Finite Group Databasebeta

← #26111217

Subgroups — #26111217

Order 12032.

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

normal subgroups only

This group has 578 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
1C2×#15689518C211yesmaximal, characteristic
2#20244360C212yesmaximal, complemented
3C2×#15688863C211yesmaximal, characteristic
4#20241269C212yesmaximal, complemented
5C47×#2379C211yesmaximal, characteristic
6C22×#10625337—21no—
7C22×#10625335—21no—
8C2×#13094024C2211yescharacteristic
9#15689518—22no—
10#15689518C414yescomplemented
11C2×#13093935C2212yes—
12C2×#13093935C2211yescharacteristic
13#15688885—22no—
14#15688885—22no—
15#15688863—22no—
16C2×#124×C47C2211yescharacteristic
17C47×#391—21no—
18C47×#388—22no—
19C47×#386—21no—
20C47×#385—22no—
21C47×#382C2212yes—
22C22×Dic94—21no—
23C2×#10625337—24no—
24C2×#10625337—42no—
25C2×#10625337—42no—
26C2×#10625335—24no—
27C2×#10625335—42no—
28C2×#10625335—42no—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—22no—
31#13094024—22no—
32#13094024—21no—
33#13094024C2×C412yescomplemented
34C2×C4⋊Dic47—22no—
35#13093935—21no—
36#13093935—22no—
37#13093935—22no—
38#13093935C2×C412yescomplemented
39C22×Q8×C47D411yescharacteristic
40C22×D4×C47D411yescharacteristic
41C23×C4×C47—21no—
42#124×C47—22no—
43#124×C47C2×C412yes—
44#124×C47D412yes—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47D412yes—
47C2×C4⋊C4×C47D412yes—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47—21no—
51C2×C42×C47—21no—
52C2×C42×C47C2311yescharacteristic
53#95×C47—22no—
54#95×C47—22no—
55#95×C47—22no—
56C24×C47—21no—
57C2×Q8×C47—21no—
58C2×Q8×C47—41no—
59C2×Q8×C47—41no—
60C2×Q8×C47C22⋊C412yes—
61C2×D4×C47—21no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47C22⋊C412yes—
65C22×C4×C47—21no—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—81no—
69C22×C4×C47—82no—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×Dic47—41no—
73C2×Dic94—22no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—81no—
82C22⋊C4×C47—24no—
83C22⋊C4×C47—24no—
84C22⋊C4×C47—81no—
85C22⋊C4×C47—82no—
86C42×C47—81no—
87C42×C47C22×C411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47#4112yes—
90#10625337—48no—
91#10625335—48no—
92C4⋊Dic47—24no—
93C4⋊Dic47—42no—
94C2×C47⋊C8—22no—
95C2×C47⋊C8—22no—
96C2×C47⋊C8—22no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—22no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—42no—
104C23×C47—42no—
105C23×C47#12011yescharacteristic
106Q8×C47—42no—
107Q8×C47—81no—
108D4×C47—42no—
109D4×C47—81no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—84no—
124C2×C4×C47C2×C22⋊C411yescharacteristic
125C2×C4×C47C4×D412yes—
126C2×C4×C47#12112yes—
127C2×C4×C47#12311yescharacteristic
128C2×Dic47—42no—
129C2×Dic47—81no—
130C2×Dic47—81no—
131Dic94—42no—
132Dic94—42no—
133Dic94—81no—
134Dic94—81no—
135C47⋊C8—24no—
136C47⋊C8—44no—
137#9003—471nomaximal, Sylow
138C22×C47—44no—
139C22×C47—42no—
140C22×C47—81no—
141C22×C47—81no—
142C22×C47—82no—
143C22×C47#38611yescharacteristic
144C22×C47#45311yescharacteristic
145C22×C47#45311yescharacteristic
146C22×C47#45512yes—
147C22×C47#45612yes—
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—21no—
151C4×C47—21no—
152C4×C47—42no—
153C4×C47—82no—
154C4×C47—82no—
155Dic47—82no—
156C2×#497—471no—
157#2379D4711yescharacteristic
