Finite Group Databasebeta

← C127×#21779 #44585452

Subgroups — C127×#21779 #44585452

Order 32512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3485C211yesmaximal, characteristic
2C127×#2888C212yesmaximal, complemented
3C127×#1915C214yesmaximal, complemented
4C127×#1891C214yesmaximal, complemented
5C127×#1829C212yesmaximal, complemented
6C127×#1785C212yesmaximal, complemented
7C2×H(5,𝔽2)×C127C2211yescharacteristic
8C22×#41×C127C2211yescharacteristic
9#568×C127C2214yes—
10#547×C127—24no—
11#546×C127—24no—
12#536×C127C2212yes—
13#534×C127C414yes—
14#525×C127C2212yes—
15C2×#123×C127C414yes—
16#518×C127C2211yescharacteristic
17#413×C127—24no—
18#413×C127C2214yes—
19C2×#101×C127C2214yescomplemented
20C2×#100×C127—24no—
21C2×#100×C127C2214yes—
22#407×C127C2214yescomplemented
23#344×C127C2218yescomplemented
24H(5,𝔽2)×C127—44no—
25C2×#41×C127—24no—
26C2×#41×C127—44no—
27C2×#41×C127D414yes—
28C2×#41×C127C2312yes—
29C22×Q8×C127C2311yescharacteristic
30C22×D4×C127C2×C414yes—
31C22×D4×C127C2×C412yes—
32C22×D4×C127D414yes—
33C22×D4×C127C2311yescharacteristic
34C23×C4×C127C2×C412yes—
35C23×C4×C127C2311yescharacteristic
36#131×C127—48no—
37C2×M16×C127—28no—
38C2×M16×C127C2318yes—
39C4⋊Q8×C127—22no—
40#124×C127—22no—
41#123×C127—28no—
42#123×C127—24no—
43#123×C127C2×C414yes—
44#123×C127C2×C418yes—
45C22⋊Q8×C127—22no—
46C22⋊Q8×C127—24no—
47#121×C127—28no—
48#121×C127—22no—
49#120×C127—24no—
50#120×C127C2×C414yes—
51C4×D4×C127—24no—
52C4×D4×C127—24no—
53#117×C127C2312yes—
54C2×C4⋊C4×C127—24no—
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127C2×C414yes—
57#101×C127—24no—
58#101×C127—44no—
59#100×C127—24no—
60#100×C127—28no—
61C22⋊C8×C127—44no—
62C24×C127C22⋊C414yes—
63C24×C127C22×C411yescharacteristic
64C24×C127C22×C412yes—
65#41×C127—28no—
66#41×C127—48no—
67#41×C127—416no—
68C2×Q8×C127—22no—
69C2×Q8×C127—24no—
70C2×Q8×C127C2×D412yes—
71C2×D4×C127—24no—
72C2×D4×C127—24no—
73C2×D4×C127—24no—
74C2×D4×C127—48no—
75C2×D4×C127—44no—
76C2×D4×C127—48no—
77C2×D4×C127C22⋊C418yes—
78C2×D4×C127C22×C414yes—
79C2×D4×C127C2×D414yes—
80C2×D4×C127C2×D412yes—
81C22×C4×C127—24no—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—24no—
85C22×C4×C127—44no—
86C22×C4×C127C22⋊C414yes—
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—28no—
98C4⋊C4×C127—22no—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—28no—
101C22⋊C4×C127—24no—
102C22⋊C4×C127—24no—
103C22⋊C4×C127C22×C414yes—
104C42×C127—22no—
105C23×C127—24no—
106C23×C127—21no—
107C23×C127—22no—
108C23×C127—44no—
109C23×C127—81no—
110C23×C127—82no—
111C23×C127—84no—
112C23×C127C2×C22⋊C414yes—
113C23×C127C2×C22⋊C414yes—
114C23×C127C23×C411yescharacteristic
115Q8×C127—24no—
116Q8×C127—44no—
117D4×C127—28no—
118D4×C127—28no—
119D4×C127—24no—
120D4×C127—48no—
121D4×C127—48no—
122D4×C127—84no—
123C2×C4×C127—28no—
124C2×C4×C127—24no—
125C2×C4×C127—21no—
126C2×C4×C127—22no—
127C2×C4×C127—22no—
128C2×C4×C127—24no—
129C2×C4×C127—44no—
130C2×C4×C127—48no—
131C2×C4×C127—44no—
132C2×C4×C127—84no—
133C2×C4×C127C2×C22⋊C414yes—
134C2×C4×C127C22×D412yes—
135C2×C4×C127H(5,𝔽2)14yes—
136C8×C127—48no—
137C22×C127—24no—
138C22×C127—24no—
139C22×C127—41no—
140C22×C127—44no—
141C22×C127—44no—
142C22×C127—42no—
143C22×C127—44no—
144C22×C127—41no—
145C22×C127—84no—
146C22×C127—164no—
147C22×C127C22×C22⋊C411yescharacteristic
148C22×C127#51812yes—
149C4×C127—24no—
150C4×C127—44no—
151C4×C127—41no—
152C4×C127—42no—
153C4×C127#53414yes—
154#21779C12711yesmaximal, characteristic, direct product factor, Sylow
155C2×C127—21no—
156C2×C127—22no—
157C2×C127—44no—
158C2×C127—44no—
159C2×C127—81no—
160C2×C127#230811yescharacteristic
161#3485C2×C12711yescharacteristic
162#2888C2×C12712yescomplemented
163#1915C2×C12714yescomplemented
164#1891C2×C12714yescomplemented
