Finite Group Databasebeta

← C2×C127×#1325 #44575119

Subgroups — C2×C127×#1325 #44575119

Order 32512.

598 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#384×C127C211yesmaximal, characteristic
2C2×#324×C127C212yesmaximal
3C127×#1325C214yesmaximal, direct product factor
4C22×C4⋊C4×C127—22no—
5C22×C4⋊C4×C127C411yescharacteristic
6C22×C4⋊C4×C127C2211yescharacteristic
7C22×C42×C127C411yescharacteristic
8C2×C22⋊C8×C127—22no—
9#384×C127—22no—
10#384×C127C412yes—
11#384×C127C2212yes—
12#324×C127—22no—
13#324×C127C2214yes—
14C23×C4×C127—21no—
15C23×C4×C127—21no—
16C23×C4×C127C2×C411yescharacteristic
17C23×C4×C127D412yes—
18C22×C8×C127—42no—
19C2×C4⋊C4×C127—24no—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127—24no—
22C2×C4⋊C4×C127—41no—
23C2×C4⋊C4×C127—41no—
24C2×C4⋊C4×C127—48no—
25C2×C4⋊C4×C127—41no—
26C2×C4⋊C4×C127—41no—
27C2×C4⋊C4×C127C2×C412yes—
28C2×C4⋊C4×C127C2×C411yescharacteristic
29C2×C4⋊C4×C127D412yes—
30C2×C4⋊C4×C127C2311yescharacteristic
31C2×C42×C127—21no—
32C2×C42×C127—22no—
33C2×C42×C127C814yescomplemented
34C2×C42×C127C2×C412yes—
35C22⋊C8×C127—24no—
36C22⋊C8×C127—42no—
37C24×C127C22⋊C411yescharacteristic
38C22×C4×C127—24no—
39C22×C4×C127—24no—
40C22×C4×C127—24no—
41C22×C4×C127—21no—
42C22×C4×C127—21no—
43C22×C4×C127—22no—
44C22×C4×C127—22no—
45C22×C4×C127—22no—
46C22×C4×C127—22no—
47C22×C4×C127—22no—
48C22×C4×C127—44no—
49C22×C4×C127—44no—
50C22×C4×C127—41no—
51C22×C4×C127—41no—
52C22×C4×C127C22⋊C411yescharacteristic
53C22×C4×C127C22⋊C412yes—
54C22×C4×C127C2×C812yes—
55C22×C4×C127M1612yes—
56C22×C4×C127C22×C411yescharacteristic
57C22×C4×C127C2×D412yes—
58C2×C8×C127—44no—
59C2×C8×C127—84no—
60C4⋊C4×C127—41no—
61C4⋊C4×C127—42no—
62C4⋊C4×C127—41no—
63C4⋊C4×C127—42no—
64C4⋊C4×C127—48no—
65C4⋊C4×C127—48no—
66C4⋊C4×C127—42no—
67C42×C127—24no—
68C42×C127—22no—
69C42×C127C2×C814yescomplemented
70C23×C127—22no—
71C23×C127—22no—
72C23×C127C22⋊C812yes—
73C23×C127#9911yescharacteristic
74C23×C127#9911yescharacteristic
75C23×C127#10112yes—
76C23×C127C2×C22⋊C411yescharacteristic
77C2×C4×C127—24no—
78C2×C4×C127—216no—
79C2×C4×C127—24no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—24no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—44no—
92C2×C4×C127—42no—
93C2×C4×C127—42no—
94C2×C4×C127—44no—
95C2×C4×C127—42no—
96C2×C4×C127—82no—
97C2×C4×C127—82no—
98C2×C4×C127—82no—
99C2×C4×C127C22⋊C812yes—
100C2×C4×C127C22×C811yescharacteristic
101C2×C4×C127C2×M1611yescharacteristic
102C8×C127—84no—
103C22×C127—24no—
104C22×C127—24no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127#32412yes—
110C22×C127#32412yes—
111C22×C127#35312yes—
112C22×C127#35612yes—
113C22×C127C2×C22⋊C811yescharacteristic
114C22×C127C2×#9911yescharacteristic
115C22×C127C2×#10111yescharacteristic
116C4×C127—28no—
117C4×C127—22no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127—42no—
121C4×C127—42no—
122C4×C127—82no—
123C2×#1325C12711yesmaximal, characteristic, direct product factor, Sylow
