Finite Group Databasebeta

← C127×#8514 #44572187

Subgroups — C127×#8514 #44572187

Order 32512.

648 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3568C211yesmaximal, characteristic
2C127×#1521C213yesmaximal, complemented
3C127×#1482C213yesmaximal, complemented
4#563×C127C413yescomplemented
5H(3,𝔽4)×C127C411yescharacteristic, complemented
6#556×C127—23no—
7#551×C127—23no—
8#542×C127—23no—
9#538×C127—23no—
10#529×C127C2213yes—
11#431×C127C2213yescomplemented
12#383×C127C2211yescharacteristic, complemented
13#328×C127—23no—
14#327×C127—23no—
15Q8⋊C8×C127—23no—
16D4⋊C8×C127—23no—
17C4⋊Q8×C127—23no—
18#126×C127—23no—
19#126×C127—21no—
20#126×C127—23no—
21#126×C127—23no—
22#126×C127—23no—
23#125×C127C2×C413yes—
24#124×C127—23no—
25#124×C127—23no—
26#124×C127—23no—
27#124×C127C2×C413yes—
28#123×C127—23no—
29#123×C127—23no—
30#123×C127—23no—
31C22⋊Q8×C127—23no—
32C22⋊Q8×C127—23no—
33C22⋊Q8×C127—23no—
34C22⋊Q8×C127D413yes—
35#121×C127—23no—
36#121×C127D413yes—
37#120×C127—21no—
38#120×C127—23no—
39C4×Q8×C127—23no—
40C4×Q8×C127D413yes—
41C4×D4×C127—23no—
42C4×D4×C127—23no—
43C4×D4×C127D413yes—
44#117×C127—23no—
45#117×C127—23no—
46C2×C22⋊C4×C127—23no—
47C2×C42×C127C2311yescharacteristic
48C4⋊C8×C127—23no—
49C22⋊C8×C127—23no—
50C4×C8×C127—23no—
51#95×C127—23no—
52C24×C127—21no—
53C2×Q8×C127—23no—
54C2×Q8×C127C22⋊C413yes—
55C2×D4×C127—23no—
56C2×D4×C127—23no—
57C2×D4×C127C22⋊C413yes—
58C22×C4×C127—23no—
59C22×C4×C127—23no—
60C22×C4×C127—41no—
61C22×C4×C127C2×D413yes—
62C2×C8×C127—43no—
63C4⋊C4×C127—23no—
64C4⋊C4×C127—23no—
65C4⋊C4×C127—23no—
66C4⋊C4×C127—23no—
67C4⋊C4×C127—23no—
68C4⋊C4×C127—23no—
69C4⋊C4×C127C22⋊C413yes—
70C4⋊C4×C127C22⋊C413yes—
71C22⋊C4×C127—23no—
72C22⋊C4×C127—23no—
73C22⋊C4×C127—23no—
74C22⋊C4×C127—23no—
75C22⋊C4×C127—23no—
76C22⋊C4×C127—23no—
77C22⋊C4×C127—23no—
78C22⋊C4×C127—23no—
79C22⋊C4×C127—23no—
80C42×C127—23no—
81C42×C127—21no—
82C42×C127—23no—
83C42×C127C22×C411yescharacteristic
84C42×C127C2×D413yes—
85C23×C127—23no—
86C23×C127—83no—
87C23×C127#12011yescharacteristic
88Q8×C127—43no—
89Q8×C127—83no—
90D4×C127—43no—
91D4×C127—83no—
92D4×C127—83no—
93C2×C4×C127—23no—
94C2×C4×C127—23no—
95C2×C4×C127—23no—
96C2×C4×C127—43no—
97C2×C4×C127—83no—
98C2×C4×C127—83no—
99C2×C4×C127—83no—
100C2×C4×C127C2×C22⋊C413yes—
101C2×C4×C127#12013yes—
102C8×C127—83no—
103C22×C127—23no—
104C22×C127—43no—
105C22×C127—43no—
106C22×C127—43no—
107C22×C127—162no—
108C22×C127#37911yescharacteristic
109C4×C127—23no—
110C4×C127—23no—
111C4×C127—43no—
112C4×C127—43no—
113C4×C127—43no—
114C4×C127—161no—
115#8514C12711yesmaximal, characteristic, direct product factor, Sylow
116C2×C127—41no—
117C2×C127—83no—
118C2×C127#179413yes—
119#3568C2×C12711yescharacteristic
