Finite Group Databasebeta

← #26127207

Subgroups — #26127207

Order 12032.

368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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#20243471C211yesmaximal, characteristic
3#20241416C212yesmaximal, complemented
4#20241395C212yesmaximal, complemented
5C47×#1458C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic
7#15689087—21no—
8#15689062C2211yescharacteristic
9#15689059C411yescharacteristic
10#15689053C2211yescharacteristic
11#15689048C411yescharacteristic
12#15689018—21no—
13#15689017—21no—
14#15689007—21no—
15#15688996—21no—
16#15688969C411yescharacteristic
17#15688892—21no—
18#15688892C2212yes—
19C2×#117×C47C2211yescharacteristic
20C47×#431—21no—
21C47×#324C2212yes—
22C47×#323—21no—
23C2×#10625329—22no—
24#13094016—41no—
25#13094016—41no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C4×Dic47—21no—
35C22⋊(C47⋊C8)—41no—
36C22⋊(C47⋊C8)—41no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45#117×C47—41no—
46#117×C47—41no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47C2311yescharacteristic
49C2×C22⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C42×C47C2×C411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C4⋊C8×C47—41no—
54C22⋊C8×C47—22no—
55C4×C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—41no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C2×C8×C47—42no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C22⋊C4×C47—41no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—21no—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C4⋊Dic47—41no—
83C4⋊Dic47—41no—
84Dic47⋊C4—42no—
85Dic47⋊C4—42no—
86Dic47⋊C4—41no—
87Dic47⋊C4—41no—
88Dic47⋊C4—42no—
89C4×Dic47—42no—
90#10625329—42no—
91#10625329—42no—
92C2×C47⋊C8—42no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47C2×C22⋊C411yescharacteristic
99C2×C4×C47—21no—
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—21no—
107C2×C4×C47—42no—
108C2×C4×C47—42no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47#9911yescharacteristic
114C2×C4×C47#9911yescharacteristic
115C2×C4×C47#13512yes—
116C2×Dic47—22no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C8×C47—82no—
129C47⋊C8—82no—
130#11853—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—42no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—81no—
138C22×C47C2×#9911yescharacteristic
139C22×C47#41611yescharacteristic
140C22×C47#42111yescharacteristic
141C4×C47—22no—
142C4×C47—21no—
143C4×C47—21no—
144C4×C47—41no—
145C4×C47—41no—
146C4×C47—41no—
147C4×C47—41no—
148Dic47—42no—
149Dic47—81no—
150Dic47—81no—
151Dic47—81no—
152#2611—471no—
153#1822—471no—
154#1458D4711yescharacteristic
155#1353—472no—
156#1346—472no—
157C2×C47—21no—
158C2×C47—21no—
159C2×C47—81no—
160C2×C47#151611yescharacteristic
161C2×C47#212311yescharacteristic
162C2×C47#213211yescharacteristic
163C2×#125—471no—
164C2×#117C2×D4711yescharacteristic
165#431—21no—
166#407—941no—
167#400—471no—
168#399—471no—
169#398—471no—
170#397—471no—
171#389—941no—
172#388—941no—
173#387—941no—
174#385—941no—
175#376—471no—
176#343—472no—
177#343—941no—
178#324C2×D4712yes—
179#323—21no—
180C47#1185311yescharacteristic, complemented, Sylow
181C23×C4D18811yescharacteristic
182C2×M16—942no—
183#125—1881no—
184#125—1881no—
185#117—41no—
186#117—41no—
187C2×C4⋊C4—471no—
188C2×C4⋊C4—471no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—941no—
193C2×C4⋊C4D18811yescharacteristic
194C2×C4⋊C4C22×D4711yescharacteristic
195C2×C22⋊C4—941no—
196C2×C22⋊C4—941no—
197C2×C22⋊C4C4×D4711yescharacteristic
198C2×C22⋊C4C22⋊D4711yescharacteristic
199C2×C42—941no—
200C2×C42C4×D4711yescharacteristic
201C2×C42C22⋊D4711yescharacteristic
202C4⋊C8—41no—
203C22⋊C8—22no—
204C22⋊C8—1881no—
205C22⋊C8—1881no—
206C4×C8—41no—
207#95—471no—
208#95—471no—
209#95—941no—
210#95—941no—
211#95—941no—
212#95—941no—
213#95—941no—
214C24(C2×D47)⋊C411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—941no—
220C22×C4—941no—
221C22×C4—941no—
222C22×C4—941no—
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4(C2×D47)⋊C411yescharacteristic
226C22×C4C2×C4×D4711yescharacteristic
227C22×C4C2×D18811yescharacteristic
228C22×C4C2×C22⋊D4711yescharacteristic
229M16—1882no—
230M16—1882no—
231C2×C8—42no—
232C2×C8—1882no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—1881no—
236C4⋊C4—1881no—
237C4⋊C4—1882no—
238C4⋊C4—1882no—
239C4⋊C4—1881no—
240C4⋊C4—1881no—
241C4⋊C4—1882no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—1882no—
245C22⋊C4—1882no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—1882no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D47)⋊C411yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—942no—
272C2×C4—1881no—
273C2×C4—1881no—
274C2×C4—1881no—
275C2×C4—1881no—
276C2×C4—1881no—
277C2×C4—1881no—
278C2×C4—1881no—
279C2×C4—1881no—
280C2×C4—1881no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4#1309393811yescharacteristic
284C2×C4#1309393811yescharacteristic
285C2×C4#1309404612yes—
286C8—82no—
287C8—3762no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1309393811yescharacteristic
296C22#1568912011yescharacteristic
297C22#1568913911yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—3761no—
306C4—3761no—
307C4—3761no—
308C4—1882no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#2024194511yescharacteristic
313C2#2024504111yescharacteristic
314C2#2024507111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11853
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #20241416C2semidirect
4. #20241395C2semidirect
180. C47#11853semidirect

Maximal subgroups

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

subgroupindex
#202490862
#202434712
#202414162
#202413952
C47×#14582
#1185347