Finite Group Databasebeta

← C127×#2432 #30635384

Subgroups — C127×#2432 #30635384

Order 16256.

836 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#41×C127C211yesmaximal, characteristic
2C2×C4⋊Q8×C127C211yesmaximal, characteristic, complemented
3C2×#124×C127C211yesmaximal, characteristic, complemented
4C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
5#394×C127C214yesmaximal, complemented
6#391×C127C211yesmaximal, characteristic, complemented
7#390×C127C211yesmaximal, characteristic, complemented
8#385×C127C214yesmaximal, complemented
9C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
10C2×#41×C127—24no—
11C2×#41×C127—24no—
12C2×#41×C127—24no—
13C22×Q8×C127C2211yescharacteristic
14C22×Q8×C127C2211yescharacteristic
15C22×D4×C127C2212yes—
16C22×D4×C127C2211yescharacteristic
17C23×C4×C127C2212yes—
18C23×C4×C127C2211yescharacteristic
19C4⋊Q8×C127—24no—
20#124×C127—24no—
21C22⋊Q8×C127—24no—
22C2×C4⋊C4×C127C2212yes—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C4⋊C4×C127C2214yes—
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2214yes—
27C2×C22⋊C4×C127C2212yes—
28C2×C22⋊C4×C127C2214yes—
29C2×C42×C127C2211yescharacteristic
30C2×C42×C127C2211yescharacteristic
31#95×C127C2212yes—
32#95×C127C2214yes—
33C24×C127C2312yes—
34C24×C127C2311yescharacteristic
35#41×C127—232no—
36C2×Q8×C127—22no—
37C2×Q8×C127—22no—
38C2×Q8×C127—22no—
39C2×Q8×C127—22no—
40C2×Q8×C127—22no—
41C2×Q8×C127D414yes—
42C2×D4×C127—24no—
43C2×D4×C127—24no—
44C2×D4×C127—24no—
45C2×D4×C127—22no—
46C2×D4×C127—22no—
47C2×D4×C127D414yes—
48C22×C4×C127—24no—
49C22×C4×C127—24no—
50C22×C4×C127—24no—
51C22×C4×C127—22no—
52C22×C4×C127—22no—
53C22×C4×C127D414yes—
54C22×C4×C127C2312yes—
55C22×C4×C127C2312yes—
56C22×C4×C127C2312yes—
57C22×C4×C127C2311yescharacteristic
58C22×C4×C127C2312yes—
59C22×C4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61C4⋊C4×C127—22no—
62C4⋊C4×C127—24no—
63C4⋊C4×C127—28no—
64C22⋊C4×C127—24no—
65C22⋊C4×C127—28no—
66C22⋊C4×C127—24no—
67C22⋊C4×C127—28no—
68C42×C127—22no—
69C42×C127—22no—
70C23×C127—22no—
71C23×C127—21no—
72C23×C127—22no—
73C23×C127—21no—
74C23×C127—22no—
75C23×C127—24no—
76C23×C127—22no—
77C23×C127—24no—
78C23×C127C2×D414yes—
79C23×C127C2×D412yes—
80C23×C127C2411yescharacteristic
81Q8×C127—28no—
82Q8×C127—44no—
83D4×C127—216no—
84D4×C127—28no—
85C2×C4×C127—216no—
86C2×C4×C127—22no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—28no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—22no—
96C2×C4×C127—22no—
97C2×C4×C127—22no—
98C2×C4×C127—24no—
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127C2×D412yes—
102C2×C4×C127C2×D414yes—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D412yes—
106C2×C4×C127#4114yescomplemented
107C22×C127—24no—
108C22×C127—22no—
109C22×C127—22no—
110C22×C127—22no—
111C22×C127—28no—
112C22×C127—42no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—44no—
117C22×C127C22×D411yescharacteristic
118C22×C127C22×D411yescharacteristic
119C22×C127C22×D411yescharacteristic
120C22×C127C2×#4112yes—
121C22×C1272−1+412yes—
122C4×C127—22no—
123C4×C127—24no—
124C4×C127—24no—
125C4×C127—42no—
126C4×C127—42no—
127C4×C127—42no—
128C4×C127#12014yes—
129C2×C127—42no—
130C2×C127—44no—
131C2×C127C2×#12011yescharacteristic
132C2×C127#53211yescharacteristic
133C2×C127#53611yescharacteristic
134C2×C127#54712yes—
135C2×C127#54812yes—
136#2432C12711yesmaximal, characteristic, direct product factor, Sylow
137C127#243211yescharacteristic, direct product factor, Sylow
