Finite Group Databasebeta

← #26104565

Subgroups — #26104565

Order 12032.

646 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 121 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#391×D47C211yesmaximal, characteristic
2#20243367C212yesmaximal
3#20241214C211yesmaximal, characteristic, complemented
4#20241213C212yesmaximal, complemented
5C47×#1272C211yesmaximal, characteristic, complemented
6C22×Q8×D47C411yescharacteristic
7C2×C4⋊C4×D47C411yescharacteristic
8C2×C42×D47C2211yescharacteristic
9C2×Dic47⋊C8—22no—
10C2×(C2×D47)⋊C8—22no—
11#15689030C411yescharacteristic
12#15689029C2211yescharacteristic
13#15689028C411yescharacteristic
14#15689027—22no—
15#95×D47—22no—
16#15688878C2212yes—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94—21no—
20C22×Dic94C2×C411yescharacteristic
21C2×Q8×D47—22no—
22C2×Q8×D47—22no—
23C2×Q8×D47—42no—
24C2×Q8×D47—42no—
25C22×C4×D47—21no—
26C22×C4×D47—21no—
27C22×C4×D47C2×C411yescharacteristic
28C22×C4×D47D412yes—
29C22×C47⋊C8—22no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C4⋊C4×D47—42no—
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47D412yes—
35C42×D47—42no—
36Dic47⋊C8—44no—
37(C2×D47)⋊C8—44no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C22×Q8×C47C2×C411yescharacteristic
42C22×C8×C47—22no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C42×C47C2311yescharacteristic
45#95×C47—22no—
46C23×D47C22⋊C411yescharacteristic
47C2×Q8×C47—21no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47M1612yes—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47C22×C411yescharacteristic
54C22×C4×C47C2×D412yes—
55C22×Dic47—21no—
56C22×Dic47—21no—
57C22×Dic47C22⋊C411yescharacteristic
58C22×Dic47C22⋊C412yes—
59Q8×D47—48no—
60Q8×D47—48no—
61C2×C4×D47—22no—
62C2×C4×D47—21no—
63C2×C4×D47—21no—
64C2×C4×D47—22no—
65C2×C4×D47—24no—
66C2×C4×D47—42no—
67C2×C4×D47—42no—
68C2×C4×D47—42no—
69C2×C4×D47—42no—
70C2×C4×D47—44no—
71C2×C4×D47—42no—
72C2×C4×D47SD1614yes—
73C2×C4×D47Q1614yes—
74C2×Dic94—24no—
75C2×Dic94—21no—
76C2×Dic94—42no—
77C2×Dic94—42no—
78C2×Dic94—41no—
79C2×Dic94—41no—
80C2×Dic94M1612yes—
81C2×C8×C47—42no—
82C2×C8×C47—42no—
83C2×C8×C47—42no—
84C4⋊C4×C47—41no—
85C42×C47—41no—
86C4⋊Dic47—41no—
87Dic47⋊C4—42no—
88C4×Dic47—24no—
89C4×Dic47—41no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—42no—
93C23×C47C2×C22⋊C411yescharacteristic
94C22×D47—21no—
95C22×D47—21no—
96C22×D47—22no—
97C22×D47#10012yes—
98C22×D47#10314yes—
99Q8×C47—42no—
100Q8×C47—42no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47C2×M1611yescharacteristic
111C2×C4×C47C2×M1611yescharacteristic
112C2×C4×C47C2×SD1612yes—
113C2×C4×C47C2×Q1612yes—
114C2×Dic47—21no—
115C2×Dic47—22no—
116C2×Dic47—21no—
117C2×Dic47—22no—
118C2×Dic47—22no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—42no—
123C2×Dic47#9912yes—
124C2×Dic47#10314yes—
125C4×D47—24no—
126C4×D47—24no—
127C4×D47—44no—
128C4×D47—44no—
129C4×D47—48no—
130C4×D47—84no—
131Dic94—48no—
132Dic94—42no—
133Dic94—42no—
134C8×C47—82no—
135C47⋊C8—82no—
136#7643—471nomaximal, Sylow
137C22×C47#40711yescharacteristic
138C22×C47C2×#9911yescharacteristic
139C22×C47C2×#10011yescharacteristic
140C22×C47C2×#10312yes—
141C22×C47#42112yes—
142C2×D47—22no—
143C2×D47—24no—
144C2×D47—24no—
145C2×D47—22no—
146C2×D47#33314yes—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—42no—
152C4×C47—81no—
153Dic47—22no—
154Dic47—24no—
155Dic47—42no—
156Dic47—81no—
