Finite Group Databasebeta

← D47×#1272 #26104219

Subgroups — D47×#1272 #26104219

Order 12032.

778 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 169 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#336×D47C212yesmaximal
3#20241214C211yesmaximal, characteristic, complemented
4#20241213C212yesmaximal, complemented
5C47×#1272C211yesmaximal, characteristic, complemented
6C22×Q8×D47C411yescharacteristic
7C22×C8×D47—22no—
8C2×C4⋊C4×D47C411yescharacteristic
9C2×C42×D47C2211yescharacteristic
10#15689030C411yescharacteristic
11#15689029C2211yescharacteristic
12#15689028C411yescharacteristic
13#15689027—22no—
14#95×D47—22no—
15#15688878C2212yes—
16#15688877—22no—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94—21no—
20C22×Dic94C2×C411yescharacteristic
21C2×Q8×D47—22no—
22C2×Q8×D47—42no—
23C2×Q8×D47—42no—
24C2×Q8×D47C814yescomplemented
25C22×C4×D47—21no—
26C22×C4×D47—21no—
27C22×C4×D47C2×C411yescharacteristic
28C22×C4×D47D412yes—
29C22×C47⋊C8—22no—
30C2×C8×D47—28no—
31C2×C8×D47—44no—
32C2×C8×D47—44no—
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×Dic47⋊C4—21no—
35C4⋊C4×D47—42no—
36C2×C4×Dic47C2×C411yescharacteristic
37C2×C4×Dic47D412yes—
38C42×D47—42no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C22×Q8×C47C2×C411yescharacteristic
43C22×C8×C47—22no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C42×C47C2311yescharacteristic
46#95×C47—22no—
47C23×D47C22⋊C411yescharacteristic
48C2×Q8×C47—21no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47C2×C812yescomplemented
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47C22×C411yescharacteristic
55C22×C4×C47C2×D412yes—
56C22×Dic47—21no—
57C22×Dic47—21no—
58C22×Dic47C22⋊C411yescharacteristic
59C22×Dic47C22⋊C412yes—
60Q8×D47—48no—
61Q8×D47—48no—
62C2×C4×D47—24no—
63C2×C4×D47—22no—
64C2×C4×D47—22no—
65C2×C4×D47—24no—
66C2×C4×D47—24no—
67C2×C4×D47—42no—
68C2×C4×D47—42no—
69C2×C4×D47—42no—
70C2×C4×D47—44no—
71C2×C4×D47C2×C812yes—
72C2×C4×D47M1612yes—
73C2×C4×D47SD1614yes—
74C2×C4×D47Q1614yes—
75C2×Dic94—21no—
76C2×Dic94—24no—
77C2×Dic94—42no—
78C2×Dic94—42no—
79C2×Dic94—41no—
80C2×Dic94—41no—
81C2×Dic94C2×C812yescomplemented
82C2×C8×C47—24no—
83C2×C8×C47—42no—
84C2×C8×C47—42no—
85C4⋊C4×C47—41no—
86C42×C47—41no—
87C4⋊Dic47—41no—
88Dic47⋊C4—42no—
89C4×Dic47—41no—
90C4×Dic47—42no—
91C2×C47⋊C8—24no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—42no—
94C8×D47—416no—
95C23×C47C2×C22⋊C411yescharacteristic
96C22×D47C22⋊C812yes—
97C22×D47#9912yes—
98C22×D47#10012yes—
99C22×D47#10314yes—
100C22×D47#10414yes—
101Q8×C47—42no—
102Q8×C47—42no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47C22×C811yescharacteristic
113C2×C4×C47C2×M1611yescharacteristic
114C2×C4×C47C2×SD1612yes—
115C2×C4×C47C2×Q1612yes—
116C2×Dic47—21no—
117C2×Dic47—22no—
118C2×Dic47—21no—
119C2×Dic47—22no—
120C2×Dic47—22no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—42no—
125C2×Dic47C22⋊C812yes—
126C2×Dic47#10314yes—
127C4×D47—24no—
128C4×D47—24no—
129C4×D47—28no—
130C4×D47—28no—
131C4×D47—48no—
132C4×D47—84no—
133Dic94—48no—
134Dic94—42no—
135Dic94—42no—
136C8×C47—44no—
137C47⋊C8—44no—
138C2×#1272—471nomaximal, Sylow
139C22×C47C2×C22⋊C811yescharacteristic
140C22×C47C2×#9911yescharacteristic
141C22×C47C2×#10011yescharacteristic
142C22×C47C2×#10312yes—
143C22×C47C2×#10412yes—
144C2×D47#32314yes—
145C2×D47Q8⋊C818yes—
146C2×D47#32818yes—
147C2×D47#32914yes—
148C2×D47#33314yes—
149C4×C47—21no—
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—42no—
154C4×C47—81no—
155Dic47—22no—
156Dic47—24no—
157Dic47—42no—
158Dic47—81no—
159C2×#391—471no—
