Finite Group Databasebeta

← C4×#15688918 #26131444

Subgroups — C4×#15688918 #26131444

Order 12032.

584 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 139 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×#15688918C212yesmaximal
2C4×#13093939C212yesmaximal
3#20241513C212yesmaximal
4C4×#124×C47C211yesmaximal, characteristic
5C2×#13093939C2214yes—
6C4×C22⋊Dic47—22no—
7#15688918C418yesdirect product factor
8#15688890—22no—
9C2×#124×C47C2211yescharacteristic
10C2×C4×Q8×C47C411yescharacteristic, complemented
11C2×C4×D4×C47C2211yescharacteristic
12C47×#391C411yescharacteristic, complemented
13C47×#390C2211yescharacteristic
14C47×#388—22no—
15C47×#382C411yescharacteristic, complemented
16C4×C22⋊C4×C47—22no—
17C43×C47C411yescharacteristic, complemented
18C2×C22⋊Dic47—24no—
19C2×C4×Dic47—42no—
20#13093939—24no—
21#13093939C2×C418yes—
22C47⋊#95—42no—
23C22×Q8×C47C2×C411yescharacteristic
24C22×D4×C47C2311yescharacteristic
25C23×C4×C47D412yes—
26#124×C47—22no—
27#124×C47C2×C414yes—
28C4×Q8×C47—41no—
29C4×Q8×C47—41no—
30C4×D4×C47—41no—
31C4×D4×C47—41no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47C2×C411yescharacteristic
34C2×C4⋊C4×C47C2×C411yescharacteristic
35C2×C22⋊C4×C47—22no—
36C2×C22⋊C4×C47—22no—
37C2×C22⋊C4×C47D412yes—
38C2×C42×C47—21no—
39C2×C42×C47—21no—
40C2×C42×C47—21no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44#95×C47—22no—
45#95×C47—22no—
46C24×C47C2×D412yes—
47C2×Q8×C47—21no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47C4212yescomplemented
51C2×D4×C47—41no—
52C2×D4×C47—41no—
53C2×D4×C47C22×C412yes—
54C2×D4×C47#4112yes—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—21no—
61C22×C4×C47—44no—
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×Dic47—44no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—42no—
69C4⋊C4×C47—41no—
70C22⋊C4×C47—28no—
71C22⋊C4×C47—42no—
72C22⋊C4×C47—44no—
73C42×C47—21no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—42no—
77C42×C47—42no—
78C42×C47—41no—
79C42×C47—41no—
80C42×C47—41no—
81C42×C47C4212yescomplemented
82C22⋊Dic47—28no—
83C22⋊Dic47—44no—
84C4×Dic47—84no—
85C23×C47—22no—
86C23×C47—22no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47C2×C22⋊C411yescharacteristic
90C23×C47C4×D414yes—
91Q8×C47—42no—
92Q8×C47—42no—
93D4×C47—42no—
94D4×C47—42no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—44no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—82no—
115C2×C4×C47#9911yescharacteristic
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47C2×C4211yescharacteristic
118C2×C4×C47#11711yescharacteristic
119C2×Dic47—48no—
120C2×Dic47—84no—
121C2×Dic47—84no—
122C4×#354—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—24no—
126C22×C47—44no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47—82no—
130C22×C47C4×C22⋊C411yescharacteristic
131C22×C47C2×#9911yescharacteristic
132C22×C47#41011yescharacteristic
133C4×C47—21no—
134C4×C47—21no—
135C4×C47—21no—
136C4×C47—42no—
137C4×C47—42no—
138C4×C47—42no—
139C4×C47—41no—
140C4×C47—42no—
141C4×C47—81no—
142C4×C47#35412yescomplemented
143Dic47—88no—
144C4×#124D4711yescharacteristic
145C2×#354—472no—
146C4×#99—472no—
147#1393—472no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47—44no—
151C2×C47C4×#9911yescharacteristic
152C2×C47C2×#35411yescharacteristic
153C2×C47#212611yescharacteristic
