Finite Group Databasebeta

← C127×#13201 #44576874

Subgroups — C127×#13201 #44576874

Order 32512.

420 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2930C211yesmaximal, characteristic
2C127×#2153C211yesmaximal, characteristic, complemented
3C127×#2117C212yesmaximal, complemented
4C2×#349×C127C212yesmaximal, complemented
5C127×#2110C211yesmaximal, characteristic, complemented
6C2×#131×C127C2211yescharacteristic
7C22×M16×C127C2211yescharacteristic
8#525×C127C411yescharacteristic
9C2×M32×C127C2212yes—
10C2×M32×C127C2212yes—
11#436×C127—22no—
12#435×C127—22no—
13#424×C127C411yescharacteristic
14#408×C127C411yescharacteristic
15#407×C127C411yescharacteristic
16#405×C127C2211yescharacteristic
17#364×C127—22no—
18#350×C127—24no—
19#349×C127—24no—
20#347×C127—22no—
21C2×#41×C127C2×C411yescharacteristic
22C23×C4×C127C2×C411yescharacteristic
23#131×C127—22no—
24#131×C127—22no—
25C2×M16×C127—22no—
26C2×M16×C127—21no—
27C2×M16×C127—21no—
28C2×M16×C127—21no—
29C2×M16×C127—22no—
30C2×M16×C127C2×C411yescharacteristic
31C2×M16×C127C2×C411yescharacteristic
32C22×C8×C127—21no—
33C22×C8×C127C2×C411yescharacteristic
34C22×C8×C127C2311yescharacteristic
35#123×C127—21no—
36C22⋊Q8×C127—21no—
37#121×C127—21no—
38#120×C127—21no—
39C4×D4×C127—22no—
40#117×C127C2×C411yescharacteristic
41M32×C127—22no—
42M32×C127—22no—
43M32×C127—22no—
44M32×C127D414yescomplemented
45C2×C16×C127—22no—
46C2×C16×C127—22no—
47C4⋊C8×C127—22no—
48C22⋊C8×C127—22no—
49C22⋊C8×C127—22no—
50#97×C127D412yes—
51C4×C8×C127D412yes—
52C24×C127M1611yescharacteristic
53#41×C127—22no—
54#41×C127—22no—
55C2×Q8×C127M1611yescharacteristic
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127M1611yescharacteristic
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—22no—
62C22×C4×C127—21no—
63C22×C4×C127—22no—
64C22×C4×C127M1611yescharacteristic
65C22×C4×C127C22×C411yescharacteristic
66M16×C127—22no—
67M16×C127—22no—
68M16×C127—42no—
69M16×C127—41no—
70M16×C127—41no—
71M16×C127—41no—
72M16×C127—41no—
73C2×C8×C127—21no—
74C2×C8×C127—21no—
75C2×C8×C127—22no—
76C2×C8×C127—42no—
77C2×C8×C127C22⋊C412yes—
78C2×C8×C127C2×D411yescharacteristic
79C2×C8×C127C2×D412yes—
80C2×C8×C127C2×D411yescharacteristic
81C2×C8×C127#4112yes—
82C4⋊C4×C127—22no—
83C4⋊C4×C127—21no—
84C22⋊C4×C127—22no—
85C22⋊C4×C127—22no—
86C22⋊C4×C127—21no—
87C42×C127C22⋊C412yes—
88C16×C127—24no—
89C16×C127—42no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—81no—
94C23×C127C2×M1611yescharacteristic
95Q8×C127—22no—
96D4×C127—22no—
97D4×C127—42no—
98D4×C127—82no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—22no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127—42no—
108C2×C4×C127C2×C22⋊C411yescharacteristic
109C2×C4×C127C4×D412yes—
110C2×C4×C127C2×M1611yescharacteristic
111C2×C4×C127#13112yes—
112C8×C127—22no—
113C8×C127—41no—
114C8×C127—41no—
115C8×C127#12112yes—
116C8×C127#12411yescharacteristic
117C8×C127#12711yescharacteristic
118C22×C127—41no—
119C22×C127—41no—
120C22×C127—42no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—82no—
125C22×C127#40711yescharacteristic
126C22×C127#43512yes—
127C4×C127—42no—
128C4×C127—41no—
129C4×C127—41no—
130C4×C127#39011yescharacteristic
131C4×C127#40811yescharacteristic
132C4×C127#43512yes—
133#13201C12711yesmaximal, characteristic, direct product factor, Sylow
134C2×C127—21no—
135C2×C127—22no—
136C2×C127—81no—
137C2×C127—81no—
138C2×C127#195511yescharacteristic
139#2930C2×C12711yescharacteristic
