Finite Group Databasebeta

← C127×#11957 #44575630

Subgroups — C127×#11957 #44575630

Order 32512.

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

normal subgroups only

This group has 406 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×#495×C127C211yesmaximal, characteristic
2C127×#2162C211yesmaximal, characteristic, complemented
3C127×#1568C211yesmaximal, characteristic, complemented
4C127×#1360C214yesmaximal, complemented
5C22×Q16×C127C411yescharacteristic
6C22×SD16×C127—21no—
7C22×D8×C127C411yescharacteristic
8C2×#124×C127—21no—
9#495×C127—21no—
10#495×C127—21no—
11#495×C127C412yes—
12#495×C127C2212yes—
13#432×C127—21no—
14C2×C4×C8×C127C2211yescharacteristic
15C4⋊C16×C127C2212yescomplemented
16C22⋊C16×C127—21no—
17C4×C16×C127—21no—
18#335×C127—21no—
19#334×C127C2212yescomplemented
20C22×Q8×C127—21no—
21C22×D4×C127—21no—
22C2×Q16×C127—21no—
23C2×Q16×C127—42no—
24C2×Q16×C127C2×C412yes—
25C2×SD16×C127—22no—
26C2×SD16×C127—22no—
27C2×SD16×C127—44no—
28C2×D8×C127—21no—
29C2×D8×C127—42no—
30C2×D8×C127C2×C412yes—
31C22×C8×C127C2×C411yescharacteristic
32C22×C8×C127D411yescharacteristic
33#124×C127—22no—
34#124×C127—22no—
35#124×C127—41no—
36#124×C127—41no—
37C2×C22⋊C4×C127—41no—
38C2×C22⋊C4×C127—41no—
39C2×C42×C127D411yescharacteristic
40C2×C16×C127—22no—
41C4⋊C8×C127—22no—
42C22⋊C8×C127—41no—
43#97×C127—41no—
44C4×C8×C127C2×C411yescharacteristic
45C4×C8×C127D411yescharacteristic
46C4×C8×C127D411yescharacteristic
47C4×C8×C127C2311yescharacteristic
48C24×C127—41no—
49C2×Q8×C127—22no—
50C2×Q8×C127—42no—
51C2×Q8×C127—42no—
52C2×Q8×C127—41no—
53C2×D4×C127—22no—
54C2×D4×C127—42no—
55C2×D4×C127—42no—
56C2×D4×C127—41no—
57C22×C4×C127—41no—
58C22×C4×C127C22⋊C411yescharacteristic
59C22×C4×C127D811yescharacteristic
60C22×C4×C127SD1611yescharacteristic
61Q16×C127—42no—
62Q16×C127—42no—
63SD16×C127—44no—
64SD16×C127—44no—
65D8×C127—42no—
66D8×C127—42no—
67C2×C8×C127—22no—
68C2×C8×C127—22no—
69C2×C8×C127—42no—
70C2×C8×C127C22⋊C411yescharacteristic
71C2×C8×C127C22⋊C411yescharacteristic
72C2×C8×C127C22×C411yescharacteristic
73C2×C8×C127C2×D411yescharacteristic
74C22⋊C4×C127—42no—
75C22⋊C4×C127—42no—
76C22⋊C4×C127—42no—
77C22⋊C4×C127—42no—
78C42×C127C22⋊C411yescharacteristic
79C42×C127D811yescharacteristic
80C42×C127SD1611yescharacteristic
81C42×C127C2×D411yescharacteristic
82C16×C127—42no—
83C23×C127—42no—
84C23×C127—82no—
85C23×C127—82no—
86C23×C127—82no—
87C23×C127#10211yescharacteristic
88Q8×C127—42no—
89Q8×C127—42no—
90Q8×C127—82no—
91D4×C127—42no—
92D4×C127—42no—
93D4×C127—82no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—41no—
98C2×C4×C127—42no—
99C2×C4×C127—82no—
100C2×C4×C127#10211yescharacteristic
101C2×C4×C127#10211yescharacteristic
102C2×C4×C127#10211yescharacteristic
103C2×C4×C127C2×C22⋊C411yescharacteristic
104C2×C4×C127C2×D811yescharacteristic
105C2×C4×C127C2×SD1611yescharacteristic
106C8×C127—21no—
107C8×C127—21no—
108C8×C127—82no—
109C8×C127#10012yes—
110C8×C127AGL(1,ℤ/8)12yes—
111C22×C127—21no—
112C22×C127—21no—
113C22×C127—82no—
114C22×C127—82no—
115C22×C127—82no—
116C22×C127—164no—
117C22×C127#35912yes—
118C22×C127C2×#10211yescharacteristic
119C4×C127—21no—
120C4×C127—21no—
121C4×C127—21no—
122C4×C127—21no—
123C4×C127—21no—
124C4×C127—82no—
125C4×C127C2×#10011yescharacteristic
126C4×C127#41811yescharacteristic
127#11957C12711yesmaximal, characteristic, direct product factor, Sylow
128C2×C127—22no—
129C2×C127—162no—
130C2×C127#154511yescharacteristic
