Finite Group Databasebeta

← C127×#21781 #44585454

Subgroups — C127×#21781 #44585454

Order 32512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3487C211yesmaximal, characteristic
2C127×#2889C212yesmaximal, complemented
3C127×#1914C214yesmaximal, complemented
4C127×#1891C214yesmaximal, complemented
5C127×#1830C212yesmaximal, complemented
6C127×#1785C212yesmaximal, complemented
7C2×2−1+4×C127C2211yescharacteristic
8C22×#41×C127C2211yescharacteristic
9#568×C127C2214yes—
10#550×C127—24no—
11#548×C127—24no—
12#535×C127C414yes—
13#535×C127C2212yes—
14#525×C127C414yes—
15#525×C127C2212yes—
16#519×C127C2211yescharacteristic
17#413×C127—24no—
18#413×C127C2214yes—
19C2×#101×C127—24no—
20C2×#101×C127C2214yes—
21C2×#100×C127C2214yes—
22#407×C127C2214yes—
23#344×C127C2218yes—
242−1+4×C127—44no—
25C2×#41×C127—24no—
26C2×#41×C127—44no—
27C2×#41×C127C2×C414yes—
28C2×#41×C127D414yes—
29C2×#41×C127C2312yes—
30C22×Q8×C127D414yes—
31C22×Q8×C127C2311yescharacteristic
32C22×D4×C127C2×C412yes—
33C22×D4×C127C2311yescharacteristic
34C23×C4×C127C2×C412yes—
35C23×C4×C127C2311yescharacteristic
36#131×C127—48no—
37C2×M16×C127—28no—
38C2×M16×C127C2318yes—
39#127×C127—24no—
40#127×C127—22no—
41#124×C127—24no—
42#124×C127—22no—
43#123×C127C2×C414yes—
44C22⋊Q8×C127—24no—
45C22⋊Q8×C127—22no—
46#121×C127—24no—
47#121×C127—22no—
48#121×C127—28no—
49#121×C127—24no—
50#120×C127C2×C418yes—
51#120×C127C2×C414yes—
52C4×Q8×C127—24no—
53C4×D4×C127—28no—
54C4×D4×C127—24no—
55#117×C127C2×C414yes—
56#117×C127C2312yes—
57#101×C127—24no—
58#101×C127—28no—
59#100×C127—24no—
60#100×C127—44no—
61C22⋊C8×C127—44no—
62C24×C127C22×C412yes—
63C24×C127C22×C411yescharacteristic
64#41×C127—28no—
65#41×C127—48no—
66#41×C127—48no—
67#41×C127—416no—
68C2×Q8×C127—24no—
69C2×Q8×C127—48no—
70C2×Q8×C127C22⋊C414yes—
71C2×Q8×C127C2×D414yes—
72C2×Q8×C127C2×D412yes—
73C2×D4×C127—28no—
74C2×D4×C127—24no—
75C2×D4×C127—24no—
76C2×D4×C127—22no—
77C2×D4×C127—24no—
78C2×D4×C127—44no—
79C2×D4×C127C22⋊C414yes—
80C2×D4×C127C22×C414yes—
81C2×D4×C127C2×D412yes—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—24no—
85C22×C4×C127—44no—
86C22×C4×C127C22⋊C418yes—
87C22×C4×C127C22×C411yescharacteristic
88C22×C4×C127C22×C412yes—
89C22×C4×C127C2×D414yes—
90C22×C4×C127C2411yescharacteristic
91M16×C127—28no—
92M16×C127—28no—
93M16×C127—48no—
94C2×C8×C127—28no—
95C2×C8×C127—48no—
96C4⋊C4×C127—24no—
97C4⋊C4×C127—24no—
98C4⋊C4×C127—22no—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—28no—
101C22⋊C4×C127—24no—
102C22⋊C4×C127C22×C414yes—
103C42×C127—24no—
104C42×C127—22no—
105C23×C127—22no—
106C23×C127—21no—
107C23×C127—22no—
108C23×C127—24no—
109C23×C127—81no—
110C23×C127—82no—
111C23×C127C2×C22⋊C414yes—
112C23×C127C23×C411yescharacteristic
113Q8×C127—28no—
114Q8×C127—24no—
115Q8×C127—44no—
116D4×C127—28no—
117D4×C127—24no—
118D4×C127—44no—
119D4×C127—48no—
120D4×C127—44no—
121D4×C127—84no—
122D4×C127—88no—
123C2×C4×C127—24no—
124C2×C4×C127—28no—
125C2×C4×C127—24no—
126C2×C4×C127—21no—
127C2×C4×C127—22no—
128C2×C4×C127—48no—
129C2×C4×C127—48no—
130C2×C4×C127—44no—
131C2×C4×C127C2×C22⋊C414yes—
132C2×C4×C127C2×C22⋊C414yes—
133C2×C4×C127C22×D412yes—
134C2×C4×C127H(5,𝔽2)14yes—
135C8×C127—48no—
136C22×C127—24no—
137C22×C127—41no—
138C22×C127—44no—
139C22×C127—42no—
140C22×C127—44no—
141C22×C127—44no—
142C22×C127—41no—
143C22×C127—42no—
144C22×C127—84no—
145C22×C127C22×C22⋊C411yescharacteristic
146C22×C127#51812yes—
147C4×C127—24no—
148C4×C127—24no—
149C4×C127—44no—
150C4×C127—41no—
151C4×C127#53414yes—
152#21781C12711yesmaximal, characteristic, direct product factor, Sylow
153C2×C127—21no—
154C2×C127—22no—
155C2×C127—44no—
156C2×C127—81no—
