Finite Group Databasebeta

← #26112954

Subgroups — #26112954

Order 12032.

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

normal subgroups only

This group has 426 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#20254529C211yesmaximal, characteristic
2#20252329C211yesmaximal, characteristic
3#20244207C211yesmaximal, characteristic
4#20242059C211yesmaximal, characteristic
5#20241246C211yesmaximal, characteristic
6#20241218C211yesmaximal, characteristic
7C47×#1518C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9#15689519C412yescomplemented
10#15689473C412yescomplemented
11#15689451—22no—
12#15689408—22no—
13#15689254—22no—
14#15689254—22no—
15#15689176C2211yescharacteristic
16#15689150—21no—
17#15689147C2211yescharacteristic
18C2×#13093947—21no—
19C2×#13093947—21no—
20#15689079—21no—
21C2×#13093935C2211yescharacteristic
22#15688868—22no—
23C47×#533C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#382C2211yescharacteristic
26C47×#328—22no—
27C22×Dic94—21no—
28C2×#10625329—21no—
29C22×C47⋊C8D411yescharacteristic
30C22×C47⋊C8D411yescharacteristic
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024C2×C412yes—
35C2×C4⋊Dic47—21no—
36C2×C4⋊Dic47—21no—
37#13093955—24no—
38C4⋊(C47⋊C8)—24no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)D412yes—
42#13093948—24no—
43#13093947—42no—
44#13093947—42no—
45C22⋊(C47⋊C8)—21no—
46#13093935—22no—
47#13093935—21no—
48#13093935—21no—
49C4×C47⋊C8—22no—
50C4×C47⋊C8C2×C412yes—
51C4⋊Q8×C47C2×C411yescharacteristic
52C4⋊Q8×C47C2×C411yescharacteristic
53#125×C47—21no—
54C22⋊Q8×C47—21no—
55C22⋊Q8×C47D412yes—
56C4×Q8×C47—21no—
57C4×Q8×C47D412yes—
58#117×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C22⋊C8×C47—21no—
63C4×C8×C47—21no—
64#95×C47—21no—
65#95×C47—21no—
66C2×Q8×C47—21no—
67C2×Q8×C47C22⋊C412yes—
68C22×C4×C47—41no—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—41no—
73C2×Dic94—22no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79C2×C8×C47—41no—
80C4⋊C4×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—21no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47C22⋊C412yes—
88C22⋊C4×C47—22no—
89C42×C47—22no—
90C42×C47C22×C411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47#4111yescharacteristic
94C4⋊Dic47—22no—
95C4⋊Dic47—22no—
96C4⋊Dic47—41no—
97C4⋊Dic47—41no—
98#10625329—41no—
99#10625329—41no—
100C2×C47⋊C8—21no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—22no—
104C2×C47⋊C8—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8SD1612yes—
108C23×C47#12111yescharacteristic
109Q8×C47—42no—
110Q8×C47—81no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—81no—
116C2×C4×C47—81no—
117C2×C4×C47—81no—
118C2×C4×C47C2×C22⋊C411yescharacteristic
119C2×C4×C47C4×D411yescharacteristic
120C2×C4×C47C4×D411yescharacteristic
121C2×C4×C47#12011yescharacteristic
122C2×C4×C47#12111yescharacteristic
123C2×C4×C47#12311yescharacteristic
124C2×C4×C47C2×SD1611yescharacteristic
125C2×C4×C47#13511yescharacteristic
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C47#13711yescharacteristic
128C2×Dic47—42no—
129C2×Dic47—81no—
130C2×Dic47—81no—
131Dic94—42no—
132Dic94—42no—
133Dic94—81no—
134Dic94—81no—
135C8×C47—81no—
136C47⋊C8—22no—
137C47⋊C8—42no—
138C47⋊C8—41no—
139C47⋊C8—42no—
140#9292—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47#38611yescharacteristic
144C22×C47#46011yescharacteristic
145C22×C47#46411yescharacteristic
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—42no—
151C4×C47—42no—
152C4×C47—161no—
153C4×C47C4×SD1611yescharacteristic
154C4×C47#44011yescharacteristic
155C4×C47#44811yescharacteristic
156C4×C47#44911yescharacteristic
157Dic47—82no—
158#3229—471no—
159#2961—471no—
