Finite Group Databasebeta

← #26112953

Subgroups — #26112953

Order 12032.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#20254532C211yesmaximal, characteristic
2#20252325C211yesmaximal, characteristic
3#20244209C211yesmaximal, characteristic
4#20241953C211yesmaximal, characteristic
5#20241248C211yesmaximal, characteristic
6#20241218C211yesmaximal, characteristic
7C47×#1546C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689519C412yescomplemented
10#15689475C412yescomplemented
11#15689453—22no—
12#15689405—22no—
13#15689252—22no—
14#15689252—22no—
15#15689176C2211yescharacteristic
16#15689151—21no—
17C2×#13093948—21no—
18C2×#13093948—21no—
19#15689081—21no—
20C2×#13093935C2211yescharacteristic
21#15688989C2211yescharacteristic
22#15688861—22no—
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C47×#332—22no—
27C22×C47⋊C8D411yescharacteristic
28C22×C47⋊C8D411yescharacteristic
29#13094024C2×C411yescharacteristic
30#13094024C2×C411yescharacteristic
31#13094016—21no—
32#13094003—21no—
33#13094003D412yes—
34C4×Dic94—21no—
35C4×Dic94D412yes—
36#13093983—21no—
37C2×C4⋊Dic47—21no—
38#13093955—24no—
39C4⋊(C47⋊C8)—21no—
40Dic47⋊C8—24no—
41#13093948—42no—
42#13093948—42no—
43#13093947—24no—
44C22⋊(C47⋊C8)—21no—
45#13093935—21no—
46#13093935—21no—
47#13093934—22no—
48C4×C47⋊C8C2×C412yes—
49C8×Dic47—22no—
50C47⋊#95—21no—
51C47⋊#95—21no—
52C22×Q8×C47—21no—
53C2×M16×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47C2×C412yes—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47D412yes—
64C22⋊C8×C47—21no—
65C4×C8×C47—21no—
66C2×Q8×C47—22no—
67C2×Q8×C47—41no—
68C2×Q8×C47—41no—
69C2×Q8×C47—41no—
70C2×Q8×C47—41no—
71C2×Q8×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47#4111yescharacteristic
76C22×Dic47—41no—
77C2×Dic94—21no—
78C2×Dic94C22⋊C412yes—
79M16×C47—41no—
80M16×C47—41no—
81C2×C8×C47—22no—
82C2×C8×C47—41no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C42×C47C22×C411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91C22⋊Dic47—22no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—41no—
96C4⋊Dic47—41no—
97C4⋊Dic47C22⋊C412yes—
98Dic47⋊C4—22no—
99Dic47⋊C4—22no—
100C4×Dic47—22no—
101C2×C47⋊C8—21no—
102C2×C47⋊C8—21no—
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—42no—
111Q8×C47—81no—
112Q8×C47—81no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—42no—
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—22no—
129C2×Dic47—22no—
130C2×Dic47—81no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133Dic94—42no—
134Dic94—81no—
135C8×C47—42no—
136C8×C47—81no—
137C47⋊C8—22no—
138C47⋊C8—42no—
139C47⋊C8—41no—
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—82no—
151C4×C47C4×SD1611yescharacteristic
152C4×C47#44011yescharacteristic
153C4×C47#44811yescharacteristic
154C4×C47#44911yescharacteristic
155Dic47—42no—
156Dic47—42no—
157Dic47—161no—
158#3229—471no—
159#2961—471no—
160#1960—471no—
161#1546D4711yescharacteristic
162#1518—471no—
163#1286—471no—
164#1275—471no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#188011yescharacteristic
168C2×C47#190211yescharacteristic
169C2×C47#190911yescharacteristic
170#533—471no—
171C2×C4⋊Q8C2×D4711yescharacteristic
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#423C2×D4711yescharacteristic
182C2×#103—941no—
183C2×#103—941no—
184#405—941no—
185C2×#97—471no—
186#382—471no—
187#332—22no—
188#328—942no—
189C47#929211yescharacteristic, complemented, Sylow
190C22×Q8—21no—
191C2×M16—21no—
192C22×C8—471no—
193C22×C8—471no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—471no—
198C4⋊Q8—471no—
199C4⋊Q8C4×D4712yes—
200#125—941no—
201C22⋊Q8—472no—
202C22⋊Q8—941no—
203C4×Q8—472no—
204C4×Q8—941no—
205#117—941no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—941no—
209C2×C42C22×D4711yescharacteristic
210#106—944no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—944no—
214C4⋊C8—941no—
215C4⋊C8C22⋊D4712yes—
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—941no—
228#95—941no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—472no—
236C2×Q8—941no—
237C22×C4—41no—
238C22×C4—1881no—
239C22×C4#1062535011yescharacteristic
240C22×C4D4×D4711yescharacteristic
241C22×C4D4×D4711yescharacteristic
242M16—41no—
243M16—41no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—472no—
247C2×C8—941no—
248C2×C8—941no—
249C2×C8—942no—
250C2×C8—941no—
251C2×C8—941no—
252C2×C8—941no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—472no—
258C4⋊C4—941no—
259C4⋊C4—941no—
260C4⋊C4—941no—
261C4⋊C4—942no—
262C4⋊C4—942no—
263C4⋊C4—1881no—
264C4⋊C4—1881no—
265C22⋊C4—942no—
266C42—942no—
267C42C2×C4×D4711yescharacteristic
268C42D4×D4711yescharacteristic
269C42#1062535211yescharacteristic
270C42C2×C22⋊D4711yescharacteristic
271C23#1309399711yescharacteristic
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8—3761no—
277Q8—1882no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—942no—
284C2×C4—942no—
285C2×C4—3761no—
286C2×C4—3761no—
287C2×C4—3761no—
288C2×C4C22⋊C4×D4711yescharacteristic
289C2×C4(C22⋊D47)⋊C411yescharacteristic
290C2×C4C4×C22⋊D4711yescharacteristic
291C2×C4#1309399211yescharacteristic
292C2×C4#1309399511yescharacteristic
293C2×C4#1309400611yescharacteristic
294C2×C4SD16×D4711yescharacteristic
295C2×C4#1309404311yescharacteristic
296C2×C4#1309404811yescharacteristic
297C2×C4#1309405311yescharacteristic
298C8—42no—
299C8—81no—
300C8—942no—
301C8—1882no—
302C8—1881no—
303C22—21no—
304C22—21no—
305C22#1568899911yescharacteristic
306C22#1568932211yescharacteristic
307C22#1568934411yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—82no—
313C4—1882no—
314C4—1882no—
315C4—7521no—
316C4#1568921411yescharacteristic
317C4#1568922011yescharacteristic
318C4#1568926011yescharacteristic
319C4#1568926511yescharacteristic
320C2—21no—
321C2—21no—
322C2#2024374011yescharacteristic
323C2#2024386411yescharacteristic
324C2#2024390711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1546C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#1546D47
hypocenterC47#9292
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
9. #15689519C4semidirect
10. #15689475C4semidirect
189. C47#9292semidirect

Maximal subgroups

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

subgroupindex
#202545322
#202523252
#202442092
#202419532
#202412482
#202412182
C47×#15462
#929247