Finite Group Databasebeta

← #26111806

Subgroups — #26111806

Order 12032.

522 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 115 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#20252473C211yesmaximal, characteristic, complemented
2#20246095C211yesmaximal, characteristic
3#20241778C212yesmaximal
4#20241216C212yesmaximal, complemented
5C47×#1461C211yesmaximal, characteristic, complemented
6C2×#13094013C411yescharacteristic
7C2×C4×D188C2211yescharacteristic
8#15689254—24no—
9#15689180—22no—
10#15689176C2212yes—
11#15689030C411yescharacteristic
12#15689024C411yescharacteristic
13#15689014—22no—
14#15688994—22no—
15#15688992C411yescharacteristic
16#15688969C2211yescharacteristic
17#15688850—24no—
18C2×C4×Q8×C47C2211yescharacteristic
19C47×#431C2212yes—
20C47×#324—22no—
21C22×D188C2×C411yescharacteristic
22C22×C4×D47—21no—
23#13094013—22no—
24#13094013—22no—
25C4×D188—22no—
26C4×D188—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—21no—
31C2×(C2×D47)⋊C4—22no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C4⋊(C47⋊C8)—24no—
35C4⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37#13093935—24no—
38C4×C47⋊C8D414yes—
39C47⋊#95—22no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C22×Q8×C47C2×C411yescharacteristic
43C4×Q8×C47—22no—
44C4×Q8×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C4⋊C4×C47D412yes—
47C2×C42×C47D412yes—
48C2×C42×C47C2311yescharacteristic
49C4⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C4×C8×C47—22no—
52C23×D47—21no—
53C2×Q8×C47—22no—
54C2×Q8×C47—21no—
55C2×Q8×C47—21no—
56C2×Q8×C47—22no—
57C22×C4×C47C22⋊C412yes—
58C22×C4×C47C22⋊C412yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47—21no—
65C2×D188—22no—
66C2×D188—22no—
67C2×D188—21no—
68C2×D188M1612yes—
69C2×C4×D47—42no—
70C2×C4×D47—82no—
71C2×C8×C47—42no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—24no—
74C42×C47—22no—
75C42×C47—22no—
76C42×C47C2×D412yes—
77C42×C47#4112yes—
78(C2×D47)⋊C4—28no—
79(C2×D47)⋊C4—41no—
80(C2×D47)⋊C4—41no—
81(C2×D47)⋊C4—82no—
82C4⋊Dic47—22no—
83Dic47⋊C4—82no—
84C4×Dic47—21no—
85C4×Dic47—22no—
86C4×Dic47M1612yes—
87C2×C47⋊C8—24no—
88C23×C47C2×C22⋊C411yescharacteristic
89C22×D47—22no—
90C22×D47—42no—
91C22×D47—42no—
92C22×D47—41no—
93C22×D47—41no—
94Q8×C47—42no—
95Q8×C47—42no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—42no—
103C2×C4×C47C4×D412yes—
104C2×C4×C47#12011yescharacteristic
105C2×C4×C47#12112yes—
106C2×C4×C47#12311yescharacteristic
107C2×C4×C47C2×M1611yescharacteristic
108C2×C4×C47C2×M1611yescharacteristic
109C2×C4×C47#13112yes—
110C2×Dic47—22no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—42no—
114C2×Dic47—41no—
115C2×Dic47—41no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118D188—24no—
119D188—42no—
120C4×D47—84no—
121C8×C47—82no—
122C47⋊C8—44no—
123#9106—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47#38611yescharacteristic
127C22×C47#40711yescharacteristic
128C22×C47#40811yescharacteristic
129C2×D47—44no—
130C2×D47—42no—
131C2×D47—82no—
132C2×D47—81no—
133C2×D47—81no—
134C4×C47—22no—
135C4×C47—21no—
136C4×C47—21no—
137C4×C47—42no—
138C4×C47—42no—
139C4×C47#43512yes—
140C4×C47#43612yes—
141Dic47—42no—
142Dic47—81no—
143Dic47—81no—
144#2983—471no—
145#2326—471no—
146#1478—472no—
