Finite Group Databasebeta

← #26114133

Subgroups — #26114133

Order 12032.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#20252469C211yesmaximal, characteristic, complemented
2#20243384C211yesmaximal, characteristic, complemented
3#20241813C212yesmaximal
4#20241246C212yesmaximal, complemented
5C47×#2536C211yesmaximal, characteristic
6Q8×C47⋊C8—24no—
7#15689176—22no—
8#15689176C2212yes—
9#15689147C2212yes—
10C2×#13093943—21no—
11#15688892—21no—
12D4⋊(C47⋊C8)—24no—
13C2×C22⋊Q8×C47—21no—
14C2×C4×Q8×C47C2211yescharacteristic
15C2×C4×D4×C47C2211yescharacteristic
16C47×#398C412yes—
17C47×#394C412yes—
18C47×#388—22no—
19C47×#382C2211yescharacteristic
20C47×#380—21no—
21C2×#10625329—22no—
22C4⋊(C47⋊C8)—24no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26#13093943—42no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—24no—
29C4×C47⋊C8D414yes—
30C22×Q8×C47D411yescharacteristic
31C22×D4×C47C2×C411yescharacteristic
32C23×C4×C47—21no—
33C22⋊Q8×C47—42no—
34C22⋊Q8×C47—42no—
35C4×Q8×C47—22no—
36C4×Q8×C47—22no—
37C4×D4×C47—22no—
38C4×D4×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C412yes—
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—21no—
47C2×C42×C47D412yes—
48C2×C42×C47C2311yescharacteristic
49#95×C47—22no—
50#95×C47—21no—
51#95×C47C2×C412yes—
52C24×C47—21no—
53C2×Q8×C47—22no—
54C2×Q8×C47—22no—
55C2×Q8×C47D812yes—
56C2×Q8×C47SD1612yes—
57C2×D4×C47—22no—
58C2×D4×C47—42no—
59C22×C4×C47—22no—
60C22×C4×C47—21no—
61C22×C4×C47—42no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C412yes—
66C22×C4×C47C22⋊C412yes—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C4⋊C4×C47—24no—
71C4⋊C4×C47—22no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—44no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C22⋊C4×C47—44no—
77C22⋊C4×C47—44no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C42×C47—22no—
81C42×C47—22no—
82C42×C47C2×D412yes—
83C42×C47#4112yes—
84#10625329—42no—
85#10625329—42no—
86C2×C47⋊C8—24no—
87C2×C47⋊C8—42no—
88C23×C47—22no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47C2×C22⋊C411yescharacteristic
96Q8×C47—24no—
97Q8×C47—24no—
98D4×C47—24no—
99D4×C47—42no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—82no—
113C2×C4×C47—82no—
114C2×C4×C47#10214yes—
115C2×C4×C47C4×D412yes—
116C2×C4×C47#12011yescharacteristic
117C2×C4×C47#12112yes—
118C2×C4×C47#12311yescharacteristic
119C2×C4×C47C2×D811yescharacteristic
120C2×C4×C47C2×SD1611yescharacteristic
121C2×C4×C47#13512yes—
122C47⋊C8—44no—
123C47⋊C8—82no—
124#9486—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—42no—
128C22×C47—42no—
129C22×C47—42no—
130C22×C47—81no—
131C22×C47—81no—
132C22×C47—81no—
133C22×C47—81no—
134C22×C47#38611yescharacteristic
135C22×C47C2×#10211yescharacteristic
136C22×C47#41611yescharacteristic
137C4×C47—22no—
138C4×C47—24no—
139C4×C47—21no—
140C4×C47—21no—
141C4×C47—42no—
142C4×C47—82no—
143C4×C47—81no—
144C4×C47#46512yes—
145C4×C47#46612yes—
146#2982—471no—
147#2536D4711yescharacteristic
148#1801—471no—
149#1486—472no—
150#1286—472no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47—82no—
154C2×C47#189111yescharacteristic
155C2×C47#189311yescharacteristic
156C2×C47#191211yescharacteristic
157C2×C22⋊Q8—21no—
158C2×C4×Q8C2×D4711yescharacteristic
159C2×C4×D4C2×D4711yescharacteristic
160C8×Q8—944no—
161#431—472no—
162#431—942no—
163#423—472no—
164C2×#101—941no—
165#398Dic4712yes—
166#394Dic4712yes—
167#388—22no—
168#382C2×D4711yescharacteristic
169#380—21no—
170#343—941no—
171D4⋊C8—944no—
172C47#948611yescharacteristic, complemented, Sylow
173C22×Q8C22⋊D4711yescharacteristic
174C22×D4C2×Dic4711yescharacteristic
175C23×C4—21no—
176C2×M16—942no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic4711yescharacteristic
186C2×C4⋊C4C2×Dic4712yes—
187C2×C4⋊C4C22⋊D4711yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C42C22⋊D4712yes—
192C2×C42C22×D4711yescharacteristic
193C4⋊C8—944no—
194C4⋊C8—942no—
195C4⋊C8—942no—
196C4⋊C8—942no—
197#101—1882no—
198C22⋊C8—942no—
199C4×C8—474no—
200C4×C8—944no—
201#95—22no—
202#95—21no—
203#95C2×Dic4712yes—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1062533412yes—
208C2×Q8#1062533512yes—
209C2×D4—22no—
210C2×D4—42no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊Dic4712yes—
218C22×C4C22⋊Dic4712yes—
219C22×C4C22×Dic4711yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221C22×C4C2×C22⋊D4711yescharacteristic
222M16—1882no—
223M16—1882no—
224C2×C8—944no—
225C2×C8—1882no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C42—22no—
237C42—22no—
238C42D4×D4712yes—
239C42#1062535212yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic4711yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—24no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#1309394514yes—
267C2×C4D4×Dic4712yes—
268C2×C4#1309399211yescharacteristic
269C2×C4#1309399612yes—
270C2×C4#1309400511yescharacteristic
271C2×C4C2×#1062533411yescharacteristic
272C2×C4C2×#1062533511yescharacteristic
273C2×C4#1309404412yes—
274C8—3762no—
275C8—1884no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#1568900311yescharacteristic
286C22C2×#1309394511yescharacteristic
287C22#1568911911yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#1568935012yes—
296C4#1568935712yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2024380711yescharacteristic
301C2#2024381711yescharacteristic
302C2#2024392611yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2536C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2536D47
hypocenterC47#9486
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #20252469C2semidirect
2. #20243384C2semidirect
4. #20241246C2semidirect
172. C47#9486semidirect

Maximal subgroups

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

subgroupindex
#202524692
#202433842
#202418132
#202412462
C47×#25362
#948647