Finite Group Databasebeta

← #26104842

Subgroups — #26104842

Order 12032.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 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#20251740C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic
3#20241210C214yesmaximal, complemented
4C47×#1817C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13094003—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C2×#13093935—21no—
9#15689029—21no—
10#15689021—21no—
11#15689021C2212yes—
12#15688993C411yescharacteristic
13#15688983—21no—
14#15688983C412yes—
15#15688878C2212yes—
16C22×C42×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#336—21no—
19C47×#336C2212yes—
20C22×Dic94—21no—
21C22×C47⋊C8—22no—
22#13094024—42no—
23#13094003—44no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C412yes—
28C2×C4⋊Dic47C2×C412yes—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C22⋊(C47⋊C8)—24no—
32#13093935—42no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C22×C8×C47—22no—
39C2×C42×C47—42no—
40C2×C42×C47C2×C411yescharacteristic
41C2×C42×C47D411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47C2311yescharacteristic
44C22⋊C8×C47—42no—
45C24×C47C22⋊C411yescharacteristic
46C22×C4×C47—22no—
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22×C411yescharacteristic
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—22no—
62C22×Dic47—22no—
63C2×Dic94—42no—
64C2×Dic94—42no—
65C2×Dic94—42no—
66C2×C8×C47—42no—
67C2×C8×C47—42no—
68C2×C8×C47—42no—
69C42×C47—22no—
70C42×C47—22no—
71C42×C47—41no—
72C42×C47—41no—
73C22⋊Dic47—44no—
74C4⋊Dic47—24no—
75C4⋊Dic47—42no—
76C4⋊Dic47—42no—
77C4⋊Dic47—42no—
78C4⋊Dic47—42no—
79Dic47⋊C4—44no—
80C2×C47⋊C8—24no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C23×C47—21no—
84C23×C47—21no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47#10112yes—
90C23×C47C2×C22⋊C411yescharacteristic
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—21no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—44no—
105C2×C4×C47—44no—
106C2×C4×C47—44no—
107C2×C4×C47#9912yes—
108C2×C4×C47#10412yes—
109C2×C4×C47#13112yes—
110C2×C4×C47AGL(1,ℤ/8)12yes—
111C2×C4×C47#13712yes—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118Dic94—84no—
119C8×C47—82no—
120C47⋊C8—44no—
121#7705—471nomaximal, Sylow
122C22×C47—22no—
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—41no—
126C22×C47—41no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47#40811yescharacteristic
130C22×C47C2×#9911yescharacteristic
131C22×C47C2×#10111yescharacteristic
132C22×C47#41811yescharacteristic
133C22×C47#41911yescharacteristic
134C22×C47C2×#10411yescharacteristic
135C22×C47#42111yescharacteristic
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—22no—
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—41no—
142Dic47—82no—
143Dic47—82no—
144#2857—471no—
145#1817D4711yescharacteristic
146#1749—471no—
147#1271—474no—
148C2×C47—42no—
149C2×C47#147211yescharacteristic
150C2×C47#148511yescharacteristic
151C2×C47#149611yescharacteristic
152C2×C47#150711yescharacteristic
153C2×C47#151111yescharacteristic
154C2×C47#153111yescharacteristic
155C2×C47#155611yescharacteristic
156C2×C4⋊Q8—471no—
157C2×C22⋊Q8—941no—
158C22×C42C2×D4711yescharacteristic
159C2×C22⋊C8—21no—
160C2×C22⋊C8—941no—
161C2×#97—941no—
162#391—941no—
163#389—472no—
164#389—941no—
165#384—471no—
166#380—472no—
167#380—941no—
168#336—21no—
169#336—472no—
170#336C2×D4712yes—
171C47#770511yescharacteristic, complemented, Sylow
172C22×Q8—941no—
173C23×C4C4×D4711yescharacteristic
174C23×C4D18811yescharacteristic
175C23×C4C22⋊D4711yescharacteristic
176C22×C8—22no—
177C22×C8—942no—
178C4⋊Q8—1882no—
179C22⋊Q8—1884no—
180C2×C4⋊C4—472no—
181C2×C4⋊C4—472no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4—941no—
185C2×C4⋊C4—942no—
186C2×C22⋊C4—942no—
187C2×C42—42no—
188C2×C42C4×D4711yescharacteristic
189C2×C42D18811yescharacteristic
190C2×C42C22⋊D4711yescharacteristic
191C2×C42C22×D4711yescharacteristic
192C22⋊C8—42no—
193C22⋊C8—944no—
194#97—1882no—
195#95—942no—
196#95—942no—
197C24(C2×D47)⋊C411yescharacteristic
198C2×Q8—1882no—
199C2×Q8—1882no—
200C2×Q8—1882no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—942no—
211C22×C4—942no—
212C22×C4(C2×D47)⋊C411yescharacteristic
213C22×C4(C2×D47)⋊C411yescharacteristic
214C22×C4(C2×D47)⋊C411yescharacteristic
215C22×C4C2×C4×D4711yescharacteristic
216C22×C4C2×D18811yescharacteristic
217C22×C4C2×C22⋊D4711yescharacteristic
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—944no—
222C2×C8—1882no—
223C2×C8—1882no—
224C4⋊C4—944no—
225C4⋊C4—1882no—
226C4⋊C4—1882no—
227C4⋊C4—1882no—
228C4⋊C4—1882no—
229C4⋊C4—1884no—
230C22⋊C4—1884no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23#1309394212yes—
242C23C2×(C2×D47)⋊C411yescharacteristic
243Q8—3764no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—1882no—
261C2×C4—1882no—
262C2×C4—1882no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4#1309393812yes—
267C2×C4#1309395012yes—
268C2×C4#1309403012yes—
269C2×C4#1309405212yes—
270C2×C4#1309405412yes—
271C8—82no—
272C8—1884no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#1568908911yescharacteristic
281C22C2×#1309393811yescharacteristic
282C22C2×#1309394211yescharacteristic
283C22#1568912911yescharacteristic
284C22#1568913111yescharacteristic
285C22C2×#1309395011yescharacteristic
286C22#1568914111yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—3762no—
294C4—3762no—
295C2—42no—
296C2#2024175111yescharacteristic
297C2#2024180811yescharacteristic
298C2#2024185811yescharacteristic
299C2#2024190511yescharacteristic
300C2#2024192011yescharacteristic
301C2#2024200411yescharacteristic
302C2#2024210411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1817D47
hypocenterC47#7705
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
3. #20241210C2semidirect
171. C47#7705semidirect

Maximal subgroups

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

subgroupindex
#202517402
#202431502
#202412102
C47×#18172
#770547