Finite Group Databasebeta

← C127×#11485 #44575158

Subgroups — C127×#11485 #44575158

Order 32512.

372 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2931C211yesmaximal, characteristic
2C127×#2115C212yesmaximal, complemented
3C127×#1327C212yesmaximal, complemented
4C127×#1321C212yesmaximal, complemented
5C2×#131×C127C411yescharacteristic
6C2×#131×C127C2211yescharacteristic
7#532×C127C411yescharacteristic
8C2×M32×C127—22no—
9#436×C127—22no—
10C8×D4×C127—22no—
11#423×C127C2211yescharacteristic
12#408×C127C411yescharacteristic
13#408×C127C2211yescharacteristic
14C2×C4×C8×C127C411yescharacteristic
15#349×C127—22no—
16#349×C127C2214yes—
17C2×#41×C127C2×C411yescharacteristic
18C2×#41×C127C2×C411yescharacteristic
19#131×C127—41no—
20#131×C127—41no—
21#131×C127—41no—
22#131×C127—41no—
23C2×M16×C127—21no—
24C2×M16×C127C2×C411yescharacteristic
25C2×M16×C127D412yes—
26C2×M16×C127C2311yescharacteristic
27C22×C8×C127—21no—
28C22×C8×C127C2×C411yescharacteristic
29C22×C8×C127C2×C411yescharacteristic
30C22×C8×C127D412yes—
31C4⋊Q8×C127C811yescharacteristic
32#127×C127C811yescharacteristic
33#124×C127C811yescharacteristic
34#124×C127C811yescharacteristic
35#121×C127—22no—
36C4×D4×C127—22no—
37C2×C42×C127C2×C411yescharacteristic
38M32×C127—24no—
39M32×C127—42no—
40C2×C16×C127—42no—
41C4⋊C8×C127—22no—
42C22⋊C8×C127—22no—
43C22⋊C8×C127—22no—
44C4×C8×C127—21no—
45C4×C8×C127—21no—
46#41×C127—41no—
47#41×C127—41no—
48#41×C127—41no—
49#41×C127—41no—
50C2×Q8×C127C2×C811yescharacteristic
51C2×Q8×C127C2×C811yescharacteristic
52C2×D4×C127—22no—
53C2×D4×C127—21no—
54C2×D4×C127C2×C811yescharacteristic
55C2×D4×C127C2×C811yescharacteristic
56C2×D4×C127M1612yes—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127C22⋊C412yes—
60C22×C4×C127C22×C411yescharacteristic
61M16×C127—42no—
62M16×C127—41no—
63M16×C127—42no—
64M16×C127—41no—
65C2×C8×C127—22no—
66C2×C8×C127—21no—
67C2×C8×C127—22no—
68C2×C8×C127—21no—
69C2×C8×C127—21no—
70C2×C8×C127—21no—
71C2×C8×C127—41no—
72C2×C8×C127—41no—
73C2×C8×C127—41no—
74C2×C8×C127C22⋊C411yescharacteristic
75C2×C8×C127C22⋊C411yescharacteristic
76C2×C8×C127C2×D412yes—
77C4⋊C4×C127—22no—
78C22⋊C4×C127—22no—
79C22⋊C4×C127—22no—
80C42×C127—21no—
81C42×C127C2×C811yescharacteristic
82C42×C127C2×C811yescharacteristic
83C16×C127—44no—
84C23×C127—21no—
85C23×C127C22⋊C812yes—
86C23×C127C2×M1611yescharacteristic
87Q8×C127—41no—
88Q8×C127—41no—
89D4×C127—41no—
90D4×C127—42no—
91D4×C127—41no—
92D4×C127—44no—
93D4×C127—42no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—21no—
98C2×C4×C127—22no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—42no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127C22⋊C812yes—
105C2×C4×C127C2×C22⋊C411yescharacteristic
106C2×C4×C127C22×C811yescharacteristic
107C8×C127—22no—
108C8×C127—21no—
109C8×C127—41no—
110C8×C127—41no—
111C8×C127#9911yescharacteristic
112C8×C127#9911yescharacteristic
113C22×C127—21no—
114C22×C127—22no—
115C22×C127—41no—
116C22×C127—42no—
117C22×C127—42no—
118C22×C127—41no—
119C22×C127C2×C22⋊C811yescharacteristic
120C4×C127—22no—
121C4×C127—22no—
122C4×C127—21no—
123C4×C127—41no—
124C4×C127—41no—
125C4×C127C2×#9911yescharacteristic
126C4×C127#41311yescharacteristic
127#11485C12711yesmaximal, characteristic, direct product factor, Sylow
128C2×C127—21no—
129C2×C127—41no—
130C2×C127—42no—
