Finite Group Databasebeta

← C127×#12628 #44576301

Subgroups — C127×#12628 #44576301

Order 32512.

628 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 162 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×#2328C211yesmaximal, characteristic
2C2×#353×C127C212yesmaximal
3C4×#99×C127C212yesmaximal, complemented
4C127×#1393C212yesmaximal, complemented
5C2×#127×C127C2211yescharacteristic
6C2×C4×D4×C127C411yescharacteristic, complemented
7C2×C4×D4×C127C2211yescharacteristic
8C2×#99×C127C2214yes—
9#390×C127C411yescharacteristic, complemented
10#390×C127C2211yescharacteristic
11#386×C127—22no—
12#386×C127—22no—
13#382×C127C411yescharacteristic, complemented
14C4×C22⋊C4×C127—22no—
15#376×C127C411yescharacteristic, complemented
16#353×C127C418yescomplemented
17#342×C127—22no—
18C22×D4×C127—22no—
19C22×D4×C127C2×C411yescharacteristic
20C22×D4×C127C2311yescharacteristic
21C23×C4×C127—21no—
22C23×C4×C127D412yes—
23#127×C127—22no—
24#127×C127C2×C414yes—
25C4×D4×C127—42no—
26C4×D4×C127—42no—
27C2×C4⋊C4×C127C2×C411yescharacteristic
28C2×C4⋊C4×C127C2×C411yescharacteristic
29C2×C22⋊C4×C127—24no—
30C2×C22⋊C4×C127—21no—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127—22no—
33C2×C22⋊C4×C127D412yes—
34C2×C42×C127—42no—
35C2×C42×C127C2×C411yescharacteristic
36C2×C42×C127C2×C411yescharacteristic
37C2×C42×C127C2×C411yescharacteristic
38#99×C127—24no—
39#99×C127C2×C418yescomplemented
40#95×C127—21no—
41#95×C127—22no—
42#95×C127—21no—
43#95×C127—42no—
44C24×C127—21no—
45C24×C127C2×D412yes—
46C2×D4×C127—28no—
47C2×D4×C127—21no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51C2×D4×C127—41no—
52C2×D4×C127—82no—
53C2×D4×C127—82no—
54C2×D4×C127C4212yescomplemented
55C2×D4×C127C22×C412yes—
56C2×D4×C127#4112yes—
57C22×C4×C127—21no—
58C22×C4×C127—22no—
59C22×C4×C127—22no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—42no—
63C22×C4×C127—44no—
64C22×C4×C127—82no—
65C22×C4×C127—82no—
66C22×C4×C127C22⋊C411yescharacteristic
67C22×C4×C127C22⋊C412yes—
68C22×C4×C127C22⋊C411yescharacteristic
69C22×C4×C127C22×C411yescharacteristic
70C4⋊C4×C127—41no—
71C4⋊C4×C127—41no—
72C22⋊C4×C127—28no—
73C22⋊C4×C127—42no—
74C22⋊C4×C127—44no—
75C22⋊C4×C127—42no—
76C22⋊C4×C127—82no—
77C22⋊C4×C127—82no—
78C42×C127—22no—
79C42×C127—21no—
80C42×C127—41no—
81C42×C127—84no—
82C42×C127C4212yescomplemented
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—22no—
86C23×C127—42no—
87C23×C127—42no—
88C23×C127—42no—
89C23×C127—42no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127C2×C22⋊C411yescharacteristic
93C23×C127C4×D414yes—
94D4×C127—42no—
95D4×C127—42no—
96D4×C127—42no—
97D4×C127—42no—
98D4×C127—88no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—22no—
102C2×C4×C127—21no—
103C2×C4×C127—22no—
104C2×C4×C127—41no—
105C2×C4×C127—44no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—42no—
109C2×C4×C127—41no—
110C2×C4×C127—48no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—42no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—82no—
117C2×C4×C127—84no—
118C2×C4×C127—84no—
119C2×C4×C127—84no—
120C2×C4×C127#9912yes—
121C2×C4×C127C2×C4211yescharacteristic
122C2×C4×C127#11711yescharacteristic
123C22×C127—21no—
124C22×C127—21no—
125C22×C127—24no—
126C22×C127—44no—
127C22×C127—44no—
128C22×C127—42no—
129C22×C127—42no—
130C22×C127—42no—
131C22×C127—82no—
132C22×C127—82no—
133C22×C127—81no—
134C22×C127—81no—
135C22×C127C4×C22⋊C411yescharacteristic
136C22×C127C2×#9911yescharacteristic
137C22×C127#41011yescharacteristic
138C4×C127—22no—
139C4×C127—21no—
140C4×C127—21no—
141C4×C127—42no—
142C4×C127—42no—
143C4×C127—41no—
144C4×C127—81no—
145C4×C127—88no—
146C4×C127—81no—
147#12628C12711yesmaximal, characteristic, direct product factor, Sylow
148C2×C127—21no—
149C2×C127—21no—
150C2×C127—44no—
151C2×C127—82no—
152C2×C127C4×#9911yescharacteristic
