Finite Group Databasebeta

← C127×#11888 #44575561

Subgroups — C127×#11888 #44575561

Order 32512.

464 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×#466×C127C211yesmaximal, characteristic
2C127×#2112C211yesmaximal, characteristic, complemented
3C127×#1807C211yesmaximal, characteristic, complemented
4C127×#1347C214yesmaximal, complemented
5C22×D8×C127C411yescharacteristic, complemented
6C23×C8×C127C2211yescharacteristic
7C2×#121×C127—21no—
8C2×M32×C127—21no—
9#466×C127—21no—
10#466×C127—21no—
11#466×C127C412yescomplemented
12#466×C127C2212yes—
13C2×#107×C127C411yescharacteristic, complemented
14C2×#106×C127—21no—
15C2×#102×C127—21no—
16#380×C127—21no—
17C22⋊C16×C127—21no—
18C22⋊C16×C127C2212yescomplemented
19#340×C127C2212yescomplemented
20C22×D4×C127—21no—
21C22×D4×C127—41no—
22C23×C4×C127D411yescharacteristic
23C2×D8×C127—21no—
24C2×D8×C127—42no—
25C2×D8×C127C2×C412yes—
26C22×C8×C127—42no—
27C22×C8×C127C2×C411yescharacteristic
28C22×C8×C127C2×C411yescharacteristic
29C22×C8×C127D411yescharacteristic
30C22×C8×C127D411yescharacteristic
31C22×C8×C127D411yescharacteristic
32C22×C8×C127C2311yescharacteristic
33#121×C127—22no—
34#121×C127—22no—
35#121×C127—41no—
36#121×C127—41no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—22no—
39C2×C22⋊C4×C127—41no—
40C2×C22⋊C4×C127—41no—
41M32×C127—42no—
42C2×C16×C127—22no—
43#107×C127—21no—
44#107×C127C2×C412yes—
45#106×C127—42no—
46#102×C127—22no—
47#102×C127—22no—
48#95×C127—41no—
49C24×C127—41no—
50C24×C127D811yescharacteristic
51C2×D4×C127—22no—
52C2×D4×C127—42no—
53C2×D4×C127—42no—
54C2×D4×C127—42no—
55C2×D4×C127—42no—
56C2×D4×C127—41no—
57C2×D4×C127—84no—
58C22×C4×C127—41no—
59C22×C4×C127—41no—
60C22×C4×C127—41no—
61C22×C4×C127—42no—
62C22×C4×C127C22⋊C411yescharacteristic
63C22×C4×C127C22⋊C411yescharacteristic
64C22×C4×C127D811yescharacteristic
65C22×C4×C127SD1611yescharacteristic
66C22×C4×C127SD1611yescharacteristic
67C22×C4×C127C2×D411yescharacteristic
68D8×C127—42no—
69D8×C127—42no—
70C2×C8×C127—22no—
71C2×C8×C127—22no—
72C2×C8×C127—22no—
73C2×C8×C127—41no—
74C2×C8×C127—41no—
75C2×C8×C127—41no—
76C2×C8×C127C22⋊C411yescharacteristic
77C2×C8×C127C22⋊C411yescharacteristic
78C2×C8×C127C22×C411yescharacteristic
79C2×C8×C127C2×D411yescharacteristic
80C4⋊C4×C127—22no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—41no—
84C22⋊C4×C127—42no—
85C22⋊C4×C127—42no—
86C22⋊C4×C127—82no—
87C16×C127—42no—
88C23×C127—22no—
89C23×C127—22no—
90C23×C127—41no—
91C23×C127—42no—
92C23×C127—82no—
93C23×C127—82no—
94C23×C127—82no—
95C23×C127#10211yescharacteristic
96C23×C127#10211yescharacteristic
97C23×C127C2×D811yescharacteristic
98D4×C127—42no—
99D4×C127—42no—
100D4×C127—82no—
101D4×C127—84no—
102D4×C127—84no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—42no—
113C2×C4×C127—82no—
114C2×C4×C127—82no—
115C2×C4×C127—82no—
116C2×C4×C127—82no—
117C2×C4×C127#10211yescharacteristic
118C2×C4×C127#10211yescharacteristic
119C2×C4×C127C2×C22⋊C411yescharacteristic
120C2×C4×C127C2×SD1611yescharacteristic
121C8×C127—21no—
122C8×C127—21no—
123C8×C127—21no—
124C8×C127#9912yes—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—21no—
130C22×C127—21no—
131C22×C127—21no—
132C22×C127—44no—
133C22×C127—82no—
134C22×C127—82no—
135C22×C127—82no—
136C22×C127—164no—
137C22×C127#36012yes—
138C22×C127C2×#10211yescharacteristic
139C22×C127#50912yes—
140C4×C127—21no—
141C4×C127—21no—
142C4×C127—21no—
143C4×C127—82no—
144C4×C127—162no—
145C4×C127C2×#9911yescharacteristic
146C4×C127#42111yescharacteristic
147#11888C12711yesmaximal, characteristic, direct product factor, Sylow
148C2×C127—22no—
149C2×C127—22no—
150C2×C127—41no—
151C2×C127—162no—
