Finite Group Databasebeta

← C127×#2600 #30635552

Subgroups — C127×#2600 #30635552

Order 16256.

604 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×C127C212yesmaximal, complemented
2#397×C127C211yesmaximal, characteristic, complemented
3#394×C127C212yesmaximal, complemented
4#392×C127C211yesmaximal, characteristic
5#388×C127C212yesmaximal, complemented
6#387×C127C212yesmaximal, complemented
7#386×C127C212yesmaximal, complemented
8#385×C127C212yesmaximal, complemented
9#381×C127C211yesmaximal, characteristic, complemented
10C22×D4×C127C2212yes—
11C22×D4×C127C2211yescharacteristic
12C23×C4×C127C2212yes—
13C23×C4×C127C2211yescharacteristic
14C4×D4×C127—28no—
15C2×C4⋊C4×C127C2212yes—
16C2×C4⋊C4×C127C2212yes—
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C22⋊C4×C127C2212yes—
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2212yes—
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2212yes—
25C2×C42×C127C2212yes—
26#95×C127C2212yes—
27#95×C127C2212yes—
28#95×C127C2212yes—
29#95×C127C2212yes—
30#95×C127C2211yescharacteristic
31#95×C127C2211yescharacteristic
32C24×C127C2311yescharacteristic
33C24×C127C2312yes—
34C2×D4×C127—24no—
35C2×D4×C127—24no—
36C2×D4×C127—24no—
37C2×D4×C127—24no—
38C2×D4×C127—41no—
39C22×C4×C127—24no—
40C22×C4×C127—41no—
41C22×C4×C127—41no—
42C22×C4×C127—42no—
43C22×C4×C127—42no—
44C22×C4×C127D414yescomplemented
45C22×C4×C127C2311yescharacteristic
46C22×C4×C127C2311yescharacteristic
47C22×C4×C127C2312yes—
48C22×C4×C127C2312yes—
49C22×C4×C127C2312yes—
50C22×C4×C127C2312yes—
51C22×C4×C127C2312yes—
52C4⋊C4×C127—24no—
53C4⋊C4×C127—24no—
54C4⋊C4×C127—41no—
55C22⋊C4×C127—24no—
56C22⋊C4×C127—24no—
57C22⋊C4×C127—24no—
58C22⋊C4×C127—24no—
59C22⋊C4×C127—41no—
60C22⋊C4×C127—42no—
61C22⋊C4×C127—41no—
62C42×C127—24no—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—22no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—22no—
72C23×C127—22no—
73C23×C127C2×D412yes—
74C23×C127#4114yes—
75C23×C127C2411yescharacteristic
76D4×C127—216no—
77D4×C127—44no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—44no—
96C2×C4×C127—48no—
97C2×C4×C127C2×D414yes—
98C2×C4×C127#4114yes—
99C22×C127—24no—
100C22×C127—24no—
101C22×C127—22no—
102C22×C127—24no—
103C22×C127—24no—
104C22×C127—42no—
105C22×C127—42no—
106C22×C127—41no—
107C22×C127—42no—
108C22×C127—42no—
109C22×C127—42no—
110C22×C127—41no—
111C22×C127C22×D411yescharacteristic
112C22×C127C2×#4112yes—
113C22×C127C2×#4112yes—
114C22×C127H(5,𝔽2)11yescharacteristic
115C22×C127H(5,𝔽2)11yescharacteristic
116C4×C127—24no—
117C4×C127—24no—
118C4×C127—42no—
119C4×C127—42no—
120C4×C127—41no—
121C4×C127—42no—
122C4×C127—41no—
123C2×C127—42no—
124C2×C127—44no—
125C2×C127#52512yes—
126C2×C127#53411yescharacteristic
127C2×C127#55111yescharacteristic
128C2×C127#55312yes—
129C2×C127#55911yescharacteristic
130#2600C12711yesmaximal, characteristic, direct product factor, Sylow
131C127#260011yescharacteristic, direct product factor, Sylow
132C2×C4×D4C2×C12712yescomplemented