131C2×C127—81no—
132C2×C127#145911yescharacteristic
133#2931C2×C12711yescharacteristic
134#2115C2×C12712yescomplemented
135#1327C2×C12712yescomplemented
136#1321C2×C12712yescomplemented
137C127#1148511yescharacteristic, direct product factor, Sylow
138C2×#131C4×C12711yescharacteristic
139C2×#131C22×C12711yescharacteristic
140#532C4×C12711yescharacteristic
141C2×M32—22no—
142#436—22no—
143C8×D4—22no—
144#423C22×C12711yescharacteristic
145#408C4×C12711yescharacteristic
146#408C22×C12711yescharacteristic
147C2×C4×C8C4×C12711yescharacteristic
148#349—22no—
149#349C22×C12714yes—
150C2×#41C2×C4×C12711yescharacteristic
151C2×#41C2×C4×C12711yescharacteristic
152#131—41no—
153#131—41no—
154#131—41no—
155#131—41no—
156C2×M16—21no—
157C2×M16C2×C4×C12711yescharacteristic
158C2×M16D4×C12712yes—
159C2×M16C23×C12711yescharacteristic
160C22×C8—21no—
161C22×C8C2×C4×C12711yescharacteristic
162C22×C8C2×C4×C12711yescharacteristic
163C22×C8D4×C12712yes—
164C4⋊Q8C8×C12711yescharacteristic
165#127C8×C12711yescharacteristic
166#124C8×C12711yescharacteristic
167#124C8×C12711yescharacteristic
168#121—22no—
169C4×D4—22no—
170C2×C42C2×C4×C12711yescharacteristic
171M32—24no—
172M32—42no—
173C2×C16—42no—
174C4⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C4×C8—21no—
178C4×C8—21no—
179#41—41no—
180#41—41no—
181#41—41no—
182#41—41no—
183C2×Q8C2×C8×C12711yescharacteristic
184C2×Q8C2×C8×C12711yescharacteristic
185C2×D4—22no—
186C2×D4—21no—
187C2×D4C2×C8×C12711yescharacteristic
188C2×D4C2×C8×C12711yescharacteristic
189C2×D4M16×C12712yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4C22⋊C4×C12712yes—
193C22×C4C22×C4×C12711yescharacteristic
194M16—42no—
195M16—41no—
196M16—42no—
197M16—41no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8C22⋊C4×C12711yescharacteristic
208C2×C8C22⋊C4×C12711yescharacteristic
209C2×C8C2×D4×C12712yes—
210C4⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C42—21no—
214C42C2×C8×C12711yescharacteristic
215C42C2×C8×C12711yescharacteristic
216C16—44no—
217C23—21no—
218C23C22⋊C8×C12712yes—
219C23C2×M16×C12711yescharacteristic
220Q8—41no—
221Q8—41no—
222D4—41no—
223D4—42no—
224D4—41no—
225D4—44no—
226D4—42no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4C22⋊C8×C12712yes—
238C2×C4C2×C22⋊C4×C12711yescharacteristic
239C2×C4C22×C8×C12711yescharacteristic
240C8—22no—
241C8—21no—
242C8—41no—
243C8—41no—
244C8#99×C12711yescharacteristic
245C8#99×C12711yescharacteristic
246C22—21no—
247C22—22no—
248C22—41no—
249C22—42no—
250C22—42no—
251C22—41no—
252C22C2×C22⋊C8×C12711yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4C2×#99×C12711yescharacteristic
259C4#413×C12711yescharacteristic
260C2—21no—
261C2—41no—
262C2—42no—
263C2—81no—
264C2C127×#145911yescharacteristic
subgroupG/H
centerC8×C127#99
commutatorC2×C4C22×C8×C127
Frattini, cosocleC2×M16C23×C127
socleC2×C127#1459
prime radical#532×C127C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8×C127 ◁ C2×C8×C127 ◁ C2×#131×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2115C2semidirect
3. C127×#1327C2semidirect
4. C127×#1321C2semidirect
127. #11485C127direct (N × H)
134. #2115C2×C127semidirect
135. #1327C2×C127semidirect
136. #1321C2×C127semidirect
137. C127#11485direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#29312
C127×#21152
C127×#13272
C127×#13212
#11485127