Finite Group Databasebeta

← C127×#3057 #30636009

Subgroups — C127×#3057 #30636009

Order 16256.

492 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SD16×C127C211yesmaximal, characteristic, complemented
2C2×#131×C127C211yesmaximal, characteristic, complemented
3#536×C127C211yesmaximal, characteristic, complemented
4#535×C127C211yesmaximal, characteristic
5#468×C127C214yesmaximal, complemented
6#465×C127C214yesmaximal, complemented
7#427×C127C211yesmaximal, characteristic, complemented
8#419×C127C211yesmaximal, characteristic, complemented
9#418×C127C211yesmaximal, characteristic, complemented
10C2×#41×C127C2211yescharacteristic
11C22×Q8×C127C2211yescharacteristic
12C22×D4×C127C2211yescharacteristic
13C2×SD16×C127—24no—
14C2×SD16×C127C2212yes—
15C2×SD16×C127C2212yes—
16#131×C127—24no—
17C2×M16×C127C2211yescharacteristic
18C2×M16×C127C2212yescomplemented
19C22×C8×C127C2211yescharacteristic
20C22×C8×C127C2212yescomplemented
21C4⋊Q8×C127—21no—
22#127×C127—21no—
23#124×C127—21no—
24#124×C127—21no—
25#123×C127—22no—
26C22⋊Q8×C127C2214yescomplemented
27#121×C127C2214yes—
28#120×C127—22no—
29#117×C127C2211yescharacteristic
30#117×C127C2211yescharacteristic
31#107×C127C2212yescomplemented
32#106×C127C2212yescomplemented
33#103×C127C2214yescomplemented
34#102×C127C2214yes—
35C24×C127—21no—
36#41×C127—22no—
37#41×C127—22no—
38C2×Q8×C127—21no—
39C2×Q8×C127—22no—
40C2×Q8×C127—22no—
41C2×Q8×C127D411yescharacteristic
42C2×Q8×C127C2312yes—
43C2×D4×C127—24no—
44C2×D4×C127—22no—
45C2×D4×C127—21no—
46C2×D4×C127—22no—
47C2×D4×C127D412yes—
48C2×D4×C127D411yescharacteristic
49C2×D4×C127C2312yes—
50C22×C4×C127—21no—
51C22×C4×C127C2312yes—
52C22×C4×C127C2311yescharacteristic
53SD16×C127—24no—
54SD16×C127—24no—
55M16×C127—22no—
56M16×C127—24no—
57C2×C8×C127—24no—
58C2×C8×C127D414yescomplemented
59C2×C8×C127C2312yes—
60C2×C8×C127C2311yescharacteristic
61C2×C8×C127C2311yescharacteristic
62C4⋊C4×C127—24no—
63C4⋊C4×C127C2312yes—
64C4⋊C4×C127C2312yes—
65C22⋊C4×C127—21no—
66C22⋊C4×C127—21no—
67C22⋊C4×C127—24no—
68C22⋊C4×C127—24no—
69C42×C127—21no—
70C42×C127—21no—
71C23×C127—22no—
72C23×C127—42no—
73C23×C127—82no—
74C23×C127C2×D412yes—
75C23×C127C2×D411yescharacteristic
76Q8×C127—22no—
77Q8×C127—22no—
78Q8×C127—22no—
79Q8×C127—42no—
80D4×C127—24no—
81D4×C127—22no—
82D4×C127—22no—
83D4×C127—22no—
84D4×C127—42no—
85D4×C127—84no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—42no—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C8×C127C2×D412yes—
99C8×C127C2×D412yes—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—24no—
103C22×C127—42no—
104C22×C127—44no—
105C22×C127—84no—
106C22×C127#12112yes—
107C22×C127C22×D411yescharacteristic
108C4×C127—22no—
109C4×C127—42no—
110C4×C127—42no—
111C4×C127—42no—
112C4×C127#12112yes—
113C4×C127C22×D411yescharacteristic
114C4×C127C2×#4111yescharacteristic
115C2×C127—22no—
116C2×C127—42no—
117C2×C127—82no—
118C2×C127C2×#12111yescharacteristic
119C2×C127#57712yes—
120#3057C12711yesmaximal, characteristic, direct product factor, Sylow
121C127#305711yescharacteristic, direct product factor, Sylow
122C22×SD16C2×C12711yescharacteristic, complemented
123C2×#131C2×C12711yescharacteristic, complemented
124#536C2×C12711yescharacteristic, complemented
125#535C2×C12711yescharacteristic
126#468C2×C12714yescomplemented
127#465C2×C12714yescomplemented
128#427C2×C12711yescharacteristic, complemented
129#419C2×C12711yescharacteristic, complemented
130#418C2×C12711yescharacteristic, complemented
