Finite Group Databasebeta

← #26134968

Subgroups — #26134968

Order 12032.

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

normal subgroups only

This group has 396 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#20248575C211yesmaximal, characteristic
2C4×#13093943C211yesmaximal, characteristic
3#20241514C212yesmaximal
4#20241514C212yesmaximal
5C47×#1801C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic
7C2×#13093943C2212yes—
8C4×#10625329—21no—
9#15689056—21no—
10#15689056—21no—
11#15689030C2211yescharacteristic
12#15689030C2211yescharacteristic
13#15689007—21no—
14#15689007—21no—
15#15688992C411yescharacteristic
16#15688988C411yescharacteristic
17C4×C4⋊Dic47C411yescharacteristic
18#15688892—21no—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#423—21no—
21C2×#101×C47C2212yes—
22C47×#343—21no—
23C2×#10625329—22no—
24#13094016—41no—
25#13094016—41no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47C2×C411yescharacteristic
35#13093943—22no—
36#13093943Q814yes—
37#13093935—41no—
38C4×C47⋊C8—41no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C22×Q8×C47C2311yescharacteristic
47C2×M16×C47—22no—
48C4×Q8×C47—41no—
49C4×Q8×C47—41no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C4⋊C4×C47D411yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C4⋊C8×C47—41no—
57C4⋊C8×C47—41no—
58#101×C47—24no—
59C2×Q8×C47—41no—
60C2×Q8×C47—41no—
61C2×Q8×C47C2×Q812yes—
62C2×Q8×C47#4112yes—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×Dic47—21no—
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74M16×C47—42no—
75M16×C47—42no—
76C2×C8×C47—42no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—21no—
86C4⋊Dic47—42no—
87C4⋊Dic47—41no—
88C4⋊Dic47—41no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93Dic47⋊C4—41no—
94C4×Dic47—41no—
95C4×Dic47—41no—
96#10625329—24no—
97#10625329—42no—
98C2×C47⋊C8—42no—
99C23×C47C2×C22⋊C411yescharacteristic
100Q8×C47—42no—
101Q8×C47—42no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47#9911yescharacteristic
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47C4×Q812yes—
118C2×C4×C47C22⋊Q812yes—
119C2×C4×C47#12411yescharacteristic
120C2×C4×C47C4⋊Q811yescharacteristic
121C2×Dic47—22no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—42no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C8×C47—82no—
133C47⋊C8—44no—
134#13303—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47#39111yescharacteristic
138C22×C47C2×#9911yescharacteristic
139C22×C47#41011yescharacteristic
140C4×C47—21no—
141C4×C47—21no—
142C4×C47—22no—
143C4×C47—21no—
144C4×C47—22no—
145C4×C47—21no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149Dic47—42no—
150Dic47—81no—
151Dic47—81no—
152Dic47—81no—
153#2565—471no—
154#1801D4711yescharacteristic
155C4×#101—471no—
156#1394—472no—
157#1394—472no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47#199611yescharacteristic
161C2×C47#213211yescharacteristic
162C2×C47#213311yescharacteristic
163C2×#125—471no—
164C2×C4×Q8C2×D4711yescharacteristic
165#423—21no—
166C2×#101—472no—
167C2×#101C2×D4712yes—
168C4×M16—941no—
169#398—941no—
170#398—941no—
171#391—471no—
172#391—471no—
173#387—941no—
174#387—941no—
175#384—471no—
176#382—471no—
177C4×C4⋊C4—471no—
178#343—21no—
179#343—941no—
180C47#1330311yescharacteristic, complemented, Sylow
181C22×Q8C22×D4711yescharacteristic
182C2×M16—22no—
183C2×M16—942no—
184#125—1881no—
185#125—1881no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—471no—
189C2×C4⋊C4—471no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4C4×D4711yescharacteristic
196C2×C4⋊C4D18811yescharacteristic
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C42—471no—
199C2×C42—471no—
200C2×C42C4×D4711yescharacteristic
201C2×C42D18811yescharacteristic
202C2×C42C22⋊D4711yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205#101—24no—
206#101—474no—
207#101—942no—
208#97—1881no—
209C4×C8—1881no—
210#95—941no—
211#95—941no—
212#95—941no—
213#95—941no—
214#95—941no—
215#95—941no—
216#95—941no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8#1062535212yes—
220C2×Q8Q8×D4712yes—
221C22×C4—941no—
222C22×C4—941no—
223C22×C4—941no—
224C22×C4—941no—
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4(C2×D47)⋊C411yescharacteristic
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4C2×C4×D4711yescharacteristic
230C22×C4C2×D18811yescharacteristic
231C22×C4C2×C22⋊D4711yescharacteristic
232M16—42no—
233M16—42no—
234M16—944no—
235M16—1882no—
236C2×C8—42no—
237C2×C8—1882no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—1881no—
242C4⋊C4—1881no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C4⋊C4—1882no—
246C4⋊C4—1882no—
247C4⋊C4—1881no—
248C4⋊C4—1881no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—1881no—
256C42—1881no—
257C23C2×(C2×D47)⋊C411yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—942no—
274C2×C4—1881no—
275C2×C4—1881no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1882no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4#1309393811yescharacteristic
285C2×C4#1309393811yescharacteristic
286C2×C4#1309398912yes—
287C2×C4#1309400412yes—
288C2×C4#1309401111yescharacteristic
289C2×C4#1309402211yescharacteristic
290C8—82no—
291C8—1884no—
292C22—21no—
293C22—21no—
294C22#1568902811yescharacteristic
295C22C2×#1309393811yescharacteristic
296C22#1568909611yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—22no—
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—3761no—
307C4—3761no—
308C4—3761no—
309C4—1882no—
310C2—21no—
311C2—21no—
312C2#2024437211yescharacteristic
313C2#2024507111yescharacteristic
314C2#2024507511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#1801C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC22×Q8×C47C23
hypercenter#1801D47
hypocenterC47#13303
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
180. C47#13303semidirect

Maximal subgroups

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

subgroupindex
#202485752
C4×#130939432
#202415142
C47×#18012
#1330347