Finite Group Databasebeta

← #26126568

Subgroups — #26126568

Order 12032.

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

normal subgroups only

This group has 448 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#20249209C211yesmaximal, characteristic
2#20243121C211yesmaximal, characteristic
3#20241410C214yesmaximal
4C47×#1458C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13093983—21no—
7#15689062—21no—
8#15689059—21no—
9#15689043C412yescomplemented
10#15689035C2212yes—
11#15689021—21no—
12#15688989C411yescharacteristic, complemented
13#15688983—21no—
14#15688890—21no—
15#15688890C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#431—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×Dic94C2×C412yes—
21#13094024—41no—
22#13094024—41no—
23#13093983—24no—
24#13093983—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—41no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C4×Dic47—41no—
33C47⋊#95—22no—
34C47⋊#95—21no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C23×C4×C47D411yescharacteristic
38#117×C47—41no—
39#117×C47—41no—
40C2×C4⋊C4×C47C2×C411yescharacteristic
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47C2311yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C4⋊C8×C47—41no—
47C22⋊C8×C47—22no—
48C4×C8×C47—41no—
49C24×C47C2×D411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47SD1612yes—
58C22×C4×C47Q1612yes—
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D411yescharacteristic
61C22×Dic47—22no—
62C22×Dic47—22no—
63C22×Dic47—42no—
64C2×Dic94—42no—
65C2×Dic94—82no—
66C2×C8×C47—42no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—41no—
69C22⋊C4×C47—41no—
70C22⋊C4×C47—41no—
71C42×C47—21no—
72C42×C47—21no—
73C42×C47—21no—
74C42×C47—21no—
75C22⋊Dic47—24no—
76C22⋊Dic47—42no—
77C22⋊Dic47—82no—
78C4⋊Dic47—42no—
79C4⋊Dic47—44no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4⋊Dic47—42no—
83Dic47⋊C4—82no—
84C4×Dic47—44no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—41no—
88C23×C47—41no—
89C23×C47C2×C22⋊C411yescharacteristic
90C23×C47C2×SD1611yescharacteristic
91C23×C47C2×Q1611yescharacteristic
92C2×C4×C47—21no—
93C2×C4×C47—21no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47—41no—
105C2×C4×C47—41no—
106C2×C4×C47#9911yescharacteristic
107C2×C4×C47#9911yescharacteristic
108C2×C4×C47#10312yes—
109C2×C4×C47#10312yes—
110C2×C4×C47#10412yes—
111C2×Dic47—44no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—42no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—82no—
119C2×Dic47—82no—
120Dic94—84no—
121C8×C47—82no—
122#11717—471nomaximal, Sylow
123C22×C47—21no—
124C22×C47—21no—
125C22×C47—21no—
126C22×C47—21no—
127C22×C47—41no—
128C22×C47—41no—
129C22×C47—41no—
130C22×C47—41no—
131C22×C47—81no—
132C22×C47C2×#9911yescharacteristic
133C22×C47C2×#10311yescharacteristic
134C22×C47C2×#10411yescharacteristic
135C4×C47—22no—
136C4×C47—41no—
137C4×C47—41no—
138C4×C47—41no—
139C4×C47—41no—
140C4×C47#32812yes—
141C4×C47#32812yes—
142Dic47—82no—
143Dic47—84no—
144Dic47—82no—
145#2628—471no—
146#1740—471no—
147#1458D4711yescharacteristic
148#1351—474no—
149C2×C47—21no—
150C2×C47—21no—
151C2×C47—41no—
152C2×C47—41no—
153C2×C47C2×#32811yescharacteristic
154C2×C47#212211yescharacteristic
155C2×C47#212911yescharacteristic
156C2×C4⋊Q8—471no—
157C2×#117—941no—
158C2×#117C2×D4711yescharacteristic
159#431—21no—
160#400—941no—
161#399—941no—
162#395—472no—
163#393—472no—
164#389—941no—
165#382—471no—
166#380—941no—
167#342—472no—
168#342—941no—
169#324—21no—
170#323C2×D4712yes—
171C47#1171711yescharacteristic, complemented, Sylow
172C22×Q8—472no—
173C23×C4D18811yescharacteristic
174C4⋊Q8—1881no—
175C4⋊Q8—1881no—
176#117—41no—
177#117—41no—
178#117—944no—
179#117—1882no—
180C2×C4⋊C4—942no—
181C2×C4⋊C4—941no—
182C2×C4⋊C4—941no—
183C2×C4⋊C4—941no—
184C2×C4⋊C4—1881no—
185C2×C4⋊C4C4×D4711yescharacteristic
186C2×C4⋊C4C22⋊D4711yescharacteristic
187C2×C22⋊C4—942no—
188C2×C22⋊C4—942no—
189C2×C22⋊C4D18811yescharacteristic
190C2×C22⋊C4C22×D4711yescharacteristic
191C2×C42—1881no—
192C2×C42C4×D4711yescharacteristic
193C2×C42C22⋊D4711yescharacteristic
194C4⋊C8—41no—
195C22⋊C8—22no—
196C4×C8—41no—
197#95—472no—
198#95—942no—
199#95—941no—
200#95—942no—
201C24C2×D18811yescharacteristic
202C2×Q8—3762no—
203C2×Q8—1882no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—942no—
208C22×C4—942no—
209C22×C4—1882no—
210C22×C4#1062532512yes—
211C22×C4Dic18812yes—
212C22×C4(C2×D47)⋊C411yescharacteristic
213C22×C4(C2×D47)⋊C411yescharacteristic
214C22×C4(C2×D47)⋊C411yescharacteristic
215C22×C4(C2×D47)⋊C411yescharacteristic
216C22×C4C2×C4×D4711yescharacteristic
217C22×C4C2×C22⋊D4711yescharacteristic
218C2×C8—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—3762no—
222C4⋊C4—1882no—
223C4⋊C4—1884no—
224C4⋊C4—1881no—
225C4⋊C4—1881no—
226C4⋊C4—1882no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—944no—
230C22⋊C4—3762no—
231C22⋊C4—1882no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1884no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×(C2×D47)⋊C411yescharacteristic
242C23C2×#1062532511yescharacteristic
243C23C2×Dic18811yescharacteristic
244Q8—3764no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—3762no—
260C2×C4—3762no—
261C2×C4—1884no—
262C2×C4—1882no—
263C2×C4—1882no—
264C2×C4—1882no—
265C2×C4—1882no—
266C2×C4—1882no—
267C2×C4—1882no—
268C2×C4#1309393811yescharacteristic
269C2×C4#1309393811yescharacteristic
270C2×C4#1309394912yes—
271C2×C4#1309394912yes—
272C2×C4#1309395012yes—
273C8—82no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1309393811yescharacteristic
284C22C2×#1309394911yescharacteristic
285C22C2×#1309395011yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—3762no—
292C4—3764no—
293C4—3762no—
294C4#1568886112yes—
295C4#1568886112yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1568886111yescharacteristic
301C2#2024503911yescharacteristic
302C2#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#11717
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. #15689043C4semidirect
12. #15688989C4semidirect
171. C47#11717semidirect

Maximal subgroups

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

subgroupindex
#202492092
#202431212
#202414102
C47×#14582
#1171747