no—
28C22×Q8×C127—21no—
29C22×D4×C127—21no—
30C22×D4×C127—41no—
31C23×C4×C127—41no—
32C23×C4×C127D411yescharacteristic
33#135×C127—28no—
34#135×C127—28no—
35#135×C127—24no—
36#135×C127—24no—
37#135×C127—24no—
38#135×C127D418yes—
39C2×Q16×C127—24no—
40C2×Q16×C127—21no—
41C2×Q16×C127C2×C412yes—
42C2×SD16×C127—28no—
43C2×SD16×C127—22no—
44C2×SD16×C127—22no—
45C2×D8×C127—24no—
46C2×D8×C127—21no—
47C2×D8×C127C2×C412yes—
48C22×C8×C127—24no—
49C22×C8×C127C2×C411yescharacteristic
50C22×C8×C127C2×C411yescharacteristic
51C22×C8×C127D411yescharacteristic
52C22×C8×C127D411yescharacteristic
53C22×C8×C127D411yescharacteristic
54C22×C8×C127C2311yescharacteristic
55C2×C4⋊C4×C127—22no—
56C2×C22⋊C4×C127—41no—
57M32×C127—24no—
58C2×C16×C127—22no—
59#106×C127—24no—
60#95×C127—41no—
61C24×C127—41no—
62C24×C127D811yescharacteristic
63#41×C127—28no—
64#41×C127—28no—
65#41×C127—48no—
66#41×C127—44no—
67#41×C127—48no—
68#41×C127—44no—
69C2×Q8×C127—24no—
70C2×Q8×C127—22no—
71C2×Q8×C127—42no—
72C2×Q8×C127—41no—
73C2×D4×C127—24no—
74C2×D4×C127—22no—
75C2×D4×C127—42no—
76C2×D4×C127—48no—
77C2×D4×C127—42no—
78C2×D4×C127—42no—
79C2×D4×C127—41no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127—48no—
83C22×C4×C127—41no—
84C22×C4×C127—42no—
85C22×C4×C127—42no—
86C22×C4×C127—42no—
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127C22⋊C411yescharacteristic
89C22×C4×C127D811yescharacteristic
90C22×C4×C127SD1611yescharacteristic
91C22×C4×C127SD1611yescharacteristic
92C22×C4×C127C2×D411yescharacteristic
93Q16×C127—24no—
94Q16×C127—22no—
95Q16×C127C22⋊C414yescomplemented
96SD16×C127—28no—
97SD16×C127—28no—
98D8×C127—24no—
99D8×C127—22no—
100D8×C127C22⋊C414yescomplemented
101C2×C8×C127—22no—
102C2×C8×C127—22no—
103C2×C8×C127—22no—
104C2×C8×C127—22no—
105C2×C8×C127—22no—
106C2×C8×C127C22⋊C411yescharacteristic
107C2×C8×C127C22⋊C411yescharacteristic
108C2×C8×C127C22×C411yescharacteristic
109C2×C8×C127C2×D414yes—
110C2×C8×C127C2×D411yescharacteristic
111C4⋊C4×C127—42no—
112C4⋊C4×C127—42no—
113C22⋊C4×C127—82no—
114C16×C127—42no—
115C23×C127—22no—
116C23×C127—22no—
117C23×C127—41no—
118C23×C127—44no—
119C23×C127—42no—
120C23×C127—82no—
121C23×C127—82no—
122C23×C127#10211yescharacteristic
123C23×C127#10211yescharacteristic
124C23×C127C2×D811yescharacteristic
125Q8×C127—24no—
126Q8×C127—44no—
127Q8×C127—42no—
128D4×C127—24no—
129D4×C127—44no—
130D4×C127—48no—
131D4×C127—48no—
132D4×C127—42no—
133C2×C4×C127—22no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—21no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—48no—
141C2×C4×C127—44no—
142C2×C4×C127—42no—
143C2×C4×C127—48no—
144C2×C4×C127—82no—
145C2×C4×C127—82no—
146C2×C4×C127—82no—
147C2×C4×C127#10211yescharacteristic
148C2×C4×C127#10211yescharacteristic
149C2×C4×C127C2×C22⋊C411yescharacteristic
150C2×C4×C127#12012yes—
151C2×C4×C127C2×SD1611yescharacteristic
152C2×C4×C127AGL(1,ℤ/8)12yes—
153C8×C127—21no—
154C8×C127—21no—
155C8×C127C2×C22⋊C412yes—
156C8×C127#12012yes—
157C22×C127—21no—
158C22×C127—21no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—44no—
165C22×C127—44no—
166C22×C127—88no—
167C22×C127—82no—
168C22×C127—82no—
169C22×C127#36012yescomplemented
170C22×C127C2×#10211yescharacteristic
171C22×C127#44712yes—
172C4×C127—21no—
173C4×C127—21no—
174C4×C127—44no—
175C4×C127—162no—
176C4×C127#37911yescharacteristic
177C4×C127#41811yescharacteristic
178C4×C127#45012yes—
179#12735C12711yesmaximal, characteristic, direct product factor, Sylow
180C2×C127—22no—
181C2×C127—22no—
182C2×C127—41no—
183C2×C127—84no—
184C2×C127#178611yescharacteristic
185C2×C127C2×#36011yescharacteristic
186C2×C127#214511yescharacteristic
187C22×#135C2×C12711yescharacteristic
188C2×#360C2×C12714yescomplemented
189#2112C2×C12711yescharacteristic, complemented
190#1807C2×C12711yescharacteristic, complemented
191C127#1273511yescharacteristic, direct product factor, Sylow
192C22×#41—21no—
193C2×#135—28no—
194C2×#135—21no—
195C2×#135—21no—
196C2×#135C4×C12712yescomplemented
197C2×#135C22×C12712yes—
198C22×Q16C4×C12711yescharacteristic, complemented
199C22×SD16—21no—