120#1521C2×C12713yescomplemented
121#1482C2×C12713yescomplemented
122C127#851411yescharacteristic, direct product factor, Sylow
123#563C4×C12713yescomplemented
124H(3,𝔽4)C4×C12711yescharacteristic, complemented
125#556—23no—
126#551—23no—
127#542—23no—
128#538—23no—
129#529C22×C12713yes—
130#431C22×C12713yescomplemented
131#383C22×C12711yescharacteristic, complemented
132#328—23no—
133#327—23no—
134Q8⋊C8—23no—
135D4⋊C8—23no—
136C4⋊Q8—23no—
137#126—23no—
138#126—21no—
139#126—23no—
140#126—23no—
141#126—23no—
142#125C2×C4×C12713yes—
143#124—23no—
144#124—23no—
145#124—23no—
146#124C2×C4×C12713yes—
147#123—23no—
148#123—23no—
149#123—23no—
150C22⋊Q8—23no—
151C22⋊Q8—23no—
152C22⋊Q8—23no—
153C22⋊Q8D4×C12713yes—
154#121—23no—
155#121D4×C12713yes—
156#120—21no—
157#120—23no—
158C4×Q8—23no—
159C4×Q8D4×C12713yes—
160C4×D4—23no—
161C4×D4—23no—
162C4×D4D4×C12713yes—
163#117—23no—
164#117—23no—
165C2×C22⋊C4—23no—
166C2×C42C23×C12711yescharacteristic
167C4⋊C8—23no—
168C22⋊C8—23no—
169C4×C8—23no—
170#95—23no—
171C24—21no—
172C2×Q8—23no—
173C2×Q8C22⋊C4×C12713yes—
174C2×D4—23no—
175C2×D4—23no—
176C2×D4C22⋊C4×C12713yes—
177C22×C4—23no—
178C22×C4—23no—
179C22×C4—41no—
180C22×C4C2×D4×C12713yes—
181C2×C8—43no—
182C4⋊C4—23no—
183C4⋊C4—23no—
184C4⋊C4—23no—
185C4⋊C4—23no—
186C4⋊C4—23no—
187C4⋊C4—23no—
188C4⋊C4C22⋊C4×C12713yes—
189C4⋊C4C22⋊C4×C12713yes—
190C22⋊C4—23no—
191C22⋊C4—23no—
192C22⋊C4—23no—
193C22⋊C4—23no—
194C22⋊C4—23no—
195C22⋊C4—23no—
196C22⋊C4—23no—
197C22⋊C4—23no—
198C22⋊C4—23no—
199C42—23no—
200C42—21no—
201C42—23no—
202C42C22×C4×C12711yescharacteristic
203C42C2×D4×C12713yes—
204C23—23no—
205C23—83no—
206C23#120×C12711yescharacteristic
207Q8—43no—
208Q8—83no—
209D4—43no—
210D4—83no—
211D4—83no—
212C2×C4—23no—
213C2×C4—23no—
214C2×C4—23no—
215C2×C4—43no—
216C2×C4—83no—
217C2×C4—83no—
218C2×C4—83no—
219C2×C4C2×C22⋊C4×C12713yes—
220C2×C4#120×C12713yes—
221C8—83no—
222C22—23no—
223C22—43no—
224C22—43no—
225C22—43no—
226C22—162no—
227C22#379×C12711yescharacteristic
228C4—23no—
229C4—23no—
230C4—43no—
231C4—43no—
232C4—43no—
233C4—161no—
234C2—41no—
235C2—83no—
236C2C127×#179413yes—
subgroupG/H
center, socleC22×C127#379
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3568C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1521C2semidirect
3. C127×#1482C2semidirect
4. #563×C127C4semidirect
5. H(3,𝔽4)×C127C4semidirect
11. #431×C127C22semidirect
12. #383×C127C22semidirect
115. #8514C127direct (N × H)
120. #1521C2×C127semidirect
121. #1482C2×C127semidirect
122. C127#8514direct (N × H)
123. #563C4×C127semidirect
124. H(3,𝔽4)C4×C127semidirect
130. #431C22×C127semidirect
131. #383C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35682
C127×#15212
C127×#14822
#8514127