Finite Group Databasebeta

← C127×#2609 #30635561

Subgroups — C127×#2609 #30635561

Order 16256.

540 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×C127C212yesmaximal, complemented
2#398×C127C211yesmaximal, characteristic, complemented
3#396×C127C212yesmaximal, complemented
4#394×C127C211yesmaximal, characteristic
5#388×C127C212yesmaximal, complemented
6#387×C127C212yesmaximal, complemented
7#386×C127C212yesmaximal
8#385×C127C212yesmaximal, complemented
9#382×C127C211yesmaximal, characteristic, complemented
10C22×D4×C127C2211yescharacteristic
11C23×C4×C127C2212yes—
12#117×C127—28no—
13C2×C4⋊C4×C127C2212yes—
14C2×C4⋊C4×C127C2212yes—
15C2×C4⋊C4×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2212yes—
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2212yes—
24C2×C42×C127C2212yes—
25C2×C42×C127C2211yescharacteristic
26#95×C127C2212yes—
27#95×C127C2212yes—
28#95×C127C2211yescharacteristic
29#95×C127C2212yes—
30#95×C127C2211yescharacteristic
31#95×C127C2212yes—
32C24×C127C2312yes—
33C2×D4×C127—24no—
34C2×D4×C127—41no—
35C22×C4×C127—24no—
36C22×C4×C127—24no—
37C22×C4×C127—24no—
38C22×C4×C127C2311yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C22×C4×C127C2312yes—
41C22×C4×C127C2312yes—
42C22×C4×C127C2312yes—
43C22×C4×C127C2312yes—
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2312yes—
46C4⋊C4×C127—24no—
47C4⋊C4×C127—24no—
48C4⋊C4×C127—24no—
49C4⋊C4×C127—41no—
50C4⋊C4×C127—41no—
51C4⋊C4×C127—41no—
52C22⋊C4×C127—24no—
53C22⋊C4×C127—24no—
54C22⋊C4×C127—24no—
55C22⋊C4×C127—24no—
56C22⋊C4×C127—42no—
57C42×C127—24no—
58C42×C127D414yescomplemented
59C23×C127—22no—
60C23×C127—22no—
61C23×C127—22no—
62C23×C127—22no—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127#4114yes—
66C23×C127C2411yescharacteristic
67D4×C127—44no—
68C2×C4×C127—216no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
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—21no—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127C2×D414yes—
89C2×C4×C127C2×D412yes—
90C2×C4×C127#4114yes—
91C22×C127—24no—
92C22×C127—24no—
93C22×C127—24no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—42no—
97C22×C127—42no—
98C22×C127C22×D411yescharacteristic
99C22×C127C2×#4112yes—
100C22×C127C2×#4112yes—
101C22×C127H(5,𝔽2)11yescharacteristic
102C22×C1272−1+411yescharacteristic
103C4×C127—24no—
104C4×C127—24no—
105C4×C127—22no—
106C4×C127—42no—
107C4×C127—42no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127—42no—
111C2×C127—44no—
112C2×C127#52512yes—
113C2×C127#53711yescharacteristic
114C2×C127#55111yescharacteristic
115C2×C127#55212yes—
116C2×C127#55511yescharacteristic
117#2609C12711yesmaximal, characteristic, direct product factor, Sylow
118C127#260911yescharacteristic, direct product factor, Sylow
119C2×#117C2×C12712yescomplemented
120#398C2×C12711yescharacteristic, complemented
121#396C2×C12712yescomplemented
122#394C2×C12711yescharacteristic
123#388C2×C12712yescomplemented
124#387C2×C12712yescomplemented
125#386C2×C12712yes—
126#385C2×C12712yescomplemented
127#382C2×C12711yescharacteristic, complemented
128C22×D4C22×C12711yescharacteristic
129C23×C4C22×C12712yes—
130#117—28no—
131C2×C4⋊C4C22×C12712yes—
132C2×C4⋊C4C22×C12712yes—
133C2×C4⋊C4C22×C12712yes—
134C2×C4⋊C4C22×C12711yescharacteristic
135C2×C4⋊C4C22×C12711yescharacteristic
136C2×C4⋊C4C22×C12711yescharacteristic
137C2×C22⋊C4C22×C12712yes—
138C2×C22⋊C4C22×C12712yes—
139C2×C22⋊C4C22×C12712yes—
140C2×C22⋊C4C22×C12712yes—
141C2×C22⋊C4C22×C12712yes—
142C2×C42C22×C12712yes—
143C2×C42C22×C12711yescharacteristic
144#95C22×C12712yes—
145#95C22×C12712yes—
146#95C22×C12711yescharacteristic
147#95C22×C12712yes—
148#95C22×C12711yescharacteristic
149#95C22×C12712yes—
150C24C23×C12712yes—
151C2×D4—24no—
152C2×D4—41no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4C23×C12711yescharacteristic
157C22×C4C23×C12711yescharacteristic
158C22×C4C23×C12712yes—
159C22×C4C23×C12712yes—
160C22×C4C23×C12712yes—
161C22×C4C23×C12712yes—
162C22×C4C23×C12711yescharacteristic
163C22×C4C23×C12712yes—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—42no—
175C42—24no—
176C42D4×C12714yescomplemented
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23#41×C12714yes—
184C23C24×C12711yescharacteristic
185D4—44no—
186C2×C4—216no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4C2×D4×C12714yes—
207C2×C4C2×D4×C12712yes—
208C2×C4#41×C12714yes—
209C22—24no—
210C22—24no—
211C22—24no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22C22×D4×C12711yescharacteristic
217C22C2×#41×C12712yes—
218C22C2×#41×C12712yes—
219C22H(5,𝔽2)×C12711yescharacteristic
220C222−1+4×C12711yescharacteristic
221C4—24no—
222C4—24no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—42no—
229C2—44no—
230C2#525×C12712yes—
231C2#537×C12711yescharacteristic
232C2#551×C12711yescharacteristic
233C2#552×C12712yes—
234C2#555×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×#117×C127C2semidirect
2. #398×C127C2semidirect
3. #396×C127C2semidirect
5. #388×C127C2semidirect
6. #387×C127C2semidirect
8. #385×C127C2semidirect
9. #382×C127C2semidirect
58. C42×C127D4semidirect
117. #2609C127direct (N × H)
118. C127#2609direct (N × H)
119. C2×#117C2×C127semidirect
120. #398C2×C127semidirect
121. #396C2×C127semidirect
123. #388C2×C127semidirect
124. #387C2×C127semidirect
126. #385C2×C127semidirect
127. #382C2×C127semidirect
176. C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#117×C1272
#398×C1272
#396×C1272
#394×C1272
#388×C1272
#387×C1272
#386×C1272
#385×C1272
#382×C1272
#2609127