Finite Group Databasebeta

← C127×#11636 #44575309

Subgroups — C127×#11636 #44575309

Order 32512.

382 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#436×C127C211yesmaximal, characteristic
2C127×#2163C211yesmaximal, characteristic, complemented
3C127×#2163C211yesmaximal, characteristic, complemented
4C127×#1330C214yesmaximal, complemented
5C22×M16×C127—21no—
6C2×C4×D4×C127C411yescharacteristic
7#436×C127—21no—
8#436×C127—21no—
9#436×C127C412yes—
10#436×C127C2212yes—
11C2×C4⋊C8×C127C411yescharacteristic
12C2×C22⋊C8×C127—21no—
13C2×C4×C8×C127C2211yescharacteristic
14C4⋊C16×C127—21no—
15C4⋊C16×C127C2212yescomplemented
16C22⋊C16×C127—21no—
17C22⋊C16×C127—21no—
18#346×C127—21no—
19#346×C127C2212yescomplemented
20C22×D4×C127C811yescharacteristic
21C23×C4×C127—21no—
22C2×M16×C127—22no—
23C2×M16×C127—22no—
24C2×M16×C127—44no—
25C22×C8×C127—21no—
26C22×C8×C127D411yescharacteristic
27C22×C8×C127D411yescharacteristic
28C4×D4×C127—21no—
29C4×D4×C127—21no—
30C4×D4×C127C812yes—
31C4×D4×C127C2×C412yes—
32C2×C4⋊C4×C127C811yescharacteristic
33C2×C22⋊C4×C127—21no—
34C2×C42×C127C2×C411yescharacteristic
35C2×C16×C127—22no—
36C2×C16×C127—22no—
37C4⋊C8×C127—21no—
38C4⋊C8×C127C2×C412yes—
39C22⋊C8×C127—22no—
40C22⋊C8×C127—22no—
41C4×C8×C127C2×C411yescharacteristic
42C4×C8×C127D411yescharacteristic
43C4×C8×C127D411yescharacteristic
44C4×C8×C127C2311yescharacteristic
45C24×C127—21no—
46C2×D4×C127—21no—
47C2×D4×C127—42no—
48C2×D4×C127C2×C812yes—
49C22×C4×C127—21no—
50C22×C4×C127—22no—
51C22×C4×C127—22no—
52C22×C4×C127—42no—
53C22×C4×C127—42no—
54C22×C4×C127C22⋊C411yescharacteristic
55C22×C4×C127C2×C811yescharacteristic
56C22×C4×C127M1611yescharacteristic
57M16×C127—44no—
58M16×C127—44no—
59C2×C8×C127—22no—
60C2×C8×C127—22no—
61C2×C8×C127—41no—
62C2×C8×C127—42no—
63C2×C8×C127C22⋊C411yescharacteristic
64C2×C8×C127C22⋊C411yescharacteristic
65C2×C8×C127C2×D411yescharacteristic
66C2×C8×C127C2×D411yescharacteristic
67C4⋊C4×C127—21no—
68C4⋊C4×C127C2×C812yes—
69C22⋊C4×C127—22no—
70C22⋊C4×C127—22no—
71C42×C127C22⋊C411yescharacteristic
72C42×C127C2×C811yescharacteristic
73C42×C127M1611yescharacteristic
74C42×C127C22×C411yescharacteristic
75C16×C127—42no—
76C16×C127—42no—
77C23×C127—22no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127—42no—
81C23×C127C22⋊C811yescharacteristic
82D4×C127—42no—
83D4×C127—42no—
84C2×C4×C127—22no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127—41no—
89C2×C4×C127—44no—
90C2×C4×C127—42no—
91C2×C4×C127—42no—
92C2×C4×C127—42no—
93C2×C4×C127C22⋊C811yescharacteristic
94C2×C4×C127C22⋊C811yescharacteristic
95C2×C4×C127C22⋊C811yescharacteristic
96C2×C4×C127C2×C22⋊C411yescharacteristic
97C2×C4×C127C22×C811yescharacteristic
98C2×C4×C127C2×M1611yescharacteristic
99C8×C127—21no—
100C8×C127—21no—
101C8×C127—21no—
102C8×C127—42no—
103C8×C127AGL(1,ℤ/8)12yes—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127—42no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—84no—
110C22×C127#34912yes—
111C22×C127C2×C22⋊C811yescharacteristic
112C4×C127—21no—
113C4×C127—21no—
114C4×C127—21no—
115C4×C127—21no—
116C4×C127—21no—
117C4×C127—42no—
118C4×C127#41811yescharacteristic
119C4×C127#42111yescharacteristic
120#11636C12711yesmaximal, characteristic, direct product factor, Sylow
121C2×C127—22no—
122C2×C127—82no—
123C2×C127#147811yescharacteristic
124C2×C127C2×#34911yescharacteristic
125C2×C127#211711yescharacteristic