157C2×C127—82no—
158C2×C127#230811yescharacteristic
159#3487C2×C12711yescharacteristic
160#2889C2×C12712yescomplemented
161#1914C2×C12714yescomplemented
162#1891C2×C12714yescomplemented
163#1830C2×C12712yescomplemented
164#1785C2×C12712yescomplemented
165C127#2178111yescharacteristic, direct product factor, Sylow
166C2×2−1+4C22×C12711yescharacteristic
167C22×#41C22×C12711yescharacteristic
168#568C22×C12714yes—
169#550—24no—
170#548—24no—
171#535C4×C12714yes—
172#535C22×C12712yes—
173#525C4×C12714yes—
174#525C22×C12712yes—
175#519C22×C12711yescharacteristic
176#413—24no—
177#413C22×C12714yes—
178C2×#101—24no—
179C2×#101C22×C12714yes—
180C2×#100C22×C12714yes—
181#407C22×C12714yes—
182#344C22×C12718yes—
1832−1+4—44no—
184C2×#41—24no—
185C2×#41—44no—
186C2×#41C2×C4×C12714yes—
187C2×#41D4×C12714yes—
188C2×#41C23×C12712yes—
189C22×Q8D4×C12714yes—
190C22×Q8C23×C12711yescharacteristic
191C22×D4C2×C4×C12712yes—
192C22×D4C23×C12711yescharacteristic
193C23×C4C2×C4×C12712yes—
194C23×C4C23×C12711yescharacteristic
195#131—48no—
196C2×M16—28no—
197C2×M16C23×C12718yes—
198#127—24no—
199#127—22no—
200#124—24no—
201#124—22no—
202#123C2×C4×C12714yes—
203C22⋊Q8—24no—
204C22⋊Q8—22no—
205#121—24no—
206#121—22no—
207#121—28no—
208#121—24no—
209#120C2×C4×C12718yes—
210#120C2×C4×C12714yes—
211C4×Q8—24no—
212C4×D4—28no—
213C4×D4—24no—
214#117C2×C4×C12714yes—
215#117C23×C12712yes—
216#101—24no—
217#101—28no—
218#100—24no—
219#100—44no—
220C22⋊C8—44no—
221C24C22×C4×C12712yes—
222C24C22×C4×C12711yescharacteristic
223#41—28no—
224#41—48no—
225#41—48no—
226#41—416no—
227C2×Q8—24no—
228C2×Q8—48no—
229C2×Q8C22⋊C4×C12714yes—
230C2×Q8C2×D4×C12714yes—
231C2×Q8C2×D4×C12712yes—
232C2×D4—28no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—24no—
237C2×D4—44no—
238C2×D4C22⋊C4×C12714yes—
239C2×D4C22×C4×C12714yes—
240C2×D4C2×D4×C12712yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—44no—
245C22×C4C22⋊C4×C12718yes—
246C22×C4C22×C4×C12711yescharacteristic
247C22×C4C22×C4×C12712yes—
248C22×C4C2×D4×C12714yes—
249C22×C4C24×C12711yescharacteristic
250M16—28no—
251M16—28no—
252M16—48no—
253C2×C8—28no—
254C2×C8—48no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C22⋊C4—24no—
259C22⋊C4—28no—
260C22⋊C4—24no—
261C22⋊C4C22×C4×C12714yes—
262C42—24no—
263C42—22no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—24no—
268C23—81no—
269C23—82no—
270C23C2×C22⋊C4×C12714yes—
271C23C23×C4×C12711yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275D4—28no—
276D4—24no—
277D4—44no—
278D4—48no—
279D4—44no—
280D4—84no—
281D4—88no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4C2×C22⋊C4×C12714yes—
291C2×C4C2×C22⋊C4×C12714yes—
292C2×C4C22×D4×C12712yes—
293C2×C4H(5,𝔽2)×C12714yes—
294C8—48no—
295C22—24no—
296C22—41no—
297C22—44no—
298C22—42no—
299C22—44no—
300C22—44no—
301C22—41no—
302C22—42no—
303C22—84no—
304C22C22×C22⋊C4×C12711yescharacteristic
305C22#518×C12712yes—
306C4—24no—
307C4—24no—
308C4—44no—
309C4—41no—
310C4#534×C12714yes—
311C2—21no—
312C2—22no—
313C2—44no—
314C2—81no—
315C2—82no—
316C2C127×#230811yescharacteristic
subgroupG/H
center, socleC2×C127#2308
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3487C2
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×#2889C2semidirect
3. C127×#1914C2semidirect
4. C127×#1891C2semidirect
5. C127×#1830C2semidirect
6. C127×#1785C2semidirect
152. #21781C127direct (N × H)
160. #2889C2×C127semidirect
161. #1914C2×C127semidirect
162. #1891C2×C127semidirect
163. #1830C2×C127semidirect
164. #1785C2×C127semidirect
165. C127#21781direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34872
C127×#28892
C127×#19142
C127×#18912
C127×#18302
C127×#17852
#21781127