Finite Group Databasebeta

← C2×C127×#1319 #44575103

Subgroups — C2×C127×#1319 #44575103

Order 32512.

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

normal subgroups only

This group has 618 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×#391×C127C211yesmaximal, characteristic
2C2×#324×C127C212yesmaximal
3C127×#1319C214yesmaximal, direct product factor
4C23×Q8×C127C411yescharacteristic
5C22×C4⋊C4×C127C2211yescharacteristic
6C22×C42×C127C411yescharacteristic
7C2×C22⋊C8×C127—22no—
8#391×C127—22no—
9#391×C127C412yes—
10#391×C127C2212yes—
11C2×#95×C127—22no—
12#324×C127—22no—
13#324×C127C2214yes—
14C22×Q8×C127—21no—
15C22×Q8×C127—22no—
16C22×Q8×C127—41no—
17C22×Q8×C127—41no—
18C22×Q8×C127—42no—
19C22×Q8×C127C814yescomplemented
20C22×Q8×C127C2×C412yes—
21C23×C4×C127—21no—
22C23×C4×C127—21no—
23C23×C4×C127C2×C411yescharacteristic
24C23×C4×C127D412yes—
25C22×C8×C127—42no—
26C2×C4⋊C4×C127—41no—
27C2×C4⋊C4×C127—41no—
28C2×C4⋊C4×C127C2×C411yescharacteristic
29C2×C4⋊C4×C127D412yes—
30C2×C4⋊C4×C127C2311yescharacteristic
31C2×C42×C127—21no—
32C2×C42×C127—42no—
33C2×C42×C127C2×C412yes—
34C22⋊C8×C127—24no—
35C22⋊C8×C127—42no—
36#95×C127—24no—
37#95×C127—24no—
38C24×C127C22⋊C411yescharacteristic
39C2×Q8×C127—24no—
40C2×Q8×C127—22no—
41C2×Q8×C127—44no—
42C2×Q8×C127—44no—
43C2×Q8×C127—42no—
44C2×Q8×C127—42no—
45C2×Q8×C127—44no—
46C2×Q8×C127—44no—
47C2×Q8×C127—42no—
48C2×Q8×C127—42no—
49C2×Q8×C127C2×C814yescomplemented
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127—22no—
53C22×C4×C127—24no—
54C22×C4×C127—24no—
55C22×C4×C127—22no—
56C22×C4×C127—22no—
57C22×C4×C127—22no—
58C22×C4×C127—22no—
59C22×C4×C127—42no—
60C22×C4×C127—41no—
61C22×C4×C127—41no—
62C22×C4×C127—42no—
63C22×C4×C127—42no—
64C22×C4×C127—44no—
65C22×C4×C127C22⋊C411yescharacteristic
66C22×C4×C127C22⋊C412yes—
67C22×C4×C127C2×C812yes—
68C22×C4×C127M1612yes—
69C22×C4×C127C22×C411yescharacteristic
70C22×C4×C127C2×D412yes—
71C2×C8×C127—44no—
72C2×C8×C127—84no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—42no—
75C4⋊C4×C127—41no—
76C42×C127—42no—
77C42×C127—42no—
78C23×C127—22no—
79C23×C127—22no—
80C23×C127C22⋊C812yes—
81C23×C127#9911yescharacteristic
82C23×C127#9911yescharacteristic
83C23×C127#10012yes—
84C23×C127C2×C22⋊C411yescharacteristic
85Q8×C127—44no—
86Q8×C127—48no—
87Q8×C127—44no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—24no—
97C2×C4×C127—24no—
98C2×C4×C127—22no—
99C2×C4×C127—22no—
100C2×C4×C127—48no—
101C2×C4×C127—42no—
102C2×C4×C127—44no—
103C2×C4×C127—44no—
104C2×C4×C127—44no—
105C2×C4×C127—42no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—42no—
109C2×C4×C127—42no—
110C2×C4×C127—82no—
111C2×C4×C127C22⋊C812yes—
112C2×C4×C127C22×C811yescharacteristic
113C2×C4×C127C2×M1611yescharacteristic
114C8×C127—84no—
115C22×C127—24no—
116C22×C127—24no—
117C22×C127—21no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—21no—
121C22×C127#32312yes—
122C22×C127#32312yes—
123C22×C127#35412yes—
124C22×C127#35612yes—
125C22×C127C2×C22⋊C811yescharacteristic
126C22×C127C2×#9911yescharacteristic
127C22×C127C2×#10011yescharacteristic
128C4×C127—22no—
129C4×C127—21no—
130C4×C127—21no—
