Finite Group Databasebeta

← #26125924

Subgroups — #26125924

Order 12032.

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

normal subgroups only

This group has 424 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#20248561C211yesmaximal, characteristic
2#20241726C211yesmaximal, characteristic
3#20241331C214yesmaximal, complemented
4C47×#1459C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6#15689087—21no—
7#15689035—21no—
8#15689035—21no—
9#15689026C412yes—
10#15688988C2212yes—
11#15688983—21no—
12#15688983—21no—
13C4×C4⋊Dic47C411yescharacteristic
14#15688849C2212yes—
15#15688847—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#324C2212yes—
19C47×#323—21no—
20C22×Dic94—21no—
21C2×#10625329—41no—
22#13094024—41no—
23#13094024—41no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C4×Dic47C2×C412yes—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—41no—
32C47⋊#95—21no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C22×D4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C22×C8×C47—41no—
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47C2311yescharacteristic
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47D411yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C22⋊C8×C47—22no—
46C22⋊C8×C47—41no—
47C24×C47C22⋊C411yescharacteristic
48C24×C47C22⋊C411yescharacteristic
49C2×D4×C47—22no—
50C2×D4×C47—22no—
51C2×D4×C47—41no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—42no—
57C22×C4×C47—42no—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—22no—
64C22×Dic47—22no—
65C2×Dic94—42no—
66C2×Dic94—82no—
67C2×C8×C47—42no—
68C2×C8×C47—82no—
69C4⋊C4×C47—41no—
70C22⋊C4×C47—41no—
71C22⋊C4×C47—41no—
72C42×C47—21no—
73C42×C47—21no—
74C22⋊Dic47—82no—
75C22⋊Dic47—82no—
76C4⋊Dic47—42no—
77C4⋊Dic47—41no—
78C4⋊Dic47—41no—
79C4⋊Dic47—42no—
80C4×Dic47—24no—
81#10625329—82no—
82C2×C47⋊C8—42no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—41no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47#10112yes—
93C23×C47C2×C22⋊C411yescharacteristic
94D4×C47—44no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—42no—
102C2×C4×C47—42no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47—81no—
107C2×C4×C47—81no—
108C2×C4×C47#9911yescharacteristic
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47#13112yes—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117Dic94—84no—
118C8×C47—82no—
119C47⋊C8—82no—
120#11569—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—21no—
123C22×C47—22no—
124C22×C47—42no—
125C22×C47—42no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47—41no—
131C22×C47—81no—
132C22×C47—81no—
133C22×C47#40811yescharacteristic
134C22×C47C2×#9911yescharacteristic
135C22×C47C2×#10111yescharacteristic
136C4×C47—21no—
137C4×C47—22no—
138C4×C47—41no—
139C4×C47—41no—
140C4×C47—41no—
141C4×C47#35412yes—
142Dic47—82no—
143Dic47—82no—
144#2563—471no—
145#1465—471no—
146#1459D4711yescharacteristic
147#1324—474no—
148C2×C47—21no—
149C2×C47—21no—
150C2×C47—42no—
151C2×C47—81no—
152C2×C47#147211yescharacteristic
153C2×C47C2×#35411yescharacteristic
154C2×C47#213411yescharacteristic
155C2×C4⋊Q8—471no—
156C2×C4×D4C2×D4711yescharacteristic
157#407—941no—
158C2×C22⋊C8—21no—
159#393—941no—
160#393—941no—
161#390—472no—
162#382—472no—
163#380—941no—
164#380—941no—
165C4×C4⋊C4—471no—
166#324—472no—
167#324C2×D4712yes—
168#323—21no—
169#323—941no—
170C47#1156911yescharacteristic, complemented, Sylow
171C22×Q8—941no—
172C22×D4C4×D4711yescharacteristic
173C23×C4D18811yescharacteristic
174C23×C4C22⋊D4711yescharacteristic
175C2×M16—1881no—
176C22×C8—41no—
177C4⋊Q8—1881no—
178C4⋊Q8—1881no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—472no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4C22×D4711yescharacteristic
185C2×C22⋊C4—942no—
186C2×C22⋊C4—942no—
187C2×C22⋊C4D18811yescharacteristic
188C2×C22⋊C4C22⋊D4711yescharacteristic
189C2×C42—472no—
190C2×C42C4×D4711yescharacteristic
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—942no—
194C22⋊C8—1881no—
195#95—941no—
196#95—942no—
197#95—942no—
198C24(C2×D47)⋊C411yescharacteristic
199C24(C2×D47)⋊C411yescharacteristic
200C2×Q8—3762no—
201C2×Q8—1882no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—942no—
212C22×C4—942no—
213C22×C4(C2×D47)⋊C411yescharacteristic
214C22×C4(C2×D47)⋊C411yescharacteristic
215C22×C4C2×C4×D4711yescharacteristic
216C22×C4C2×D18811yescharacteristic
217C22×C4C2×C22⋊D4711yescharacteristic
218M16—3762no—
219C2×C8—42no—
220C2×C8—82no—
221C2×C8—1882no—
222C4⋊C4—41no—
223C4⋊C4—1882no—
224C4⋊C4—1882no—
225C4⋊C4—1881no—
226C4⋊C4—1881no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—3762no—
230C22⋊C4—3762no—
231C42—21no—
232C42—21no—
233C42—944no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1309394212yes—
244C23C2×(C2×D47)⋊C411yescharacteristic
245Q8—3764no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—1882no—
261C2×C4—1882no—
262C2×C4—1882no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4#1309393811yescharacteristic
267C2×C4#1309393811yescharacteristic
268C2×C4#1309403012yes—
269C8—82no—
270C8—3762no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22#1568908911yescharacteristic
284C22C2×#1309393811yescharacteristic
285C22C2×#1309394211yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—3762no—
292C4—3762no—
293C4#1568891712yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2024175111yescharacteristic
299C2C2×#1568891711yescharacteristic
300C2#2024507911yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1459C2
socleC22×C47C2×#99
prime radicalC22×D4×C47C2×C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1459D47
hypocenterC47#11569
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20241331C2semidirect
170. C47#11569semidirect

Maximal subgroups

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

subgroupindex
#202485612
#202417262
#202413312
C47×#14592
#1156947