Finite Group Databasebeta

← #26112955

Subgroups — #26112955

Order 12032.

426 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 109 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#20254534C211yesmaximal, characteristic
2#20252328C211yesmaximal, characteristic
3#20244208C211yesmaximal, characteristic
4#20242059C211yesmaximal, characteristic
5#20241953C211yesmaximal, characteristic
6#20241247C211yesmaximal, characteristic
7C47×#1275C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9C2×#13094024C2211yescharacteristic
10#15689521C412yescomplemented
11#15689474C412yescomplemented
12#15689454—22no—
13#15689409—22no—
14#15689251—22no—
15#15689251—22no—
16#15689149—21no—
17#15689147C2211yescharacteristic
18C2×#13093949—21no—
19C2×#13093949—21no—
20#15689078—21no—
21#15688989C2211yescharacteristic
22#15688868—22no—
23#15688861—22no—
24C47×#431C2211yescharacteristic
25C47×#431C2211yescharacteristic
26C2×#97×C47C2211yescharacteristic
27C22×Dic94—21no—
28C2×#10625329—21no—
29#13094024—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024C2×C412yes—
33#13094024C2×C411yescharacteristic
34#13094024C2×C411yescharacteristic
35#13094016—21no—
36#13094003—21no—
37#13094003D412yes—
38C4×Dic94—21no—
39C4×Dic94D412yes—
40#13093983—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47—21no—
43C2×C4⋊Dic47—21no—
44#13093956—24no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)D412yes—
47Dic47⋊C8—24no—
48#13093949—24no—
49#13093949—42no—
50#13093949—42no—
51#13093934—22no—
52C8×Dic47—22no—
53C47⋊#95—21no—
54C47⋊#95—21no—
55C22×C8×C47D411yescharacteristic
56C22×C8×C47D411yescharacteristic
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C22⋊C8×C47—21no—
61C22⋊C8×C47—21no—
62#97×C47—21no—
63#97×C47—21no—
64C4×C8×C47—21no—
65C4×C8×C47C2×C412yes—
66C22×C4×C47C2×D411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47#4111yescharacteristic
69C22×Dic47—41no—
70C22×Dic47—41no—
71C2×Dic94—22no—
72C2×Dic94—21no—
73C2×Dic94—41no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94C22⋊C412yes—
79C2×C8×C47—21no—
80C2×C8×C47—21no—
81C2×C8×C47—22no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—41no—
86C2×C8×C47SD1612yes—
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—22no—
96C4⋊Dic47—22no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101C4⋊Dic47C22⋊C412yes—
102Dic47⋊C4—22no—
103Dic47⋊C4—22no—
104C4×Dic47—22no—
105#10625329—41no—
106#10625329—41no—
107C2×C47⋊C8—22no—
108C23×C47#12111yescharacteristic
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47C2×C22⋊C411yescharacteristic
112C2×C4×C47C4×D411yescharacteristic
113C2×C4×C47C4×D411yescharacteristic
114C2×C4×C47#12011yescharacteristic
115C2×C4×C47#12111yescharacteristic
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47C2×SD1611yescharacteristic
118C2×C4×C47#13511yescharacteristic
119C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C47#13711yescharacteristic
121C2×Dic47—22no—
122C2×Dic47—22no—
123C2×Dic47—42no—
124C2×Dic47—81no—
125C2×Dic47—81no—
126C2×Dic47—81no—
127C2×Dic47—81no—
128C2×Dic47—81no—
129Dic94—42no—
130Dic94—42no—
131Dic94—42no—
132Dic94—81no—
133Dic94—81no—
134Dic94—81no—
135C8×C47—22no—
136C8×C47—42no—
137C8×C47—41no—
138C8×C47—81no—
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×C47C4×SD1611yescharacteristic
151C4×C47#44011yescharacteristic
152C4×C47#44811yescharacteristic
153C4×C47#44911yescharacteristic
154Dic47—42no—
155Dic47—42no—
156Dic47—82no—
157Dic47—161no—
158#3229—471no—
159#2961—471no—
160#1960—471no—
161#1546—471no—
162#1518—471no—
163#1286—471no—
164#1275D4711yescharacteristic
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#188011yescharacteristic
168C2×C47#190211yescharacteristic
