Finite Group Databasebeta

← C127×#12159 #44575832

Subgroups — C127×#12159 #44575832

Order 32512.

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

normal subgroups only

This group has 390 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×#3249C211yesmaximal, characteristic, complemented
2C127×#2187C212yesmaximal
3C127×#2054C211yesmaximal, characteristic
4C127×#1348C212yesmaximal, complemented
5C127×#1322C211yesmaximal, characteristic, complemented
6C22×D8×C127C2211yescharacteristic
7#533×C127—21no—
8C2×SD32×C127—22no—
9#477×C127—22no—
10#467×C127—21no—
11#467×C127C2212yes—
12#462×C127—22no—
13C2×C4⋊C8×C127C2211yescharacteristic
14C2×#102×C127—21no—
15#390×C127—21no—
16#357×C127—22no—
17C22⋊C16×C127C2212yes—
18#340×C127C2211yescharacteristic
19C22×D4×C127—21no—
20C2×Q16×C127—21no—
21C2×Q16×C127D412yes—
22C2×D8×C127—22no—
23C2×D8×C127—42no—
24C22×C8×C127D411yescharacteristic
25C22×C8×C127C2311yescharacteristic
26C4⋊Q8×C127—21no—
27C4⋊Q8×C127—21no—
28#125×C127—41no—
29C22⋊Q8×C127—22no—
30C22⋊Q8×C127—41no—
31C4×Q8×C127—22no—
32C4×Q8×C127—41no—
33#117×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C22⋊C4×C127—41no—
36C2×C22⋊C4×C127—41no—
37C2×C42×C127D411yescharacteristic
38SD32×C127—44no—
39C2×C16×C127—22no—
40#107×C127—21no—
41#107×C127D412yes—
42C4⋊C8×C127—22no—
43#103×C127—22no—
44#103×C127—22no—
45#102×C127—42no—
46C24×C127—41no—
47C2×Q8×C127—22no—
48C2×Q8×C127—41no—
49C2×D4×C127—22no—
50C2×D4×C127—41no—
51C2×D4×C127—41no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127—41no—
55C22×C4×C127—41no—
56C22×C4×C127D811yescharacteristic
57C22×C4×C127D811yescharacteristic
58C22×C4×C127C2×D411yescharacteristic
59Q16×C127—42no—
60Q16×C127—81no—
61D8×C127—24no—
62D8×C127—42no—
63C2×C8×C127—21no—
64C2×C8×C127—41no—
65C2×C8×C127—41no—
66C2×C8×C127C2×D412yes—
67C4⋊C4×C127—22no—
68C4⋊C4×C127—41no—
69C4⋊C4×C127—41no—
70C4⋊C4×C127—42no—
71C4⋊C4×C127—41no—
72C4⋊C4×C127—41no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—42no—
75C22⋊C4×C127—42no—
76C22⋊C4×C127—82no—
77C22⋊C4×C127—82no—
78C42×C127—21no—
79C42×C127—21no—
80C42×C127—42no—
81C16×C127—42no—
82C23×C127—42no—
83C23×C127—81no—
84C23×C127—81no—
85C23×C127—81no—
86C23×C127—81no—
87C23×C127—81no—
88C23×C127—81no—
89C23×C127C2×D811yescharacteristic
90Q8×C127—42no—
91Q8×C127—42no—
92Q8×C127—82no—
93D4×C127—42no—
94D4×C127—42no—
95D4×C127—81no—
96D4×C127—81no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—41no—
103C2×C4×C127—42no—
104C2×C4×C127—42no—
105C2×C4×C127—81no—
106C2×C4×C127—81no—
107C2×C4×C127—81no—
108C2×C4×C127SD3212yescomplemented
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—82no—
116C22×C127—82no—
117C22×C127—82no—
118C22×C127—161no—
119C22×C127—161no—
120C22×C127—161no—
121C22×C127—161no—
122C22×C127#44711yescharacteristic
123C22×C127C2×SD3211yescharacteristic
124C22×C127#51011yescharacteristic
125C4×C127—21no—
126C4×C127—22no—
127C4×C127—41no—
128C4×C127—82no—
129C4×C127—82no—
130C4×C127—161no—
131C4×C127#45411yescharacteristic