160C2×#336—472no—
161#1272—471no—
162#1272—472no—
163#1272D4711yescharacteristic, direct product factor
164C2×C47C2×#32311yescharacteristic
165C2×C47C2×Q8⋊C812yes—
166C2×C47C2×#32812yes—
167C2×C47C2×#32911yescharacteristic
168C2×C47C2×#33311yescharacteristic
169D47#127218yesdirect product factor
170C23×Q8—471no—
171C23×C8—942no—
172C22×C4⋊C4—471no—
173C22×C42—471no—
174#391—471no—
175#391—471no—
176#391—471no—
177#391—942no—
178#391C2×D4711yescharacteristic
179C2×#95—942no—
180#336—472no—
181#336—942no—
182#336C2×D4712yes—
183C47C2×#127211yescharacteristic, complemented, Sylow
184C22×Q8—474no—
185C22×Q8—471no—
186C22×Q8—941no—
187C22×Q8—942no—
188C22×Q8—1882no—
189C22×Q8—1882no—
190C22×Q8C4×D4711yescharacteristic
191C23×C4—471no—
192C23×C4—472no—
193C23×C4—941no—
194C23×C4—941no—
195C22×C8—22no—
196C22×C8—948no—
197C22×C8—942no—
198C22×C8—1884no—
199C22×C8—1884no—
200C2×C4⋊C4—471no—
201C2×C4⋊C4—941no—
202C2×C4⋊C4—1882no—
203C2×C4⋊C4C4×D4711yescharacteristic
204C2×C42—471no—
205C2×C42—472no—
206C2×C42—1882no—
207C2×C42C22×D4711yescharacteristic
208#95—22no—
209#95—942no—
210#95—942no—
211#95—942no—
212C24—471no—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—472no—
217C2×Q8—944no—
218C2×Q8—941no—
219C2×Q8—1881no—
220C2×Q8—1881no—
221C2×Q8—1882no—
222C2×Q8—1882no—
223C2×Q8—1888no—
224C2×Q8—1888no—
225C2×Q8C8×D4712yescomplemented
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—472no—
229C22×C4—471no—
230C22×C4—472no—
231C22×C4—472no—
232C22×C4—474no—
233C22×C4—474no—
234C22×C4—944no—
235C22×C4—942no—
236C22×C4—942no—
237C22×C4—944no—
238C22×C4—944no—
239C22×C4—941no—
240C22×C4—941no—
241C22×C4—1882no—
242C22×C4—1884no—
243C22×C4—1882no—
244C22×C4—1882no—
245C22×C4C2×C4×D4711yescharacteristic
246C22×C4D4×D4712yes—
247C2×C8—24no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—944no—
251C2×C8—18816no—
252C2×C8—1882no—
253C2×C8—1882no—
254C4⋊C4—41no—
255C4⋊C4—1881no—
256C4⋊C4—1882no—
257C42—41no—
258C42—1882no—
259C42—1881no—
260C23—472no—
261C23—472no—
262C23—472no—
263C23—474no—
264C23—474no—
265C23C22⋊C4×D4711yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—1882no—
269Q8—1882no—
270Q8—1888no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—472no—
281C2×C4—474no—
282C2×C4—948no—
283C2×C4—948no—
284C2×C4—944no—
285C2×C4—944no—
286C2×C4—942no—
287C2×C4—941no—
288C2×C4—941no—
289C2×C4—942no—
290C2×C4—942no—
291C2×C4—3764no—
292C2×C4—1888no—
293C2×C4—1881no—
294C2×C4—1882no—
295C2×C4—1881no—
296C2×C4—1881no—
297C2×C4C2×C8×D4711yescharacteristic
298C2×C4M16×D4711yescharacteristic
299C2×C4SD16×D4712yes—
300C2×C4Q16×D4712yes—
301C8—44no—
302C8—1884no—
303C22—474no—
304C22—474no—
305C22—478no—
306C22—474no—
307C22—478no—
308C22C22⋊C8×D4711yescharacteristic
309C22#99×D4711yescharacteristic
310C22#100×D4711yescharacteristic
311C22#103×D4712yes—
312C22#104×D4712yes—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—42no—
318C4—81no—
319C4—942no—
320C4—944no—
321C4—3761no—
322C4—1882no—
323C2—478no—
324C2#323×D4711yescharacteristic
325C2Q8⋊C8×D4712yes—
326C2#328×D4712yes—
327C2#329×D4711yescharacteristic
328C2#333×D4711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47C2×#1272
LCSG ▷ C2×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
24. C2×Q8×D47C8semidirect
51. C2×Q8×C47C2×C8semidirect
81. C2×Dic94C2×C8semidirect
163. #1272D47direct (N × H)
169. D47#1272direct (N × H)
183. C47C2×#1272semidirect
225. C2×Q8C8×D47semidirect

Maximal subgroups

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

subgroupindex
#391×D472
#336×D472
#202412142
#202412132
C47×#12722
C2×#127247