131C4×C127—44no—
132C4×C127—42no—
133C4×C127—42no—
134C4×C127—44no—
135C2×#1319C12711yesmaximal, characteristic, direct product factor, Sylow
136C2×C127—21no—
137C2×C127—22no—
138C2×C127—21no—
139C2×C127#131914yesdirect product factor
140C2×C127C2×#32311yescharacteristic
141C2×C127C2×#35411yescharacteristic
142C2×C127C2×#35611yescharacteristic
143C2×#391C2×C12711yescharacteristic
144C2×#324C2×C12712yes—
145#1319C2×C12714yesdirect product factor
146C127C2×#131911yescharacteristic, direct product factor, Sylow
147C23×Q8C4×C12711yescharacteristic
148C22×C4⋊C4C22×C12711yescharacteristic
149C22×C42C4×C12711yescharacteristic
150C2×C22⋊C8—22no—
151#391—22no—
152#391C4×C12712yes—
153#391C22×C12712yes—
154C2×#95—22no—
155#324—22no—
156#324C22×C12714yes—
157C22×Q8—21no—
158C22×Q8—22no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8C8×C12714yescomplemented
163C22×Q8C2×C4×C12712yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2×C4×C12711yescharacteristic
167C23×C4D4×C12712yes—
168C22×C8—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C2×C4×C12711yescharacteristic
172C2×C4⋊C4D4×C12712yes—
173C2×C4⋊C4C23×C12711yescharacteristic
174C2×C42—21no—
175C2×C42—42no—
176C2×C42C2×C4×C12712yes—
177C22⋊C8—24no—
178C22⋊C8—42no—
179#95—24no—
180#95—24no—
181C24C22⋊C4×C12711yescharacteristic
182C2×Q8—24no—
183C2×Q8—22no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8C2×C8×C12714yescomplemented
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4C22⋊C4×C12711yescharacteristic
209C22×C4C22⋊C4×C12712yes—
210C22×C4C2×C8×C12712yes—
211C22×C4M16×C12712yes—
212C22×C4C22×C4×C12711yescharacteristic
213C22×C4C2×D4×C12712yes—
214C2×C8—44no—
215C2×C8—84no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C42—42no—
220C42—42no—
221C23—22no—
222C23—22no—
223C23C22⋊C8×C12712yes—
224C23#99×C12711yescharacteristic
225C23#99×C12711yescharacteristic
226C23#100×C12712yes—
227C23C2×C22⋊C4×C12711yescharacteristic
228Q8—44no—
229Q8—48no—
230Q8—44no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—48no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4C22⋊C8×C12712yes—
255C2×C4C22×C8×C12711yescharacteristic
256C2×C4C2×M16×C12711yescharacteristic
257C8—84no—
258C22—24no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22#323×C12712yes—
265C22#323×C12712yes—
266C22#354×C12712yes—
267C22#356×C12712yes—
268C22C2×C22⋊C8×C12711yescharacteristic
269C22C2×#99×C12711yescharacteristic
270C22C2×#100×C12711yescharacteristic
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—42no—
276C4—42no—
277C4—44no—
278C2—21no—
279C2—22no—
280C2—21no—
281C2C127×#131914yesdirect product factor
282C2C2×#323×C12711yescharacteristic
283C2C2×#354×C12711yescharacteristic
284C2C2×#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×#1319C2direct (N × H)
19. C22×Q8×C127C8semidirect
49. C2×Q8×C127C2×C8semidirect
135. C2×#1319C127direct (N × H)
139. C2×C127#1319direct (N × H)
145. #1319C2×C127direct (N × H)
146. C127C2×#1319direct (N × H)
162. C22×Q8C8×C127semidirect
192. C2×Q8C2×C8×C127semidirect
281. C2C127×#1319direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#391×C1272
C2×#324×C1272
C127×#13192
C2×#1319127