Finite Group Databasebeta

← C2×#20241318 #26125360

Subgroups — C2×#20241318 #26125360

Order 12032.

726 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 147 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×#15688847C212yesmaximal
2#20241318C214yesmaximal, direct product factor
3C2×C47×#381C211yesmaximal, characteristic
4C2×C22⋊(C47⋊C8)—22no—
5#15688847—22no—
6#15688847C2214yes—
7C24×C4×C47C411yescharacteristic
8C22×C22⋊C4×C47C411yescharacteristic
9C22×C22⋊C4×C47C2211yescharacteristic
10C47×#381—22no—
11C47×#381C412yes—
12C47×#381C2212yes—
13C2×#95×C47—22no—
14C22×C47⋊C8—42no—
15C22⋊(C47⋊C8)—24no—
16C22⋊(C47⋊C8)—42no—
17C25×C47C2×C411yescharacteristic
18C23×C4×C47—21no—
19C23×C4×C47—21no—
20C23×C4×C47—22no—
21C23×C4×C47—21no—
22C23×C4×C47—41no—
23C23×C4×C47—41no—
24C23×C4×C47—42no—
25C23×C4×C47C814yescomplemented
26C23×C4×C47C2×C412yes—
27C23×C4×C47D412yes—
28C2×C22⋊C4×C47—21no—
29C2×C22⋊C4×C47—41no—
30C2×C22⋊C4×C47—41no—
31C2×C22⋊C4×C47—42no—
32C2×C22⋊C4×C47C2×C412yes—
33C2×C22⋊C4×C47C2×C411yescharacteristic
34C2×C22⋊C4×C47D412yes—
35C2×C22⋊C4×C47C2311yescharacteristic
36#95×C47—24no—
37#95×C47—24no—
38C24×C47—21no—
39C24×C47—21no—
40C24×C47—22no—
41C24×C47—44no—
42C24×C47C22⋊C411yescharacteristic
43C24×C47C22⋊C411yescharacteristic
44C24×C47C2×C812yes—
45C24×C47M1612yes—
46C24×C47C22×C411yescharacteristic
47C22×C4×C47—24no—
48C22×C4×C47—24no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—44no—
55C22×C4×C47—44no—
56C22×C4×C47—42no—
57C22×C4×C47—42no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—42no—
61C22×C4×C47—44no—
62C22×C4×C47—44no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47—44no—
66C22×C4×C47—44no—
67C22×C4×C47—42no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C2×C814yescomplemented
70C22×C4×C47C2×D412yes—
71C22⋊C4×C47—41no—
72C22⋊C4×C47—42no—
73C22⋊C4×C47—41no—
74C22⋊C4×C47—42no—
75C22⋊C4×C47—42no—
76C2×C47⋊C8—44no—
77C2×C47⋊C8—84no—
78C23×C47—24no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—22no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—42no—
88C23×C47—44no—
89C23×C47—44no—
90C23×C47—42no—
91C23×C47—48no—
92C23×C47—44no—
93C23×C47—44no—
94C23×C47C22⋊C812yes—
95C23×C47C22⋊C812yes—
96C23×C47#9911yescharacteristic
97C23×C47#9911yescharacteristic
98C23×C47#10012yes—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47C22×C811yescharacteristic
101C23×C47C2×M1611yescharacteristic
102C2×C4×C47—24no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—48no—
106C2×C4×C47—44no—
107C2×C4×C47—44no—
108C2×C4×C47—44no—
109C2×C4×C47—44no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—44no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—44no—
116C2×C4×C47—42no—
117C2×C4×C47—82no—
118C2×C4×C47—84no—
119C47⋊C8—84no—
120C2×#1317—471nomaximal, Sylow
121C22×C47—24no—
122C22×C47—24no—
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—22no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—42no—
129C22×C47—41no—
130C22×C47—41no—
131C22×C47—42no—
132C22×C47—42no—
133C22×C47—44no—
134C22×C47—44no—
135C22×C47—44no—
136C22×C47—44no—
137C22×C47—42no—
138C22×C47—42no—
139C22×C47#32312yes—
140C22×C47#32312yes—
141C22×C47#35214yes—
142C22×C47C2×C22⋊C811yescharacteristic
