Finite Group Databasebeta

← #26113208

Subgroups — #26113208

Order 12032.

444 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 103 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#20253518C211yesmaximal, characteristic
2#20244331C211yesmaximal, characteristic
3#20241899C211yesmaximal, characteristic
4#20241835C211yesmaximal, characteristic
5#20241281C211yesmaximal, characteristic
6#20241244C211yesmaximal, characteristic
7C47×#2320C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689453—21no—
10#15689408C412yescomplemented
11#15689333C412yescomplemented
12#15689279—21no—
13#15689228—22no—
14#15689180C2211yescharacteristic
15#15689180C2211yescharacteristic
16C4×#10625329C2211yescharacteristic
17#15688883—22no—
18#15688859—21no—
19#15688859—21no—
20Q8⋊(C47⋊C8)—22no—
21C2×C4×Q8×C47C2211yescharacteristic
22C47×#391—21no—
23C47×#389—21no—
24C47×#387—22no—
25C47×#383—21no—
26C47×#382—21no—
27C4×C4⋊C4×C47C2211yescharacteristic
28C47×#376C2211yescharacteristic
29C2×#10625337—22no—
30C2×#10625329—21no—
31#13094024C2×C411yescharacteristic
32#13094024C2×C411yescharacteristic
33#13094016—21no—
34#13094003—21no—
35#13094003—21no—
36C4×Dic94—21no—
37C4×Dic94—21no—
38#13093983—21no—
39#13093955—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)C2×C412yes—
42#13093948—22no—
43#13093947—22no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46#13093935—21no—
47#13093935—21no—
48#13093935—21no—
49C4×C47⋊C8—21no—
50C22×Q8×C47D411yescharacteristic
51C4×Q8×C47—21no—
52C4×Q8×C47—21no—
53C4×Q8×C47C2×C412yes—
54C4×Q8×C47D412yes—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—22no—
58C2×C4⋊C4×C47D411yescharacteristic
59C2×C42×C47—21no—
60C2×C42×C47—21no—
61C2×C42×C47D412yes—
62C2×C42×C47C2311yescharacteristic
63#95×C47—22no—
64#95×C47—22no—
65#95×C47—22no—
66#95×C47—21no—
67#95×C47—21no—
68#95×C47D411yescharacteristic
69#95×C47D411yescharacteristic
70C2×Q8×C47—21no—
71C2×Q8×C47—42no—
72C2×Q8×C47C22⋊C412yes—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—22no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C2×Dic94—21no—
82C2×Dic94—21no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—42no—
87C4⋊C4×C47—82no—
88C4⋊C4×C47C22⋊C412yes—
89C42×C47—22no—
90C42×C47—22no—
91C42×C47—42no—
92C42×C47—42no—
93C42×C47C22×C411yescharacteristic
94C42×C47C2×D411yescharacteristic
95C42×C47#4111yescharacteristic
96C42×C47#4111yescharacteristic
97C22⋊Dic47—41no—
98#10625337—44no—
99C4⋊Dic47—21no—
100C4⋊Dic47—21no—
101C4⋊Dic47—21no—
102C4⋊Dic47—21no—
103Dic47⋊C4—41no—
104Dic47⋊C4—41no—
105C4×Dic47—41no—
106#10625329—22no—
107#10625329—41no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C23×C47#12011yescharacteristic
112Q8×C47—22no—
113Q8×C47—22no—
114Q8×C47—42no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47#12311yescharacteristic
139C2×Dic47—41no—
140C2×Dic47—41no—
141Dic94—42no—
142Dic94—81no—
143C47⋊C8—42no—
144C47⋊C8—42no—
145C47⋊C8—81no—
146#9328—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—21no—
149C22×C47#38611yescharacteristic
150C22×C47#45511yescharacteristic
151C22×C47#45611yescharacteristic
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—42no—
158C4×C47—42no—
159C4×C47—81no—
160C4×C47—82no—
161C4×C47—81no—
162C4×C47#42112yes—
163Dic47—81no—
164Dic47—81no—
165#3113—471no—
166#2320D4711yescharacteristic
167#1986—471no—
168#1506—471no—
169#1491—471no—
170#1300—471no—
