Finite Group Databasebeta

← #26126957

Subgroups — #26126957

Order 12032.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#20243465C211yesmaximal, characteristic
3#20241389C214yesmaximal
4C47×#1464C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6C2×#13094003—21no—
7#15689053—21no—
8#15689053C412yescomplemented
9#15689048—21no—
10#15689048C2212yes—
11#15689029—21no—
12#15688989C411yescharacteristic, complemented
13#15688986—21no—
14#15688890—21no—
15#15688890C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#432—21no—
18C47×#324—21no—
19C47×#323C2212yes—
20C22×Dic94—21no—
21#13094024—41no—
22#13094024—41no—
23#13094003—44no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C412yes—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C47⋊#95—21no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C47⋊#95—41no—
34C47⋊#95—41no—
35C47⋊#95C2×C412yes—
36C23×C4×C47D411yescharacteristic
37#117×C47—41no—
38#117×C47—41no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47C2311yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C4⋊C8×C47—41no—
46C22⋊C8×C47—22no—
47#97×C47—41no—
48C24×C47C2×D411yescharacteristic
49C22×C4×C47—22no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—41no—
53C22×C4×C47C22⋊C411yescharacteristic
54C22×C4×C47C22⋊C411yescharacteristic
55C22×C4×C47C22⋊C411yescharacteristic
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22×C411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—22no—
60C22×Dic47—22no—
61C22×Dic47—42no—
62C2×Dic94—42no—
63C2×Dic94—42no—
64C2×Dic94—41no—
65C2×Dic94—41no—
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—44no—
76C22⋊Dic47—42no—
77C22⋊Dic47—42no—
78C4⋊Dic47—42no—
79C4⋊Dic47—42no—
80C4⋊Dic47—42no—
81Dic47⋊C4—44no—
82C23×C47—21no—
83C23×C47—21no—
84C23×C47—41no—
85C23×C47—41no—
86C23×C47C2×C22⋊C411yescharacteristic
87C23×C47AGL(1,ℤ/8)12yes—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
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—42no—
98C2×C4×C47—42no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47#9911yescharacteristic
104C2×C4×C47#9911yescharacteristic
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—82no—
112C2×Dic47—82no—
113C2×Dic47—82no—
114Dic94—82no—
115Dic94—82no—
116C8×C47—82no—
117#11801—471nomaximal, Sylow
118C22×C47—21no—
119C22×C47—21no—
120C22×C47—22no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47—41no—
124C22×C47—81no—
125C22×C47C2×#9911yescharacteristic
126C22×C47#41811yescharacteristic
127C22×C47#42111yescharacteristic
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—41no—
131C4×C47—41no—
132C4×C47—41no—
133C4×C47—41no—
134C4×C47—41no—
135Dic47—82no—
136Dic47—82no—
137Dic47—162no—
138#2621—471no—
139#1820—471no—
140#1464D4711yescharacteristic
141#1344—474no—
142C2×C47—21no—
143C2×C47—21no—
144C2×C47—42no—
145C2×C47#151011yescharacteristic
146C2×C47#212311yescharacteristic
147C2×C47#212911yescharacteristic
148C2×C4⋊Q8—471no—
149C2×C22⋊Q8—941no—
150C2×#117C2×D4711yescharacteristic
151#432—21no—
152#398—472no—
153#398—941no—
154#397—472no—
155#397—941no—
156#391—941no—
157#382—471no—
158#381—941no—
159#342—472no—
160#342—941no—
161#324—21no—
162#323C2×D4712yes—
163C47#1180111yescharacteristic, complemented, Sylow
164C22×Q8—941no—
165C23×C4D18811yescharacteristic
166C4⋊Q8—1881no—
167C4⋊Q8—1881no—
168C22⋊Q8—1884no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—472no—
172C2×C4⋊C4—942no—
173C2×C4⋊C4—941no—
174C2×C4⋊C4—941no—
175C2×C4⋊C4C4×D4711yescharacteristic
176C2×C4⋊C4C22⋊D4711yescharacteristic
177C2×C22⋊C4—942no—
178C2×C22⋊C4—942no—
179C2×C22⋊C4D18811yescharacteristic
180C2×C22⋊C4C22×D4711yescharacteristic
181C2×C42C4×D4711yescharacteristic
182C2×C42C22⋊D4711yescharacteristic
183C4⋊C8—41no—
184C22⋊C8—22no—
185#97—41no—
186#95—472no—
187#95—941no—
188#95—942no—
189#95—942no—
190#95—1881no—
191#95—1881no—
192C24C2×D18811yescharacteristic
193C2×Q8—1882no—
194C2×Q8—1882no—
195C2×Q8—1881no—
196C2×Q8—1881no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—942no—
202C22×C4—942no—
203C22×C4—1882no—
204C22×C4(C2×D47)⋊C411yescharacteristic
205C22×C4(C2×D47)⋊C411yescharacteristic
206C22×C4(C2×D47)⋊C411yescharacteristic
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4C2×C4×D4711yescharacteristic
209C22×C4C2×C22⋊D4711yescharacteristic
210C2×C8—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—1882no—
214C4⋊C4—1884no—
215C4⋊C4—1882no—
216C4⋊C4—1882no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—1882no—
220C22⋊C4—1882no—
221C22⋊C4—1884no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×(C2×D47)⋊C411yescharacteristic
231C23#1309405212yes—
232Q8—3762no—
233Q8—3762no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—3762no—
250C2×C4—3762no—
251C2×C4—3762no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4#1309393811yescharacteristic
259C2×C4#1309393811yescharacteristic
260C8—82no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1309393811yescharacteristic
269C22#1568912911yescharacteristic
270C22#1568913911yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—3762no—
279C4—3762no—
280C4—7522no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2024191811yescharacteristic
285C2#2024504311yescharacteristic
286C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1464C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1464D47
hypocenterC47#11801
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
5. C2×#13094024C4semidirect
8. #15689053C4semidirect
12. #15688989C4semidirect
163. C47#11801semidirect

Maximal subgroups

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

subgroupindex
#202491542
#202434652
#202413892
C47×#14642
#1180147