131C2×C127C2×#35911yescharacteristic
132C2×C127#214511yescharacteristic
133C2×#495C2×C12711yescharacteristic
134#2162C2×C12711yescharacteristic, complemented
135#1568C2×C12711yescharacteristic, complemented
136#1360C2×C12714yescomplemented
137C127#1195711yescharacteristic, direct product factor, Sylow
138C22×Q16C4×C12711yescharacteristic
139C22×SD16—21no—
140C22×D8C4×C12711yescharacteristic
141C2×#124—21no—
142#495—21no—
143#495—21no—
144#495C4×C12712yes—
145#495C22×C12712yes—
146#432—21no—
147C2×C4×C8C22×C12711yescharacteristic
148C4⋊C16C22×C12712yescomplemented
149C22⋊C16—21no—
150C4×C16—21no—
151#335—21no—
152#334C22×C12712yescomplemented
153C22×Q8—21no—
154C22×D4—21no—
155C2×Q16—21no—
156C2×Q16—42no—
157C2×Q16C2×C4×C12712yes—
158C2×SD16—22no—
159C2×SD16—22no—
160C2×SD16—44no—
161C2×D8—21no—
162C2×D8—42no—
163C2×D8C2×C4×C12712yes—
164C22×C8C2×C4×C12711yescharacteristic
165C22×C8D4×C12711yescharacteristic
166#124—22no—
167#124—22no—
168#124—41no—
169#124—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42D4×C12711yescharacteristic
173C2×C16—22no—
174C4⋊C8—22no—
175C22⋊C8—41no—
176#97—41no—
177C4×C8C2×C4×C12711yescharacteristic
178C4×C8D4×C12711yescharacteristic
179C4×C8D4×C12711yescharacteristic
180C4×C8C23×C12711yescharacteristic
181C24—41no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—41no—
190C22×C4—41no—
191C22×C4C22⋊C4×C12711yescharacteristic
192C22×C4D8×C12711yescharacteristic
193C22×C4SD16×C12711yescharacteristic
194Q16—42no—
195Q16—42no—
196SD16—44no—
197SD16—44no—
198D8—42no—
199D8—42no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—42no—
203C2×C8C22⋊C4×C12711yescharacteristic
204C2×C8C22⋊C4×C12711yescharacteristic
205C2×C8C22×C4×C12711yescharacteristic
206C2×C8C2×D4×C12711yescharacteristic
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42C22⋊C4×C12711yescharacteristic
212C42D8×C12711yescharacteristic
213C42SD16×C12711yescharacteristic
214C42C2×D4×C12711yescharacteristic
215C16—42no—
216C23—42no—
217C23—82no—
218C23—82no—
219C23—82no—
220C23#102×C12711yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—82no—
224D4—42no—
225D4—42no—
226D4—82no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4#102×C12711yescharacteristic
234C2×C4#102×C12711yescharacteristic
235C2×C4#102×C12711yescharacteristic
236C2×C4C2×C22⋊C4×C12711yescharacteristic
237C2×C4C2×D8×C12711yescharacteristic
238C2×C4C2×SD16×C12711yescharacteristic
239C8—21no—
240C8—21no—
241C8—82no—
242C8#100×C12712yes—
243C8AGL(1,ℤ/8)×C12712yes—
244C22—21no—
245C22—21no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—164no—
250C22#359×C12712yes—
251C22C2×#102×C12711yescharacteristic
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—82no—
258C4C2×#100×C12711yescharacteristic
259C4#418×C12711yescharacteristic
260C2—22no—
261C2—162no—
262C2C127×#154511yescharacteristic
263C2C2×#359×C12711yescharacteristic
264C2C127×#214511yescharacteristic
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC4×C8C23×C127
prime radicalC22×D8×C127C4
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2162C2semidirect
3. C127×#1568C2semidirect
4. C127×#1360C2semidirect
15. C4⋊C16×C127C22semidirect
19. #334×C127C22semidirect
127. #11957C127direct (N × H)
134. #2162C2×C127semidirect
135. #1568C2×C127semidirect
136. #1360C2×C127semidirect
137. C127#11957direct (N × H)
148. C4⋊C16C22×C127semidirect
152. #334C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#495×C1272
C127×#21622
C127×#15682
C127×#13602
#11957127