Finite Group Databasebeta

← #26126468

Subgroups — #26126468

Order 12032.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 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#20249154C211yesmaximal, characteristic
2#20243121C211yesmaximal, characteristic
3#20241389C214yesmaximal
4C47×#1458C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094003—21no—
7C2×#13093983—21no—
8#15689053—21no—
9#15689053C412yescomplemented
10#15689048—21no—
11#15689048C2212yes—
12#15689029—21no—
13#15688989C411yescharacteristic, complemented
14#15688890—21no—
15#15688890C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#431—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23#13094003—42no—
24#13094003—42no—
25#13093983—24no—
26#13093983—42no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—41no—
30C2×C4⋊Dic47C2×C412yes—
31C2×Dic47⋊C4—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C4×Dic47—41no—
35C47⋊#95—21no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95C2×C412yes—
39C23×C4×C47D411yescharacteristic
40#117×C47—41no—
41#117×C47—41no—
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2311yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C4⋊C8×C47—41no—
49C22⋊C8×C47—22no—
50C4×C8×C47—41no—
51C24×C47C2×D411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47D812yes—
60C22×C4×C47SD1612yes—
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—22no—
64C22×Dic47—22no—
65C22×Dic47—42no—
66C2×Dic94—41no—
67C2×Dic94—41no—
68C2×Dic94—42no—
69C2×Dic94—42no—
70C2×C8×C47—42no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47—41no—
73C22⋊C4×C47—41no—
74C22⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—21no—
79C22⋊Dic47—24no—
80C22⋊Dic47—42no—
81C22⋊Dic47—42no—
82C22⋊Dic47—42no—
83C4⋊Dic47—42no—
84C4⋊Dic47—44no—
85C4⋊Dic47—42no—
86C4⋊Dic47—42no—
87Dic47⋊C4—44no—
88C4×Dic47—44no—
89C23×C47—21no—
90C23×C47—21no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47C2×D811yescharacteristic
95C23×C47C2×SD1611yescharacteristic
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47—41no—
109C2×C4×C47—41no—
110C2×C4×C47#9911yescharacteristic
111C2×C4×C47#9911yescharacteristic
112C2×C4×C47#10212yes—
113C2×C4×C47#10212yes—
114C2×C4×C47#10412yes—
115C2×Dic47—44no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—42no—
121C2×Dic47—42no—
122C2×Dic47—82no—
123C2×Dic47—82no—
124Dic94—82no—
125Dic94—82no—
126C8×C47—82no—
127#11694—471nomaximal, Sylow
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47—21no—
132C22×C47—41no—
133C22×C47—41no—
134C22×C47—41no—
135C22×C47—41no—
136C22×C47—81no—
137C22×C47C2×#9911yescharacteristic
138C22×C47C2×#10211yescharacteristic
139C22×C47C2×#10411yescharacteristic
140C4×C47—22no—
141C4×C47—41no—
142C4×C47—41no—
143C4×C47—41no—
144C4×C47—41no—
145C4×C47#32712yes—
146C4×C47#32712yes—
147Dic47—82no—
148Dic47—84no—
149Dic47—82no—
150#2621—471no—
151#1740—471no—
152#1458D4711yescharacteristic
153#1344—474no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47—41no—
157C2×C47—41no—
158C2×C47C2×#32711yescharacteristic
159C2×C47#212311yescharacteristic
160C2×C47#212911yescharacteristic
161C2×C4⋊Q8—471no—
162C2×C22⋊Q8—941no—
163C2×#117—941no—
164C2×#117C2×D4711yescharacteristic
165#431—21no—
166#398—472no—
167#398—941no—
168#397—472no—
169#397—941no—
170#391—941no—
171#382—471no—
172#342—472no—
173#342—941no—
174#324—21no—
175#323C2×D4712yes—
176C47#1169411yescharacteristic, complemented, Sylow
177C22×Q8—941no—
178C23×C4D18811yescharacteristic
179C4⋊Q8—1881no—
180C4⋊Q8—1881no—
181C22⋊Q8—1882no—
182C22⋊Q8—1882no—
183#117—41no—
184#117—41no—
185#117—944no—
186#117—1882no—
187C2×C4⋊C4—472no—
188C2×C4⋊C4—942no—
189C2×C4⋊C4—941no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4—1881no—
192C2×C4⋊C4C4×D4711yescharacteristic
193C2×C4⋊C4C22⋊D4711yescharacteristic
194C2×C22⋊C4—942no—
195C2×C22⋊C4—942no—
196C2×C22⋊C4D18811yescharacteristic
197C2×C22⋊C4C22×D4711yescharacteristic
198C2×C42—1881no—
199C2×C42C4×D4711yescharacteristic
200C2×C42C22⋊D4711yescharacteristic
201C4⋊C8—41no—
202C22⋊C8—22no—
203C4×C8—41no—
204#95—472no—
205#95—941no—
206#95—942no—
207#95—942no—
208C24C2×D18811yescharacteristic
209C2×Q8—1881no—
210C2×Q8—1881no—
211C2×Q8—1882no—
212C2×Q8—1882no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—942no—
217C22×C4—942no—
218C22×C4—1882no—
219C22×C4#1062532512yes—
220C22×C4D37612yes—
221C22×C4(C2×D47)⋊C411yescharacteristic
222C22×C4(C2×D47)⋊C411yescharacteristic
223C22×C4(C2×D47)⋊C411yescharacteristic
224C22×C4(C2×D47)⋊C411yescharacteristic
225C22×C4C2×C4×D4711yescharacteristic
226C22×C4C2×C22⋊D4711yescharacteristic
227C2×C8—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—1882no—
231C4⋊C4—1882no—
232C4⋊C4—1884no—
233C4⋊C4—1884no—
234C4⋊C4—1882no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—944no—
238C22⋊C4—1882no—
239C22⋊C4—1882no—
240C22⋊C4—1882no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—1884no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×(C2×D47)⋊C411yescharacteristic
251C23C2×D37611yescharacteristic
252C23C2×#1062532511yescharacteristic
253Q8—3762no—
254Q8—3762no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—3762no—
270C2×C4—3762no—
271C2×C4—1884no—
272C2×C4—1882no—
273C2×C4—1882no—
274C2×C4—1882no—
275C2×C4—1882no—
276C2×C4—1882no—
277C2×C4—1882no—
278C2×C4#1309393811yescharacteristic
279C2×C4#1309393811yescharacteristic
280C2×C4#1309394612yes—
281C2×C4#1309394612yes—
282C2×C4#1309395012yes—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#1309393811yescharacteristic
294C22C2×#1309394611yescharacteristic
295C22C2×#1309395011yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—3762no—
302C4—3764no—
303C4—3762no—
304C4#1568885812yes—
305C4#1568885812yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#1568885811yescharacteristic
311C2#2024504311yescharacteristic
312C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1458C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1458D47
hypocenterC47#11694
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
5. C2×#13094024C4semidirect
9. #15689053C4semidirect
13. #15688989C4semidirect
176. C47#11694semidirect

Maximal subgroups

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

subgroupindex
#202491542
#202431212
#202413892
C47×#14582
#1169447