143C22×C47C2×#9911yescharacteristic
144C22×C47C2×#10011yescharacteristic
145C4×C47—42no—
146C4×C47—42no—
147C4×C47—44no—
148C4×C47—82no—
149C2×#381D4711yescharacteristic
150C2×#323—472no—
151#1317—474no—
152C2×C47—21no—
153C2×C47—22no—
154C2×C47—21no—
155C2×C47—42no—
156C2×C47—41no—
157C2×C47—41no—
158C2×C47#131714yescomplemented
159C2×C47C2×#32311yescharacteristic
160C2×C47C2×#35212yes—
161C24×C4Dic4711yescharacteristic
162C22×C22⋊C4Dic4711yescharacteristic
163C22×C22⋊C4C2×D4711yescharacteristic
164C2×C22⋊C8—942no—
165#381—22no—
166#381Dic4712yes—
167#381C2×D4712yes—
168C2×#95—22no—
169#323—474no—
170#323—942no—
171C47C2×#131711yescharacteristic, complemented, Sylow
172C25C2×Dic4711yescharacteristic
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—42no—
180C23×C4C47⋊C814yescomplemented
181C23×C4C2×Dic4712yes—
182C23×C4C22⋊D4712yes—
183C22×C8—1882no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4C2×Dic4712yes—
189C2×C22⋊C4C2×Dic4711yescharacteristic
190C2×C22⋊C4C22⋊D4712yes—
191C2×C22⋊C4C22×D4711yescharacteristic
192C22⋊C8—944no—
193C22⋊C8—1882no—
194#95—24no—
195#95—24no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—44no—
200C24C2×C47⋊C812yes—
201C24#1062532912yes—
202C24C22⋊Dic4711yescharacteristic
203C24C22⋊Dic4711yescharacteristic
204C24C22×Dic4711yescharacteristic
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4C2×C47⋊C814yescomplemented
227C22×C4C22⋊Dic4712yes—
228C22×C4C2×C22⋊D4712yes—
229C2×C8—3764no—
230C2×C8—1884no—
231C22⋊C4—41no—
232C22⋊C4—42no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C23—24no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—44no—
246C23—44no—
247C23—42no—
248C23—42no—
249C23—44no—
250C23—48no—
251C23—44no—
252C23C22⋊(C47⋊C8)12yes—
253C23C22⋊(C47⋊C8)12yes—
254C23#1309393911yescharacteristic
255C23#1309393911yescharacteristic
256C23#1309394112yes—
257C23C2×C22⋊Dic4711yescharacteristic
258C23C22×C47⋊C811yescharacteristic
259C23C2×#1062532911yescharacteristic
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—84no—
277C8—3764no—
278C22—22no—
279C22—22no—
280C22—24no—
281C22—24no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—44no—
295C22—44no—
296C22#1568884712yes—
297C22#1568884712yes—
298C22#1568891314yes—
299C22C2×C22⋊(C47⋊C8)11yescharacteristic
300C22C2×#1309393911yescharacteristic
301C22C2×#1309394111yescharacteristic
302C4—44no—
303C4—42no—
304C4—42no—
305C4—82no—
306C2—21no—
307C2—21no—
308C2—22no—
309C2—42no—
310C2—41no—
311C2—41no—
312C2#2024131814yesdirect product factor
313C2C2×#1568884711yescharacteristic
314C2C2×#1568891312yes—
subgroupG/H
centerC23#13093939
commutatorC23×C47C22×C8
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#381C2
socleC23×C47#99
prime radicalC25×C47C2×C4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#381D47
hypocenterC47C2×#1317
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
2. #20241318C2direct (N × H)
25. C23×C4×C47C8semidirect
69. C22×C4×C47C2×C8semidirect
158. C2×C47#1317semidirect
171. C47C2×#1317semidirect
180. C23×C4C47⋊C8semidirect
226. C22×C4C2×C47⋊C8semidirect
312. C2#20241318direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156888472
#202413182
C2×C47×#3812
C2×#131747