165#1829C2×C12712yescomplemented
166#1785C2×C12712yescomplemented
167C127#2177911yescharacteristic, direct product factor, Sylow
168C2×H(5,𝔽2)C22×C12711yescharacteristic
169C22×#41C22×C12711yescharacteristic
170#568C22×C12714yes—
171#547—24no—
172#546—24no—
173#536C22×C12712yes—
174#534C4×C12714yes—
175#525C22×C12712yes—
176C2×#123C4×C12714yes—
177#518C22×C12711yescharacteristic
178#413—24no—
179#413C22×C12714yes—
180C2×#101C22×C12714yescomplemented
181C2×#100—24no—
182C2×#100C22×C12714yes—
183#407C22×C12714yescomplemented
184#344C22×C12718yescomplemented
185H(5,𝔽2)—44no—
186C2×#41—24no—
187C2×#41—44no—
188C2×#41D4×C12714yes—
189C2×#41C23×C12712yes—
190C22×Q8C23×C12711yescharacteristic
191C22×D4C2×C4×C12714yes—
192C22×D4C2×C4×C12712yes—
193C22×D4D4×C12714yes—
194C22×D4C23×C12711yescharacteristic
195C23×C4C2×C4×C12712yes—
196C23×C4C23×C12711yescharacteristic
197#131—48no—
198C2×M16—28no—
199C2×M16C23×C12718yes—
200C4⋊Q8—22no—
201#124—22no—
202#123—28no—
203#123—24no—
204#123C2×C4×C12714yes—
205#123C2×C4×C12718yes—
206C22⋊Q8—22no—
207C22⋊Q8—24no—
208#121—28no—
209#121—22no—
210#120—24no—
211#120C2×C4×C12714yes—
212C4×D4—24no—
213C4×D4—24no—
214#117C23×C12712yes—
215C2×C4⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4C2×C4×C12714yes—
218#101—24no—
219#101—44no—
220#100—24no—
221#100—28no—
222C22⋊C8—44no—
223C24C22⋊C4×C12714yes—
224C24C22×C4×C12711yescharacteristic
225C24C22×C4×C12712yes—
226#41—28no—
227#41—48no—
228#41—416no—
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8C2×D4×C12712yes—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—48no—
236C2×D4—44no—
237C2×D4—48no—
238C2×D4C22⋊C4×C12718yes—
239C2×D4C22×C4×C12714yes—
240C2×D4C2×D4×C12714yes—
241C2×D4C2×D4×C12712yes—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—44no—
247C22×C4C22⋊C4×C12714yes—
248C22×C4C22×C4×C12711yescharacteristic
249C22×C4C22×C4×C12712yes—
250C22×C4C2×D4×C12714yes—
251C22×C4C24×C12711yescharacteristic
252M16—28no—
253M16—28no—
254M16—48no—
255C2×C8—28no—
256C2×C8—48no—
257C4⋊C4—24no—
258C4⋊C4—28no—
259C4⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—28no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4C22×C4×C12714yes—
265C42—22no—
266C23—24no—
267C23—21no—
268C23—22no—
269C23—44no—
270C23—81no—
271C23—82no—
272C23—84no—
273C23C2×C22⋊C4×C12714yes—
274C23C2×C22⋊C4×C12714yes—
275C23C23×C4×C12711yescharacteristic
276Q8—24no—
277Q8—44no—
278D4—28no—
279D4—28no—
280D4—24no—
281D4—48no—
282D4—48no—
283D4—84no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—84no—
294C2×C4C2×C22⋊C4×C12714yes—
295C2×C4C22×D4×C12712yes—
296C2×C4H(5,𝔽2)×C12714yes—
297C8—48no—
298C22—24no—
299C22—24no—
300C22—41no—
301C22—44no—
302C22—44no—
303C22—42no—
304C22—44no—
305C22—41no—
306C22—84no—
307C22—164no—
308C22C22×C22⋊C4×C12711yescharacteristic
309C22#518×C12712yes—
310C4—24no—
311C4—44no—
312C4—41no—
313C4—42no—
314C4#534×C12714yes—
315C2—21no—
316C2—22no—
317C2—44no—
318C2—44no—
319C2—81no—
320C2C127×#230811yescharacteristic
subgroupG/H
center, socleC2×C127#2308
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC127×#3485C2
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×#1915C2semidirect
4. C127×#1891C2semidirect
5. C127×#1829C2semidirect
6. C127×#1785C2semidirect
19. C2×#101×C127C22semidirect
22. #407×C127C22semidirect
23. #344×C127C22semidirect
154. #21779C127direct (N × H)
162. #2888C2×C127semidirect
163. #1915C2×C127semidirect
164. #1891C2×C127semidirect
165. #1829C2×C127semidirect
166. #1785C2×C127semidirect
167. C127#21779direct (N × H)
180. C2×#101C22×C127semidirect
183. #407C22×C127semidirect
184. #344C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34852
C127×#28882
C127×#19152
C127×#18912
C127×#18292
C127×#17852
#21779127