Finite Group Databasebeta

← #26112001

Subgroups — #26112001

Order 12032.

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

normal subgroups only

This group has 694 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#20251727C211yesmaximal, characteristic
2C2×#15688969C211yesmaximal, characteristic
3C2×#15688861C212yesmaximal
4#20241268C212yesmaximal
5C47×#1817C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C22×C4×Dic47—21no—
8#15689029C412yescomplemented
9#15689021C412yescomplemented
10#15689007—22no—
11#15688996—21no—
12#15688986—21no—
13#15688983C2211yescharacteristic
14#15688969—24no—
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2212yes—
17#15688861—24no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#336—22no—
21C22×Dic94C2×C411yescharacteristic
22C23×Dic47—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—21no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C4×Dic47—24no—
32C2×C4×Dic47—28no—
33C47⋊#95—24no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—21no—
39C47⋊#95D414yes—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C22×C8×C47—22no—
44C2×C42×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47C2×C412yes—
47C2×C42×C47D412yes—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C24×C47C2×D411yescharacteristic
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—24no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—42no—
59C22×C4×C47C22⋊C412yes—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×C4×C47#4112yes—
65C22×Dic47—28no—
66C22×Dic47—24no—
67C22×Dic47—22no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—44no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73C2×Dic94—22no—
74C2×Dic94—42no—
75C2×C8×C47—44no—
76C2×C8×C47—42no—
77C42×C47—24no—
78C42×C47—24no—
79C22⋊Dic47—41no—
80C22⋊Dic47—41no—
81C22⋊Dic47—42no—
82C4⋊Dic47—22no—
83C4⋊Dic47—42no—
84Dic47⋊C4—44no—
85Dic47⋊C4—42no—
86C4×Dic47—216no—
87C23×C47—22no—
88C23×C47—22no—
89C23×C47C2×C22⋊C411yescharacteristic
90C23×C47#12011yescharacteristic
91C23×C47#12311yescharacteristic
92C23×C47#13512yes—
93C23×C47#13712yes—
94C2×C4×C47—24no—
95C2×C4×C47—24no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47#10414yes—
108C2×C4×C47C4×D412yes—
109C2×C4×C47#12112yes—
110C2×Dic47—28no—
111C2×Dic47—44no—
112C2×Dic47—48no—
113C2×Dic47—48no—
114C2×Dic47—44no—
115C2×Dic47—41no—
116C2×Dic47—41no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122Dic94—82no—
123C8×C47—82no—
124#9145—471nomaximal, Sylow
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—24no—
130C22×C47—24no—
131C22×C47#38611yescharacteristic
132C22×C47#41012yes—
133C22×C47#41611yescharacteristic
134C22×C47#41911yescharacteristic
135C22×C47C2×#10412yes—
136C22×C47#45212yes—
137C22×C47#48512yes—
138C4×C47—24no—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—42no—
142Dic47—48no—
143Dic47—81no—
144Dic47—81no—
145Dic47—82no—
146#2855—471no—
147C2×#376—471no—
148#1817D4711yescharacteristic
149C2×#328—472no—
150#1294—472no—
151C2×C47—22no—
152C2×C47—22no—
153C2×C47#150012yes—
154C2×C47#150612yes—
155C2×C47#187311yescharacteristic
156C2×C47#189611yescharacteristic
157C2×C47#189911yescharacteristic
158C2×C22⋊Q8—471no—
159C22×C42—941no—
160C22×C42C2×D4711yescharacteristic
161C2×C22⋊C8C2×D4712yes—
162#391—472no—
163#389—472no—
164#387—942no—
165#385—941no—
166#381—941no—
167#380—471no—
168#376—944no—
169C2×#95—472no—
170C2×#95—941no—
171#336—22no—
172#328—944no—
173C47#914511yescharacteristic, complemented, Sylow
174C22×Q8—471no—
175C23×C4—942no—
176C23×C4D18811yescharacteristic
177C23×C4D18811yescharacteristic
178C23×C4C22×D4711yescharacteristic
179C22×C8—22no—
180C22⋊Q8—942no—
181C22⋊Q8—942no—
182C2×C4⋊C4—471no—
183C2×C4⋊C4—472no—
184C2×C4⋊C4—942no—
185C2×C4⋊C4—941no—
186C2×C22⋊C4—941no—
187C2×C22⋊C4—941no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—944no—
191C2×C42—948no—
192C2×C42C4×D4712yes—
193C2×C42C22⋊D4712yes—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—474no—
197#95—944no—
198#95—942no—
199#95—942no—
200#95—942no—
201#95—942no—
202#95—941no—
203C24C2×D18811yescharacteristic
204C2×Q8—942no—
205C2×Q8—1882no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—948no—
215C22×C4—944no—
216C22×C4—942no—
217C22×C4—941no—
218C22×C4—941no—
219C22×C4—1884no—
220C22×C4—1882no—
221C22×C4—1882no—
222C22×C4(C2×D47)⋊C412yes—
223C22×C4(C2×D47)⋊C412yes—
224C22×C4C2×C4×D4711yescharacteristic
225C22×C4D4×D4712yes—
226C22×C4#1062535212yes—
227C22×C4C2×C22⋊D4711yescharacteristic
228C2×C8—44no—
229C2×C8—42no—
230C4⋊C4—942no—
231C4⋊C4—1882no—
232C4⋊C4—1884no—
233C4⋊C4—1882no—
234C22⋊C4—1881no—
235C22⋊C4—1881no—
236C22⋊C4—1882no—
237C42—24no—
238C42—24no—
239C42—9416no—
240C23—22no—
241C23—22no—
242C23C2×(C2×D47)⋊C411yescharacteristic
243C23#1309399411yescharacteristic
244C23#1309400811yescharacteristic
245C23#1309404612yes—
246C23#1309405712yes—
247Q8—3762no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
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—948no—
262C2×C4—1881no—
263C2×C4—1881no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4—1882no—
269C2×C4—1884no—
270C2×C4—1888no—
271C2×C4—1888no—
272C2×C4—1884no—
273C2×C4#1309395014yes—
274C2×C4(C22⋊D47)⋊C412yes—
275C2×C4#1309399612yes—
276C8—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—24no—
283C22#1568900011yescharacteristic
284C22#1568909612yes—
285C22#1568912011yescharacteristic
286C22#1568913311yescharacteristic
287C22C2×#1309395012yes—
288C22#1568928712yes—
289C22#1568946512yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—3761no—
295C4—3761no—
296C4—3762no—
297C4—1888no—
298C2—22no—
299C2—22no—
300C2#2024187512yes—
301C2#2024190112yes—
302C2#2024369811yescharacteristic
303C2#2024382911yescharacteristic
304C2#2024384411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1817D47
hypocenterC47#9145
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #15689029C4semidirect
9. #15689021C4semidirect
173. C47#9145semidirect

Maximal subgroups

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

subgroupindex
#202517272
C2×#156889692
C2×#156888612
#202412682
C47×#18172
#914547