Finite Group Databasebeta

← #26114025

Subgroups — #26114025

Order 12032.

554 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 99 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#20247876C211yesmaximal, characteristic
2#20245991C211yesmaximal, characteristic, complemented
3#20241918C212yesmaximal
4#20241242C212yesmaximal, complemented
5C47×#1821C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7#15689049—22no—
8#15689024C412yescomplemented
9#15689009—24no—
10#15689002—21no—
11#15689002—21no—
12#15688988C412yescomplemented
13#15688986C2212yes—
14#15688983—21no—
15C4×C22⋊Dic47—21no—
16C4×C22⋊Dic47C2211yescharacteristic
17#15688858—24no—
18C22×C4⋊C4×C47C2211yescharacteristic
19C47×#407C2212yes—
20C47×#343—22no—
21C22×D188C2×C411yescharacteristic
22C22×C22⋊D47—21no—
23#13093998—22no—
24#13093998—22no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—21no—
29C2×(C2×D47)⋊C4—21no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×C4×Dic47—22no—
33C2×C4×Dic47C2×C412yes—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C47⋊#95D414yes—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C2×M16×C47—22no—
45C2×C4⋊C4×C47—22no—
46C2×C4⋊C4×C47—22no—
47C2×C4⋊C4×C47C2×C412yes—
48C2×C4⋊C4×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47C2×D411yescharacteristic
52C23×D47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—42no—
58C22×C4×C47—81no—
59C22×C4×C47—81no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×C4×C47#4112yes—
66C2×C22⋊D47—44no—
67C2×C22⋊D47—41no—
68C2×C22⋊D47—41no—
69C22×Dic47—21no—
70C22×Dic47—22no—
71C22×Dic47—24no—
72C2×D188—22no—
73C2×D188—42no—
74M16×C47—44no—
75C2×C8×C47—42no—
76C4⋊C4×C47—24no—
77C4⋊C4×C47—24no—
78C22⋊Dic47—44no—
79C22⋊Dic47—42no—
80(C2×D47)⋊C4—42no—
81(C2×D47)⋊C4—82no—
82(C2×D47)⋊C4—82no—
83C4⋊Dic47—22no—
84C4⋊Dic47—82no—
85C4×Dic47—42no—
86C4×Dic47—82no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47—41no—
90C23×C47C2×C22⋊C411yescharacteristic
91C23×C47#12011yescharacteristic
92C23×C47#12311yescharacteristic
93C22×D47—42no—
94C22×D47—42no—
95C22×D47—41no—
96C22×D47—42no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—81no—
108C2×C4×C47—81no—
109C2×C4×C47C4×D412yes—
110C2×C4×C47#12112yes—
111C22⋊D47—82no—
112C22⋊D47—82no—
113C2×Dic47—44no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—44no—
117C2×Dic47—42no—
118C2×Dic47—41no—
119C2×Dic47—42no—
120C2×Dic47—44no—
121D188—82no—
122C8×C47—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#38611yescharacteristic
133C22×C47#41012yes—
134C22×C47#45412yes—
135C2×D47—84no—
136C2×D47—81no—
137C2×D47—81no—
138C2×D47—81no—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—42no—
142C4×C47—42no—
143Dic47—84no—
144Dic47—82no—
145Dic47—81no—
146#2504—471no—
147#2313—471no—
148#1821D4711yescharacteristic, complemented
149#1510—472no—
150#1284—472no—
151C2×C47—22no—
152C2×C47—22no—
153C2×C47—41no—
154C2×C47#188811yescharacteristic
155C2×C47#189911yescharacteristic
156C2×C47#191111yescharacteristic
157D47—161no—
158C2×#121—471no—
159C22×C4⋊C4C2×D4711yescharacteristic
160#407C2×D4712yes—
161#397—942no—
162#390—472no—
163#387—944no—
164#386—941no—
165#386—941no—
166#382—472no—
167#381—472no—
168#380—941no—
169C4×C22⋊C4—471no—
170C4×C22⋊C4—941no—
171#343—22no—
172#327—944no—
173C47#946511yescharacteristic, complemented, Sylow
174C22×D4—471no—
175C22×D4—941no—
176C23×C4D18811yescharacteristic
177C23×C4D18811yescharacteristic
178C23×C4C22×D4711yescharacteristic
179C2×M16—22no—
180#121—942no—
181#121—942no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4—942no—
186C2×C4⋊C4C4×D4712yes—
187C2×C4⋊C4C22⋊D4712yes—
188C2×C22⋊C4—942no—
189C2×C22⋊C4—942no—
190C2×C22⋊C4—941no—
191C2×C22⋊C4—942no—
192C2×C22⋊C4—941no—
193C2×C42—472no—
194C2×C42—942no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—474no—
198#95—942no—
199#95—942no—
200#95—942no—
201#95—942no—
202#95—942no—
203#95—941no—
204C24—941no—
205C24C2×D18811yescharacteristic
206C2×D4—942no—
207C2×D4—1882no—
208C2×D4—1884no—
209C2×D4—1881no—
210C2×D4—1881no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—941no—
219C22×C4—942no—
220C22×C4—944no—
221C22×C4(C2×D47)⋊C412yes—
222C22×C4(C2×D47)⋊C412yes—
223C22×C4C2×C4×D4711yescharacteristic
224C22×C4D4×D4712yes—
225C22×C4#1062535212yes—
226C22×C4C2×C22⋊D4711yescharacteristic
227M16—44no—
228C2×C8—42no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—942no—
232C4⋊C4—3762no—
233C22⋊C4—3762no—
234C22⋊C4—3762no—
235C22⋊C4—1882no—
236C22⋊C4—1884no—
237C22⋊C4—1882no—
238C42—3762no—
239C42—1882no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—1882no—
244C23—1882no—
245C23—1881no—
246C23—1882no—
247C23C2×(C2×D47)⋊C411yescharacteristic
248C23#1309399411yescharacteristic
249C23#1309400811yescharacteristic
250D4—3762no—
251D4—3762no—
252D4—3762no—
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—81no—
264C2×C4—81no—
265C2×C4—1884no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4—1884no—
269C2×C4—1882no—
270C2×C4—1881no—
271C2×C4—1884no—
272C2×C4—1882no—
273C2×C4(C22⋊D47)⋊C412yes—
274C2×C4#1309399612yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—3764no—
285C22—3761no—
286C22—3761no—
287C22—3761no—
288C22#1568900011yescharacteristic
289C22#1568909612yes—
290C22#1568929812yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—3764no—
296C4—3762no—
297C4—3761no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—7521no—
302C2#2024378911yescharacteristic
303C2#2024384411yescharacteristic
304C2#2024391811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1821C2
socleC22×C47#386
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1821D47
hypocenterC47#9465
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #20245991C2semidirect
4. #20241242C2semidirect
5. C47×#1821C2semidirect
8. #15689024C4semidirect
12. #15688988C4semidirect
148. #1821D47semidirect
173. C47#9465semidirect

Maximal subgroups

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

subgroupindex
#202478762
#202459912
#202419182
#202412422
C47×#18212
#946547