Finite Group Databasebeta

← #26111273

Subgroups — #26111273

Order 12032.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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#20245991C211yesmaximal, characteristic
2#20243323C212yesmaximal
3C2×#15688849C211yesmaximal, characteristic
4#20241267C212yesmaximal
5C2×C47×#395C211yesmaximal, characteristic
6C2×#13093948—24no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689009—24no—
9#15688986—21no—
10#15688983C2212yes—
11C4×C22⋊Dic47—21no—
12#15688849—22no—
13C23×Q8×C47C2211yescharacteristic
14C22×C4⋊C4×C47—21no—
15C22×C4⋊C4×C47C2211yescharacteristic
16C47×#395—22no—
17C47×#395C414yescomplemented
18C2×#95×C47—21no—
19C2×#95×C47C2211yescharacteristic
20C22×C47⋊C8—22no—
21C2×C4⋊Dic47D414yes—
22C2×C22⋊Dic47—22no—
23C2×C4×Dic47—22no—
24#13093948—48no—
25C22⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—24no—
29C47⋊#95—22no—
30C22×Q8×C47—21no—
31C22×Q8×C47—22no—
32C22×Q8×C47—22no—
33C22×Q8×C47—44no—
34C22×Q8×C47C2×C412yes—
35C23×C4×C47—21no—
36C23×C4×C47—21no—
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47—24no—
43C2×C4⋊C4×C47—44no—
44C2×C4⋊C4×C47C2×C412yes—
45C2×C4⋊C4×C47D412yes—
46#95×C47—24no—
47#95×C47—21no—
48#95×C47C2×C412yes—
49C24×C47C2×D411yescharacteristic
50C2×Q8×C47—28no—
51C2×Q8×C47—24no—
52C2×Q8×C47—44no—
53C2×Q8×C47—44no—
54C2×Q8×C47—44no—
55C2×Q8×C47—48no—
56C2×Q8×C47—42no—
57C22×C4×C47—24no—
58C22×C4×C47—24no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—21no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—42no—
73C22×C4×C47—42no—
74C22×C4×C47C22⋊C412yes—
75C22×C4×C47C22⋊C412yes—
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D412yes—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4112yes—
80C22×Dic47—24no—
81C4⋊C4×C47—24no—
82C4⋊C4×C47—24no—
83C4⋊C4×C47—48no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—44no—
87C4×Dic47—82no—
88C2×C47⋊C8—44no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47C2×C22⋊C411yescharacteristic
93C23×C47#12011yescharacteristic
94C23×C47#12311yescharacteristic
95C23×C47#13512yes—
96C23×C47#13712yes—
97Q8×C47—28no—
98Q8×C47—48no—
99Q8×C47—44no—
100C2×C4×C47—24no—
101C2×C4×C47—24no—
102C2×C4×C47—24no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—48no—
112C2×C4×C47—42no—
113C2×C4×C47—44no—
114C2×C4×C47—42no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—44no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—81no—
122C2×C4×C47C4×D412yes—
123C2×C4×C47#12112yes—
124C2×Dic47—44no—
125C2×Dic47—44no—
126C2×Dic47—44no—
127C47⋊C8—82no—
128#9014—471nomaximal, Sylow
129C22×C47—24no—
130C22×C47—24no—
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47#38611yescharacteristic
136C22×C47#41611yescharacteristic
137C22×C47#41911yescharacteristic
138C22×C47#45212yes—
139C22×C47#45612yes—
140C22×C47B(4,𝔽2)12yes—
141C22×C47#48512yes—
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—44no—
145C4×C47—42no—
146C4×C47—44no—
147C4×C47—42no—
148Dic47—84no—
149C2×#395D4711yescharacteristic
150#2313—471no—
151#1787—472no—
152C2×#324—471no—
153#1294—472no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47—22no—
157C2×C47#160314yes—
158C2×C47#187311yescharacteristic
159C2×C47#188611yescharacteristic
160C2×C47#188911yescharacteristic
161C23×Q8C2×D4711yescharacteristic
162C22×C4⋊C4—21no—
163C22×C4⋊C4C2×D4711yescharacteristic
164C2×#103—944no—
165C2×C22⋊C8—472no—
166#395—22no—
167#395Dic4714yescomplemented
168#387—944no—
169#381—941no—
170#380—472no—
171C4×C22⋊C4—941no—
172C2×#95—21no—
173C2×#95C2×D4711yescharacteristic
174#324—942no—
175C47#901411yescharacteristic, complemented, Sylow
176C22×Q8—21no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—44no—
180C22×Q8C2×Dic4712yes—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊D4711yescharacteristic
184C23×C4C22⋊D4711yescharacteristic
185C23×C4C22×D4711yescharacteristic
186C22×C8—942no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—474no—
192C2×C4⋊C4C2×Dic4712yes—
193C2×C4⋊C4C22⋊D4712yes—
194C2×C22⋊C4—942no—
195C2×C42—942no—
196#103—1888no—
197C22⋊C8—942no—
198C22⋊C8—942no—
199#95—24no—
200#95—21no—
201#95—942no—
202#95—944no—
203#95—942no—
204#95C2×Dic4712yes—
205C24C2×C22⋊D4711yescharacteristic
206C2×Q8—28no—
207C2×Q8—24no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—48no—
212C2×Q8—42no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—944no—
231C22×C4C22⋊Dic4712yes—
232C22×C4C22⋊Dic4712yes—
233C22×C4D4×D4712yes—
234C22×C4#1062535212yes—
235C22×C4C22×Dic4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—1884no—
238C2×C8—1882no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—48no—
242C4⋊C4—1884no—
243C22⋊C4—1882no—
244C22⋊C4—1882no—
245C42—3762no—
246C23—22no—
247C23—22no—
248C23C2×C22⋊Dic4711yescharacteristic
249C23#1309399211yescharacteristic
250C23#1309400511yescharacteristic
251C23#1309404412yes—
252C23#1309405512yes—
253Q8—28no—
254Q8—48no—
255Q8—44no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—48no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—1884no—
279C2×C4—1884no—
280C2×C4—1884no—
281C2×C4D4×Dic4712yes—
282C2×C4#1309399612yes—
283C8—3762no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1568900311yescharacteristic
291C22#1568911911yescharacteristic
292C22#1568913211yescharacteristic
293C22#1568928312yes—
294C22#1568930512yes—
295C22#1568930912yes—
296C22#1568946112yes—
297C4—22no—
298C4—22no—
299C4—44no—
300C4—44no—
301C4—42no—
302C4—42no—
303C4—3764no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#2024234914yes—
308C2#2024370111yescharacteristic
309C2#2024378211yescharacteristic
310C2#2024379711yescharacteristic
subgroupG/H
centerC23#13093992
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#395C2
socleC23×C47#120
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenterC2×#395D47
hypocenterC47#9014
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
17. C47×#395C4semidirect
167. #395Dic47semidirect
175. C47#9014semidirect

Maximal subgroups

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

subgroupindex
#202459912
#202433232
C2×#156888492
#202412672
C2×C47×#3952
#901447