Finite Group Databasebeta

← #26113628

Subgroups — #26113628

Order 12032.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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#20249086C211yesmaximal, characteristic
2#20241989C212yesmaximal, complemented
3#20241722C211yesmaximal, characteristic
4#20241282C212yesmaximal, complemented
5C47×#2921C211yesmaximal, characteristic
6C2×#13094016C2211yescharacteristic
7#15689180C2212yes—
8#15689062C411yescharacteristic
9#15689059C411yescharacteristic
10#15689053C411yescharacteristic
11#15689048C411yescharacteristic
12#15689018—21no—
13#15689017—21no—
14#15689007—21no—
15#15688996—21no—
16#15688969C2211yescharacteristic
17#15688864—24no—
18#15688849—21no—
19#15688847—21no—
20C2×#117×C47C2211yescharacteristic
21C47×#433—24no—
22C47×#423—21no—
23C47×#407—21no—
24C4×M16×C47C2212yes—
25#13094016—22no—
26#13094016—22no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38#13093935—22no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic
45C47⋊#95C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C2×M16×C47—22no—
48#117×C47—22no—
49#117×C47—22no—
50C2×C4⋊C4×C47C2×C411yescharacteristic
51C2×C4⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—24no—
57C22⋊C8×C47—24no—
58#97×C47D414yes—
59C4×C8×C47—22no—
60C24×C47C22⋊C411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75M16×C47—42no—
76M16×C47—42no—
77C2×C8×C47—24no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C22⋊C4×C47—22no—
81C22⋊C4×C47—22no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47C2×D412yes—
85C42×C47#4112yes—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C4⋊Dic47—22no—
89C4⋊Dic47—22no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93Dic47⋊C4—42no—
94Dic47⋊C4—41no—
95C4×Dic47—42no—
96C2×C47⋊C8—42no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—41no—
101C23×C47—41no—
102C23×C47C2×C22⋊C411yescharacteristic
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—21no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12112yes—
119C2×C4×C47#12311yescharacteristic
120C2×Dic47—22no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C8×C47—44no—
133C47⋊C8—82no—
134#9390—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47#38611yescharacteristic
143C22×C47#42111yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152Dic47—42no—
153Dic47—81no—
154Dic47—81no—
155Dic47—81no—
156#2921D4711yescharacteristic
157#2611—471no—
158#1528—472no—
159#1464—471no—
160#1300—472no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—81no—
164C2×C47#188411yescharacteristic
165C2×C47#189711yescharacteristic
166C2×C47#191511yescharacteristic
167C2×#125—471no—
168C2×#117C2×D4711yescharacteristic
169#433—24no—
170#432—472no—
171#423—21no—
172#407—21no—
173C4×M16C2×D4712yes—
174#400—471no—
175#399—471no—
176#398—471no—
177#397—471no—
178#389—941no—
179#388—941no—
180#387—941no—
181#385—941no—
182#376—471no—
183#330—944no—
184#324—941no—
185#323—941no—
186C47#939011yescharacteristic, complemented, Sylow
187C23×C4C22⋊D4711yescharacteristic
188C2×M16—22no—
189#125—942no—
190#125—942no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—471no—
194C2×C4⋊C4—471no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4C4×D4711yescharacteristic
200C2×C4⋊C4D18811yescharacteristic
201C2×C22⋊C4—941no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4C4×D4711yescharacteristic
204C2×C22⋊C4D18811yescharacteristic
205C2×C42—941no—
206C2×C42C22⋊D4711yescharacteristic
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—942no—
210C22⋊C8—24no—
211C22⋊C8—942no—
212#97—942no—
213#97C22⋊D4714yes—
214C4×C8—22no—
215#95—471no—
216#95—471no—
217#95—941no—
218#95—941no—
219#95—941no—
220#95—941no—
221#95—941no—
222C24(C2×D47)⋊C411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—941no—
228C22×C4—941no—
229C22×C4—941no—
230C22×C4—941no—
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4(C2×D47)⋊C411yescharacteristic
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4C2×C4×D4711yescharacteristic
235C22×C4C2×D18811yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—1882no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—942no—
244C4⋊C4—942no—
245C4⋊C4—1882no—
246C4⋊C4—1882no—
247C4⋊C4—1881no—
248C4⋊C4—1882no—
249C4⋊C4—1881no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—1882no—
253C22⋊C4—1882no—
254C42—21no—
255C42—21no—
256C42—1882no—
257C42#1062535012yes—
258C42C2×C22⋊D4712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—942no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4C4×C22⋊D4712yes—
291C2×C4#1309399311yescharacteristic
292C2×C4#1309399812yes—
293C2×C4#1309400711yescharacteristic
294C8—44no—
295C8—3762no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1568900211yescharacteristic
304C22#1568914111yescharacteristic
305C22#1568914111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3761no—
314C4—3761no—
315C4—3761no—
316C4—1882no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2024377111yescharacteristic
321C2#2024383611yescharacteristic
322C2#2024394011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2921C2
socleC22×C47#386
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2921D47
hypocenterC47#9390
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #20241989C2semidirect
4. #20241282C2semidirect
186. C47#9390semidirect

Maximal subgroups

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

subgroupindex
#202490862
#202419892
#202417222
#202412822
C47×#29212
#939047