Finite Group Databasebeta

← #26126870

Subgroups — #26126870

Order 12032.

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

normal subgroups only

This group has 482 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#20248387C211yesmaximal, characteristic
2C4×#13093943C211yesmaximal, characteristic, complemented
3#20241392C214yesmaximal, complemented
4C47×#1467C211yesmaximal, characteristic, complemented
5C2×#13094021C411yescharacteristic
6C2×#13094013—21no—
7C2×#13093943—21no—
8C4×#10625329—21no—
9#15689040C412yes—
10#15689040C2212yes—
11#15689025—21no—
12#15689025—21no—
13#15689024—21no—
14C4×C4⋊Dic47C411yescharacteristic
15#15688892C2212yes—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324—21no—
19C47×#324C2212yes—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23C2×#10625329—22no—
24#13094021—41no—
25#13094021—41no—
26#13094013—44no—
27C2×C4⋊Dic47—21no—
28C2×(C2×D47)⋊C4—22no—
29C2×(C2×D47)⋊C4—22no—
30C2×(C2×D47)⋊C4—22no—
31C2×(C2×D47)⋊C4—22no—
32C2×C4×Dic47—21no—
33#13093943—24no—
34#13093935—41no—
35C4×C47⋊C8—41no—
36C47⋊#95C2×C412yes—
37C22×Q8×C47D411yescharacteristic
38C4×Q8×C47—41no—
39C4×Q8×C47—41no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C4⋊C4×C47C2311yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C4⋊C8×C47—41no—
47C22⋊C8×C47—22no—
48#97×C47—41no—
49C23×D47—22no—
50C2×Q8×C47—22no—
51C2×Q8×C47—22no—
52C2×Q8×C47—41no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22×C411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×Dic47—22no—
61C2×D188—42no—
62C2×D188—42no—
63C2×D188—42no—
64C2×D188—42no—
65C2×D188—42no—
66C2×D188—41no—
67C2×D188—41no—
68C2×D188—42no—
69C2×D188—82no—
70C2×C8×C47—42no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C42×C47—21no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—21no—
80(C2×D47)⋊C4—44no—
81(C2×D47)⋊C4—44no—
82(C2×D47)⋊C4—82no—
83(C2×D47)⋊C4—82no—
84C4⋊Dic47—42no—
85C4×Dic47—42no—
86#10625329—24no—
87#10625329—42no—
88C2×C47⋊C8—42no—
89C23×C47C2×C22⋊C411yescharacteristic
90C22×D47—42no—
91C22×D47—42no—
92C22×D47—42no—
93C22×D47—42no—
94C22×D47—42no—
95C22×D47—42no—
96C22×D47—42no—
97Q8×C47—44no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
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—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47#9911yescharacteristic
113C2×C4×C47#9911yescharacteristic
114C2×C4×C47#10412yes—
115C2×C4×C47AGL(1,ℤ/8)12yes—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119D188—82no—
120D188—84no—
121D188—84no—
122D188—82no—
123C8×C47—82no—
124C47⋊C8—44no—
125#11781—471nomaximal, Sylow
126C22×C47—21no—
127C22×C47—21no—
128C22×C47C2×#9911yescharacteristic
129C22×C47#41811yescharacteristic
130C22×C47C2×#10411yescharacteristic
131C2×D47—82no—
132C2×D47—82no—
133C2×D47—82no—
134C2×D47—82no—
135C2×D47—82no—
136C2×D47—82no—
137C2×D47—82no—
138C4×C47—21no—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—41no—
142C4×C47—41no—
143C4×C47—41no—
144C4×C47—41no—
145C4×C47#35512yes—
146Dic47—82no—
147#2548—471no—
148C4×#101—471no—
149#1467D4711yescharacteristic, complemented
150#1345—474no—
151C2×C47—21no—
152C2×C47—21no—
153C2×C47#150711yescharacteristic
