Finite Group Databasebeta

← #26194270

Subgroups — #26194270

Order 12032.

812 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 179 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#20254779C212yesmaximal
2#20253898C211yesmaximal, characteristic
3#20242665C214yesmaximal, complemented
4#20242630C214yesmaximal, complemented
5#20242126C211yesmaximal, characteristic
6#20241843C212yesmaximal
7C47×#3556C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689526C2212yes—
10#15689522—22no—
11#15689519C2212yes—
12#15689408C2214yes—
13#15689405—22no—
14#15689379—24no—
15#15689333C2214yes—
16#15689330—22no—
17#15689159—22no—
18#15689147C2212yes—
19C2×C4×C47⋊C8C2211yescharacteristic
20#15688869C2214yes—
21Q8⋊(C47⋊C8)C2214yes—
22Q8⋊(C47⋊C8)C2214yes—
23C47×#564—22no—
24C47×#557—22no—
25C47×#557—22no—
26C47×#557C2214yes—
27C47×#541—24no—
28C47×#541—22no—
29C47×#533C2212yes—
30C47×#533C2211yescharacteristic
31C2×#10625337—24no—
32C2×#10625329—22no—
33C22×C47⋊C8—21no—
34#13094024C2312yes—
35#13094016—21no—
36#13094003—22no—
37C4×Dic94—22no—
38#13093983—21no—
39#13093957—24no—
40#13093955—24no—
41#13093955—24no—
42C4⋊(C47⋊C8)—22no—
43C4⋊(C47⋊C8)C2314yes—
44#13093948—24no—
45#13093947—24no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8C2311yescharacteristic
48C4×C47⋊C8C2311yescharacteristic
49C4⋊Q8×C47—22no—
50C4⋊Q8×C47—24no—
51C4⋊Q8×C47D412yes—
52C4⋊Q8×C47D412yes—
53C4⋊Q8×C47C2312yes—
54#126×C47—28no—
55#125×C47—22no—
56#125×C47—28no—
57#125×C47—28no—
58#125×C47—24no—
59#125×C47—28no—
60#125×C47—22no—
61#125×C47D412yes—
62C22⋊Q8×C47—22no—
63C22⋊Q8×C47—22no—
64C22⋊Q8×C47D414yes—
65C4×Q8×C47—24no—
66C4×Q8×C47—24no—
67C4×Q8×C47—24no—
68C4×Q8×C47—22no—
69C4×Q8×C47—22no—
70C4×Q8×C47C2314yes—
71#117×C47—22no—
72#117×C47D412yes—
73C2×C42×C47C2311yescharacteristic
74C2×Q8×C47—22no—
75C2×Q8×C47—22no—
76C2×Q8×C47C2×D414yes—
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C2×Dic94—22no—
80C4⋊C4×C47—28no—
81C4⋊C4×C47—28no—
82C4⋊C4×C47—28no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—24no—
87C4⋊C4×C47—24no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—24no—
90C4⋊C4×C47C2×D414yes—
91C4⋊C4×C47C2×D414yes—
92C22⋊C4×C47—24no—
93C22⋊C4×C47—22no—
94C42×C47—28no—
95C42×C47—22no—
96C42×C47—24no—
97C42×C47C2×D412yes—
98C42×C47C2×D411yescharacteristic
99C42×C47C2411yescharacteristic
100C22⋊Dic47—41no—
101#10625337—84no—
102C4⋊Dic47—22no—
103C4⋊Dic47—22no—
104Dic47⋊C4—42no—
105C4×Dic47—41no—
106#10625329—82no—
107C2×C47⋊C8—24no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—24no—
111C23×C47#12011yescharacteristic
112Q8×C47—24no—
113Q8×C47—24no—
114Q8×C47—42no—
115Q8×C47—42no—
116Q8×C47—44no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—22no—
120C2×C4×C47—24no—
121C2×C4×C47—24no—
122C2×C4×C47—24no—
123C2×C4×C47#12012yes—
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47C22×D411yescharacteristic
126C2×C4×C47C22×D412yes—
127C2×Dic47—42no—
128Dic94—42no—
129Dic94—42no—
130C47⋊C8—22no—
131C47⋊C8—22no—
132C47⋊C8—44no—
133#18703—471nomaximal, Sylow
134C22×C47—22no—
135C22×C47—21no—
136C22×C47C2×#12011yescharacteristic
137C4×C47—22no—
138C4×C47—21no—
139C4×C47—44no—
140C4×C47—44no—
