Finite Group Databasebeta

← #26114028

Subgroups — #26114028

Order 12032.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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#20245991C211yesmaximal, characteristic, complemented
2#20243468C211yesmaximal, characteristic, complemented
3#20241917C212yesmaximal
4#20241241C212yesmaximal, complemented
5C47×#2504C211yesmaximal, characteristic
6#15689087C2212yes—
7#15689009—24no—
8#15688986C2212yes—
9#15688983—21no—
10C4×C22⋊Dic47—21no—
11#15688892—22no—
12#15688857—24no—
13C2×#121×C47C2211yescharacteristic
14C22×C4⋊C4×C47C2211yescharacteristic
15C47×#397—22no—
16C47×#390C412yescomplemented
17C47×#386—21no—
18C47×#386—21no—
19C47×#382C412yescomplemented
20C4×C22⋊C4×C47C2211yescharacteristic
21C2×#10625329—22no—
22C2×C4⋊Dic47—22no—
23C2×C22⋊Dic47—22no—
24C2×C4×Dic47—22no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C47⋊#95D414yes—
31C22×D4×C47—21no—
32C22×D4×C47C2×C411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47C2311yescharacteristic
36#121×C47—22no—
37#121×C47—22no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47C2×C412yes—
42C2×C4⋊C4×C47D412yes—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C42×C47C2×C412yes—
48#95×C47—22no—
49#95×C47—22no—
50#95×C47—21no—
51C24×C47—21no—
52C24×C47C2×D411yescharacteristic
53C2×D4×C47—22no—
54C2×D4×C47—42no—
55C2×D4×C47—44no—
56C2×D4×C47—41no—
57C2×D4×C47—41no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—22no—
63C22×C4×C47—21no—
64C22×C4×C47—42no—
65C22×C4×C47—81no—
66C22×C4×C47—81no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×C4×C47#4112yes—
73C22×Dic47—24no—
74C4⋊C4×C47—24no—
75C4⋊C4×C47—24no—
76C4⋊C4×C47—22no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—82no—
80C22⋊C4×C47—82no—
81C42×C47—42no—
82C22⋊Dic47—44no—
83C4⋊Dic47—82no—
84C4×Dic47—82no—
85#10625329—44no—
86C2×C47⋊C8—42no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—41no—
90C23×C47—42no—
91C23×C47—42no—
92C23×C47—41no—
93C23×C47—42no—
94C23×C47C2×C22⋊C411yescharacteristic
95C23×C47#12011yescharacteristic
96C23×C47#12311yescharacteristic
97D4×C47—82no—
98D4×C47—82no—
99D4×C47—82no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—81no—
116C2×C4×C47—81no—
117C2×C4×C47C4×D412yes—
118C2×C4×C47#12112yes—
119C2×Dic47—44no—
120C2×Dic47—44no—
121C2×Dic47—44no—
122C47⋊C8—82no—
123#9465—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—44no—
132C22×C47—84no—
133C22×C47—81no—
134C22×C47—81no—
135C22×C47—81no—
136C22×C47#38611yescharacteristic
137C22×C47#41012yes—
138C22×C47#45412yes—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—42no—
142C4×C47—42no—
143C4×C47—82no—
144C4×C47—81no—
145Dic47—84no—
146#2504D4711yescharacteristic
147#2313—471no—
148#1821—471no—
149#1510—472no—
150#1284—472no—
151C2×C47—22no—
152C2×C47—22no—
153C2×C47—41no—
154C2×C47—161no—
155C2×C47#188811yescharacteristic
156C2×C47#189911yescharacteristic
157C2×C47#191111yescharacteristic
158C2×#121C2×D4711yescharacteristic
159C22×C4⋊C4C2×D4711yescharacteristic
160#407—472no—
161#397—22no—
162#390Dic4712yescomplemented
163#387—944no—
164#386—21no—
165#386—21no—
166#382Dic4712yescomplemented
167#381—472no—
168#380—941no—
169C4×C22⋊C4—941no—
170C4×C22⋊C4C2×D4711yescharacteristic
171#343—942no—
172#327—944no—
173C47#946511yescharacteristic, complemented, Sylow
174C22×D4—21no—
175C22×D4C2×Dic4711yescharacteristic
176C23×C4C22⋊D4711yescharacteristic
177C23×C4C22⋊D4711yescharacteristic
178C23×C4C22×D4711yescharacteristic
179C2×M16—942no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4C2×Dic4711yescharacteristic
186C2×C4⋊C4C2×Dic4712yes—
187C2×C4⋊C4C22⋊D4712yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—942no—
193C2×C42—942no—
194C2×C42C2×Dic4712yes—
195C22⋊C8—942no—
196C22⋊C8—942no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95—474no—
201#95—942no—
202#95—942no—
203#95—942no—
204C24—21no—
205C24C2×C22⋊D4711yescharacteristic
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—944no—
221C22×C4C22⋊Dic4712yes—
222C22×C4C22⋊Dic4712yes—
223C22×C4D4×D4712yes—
224C22×C4#1062535212yes—
225C22×C4C22×Dic4711yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—1884no—
228C2×C8—1882no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—3762no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—1884no—
238C42—42no—
239C42—3762no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23C2×C22⋊Dic4711yescharacteristic
248C23#1309399211yescharacteristic
249C23#1309400511yescharacteristic
250D4—82no—
251D4—82no—
252D4—82no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—1884no—
271C2×C4—1884no—
272C2×C4—1884no—
273C2×C4D4×Dic4712yes—
274C2×C4#1309399612yes—
275C8—3762no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22#1568900311yescharacteristic
289C22#1568909712yes—
290C22#1568929512yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—82no—
296C4—81no—
297C4—3764no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#2024379211yescharacteristic
303C2#2024384711yescharacteristic
304C2#2024392111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2504C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#2504D47
hypocenterC47#9465
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. #20245991C2semidirect
2. #20243468C2semidirect
4. #20241241C2semidirect
16. C47×#390C4semidirect
19. C47×#382C4semidirect
162. #390Dic47semidirect
166. #382Dic47semidirect
173. C47#9465semidirect

Maximal subgroups

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

subgroupindex
#202459912
#202434682
#202419172
#202412412
C47×#25042
#946547