131C2×#41C22×C12711yescharacteristic
132C22×Q8C22×C12711yescharacteristic
133C22×D4C22×C12711yescharacteristic
134C2×SD16—24no—
135C2×SD16C22×C12712yes—
136C2×SD16C22×C12712yes—
137#131—24no—
138C2×M16C22×C12711yescharacteristic
139C2×M16C22×C12712yescomplemented
140C22×C8C22×C12711yescharacteristic
141C22×C8C22×C12712yescomplemented
142C4⋊Q8—21no—
143#127—21no—
144#124—21no—
145#124—21no—
146#123—22no—
147C22⋊Q8C22×C12714yescomplemented
148#121C22×C12714yes—
149#120—22no—
150#117C22×C12711yescharacteristic
151#117C22×C12711yescharacteristic
152#107C22×C12712yescomplemented
153#106C22×C12712yescomplemented
154#103C22×C12714yescomplemented
155#102C22×C12714yes—
156C24—21no—
157#41—22no—
158#41—22no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8D4×C12711yescharacteristic
163C2×Q8C23×C12712yes—
164C2×D4—24no—
165C2×D4—22no—
166C2×D4—21no—
167C2×D4—22no—
168C2×D4D4×C12712yes—
169C2×D4D4×C12711yescharacteristic
170C2×D4C23×C12712yes—
171C22×C4—21no—
172C22×C4C23×C12712yes—
173C22×C4C23×C12711yescharacteristic
174SD16—24no—
175SD16—24no—
176M16—22no—
177M16—24no—
178C2×C8—24no—
179C2×C8D4×C12714yescomplemented
180C2×C8C23×C12712yes—
181C2×C8C23×C12711yescharacteristic
182C2×C8C23×C12711yescharacteristic
183C4⋊C4—24no—
184C4⋊C4C23×C12712yes—
185C4⋊C4C23×C12712yes—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C42—21no—
191C42—21no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23C2×D4×C12712yes—
196C23C2×D4×C12711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—24no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—42no—
206D4—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4C2×D4×C12712yes—
214C2×C4C2×D4×C12711yescharacteristic
215C2×C4C2×D4×C12712yes—
216C2×C4#41×C12712yes—
217C2×C4C24×C12711yescharacteristic
218C8—22no—
219C8C2×D4×C12712yes—
220C8C2×D4×C12712yes—
221C22—22no—
222C22—22no—
223C22—24no—
224C22—42no—
225C22—44no—
226C22—84no—
227C22#121×C12712yes—
228C22C22×D4×C12711yescharacteristic
229C4—22no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4#121×C12712yes—
234C4C22×D4×C12711yescharacteristic
235C4C2×#41×C12711yescharacteristic
236C2—22no—
237C2—42no—
238C2—82no—
239C2C2×#121×C12711yescharacteristic
240C2#577×C12712yes—
subgroupG/H
center, socleC22×C127C22×D4
commutator, Frattini, cosocleC2×C4C24×C127
prime radical#535×C127C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×SD16×C127C2semidirect
2. C2×#131×C127C2semidirect
3. #536×C127C2semidirect
5. #468×C127C2semidirect
6. #465×C127C2semidirect
7. #427×C127C2semidirect
8. #419×C127C2semidirect
9. #418×C127C2semidirect
18. C2×M16×C127C22semidirect
20. C22×C8×C127C22semidirect
26. C22⋊Q8×C127C22semidirect
31. #107×C127C22semidirect
32. #106×C127C22semidirect
33. #103×C127C22semidirect
58. C2×C8×C127D4semidirect
120. #3057C127direct (N × H)
121. C127#3057direct (N × H)
122. C22×SD16C2×C127semidirect
123. C2×#131C2×C127semidirect
124. #536C2×C127semidirect
126. #468C2×C127semidirect
127. #465C2×C127semidirect
128. #427C2×C127semidirect
129. #419C2×C127semidirect
130. #418C2×C127semidirect
139. C2×M16C22×C127semidirect
141. C22×C8C22×C127semidirect
147. C22⋊Q8C22×C127semidirect
152. #107C22×C127semidirect
153. #106C22×C127semidirect
154. #103C22×C127semidirect
179. C2×C8D4×C127semidirect

Maximal subgroups

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

subgroupindex
C22×SD16×C1272
C2×#131×C1272
#536×C1272
#535×C1272
#468×C1272
#465×C1272
#427×C1272
#419×C1272
#418×C1272
#3057127