Finite Group Databasebeta

← #26104949

Subgroups — #26104949

Order 12032.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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#20243368C211yesmaximal, characteristic
3#20241210C214yesmaximal, complemented
4C47×#1817C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13094003—21no—
7C2×C4⋊(C47⋊C8)—21no—
8C2×C22⋊(C47⋊C8)—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—42no—
24#13094003—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C412yes—
29C2×C4⋊Dic47C2×C412yes—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C4⋊(C47⋊C8)—42no—
33C22⋊(C47⋊C8)—42no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C23×C4×C47C2×C411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C22×C8×C47—22no—
40C2×C42×C47—42no—
41C2×C42×C47C2×C411yescharacteristic
42C2×C42×C47D411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47C2311yescharacteristic
45C22⋊C8×C47—42no—
46C24×C47C22⋊C411yescharacteristic
47C22×C4×C47—22no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—41no—
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×C47D812yes—
59C22×C4×C47SD1612yes—
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—22no—
64C22×Dic47—22no—
65C2×Dic94—42no—
66C2×Dic94—42no—
67C2×Dic94—42no—
68C2×C8×C47—42no—
69C2×C8×C47—42no—
70C2×C8×C47—42no—
71C42×C47—22no—
72C42×C47—22no—
73C42×C47—41no—
74C42×C47—41no—
75C22⋊Dic47—44no—
76C4⋊Dic47—24no—
77C4⋊Dic47—42no—
78C4⋊Dic47—42no—
79C4⋊Dic47—42no—
80C4⋊Dic47—42no—
81Dic47⋊C4—42no—
82Dic47⋊C4—42no—
83C2×C47⋊C8—42no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—42no—
86C23×C47—21no—
87C23×C47—21no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47#10212yes—
93C23×C47C2×C22⋊C411yescharacteristic
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—21no—
105C2×C4×C47—21no—
106C2×C4×C47—42no—
107C2×C4×C47—42no—
108C2×C4×C47—44no—
109C2×C4×C47—44no—
110C2×C4×C47#9912yes—
111C2×C4×C47#10112yes—
112C2×C4×C47#10212yes—
113C2×C4×C47#13112yes—
114C2×C4×C47C2×D811yescharacteristic
115C2×C4×C47C2×SD1611yescharacteristic
116C2×C4×C47#13712yes—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—42no—
123Dic94—84no—
124C8×C47—82no—
125C47⋊C8—82no—
126#7728—471nomaximal, Sylow
127C22×C47—22no—
128C22×C47—22no—
129C22×C47—22no—
130C22×C47—41no—
131C22×C47—41no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47#40811yescharacteristic
135C22×C47C2×#9911yescharacteristic
136C22×C47C2×#10111yescharacteristic
137C22×C47C2×#10211yescharacteristic
138C22×C47#41911yescharacteristic
139C22×C47#42111yescharacteristic
140C22×C47#42111yescharacteristic
141C4×C47—22no—
142C4×C47—22no—
143C4×C47—22no—
144C4×C47—22no—
145C4×C47—21no—
146C4×C47—22no—
147C4×C47—21no—
148Dic47—82no—
149Dic47—82no—
150#2857—471no—
151#1817D4711yescharacteristic
152#1797—471no—
153#1271—474no—
154C2×C47—42no—
155C2×C47#147211yescharacteristic
156C2×C47#148011yescharacteristic
157C2×C47#149711yescharacteristic
158C2×C47#151111yescharacteristic
159C2×C47#151911yescharacteristic
160C2×C47#153711yescharacteristic
161C2×C47#154611yescharacteristic
162C2×C4⋊Q8—471no—
163C2×C22⋊Q8—941no—
164C22×C42C2×D4711yescharacteristic
165C2×C4⋊C8—941no—
166C2×C22⋊C8—21no—
167C2×C22⋊C8—941no—
168#391—941no—
169#389—472no—
170#389—941no—
171#384—471no—
172#380—472no—
173#380—941no—
174#336—21no—
175#336—472no—
176#336C2×D4712yes—
177C47#772811yescharacteristic, complemented, Sylow
178C22×Q8—941no—
179C23×C4C4×D4711yescharacteristic
180C23×C4D18811yescharacteristic
181C23×C4C22⋊D4711yescharacteristic
182C22×C8—22no—
183C22×C8—942no—
184C4⋊Q8—1882no—
185C22⋊Q8—1882no—
186C22⋊Q8—1882no—
187C2×C4⋊C4—472no—
188C2×C4⋊C4—472no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—941no—
192C2×C4⋊C4—942no—
193C2×C22⋊C4—942no—
194C2×C42—42no—
195C2×C42C4×D4711yescharacteristic
196C2×C42D18811yescharacteristic
197C2×C42C22⋊D4711yescharacteristic
198C2×C42C22×D4711yescharacteristic
199C4⋊C8—1882no—
200C22⋊C8—42no—
201C22⋊C8—1882no—
202#95—942no—
203#95—942no—
204C24(C2×D47)⋊C411yescharacteristic
205C2×Q8—1882no—
206C2×Q8—1882no—
207C2×Q8—1882no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—942no—
217C22×C4—942no—
218C22×C4#1062532512yes—
219C22×C4D37612yes—
220C22×C4(C2×D47)⋊C411yescharacteristic
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4C2×C4×D4711yescharacteristic
224C22×C4C2×D18811yescharacteristic
225C22×C4C2×C22⋊D4711yescharacteristic
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—1882no—
230C2×C8—1882no—
231C2×C8—1882no—
232C4⋊C4—944no—
233C4⋊C4—1882no—
234C4⋊C4—1882no—
235C4⋊C4—1882no—
236C4⋊C4—1882no—
237C4⋊C4—1882no—
238C4⋊C4—1882no—
239C22⋊C4—1884no—
240C42—22no—
241C42—22no—
242C42—41no—
243C42—41no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23#1309394612yes—
251C23C2×(C2×D47)⋊C411yescharacteristic
252Q8—3764no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
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—44no—
268C2×C4—44no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4—1882no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4#1309393812yes—
276C2×C4#1309394212yes—
277C2×C4#1309394612yes—
278C2×C4#1309403212yes—
279C2×C4C2×D37611yescharacteristic
280C2×C4C2×#1062532511yescharacteristic
281C2×C4#1309405412yes—
282C8—82no—
283C8—3762no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22#1568909111yescharacteristic
292C22C2×#1309393811yescharacteristic
293C22C2×#1309394211yescharacteristic
294C22C2×#1309394611yescharacteristic
295C22#1568913111yescharacteristic
296C22#1568913911yescharacteristic
297C22#1568914111yescharacteristic
298C4—22no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—22no—
304C4—22no—
305C4—3762no—
306C4—3762no—
307C2—42no—
308C2#2024175311yescharacteristic
309C2#2024178811yescharacteristic
310C2#2024186011yescharacteristic
311C2#2024192011yescharacteristic
312C2#2024195711yescharacteristic
313C2#2024202511yescharacteristic
314C2#2024205911yescharacteristic
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#7728
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
177. C47#7728semidirect

Maximal subgroups

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

subgroupindex
#202517402
#202433682
#202412102
C47×#18172
#772847