Finite Group Databasebeta

← C127×#12331 #44576004

Subgroups — C127×#12331 #44576004

Order 32512.

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

normal subgroups only

This group has 328 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×#3393C211yesmaximal, characteristic, complemented
2C127×#2248C211yesmaximal, characteristic, complemented
3C127×#2206C211yesmaximal, characteristic
4C127×#1668C211yesmaximal, characteristic, complemented
5C127×#1362C211yesmaximal, characteristic, complemented
6C127×Q8⋊C16C211yesmaximal, characteristic, complemented
7C127×#1331C211yesmaximal, characteristic, complemented
8#556×C127—21no—
9C2×SD32×C127—22no—
10C2×D16×C127—21no—
11#495×C127—21no—
12#494×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#464×C127—22no—
15#461×C127—21no—
16#460×C127—21no—
17C8×Q8×C127C2211yescharacteristic
18C4×Q16×C127—21no—
19C4×D8×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#357×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C2×Q16×C127—21no—
26C2×Q16×C127D412yes—
27C2×SD16×C127—22no—
28C2×SD16×C127—41no—
29C2×D8×C127—21no—
30C2×D8×C127—21no—
31C2×D8×C127D411yescharacteristic
32C4⋊Q8×C127—21no—
33#127×C127—21no—
34#126×C127—42no—
35#124×C127—22no—
36C22⋊Q8×C127—41no—
37C4×Q8×C127—22no—
38C4×Q8×C127D411yescharacteristic
39C4×D4×C127—21no—
40#117×C127—41no—
41SD32×C127—44no—
42D16×C127—42no—
43C2×C16×C127—21no—
44C2×C16×C127—21no—
45#107×C127D411yescharacteristic
46C4⋊C8×C127—21no—
47C4⋊C8×C127—21no—
48C4⋊C8×C127D411yescharacteristic
49#103×C127—22no—
50#102×C127—21no—
51#102×C127—41no—
52C4×C8×C127—21no—
53C4×C8×C127C2311yescharacteristic
54C2×Q8×C127—22no—
55C2×Q8×C127D811yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127—41no—
60C22×C4×C127—41no—
61Q16×C127—42no—
62Q16×C127—41no—
63Q16×C127—41no—
64SD16×C127—82no—
65SD16×C127—82no—
66D8×C127—21no—
67D8×C127—21no—
68D8×C127—42no—
69D8×C127—41no—
70D8×C127—41no—
71C2×C8×C127—21no—
72C2×C8×C127—41no—
73C2×C8×C127C2×D411yescharacteristic
74C2×C8×C127C2×D411yescharacteristic
75C4⋊C4×C127—21no—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—42no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127D811yescharacteristic
81C22⋊C4×C127—42no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—41no—
84C42×C127—21no—
85C42×C127—42no—
86C42×C127C2×D411yescharacteristic
87C16×C127—22no—
88C16×C127—41no—
89C23×C127—41no—
90C23×C127—81no—
91Q8×C127—21no—
92Q8×C127—21no—
93Q8×C127—42no—
94Q8×C127—42no—
95D4×C127—41no—
96D4×C127—41no—
97D4×C127—81no—
98D4×C127—81no—
99D4×C127—81no—
100D4×C127—81no—
101D4×C127—161no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—41no—
105C2×C4×C127—42no—
106C2×C4×C127—81no—
107C2×C4×C127—81no—
108C2×C4×C127#12011yescharacteristic
109C2×C4×C127C2×D811yescharacteristic
110C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
111C8×C127—21no—
112C8×C127—41no—
113C8×C127—81no—
114C8×C127#13511yescharacteristic
115C8×C127AGL(1,ℤ/8)11yescharacteristic
116C22×C127—81no—
117C22×C127—81no—
118C22×C127—81no—
119C22×C127—161no—
120C22×C127—161no—
121C22×C127—161no—
122C22×C127#44711yescharacteristic
123C4×C127—21no—
124C4×C127—41no—
125C4×C127—41no—
126C4×C127—82no—
127C4×C127—81no—
128C4×C127#44911yescharacteristic
129C4×C127#45311yescharacteristic
130C4×C127#50811yescharacteristic
131C4×C127#50911yescharacteristic
132#12331C12711yesmaximal, characteristic, direct product factor, Sylow
133C2×C127—161no—
134C2×C127—321no—
135C2×C127#162111yescharacteristic