158#1993—472no—
159C2×#329—471no—
160#1295—472no—
161C2×C47—82no—
162C2×C47#165911yescharacteristic
163C2×C47#166011yescharacteristic
164C2×C47#166112yes—
165C2×C47#188212yes—
166C2×C47#188711yescharacteristic
167C22×Q16—941no—
168C22×SD16—941no—
169C2×C4⋊Q8—471no—
170C2×#124C2×D4711yescharacteristic
171#497—474no—
172#497—942no—
173C2×#97—471no—
174C2×#97—472no—
175#391—21no—
176#388—22no—
177#386—21no—
178#385—22no—
179#382C2×D4712yes—
180#340—942no—
181#340—942no—
182#329—942no—
183C47#900311yescharacteristic, complemented, Sylow
184C22×Q8—941no—
185C22×Q8C22⋊D4711yescharacteristic
186C22×D4C22⋊D4711yescharacteristic
187C23×C4—21no—
188C2×Q16—944no—
189C2×Q16—1882no—
190C2×Q16—1882no—
191C2×SD16—944no—
192C2×SD16—1882no—
193C2×SD16—1882no—
194C22×C8—941no—
195C22×C8—942no—
196C4⋊Q8—472no—
197C4⋊Q8—941no—
198C4⋊Q8—942no—
199#124—22no—
200#124C4×D4712yes—
201#124D18812yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—942no—
204C2×C4⋊C4C22⋊D4712yes—
205C2×C4⋊C4C22⋊D4712yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C22×D4711yescharacteristic
211#97—472no—
212#97—941no—
213#97—942no—
214#97—942no—
215#95—22no—
216#95—22no—
217#95—22no—
218C24—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—942no—
223C2×Q8—1881no—
224C2×Q8—1881no—
225C2×Q8—1881no—
226C2×Q8—1881no—
227C2×Q8—1881no—
228C2×Q8(C2×D47)⋊C412yes—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4(C2×D47)⋊C412yes—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—1881no—
239C22×C4C2×C22⋊D4711yescharacteristic
240C22×C4C2×C22⋊D4712yes—
241Q16—1888no—
242SD16—1888no—
243C2×C8—942no—
244C2×C8—942no—
245C2×C8—942no—
246C2×C8—944no—
247C2×C8—1882no—
248C2×C8—1882no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4—944no—
253C4⋊C4—1882no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—81no—
257C22⋊C4—82no—
258C42—81no—
259C42C2×C4×D4711yescharacteristic
260C42C2×D18811yescharacteristic
261C42#1062535012yes—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23#1309399311yescharacteristic
268Q8—42no—
269Q8—81no—
270Q8—3761no—
271Q8—3761no—
272Q8—1882no—
273Q8—1882no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4—3761no—
291C2×C4—3761no—
292C2×C4—1882no—
293C2×C4C2×(C2×D47)⋊C411yescharacteristic
294C2×C4C4×C22⋊D4712yes—
295C2×C4#1309399812yes—
296C2×C4#1309400711yescharacteristic
297C8—944no—
298C8—1884no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#1568900211yescharacteristic
305C22#1568929111yescharacteristic
306C22#1568929111yescharacteristic
307C22#1568930112yes—
308C22#1568930612yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—3762no—
317C2—82no—
318C2#2024269411yescharacteristic
319C2#2024269911yescharacteristic
320C2#2024270512yes—
321C2#2024375712yes—
322C2#2024378611yescharacteristic
subgroupG/H
centerC23#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2379C2
socleC23×C47#120
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2379D47
hypocenterC47#9003
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #20244360C2semidirect
4. #20241269C2semidirect
10. #15689518C4semidirect
33. #13094024C2×C4semidirect
38. #13093935C2×C4semidirect
183. C47#9003semidirect

Maximal subgroups

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

subgroupindex
C2×#156895182
#202443602
C2×#156888632
#202412692
C47×#23792
#900347