Finite Group Databasebeta

← C127×#2727 #30635679

Subgroups — C127×#2727 #30635679

Order 16256.

476 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×C127C212yesmaximal, complemented
2#399×C127C212yesmaximal, complemented
3#396×C127C212yesmaximal, complemented
4#394×C127C212yesmaximal
5#390×C127C211yesmaximal, characteristic
6#389×C127C212yesmaximal, complemented
7#389×C127C211yesmaximal, characteristic, complemented
8#389×C127C211yesmaximal, characteristic, complemented
9#388×C127C212yesmaximal, complemented
10C22×D4×C127C2211yescharacteristic
11#126×C127—28no—
12C2×C4⋊C4×C127C2212yes—
13C2×C4⋊C4×C127C2212yes—
14C2×C4⋊C4×C127C2212yes—
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2212yes—
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2212yes—
24C2×C22⋊C4×C127C2212yes—
25C2×C22⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C42×C127C2212yes—
28C2×C42×C127C2211yescharacteristic
29#95×C127C2212yes—
30#95×C127C2212yes—
31#95×C127C2212yes—
32C24×C127C2312yes—
33C2×D4×C127—22no—
34C2×D4×C127—41no—
35C2×D4×C127—41no—
36C22×C4×C127C2312yes—
37C22×C4×C127C2312yes—
38C22×C4×C127C2312yes—
39C22×C4×C127C2312yes—
40C22×C4×C127C2312yes—
41C22×C4×C127C2311yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—24no—
45C4⋊C4×C127—24no—
46C4⋊C4×C127—24no—
47C4⋊C4×C127—22no—
48C4⋊C4×C127—41no—
49C4⋊C4×C127—41no—
50C4⋊C4×C127—41no—
51C4⋊C4×C127—42no—
52C4⋊C4×C127D414yescomplemented
53C22⋊C4×C127—24no—
54C22⋊C4×C127—24no—
55C22⋊C4×C127—24no—
56C22⋊C4×C127—24no—
57C22⋊C4×C127—42no—
58C22⋊C4×C127—42no—
59C42×C127—24no—
60C42×C127—41no—
61C23×C127—22no—
62C23×C127—22no—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127—22no—
68C23×C127C2411yescharacteristic
69D4×C127—44no—
70C2×C4×C127—22no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
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×C127C2×D414yes—
91C2×C4×C127C2×D412yes—
92C2×C4×C127#4112yes—
93C2×C4×C127#4112yes—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127—42no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127C22×D411yescharacteristic
102C22×C127C2×#4111yescharacteristic
103C22×C127C2×#4111yescharacteristic
104C22×C127H(5,𝔽2)11yescharacteristic
105C22×C127H(5,𝔽2)11yescharacteristic
106C22×C1272−1+411yescharacteristic
107C22×C1272−1+411yescharacteristic
108C4×C127—22no—
109C4×C127—22no—
110C4×C127—22no—
111C4×C127—24no—
112C4×C127—42no—
113C4×C127—42no—
114C4×C127—42no—
115C4×C127—42no—
116C2×C127—82no—
117C2×C127#53711yescharacteristic
118C2×C127#53811yescharacteristic
119C2×C127#53911yescharacteristic
120C2×C127#54711yescharacteristic
121C2×C127#55011yescharacteristic
122C2×C127#55611yescharacteristic
123C2×C127#56311yescharacteristic
124#2727C12711yesmaximal, characteristic, direct product factor, Sylow
125C127#272711yescharacteristic, direct product factor, Sylow
126C2×#126C2×C12712yescomplemented
127#399C2×C12712yescomplemented
128#396C2×C12712yescomplemented
129#394C2×C12712yes—
130#390C2×C12711yescharacteristic
131#389C2×C12712yescomplemented
132#389C2×C12711yescharacteristic, complemented
133#389C2×C12711yescharacteristic, complemented
134#388C2×C12712yescomplemented
135C22×D4C22×C12711yescharacteristic
136#126—28no—
137C2×C4⋊C4C22×C12712yes—
138C2×C4⋊C4C22×C12712yes—
139C2×C4⋊C4C22×C12712yes—
140C2×C4⋊C4C22×C12711yescharacteristic
141C2×C4⋊C4C22×C12711yescharacteristic
142C2×C4⋊C4C22×C12711yescharacteristic
143C2×C4⋊C4C22×C12711yescharacteristic
144C2×C4⋊C4C22×C12712yes—
145C2×C4⋊C4C22×C12711yescharacteristic
146C2×C22⋊C4C22×C12712yes—
147C2×C22⋊C4C22×C12712yes—
148C2×C22⋊C4C22×C12712yes—
149C2×C22⋊C4C22×C12712yes—
150C2×C22⋊C4C22×C12712yes—
151C2×C22⋊C4C22×C12712yes—
152C2×C42C22×C12712yes—
153C2×C42C22×C12711yescharacteristic
154#95C22×C12712yes—
155#95C22×C12712yes—
156#95C22×C12712yes—
157C24C23×C12712yes—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—41no—
161C22×C4C23×C12712yes—
162C22×C4C23×C12712yes—
163C22×C4C23×C12712yes—
164C22×C4C23×C12712yes—
165C22×C4C23×C12712yes—
166C22×C4C23×C12711yescharacteristic
167C22×C4C23×C12711yescharacteristic
168C22×C4C23×C12711yescharacteristic
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4D4×C12714yescomplemented
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C42—24no—
185C42—41no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23C24×C12711yescharacteristic
194D4—44no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4C2×D4×C12714yes—
216C2×C4C2×D4×C12712yes—
217C2×C4#41×C12712yes—
218C2×C4#41×C12712yes—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22C22×D4×C12711yescharacteristic
227C22C2×#41×C12711yescharacteristic
228C22C2×#41×C12711yescharacteristic
229C22H(5,𝔽2)×C12711yescharacteristic
230C22H(5,𝔽2)×C12711yescharacteristic
231C222−1+4×C12711yescharacteristic
232C222−1+4×C12711yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—24no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—42no—
241C2—82no—
242C2#537×C12711yescharacteristic
243C2#538×C12711yescharacteristic
244C2#539×C12711yescharacteristic
245C2#547×C12711yescharacteristic
246C2#550×C12711yescharacteristic
247C2#556×C12711yescharacteristic
248C2#563×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC22×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#126×C127C2semidirect
2. #399×C127C2semidirect
3. #396×C127C2semidirect
6. #389×C127C2semidirect
7. #389×C127C2semidirect
8. #389×C127C2semidirect
9. #388×C127C2semidirect
52. C4⋊C4×C127D4semidirect
124. #2727C127direct (N × H)
125. C127#2727direct (N × H)
126. C2×#126C2×C127semidirect
127. #399C2×C127semidirect
128. #396C2×C127semidirect
131. #389C2×C127semidirect
132. #389C2×C127semidirect
133. #389C2×C127semidirect
134. #388C2×C127semidirect
177. C4⋊C4D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#126×C1272
#399×C1272
#396×C1272
#394×C1272
#390×C1272
#389×C1272
#388×C1272
#2727127