169C2×C47#190911yescharacteristic
170#533—471no—
171C2×C4⋊Q8—471no—
172#498—472no—
173#487—472no—
174#483—942no—
175#475—942no—
176#446—942no—
177#446—942no—
178#431C2×D4711yescharacteristic
179#431C2×D4711yescharacteristic
180#424—941no—
181#423—471no—
182C2×#103—941no—
183C2×#103—941no—
184#405—941no—
185C2×#97C2×D4711yescharacteristic
186#382—471no—
187#332—942no—
188#328—942no—
189C47#929211yescharacteristic, complemented, Sylow
190C22×Q8—941no—
191C2×M16—941no—
192C22×C8D18811yescharacteristic
193C22×C8D18811yescharacteristic
194C4⋊Q8—471no—
195C4⋊Q8—471no—
196C4⋊Q8—472no—
197C4⋊Q8—941no—
198C4⋊Q8—941no—
199C4⋊Q8—941no—
200#125—941no—
201C22⋊Q8—472no—
202C22⋊Q8—941no—
203C4×Q8—472no—
204C4×Q8—941no—
205#117—941no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4—941no—
208C2×C4⋊C4—941no—
209C2×C42C22×D4711yescharacteristic
210#106—944no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—472no—
214C4⋊C8—944no—
215C4⋊C8—941no—
216#103—944no—
217#103—1882no—
218#103—1882no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221#97—21no—
222#97—21no—
223#97—942no—
224C4×C8—21no—
225C4×C8—942no—
226C4×C8C4×D4712yes—
227#95—941no—
228#95—941no—
229C2×Q8—472no—
230C2×Q8—941no—
231C2×Q8—942no—
232C2×Q8—1881no—
233C2×Q8—1881no—
234C2×Q8—1881no—
235C2×Q8—1881no—
236C2×Q8—1881no—
237C22×C4—1881no—
238C22×C4—1881no—
239C22×C4C2×D18811yescharacteristic
240C22×C4D4×D4711yescharacteristic
241C22×C4#1062535411yescharacteristic
242M16—1881no—
243M16—1881no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—22no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—942no—
252C2×C8#1062532512yes—
253C4⋊C4—472no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C4⋊C4—941no—
257C4⋊C4—942no—
258C4⋊C4—942no—
259C4⋊C4—942no—
260C4⋊C4—942no—
261C4⋊C4—1881no—
262C4⋊C4—1881no—
263C4⋊C4—1881no—
264C4⋊C4—1881no—
265C22⋊C4—942no—
266C42—942no—
267C42C2×C4×D4711yescharacteristic
268C42#1062535011yescharacteristic
269C42D4×D4711yescharacteristic
270C42D4×D4711yescharacteristic
271C23#1309399911yescharacteristic
272Q8—3761no—
273Q8—3761no—
274Q8—3761no—
275Q8—1882no—
276Q8—1882no—
277Q8—1882no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—942no—
281C2×C4—942no—
282C2×C4—3761no—
283C2×C4—3761no—
284C2×C4—3761no—
285C2×C4—3761no—
286C2×C4—3761no—
287C2×C4—1882no—
288C2×C4C22⋊C4×D4711yescharacteristic
289C2×C4#1309398611yescharacteristic
290C2×C4C4×D18811yescharacteristic
291C2×C4#1309399411yescharacteristic
292C2×C4#1309399711yescharacteristic
293C2×C4#1309400911yescharacteristic
294C2×C4C2×#1062532511yescharacteristic
295C2×C4#1309404611yescharacteristic
296C2×C4#1309405211yescharacteristic
297C2×C4#1309405711yescharacteristic
298C8—22no—
299C8—42no—
300C8—41no—
301C8—81no—
302C8—1882no—
303C22—21no—
304C22—21no—
305C22#1568900111yescharacteristic
306C22#1568932411yescharacteristic
307C22#1568934611yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—3762no—
313C4—1882no—
314C4—1882no—
315C4—7521no—
316C4C4×#1062532511yescharacteristic
317C4#1568922511yescharacteristic
318C4#1568926411yescharacteristic
319C4#1568927011yescharacteristic
320C2—21no—
321C2—21no—
322C2#2024374211yescharacteristic
323C2#2024386211yescharacteristic
324C2#2024390911yescharacteristic
subgroupG/H
centerC22#15689001
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1275C2
socleC22×C47#386
prime radicalC23×C47#121
cosocleC2×C42×C47C23
hypercenter#1275D47
hypocenterC47#9292
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
10. #15689521C4semidirect
11. #15689474C4semidirect
189. C47#9292semidirect

Maximal subgroups

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

subgroupindex
#202545342
#202523282
#202442082
#202420592
#202419532
#202412472
C47×#12752
#929247