141C4×C47—44no—
142C4×C47#45612yes—
143C4×C47#57712yes—
144C4×C47#57812yes—
145Dic47—82no—
146#3556D4711yescharacteristic
147#3255—472no—
148#3160—471no—
149#1653—474no—
150#1648—474no—
151#1561—471no—
152#1493—472no—
153C2×C47—41no—
154C2×C47#301312yes—
155C2×C47C2×#45611yescharacteristic
156#564—22no—
157#557—22no—
158#557—22no—
159#557C2×D4714yes—
160#541—24no—
161#541—22no—
162#533—471no—
163#533C2×D4712yes—
164#533C2×D4711yescharacteristic
165#500—472no—
166#499—942no—
167#498—472no—
168#475—474no—
169#475—942no—
170#470—944no—
171#462—474no—
172#462—942no—
173#427—942no—
174#423—472no—
175C2×C4×C8—471no—
176#332—474no—
177Q8⋊C8—474no—
178Q8⋊C8—474no—
179C47#1870311yescharacteristic, complemented, Sylow
180C2×Q16—944no—
181C2×M16—942no—
182C22×C8—941no—
183C4⋊Q8—22no—
184C4⋊Q8—24no—
185C4⋊Q8—472no—
186C4⋊Q8C22⋊D4712yes—
187C4⋊Q8C22⋊D4712yes—
188C4⋊Q8C22×D4712yes—
189#126—28no—
190#125—22no—
191#125—28no—
192#125—28no—
193#125—24no—
194#125—28no—
195#125—22no—
196#125—941no—
197#125C22⋊D4712yes—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—942no—
201C22⋊Q8C22⋊D4714yes—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—942no—
208C4×Q8C22×D4714yes—
209#117—22no—
210#117—941no—
211#117C22⋊D4712yes—
212C2×C42C22×D4711yescharacteristic
213#107—944no—
214#106—944no—
215#106—944no—
216C4⋊C8—474no—
217C4⋊C8—942no—
218#103—944no—
219#103—944no—
220C4×C8—471no—
221C4×C8—471no—
222C4×C8—941no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—942no—
226C2×Q8C2×C22⋊D4714yes—
227C22×C4C2×C22⋊D4712yes—
228C22×C4C2×C22⋊D4711yescharacteristic
229Q16—3764no—
230M16—3762no—
231C2×C8—944no—
232C2×C8—941no—
233C2×C8—941no—
234C2×C8—944no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—942no—
246C4⋊C4—942no—
247C4⋊C4—1882no—
248C4⋊C4C2×C22⋊D4714yes—
249C4⋊C4C2×C22⋊D4714yes—
250C22⋊C4—24no—
251C22⋊C4—22no—
252C22⋊C4—1881no—
253C42—28no—
254C42—22no—
255C42—24no—
256C42—1881no—
257C42C2×C22⋊D4711yescharacteristic
258C42C2×C22⋊D4712yes—
259C42C23×D4711yescharacteristic
260C23#1309399311yescharacteristic
261Q8—24no—
262Q8—24no—
263Q8—44no—
264Q8—42no—
265Q8—42no—
266Q8—1882no—
267Q8—1882no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—1882no—
275C2×C4#1309399312yes—
276C2×C4#1309399311yescharacteristic
277C2×C4C22×C22⋊D4711yescharacteristic
278C2×C4C22×C22⋊D4712yes—
279C8—942no—
280C8—942no—
281C8—1884no—
282C22—22no—
283C22—21no—
284C22C2×#1309399311yescharacteristic
285C4—22no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—44no—
290C4—3762no—
291C4#1568930612yes—
292C4#1568993112yes—
293C4#1568993812yes—
294C2—41no—
295C2#2025270312yes—
296C2C2×#1568930611yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3556C2
socleC22×C47C2×#120
prime radicalC23×C47#120
hypercenter#3556D47
hypocenterC47#18703
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3556

Complements

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

NH ≅ G/Ntype
3. #20242665C2semidirect
4. #20242630C2semidirect
179. C47#18703semidirect

Maximal subgroups

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

subgroupindex
#202547792
#202538982
#202426652
#202426302
#202421262
#202418432
C47×#35562
#1870347