Finite Group Databasebeta

← #26127151

Subgroups — #26127151

Order 12032.

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

normal subgroups only

This group has 390 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#20241409C212yesmaximal
4#20241388C212yesmaximal
5C47×#2599C211yesmaximal, characteristic
6#15689176—21no—
7#15688986—21no—
8#15688890—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×C4×Q8×C47C411yescharacteristic, complemented
13C2×#117×C47C2211yescharacteristic
14C47×#398C411yescharacteristic, complemented
15C47×#397C2211yescharacteristic
16C47×#395C411yescharacteristic, complemented
17C47×#393C2211yescharacteristic
18C47×#388—21no—
19C47×#387—21no—
20C47×#387—21no—
21C47×#385—21no—
22C47×#383C411yescharacteristic, complemented
23C2×C22⋊Dic47—22no—
24C4⋊(C47⋊C8)—41no—
25C22⋊(C47⋊C8)—22no—
26C4×C47⋊C8—41no—
27C47⋊#95—41no—
28C47⋊#95—41no—
29C22×Q8×C47C2×C411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C4×Q8×C47—41no—
32C4×Q8×C47—41no—
33#117×C47—41no—
34#117×C47—41no—
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×C47D411yescharacteristic
41C2×C22⋊C4×C47—21no—
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47C2311yescharacteristic
45C2×C42×C47—21no—
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47D411yescharacteristic
48#95×C47—21no—
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
57C2×Q8×C47—42no—
58C2×Q8×C47—41no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—41no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—42no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—42no—
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—
99Q8×C47—42no—
100Q8×C47—42no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—22no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
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—41no—
127C2×C4×C47—41no—
128C2×C4×C47#9911yescharacteristic
129C2×C4×C47#9911yescharacteristic
130C2×Dic47—82no—
131C2×Dic47—82no—
132C2×Dic47—82no—
133C47⋊C8—82no—
134#11842—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—22no—
138C22×C47—42no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47C2×#9911yescharacteristic
143C22×C47#41611yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—41no—
148C4×C47—42no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—81no—
154C4×C47—81no—
155C4×C47—81no—
156Dic47—162no—
157#2599D4711yescharacteristic
158#1820—471no—
159#1458—471no—
160#1351—472no—
161#1344—472no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47—42no—
165C2×C47#151611yescharacteristic
166C2×C47#212011yescharacteristic
167C2×C47#212911yescharacteristic
168C2×C4×Q8Dic4711yescharacteristic, complemented
169C2×#117C2×D4711yescharacteristic
170#431—941no—
171#398Dic4711yescharacteristic, complemented
172#397C2×D4711yescharacteristic
173#395Dic4711yescharacteristic, complemented
174#393C2×D4711yescharacteristic
175#388—21no—
176#387—21no—
177#387—21no—
178#385—21no—
179#383Dic4711yescharacteristic, complemented
180#381—941no—
181#342—472no—
182#342—941no—
183#324—941no—
184#323—472no—
185C47#1184211yescharacteristic, complemented, Sylow
186C22×Q8C2×Dic4711yescharacteristic
187C23×C4C22⋊D4711yescharacteristic
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic4711yescharacteristic
196C2×C4⋊C4C2×Dic4711yescharacteristic
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4C22⋊D4711yescharacteristic
202C2×C22⋊C4C22×D4711yescharacteristic
203C2×C42—21no—
204C2×C42C2×Dic4711yescharacteristic
205C2×C42C22⋊D4711yescharacteristic
206C4⋊C8—1881no—
207C22⋊C8—942no—
208C4×C8—1881no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—1881no—
216#95—1881no—
217#95C2×Dic4711yescharacteristic
218#95C2×Dic4711yescharacteristic
219C24C2×C22⋊D4711yescharacteristic
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—1882no—
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22⋊Dic4711yescharacteristic
234C22×C4C22⋊Dic4711yescharacteristic
235C22×C4C22×Dic4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—1882no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—1882no—
250C22⋊C4—1882no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×C22⋊Dic4711yescharacteristic
261C23#1309404412yes—
262Q8—42no—
263Q8—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
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—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—3762no—
292C2×C4—3762no—
293C2×C4—3762no—
294C2×C4#1309393911yescharacteristic
295C2×C4#1309393911yescharacteristic
296C8—3762no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1309393911yescharacteristic
305C22#1568911911yescharacteristic
306C22#1568914011yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—7522no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2024194411yescharacteristic
323C2#2024502911yescharacteristic
324C2#2024506311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#2599C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#2599D47
hypocenterC47#11842
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
12. C2×C4×Q8×C47C4semidirect
14. C47×#398C4semidirect
16. C47×#395C4semidirect
22. C47×#383C4semidirect
168. C2×C4×Q8Dic47semidirect
171. #398Dic47semidirect
173. #395Dic47semidirect
179. #383Dic47semidirect
185. C47#11842semidirect

Maximal subgroups

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

subgroupindex
#202434652
#202417022
#202414092
#202413882
C47×#25992
#1184247