126C2×#436C2×C12711yescharacteristic
127#2163C2×C12711yescharacteristic, complemented
128#2163C2×C12711yescharacteristic, complemented
129#1330C2×C12714yescomplemented
130C127#1163611yescharacteristic, direct product factor, Sylow
131C22×M16—21no—
132C2×C4×D4C4×C12711yescharacteristic
133#436—21no—
134#436—21no—
135#436C4×C12712yes—
136#436C22×C12712yes—
137C2×C4⋊C8C4×C12711yescharacteristic
138C2×C22⋊C8—21no—
139C2×C4×C8C22×C12711yescharacteristic
140C4⋊C16—21no—
141C4⋊C16C22×C12712yescomplemented
142C22⋊C16—21no—
143C22⋊C16—21no—
144#346—21no—
145#346C22×C12712yescomplemented
146C22×D4C8×C12711yescharacteristic
147C23×C4—21no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—44no—
151C22×C8—21no—
152C22×C8D4×C12711yescharacteristic
153C22×C8D4×C12711yescharacteristic
154C4×D4—21no—
155C4×D4—21no—
156C4×D4C8×C12712yes—
157C4×D4C2×C4×C12712yes—
158C2×C4⋊C4C8×C12711yescharacteristic
159C2×C22⋊C4—21no—
160C2×C42C2×C4×C12711yescharacteristic
161C2×C16—22no—
162C2×C16—22no—
163C4⋊C8—21no—
164C4⋊C8C2×C4×C12712yes—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C4×C8C2×C4×C12711yescharacteristic
168C4×C8D4×C12711yescharacteristic
169C4×C8D4×C12711yescharacteristic
170C4×C8C23×C12711yescharacteristic
171C24—21no—
172C2×D4—21no—
173C2×D4—42no—
174C2×D4C2×C8×C12712yes—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4C22⋊C4×C12711yescharacteristic
181C22×C4C2×C8×C12711yescharacteristic
182C22×C4M16×C12711yescharacteristic
183M16—44no—
184M16—44no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8C22⋊C4×C12711yescharacteristic
190C2×C8C22⋊C4×C12711yescharacteristic
191C2×C8C2×D4×C12711yescharacteristic
192C2×C8C2×D4×C12711yescharacteristic
193C4⋊C4—21no—
194C4⋊C4C2×C8×C12712yes—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C42C22⋊C4×C12711yescharacteristic
198C42C2×C8×C12711yescharacteristic
199C42M16×C12711yescharacteristic
200C42C22×C4×C12711yescharacteristic
201C16—42no—
202C16—42no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23C22⋊C8×C12711yescharacteristic
208D4—42no—
209D4—42no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4C22⋊C8×C12711yescharacteristic
220C2×C4C22⋊C8×C12711yescharacteristic
221C2×C4C22⋊C8×C12711yescharacteristic
222C2×C4C2×C22⋊C4×C12711yescharacteristic
223C2×C4C22×C8×C12711yescharacteristic
224C2×C4C2×M16×C12711yescharacteristic
225C8—21no—
226C8—21no—
227C8—21no—
228C8—42no—
229C8AGL(1,ℤ/8)×C12712yes—
230C22—21no—
231C22—21no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—84no—
236C22#349×C12712yes—
237C22C2×C22⋊C8×C12711yescharacteristic
238C4—21no—
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4#418×C12711yescharacteristic
245C4#421×C12711yescharacteristic
246C2—22no—
247C2—82no—
248C2C127×#147811yescharacteristic
249C2C2×#349×C12711yescharacteristic
250C2C127×#211711yescharacteristic
subgroupG/H
centerC2×C4×C127C2×C22⋊C4
commutatorC2×C4C22×C8×C127
Frattini, cosocleC4×C8C23×C127
socleC22×C127C2×C22⋊C8
prime radicalC22×D4×C127C8
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2163C2semidirect
3. C127×#2163C2semidirect
4. C127×#1330C2semidirect
15. C4⋊C16×C127C22semidirect
19. #346×C127C22semidirect
120. #11636C127direct (N × H)
127. #2163C2×C127semidirect
128. #2163C2×C127semidirect
129. #1330C2×C127semidirect
130. C127#11636direct (N × H)
141. C4⋊C16C22×C127semidirect
145. #346C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#436×C1272
C127×#21632
C127×#13302
#11636127