153C2×C127#212711yescharacteristic
154C2×C127#212811yescharacteristic
155#2328C2×C12711yescharacteristic
156C2×#353C2×C12712yes—
157C4×#99C2×C12712yescomplemented
158#1393C2×C12712yescomplemented
159C127#1262811yescharacteristic, direct product factor, Sylow
160C2×#127C22×C12711yescharacteristic
161C2×C4×D4C4×C12711yescharacteristic, complemented
162C2×C4×D4C22×C12711yescharacteristic
163C2×#99C22×C12714yes—
164#390C4×C12711yescharacteristic, complemented
165#390C22×C12711yescharacteristic
166#386—22no—
167#386—22no—
168#382C4×C12711yescharacteristic, complemented
169C4×C22⋊C4—22no—
170#376C4×C12711yescharacteristic, complemented
171#353C4×C12718yescomplemented
172#342—22no—
173C22×D4—22no—
174C22×D4C2×C4×C12711yescharacteristic
175C22×D4C23×C12711yescharacteristic
176C23×C4—21no—
177C23×C4D4×C12712yes—
178#127—22no—
179#127C2×C4×C12714yes—
180C4×D4—42no—
181C4×D4—42no—
182C2×C4⋊C4C2×C4×C12711yescharacteristic
183C2×C4⋊C4C2×C4×C12711yescharacteristic
184C2×C22⋊C4—24no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4D4×C12712yes—
189C2×C42—42no—
190C2×C42C2×C4×C12711yescharacteristic
191C2×C42C2×C4×C12711yescharacteristic
192C2×C42C2×C4×C12711yescharacteristic
193#99—24no—
194#99C2×C4×C12718yescomplemented
195#95—21no—
196#95—21no—
197#95—22no—
198#95—42no—
199C24—21no—
200C24C2×D4×C12712yes—
201C2×D4—28no—
202C2×D4—21no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—82no—
208C2×D4—82no—
209C2×D4C42×C12712yescomplemented
210C2×D4C22×C4×C12712yes—
211C2×D4#41×C12712yes—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4C22⋊C4×C12711yescharacteristic
222C22×C4C22⋊C4×C12712yes—
223C22×C4C22⋊C4×C12711yescharacteristic
224C22×C4C22×C4×C12711yescharacteristic
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—28no—
228C22⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C42—22no—
234C42—21no—
235C42—41no—
236C42—84no—
237C42C42×C12712yescomplemented
238C23—22no—
239C23—22no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊C4×C12711yescharacteristic
248C23C4×D4×C12714yes—
249D4—42no—
250D4—42no—
251D4—42no—
252D4—42no—
253D4—88no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—44no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—48no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—82no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4#99×C12712yes—
276C2×C4C2×C42×C12711yescharacteristic
277C2×C4#117×C12711yescharacteristic
278C22—21no—
279C22—21no—
280C22—24no—
281C22—44no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—82no—
287C22—82no—
288C22—81no—
289C22—81no—
290C22C4×C22⋊C4×C12711yescharacteristic
291C22C2×#99×C12711yescharacteristic
292C22#410×C12711yescharacteristic
293C4—22no—
294C4—21no—
295C4—21no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—81no—
300C4—88no—
301C4—81no—
302C2—21no—
303C2—21no—
304C2—44no—
305C2—82no—
306C2C4×#99×C12711yescharacteristic
307C2C127×#212711yescharacteristic
308C2C127×#212811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC2×C4C2×C42×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC2×#127×C127C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#99×C127C2semidirect
4. C127×#1393C2semidirect
6. C2×C4×D4×C127C4semidirect
9. #390×C127C4semidirect
13. #382×C127C4semidirect
15. #376×C127C4semidirect
16. #353×C127C4semidirect
39. #99×C127C2×C4semidirect
54. C2×D4×C127C42semidirect
82. C42×C127C42semidirect
147. #12628C127direct (N × H)
157. C4×#99C2×C127semidirect
158. #1393C2×C127semidirect
159. C127#12628direct (N × H)
161. C2×C4×D4C4×C127semidirect
164. #390C4×C127semidirect
168. #382C4×C127semidirect
170. #376C4×C127semidirect
171. #353C4×C127semidirect
194. #99C2×C4×C127semidirect
209. C2×D4C42×C127semidirect
237. C42C42×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23282
C2×#353×C1272
C4×#99×C1272
C127×#13932
#12628127