Finite Group Databasebeta

← #26113958

Subgroups — #26113958

Order 12032.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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#20252417C211yesmaximal, characteristic
2#20242114C212yesmaximal, complemented
3#20241729C211yesmaximal, characteristic
4#20241244C212yesmaximal, complemented
5C47×#2547C211yesmaximal, characteristic
6#15689195—24no—
7#15689180C2212yes—
8#15689180C2212yes—
9#15689087—21no—
10C2×C22⋊(C47⋊C8)—21no—
11#15688872—24no—
12#15688847—21no—
13#15688847—21no—
14C2×C4⋊Q8×C47C2211yescharacteristic
15C2×#124×C47C412yes—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#397—21no—
18C47×#397—21no—
19C47×#391C412yes—
20C47×#380—21no—
21C47×#380—21no—
22C4×C4⋊C4×C47C2211yescharacteristic
23C2×#10625329—22no—
24C22×C47⋊C8—22no—
25C4⋊(C47⋊C8)—22no—
26C4⋊(C47⋊C8)D414yes—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—22no—
33C22×Q8×C47C2×C412yes—
34C22×D4×C47C2×C411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C4⋊Q8×C47—22no—
38C4⋊Q8×C47—22no—
39#124×C47—44no—
40C4×D4×C47—22no—
41C4×D4×C47—22no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C42×C47C2×C412yes—
51C2×C42×C47C2311yescharacteristic
52#95×C47—22no—
53#95×C47—22no—
54#95×C47—21no—
55C24×C47C22⋊C411yescharacteristic
56C24×C47C22⋊C411yescharacteristic
57C2×Q8×C47—24no—
58C2×Q8×C47—42no—
59C2×Q8×C47—42no—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—41no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—42no—
70C22×C4×C47—42no—
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22×C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C4⋊C4×C47—22no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—41no—
81C22⋊C4×C47—22no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—44no—
84C22⋊C4×C47—44no—
85C42×C47—42no—
86C42×C47C2×D412yes—
87C42×C47#4112yes—
88#10625329—82no—
89C2×C47⋊C8—24no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—82no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—41no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47—41no—
101C23×C47#10012yes—
102C23×C47C2×C22⋊C411yescharacteristic
103Q8×C47—84no—
104D4×C47—44no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—81no—
122C2×C4×C47—81no—
123C2×C4×C47C4×D412yes—
124C2×C4×C47#12011yescharacteristic
125C2×C4×C47#12112yes—
126C2×C4×C47#12311yescharacteristic
127C47⋊C8—44no—
128C47⋊C8—82no—
129#9451—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—22no—
133C22×C47—42no—
134C22×C47—42no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—81no—
141C22×C47—81no—
142C22×C47#38611yescharacteristic
143C22×C47C2×#10011yescharacteristic
144C22×C47#41311yescharacteristic
145C4×C47—21no—
146C4×C47—22no—
147C4×C47—22no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—82no—
151C4×C47—82no—
152C4×C47#45512yes—
153#2973—471no—
154#2547D4711yescharacteristic
155#1558—472no—
156#1466—471no—
157#1285—472no—
158C2×C47—21no—
159C2×C47—21no—
160C2×C47—42no—
161C2×C47—81no—
162C2×C47#188511yescharacteristic
163C2×C47#190011yescharacteristic
164C2×C47#190911yescharacteristic
165C2×C4⋊Q8C2×D4711yescharacteristic
166C2×#124Dic4712yes—
167C2×C4×D4C2×D4711yescharacteristic
168#435—944no—
169#432—472no—
170#432—472no—
171#407—941no—
172C2×C22⋊C8—941no—
173#397—21no—
174#397—21no—
175#391Dic4712yes—
176#380—21no—
177#380—21no—
178C4×C4⋊C4C2×D4711yescharacteristic
179#333—944no—
180#323—941no—
181#323—941no—
182C47#945111yescharacteristic, complemented, Sylow
183C22×Q8C2×Dic4712yes—
184C22×D4C2×Dic4711yescharacteristic
185C23×C4C22⋊D4711yescharacteristic
186C23×C4C22⋊D4711yescharacteristic
187C2×M16—942no—
188C22×C8—942no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191#124—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic4711yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4C22⋊D4711yescharacteristic
202C2×C42C2×Dic4712yes—
203C2×C42C22×D4711yescharacteristic
204C4⋊C8—474no—
205C4⋊C8—942no—
206C22⋊C8—942no—
207C22⋊C8—942no—
208C22⋊C8—942no—
209C22⋊C8—942no—
210#97—942no—
211#97—942no—
212#95—22no—
213#95—22no—
214#95—21no—
215C24C22⋊Dic4711yescharacteristic
216C24C22⋊Dic4711yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22×Dic4711yescharacteristic
234C22×C4C2×C22⋊D4711yescharacteristic
235C22×C4C2×C22⋊D4711yescharacteristic
236M16—3762no—
237C2×C8—944no—
238C2×C8—3762no—
239C2×C8—1882no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42D4×D4712yes—
251C42#1062535212yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1309394112yes—
262C23C2×C22⋊Dic4711yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4D4×Dic4712yes—
284C2×C4#1309399211yescharacteristic
285C2×C4#1309399612yes—
286C2×C4#1309400511yescharacteristic
287C8—3762no—
288C8—1884no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1568900311yescharacteristic
302C22C2×#1309394111yescharacteristic
303C22#1568910711yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4#1568930012yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2024377711yescharacteristic
317C2#2024385211yescharacteristic
318C2#2024391111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2547C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#2547D47
hypocenterC47#9451
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. #20242114C2semidirect
4. #20241244C2semidirect
182. C47#9451semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202421142
#202417292
#202412442
C47×#25472
#945147