Finite Group Databasebeta

← #26112307

Subgroups — #26112307

Order 12032.

478 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 119 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#20252026C211yesmaximal, characteristic
2#20242060C212yesmaximal, complemented
3#20241702C211yesmaximal, characteristic
4#20241246C212yesmaximal, complemented
5C47×#2621C211yesmaximal, characteristic
6#15689184—24no—
7#15689176C2212yes—
8#15689147C2212yes—
9#15689087—21no—
10C4×#10625329—21no—
11#15688869—24no—
12#15688849—21no—
13#15688847—21no—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C2×C22⋊Q8×C47—21no—
16C2×#117×C47C2211yescharacteristic
17C47×#398—21no—
18C47×#398C412yes—
19C47×#397—21no—
20C47×#397C412yes—
21C47×#391—21no—
22C47×#382C2211yescharacteristic
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)D414yes—
27C22⋊(C47⋊C8)—24no—
28C22⋊(C47⋊C8)—22no—
29#13093935—22no—
30C4×C47⋊C8—22no—
31C4×C47⋊C8—22no—
32C22×Q8×C47—21no—
33C23×C4×C47D411yescharacteristic
34C4⋊Q8×C47—22no—
35C4⋊Q8×C47—22no—
36C22⋊Q8×C47—44no—
37#117×C47—22no—
38#117×C47—22no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47C2×C412yes—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47C2×C411yescharacteristic
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47C2311yescharacteristic
50#95×C47—22no—
51#95×C47—22no—
52#95×C47—21no—
53#95×C47C2×C412yes—
54C24×C47C22⋊C411yescharacteristic
55C2×Q8×C47—41no—
56C2×Q8×C47—41no—
57C2×Q8×C47—42no—
58C2×Q8×C47—42no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C4⋊C4×C47—22no—
72C4⋊C4×C47—22no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—44no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—42no—
77C22⋊C4×C47—22no—
78C22⋊C4×C47—22no—
79C22⋊C4×C47—44no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47C2×D412yes—
83C42×C47#4112yes—
84#10625329—42no—
85#10625329—42no—
86C2×C47⋊C8—24no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94Q8×C47—82no—
95Q8×C47—82no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47C4×D412yes—
114C2×C4×C47#12011yescharacteristic
115C2×C4×C47#12112yes—
116C2×C4×C47#12311yescharacteristic
117C2×C4×C47#13512yes—
118C2×C4×C47AGL(1,ℤ/8)12yes—
119C47⋊C8—44no—
120C47⋊C8—82no—
121#9196—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—21no—
124C22×C47—42no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—81no—
129C22×C47#38611yescharacteristic
130C22×C47#41611yescharacteristic
131C22×C47#41811yescharacteristic
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47—82no—
138C4×C47—82no—
139C4×C47#44912yes—
140C4×C47#48212yes—
141#2921—471no—
142#2621D4711yescharacteristic
143#1546—472no—
144#1458—471no—
145#1286—472no—
146C2×C47—21no—
147C2×C47—21no—
148C2×C47—81no—
149C2×C47#187511yescharacteristic
150C2×C47#190911yescharacteristic
151C2×C47#191511yescharacteristic
152C2×C4⋊Q8C2×D4711yescharacteristic
153C2×C22⋊Q8—21no—
154C2×#117C2×D4711yescharacteristic
155#433—944no—
156#431—472no—
157#423—472no—
158#407—941no—
159C4×M16—941no—
160#398—21no—
161#398Dic4712yes—
162#397—21no—
163#397Dic4712yes—
164#391—21no—
165#382C2×D4711yescharacteristic
166#332—944no—
167#324—941no—
168#323—941no—
169C47#919611yescharacteristic, complemented, Sylow
170C22×Q8—21no—
171C23×C4C22⋊D4711yescharacteristic
172C2×M16—942no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C22⋊Q8—44no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×Dic4712yes—
182C2×C4⋊C4C2×Dic4711yescharacteristic
183C2×C4⋊C4C22⋊D4711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic4711yescharacteristic
186C2×C22⋊C4C22⋊D4711yescharacteristic
187C2×C42C22⋊D4711yescharacteristic
188C2×C42C22×D4711yescharacteristic
189C4⋊C8—474no—
190C4⋊C8—942no—
191C4⋊C8—942no—
192C22⋊C8—944no—
193C22⋊C8—942no—
194#97—942no—
195C4×C8—942no—
196C4×C8—942no—
197#95—22no—
198#95—22no—
199#95—21no—
200#95C2×Dic4712yes—
201C24C22⋊Dic4711yescharacteristic
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4C22⋊Dic4711yescharacteristic
213C22×C4C22⋊Dic4711yescharacteristic
214C22×C4C22⋊Dic4711yescharacteristic
215C22×C4C22×Dic4711yescharacteristic
216C22×C4C2×C22⋊D4711yescharacteristic
217C22×C4C2×C22⋊D4711yescharacteristic
218M16—1882no—
219M16—1882no—
220C2×C8—944no—
221C2×C8—1882no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—44no—
231C42—21no—
232C42—21no—
233C42D4×D4712yes—
234C42#1062535212yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×C22⋊Dic4711yescharacteristic
241Q8—82no—
242Q8—82no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4D4×Dic4712yes—
261C2×C4#1309399211yescharacteristic
262C2×C4#1309399612yes—
263C2×C4#1309400511yescharacteristic
264C2×C4#1309404412yes—
265C2×C4#1309405012yes—
266C8—3762no—
267C8—1884no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1568900311yescharacteristic
276C22#1568911911yescharacteristic
277C22#1568912811yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1568926612yes—
286C4#1568944412yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#2024371111yescharacteristic
291C2#2024391111yescharacteristic
292C2#2024394111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2621C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2621D47
hypocenterC47#9196
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. #20242060C2semidirect
4. #20241246C2semidirect
169. C47#9196semidirect

Maximal subgroups

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

subgroupindex
#202520262
#202420602
#202417022
#202412462
C47×#26212
#919647