Finite Group Databasebeta

← #26127094

Subgroups — #26127094

Order 12032.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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#20243384C211yesmaximal, characteristic
2#20241732C211yesmaximal, characteristic
3#20241418C212yesmaximal
4#20241394C212yesmaximal
5C47×#2399C211yesmaximal, characteristic
6#15689180—21no—
7#15689147—21no—
8C2×#13093943—21no—
9#15688892C2212yes—
10#15688849—21no—
11#15688849C2212yes—
12C2×C4×Q8×C47C411yescharacteristic
13C2×C4×Q8×C47C2211yescharacteristic
14C47×#398—21no—
15C47×#398C411yescharacteristic
16C47×#398C2211yescharacteristic
17C47×#395—21no—
18C47×#395C411yescharacteristic
19C47×#395C2211yescharacteristic
20C4×C4⋊C4×C47—21no—
21C4×C4⋊C4×C47—21no—
22C47×#376C411yescharacteristic
23C2×#10625329—22no—
24C4⋊(C47⋊C8)—41no—
25C4⋊(C47⋊C8)—41no—
26C4⋊(C47⋊C8)—41no—
27#13093943—42no—
28C22⋊(C47⋊C8)—22no—
29#13093935—41no—
30C22×Q8×C47C2×C411yescharacteristic
31C22×Q8×C47D411yescharacteristic
32C4×Q8×C47—41no—
33C4×Q8×C47—41no—
34C4×Q8×C47—41no—
35C4×Q8×C47—41no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C4⋊C4×C47C2311yescharacteristic
45C2×C42×C47—21no—
46C2×C42×C47—21no—
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50#95×C47—21no—
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C2×Q8×C47—21no—
58C2×Q8×C47—21no—
59C2×Q8×C47—41no—
60C2×Q8×C47—41no—
61C2×Q8×C47—42no—
62C2×Q8×C47—41no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
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×C4×C47C2×D411yescharacteristic
74C4⋊C4×C47—41no—
75C4⋊C4×C47—41no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—42no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C42×C47—24no—
85C42×C47—21no—
86C42×C47—21no—
87C42×C47—21no—
88C42×C47—21no—
89C42×C47—21no—
90C42×C47—21no—
91C42×C47—41no—
92C42×C47—41no—
93#10625329—42no—
94#10625329—42no—
95C2×C47⋊C8—42no—
96C2×C47⋊C8—42no—
97C23×C47C2×C22⋊C411yescharacteristic
98Q8×C47—42no—
99Q8×C47—42no—
100Q8×C47—42no—
101Q8×C47—42no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
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—41no—
129C2×C4×C47#9911yescharacteristic
130C2×C4×C47#9911yescharacteristic
131C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C47#13711yescharacteristic
133C47⋊C8—82no—
134C47⋊C8—82no—
135#11831—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47C2×#9911yescharacteristic
139C22×C47#41711yescharacteristic
140C22×C47#42111yescharacteristic
141C4×C47—21no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—42no—
145C4×C47—42no—
146C4×C47—41no—
147C4×C47—41no—
148C4×C47—42no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—81no—
153C4×C47#35612yes—
154#2399D4711yescharacteristic
155#1801—471no—
156#1467—471no—
157#1354—472no—
158#1346—472no—
159C2×C47—21no—
160C2×C47—21no—
161C2×C47#151311yescharacteristic
162C2×C47#212011yescharacteristic
163C2×C47C2×#35611yescharacteristic
164C2×C4×Q8Dic4711yescharacteristic
165C2×C4×Q8C2×D4711yescharacteristic
166#432—941no—
167#423—941no—
168C2×#101—941no—
169#398—21no—
170#398Dic4711yescharacteristic
171#398C2×D4711yescharacteristic
172#395—21no—
173#395Dic4711yescharacteristic
174#395C2×D4711yescharacteristic
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177#376Dic4711yescharacteristic
178#343—472no—
179#324—472no—
180#324—941no—
181C47#1183111yescharacteristic, complemented, Sylow
182C22×Q8C2×Dic4711yescharacteristic
183C22×Q8C22⋊D4711yescharacteristic
184C2×M16—942no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×Dic4711yescharacteristic
195C2×C4⋊C4C2×Dic4711yescharacteristic
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C4⋊C4C22×D4711yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×Dic4711yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C2×C42C22⋊D4711yescharacteristic
203C4⋊C8—1881no—
204C4⋊C8—1881no—
205C4⋊C8—1881no—
206#101—1882no—
207C22⋊C8—942no—
208#97—1881no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C2×Dic4711yescharacteristic
215#95C2×Dic4711yescharacteristic
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4C22⋊Dic4711yescharacteristic
227C22×C4C22⋊Dic4711yescharacteristic
228C22×C4C22⋊Dic4711yescharacteristic
229C22×C4C22⋊Dic4711yescharacteristic
230C22×C4C22×Dic4711yescharacteristic
231C22×C4C2×C22⋊D4711yescharacteristic
232C22×C4C2×C22⋊D4711yescharacteristic
233M16—1882no—
234M16—1882no—
235C2×C8—1882no—
236C2×C8—1882no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C23C2×C22⋊Dic4711yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
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—21no—
275C2×C4—21no—
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#1309393911yescharacteristic
289C2×C4#1309393911yescharacteristic
290C2×C4#1309404911yescharacteristic
291C2×C4#1309405411yescharacteristic
292C8—3762no—
293C8—3762no—
294C22—21no—
295C22—21no—
296C22C2×#1309393911yescharacteristic
297C22#1568912411yescharacteristic
298C22#1568914011yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4#1568892212yes—
312C2—21no—
313C2—21no—
314C2#2024193111yescharacteristic
315C2#2024502911yescharacteristic
316C2C2×#1568892211yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2399C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2399D47
hypocenterC47#11831
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2399

Complements

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

NH ≅ G/Ntype
181. C47#11831semidirect

Maximal subgroups

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

subgroupindex
#202433842
#202417322
#202414182
#202413942
C47×#23992
#1183147