Finite Group Databasebeta

← C127×#12281 #44575954

Subgroups — C127×#12281 #44575954

Order 32512.

396 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3186C211yesmaximal, characteristic
2C127×#2233C212yesmaximal, complemented
3C127×#2076C211yesmaximal, characteristic, complemented
4C127×#1347C212yesmaximal, complemented
5C127×#1314C211yesmaximal, characteristic, complemented
6C2×#121×C127—21no—
7#480×C127—22no—
8#466×C127—21no—
9#466×C127C2212yes—
10#447×C127—22no—
11C2×#107×C127C2211yescharacteristic, complemented
12C2×C22⋊C8×C127C2211yescharacteristic
13#380×C127—21no—
14#367×C127—22no—
15#357×C127—22no—
16C22⋊C16×C127C2212yescomplemented
17#340×C127—21no—
18#340×C127C2211yescharacteristic, complemented
19C22×D4×C127—21no—
20C22×D4×C127—41no—
21C23×C4×C127D411yescharacteristic
22C2×D8×C127—21no—
23C2×D8×C127D412yes—
24C22×C8×C127D411yescharacteristic
25C22×C8×C127C2311yescharacteristic
26#121×C127—22no—
27#121×C127—22no—
28#121×C127—41no—
29#121×C127—41no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—21no—
33C2×C22⋊C4×C127—41no—
34C2×C22⋊C4×C127—41no—
35C2×C16×C127—22no—
36#107×C127—21no—
37#107×C127—22no—
38#107×C127D412yes—
39#102×C127—22no—
40#102×C127—22no—
41C22⋊C8×C127—22no—
42#95×C127—41no—
43C24×C127—41no—
44C24×C127D811yescharacteristic
45C2×D4×C127—22no—
46C2×D4×C127—41no—
47C2×D4×C127—42no—
48C2×D4×C127—42no—
49C2×D4×C127—81no—
50C2×D4×C127—81no—
51C2×D4×C127—82no—
52C2×D4×C127—82no—
53C22×C4×C127—21no—
54C22×C4×C127—21no—
55C22×C4×C127—41no—
56C22×C4×C127—41no—
57C22×C4×C127—41no—
58C22×C4×C127—41no—
59C22×C4×C127—41no—
60C22×C4×C127D811yescharacteristic
61C22×C4×C127C2×D411yescharacteristic
62D8×C127—42no—
63D8×C127—81no—
64C2×C8×C127—21no—
65C2×C8×C127—41no—
66C2×C8×C127—41no—
67C2×C8×C127C2×D412yes—
68C4⋊C4×C127—22no—
69C4⋊C4×C127—41no—
70C4⋊C4×C127—41no—
71C4⋊C4×C127—41no—
72C4⋊C4×C127—41no—
73C4⋊C4×C127—41no—
74C22⋊C4×C127—42no—
75C22⋊C4×C127—42no—
76C22⋊C4×C127—82no—
77C16×C127—42no—
78C23×C127—21no—
79C23×C127—21no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—42no—
83C23×C127—82no—
84C23×C127—82no—
85C23×C127—82no—
86C23×C127SD3212yes—
87C23×C127C2×D811yescharacteristic
88D4×C127—42no—
89D4×C127—42no—
90D4×C127—82no—
91D4×C127—84no—
92D4×C127—82no—
93D4×C127—82no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—42no—
101C2×C4×C127—81no—
102C2×C4×C127—82no—
103C2×C4×C127—81no—
104C2×C4×C127—81no—
105C2×C4×C127—81no—
106C2×C4×C127—81no—
107C2×C4×C127—81no—
108C2×C4×C127—81no—
109C2×C4×C127#12011yescharacteristic
110C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
111C8×C127—22no—
112C8×C127—41no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—22no—
116C22×C127—42no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127—81no—
120C22×C127—82no—
121C22×C127—82no—
122C22×C127—82no—
123C22×C127—164no—
124C22×C127#44711yescharacteristic
125C22×C127C2×SD3211yescharacteristic
126C22×C127#50911yescharacteristic
127C4×C127—21no—
128C4×C127—41no—
129C4×C127—82no—
130C4×C127—161no—
131C4×C127—161no—
132C4×C127#45511yescharacteristic
133C4×C127#45811yescharacteristic
134#12281C12711yesmaximal, characteristic, direct product factor, Sylow
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—42no—
138C2×C127—162no—
139C2×C127#160711yescharacteristic
140C2×C127#218711yescharacteristic
141C2×C127#219011yescharacteristic
142#3186C2×C12711yescharacteristic
143#2233C2×C12712yescomplemented
144#2076C2×C12711yescharacteristic, complemented
145#1347C2×C12712yescomplemented
146#1314C2×C12711yescharacteristic, complemented
147C127#1228111yescharacteristic, direct product factor, Sylow
148C2×#121—21no—
149#480—22no—
150#466—21no—
151#466C22×C12712yes—
152#447—22no—
153C2×#107C22×C12711yescharacteristic, complemented
154C2×C22⋊C8C22×C12711yescharacteristic
155#380—21no—
156#367—22no—
157#357—22no—
158C22⋊C16C22×C12712yescomplemented
159#340—21no—
160#340C22×C12711yescharacteristic, complemented
161C22×D4—21no—
162C22×D4—41no—
163C23×C4D4×C12711yescharacteristic
164C2×D8—21no—
165C2×D8D4×C12712yes—
166C22×C8D4×C12711yescharacteristic
167C22×C8C23×C12711yescharacteristic
168#121—22no—
169#121—22no—
170#121—41no—
171#121—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C16—22no—
178#107—21no—
179#107—22no—
180#107D4×C12712yes—
181#102—22no—
182#102—22no—
183C22⋊C8—22no—
184#95—41no—
185C24—41no—
186C24D8×C12711yescharacteristic
187C2×D4—22no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—82no—
194C2×D4—82no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4D8×C12711yescharacteristic
203C22×C4C2×D4×C12711yescharacteristic
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×D4×C12712yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—82no—
219C16—42no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23SD32×C12712yes—
229C23C2×D8×C12711yescharacteristic
230D4—42no—
231D4—42no—
232D4—82no—
233D4—84no—
234D4—82no—
235D4—82no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4#120×C12711yescharacteristic
252C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—164no—
266C22#447×C12711yescharacteristic
267C22C2×SD32×C12711yescharacteristic
268C22#509×C12711yescharacteristic
269C4—21no—
270C4—41no—
271C4—82no—
272C4—161no—
273C4—161no—
274C4#455×C12711yescharacteristic
275C4#458×C12711yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—162no—
280C2C127×#160711yescharacteristic
281C2C127×#218711yescharacteristic
282C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3186C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2233C2semidirect
3. C127×#2076C2semidirect
4. C127×#1347C2semidirect
5. C127×#1314C2semidirect
11. C2×#107×C127C22semidirect
16. C22⋊C16×C127C22semidirect
18. #340×C127C22semidirect
134. #12281C127direct (N × H)
143. #2233C2×C127semidirect
144. #2076C2×C127semidirect
145. #1347C2×C127semidirect
146. #1314C2×C127semidirect
147. C127#12281direct (N × H)
153. C2×#107C22×C127semidirect
158. C22⋊C16C22×C127semidirect
160. #340C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31862
C127×#22332
C127×#20762
C127×#13472
C127×#13142
#12281127