Finite Group Databasebeta

← C127×#8464 #44572137

Subgroups — C127×#8464 #44572137

Order 32512.

1002 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3487C211yesmaximal, characteristic
2C127×#1782C211yesmaximal, characteristic, complemented
3C127×#1780C211yesmaximal, characteristic, complemented
4C127×#1516C214yesmaximal, complemented
5C2×2−1+4×C127C411yescharacteristic, complemented
6C22×#41×C127C2211yescharacteristic
7#550×C127—24no—
8#548×C127—24no—
9#535×C127—22no—
10#535×C127C412yescomplemented
11#525×C127—22no—
12#525×C127C2212yes—
13#519×C127C411yescharacteristic, complemented
14C2×#99×C127—22no—
15C2×C22⋊C8×C127—22no—
16C2×C22⋊C8×C127C2212yes—
17#381×C127C2212yes—
18#328×C127—24no—
19#327×C127—24no—
202−1+4×C127—44no—
21C2×#41×C127—22no—
22C2×#41×C127—22no—
23C2×#41×C127—24no—
24C2×#41×C127—44no—
25C2×#41×C127C2×C412yes—
26C22×Q8×C127—22no—
27C22×Q8×C127C2×C411yescharacteristic
28C22×D4×C127C2×C411yescharacteristic
29C22×D4×C127D412yes—
30C23×C4×C127D412yes—
31C23×C4×C127C2311yescharacteristic
32C22×C8×C127—22no—
33#127×C127—24no—
34#127×C127—22no—
35#124×C127—24no—
36#124×C127—22no—
37#123×C127—22no—
38C22⋊Q8×C127—24no—
39C22⋊Q8×C127D414yes—
40#121×C127—24no—
41#121×C127—28no—
42#121×C127—24no—
43#121×C127D414yes—
44#120×C127—24no—
45#120×C127—22no—
46C4×Q8×C127—24no—
47C4×D4×C127—28no—
48C4×D4×C127—24no—
49#117×C127—22no—
50#117×C127C2×C412yes—
51C2×C22⋊C4×C127—22no—
52#99×C127—44no—
53C22⋊C8×C127—24no—
54C22⋊C8×C127—44no—
55#95×C127—24no—
56C24×C127C22⋊C412yes—
57C24×C127C2×D411yescharacteristic
58#41×C127—44no—
59#41×C127—44no—
60#41×C127—416no—
61#41×C127—48no—
62#41×C127—44no—
63C2×Q8×C127—22no—
64C2×Q8×C127—24no—
65C2×Q8×C127—22no—
66C2×Q8×C127—48no—
67C2×Q8×C127C22⋊C412yes—
68C2×D4×C127—22no—
69C2×D4×C127—22no—
70C2×D4×C127—24no—
71C2×D4×C127—28no—
72C2×D4×C127—24no—
73C2×D4×C127—24no—
74C2×D4×C127—22no—
75C2×D4×C127—42no—
76C2×D4×C127—42no—
77C2×D4×C127C22⋊C412yes—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—22no—
81C22×C4×C127—24no—
82C22×C4×C127—42no—
83C22×C4×C127—42no—
84C22×C4×C127—42no—
85C22×C4×C127C22⋊C412yes—
86C22×C4×C127C22×C411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D414yes—
89C2×C8×C127—44no—
90C2×C8×C127—42no—
91C4⋊C4×C127—24no—
92C4⋊C4×C127—24no—
93C4⋊C4×C127C22⋊C414yes—
94C22⋊C4×C127—28no—
95C22⋊C4×C127—22no—
96C22⋊C4×C127—24no—
97C22⋊C4×C127—24no—
98C22⋊C4×C127—44no—
99C42×C127—24no—
100C42×C127—22no—
101C23×C127—22no—
102C23×C127—22no—
103C23×C127—22no—
104C23×C127—22no—
105C23×C127—22no—
106C23×C127—42no—
107C23×C127—44no—
108C23×C127C2×C22⋊C411yescharacteristic
109C23×C127#12012yes—
110Q8×C127—28no—
111Q8×C127—42no—
112Q8×C127—44no—
113D4×C127—48no—
114D4×C127—42no—
115D4×C127—44no—
116D4×C127—44no—
117D4×C127—44no—
118D4×C127—84no—
119D4×C127—88no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—24no—
125C2×C4×C127—22no—
126C2×C4×C127—44no—
127C2×C4×C127—48no—
128C2×C4×C127—48no—
129C2×C4×C127—44no—
130C2×C4×C127—82no—
131C2×C4×C127—84no—
132C2×C4×C127#9914yes—
133C2×C4×C127C2×C22⋊C412yes—
134C2×C4×C127#12012yes—
135C8×C127—82no—
136C22×C127—22no—
137C22×C127—24no—
138C22×C127—22no—
139C22×C127—44no—
140C22×C127—44no—
141C22×C127—44no—
142C22×C127—42no—
143C22×C127—42no—
144C22×C127—82no—
145C22×C127—82no—
146C22×C127#37911yescharacteristic
147C22×C127C2×#9912yes—
148C4×C127—22no—
149C4×C127—24no—
150C4×C127—22no—
151C4×C127—44no—
152C4×C127—42no—
153C4×C127—162no—