154C2×C47#212211yescharacteristic
155C2×C47C2×#35511yescharacteristic
156D47—162no—
157C2×#127—471no—
158C2×#124—941no—
159C2×C4×Q8C2×D4711yescharacteristic
160#432—21no—
161C2×#101—941no—
162C4×M16—941no—
163#394—472no—
164#394—472no—
165#390—941no—
166#390—941no—
167#390—941no—
168C4×C4⋊C4—471no—
169#343—472no—
170#324—21no—
171#324C2×D4712yes—
172C47#1178111yescharacteristic, complemented, Sylow
173C22×Q8D18811yescharacteristic
174C22×D4—472no—
175C22×D4—941no—
176C22×D4—941no—
177C2×M16—942no—
178#127—1881no—
179#127—1881no—
180#124—1884no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4C4×D4711yescharacteristic
185C2×C4⋊C4C22⋊D4711yescharacteristic
186C2×C4⋊C4C22×D4711yescharacteristic
187C2×C22⋊C4—942no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4—942no—
190C2×C22⋊C4—942no—
191C2×C42—941no—
192C2×C42C4×D4711yescharacteristic
193C2×C42D18811yescharacteristic
194C2×C42C22⋊D4711yescharacteristic
195C4⋊C8—41no—
196#101—944no—
197C22⋊C8—22no—
198#97—41no—
199#97—1881no—
200C4×C8—1881no—
201#95—472no—
202C24—942no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—3762no—
207C2×D4—1882no—
208C2×D4—1882no—
209C2×D4—1882no—
210C2×D4—1882no—
211C2×D4—1882no—
212C2×D4—1881no—
213C2×D4—1881no—
214C2×D4—1882no—
215C22×C4—942no—
216C22×C4(C2×D47)⋊C411yescharacteristic
217C22×C4(C2×D47)⋊C411yescharacteristic
218C22×C4(C2×D47)⋊C411yescharacteristic
219C22×C4(C2×D47)⋊C411yescharacteristic
220C22×C4C2×C4×D4711yescharacteristic
221C22×C4C2×D18811yescharacteristic
222C22×C4C2×C22⋊D4711yescharacteristic
223M16—944no—
224M16—1882no—
225C2×C8—42no—
226C2×C8—1882no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—1882no—
231C22⋊C4—3762no—
232C22⋊C4—3762no—
233C22⋊C4—1884no—
234C22⋊C4—1884no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—1882no—
242C23—1882no—
243C23—1882no—
244C23—1882no—
245C23—1882no—
246C23—1882no—
247C23—1882no—
248C23—1882no—
249C23C2×(C2×D47)⋊C411yescharacteristic
250Q8—44no—
251D4—3762no—
252D4—3764no—
253D4—3764no—
254D4—3762no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4#1309393811yescharacteristic
273C2×C4#1309393811yescharacteristic
274C2×C4#1309395012yes—
275C2×C4#1309405212yes—
276C8—82no—
277C8—1884no—
278C22—21no—
279C22—21no—
280C22—3762no—
281C22—3762no—
282C22—3762no—
283C22—3762no—
284C22—3762no—
285C22—3762no—
286C22—3762no—
287C22C2×#1309393811yescharacteristic
288C22#1568912911yescharacteristic
289C22C2×#1309395011yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—3762no—
298C4#1568891912yes—
299C2—21no—
300C2—21no—
301C2—7522no—
302C2#2024190511yescharacteristic
303C2#2024503711yescharacteristic
304C2C2×#1568891911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1467C2
socleC22×C47C2×#99
prime radicalC2×#13094021C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1467D47
hypocenterC47#11781
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C4×#13093943C2semidirect
3. #20241392C2semidirect
4. C47×#1467C2semidirect
149. #1467D47semidirect
172. C47#11781semidirect

Maximal subgroups

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

subgroupindex
#202483872
C4×#130939432
#202413922
C47×#14672
#1178147