Finite Group Databasebeta

← C127×#21778 #44585451

Subgroups — C127×#21778 #44585451

Order 32512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3484C211yesmaximal, characteristic
2C127×#2888C212yesmaximal, complemented
3C127×#1914C214yesmaximal, complemented
4C127×#1891C214yesmaximal, complemented
5C127×#1829C212yesmaximal, complemented
6C127×#1783C212yesmaximal, complemented
7C2×H(5,𝔽2)×C127C2211yescharacteristic
8C23×D4×C127C2211yescharacteristic
9#568×C127C2214yescomplemented
10#546×C127—24no—
11D42×C127—24no—
12#535×C127C2212yes—
13#534×C127C414yes—
14#525×C127C2212yes—
15C2×#120×C127C414yes—
16#518×C127C2211yescharacteristic
17#413×C127—24no—
18#413×C127C2214yescomplemented
19C2×#100×C127—24no—
20C2×#100×C127C2214yes—
21C2×#100×C127C2214yescomplemented
22#407×C127C2214yescomplemented
23#344×C127C2218yescomplemented
24C25×C127C2×C412yes—
25H(5,𝔽2)×C127—44no—
26C2×#41×C127D414yes—
27C2×#41×C127C2312yes—
28C22×D4×C127—24no—
29C22×D4×C127—24no—
30C22×D4×C127—44no—
31C22×D4×C127C2×C414yes—
32C22×D4×C127C2×C412yes—
33C22×D4×C127D414yes—
34C22×D4×C127C2311yescharacteristic
35C22×D4×C127C2311yescharacteristic
36C23×C4×C127C2311yescharacteristic
37#131×C127—48no—
38C2×M16×C127—28no—
39C2×M16×C127C2318yes—
40#127×C127—22no—
41#124×C127—22no—
42#123×C127—24no—
43#123×C127C2×C414yes—
44C22⋊Q8×C127—22no—
45#121×C127—28no—
46#121×C127—22no—
47#121×C127—24no—
48#120×C127—24no—
49#120×C127—28no—
50#120×C127C2×C414yes—
51#120×C127C2×C418yes—
52C4×D4×C127—24no—
53C4×D4×C127—24no—
54#117×C127C2312yes—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127C2×C414yes—
57#101×C127—24no—
58#100×C127—24no—
59#100×C127—28no—
60#100×C127—44no—
61C22⋊C8×C127—44no—
62C24×C127—24no—
63C24×C127—44no—
64C24×C127—84no—
65C24×C127C22⋊C414yes—
66C24×C127C22×C411yescharacteristic
67C24×C127C22×C412yes—
68C24×C127C22×C412yes—
69#41×C127—28no—
70#41×C127—48no—
71C2×Q8×C127C2×D412yes—
72C2×D4×C127—24no—
73C2×D4×C127—24no—
74C2×D4×C127—28no—
75C2×D4×C127—22no—
76C2×D4×C127—24no—
77C2×D4×C127—24no—
78C2×D4×C127—48no—
79C2×D4×C127—44no—
80C2×D4×C127—48no—
81C2×D4×C127—416no—
82C2×D4×C127—88no—
83C2×D4×C127C22⋊C418yes—
84C2×D4×C127C22×C414yes—
85C2×D4×C127C2×D414yes—
86C2×D4×C127C2×D412yes—
87C22×C4×C127—24no—
88C22×C4×C127—22no—
89C22×C4×C127—24no—
90C22×C4×C127C22⋊C414yes—
91C22×C4×C127C22×C411yescharacteristic
92C22×C4×C127C2×D414yes—
93C22×C4×C127C2411yescharacteristic
94M16×C127—28no—
95M16×C127—28no—
96M16×C127—48no—
97C2×C8×C127—28no—
98C2×C8×C127—48no—
99C4⋊C4×C127—24no—
100C4⋊C4×C127—22no—
101C22⋊C4×C127—24no—
102C22⋊C4×C127—24no—
103C22⋊C4×C127—24no—
104C22⋊C4×C127—28no—
105C22⋊C4×C127C22×C414yes—
106C42×C127—22no—
107C23×C127—21no—
108C23×C127—22no—
109C23×C127—24no—
110C23×C127—22no—
111C23×C127—24no—