154#8464C12711yesmaximal, characteristic, direct product factor, Sylow
155C2×C127—22no—
156C2×C127—42no—
157C2×C127—82no—
158C2×C127—82no—
159C2×C127#178111yescharacteristic
160C2×C127#179412yes—
161#3487C2×C12711yescharacteristic
162#1782C2×C12711yescharacteristic, complemented
163#1780C2×C12711yescharacteristic, complemented
164#1516C2×C12714yescomplemented
165C127#846411yescharacteristic, direct product factor, Sylow
166C2×2−1+4C4×C12711yescharacteristic, complemented
167C22×#41C22×C12711yescharacteristic
168#550—24no—
169#548—24no—
170#535—22no—
171#535C4×C12712yescomplemented
172#525—22no—
173#525C22×C12712yes—
174#519C4×C12711yescharacteristic, complemented
175C2×#99—22no—
176C2×C22⋊C8—22no—
177C2×C22⋊C8C22×C12712yes—
178#381C22×C12712yes—
179#328—24no—
180#327—24no—
1812−1+4—44no—
182C2×#41—22no—
183C2×#41—22no—
184C2×#41—24no—
185C2×#41—44no—
186C2×#41C2×C4×C12712yes—
187C22×Q8—22no—
188C22×Q8C2×C4×C12711yescharacteristic
189C22×D4C2×C4×C12711yescharacteristic
190C22×D4D4×C12712yes—
191C23×C4D4×C12712yes—
192C23×C4C23×C12711yescharacteristic
193C22×C8—22no—
194#127—24no—
195#127—22no—
196#124—24no—
197#124—22no—
198#123—22no—
199C22⋊Q8—24no—
200C22⋊Q8D4×C12714yes—
201#121—24no—
202#121—28no—
203#121—24no—
204#121D4×C12714yes—
205#120—24no—
206#120—22no—
207C4×Q8—24no—
208C4×D4—28no—
209C4×D4—24no—
210#117—22no—
211#117C2×C4×C12712yes—
212C2×C22⋊C4—22no—
213#99—44no—
214C22⋊C8—24no—
215C22⋊C8—44no—
216#95—24no—
217C24C22⋊C4×C12712yes—
218C24C2×D4×C12711yescharacteristic
219#41—44no—
220#41—44no—
221#41—416no—
222#41—48no—
223#41—44no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—22no—
227C2×Q8—48no—
228C2×Q8C22⋊C4×C12712yes—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—28no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4C22⋊C4×C12712yes—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4C22⋊C4×C12712yes—
247C22×C4C22×C4×C12711yescharacteristic
248C22×C4C2×D4×C12711yescharacteristic
249C22×C4C2×D4×C12714yes—
250C2×C8—44no—
251C2×C8—42no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4C22⋊C4×C12714yes—
255C22⋊C4—28no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—44no—
260C42—24no—
261C42—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23C2×C22⋊C4×C12711yescharacteristic
270C23#120×C12712yes—
271Q8—28no—
272Q8—42no—
273Q8—44no—
274D4—48no—
275D4—42no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—84no—
280D4—88no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4#99×C12714yes—
294C2×C4C2×C22⋊C4×C12712yes—
295C2×C4#120×C12712yes—
296C8—82no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—44no—
304C22—44no—
305C22—82no—
306C22—82no—
307C22#379×C12711yescharacteristic
308C22C2×#99×C12712yes—
309C4—22no—
310C4—24no—
311C4—22no—
312C4—42no—
313C4—44no—
314C4—162no—
315C2—22no—
316C2—42no—
317C2—82no—
318C2—82no—
319C2C127×#178111yescharacteristic
320C2C127×#179412yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3487C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1782C2semidirect
3. C127×#1780C2semidirect
4. C127×#1516C2semidirect
5. C2×2−1+4×C127C4semidirect
10. #535×C127C4semidirect
13. #519×C127C4semidirect
154. #8464C127direct (N × H)
162. #1782C2×C127semidirect
163. #1780C2×C127semidirect
164. #1516C2×C127semidirect
165. C127#8464direct (N × H)
166. C2×2−1+4C4×C127semidirect
171. #535C4×C127semidirect
174. #519C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34872
C127×#17822
C127×#17802
C127×#15162
#8464127