132C4×C127B(4,𝔽2)11yescharacteristic
133#12159C12711yesmaximal, characteristic, direct product factor, Sylow
134C2×C127—21no—
135C2×C127—21no—
136C2×C127—162no—
137C2×C127#159811yescharacteristic
138C2×C127#218511yescharacteristic
139C2×C127#219011yescharacteristic
140#3249C2×C12711yescharacteristic, complemented
141#2187C2×C12712yes—
142#2054C2×C12711yescharacteristic
143#1348C2×C12712yescomplemented
144#1322C2×C12711yescharacteristic, complemented
145C127#1215911yescharacteristic, direct product factor, Sylow
146C22×D8C22×C12711yescharacteristic
147#533—21no—
148C2×SD32—22no—
149#477—22no—
150#467—21no—
151#467C22×C12712yes—
152#462—22no—
153C2×C4⋊C8C22×C12711yescharacteristic
154C2×#102—21no—
155#390—21no—
156#357—22no—
157C22⋊C16C22×C12712yes—
158#340C22×C12711yescharacteristic
159C22×D4—21no—
160C2×Q16—21no—
161C2×Q16D4×C12712yes—
162C2×D8—22no—
163C2×D8—42no—
164C22×C8D4×C12711yescharacteristic
165C22×C8C23×C12711yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168#125—41no—
169C22⋊Q8—22no—
170C22⋊Q8—41no—
171C4×Q8—22no—
172C4×Q8—41no—
173#117—41no—
174C2×C4⋊C4—21no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42D4×C12711yescharacteristic
178SD32—44no—
179C2×C16—22no—
180#107—21no—
181#107D4×C12712yes—
182C4⋊C8—22no—
183#103—22no—
184#103—22no—
185#102—42no—
186C24—41no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×D4—22no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C22×C4—41no—
196C22×C4D8×C12711yescharacteristic
197C22×C4D8×C12711yescharacteristic
198C22×C4C2×D4×C12711yescharacteristic
199Q16—42no—
200Q16—81no—
201D8—24no—
202D8—42no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8C2×D4×C12712yes—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—21no—
219C42—21no—
220C42—42no—
221C16—42no—
222C23—42no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23C2×D8×C12711yescharacteristic
230Q8—42no—
231Q8—42no—
232Q8—82no—
233D4—42no—
234D4—42no—
235D4—81no—
236D4—81no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4SD32×C12712yescomplemented
249C2×C4#120×C12711yescharacteristic
250C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—161no—
259C22—161no—
260C22—161no—
261C22—161no—
262C22#447×C12711yescharacteristic
263C22C2×SD32×C12711yescharacteristic
264C22#510×C12711yescharacteristic
265C4—21no—
266C4—22no—
267C4—41no—
268C4—82no—
269C4—82no—
270C4—161no—
271C4#454×C12711yescharacteristic
272C4B(4,𝔽2)×C12711yescharacteristic
273C2—21no—
274C2—21no—
275C2—162no—
276C2C127×#159811yescharacteristic
277C2C127×#218511yescharacteristic
278C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC22×D8×C127C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3249C2semidirect
4. C127×#1348C2semidirect
5. C127×#1322C2semidirect
108. C2×C4×C127SD32semidirect
133. #12159C127direct (N × H)
140. #3249C2×C127semidirect
143. #1348C2×C127semidirect
144. #1322C2×C127semidirect
145. C127#12159direct (N × H)
248. C2×C4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32492
C127×#21872
C127×#20542
C127×#13482
C127×#13222
#12159127