160#1960—471no—
161#1546—471no—
162#1518D4711yescharacteristic
163#1286—471no—
164#1275—471no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#188011yescharacteristic
168C2×C47#190211yescharacteristic
169C2×C47#190911yescharacteristic
170#533C2×D4711yescharacteristic
171C2×C4⋊Q8—471no—
172#498—472no—
173#487—472no—
174#483—942no—
175#475—942no—
176#446—942no—
177#446—942no—
178#431—471no—
179#431C2×D4711yescharacteristic
180#424—941no—
181#423—471no—
182C2×#103—941no—
183C2×#103—941no—
184#405—941no—
185C2×#97—471no—
186#382C2×D4711yescharacteristic
187#332—942no—
188#328—22no—
189C47#929211yescharacteristic, complemented, Sylow
190C22×Q8—941no—
191C2×M16—941no—
192C22×C8—471no—
193C22×C8—471no—
194C4⋊Q8—472no—
195C4⋊Q8—941no—
196C4⋊Q8—941no—
197C4⋊Q8—941no—
198C4⋊Q8C4×D4711yescharacteristic
199C4⋊Q8C4×D4711yescharacteristic
200#125—21no—
201C22⋊Q8—21no—
202C22⋊Q8C22⋊D4712yes—
203C4×Q8—21no—
204C4×Q8D18812yes—
205#117—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—941no—
208C2×C4⋊C4—941no—
209C2×C42C22×D4711yescharacteristic
210#106—944no—
211C4⋊C8—21no—
212C4⋊C8—472no—
213C4⋊C8—944no—
214C4⋊C8—941no—
215C4⋊C8—941no—
216#103—944no—
217#103—1882no—
218#103—1882no—
219C22⋊C8—21no—
220C22⋊C8—941no—
221#97—942no—
222#97—941no—
223#97—941no—
224C4×C8—21no—
225C4×C8—472no—
226C4×C8—942no—
227#95—21no—
228#95—21no—
229C2×Q8—21no—
230C2×Q8—942no—
231C2×Q8—1881no—
232C2×Q8—1881no—
233C2×Q8—1881no—
234C2×Q8—1881no—
235C2×Q8—1881no—
236C2×Q8(C2×D47)⋊C412yes—
237C22×C4—41no—
238C22×C4—1881no—
239C22×C4D4×D4711yescharacteristic
240C22×C4#1062535211yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242M16—1881no—
243M16—1881no—
244C2×C8—41no—
245C2×C8—472no—
246C2×C8—941no—
247C2×C8—941no—
248C2×C8—942no—
249C2×C8—942no—
250C2×C8—941no—
251C2×C8—941no—
252C2×C8—941no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—942no—
261C4⋊C4—942no—
262C4⋊C4—1881no—
263C4⋊C4—1881no—
264C4⋊C4(C2×D47)⋊C412yes—
265C22⋊C4—22no—
266C42—22no—
267C42C2×C4×D4711yescharacteristic
268C42C2×D18811yescharacteristic
269C42D4×D4711yescharacteristic
270C42#1062535211yescharacteristic
271C23#1309399611yescharacteristic
272Q8—42no—
273Q8—81no—
274Q8—3761no—
275Q8—3761no—
276Q8—1882no—
277Q8—1882no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—3761no—
286C2×C4—3761no—
287C2×C4—1882no—
288C2×C4C2×(C2×D47)⋊C411yescharacteristic
289C2×C4(C22⋊D47)⋊C411yescharacteristic
290C2×C4(C22⋊D47)⋊C411yescharacteristic
291C2×C4#1309399411yescharacteristic
292C2×C4#1309399611yescharacteristic
293C2×C4#1309400811yescharacteristic
294C2×C4SD16×D4711yescharacteristic
295C2×C4#1309404711yescharacteristic
296C2×C4#1309405111yescharacteristic
297C2×C4#1309405611yescharacteristic
298C8—81no—
299C8—942no—
300C8—1882no—
301C8—1881no—
302C8—1882no—
303C22—21no—
304C22—21no—
305C22#1568900011yescharacteristic
306C22#1568932111yescharacteristic
307C22#1568934711yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—42no—
313C4—42no—
314C4—161no—
315C4—3762no—
316C4#1568920811yescharacteristic
317C4#1568922611yescharacteristic
318C4#1568926311yescharacteristic
319C4#1568927111yescharacteristic
320C2—21no—
321C2—21no—
322C2#2024374111yescharacteristic
323C2#2024385911yescharacteristic
324C2#2024390811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1518C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#1518D47
hypocenterC47#9292
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
9. #15689519C4semidirect
10. #15689473C4semidirect
189. C47#9292semidirect

Maximal subgroups

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

subgroupindex
#202545292
#202523292
#202442072
#202420592
#202412462
#202412182
C47×#15182
#929247