Finite Group Databasebeta

← #26118989

Subgroups — #26118989

Order 12032.

588 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 113 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#20254539C211yesmaximal, characteristic
2#20254336C211yesmaximal, characteristic
3#20244365C211yesmaximal, characteristic
4#20242073C211yesmaximal, characteristic
5#20242069C211yesmaximal, characteristic
6#20241221C211yesmaximal, characteristic
7C47×#2868C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689526—21no—
10#15689502—21no—
11#15689475—21no—
12#15689473—21no—
13#15689458—22no—
14#15689415—22no—
15#15689330—22no—
16#15689279—22no—
17#15689176C2211yescharacteristic
18#15689176C2211yescharacteristic
19#15689147C2211yescharacteristic
20#15688883—22no—
21#15688872—22no—
22#15688869—22no—
23C2×C4⋊Q8×C47—22no—
24C2×C4⋊Q8×C47C2211yescharacteristic
25C2×C4⋊Q8×C47C2211yescharacteristic
26C47×#384—22no—
27C47×#384—22no—
28C43×C47C2211yescharacteristic
29C2×#10625337—24no—
30C2×#10625329—21no—
31#13094024D411yescharacteristic
32#13094024D411yescharacteristic
33#13094016—21no—
34#13094003—21no—
35#13094003—21no—
36C4×Dic94—21no—
37C4×Dic94—21no—
38#13093983—21no—
39#13093957—24no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)D412yes—
43C4⋊(C47⋊C8)D412yes—
44#13093948—24no—
45#13093947—24no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8—21no—
50C22×Q8×C47—21no—
51C22×Q8×C47D412yes—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47—21no—
59C4⋊Q8×C47—21no—
60C4⋊Q8×C47—48no—
61C2×C4⋊C4×C47—28no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47D412yes—
66C2×C42×C47—21no—
67C2×C42×C47—21no—
68C2×C42×C47D412yes—
69C2×C42×C47C2311yescharacteristic
70C2×Q8×C47—22no—
71C2×Q8×C47—22no—
72C2×Q8×C47—44no—
73C2×Q8×C47—44no—
74C2×Q8×C47—41no—
75C2×Q8×C47—41no—
76C22×C4×C47—22no—
77C22×C4×C47—22no—
78C22×C4×C47—22no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C2×Dic94—21no—
83C2×Dic94—21no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—416no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C42×C47—24no—
94C42×C47—24no—
95C42×C47—22no—
96C42×C47Q1614yescomplemented
97C42×C47C2×D411yescharacteristic
98C42×C47C2×D411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47#4111yescharacteristic
101C22⋊Dic47—41no—
102#10625337—84no—
103C4⋊Dic47—21no—
104C4⋊Dic47—21no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107Dic47⋊C4—41no—
108Dic47⋊C4—41no—
109C4×Dic47—41no—
110#10625329—42no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—41no—
114C23×C47#12011yescharacteristic
115Q8×C47—44no—
116Q8×C47—82no—
117Q8×C47—82no—
118Q8×C47—82no—
119C2×C4×C47—24no—
120C2×C4×C47—24no—
121C2×C4×C47—24no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—44no—
125C2×C4×C47—44no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47#12011yescharacteristic
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12111yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47#12711yescharacteristic
134C2×C4×C47C2×Q1612yes—
135C2×C4×C47#13712yes—
136C2×Dic47—41no—
137C2×Dic47—41no—
138Dic94—42no—
139Dic94—42no—
140C47⋊C8—42no—
141C47⋊C8—42no—
142C47⋊C8—81no—
143#10293—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47#39211yescharacteristic
147C22×C47#45112yes—
148C4×C47—28no—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—82no—
152C4×C47—82no—
153C4×C47#45312yes—
154C4×C47#45612yes—
155C4×C47#46212yes—
156C4×C47#49411yescharacteristic
