Finite Group Databasebeta

← #26126833

Subgroups — #26126833

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#20247696C211yesmaximal, characteristic
2#20241729C211yesmaximal, characteristic
3#20241387C214yesmaximal, complemented
4C4×#99×C47C211yesmaximal, characteristic
5C2×#13094005—21no—
6C22×C4⋊Dic47C411yescharacteristic, complemented
7#15689180—21no—
8#15689053C412yescomplemented
9#15689048C2212yes—
10#15689021—21no—
11#15688984—21no—
12#15688983—21no—
13#15688983C411yescharacteristic, complemented
14#15688847—21no—
15#15688847C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C2×#99×C47—21no—
18C4×C22⋊C4×C47—21no—
19C47×#342C2212yes—
20C23×Dic47—21no—
21#13094005—44no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—21no—
29C2×C22⋊Dic47—22no—
30C4⋊(C47⋊C8)—41no—
31C22⋊(C47⋊C8)—22no—
32#13093935—41no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95C2×C412yes—
36C22×D4×C47D411yescharacteristic
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47C2311yescharacteristic
45C2×C42×C47—41no—
46C2×C42×C47D411yescharacteristic
47#99×C47—24no—
48#95×C47—41no—
49C24×C47C22⋊C411yescharacteristic
50C24×C47C2×D411yescharacteristic
51C2×D4×C47—22no—
52C2×D4×C47—22no—
53C2×D4×C47—41no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47—42no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—22no—
67C22×Dic47—22no—
68C22×Dic47—42no—
69C22×Dic47—42no—
70C22×Dic47—42no—
71C4⋊C4×C47—41no—
72C22⋊C4×C47—24no—
73C22⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C22⋊C4×C47—42no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—82no—
79C22⋊Dic47—42no—
80C22⋊Dic47—44no—
81C4⋊Dic47—42no—
82C4⋊Dic47—41no—
83C4⋊Dic47—41no—
84C4⋊Dic47—42no—
85Dic47⋊C4—44no—
86C2×C47⋊C8—42no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47#10412yes—
96C23×C47C2×C22⋊C411yescharacteristic
97C23×C47AGL(1,ℤ/8)12yes—
98D4×C47—44no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—44no—
113C2×C4×C47—81no—
114C2×C4×C47—82no—
115C2×C4×C47—82no—
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#9911yescharacteristic
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—84no—
125C47⋊C8—82no—
126#11773—471nomaximal, Sylow
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—22no—
130C22×C47—22no—
131C22×C47—42no—
132C22×C47—42no—
133C22×C47—42no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—41no—
138C22×C47—81no—
139C22×C47C2×#9911yescharacteristic
140C22×C47#41811yescharacteristic
141C22×C47C2×#10411yescharacteristic
142C4×C47—21no—
143C4×C47—41no—
144C4×C47—41no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47—84no—
148C4×C47#35212yes—
149Dic47—82no—
150Dic47—82no—
151#2486—471no—
152C4×#99D4711yescharacteristic
153#1466—471no—
154#1344—474no—
155C2×C47—21no—
156C2×C47—21no—
157C2×C47—42no—
158C2×C47—42no—
159C2×C47#150711yescharacteristic
160C2×C47C2×#35211yescharacteristic
161C2×C47#212911yescharacteristic
162C2×#123—941no—
163C2×C4×D4C2×D4711yescharacteristic
164C22×C4⋊C4—471no—
165#432—941no—
166C2×#99—21no—
167#398—472no—
168#397—472no—
169#389—941no—
170#381—941no—
171#380—471no—
172#380—941no—
173C4×C22⋊C4—21no—
174#342C2×D4712yes—
175#323—472no—
176#323—941no—
177C47#1177311yescharacteristic, complemented, Sylow
178C22×D4C22⋊D4711yescharacteristic
179C23×C4—941no—
180C23×C4C4×D4711yescharacteristic
181C23×C4C22⋊D4711yescharacteristic
182#123—1884no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—472no—
186C2×C4⋊C4—942no—
187C2×C4⋊C4—941no—
188C2×C4⋊C4—941no—
189C2×C4⋊C4—1881no—
190C2×C4⋊C4—1881no—
191C2×C4⋊C4C4×D4711yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—941no—
194C2×C22⋊C4—942no—
195C2×C22⋊C4D18811yescharacteristic
196C2×C22⋊C4C22×D4711yescharacteristic
197C2×C42—41no—
198C2×C42D18811yescharacteristic
199C4⋊C8—1881no—
200#99—24no—
201C22⋊C8—942no—
202#97—1881no—
203#95—41no—
204#95—472no—
205#95—942no—
206#95—942no—
207C24(C2×D47)⋊C411yescharacteristic
208C24C2×C22⋊D4711yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—942no—
220C22×C4—942no—
221C22×C4—1882no—
222C22×C4—1882no—
223C22×C4—1882no—
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4(C2×D47)⋊C411yescharacteristic
227C22×C4C2×C4×D4711yescharacteristic
228C22×C4C2×D18811yescharacteristic
229C2×C8—1882no—
230C4⋊C4—41no—
231C4⋊C4—1882no—
232C4⋊C4—1884no—
233C4⋊C4—1882no—
234C4⋊C4—1881no—
235C4⋊C4—1881no—
236C22⋊C4—24no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—1882no—
241C22⋊C4—1884no—
242C42—21no—
243C42—21no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1309395212yes—
254C23C2×(C2×D47)⋊C411yescharacteristic
255C23#1309404912yes—
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—3764no—
275C2×C4—1882no—
276C2×C4—1882no—
277C2×C4—1882no—
278C2×C4—1882no—
279C2×C4—1882no—
280C2×C4—1882no—
281C2×C4#1309393811yescharacteristic
282C2×C4#1309393811yescharacteristic
283C8—3762no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#1309393811yescharacteristic
297C22#1568912711yescharacteristic
298C22C2×#1309395211yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—84no—
305C4—3762no—
306C4—3762no—
307C4#1568891212yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2024190311yescharacteristic
313C2C2×#1568891211yescharacteristic
314C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC4×#99×C47C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenterC4×#99D47
hypocenterC47#11773
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20241387C2semidirect
6. C22×C4⋊Dic47C4semidirect
8. #15689053C4semidirect
13. #15688983C4semidirect
177. C47#11773semidirect

Maximal subgroups

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

subgroupindex
#202476962
#202417292
#202413872
C4×#99×C472
#1177347