Finite Group Databasebeta

← C127×#12861 #44576534

Subgroups — C127×#12861 #44576534

Order 32512.

642 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2297C211yesmaximal, characteristic
2C127×#1785C212yesmaximal
3C127×#1394C214yesmaximal, complemented
4C22×#41×C127C2211yescharacteristic
5C2×#117×C127C411yescharacteristic
6C22×C42×C127C411yescharacteristic
7C2×#101×C127C2214yes—
8C2×#100×C127—22no—
9#407×C127—22no—
10#391×C127C2212yes—
11#390×C127C412yes—
12#386×C127—22no—
13C4×C22⋊C4×C127—22no—
14C2×#41×C127—24no—
15C2×#41×C127—42no—
16C2×#41×C127—42no—
17C22×Q8×C127C2311yescharacteristic
18C22×D4×C127C2×C411yescharacteristic
19C22×D4×C127D412yes—
20C23×C4×C127—21no—
21C23×C4×C127C2×C411yescharacteristic
22C23×C4×C127D412yes—
23C2×M16×C127—24no—
24#117×C127—42no—
25C2×C4⋊C4×C127C2×C412yes—
26C2×C22⋊C4×C127—21no—
27C2×C22⋊C4×C127—24no—
28C2×C42×C127—21no—
29C2×C42×C127—21no—
30C2×C42×C127—42no—
31C2×C42×C127C2×C412yes—
32#101×C127—28no—
33#100×C127—44no—
34C22⋊C8×C127—44no—
35#95×C127—24no—
36C24×C127C22⋊C412yes—
37C24×C127C22⋊C411yescharacteristic
38#41×C127—416no—
39C2×Q8×C127—41no—
40C2×Q8×C127—41no—
41C2×Q8×C127#4114yes—
42C2×D4×C127—24no—
43C2×D4×C127—22no—
44C2×D4×C127—24no—
45C2×D4×C127—41no—
46C2×D4×C127—41no—
47C2×D4×C127—42no—
48C22×C4×C127—22no—
49C22×C4×C127—22no—
50C22×C4×C127—22no—
51C22×C4×C127—22no—
52C22×C4×C127—24no—
53C22×C4×C127—24no—
54C22×C4×C127—22no—
55C22×C4×C127—42no—
56C22×C4×C127—44no—
57C22×C4×C127—42no—
58C22×C4×C127C22⋊C411yescharacteristic
59C22×C4×C127C22×C411yescharacteristic
60C22×C4×C127C2×D412yes—
61M16×C127—44no—
62M16×C127—44no—
63C2×C8×C127—44no—
64C4⋊C4×C127—42no—
65C22⋊C4×C127—48no—
66C22⋊C4×C127—42no—
67C42×C127—42no—
68C42×C127—42no—
69C42×C127—42no—
70C23×C127—21no—
71C23×C127—21no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—42no—
76C23×C127—44no—
77C23×C127#9912yes—
78C23×C127C2×C22⋊C411yescharacteristic
79Q8×C127—44no—
80D4×C127—48no—
81D4×C127—44no—
82C2×C4×C127—28no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—21no—
88C2×C4×C127—22no—
89C2×C4×C127—24no—
90C2×C4×C127—48no—
91C2×C4×C127—42no—
92C2×C4×C127—48no—
93C2×C4×C127—42no—
94C2×C4×C127—44no—
95C2×C4×C127—42no—
96C2×C4×C127#9912yes—
97C2×C4×C127#11712yes—
98C2×C4×C127#12314yes—
99C8×C127—84no—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—44no—
105C22×C127—42no—
106C22×C127—42no—
107C22×C127—44no—
108C22×C127—82no—
109C22×C127#38111yescharacteristic
110C22×C127C2×#9911yescharacteristic
111C22×C127C2×#9911yescharacteristic
112C4×C127—24no—
113C4×C127—22no—
114C4×C127—22no—
115C4×C127—44no—
116C4×C127—82no—
117#12861C12711yesmaximal, characteristic, direct product factor, Sylow
118C2×C127—21no—
119C2×C127—21no—
120C2×C127—42no—
121C2×C127—82no—
122C2×C127#181911yescharacteristic
123C2×C127#213212yes—
124#2297C2×C12711yescharacteristic
125#1785C2×C12712yes—
126#1394C2×C12714yescomplemented
127C127#1286111yescharacteristic, direct product factor, Sylow
128C22×#41C22×C12711yescharacteristic
129C2×#117C4×C12711yescharacteristic
130C22×C42C4×C12711yescharacteristic
131C2×#101C22×C12714yes—
132C2×#100—22no—
133#407—22no—
134#391C22×C12712yes—
135#390C4×C12712yes—
136#386—22no—
137C4×C22⋊C4—22no—
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8C23×C12711yescharacteristic
142C22×D4C2×C4×C12711yescharacteristic
143C22×D4D4×C12712yes—
144C23×C4—21no—
145C23×C4C2×C4×C12711yescharacteristic
146C23×C4D4×C12712yes—
147C2×M16—24no—
148#117—42no—
149C2×C4⋊C4C2×C4×C12712yes—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—24no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—42no—
155C2×C42C2×C4×C12712yes—
156#101—28no—
157#100—44no—
158C22⋊C8—44no—
159#95—24no—
160C24C22⋊C4×C12712yes—
161C24C22⋊C4×C12711yescharacteristic
162#41—416no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#41×C12714yes—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4C22⋊C4×C12711yescharacteristic
183C22×C4C22×C4×C12711yescharacteristic
184C22×C4C2×D4×C12712yes—
185M16—44no—
186M16—44no—
187C2×C8—44no—
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—44no—
201C23#99×C12712yes—
202C23C2×C22⋊C4×C12711yescharacteristic
203Q8—44no—
204D4—48no—
205D4—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4#99×C12712yes—
221C2×C4#117×C12712yes—
222C2×C4#123×C12714yes—
223C8—84no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—82no—
233C22#381×C12711yescharacteristic
234C22C2×#99×C12711yescharacteristic
235C22C2×#99×C12711yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2C127×#181911yescharacteristic
246C2C127×#213212yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1394C2semidirect
117. #12861C127direct (N × H)
126. #1394C2×C127semidirect
127. C127#12861direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#22972
C127×#17852
C127×#13942
#12861127