112C23×C127—42no—
113C23×C127—44no—
114C23×C127—48no—
115C23×C127—48no—
116C23×C127—88no—
117C23×C127—82no—
118C23×C127—82no—
119C23×C127—84no—
120C23×C127—84no—
121C23×C127—88no—
122C23×C127C2×C22⋊C414yes—
123C23×C127C2×C22⋊C414yes—
124C23×C127C23×C411yescharacteristic
125Q8×C127—24no—
126D4×C127—28no—
127D4×C127—28no—
128D4×C127—24no—
129D4×C127—48no—
130D4×C127—48no—
131D4×C127—48no—
132D4×C127—44no—
133D4×C127—88no—
134C2×C4×C127—28no—
135C2×C4×C127—24no—
136C2×C4×C127—21no—
137C2×C4×C127—22no—
138C2×C4×C127—44no—
139C2×C4×C127—44no—
140C2×C4×C127—84no—
141C2×C4×C127C2×C22⋊C414yes—
142C2×C4×C127C22×D412yes—
143C2×C4×C127H(5,𝔽2)14yes—
144C8×C127—48no—
145C22×C127—22no—
146C22×C127—24no—
147C22×C127—24no—
148C22×C127—24no—
149C22×C127—41no—
150C22×C127—44no—
151C22×C127—44no—
152C22×C127—41no—
153C22×C127—42no—
154C22×C127—44no—
155C22×C127—41no—
156C22×C127—42no—
157C22×C127—84no—
158C22×C127—84no—
159C22×C127—816no—
160C22×C127—164no—
161C22×C127C22×C22⋊C411yescharacteristic
162C22×C127#51812yes—
163C4×C127—24no—
164C4×C127—44no—
165C4×C127—41no—
166C4×C127#53414yes—
167#21778C12711yesmaximal, characteristic, direct product factor, Sylow
168C2×C127—21no—
169C2×C127—22no—
170C2×C127—42no—
171C2×C127—44no—
172C2×C127—44no—
173C2×C127—82no—
174C2×C127#230811yescharacteristic
175#3484C2×C12711yescharacteristic
176#2888C2×C12712yescomplemented
177#1914C2×C12714yescomplemented
178#1891C2×C12714yescomplemented
179#1829C2×C12712yescomplemented
180#1783C2×C12712yescomplemented
181C127#2177811yescharacteristic, direct product factor, Sylow
182C2×H(5,𝔽2)C22×C12711yescharacteristic
183C23×D4C22×C12711yescharacteristic
184#568C22×C12714yescomplemented
185#546—24no—
186D42—24no—
187#535C22×C12712yes—
188#534C4×C12714yes—
189#525C22×C12712yes—
190C2×#120C4×C12714yes—
191#518C22×C12711yescharacteristic
192#413—24no—
193#413C22×C12714yescomplemented
194C2×#100—24no—
195C2×#100C22×C12714yes—
196C2×#100C22×C12714yescomplemented
197#407C22×C12714yescomplemented
198#344C22×C12718yescomplemented
199C25C2×C4×C12712yes—
200H(5,𝔽2)—44no—
201C2×#41D4×C12714yes—
202C2×#41C23×C12712yes—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—44no—
206C22×D4C2×C4×C12714yes—
207C22×D4C2×C4×C12712yes—
208C22×D4D4×C12714yes—
209C22×D4C23×C12711yescharacteristic
210C22×D4C23×C12711yescharacteristic
211C23×C4C23×C12711yescharacteristic
212#131—48no—
213C2×M16—28no—
214C2×M16C23×C12718yes—
215#127—22no—
216#124—22no—
217#123—24no—
218#123C2×C4×C12714yes—
219C22⋊Q8—22no—
220#121—28no—
221#121—22no—
222#121—24no—
223#120—24no—
224#120—28no—
225#120C2×C4×C12714yes—
226#120C2×C4×C12718yes—
227C4×D4—24no—
228C4×D4—24no—
229#117C23×C12712yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4C2×C4×C12714yes—
