Finite Group Databasebeta

← #26113955

Subgroups — #26113955

Order 12032.

518 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 103 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#20248380C211yesmaximal, characteristic
3#20242113C212yesmaximal, complemented
4#20241243C212yesmaximal, complemented
5C47×#1466C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7C2×#13094011C412yes—
8#15689195—24no—
9#15689180C2212yes—
10#15689087—21no—
11C2×C22⋊(C47⋊C8)—21no—
12#15689048—21no—
13#15689048—21no—
14#15689028C412yes—
15#15688983—21no—
16#15688983—21no—
17C4×C4⋊Dic47C2211yescharacteristic
18#15688871—24no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#432C2212yes—
21C47×#323—21no—
22C47×#323—21no—
23C22×Dic94C2×C412yes—
24C2×#10625329—22no—
25C22×C47⋊C8—22no—
26#13094024—22no—
27#13094024—22no—
28#13094011—44no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C4×Dic47C2×C412yes—
35C4⋊(C47⋊C8)D414yes—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39#13093935—22no—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C47⋊#95—21no—
43C22×D4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C4×D4×C47—22no—
47C4×D4×C47—22no—
48C2×C4⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47D411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54#97×C47—22no—
55C24×C47C22⋊C411yescharacteristic
56C24×C47C22⋊C411yescharacteristic
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—41no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—22no—
72C22×Dic47—22no—
73C2×Dic94—24no—
74C2×Dic94—42no—
75C2×Dic94—42no—
76C2×C8×C47—42no—
77C4⋊C4×C47—22no—
78C22⋊C4×C47—22no—
79C22⋊C4×C47—22no—
80C42×C47C2×D412yes—
81C42×C47#4112yes—
82C22⋊Dic47—44no—
83C22⋊Dic47—44no—
84C4⋊Dic47—42no—
85C4⋊Dic47—41no—
86C4⋊Dic47—42no—
87C4⋊Dic47—41no—
88C4×Dic47—42no—
89#10625329—82no—
90C2×C47⋊C8—24no—
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
103D4×C47—44no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—41no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—81no—
115C2×C4×C47—81no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12112yes—
119C2×C4×C47#12311yescharacteristic
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126Dic94—84no—
127C8×C47—82no—
128C47⋊C8—44no—
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#45512yes—
151Dic47—82no—
152Dic47—82no—
153#2973—471no—
154#2547—471no—
155#1558—472no—
156#1466D4711yescharacteristic
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⋊Q8—471no—
166C2×#124—472no—
167C2×C4×D4C2×D4711yescharacteristic
168#435—944no—
169#432—472no—
170#432C2×D4712yes—
171#407—941no—
172C2×C22⋊C8—941no—
173#397—941no—
174#397—941no—
175#391—472no—
176#380—941no—
177#380—941no—
178C4×C4⋊C4—471no—
179#333—944no—
180#323—21no—
181#323—21no—
182C47#945111yescharacteristic, complemented, Sylow
183C22×Q8—472no—
184C22×D4C4×D4711yescharacteristic
185C23×C4D18811yescharacteristic
186C23×C4D18811yescharacteristic
187C2×M16—942no—
188C22×C8—942no—
189C4⋊Q8—942no—
190C4⋊Q8—942no—
191#124—1884no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4C4×D4711yescharacteristic
198C2×C22⋊C4—942no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4C22⋊D4711yescharacteristic
202C2×C42—472no—
203C2×C42C22×D4711yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—474no—
206C22⋊C8—22no—
207C22⋊C8—942no—
208C22⋊C8—942no—
209C22⋊C8—942no—
210#97—22no—
211#97—942no—
212#95—942no—
213#95—942no—
214#95—941no—
215C24(C2×D47)⋊C411yescharacteristic
216C24(C2×D47)⋊C411yescharacteristic
217C2×Q8—944no—
218C2×Q8—1882no—
219C2×Q8—1882no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—942no—
230C22×C4—942no—
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4C2×C4×D4711yescharacteristic
234C22×C4C2×D18811yescharacteristic
235C22×C4C2×C22⋊D4711yescharacteristic
236M16—3762no—
237C2×C8—42no—
238C2×C8—944no—
239C2×C8—3762no—
240C4⋊C4—22no—
241C4⋊C4—1882no—
242C4⋊C4—1881no—
243C4⋊C4—1882no—
244C4⋊C4—1881no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—1884no—
248C22⋊C4—1884no—
249C42—1882no—
250C42D4×D4712yes—
251C42#1062535212yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1309394012yes—
262C23C2×(C2×D47)⋊C411yescharacteristic
263Q8—3764no—
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—81no—
276C2×C4—81no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1882no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4(C22⋊D47)⋊C412yes—
284C2×C4#1309399411yescharacteristic
285C2×C4#1309399612yes—
286C2×C4#1309400811yescharacteristic
287C8—82no—
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#1568900011yescharacteristic
302C22C2×#1309394011yescharacteristic
303C22#1568910511yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—3762no—
310C4—3762no—
311C4#1568930312yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2024377411yescharacteristic
317C2#2024384911yescharacteristic
318C2#2024390811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1466C2
socleC22×C47#386
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenter#1466D47
hypocenterC47#9451
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #20242113C2semidirect
4. #20241243C2semidirect
182. C47#9451semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202483802
#202421132
#202412432
C47×#14662
#945147