152C2×C127#151911yescharacteristic
153C2×C127#214011yescharacteristic
154C2×C127C2×#36011yescharacteristic
155C2×#466C2×C12711yescharacteristic
156#2112C2×C12711yescharacteristic, complemented
157#1807C2×C12711yescharacteristic, complemented
158#1347C2×C12714yescomplemented
159C127#1188811yescharacteristic, direct product factor, Sylow
160C22×D8C4×C12711yescharacteristic, complemented
161C23×C8C22×C12711yescharacteristic
162C2×#121—21no—
163C2×M32—21no—
164#466—21no—
165#466—21no—
166#466C4×C12712yescomplemented
167#466C22×C12712yes—
168C2×#107C4×C12711yescharacteristic, complemented
169C2×#106—21no—
170C2×#102—21no—
171#380—21no—
172C22⋊C16—21no—
173C22⋊C16C22×C12712yescomplemented
174#340C22×C12712yescomplemented
175C22×D4—21no—
176C22×D4—41no—
177C23×C4D4×C12711yescharacteristic
178C2×D8—21no—
179C2×D8—42no—
180C2×D8C2×C4×C12712yes—
181C22×C8—42no—
182C22×C8C2×C4×C12711yescharacteristic
183C22×C8C2×C4×C12711yescharacteristic
184C22×C8D4×C12711yescharacteristic
185C22×C8D4×C12711yescharacteristic
186C22×C8D4×C12711yescharacteristic
187C22×C8C23×C12711yescharacteristic
188#121—22no—
189#121—22no—
190#121—41no—
191#121—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196M32—42no—
197C2×C16—22no—
198#107—21no—
199#107C2×C4×C12712yes—
200#106—42no—
201#102—22no—
202#102—22no—
203#95—41no—
204C24—41no—
205C24D8×C12711yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—84no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊C4×C12711yescharacteristic
218C22×C4C22⋊C4×C12711yescharacteristic
219C22×C4D8×C12711yescharacteristic
220C22×C4SD16×C12711yescharacteristic
221C22×C4SD16×C12711yescharacteristic
222C22×C4C2×D4×C12711yescharacteristic
223D8—42no—
224D8—42no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8C22⋊C4×C12711yescharacteristic
232C2×C8C22⋊C4×C12711yescharacteristic
233C2×C8C22×C4×C12711yescharacteristic
234C2×C8C2×D4×C12711yescharacteristic
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—82no—
242C16—42no—
243C23—22no—
244C23—22no—
245C23—41no—
246C23—42no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23#102×C12711yescharacteristic
251C23#102×C12711yescharacteristic
252C23C2×D8×C12711yescharacteristic
253D4—42no—
254D4—42no—
255D4—82no—
256D4—84no—
257D4—84no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4#102×C12711yescharacteristic
273C2×C4#102×C12711yescharacteristic
274C2×C4C2×C22⋊C4×C12711yescharacteristic
275C2×C4C2×SD16×C12711yescharacteristic
276C8—21no—
277C8—21no—
278C8—21no—
279C8#99×C12712yes—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—44no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—164no—
292C22#360×C12712yes—
293C22C2×#102×C12711yescharacteristic
294C22#509×C12712yes—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—162no—
300C4C2×#99×C12711yescharacteristic
301C4#421×C12711yescharacteristic
302C2—22no—
303C2—22no—
304C2—41no—
305C2—162no—
306C2C127×#151911yescharacteristic
307C2C127×#214011yescharacteristic
308C2C2×#360×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC2×#466×C127C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2112C2semidirect
3. C127×#1807C2semidirect
4. C127×#1347C2semidirect
5. C22×D8×C127C4semidirect
11. #466×C127C4semidirect
13. C2×#107×C127C4semidirect
18. C22⋊C16×C127C22semidirect
19. #340×C127C22semidirect
147. #11888C127direct (N × H)
156. #2112C2×C127semidirect
157. #1807C2×C127semidirect
158. #1347C2×C127semidirect
159. C127#11888direct (N × H)
160. C22×D8C4×C127semidirect
166. #466C4×C127semidirect
168. C2×#107C4×C127semidirect
173. C22⋊C16C22×C127semidirect
174. #340C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#466×C1272
C127×#21122
C127×#18072
C127×#13472
#11888127