Finite Group Databasebeta

← #26113990

Subgroups — #26113990

Order 12032.

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

normal subgroups only

This group has 536 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#20252473C211yesmaximal, characteristic, complemented
2#20248387C211yesmaximal, characteristic
3#20242101C212yesmaximal
4#20241243C212yesmaximal, complemented
5C47×#1467C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7C2×#13094013C412yes—
8#15689254—24no—
9#15689180—21no—
10#15689180C2212yes—
11#15689176—21no—
12#15689040—21no—
13#15689040—21no—
14#15689025—21no—
15#15689025—21no—
16#15689024C412yes—
17C4×C4⋊Dic47C2211yescharacteristic
18#15688866—24no—
19C2×C4×Q8×C47C2211yescharacteristic
20C47×#432C2212yes—
21C47×#324—21no—
22C47×#324—21no—
23C22×D188—21no—
24C22×D188—21no—
25C22×D188C2×C412yes—
26#13094021—22no—
27#13094021—22no—
28#13094013—44no—
29C2×C4⋊Dic47—21no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×C4×Dic47C2×C412yes—
35C4⋊(C47⋊C8)—22no—
36C4⋊(C47⋊C8)—22no—
37C4⋊(C47⋊C8)D414yes—
38C22⋊(C47⋊C8)—22no—
39#13093935—22no—
40#13093935—22no—
41C4×C47⋊C8—22no—
42C47⋊#95—21no—
43C22×Q8×C47C2×C411yescharacteristic
44C4×Q8×C47—22no—
45C4×Q8×C47—22no—
46C2×C4⋊C4×C47C2×C411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C4⋊C8×C47—22no—
53C22⋊C8×C47—22no—
54#97×C47—22no—
55C23×D47—22no—
56C2×Q8×C47—22no—
57C2×Q8×C47—22no—
58C2×Q8×C47—41no—
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—22no—
67C2×D188—24no—
68C2×D188—42no—
69C2×D188—42no—
70C2×D188—42no—
71C2×D188—42no—
72C2×D188—41no—
73C2×D188—42no—
74C2×D188—41no—
75C2×D188—82no—
76C2×C8×C47—42no—
77C4⋊C4×C47—22no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47C2×D412yes—
85C42×C47#4112yes—
86(C2×D47)⋊C4—44no—
87(C2×D47)⋊C4—44no—
88(C2×D47)⋊C4—82no—
89(C2×D47)⋊C4—82no—
90C4⋊Dic47—42no—
91C4×Dic47—42no—
92C2×C47⋊C8—24no—
93C23×C47C2×C22⋊C411yescharacteristic
94C22×D47—42no—
95C22×D47—42no—
96C22×D47—42no—
97C22×D47—42no—
98C22×D47—42no—
99C22×D47—42no—
100C22×D47—42no—
101Q8×C47—44no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47#10012yes—
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—
123D188—84no—
124D188—82no—
125D188—84no—
126D188—82no—
127C8×C47—82no—
128C47⋊C8—44no—
129#9458—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47#38611yescharacteristic
133C22×C47C2×#10011yescharacteristic
134C22×C47#41311yescharacteristic
135C2×D47—82no—
136C2×D47—82no—
137C2×D47—82no—
138C2×D47—82no—
139C2×D47—82no—
140C2×D47—82no—
141C2×D47—82no—
142C4×C47—21no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—22no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47#45412yes—
150Dic47—82no—
151#2983—471no—
152#2548—471no—
153#1555—472no—
154#1467D4711yescharacteristic, complemented
155#1285—472no—
156C2×C47—21no—
157C2×C47—21no—
158C2×C47#188811yescharacteristic
159C2×C47#190011yescharacteristic
160C2×C47#191011yescharacteristic
161D47—162no—
162C2×#127—471no—
163C2×#124—472no—
164C2×C4×Q8C2×D4711yescharacteristic
165#446—944no—
166#432—472no—
167#432—941no—
168#432C2×D4712yes—
169#431—941no—
170#394—941no—
171#394—941no—
172#390—472no—
173#390—941no—
174#390—941no—
175C4×C4⋊C4—471no—
176#331—944no—
177#324—21no—
178#324—21no—
179C47#945811yescharacteristic, complemented, Sylow
180C22×Q8C4×D4711yescharacteristic
181C22×D4—472no—
182C22×D4—941no—
183C22×D4—941no—
184#127—942no—
185#127—942no—
186#124—1884no—
187C4×Q8—22no—
188C4×Q8—22no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4C4×D4711yescharacteristic
191C2×C4⋊C4C22⋊D4711yescharacteristic
192C2×C4⋊C4C22⋊D4711yescharacteristic
193C2×C22⋊C4—942no—
194C2×C22⋊C4—942no—
195C2×C22⋊C4—942no—
196C2×C22⋊C4—942no—
197C2×C42—472no—
198C2×C42D18811yescharacteristic
199C2×C42D18811yescharacteristic
200C2×C42C22×D4711yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—474no—
203C4⋊C8—942no—
204C4⋊C8—942no—
205C22⋊C8—22no—
206C22⋊C8—942no—
207#97—22no—
208#97—942no—
209#97—942no—
210C4×C8—942no—
211#95—941no—
212C24—942no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—944no—
217C2×D4—3762no—
218C2×D4—1882no—
219C2×D4—1882no—
220C2×D4—1882no—
221C2×D4—1882no—
222C2×D4—1881no—
223C2×D4—1882no—
224C2×D4—1881no—
225C22×C4—942no—
226C22×C4(C2×D47)⋊C411yescharacteristic
227C22×C4(C2×D47)⋊C411yescharacteristic
228C22×C4(C2×D47)⋊C411yescharacteristic
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4C2×C4×D4711yescharacteristic
231C22×C4C2×D18811yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—42no—
234C2×C8—944no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—1882no—
239C22⋊C4—3762no—
240C22⋊C4—3762no—
241C22⋊C4—1884no—
242C22⋊C4—1884no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—1882no—
248C42D4×D4712yes—
249C42#1062535212yes—
250C23—1882no—
251C23—1882no—
252C23—1882no—
253C23—1882no—
254C23—1882no—
255C23—1882no—
256C23—1882no—
257C23C2×(C2×D47)⋊C411yescharacteristic
258Q8—44no—
259D4—3764no—
260D4—3764no—
261D4—3762no—
262D4—3762no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—1882no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4#1309394012yes—
280C2×C4(C22⋊D47)⋊C412yes—
281C2×C4#1309399411yescharacteristic
282C2×C4#1309399612yes—
283C2×C4#1309400811yescharacteristic
284C8—82no—
285C8—1884no—
286C22—21no—
287C22—21no—
288C22—3762no—
289C22—3762no—
290C22—3762no—
291C22—3762no—
292C22—3762no—
293C22—3762no—
294C22—3762no—
295C22#1568900011yescharacteristic
296C22C2×#1309394011yescharacteristic
297C22#1568910511yescharacteristic
298C4—21no—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—3762no—
306C4#1568929812yes—
307C2—21no—
308C2—21no—
309C2—7522no—
310C2#2024378911yescharacteristic
311C2#2024384911yescharacteristic
312C2#2024391311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1467C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1467D47
hypocenterC47#9458
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
4. #20241243C2semidirect
5. C47×#1467C2semidirect
154. #1467D47semidirect
179. C47#9458semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202483872
#202421012
#202412432
C47×#14672
#945847