Finite Group Databasebeta

← #26127003

Subgroups — #26127003

Order 12032.

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

normal subgroups only

This group has 382 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#20249209C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241408C214yesmaximal
4C47×#1820C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic, complemented
6#15689180—21no—
7#15689062—21no—
8#15689059—21no—
9#15689043C412yescomplemented
10#15689035C2212yes—
11#15689021—21no—
12#15688989C411yescharacteristic, complemented
13#15688983—21no—
14#15688849—21no—
15#15688847C2212yes—
16C2×#117×C47C2211yescharacteristic
17C47×#381—21no—
18C47×#342—21no—
19C47×#342C2212yes—
20C22×Dic94C2×C412yes—
21#13094024—41no—
22#13094024—41no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×Dic47⋊C4—22no—
27C2×C22⋊Dic47—22no—
28C4⋊(C47⋊C8)—41no—
29C22⋊(C47⋊C8)—22no—
30#13093935—41no—
31C47⋊#95—22no—
32C47⋊#95—21no—
33C47⋊#95—22no—
34C47⋊#95C2×C412yes—
35C23×C4×C47D411yescharacteristic
36#117×C47—41no—
37#117×C47—41no—
38C2×C4⋊C4×C47C2×C411yescharacteristic
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47C2311yescharacteristic
43C2×C42×C47C2×C411yescharacteristic
44C2×C42×C47D411yescharacteristic
45#95×C47—41no—
46#95×C47—41no—
47C24×C47C2×D411yescharacteristic
48C22×C4×C47—22no—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—41no—
52C22×C4×C47—42no—
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—
61C2×Dic94—42no—
62C2×Dic94—82no—
63C4⋊C4×C47—41no—
64C4⋊C4×C47—41no—
65C22⋊C4×C47—41no—
66C22⋊C4×C47—41no—
67C22⋊C4×C47—42no—
68C22⋊C4×C47—42no—
69C42×C47—21no—
70C42×C47—21no—
71C42×C47—21no—
72C42×C47—21no—
73C22⋊Dic47—82no—
74C4⋊Dic47—42no—
75C4⋊Dic47—42no—
76C4⋊Dic47—41no—
77C4⋊Dic47—41no—
78Dic47⋊C4—82no—
79C2×C47⋊C8—42no—
80C23×C47—21no—
81C23×C47—21no—
82C23×C47—41no—
83C23×C47—41no—
84C23×C47C2×C22⋊C411yescharacteristic
85C23×C47#13712yes—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
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—42no—
96C2×C4×C47—42no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47—41no—
101C2×C4×C47—82no—
102C2×C4×C47—82no—
103C2×C4×C47—82no—
104C2×C4×C47#9911yescharacteristic
105C2×C4×C47#9911yescharacteristic
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112Dic94—84no—
113C47⋊C8—82no—
114#11812—471nomaximal, Sylow
115C22×C47—21no—
116C22×C47—21no—
117C22×C47—22no—
118C22×C47—42no—
119C22×C47—41no—
120C22×C47—41no—
121C22×C47—81no—
122C22×C47C2×#9911yescharacteristic
123C22×C47#41911yescharacteristic
124C22×C47#42111yescharacteristic
125C4×C47—21no—
126C4×C47—21no—
127C4×C47—41no—
128C4×C47—41no—
129C4×C47—41no—
130C4×C47—41no—
131C4×C47—41no—
132C4×C47—162no—
133Dic47—82no—
134Dic47—82no—
135#2628—471no—
136#1820D4711yescharacteristic
137#1464—471no—
138#1351—474no—
139C2×C47—21no—
140C2×C47—21no—
141C2×C47—42no—
142C2×C47#151111yescharacteristic
143C2×C47#212211yescharacteristic
144C2×C47#212911yescharacteristic
145C2×C4⋊Q8—471no—
146C2×#117C2×D4711yescharacteristic
147#432—941no—
148#400—941no—
149#399—941no—
150#395—472no—
151#393—472no—
152#389—941no—
153#382—471no—
154#381—21no—
155#380—941no—
156#342—21no—
157#342C2×D4712yes—
158#324—941no—
159#323—472no—
160C47#1181211yescharacteristic, complemented, Sylow
161C22×Q8—472no—
162C23×C4C22⋊D4711yescharacteristic
163C4⋊Q8—1881no—
164C4⋊Q8—1881no—
165#117—41no—
166#117—41no—
167C2×C4⋊C4—942no—
168C2×C4⋊C4—941no—
169C2×C4⋊C4—941no—
170C2×C4⋊C4—941no—
171C2×C4⋊C4C4×D4711yescharacteristic
172C2×C4⋊C4D18811yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4—942no—
175C2×C22⋊C4C22⋊D4711yescharacteristic
176C2×C22⋊C4C22×D4711yescharacteristic
177C2×C42C4×D4711yescharacteristic
178C2×C42D18811yescharacteristic
179C4⋊C8—1881no—
180C22⋊C8—942no—
181#97—1881no—
182#95—41no—
183#95—41no—
184#95—472no—
185#95—942no—
186#95—941no—
187#95—942no—
188C24C2×C22⋊D4711yescharacteristic
189C2×Q8—3762no—
190C2×Q8—1882no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—942no—
197C22×C4—942no—
198C22×C4(C2×D47)⋊C411yescharacteristic
199C22×C4(C2×D47)⋊C411yescharacteristic
200C22×C4(C2×D47)⋊C411yescharacteristic
201C22×C4(C2×D47)⋊C411yescharacteristic
202C22×C4C2×C4×D4711yescharacteristic
203C22×C4C2×D18811yescharacteristic
204C2×C8—1882no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—3762no—
208C4⋊C4—1882no—
209C4⋊C4—1882no—
210C4⋊C4—1881no—
211C4⋊C4—1881no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—3762no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C23—21no—
222C23—21no—
223C23—41no—
224C23—41no—
225C23C2×(C2×D47)⋊C411yescharacteristic
226C23#1309405412yes—
227Q8—3764no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4—1882no—
252C2×C4#1309393811yescharacteristic
253C2×C4#1309393811yescharacteristic
254C8—3762no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×#1309393811yescharacteristic
263C22#1568913111yescharacteristic
264C22#1568914111yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—162no—
273C4—3762no—
274C4—3762no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2#2024192011yescharacteristic
279C2#2024503711yescharacteristic
280C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1820C2
socleC22×C47C2×#99
prime radicalC24×C47C2×D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1820D47
hypocenterC47#11812
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
5. C2×#13094024C4semidirect
9. #15689043C4semidirect
12. #15688989C4semidirect
160. C47#11812semidirect

Maximal subgroups

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

subgroupindex
#202492092
#202417222
#202414082
C47×#18202
#1181247