Finite Group Databasebeta

← #26127171

Subgroups — #26127171

Order 12032.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#20243465C211yesmaximal, characteristic
2#20241702C211yesmaximal, characteristic
3#20241412C212yesmaximal
4#20241388C212yesmaximal
5C47×#2611C211yesmaximal, characteristic
6#15689176—21no—
7#15688986—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×#125×C47C411yescharacteristic, complemented
13C2×#117×C47C2211yescharacteristic
14C47×#400C411yescharacteristic, complemented
15C47×#399C2211yescharacteristic
16C47×#398C411yescharacteristic, complemented
17C47×#397C2211yescharacteristic
18C47×#389—21no—
19C47×#388—21no—
20C47×#387—21no—
21C47×#385—21no—
22C47×#376C411yescharacteristic, complemented
23C2×C22⋊Dic47—22no—
24C4⋊(C47⋊C8)—41no—
25C22⋊(C47⋊C8)—22no—
26C4×C47⋊C8—41no—
27C47⋊#95—41no—
28C47⋊#95—41no—
29C23×C4×C47D411yescharacteristic
30#125×C47—41no—
31#125×C47—41no—
32#117×C47—41no—
33#117×C47—41no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2311yescharacteristic
46C2×C42×C47—21no—
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49#95×C47—21no—
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47C2×C411yescharacteristic
55#95×C47C2×C411yescharacteristic
56C24×C47C2×D411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—22no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—41no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—42no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—42no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—41no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C22⋊C4×C47—41no—
82C22⋊C4×C47—41no—
83C22⋊C4×C47—42no—
84C22⋊C4×C47—42no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—21no—
89C42×C47—42no—
90C22⋊Dic47—42no—
91C22⋊Dic47—42no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47C2×C22⋊C411yescharacteristic
98C23×C47#13512yes—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#9911yescharacteristic
128C2×Dic47—82no—
129C2×Dic47—82no—
130C2×Dic47—82no—
131C47⋊C8—82no—
132#11846—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—22no—
136C22×C47—42no—
137C22×C47—41no—
138C22×C47—41no—
139C22×C47—81no—
140C22×C47C2×#9911yescharacteristic
141C22×C47#41611yescharacteristic
142C22×C47#42111yescharacteristic
143C4×C47—21no—
144C4×C47—21no—
145C4×C47—42no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—81no—
152C4×C47—81no—
153C4×C47—81no—
154Dic47—162no—
155#2611D4711yescharacteristic
156#1820—471no—
157#1458—471no—
158#1352—472no—
159#1344—472no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47—42no—
163C2×C47#151611yescharacteristic
164C2×C47#212311yescharacteristic
165C2×C47#212611yescharacteristic
166C2×#125Dic4711yescharacteristic, complemented
167C2×#117C2×D4711yescharacteristic
168#431—941no—
169#400Dic4711yescharacteristic, complemented
170#399C2×D4711yescharacteristic
171#398Dic4711yescharacteristic, complemented
172#397C2×D4711yescharacteristic
173#389—21no—
174#388—21no—
175#387—21no—
176#385—21no—
177#381—941no—
178#376Dic4711yescharacteristic, complemented
179#342—472no—
180#342—941no—
181#324—941no—
182#323—472no—
183C47#1184611yescharacteristic, complemented, Sylow
184C23×C4C22⋊D4711yescharacteristic
185#125—41no—
186#125—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic4711yescharacteristic
194C2×C4⋊C4C2×Dic4711yescharacteristic
195C2×C4⋊C4C2×Dic4711yescharacteristic
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4C22⋊D4711yescharacteristic
201C2×C22⋊C4C22×D4711yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic4711yescharacteristic
204C2×C42C22⋊D4711yescharacteristic
205C4⋊C8—1881no—
206C22⋊C8—942no—
207C4×C8—1881no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—1881no—
214#95—1881no—
215#95C2×Dic4711yescharacteristic
216#95C2×Dic4711yescharacteristic
217C24C2×C22⋊D4711yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—1882no—
227C22×C4C22⋊Dic4711yescharacteristic
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22⋊Dic4711yescharacteristic
231C22×C4C22×Dic4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233C2×C8—1882no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—1882no—
248C22⋊C4—1882no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic4711yescharacteristic
259C23#1309404412yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—3762no—
288C2×C4—3762no—
289C2×C4—3762no—
290C2×C4#1309393911yescharacteristic
291C2×C4#1309393911yescharacteristic
292C8—3762no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1309393911yescharacteristic
301C22#1568911911yescharacteristic
302C22#1568914011yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—7522no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2024194411yescharacteristic
319C2#2024504211yescharacteristic
320C2#2024505311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2611C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2611D47
hypocenterC47#11846
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C47C4semidirect
14. C47×#400C4semidirect
16. C47×#398C4semidirect
22. C47×#376C4semidirect
166. C2×#125Dic47semidirect
169. #400Dic47semidirect
171. #398Dic47semidirect
178. #376Dic47semidirect
183. C47#11846semidirect

Maximal subgroups

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

subgroupindex
#202434652
#202417022
#202414122
#202413882
C47×#26112
#1184647