Finite Group Databasebeta

← C127×#2604 #30635556

Subgroups — C127×#2604 #30635556

Order 16256.

470 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#126×C127C211yesmaximal, characteristic
2C2×C4×Q8×C127C211yesmaximal, characteristic, complemented
3#400×C127C211yesmaximal, characteristic, complemented
4#399×C127C211yesmaximal, characteristic
5#395×C127C211yesmaximal, characteristic, complemented
6#393×C127C211yesmaximal, characteristic
7#389×C127C212yesmaximal, complemented
8#388×C127C212yesmaximal
9#388×C127C212yesmaximal
10#387×C127C212yesmaximal, complemented
11#376×C127C211yesmaximal, characteristic, complemented
12C22×Q8×C127C2211yescharacteristic
13#126×C127—24no—
14C4×Q8×C127—24no—
15C2×C4⋊C4×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2212yes—
21C2×C4⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2211yescharacteristic
23C2×C22⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C42×C127C2212yes—
28C2×C42×C127C2212yes—
29C2×C42×C127C2211yescharacteristic
30#95×C127C2212yes—
31#95×C127C2211yescharacteristic
32#95×C127C2211yescharacteristic
33#95×C127C2212yes—
34#95×C127C2212yes—
35#95×C127C2212yes—
36#95×C127C2212yes—
37C24×C127C2311yescharacteristic
38C2×Q8×C127—22no—
39C2×Q8×C127—24no—
40C22×C4×C127C2312yes—
41C22×C4×C127C2311yescharacteristic
42C22×C4×C127C2312yes—
43C22×C4×C127C2311yescharacteristic
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2311yescharacteristic
46C22×C4×C127C2311yescharacteristic
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2312yes—
49C22×C4×C127C2312yes—
50C4⋊C4×C127—24no—
51C4⋊C4×C127—22no—
52C4⋊C4×C127—22no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127—22no—
55C4⋊C4×C127—24no—
56C4⋊C4×C127—24no—
57C22⋊C4×C127—22no—
58C22⋊C4×C127—22no—
59C22⋊C4×C127—22no—
60C22⋊C4×C127—24no—
61C22⋊C4×C127—24no—
62C42×C127—24no—
63C42×C127—24no—
64C42×C127D414yes—
65C23×C127—21no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127—21no—
69C23×C127—21no—
70C23×C127C2411yescharacteristic
71Q8×C127—28no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
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—21no—
91C2×C4×C127—21no—
92C2×C4×C127—22no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D412yes—
96C2×C4×C127#4114yes—
97C2×C4×C127#4114yes—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127C22×D411yescharacteristic
104C22×C127C2×#4112yes—
105C22×C127C2×#4112yes—
106C22×C1272−1+411yescharacteristic
107C22×C1272−1+411yescharacteristic
108C4×C127—22no—
109C4×C127—24no—
110C4×C127—22no—
111C4×C127—22no—
112C4×C127—24no—
113C4×C127—41no—
114C4×C127—42no—
115C4×C127—41no—
116C4×C127—41no—
117C4×C127—42no—
118C2×C127—81no—
119C2×C127#52511yescharacteristic
120C2×C127#53211yescharacteristic
121C2×C127#53611yescharacteristic
122C2×C127#55212yes—
123C2×C127#55612yes—
124#2604C12711yesmaximal, characteristic, direct product factor, Sylow
125C127#260411yescharacteristic, direct product factor, Sylow
126C2×#126C2×C12711yescharacteristic
127C2×C4×Q8C2×C12711yescharacteristic, complemented
128#400C2×C12711yescharacteristic, complemented
129#399C2×C12711yescharacteristic
130#395C2×C12711yescharacteristic, complemented
131#393C2×C12711yescharacteristic
132#389C2×C12712yescomplemented
133#388C2×C12712yes—
134#388C2×C12712yes—
135#387C2×C12712yescomplemented
136#376C2×C12711yescharacteristic, complemented
137C22×Q8C22×C12711yescharacteristic
138#126—24no—
139C4×Q8—24no—
140C2×C4⋊C4C22×C12712yes—
141C2×C4⋊C4C22×C12711yescharacteristic
142C2×C4⋊C4C22×C12711yescharacteristic
143C2×C4⋊C4C22×C12711yescharacteristic
144C2×C4⋊C4C22×C12711yescharacteristic
145C2×C4⋊C4C22×C12712yes—
146C2×C4⋊C4C22×C12712yes—
147C2×C22⋊C4C22×C12711yescharacteristic
148C2×C22⋊C4C22×C12711yescharacteristic
149C2×C22⋊C4C22×C12711yescharacteristic
150C2×C22⋊C4C22×C12712yes—
151C2×C22⋊C4C22×C12712yes—
152C2×C42C22×C12712yes—
153C2×C42C22×C12712yes—
154C2×C42C22×C12711yescharacteristic
155#95C22×C12712yes—
156#95C22×C12711yescharacteristic
157#95C22×C12711yescharacteristic
158#95C22×C12712yes—
159#95C22×C12712yes—
160#95C22×C12712yes—
161#95C22×C12712yes—
162C24C23×C12711yescharacteristic
163C2×Q8—22no—
164C2×Q8—24no—
165C22×C4C23×C12712yes—
166C22×C4C23×C12711yescharacteristic
167C22×C4C23×C12712yes—
168C22×C4C23×C12711yescharacteristic
169C22×C4C23×C12711yescharacteristic
170C22×C4C23×C12711yescharacteristic
171C22×C4C23×C12711yescharacteristic
172C22×C4C23×C12711yescharacteristic
173C22×C4C23×C12712yes—
174C22×C4C23×C12712yes—
175C4⋊C4—24no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C42—24no—
188C42—24no—
189C42D4×C12714yes—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23C24×C12711yescharacteristic
196Q8—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4C2×D4×C12712yes—
219C2×C4C2×D4×C12712yes—
220C2×C4C2×D4×C12712yes—
221C2×C4#41×C12714yes—
222C2×C4#41×C12714yes—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22C22×D4×C12711yescharacteristic
229C22C2×#41×C12712yes—
230C22C2×#41×C12712yes—
231C222−1+4×C12711yescharacteristic
232C222−1+4×C12711yescharacteristic
233C4—22no—
234C4—24no—
235C4—22no—
236C4—22no—
237C4—24no—
238C4—41no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—42no—
243C2—81no—
244C2#525×C12711yescharacteristic
245C2#532×C12711yescharacteristic
246C2#536×C12711yescharacteristic
247C2#552×C12712yes—
248C2#556×C12712yes—
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC24×C127C23
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. #400×C127C2semidirect
5. #395×C127C2semidirect
7. #389×C127C2semidirect
10. #387×C127C2semidirect
11. #376×C127C2semidirect
124. #2604C127direct (N × H)
125. C127#2604direct (N × H)
127. C2×C4×Q8C2×C127semidirect
128. #400C2×C127semidirect
130. #395C2×C127semidirect
132. #389C2×C127semidirect
135. #387C2×C127semidirect
136. #376C2×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#126×C1272
C2×C4×Q8×C1272
#400×C1272
#399×C1272
#395×C1272
#393×C1272
#389×C1272
#388×C1272
#387×C1272
#376×C1272
#2604127