200C22×D8C4×C12711yescharacteristic, complemented
201C23×C8C22×C12711yescharacteristic
202C2×M32C22×C12712yescomplemented
203C2×#106C22×C12712yescomplemented
204#380—21no—
205#360—28no—
206C22⋊C16—21no—
207C22⋊C16—21no—
208#340—21no—
209C2×#41—28no—
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—44no—
213C2×#41—41no—
214C2×#41—41no—
215C22×Q8—21no—
216C22×D4—21no—
217C22×D4—41no—
218C23×C4—41no—
219C23×C4D4×C12711yescharacteristic
220#135—28no—
221#135—28no—
222#135—24no—
223#135—24no—
224#135—24no—
225#135D4×C12718yes—
226C2×Q16—24no—
227C2×Q16—21no—
228C2×Q16C2×C4×C12712yes—
229C2×SD16—28no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×D8—24no—
233C2×D8—21no—
234C2×D8C2×C4×C12712yes—
235C22×C8—24no—
236C22×C8C2×C4×C12711yescharacteristic
237C22×C8C2×C4×C12711yescharacteristic
238C22×C8D4×C12711yescharacteristic
239C22×C8D4×C12711yescharacteristic
240C22×C8D4×C12711yescharacteristic
241C22×C8C23×C12711yescharacteristic
242C2×C4⋊C4—22no—
243C2×C22⋊C4—41no—
244M32—24no—
245C2×C16—22no—
246#106—24no—
247#95—41no—
248C24—41no—
249C24D8×C12711yescharacteristic
250#41—28no—
251#41—28no—
252#41—48no—
253#41—44no—
254#41—48no—
255#41—44no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—42no—
263C2×D4—48no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—48no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4C22⋊C4×C12711yescharacteristic
275C22×C4C22⋊C4×C12711yescharacteristic
276C22×C4D8×C12711yescharacteristic
277C22×C4SD16×C12711yescharacteristic
278C22×C4SD16×C12711yescharacteristic
279C22×C4C2×D4×C12711yescharacteristic
280Q16—24no—
281Q16—22no—
282Q16C22⋊C4×C12714yescomplemented
283SD16—28no—
284SD16—28no—
285D8—24no—
286D8—22no—
287D8C22⋊C4×C12714yescomplemented
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8C22⋊C4×C12711yescharacteristic
294C2×C8C22⋊C4×C12711yescharacteristic
295C2×C8C22×C4×C12711yescharacteristic
296C2×C8C2×D4×C12714yes—
297C2×C8C2×D4×C12711yescharacteristic
298C4⋊C4—42no—
299C4⋊C4—42no—
300C22⋊C4—82no—
301C16—42no—
302C23—22no—
303C23—22no—
304C23—41no—
305C23—44no—
306C23—42no—
307C23—82no—
308C23—82no—
309C23#102×C12711yescharacteristic
310C23#102×C12711yescharacteristic
311C23C2×D8×C12711yescharacteristic
312Q8—24no—
313Q8—44no—
314Q8—42no—
315D4—24no—
316D4—44no—
317D4—48no—
318D4—48no—
319D4—42no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—48no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4#102×C12711yescharacteristic
335C2×C4#102×C12711yescharacteristic
336C2×C4C2×C22⋊C4×C12711yescharacteristic
337C2×C4#120×C12712yes—
338C2×C4C2×SD16×C12711yescharacteristic
339C2×C4AGL(1,ℤ/8)×C12712yes—
340C8—21no—
341C8—21no—
342C8C2×C22⋊C4×C12712yes—
343C8#120×C12712yes—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—44no—
353C22—88no—
354C22—82no—
355C22—82no—
356C22#360×C12712yescomplemented
357C22C2×#102×C12711yescharacteristic
358C22#447×C12712yes—
359C4—21no—
360C4—21no—
361C4—44no—
362C4—162no—
363C4#379×C12711yescharacteristic
364C4#418×C12711yescharacteristic
365C4#450×C12712yes—
366C2—22no—
367C2—22no—
368C2—41no—
369C2—84no—
370C2C127×#178611yescharacteristic
371C2C2×#360×C12711yescharacteristic
372C2C127×#214511yescharacteristic
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC22×#135×C127C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#360×C127C2semidirect
3. C127×#2112C2semidirect
4. C127×#1807C2semidirect
9. C2×#135×C127C4semidirect
11. C22×Q16×C127C4semidirect
13. C22×D8×C127C4semidirect
15. C2×M32×C127C22semidirect
16. C2×#106×C127C22semidirect
95. Q16×C127C22⋊C4semidirect
100. D8×C127C22⋊C4semidirect
169. C22×C127#360semidirect
179. #12735C127direct (N × H)
188. C2×#360C2×C127semidirect
189. #2112C2×C127semidirect
190. #1807C2×C127semidirect
191. C127#12735direct (N × H)
196. C2×#135C4×C127semidirect
198. C22×Q16C4×C127semidirect
200. C22×D8C4×C127semidirect
202. C2×M32C22×C127semidirect
203. C2×#106C22×C127semidirect
282. Q16C22⋊C4×C127semidirect
287. D8C22⋊C4×C127semidirect
356. C22#360×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#135×C1272
C2×#360×C1272
C127×#21122
C127×#18072
#12735127