124C2×C127—21no—
125C2×C127—22no—
126C2×C127—21no—
127C2×C127#132514yesdirect product factor
128C2×C127C2×#32411yescharacteristic
129C2×C127C2×#35311yescharacteristic
130C2×C127C2×#35611yescharacteristic
131C2×#384C2×C12711yescharacteristic
132C2×#324C2×C12712yes—
133#1325C2×C12714yesdirect product factor
134C127C2×#132511yescharacteristic, direct product factor, Sylow
135C22×C4⋊C4—22no—
136C22×C4⋊C4C4×C12711yescharacteristic
137C22×C4⋊C4C22×C12711yescharacteristic
138C22×C42C4×C12711yescharacteristic
139C2×C22⋊C8—22no—
140#384—22no—
141#384C4×C12712yes—
142#384C22×C12712yes—
143#324—22no—
144#324C22×C12714yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×C4×C12711yescharacteristic
148C23×C4D4×C12712yes—
149C22×C8—42no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C2×C4×C12712yes—
159C2×C4⋊C4C2×C4×C12711yescharacteristic
160C2×C4⋊C4D4×C12712yes—
161C2×C4⋊C4C23×C12711yescharacteristic
162C2×C42—21no—
163C2×C42—22no—
164C2×C42C8×C12714yescomplemented
165C2×C42C2×C4×C12712yes—
166C22⋊C8—24no—
167C22⋊C8—42no—
168C24C22⋊C4×C12711yescharacteristic
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4C22⋊C4×C12711yescharacteristic
184C22×C4C22⋊C4×C12712yes—
185C22×C4C2×C8×C12712yes—
186C22×C4M16×C12712yes—
187C22×C4C22×C4×C12711yescharacteristic
188C22×C4C2×D4×C12712yes—
189C2×C8—44no—
190C2×C8—84no—
191C4⋊C4—41no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—48no—
196C4⋊C4—48no—
197C4⋊C4—42no—
198C42—24no—
199C42—22no—
200C42C2×C8×C12714yescomplemented
201C23—22no—
202C23—22no—
203C23C22⋊C8×C12712yes—
204C23#99×C12711yescharacteristic
205C23#99×C12711yescharacteristic
206C23#101×C12712yes—
207C23C2×C22⋊C4×C12711yescharacteristic
208C2×C4—24no—
209C2×C4—216no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4C22⋊C8×C12712yes—
231C2×C4C22×C8×C12711yescharacteristic
232C2×C4C2×M16×C12711yescharacteristic
233C8—84no—
234C22—24no—
235C22—24no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#324×C12712yes—
241C22#324×C12712yes—
242C22#353×C12712yes—
243C22#356×C12712yes—
244C22C2×C22⋊C8×C12711yescharacteristic
245C22C2×#99×C12711yescharacteristic
246C22C2×#101×C12711yescharacteristic
247C4—28no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C2—21no—
255C2—22no—
256C2—21no—
257C2C127×#132514yesdirect product factor
258C2C2×#324×C12711yescharacteristic
259C2C2×#353×C12711yescharacteristic
260C2C2×#356×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#99
commutatorC2×C4C22×C8×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1325C2direct (N × H)
33. C2×C42×C127C8semidirect
69. C42×C127C2×C8semidirect
123. C2×#1325C127direct (N × H)
127. C2×C127#1325direct (N × H)
133. #1325C2×C127direct (N × H)
134. C127C2×#1325direct (N × H)
164. C2×C42C8×C127semidirect
200. C42C2×C8×C127semidirect
257. C2C127×#1325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#384×C1272
C2×#324×C1272
C127×#13252
C2×#1325127