138C22×#41C2×C12711yescharacteristic
139C2×C4⋊Q8C2×C12711yescharacteristic, complemented
140C2×#124C2×C12711yescharacteristic, complemented
141C2×C22⋊Q8C2×C12711yescharacteristic, complemented
142#394C2×C12714yescomplemented
143#391C2×C12711yescharacteristic, complemented
144#390C2×C12711yescharacteristic, complemented
145#385C2×C12714yescomplemented
146C4×C22⋊C4C2×C12711yescharacteristic, complemented
147C2×#41—24no—
148C2×#41—24no—
149C2×#41—24no—
150C22×Q8C22×C12711yescharacteristic
151C22×Q8C22×C12711yescharacteristic
152C22×D4C22×C12712yes—
153C22×D4C22×C12711yescharacteristic
154C23×C4C22×C12712yes—
155C23×C4C22×C12711yescharacteristic
156C4⋊Q8—24no—
157#124—24no—
158C22⋊Q8—24no—
159C2×C4⋊C4C22×C12712yes—
160C2×C4⋊C4C22×C12711yescharacteristic
161C2×C4⋊C4C22×C12714yes—
162C2×C22⋊C4C22×C12712yes—
163C2×C22⋊C4C22×C12714yes—
164C2×C22⋊C4C22×C12712yes—
165C2×C22⋊C4C22×C12714yes—
166C2×C42C22×C12711yescharacteristic
167C2×C42C22×C12711yescharacteristic
168#95C22×C12712yes—
169#95C22×C12714yes—
170C24C23×C12712yes—
171C24C23×C12711yescharacteristic
172#41—232no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8D4×C12714yes—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4D4×C12714yes—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4D4×C12714yes—
191C22×C4C23×C12712yes—
192C22×C4C23×C12712yes—
193C22×C4C23×C12712yes—
194C22×C4C23×C12711yescharacteristic
195C22×C4C23×C12712yes—
196C22×C4C23×C12712yes—
197C22×C4C23×C12711yescharacteristic
198C4⋊C4—22no—
199C4⋊C4—24no—
200C4⋊C4—28no—
201C22⋊C4—24no—
202C22⋊C4—28no—
203C22⋊C4—24no—
204C22⋊C4—28no—
205C42—22no—
206C42—22no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—24no—
213C23—22no—
214C23—24no—
215C23C2×D4×C12714yes—
216C23C2×D4×C12712yes—
217C23C24×C12711yescharacteristic
218Q8—28no—
219Q8—44no—
220D4—216no—
221D4—28no—
222C2×C4—216no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—28no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4C2×D4×C12712yes—
239C2×C4C2×D4×C12714yes—
240C2×C4C2×D4×C12712yes—
241C2×C4C2×D4×C12712yes—
242C2×C4C2×D4×C12712yes—
243C2×C4#41×C12714yescomplemented
244C22—24no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—28no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—44no—
254C22C22×D4×C12711yescharacteristic
255C22C22×D4×C12711yescharacteristic
256C22C22×D4×C12711yescharacteristic
257C22C2×#41×C12712yes—
258C222−1+4×C12712yes—
259C4—22no—
260C4—24no—
261C4—24no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4#120×C12714yes—
266C2—42no—
267C2—44no—
268C2C2×#120×C12711yescharacteristic
269C2#532×C12711yescharacteristic
270C2#536×C12711yescharacteristic
271C2#547×C12712yes—
272C2#548×C12712yes—
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC22×#41×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊Q8×C127C2semidirect
3. C2×#124×C127C2semidirect
4. C2×C22⋊Q8×C127C2semidirect
5. #394×C127C2semidirect
6. #391×C127C2semidirect
7. #390×C127C2semidirect
8. #385×C127C2semidirect
9. C4×C22⋊C4×C127C2semidirect
106. C2×C4×C127#41semidirect
136. #2432C127direct (N × H)
137. C127#2432direct (N × H)
139. C2×C4⋊Q8C2×C127semidirect
140. C2×#124C2×C127semidirect
141. C2×C22⋊Q8C2×C127semidirect
142. #394C2×C127semidirect
143. #391C2×C127semidirect
144. #390C2×C127semidirect
145. #385C2×C127semidirect
146. C4×C22⋊C4C2×C127semidirect
243. C2×C4#41×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#41×C1272
C2×C4⋊Q8×C1272
C2×#124×C1272
C2×C22⋊Q8×C1272
#394×C1272
#391×C1272
#390×C1272
#385×C1272
C4×C22⋊C4×C1272
#2432127