232#101—24no—
233#100—24no—
234#100—28no—
235#100—44no—
236C22⋊C8—44no—
237C24—24no—
238C24—44no—
239C24—84no—
240C24C22⋊C4×C12714yes—
241C24C22×C4×C12711yescharacteristic
242C24C22×C4×C12712yes—
243C24C22×C4×C12712yes—
244#41—28no—
245#41—48no—
246C2×Q8C2×D4×C12712yes—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—28no—
250C2×D4—22no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—416no—
257C2×D4—88no—
258C2×D4C22⋊C4×C12718yes—
259C2×D4C22×C4×C12714yes—
260C2×D4C2×D4×C12714yes—
261C2×D4C2×D4×C12712yes—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4C22⋊C4×C12714yes—
266C22×C4C22×C4×C12711yescharacteristic
267C22×C4C2×D4×C12714yes—
268C22×C4C24×C12711yescharacteristic
269M16—28no—
270M16—28no—
271M16—48no—
272C2×C8—28no—
273C2×C8—48no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—28no—
280C22⋊C4C22×C4×C12714yes—
281C42—22no—
282C23—21no—
283C23—22no—
284C23—24no—
285C23—22no—
286C23—24no—
287C23—42no—
288C23—44no—
289C23—48no—
290C23—48no—
291C23—88no—
292C23—82no—
293C23—82no—
294C23—84no—
295C23—84no—
296C23—88no—
297C23C2×C22⋊C4×C12714yes—
298C23C2×C22⋊C4×C12714yes—
299C23C23×C4×C12711yescharacteristic
300Q8—24no—
301D4—28no—
302D4—28no—
303D4—24no—
304D4—48no—
305D4—48no—
306D4—48no—
307D4—44no—
308D4—88no—
309C2×C4—28no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—84no—
316C2×C4C2×C22⋊C4×C12714yes—
317C2×C4C22×D4×C12712yes—
318C2×C4H(5,𝔽2)×C12714yes—
319C8—48no—
320C22—22no—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—41no—
325C22—44no—
326C22—44no—
327C22—41no—
328C22—42no—
329C22—44no—
330C22—41no—
331C22—42no—
332C22—84no—
333C22—84no—
334C22—816no—
335C22—164no—
336C22C22×C22⋊C4×C12711yescharacteristic
337C22#518×C12712yes—
338C4—24no—
339C4—44no—
340C4—41no—
341C4#534×C12714yes—
342C2—21no—
343C2—22no—
344C2—42no—
345C2—44no—
346C2—44no—
347C2—82no—
348C2C127×#230811yescharacteristic
subgroupG/H
center, socleC2×C127#2308
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3484C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2888C2semidirect
3. C127×#1914C2semidirect
4. C127×#1891C2semidirect
5. C127×#1829C2semidirect
6. C127×#1783C2semidirect
9. #568×C127C22semidirect
18. #413×C127C22semidirect
21. C2×#100×C127C22semidirect
22. #407×C127C22semidirect
23. #344×C127C22semidirect
167. #21778C127direct (N × H)
176. #2888C2×C127semidirect
177. #1914C2×C127semidirect
178. #1891C2×C127semidirect
179. #1829C2×C127semidirect
180. #1783C2×C127semidirect
181. C127#21778direct (N × H)
184. #568C22×C127semidirect
193. #413C22×C127semidirect
196. C2×#100C22×C127semidirect
197. #407C22×C127semidirect
198. #344C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34842
C127×#28882
C127×#19142
C127×#18912
C127×#18292
C127×#17832
#21778127