140#2153C2×C12711yescharacteristic, complemented
141#2117C2×C12712yescomplemented
142C2×#349C2×C12712yescomplemented
143#2110C2×C12711yescharacteristic, complemented
144C127#1320111yescharacteristic, direct product factor, Sylow
145C2×#131C22×C12711yescharacteristic
146C22×M16C22×C12711yescharacteristic
147#525C4×C12711yescharacteristic
148C2×M32C22×C12712yes—
149C2×M32C22×C12712yes—
150#436—22no—
151#435—22no—
152#424C4×C12711yescharacteristic
153#408C4×C12711yescharacteristic
154#407C4×C12711yescharacteristic
155#405C22×C12711yescharacteristic
156#364—22no—
157#350—24no—
158#349—24no—
159#347—22no—
160C2×#41C2×C4×C12711yescharacteristic
161C23×C4C2×C4×C12711yescharacteristic
162#131—22no—
163#131—22no—
164C2×M16—22no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—22no—
169C2×M16C2×C4×C12711yescharacteristic
170C2×M16C2×C4×C12711yescharacteristic
171C22×C8—21no—
172C22×C8C2×C4×C12711yescharacteristic
173C22×C8C23×C12711yescharacteristic
174#123—21no—
175C22⋊Q8—21no—
176#121—21no—
177#120—21no—
178C4×D4—22no—
179#117C2×C4×C12711yescharacteristic
180M32—22no—
181M32—22no—
182M32—22no—
183M32D4×C12714yescomplemented
184C2×C16—22no—
185C2×C16—22no—
186C4⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189#97D4×C12712yes—
190C4×C8D4×C12712yes—
191C24M16×C12711yescharacteristic
192#41—22no—
193#41—22no—
194C2×Q8M16×C12711yescharacteristic
195C2×D4—21no—
196C2×D4—22no—
197C2×D4M16×C12711yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4M16×C12711yescharacteristic
204C22×C4C22×C4×C12711yescharacteristic
205M16—22no—
206M16—22no—
207M16—42no—
208M16—41no—
209M16—41no—
210M16—41no—
211M16—41no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—42no—
216C2×C8C22⋊C4×C12712yes—
217C2×C8C2×D4×C12711yescharacteristic
218C2×C8C2×D4×C12712yes—
219C2×C8C2×D4×C12711yescharacteristic
220C2×C8#41×C12712yes—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—21no—
226C42C22⋊C4×C12712yes—
227C16—24no—
228C16—42no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—81no—
233C23C2×M16×C12711yescharacteristic
234Q8—22no—
235D4—22no—
236D4—42no—
237D4—82no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4C2×C22⋊C4×C12711yescharacteristic
248C2×C4C4×D4×C12712yes—
249C2×C4C2×M16×C12711yescharacteristic
250C2×C4#131×C12712yes—
251C8—22no—
252C8—41no—
253C8—41no—
254C8#121×C12712yes—
255C8#124×C12711yescharacteristic
256C8#127×C12711yescharacteristic
257C22—41no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—82no—
264C22#407×C12711yescharacteristic
265C22#435×C12712yes—
266C4—42no—
267C4—41no—
268C4—41no—
269C4#390×C12711yescharacteristic
270C4#408×C12711yescharacteristic
271C4#435×C12712yes—
272C2—21no—
273C2—22no—
274C2—81no—
275C2—81no—
276C2C127×#195511yescharacteristic
subgroupG/H
centerC4×C127#390
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×C8C23×C127
socleC2×C127#1955
prime radical#525×C127C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C22×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2153C2semidirect
3. C127×#2117C2semidirect
4. C2×#349×C127C2semidirect
5. C127×#2110C2semidirect
44. M32×C127D4semidirect
133. #13201C127direct (N × H)
140. #2153C2×C127semidirect
141. #2117C2×C127semidirect
142. C2×#349C2×C127semidirect
143. #2110C2×C127semidirect
144. C127#13201direct (N × H)
183. M32D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#29302
C127×#21532
C127×#21172
C2×#349×C1272
C127×#21102
#13201127