136C2×C127#218611yescharacteristic
137C2×C127#219311yescharacteristic
138#3393C2×C12711yescharacteristic, complemented
139#2248C2×C12711yescharacteristic, complemented
140#2206C2×C12711yescharacteristic
141#1668C2×C12711yescharacteristic, complemented
142#1362C2×C12711yescharacteristic, complemented
143Q8⋊C16C2×C12711yescharacteristic, complemented
144#1331C2×C12711yescharacteristic, complemented
145C127#1233111yescharacteristic, direct product factor, Sylow
146#556—21no—
147C2×SD32—22no—
148C2×D16—21no—
149#495—21no—
150#494C22×C12711yescharacteristic
151#493C22×C12711yescharacteristic
152#464—22no—
153#461—21no—
154#460—21no—
155C8×Q8C22×C12711yescharacteristic
156C4×Q16—21no—
157C4×D8C22×C12711yescharacteristic
158C4⋊C16C22×C12711yescharacteristic
159#357—21no—
160C4×C16C22×C12711yescharacteristic
161#335C22×C12711yescharacteristic
162#331—21no—
163C2×Q16—21no—
164C2×Q16D4×C12712yes—
165C2×SD16—22no—
166C2×SD16—41no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8D4×C12711yescharacteristic
170C4⋊Q8—21no—
171#127—21no—
172#126—42no—
173#124—22no—
174C22⋊Q8—41no—
175C4×Q8—22no—
176C4×Q8D4×C12711yescharacteristic
177C4×D4—21no—
178#117—41no—
179SD32—44no—
180D16—42no—
181C2×C16—21no—
182C2×C16—21no—
183#107D4×C12711yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8D4×C12711yescharacteristic
187#103—22no—
188#102—21no—
189#102—41no—
190C4×C8—21no—
191C4×C8C23×C12711yescharacteristic
192C2×Q8—22no—
193C2×Q8D8×C12711yescharacteristic
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C22×C4—41no—
199Q16—42no—
200Q16—41no—
201Q16—41no—
202SD16—82no—
203SD16—82no—
204D8—21no—
205D8—21no—
206D8—42no—
207D8—41no—
208D8—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8C2×D4×C12711yescharacteristic
212C2×C8C2×D4×C12711yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4D8×C12711yescharacteristic
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42C2×D4×C12711yescharacteristic
225C16—22no—
226C16—41no—
227C23—41no—
228C23—81no—
229Q8—21no—
230Q8—21no—
231Q8—42no—
232Q8—42no—
233D4—41no—
234D4—41no—
235D4—81no—
236D4—81no—
237D4—81no—
238D4—81no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#120×C12711yescharacteristic
247C2×C4C2×D8×C12711yescharacteristic
248C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
249C8—21no—
250C8—41no—
251C8—81no—
252C8#135×C12711yescharacteristic
253C8AGL(1,ℤ/8)×C12711yescharacteristic
254C22—81no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—161no—
259C22—161no—
260C22#447×C12711yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4—81no—
266C4#449×C12711yescharacteristic
267C4#453×C12711yescharacteristic
268C4#508×C12711yescharacteristic
269C4#509×C12711yescharacteristic
270C2—161no—
271C2—321no—
272C2C127×#162111yescharacteristic
273C2C127×#218611yescharacteristic
274C2C127×#219311yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#2206C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3393C2semidirect
2. C127×#2248C2semidirect
4. C127×#1668C2semidirect
5. C127×#1362C2semidirect
6. C127×Q8⋊C16C2semidirect
7. C127×#1331C2semidirect
132. #12331C127direct (N × H)
138. #3393C2×C127semidirect
139. #2248C2×C127semidirect
141. #1668C2×C127semidirect
142. #1362C2×C127semidirect
143. Q8⋊C16C2×C127semidirect
144. #1331C2×C127semidirect
145. C127#12331direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#33932
C127×#22482
C127×#22062
C127×#16682
C127×#13622
C127×Q8⋊C162
C127×#13312
#12331127