171#1285—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47#188211yescharacteristic
175C2×C47#189711yescharacteristic
176C2×C47#190511yescharacteristic
177#533—471no—
178C2×C4×Q8C2×D4711yescharacteristic
179#483—941no—
180#475—472no—
181#462—472no—
182#451—941no—
183#441—942no—
184#432—471no—
185#432—471no—
186C4×M16—471no—
187#391—21no—
188#389—21no—
189#387—22no—
190#383—21no—
191#382—21no—
192C4×C4⋊C4C2×D4711yescharacteristic
193#376C2×D4711yescharacteristic
194#339—942no—
195#328—941no—
196#328—941no—
197Q8⋊C8—942no—
198C47#932811yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D4711yescharacteristic
200C2×Q16—942no—
201C2×M16—941no—
202C4⋊Q8—471no—
203C4⋊Q8—471no—
204#125—941no—
205C22⋊Q8—941no—
206C22⋊Q8—941no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—941no—
210C4×Q8—941no—
211C4×Q8C4×D4712yes—
212C4×Q8D18812yes—
213#117—941no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22⋊D4711yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22⋊D4712yes—
221C2×C42C22×D4711yescharacteristic
222#106—942no—
223C4⋊C8—472no—
224C4⋊C8—941no—
225#103—942no—
226#103—942no—
227C22⋊C8—941no—
228C22⋊C8—941no—
229#97—941no—
230#97—941no—
231#97—941no—
232C4×C8—941no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95C22⋊D4711yescharacteristic
239#95C22⋊D4711yescharacteristic
240C2×Q8—21no—
241C2×Q8—42no—
242C2×Q8—941no—
243C2×Q8—941no—
244C2×Q8(C2×D47)⋊C412yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4C2×C22⋊D4711yescharacteristic
251C22×C4C2×C22⋊D4711yescharacteristic
252C22×C4C2×C22⋊D4711yescharacteristic
253Q16—1884no—
254M16—942no—
255M16—1881no—
256C2×C8—942no—
257C2×C8—1881no—
258C2×C8—1881no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—941no—
265C4⋊C4—941no—
266C4⋊C4—941no—
267C4⋊C4—941no—
268C4⋊C4—1881no—
269C4⋊C4—1881no—
270C4⋊C4(C2×D47)⋊C412yes—
271C22⋊C4—1881no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—42no—
276C42—1881no—
277C42C2×C4×D4711yescharacteristic
278C42C2×D18811yescharacteristic
279C42#1062535011yescharacteristic
280C42#1062535011yescharacteristic
281C23#1309399311yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—3761no—
286Q8—1882no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—1881no—
306C2×C4—1881no—
307C2×C4C2×(C2×D47)⋊C411yescharacteristic
308C2×C4C4×C22⋊D4711yescharacteristic
309C2×C4C4×C22⋊D4711yescharacteristic
310C2×C4#1309399811yescharacteristic
311C2×C4#1309399811yescharacteristic
312C2×C4#1309400711yescharacteristic
313C8—3761no—
314C8—1882no—
315C8—1882no—
316C22—21no—
317C22—21no—
318C22#1568900211yescharacteristic
319C22#1568930111yescharacteristic
320C22#1568930611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—82no—
330C4—81no—
331C4—3761no—
332C4—3761no—
333C4#1568914112yes—
334C2—21no—
335C2—21no—
336C2#2024375711yescharacteristic
337C2#2024383611yescharacteristic
338C2#2024388211yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2320C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#2320D47
hypocenterC47#9328
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2320

Complements

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

NH ≅ G/Ntype
10. #15689408C4semidirect
11. #15689333C4semidirect
198. C47#9328semidirect

Maximal subgroups

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

subgroupindex
#202535182
#202443312
#202418992
#202418352
#202412812
#202412442
C47×#23202
#932847