157C2×#391—471no—
158#1797—472no—
159#1272—471no—
160#1272—472no—
161#1272D4711yescharacteristic, complemented
162C2×C47#146211yescharacteristic
163C2×C47#148712yes—
164C2×C47#152012yes—
165C2×C47#153311yescharacteristic
166C2×C47C2×#33311yescharacteristic
167D47—24no—
168C23×Q8—471no—
169C22×C4⋊C4—471no—
170C22×C42—471no—
171C2×C4⋊C8—942no—
172C2×C22⋊C8—942no—
173#391—471no—
174#391—471no—
175#391—471no—
176#391—942no—
177#391C2×D4711yescharacteristic
178C2×#95—942no—
179#336—472no—
180#336C2×D4712yes—
181C47#764311yescharacteristic, complemented, Sylow
182C22×Q8—471no—
183C22×Q8—941no—
184C22×Q8—942no—
185C22×Q8—942no—
186C22×Q8—1882no—
187C22×Q8—1882no—
188C22×Q8C4×D4711yescharacteristic
189C23×C4—472no—
190C23×C4—471no—
191C23×C4—941no—
192C23×C4—941no—
193C22×C8—22no—
194C22×C8—942no—
195C2×C4⋊C4—471no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—1882no—
198C2×C4⋊C4C4×D4711yescharacteristic
199C2×C42—472no—
200C2×C42—471no—
201C2×C42—1882no—
202C2×C42C22×D4711yescharacteristic
203C4⋊C8—1884no—
204C22⋊C8—1884no—
205#95—22no—
206#95—942no—
207#95—942no—
208#95—942no—
209C24—471no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—472no—
214C2×Q8—944no—
215C2×Q8—941no—
216C2×Q8—1881no—
217C2×Q8—1881no—
218C2×Q8—1882no—
219C2×Q8—1882no—
220C2×Q8—1888no—
221C2×Q8—1888no—
222C2×Q8#1062532412yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—472no—
226C22×C4—471no—
227C22×C4—474no—
228C22×C4—474no—
229C22×C4—942no—
230C22×C4—942no—
231C22×C4—944no—
232C22×C4—941no—
233C22×C4—941no—
234C22×C4—941no—
235C22×C4—941no—
236C22×C4—1882no—
237C22×C4—1882no—
238C22×C4—1882no—
239C22×C4—1884no—
240C22×C4—1882no—
241C22×C4—1882no—
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4D4×D4712yes—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—1882no—
248C2×C8—1882no—
249C2×C8—1882no—
250C4⋊C4—41no—
251C4⋊C4—1881no—
252C4⋊C4—1882no—
253C42—41no—
254C42—944no—
255C42—1881no—
256C23—472no—
257C23—474no—
258C23—941no—
259C23—941no—
260C23—942no—
261C23C22⋊C4×D4711yescharacteristic
262Q8—42no—
263Q8—42no—
264Q8—1882no—
265Q8—1882no—
266Q8—1888no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—472no—
277C2×C4—474no—
278C2×C4—944no—
279C2×C4—944no—
280C2×C4—941no—
281C2×C4—942no—
282C2×C4—941no—
283C2×C4—942no—
284C2×C4—942no—
285C2×C4—3764no—
286C2×C4—1884no—
287C2×C4—1884no—
288C2×C4—1888no—
289C2×C4—1881no—
290C2×C4—1882no—
291C2×C4—1881no—
292C2×C4—1881no—
293C2×C4M16×D4711yescharacteristic
294C2×C4C2×#1062532411yescharacteristic
295C2×C4SD16×D4712yes—
296C2×C4Q16×D4712yes—
297C8—82no—
298C8—3762no—
299C22—474no—
300C22—944no—
301C22—942no—
302C22—942no—
303C22—944no—
304C22#1568908511yescharacteristic
305C22#99×D4711yescharacteristic
306C22#100×D4711yescharacteristic
307C22#103×D4712yes—
308C22#1568913812yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—81no—
315C4—944no—
316C4—942no—
317C4—3761no—
318C4—1882no—
319C2—944no—
320C2#2024171311yescharacteristic
321C2#2024181512yes—
322C2#2024195812yes—
323C2#2024201011yescharacteristic
324C2#333×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7643
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
3. #20241214C2semidirect
4. #20241213C2semidirect
5. C47×#1272C2semidirect
161. #1272D47semidirect
181. C47#7643semidirect

Maximal subgroups

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

subgroupindex
#391×D472
#202433672
#202412142
#202412132
C47×#12722
#764347