133#397C2×C12711yescharacteristic, complemented
134#394C2×C12712yescomplemented
135#392C2×C12711yescharacteristic
136#388C2×C12712yescomplemented
137#387C2×C12712yescomplemented
138#386C2×C12712yescomplemented
139#385C2×C12712yescomplemented
140#381C2×C12711yescharacteristic, complemented
141C22×D4C22×C12712yes—
142C22×D4C22×C12711yescharacteristic
143C23×C4C22×C12712yes—
144C23×C4C22×C12711yescharacteristic
145C4×D4—28no—
146C2×C4⋊C4C22×C12712yes—
147C2×C4⋊C4C22×C12712yes—
148C2×C4⋊C4C22×C12711yescharacteristic
149C2×C22⋊C4C22×C12712yes—
150C2×C22⋊C4C22×C12712yes—
151C2×C22⋊C4C22×C12712yes—
152C2×C22⋊C4C22×C12712yes—
153C2×C22⋊C4C22×C12711yescharacteristic
154C2×C22⋊C4C22×C12711yescharacteristic
155C2×C22⋊C4C22×C12712yes—
156C2×C42C22×C12712yes—
157#95C22×C12712yes—
158#95C22×C12712yes—
159#95C22×C12712yes—
160#95C22×C12712yes—
161#95C22×C12711yescharacteristic
162#95C22×C12711yescharacteristic
163C24C23×C12711yescharacteristic
164C24C23×C12712yes—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—41no—
170C22×C4—24no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4D4×C12714yescomplemented
176C22×C4C23×C12711yescharacteristic
177C22×C4C23×C12711yescharacteristic
178C22×C4C23×C12712yes—
179C22×C4C23×C12712yes—
180C22×C4C23×C12712yes—
181C22×C4C23×C12712yes—
182C22×C4C23×C12712yes—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—41no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—41no—
191C22⋊C4—42no—
192C22⋊C4—41no—
193C42—24no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23C2×D4×C12712yes—
205C23#41×C12714yes—
206C23C24×C12711yescharacteristic
207D4—216no—
208D4—44no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4C2×D4×C12714yes—
229C2×C4#41×C12714yes—
230C22—24no—
231C22—24no—
232C22—22no—
233C22—24no—
234C22—24no—
235C22—42no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22C22×D4×C12711yescharacteristic
243C22C2×#41×C12712yes—
244C22C2×#41×C12712yes—
245C22H(5,𝔽2)×C12711yescharacteristic
246C22H(5,𝔽2)×C12711yescharacteristic
247C4—24no—
248C4—24no—
249C4—42no—
250C4—42no—
251C4—41no—
252C4—42no—
253C4—41no—
254C2—42no—
255C2—44no—
256C2#525×C12712yes—
257C2#534×C12711yescharacteristic
258C2#551×C12711yescharacteristic
259C2#553×C12712yes—
260C2#559×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radical#392×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C4×D4×C127C2semidirect
2. #397×C127C2semidirect
3. #394×C127C2semidirect
5. #388×C127C2semidirect
6. #387×C127C2semidirect
7. #386×C127C2semidirect
8. #385×C127C2semidirect
9. #381×C127C2semidirect
44. C22×C4×C127D4semidirect
130. #2600C127direct (N × H)
131. C127#2600direct (N × H)
132. C2×C4×D4C2×C127semidirect
133. #397C2×C127semidirect
134. #394C2×C127semidirect
136. #388C2×C127semidirect
137. #387C2×C127semidirect
138. #386C2×C127semidirect
139. #385C2×C127semidirect
140. #381C2×C127semidirect
175. C22×C4D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×C1272
#397×C1272
#394×C1272
#392×C1272
#388×C1272
#387×C1272
#386×C1272
#385×C1272
#381×C1272
#2600127