157C4×C47#49711yescharacteristic
158Dic47—81no—
159Dic47—81no—
160#3230—471no—
161#3209—471no—
162#2868D4711yescharacteristic
163#1994—471no—
164#1549—471no—
165#1548—471no—
166#1276—471no—
167C2×C47—22no—
168C2×C47#200211yescharacteristic
169C2×C47#200511yescharacteristic
170C2×C47#201611yescharacteristic
171#533—471no—
172C2×C4⋊Q8—22no—
173C2×C4⋊Q8C2×D4711yescharacteristic
174C2×C4⋊Q8C2×D4711yescharacteristic
175#500—941no—
176#494—941no—
177#487—941no—
178#487—941no—
179#484—942no—
180#477—942no—
181#462—942no—
182#451—942no—
183#431—471no—
184#431—471no—
185#423—471no—
186#384—22no—
187#384—22no—
188C43C2×D4711yescharacteristic
189#339—942no—
190#333—942no—
191#332—942no—
192C47#1029311yescharacteristic, complemented, Sylow
193C22×Q8—21no—
194C22×Q8C22⋊D4712yes—
195C2×Q16—944no—
196C2×M16—941no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—48no—
206C4⋊Q8—471no—
207C4⋊Q8—471no—
208#125—941no—
209C22⋊Q8—941no—
210C22⋊Q8—941no—
211C4×Q8—941no—
212C4×Q8—941no—
213#117—941no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22⋊D4712yes—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C22⋊D4712yes—
222C2×C42C22×D4711yescharacteristic
223#107—944no—
224C4⋊C8—472no—
225C4⋊C8—472no—
226C4⋊C8—941no—
227C4⋊C8—941no—
228#103—944no—
229#103—944no—
230C22⋊C8—941no—
231C22⋊C8—941no—
232C4×C8—941no—
233C4×C8—941no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—941no—
241C2×Q8—941no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4C2×C22⋊D4711yescharacteristic
246C22×C4C2×C22⋊D4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248Q16—3764no—
249M16—1882no—
250C2×C8—942no—
251C2×C8—942no—
252C2×C8—1881no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—416no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—941no—
263C4⋊C4—941no—
264C4⋊C4—941no—
265C4⋊C4—941no—
266C4⋊C4—1881no—
267C4⋊C4—1881no—
268C22⋊C4—1881no—
269C42—24no—
270C42—24no—
271C42—22no—
272C42—1881no—
273C42#1062533714yescomplemented
274C42D4×D4711yescharacteristic
275C42D4×D4711yescharacteristic
276C42D4×D4711yescharacteristic
277C42#1062535211yescharacteristic
278C23#1309399311yescharacteristic
279Q8—44no—
280Q8—82no—
281Q8—82no—
282Q8—82no—
283Q8—1882no—
284Q8—1882no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—1881no—
295C2×C4—1881no—
296C2×C4#1309399211yescharacteristic
297C2×C4#1309399211yescharacteristic
298C2×C4#1309399511yescharacteristic
299C2×C4#1309399611yescharacteristic
300C2×C4#1309399611yescharacteristic
301C2×C4#1309402011yescharacteristic
302C2×C4C2×#1062533712yes—
303C2×C4#1309405512yes—
304C8—3761no—
305C8—1882no—
306C8—1882no—
307C22—21no—
308C22—21no—
309C22#1568903111yescharacteristic
310C22#1568927912yes—
311C4—28no—
312C4—21no—
313C4—21no—
314C4—82no—
315C4—82no—
316C4—3761no—
317C4—3761no—
318C4#1568929012yes—
319C4#1568930512yes—
320C4#1568933012yes—
321C4#1568950211yescharacteristic
322C4#1568951511yescharacteristic
323C2—22no—
324C2#2024440311yescharacteristic
325C2#2024442411yescharacteristic
326C2#2024448811yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2868C2
socleC22×C47#392
prime radicalC23×C47#120
hypercenter#2868D47
hypocenterC47#10293
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
96. C42×C47Q16semidirect
192. C47#10293semidirect
273. C42#10625337semidirect

Maximal subgroups

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

subgroupindex
#202545392
#202543362
#202443652
#202420732
#202420692
#202412212
C47×#28682
#1029347