147#1461D4711yescharacteristic, complemented
148#1274—472no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47#187011yescharacteristic
152C2×C47#191011yescharacteristic
153C2×C47#191211yescharacteristic
154D47—82no—
155C2×#124—471no—
156C2×C4×Q8C2×D4711yescharacteristic
157C2×C4×D4—471no—
158#446—944no—
159#432—942no—
160#431—472no—
161#431C2×D4712yes—
162#391—471no—
163#390—471no—
164#388—942no—
165#385—942no—
166#384—471no—
167#376—471no—
168D4⋊C8—944no—
169#324—22no—
170C47#910611yescharacteristic, complemented, Sylow
171C22×Q8C4×D4711yescharacteristic
172C22×D4—471no—
173C23×C4—941no—
174#124—942no—
175#124—942no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—942no—
179C4×D4—942no—
180C2×C4⋊C4—471no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4C4×D4711yescharacteristic
183C2×C4⋊C4C22⋊D4712yes—
184C2×C22⋊C4—942no—
185C2×C22⋊C4—942no—
186C2×C22⋊C4—941no—
187C2×C42—471no—
188C2×C42—471no—
189C2×C42D18812yes—
190C2×C42C22×D4711yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—944no—
193C4⋊C8—942no—
194C22⋊C8—22no—
195C22⋊C8—942no—
196#97—944no—
197C4×C8—22no—
198C4×C8—474no—
199#95—942no—
200#95—941no—
201#95—941no—
202C24—941no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×D4—472no—
208C2×D4—942no—
209C2×D4—941no—
210C2×D4—942no—
211C22×C4—941no—
212C22×C4—941no—
213C22×C4—941no—
214C22×C4—3762no—
215C22×C4—1882no—
216C22×C4(C2×D47)⋊C412yes—
217C22×C4(C2×D47)⋊C412yes—
218C22×C4C2×C4×D4711yescharacteristic
219C22×C4C2×D18811yescharacteristic
220C22×C4C2×C22⋊D4711yescharacteristic
221C2×C8—42no—
222C2×C8—944no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—942no—
226C4⋊C4—3762no—
227C22⋊C4—948no—
228C22⋊C4—3762no—
229C22⋊C4—1881no—
230C22⋊C4—1881no—
231C42—22no—
232C42—22no—
233C42—472no—
234C42—941no—
235C42—942no—
236C42D4×D4712yes—
237C42#1062535212yes—
238C23—942no—
239C23—1882no—
240C23—1882no—
241C23—1881no—
242C23—1881no—
243C23C2×(C2×D47)⋊C411yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—944no—
247D4—1882no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—942no—
256C2×C4—3764no—
257C2×C4—1881no—
258C2×C4—1882no—
259C2×C4—1881no—
260C2×C4—1881no—
261C2×C4—1881no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4(C22⋊D47)⋊C412yes—
265C2×C4#1309399411yescharacteristic
266C2×C4#1309399612yes—
267C2×C4#1309400811yescharacteristic
268C2×C4M16×D4711yescharacteristic
269C2×C4M16×D4711yescharacteristic
270C2×C4#1309403012yes—
271C8—82no—
272C8—1884no—
273C22—21no—
274C22—21no—
275C22—3762no—
276C22—3761no—
277C22—3761no—
278C22—1884no—
279C22—1882no—
280C22#1568900011yescharacteristic
281C22#1568908611yescharacteristic
282C22#1568908911yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—3761no—
289C4—3761no—
290C4—1882no—
291C4#1568919712yes—
292C4#1568919812yes—
293C2—21no—
294C2—21no—
295C2—3762no—
296C2#2024368311yescharacteristic
297C2#2024391311yescharacteristic
298C2#2024392311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1461D47
hypocenterC47#9106
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
4. #20241216C2semidirect
5. C47×#1461C2semidirect
147. #1461D47semidirect
170. C47#9106semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202460952
#202417782
#202412162
C47×#14612
#910647