154C2×#124C2×D4711yescharacteristic
155C2×C4×Q8Dic4711yescharacteristic, complemented
156C2×C4×D4C2×D4711yescharacteristic
157C2×#99—474no—
158#391Dic4711yescharacteristic, complemented
159#390C2×D4711yescharacteristic
160#388—22no—
161#382Dic4711yescharacteristic, complemented
162C4×C22⋊C4—22no—
163C4×C22⋊C4—942no—
164C43Dic4711yescharacteristic, complemented
165#354—478no—
166#342—942no—
167C47C4×#35411yescharacteristic, complemented, Sylow
168C22×Q8C2×Dic4711yescharacteristic
169C22×D4C22×D4711yescharacteristic
170C23×C4C22⋊D4712yes—
171#124—22no—
172#124C4×D4714yes—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C2×Dic4711yescharacteristic
179C2×C4⋊C4C2×Dic4711yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—944no—
183C2×C22⋊C4C22⋊D4712yes—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—1882no—
188C2×C42C2×Dic4711yescharacteristic
189C2×C42C2×Dic4711yescharacteristic
190C2×C42C2×Dic4711yescharacteristic
191#99—478no—
192#99—944no—
193#95—22no—
194#95—22no—
195#95—1882no—
196C24C2×C22⋊D4712yes—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4×Dic4712yescomplemented
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C2×C4×D4712yes—
204C2×D4#1062535012yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—44no—
212C22×C4—1884no—
213C22×C4C22⋊Dic4712yes—
214C22×C4C22⋊Dic4711yescharacteristic
215C22×C4C22⋊Dic4711yescharacteristic
216C22×C4C22×Dic4711yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C22⋊C4—28no—
221C22⋊C4—42no—
222C22⋊C4—44no—
223C22⋊C4—948no—
224C22⋊C4—1884no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—41no—
229C42—42no—
230C42—42no—
231C42—41no—
232C42—41no—
233C42—3764no—
234C42C4×Dic4712yescomplemented
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23C2×C22⋊Dic4711yescharacteristic
240C23C4×C22⋊D4714yes—
241Q8—42no—
242Q8—42no—
243D4—42no—
244D4—42no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—3764no—
266C2×C4—3764no—
267C2×C4—1888no—
268C2×C4#1309393911yescharacteristic
269C2×C4#1309393911yescharacteristic
270C2×C4C2×C4×Dic4711yescharacteristic
271C2×C4#1309398311yescharacteristic
272C22—21no—
273C22—21no—
274C22—24no—
275C22—44no—
276C22—42no—
277C22—42no—
278C22—82no—
279C22C4×C22⋊Dic4711yescharacteristic
280C22C2×#1309393911yescharacteristic
281C22#1568909711yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—81no—
291C4—3768no—
292C4#1568891812yesdirect product factor
293C2—21no—
294C2—21no—
295C2—44no—
296C2C4×#1309393911yescharacteristic
297C2C2×#1568891811yescharacteristic
298C2#2024505311yescharacteristic
subgroupG/H
centerC2×C4#13093939
commutatorC2×C4×C47C2×C42
FrattiniC22×D4C22×D47
FittingC4×#124×C47C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenterC4×#124D47
hypocenterC47C4×#354
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
7. #15688918C4direct (N × H)
10. C2×C4×Q8×C47C4semidirect
12. C47×#391C4semidirect
15. C47×#382C4semidirect
17. C43×C47C4semidirect
50. C2×Q8×C47C42semidirect
81. C42×C47C42semidirect
142. C4×C47#354semidirect
155. C2×C4×Q8Dic47semidirect
158. #391Dic47semidirect
161. #382Dic47semidirect
164. C43Dic47semidirect
167. C47C4×#354semidirect
200. C2×Q8C4×Dic47semidirect
234. C42C4×Dic47semidirect
292. C4#15688918direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156889182
C4×#130939392
#202415132
C4×#124×C472
C4×#35447