Finite Group Databasebeta

← #26109611

Subgroups — #26109611

Order 12032.

786 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#15688986C211yesmaximal, characteristic
2#20243298C211yesmaximal, characteristic
3#20241230C214yesmaximal, complemented
4C47×#1779C211yesmaximal, characteristic
5C22×C22⋊Dic47C412yes—
6C2×C22⋊(C47⋊C8)—22no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688986—22no—
9#15688986—22no—
10C2×C47⋊#95C411yescharacteristic
11C2×C47⋊#95C411yescharacteristic
12C2×C47⋊#95C2212yes—
13C24×C4×C47C2211yescharacteristic
14C2×C22⋊C8×C47—22no—
15C2×C22⋊C8×C47C2212yes—
16C23×Dic47C2×C412yes—
17C23×Dic47C2×C412yes—
18C22×C47⋊C8—22no—
19C2×C22⋊Dic47—22no—
20C2×C22⋊Dic47—22no—
21C2×C22⋊Dic47—44no—
22C22⋊(C47⋊C8)—24no—
23C22⋊(C47⋊C8)—44no—
24C47⋊#95—22no—
25C47⋊#95—22no—
26C47⋊#95—22no—
27C47⋊#95—22no—
28C25×C47C2×C411yescharacteristic
29C23×C4×C47—24no—
30C23×C4×C47—44no—
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47D412yes—
33C23×C4×C47D412yes—
34C23×C4×C47C2311yescharacteristic
35C22×C8×C47—22no—
36C22⋊C8×C47—24no—
37C22⋊C8×C47—44no—
38C24×C47—24no—
39C24×C47—22no—
40C24×C47—22no—
41C24×C47—22no—
42C24×C47—42no—
43C24×C47C22⋊C412yes—
44C24×C47C22×C411yescharacteristic
45C22×C4×C47—24no—
46C22×C4×C47—24no—
47C22×C4×C47—22no—
48C22×C4×C47—24no—
49C22×C4×C47—24no—
50C22×C4×C47—42no—
51C22×C4×C47—42no—
52C22×C4×C47—42no—
53C22×C4×C47—42no—
54C22×C4×C47—416no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47#4114yes—
59C22×Dic47—22no—
60C22×Dic47—22no—
61C22×Dic47—24no—
62C22×Dic47—24no—
63C22×Dic47—42no—
64C22×Dic47—42no—
65C22×Dic47—42no—
66C22×Dic47—42no—
67C2×C8×C47—44no—
68C2×C8×C47—42no—
69C22⋊Dic47—44no—
70C22⋊Dic47—44no—
71C2×C47⋊C8—44no—
72C2×C47⋊C8—42no—
73C23×C47—22no—
74C23×C47—22no—
75C23×C47—28no—
76C23×C47—22no—
77C23×C47—22no—
78C23×C47—22no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—48no—
83C23×C47—48no—
84C23×C47—44no—
85C23×C47—44no—
86C23×C47#9914yes—
87C23×C47C2×C22⋊C411yescharacteristic
88C23×C47#11712yes—
89C23×C47#13112yes—
90C23×C47#13112yes—
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—22no—
96C2×C4×C47—44no—
97C2×C4×C47—48no—
98C2×C4×C47—48no—
99C2×C4×C47—44no—
100C2×C4×C47#12312yes—
101C2×C4×C47#12312yes—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—42no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107C2×Dic47—42no—
108C2×Dic47—84no—
109C2×Dic47—84no—
110C8×C47—82no—
111C47⋊C8—82no—
112#8702—471nomaximal, Sylow
113C22×C47—22no—
114C22×C47—22no—
115C22×C47—24no—
116C22×C47—24no—
117C22×C47—24no—
118C22×C47—24no—
119C22×C47—24no—
120C22×C47—416no—
121C22×C47—42no—
122C22×C47—42no—
123C22×C47—42no—
124C22×C47—42no—
125C22×C47#38111yescharacteristic
126C22×C47#40811yescharacteristic
127C22×C47#40811yescharacteristic
128C22×C47C2×#9912yes—
129C22×C47#41312yes—
130C22×C47#43314yes—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—42no—
134Dic47—82no—
135Dic47—82no—
136C2×#381—471no—
137#1779—471no—
138#1779D4711yescharacteristic
139#1280—474no—
140C2×C47—24no—
141C2×C47—44no—
142C2×C47#146812yes—
143C2×C47#146812yes—
144C2×C47#181811yescharacteristic
145C2×C47#181911yescharacteristic
146C2×C47#182211yescharacteristic
147C24×C4C2×D4711yescharacteristic
148C22×C22⋊C4—472no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8—472no—
151C2×C22⋊C8—942no—
152C2×C22⋊C8C2×D4712yes—
153#381—942no—
154#381—942no—
155C2×#95—471no—
156C2×#95—471no—
157C2×#95—472no—
158C47#870211yescharacteristic, complemented, Sylow
159C25C4×D4711yescharacteristic
160C23×C4—24no—
161C23×C4—44no—
162C23×C4—472no—
163C23×C4—472no—
164C23×C4C4×D4711yescharacteristic
165C23×C4D18812yes—
166C23×C4C22⋊D4712yes—
167C23×C4C22×D4711yescharacteristic
168C22×C8—22no—
169C22×C8—942no—
170C2×C22⋊C4—942no—
171C2×C22⋊C4—942no—
172C2×C22⋊C4—1884no—
173C22⋊C8—24no—
174C22⋊C8—44no—
175C22⋊C8—944no—
176C22⋊C8—1884no—
177#95—942no—
178#95—942no—
179#95—942no—
180#95—942no—
181C24—24no—
182C24—22no—
183C24—22no—
184C24—22no—
185C24—42no—
186C24(C2×D47)⋊C412yes—
187C24C2×C4×D4711yescharacteristic
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—416no—
198C22×C4—944no—
199C22×C4—944no—
200C22×C4—942no—
201C22×C4—942no—
202C22×C4—1882no—
203C22×C4—1882no—
204C22×C4—1882no—
205C22×C4—1882no—
206C22×C4(C2×D47)⋊C412yes—
207C22×C4C2×D18811yescharacteristic
208C22×C4#1062535214yes—
209C22×C4C2×C22⋊D4711yescharacteristic
210C2×C8—44no—
211C2×C8—42no—
212C2×C8—1884no—
213C2×C8—1882no—
214C22⋊C4—1884no—
215C22⋊C4—1884no—
216C23—22no—
217C23—22no—
218C23—28no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—48no—
226C23—48no—
227C23—44no—
228C23—44no—
229C23#1309393814yes—
230C23C2×(C2×D47)⋊C411yescharacteristic
231C23#1309398012yes—
232C23#1309403012yes—
233C23#1309403212yes—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—48no—
241C2×C4—48no—
242C2×C4—44no—
243C2×C4—3764no—
244C2×C4—3764no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4#1309400512yes—
252C2×C4#1309400812yes—
253C8—82no—
254C8—3762no—
255C22—22no—
256C22—22no—
257C22—24no—
258C22—24no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—416no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22#1568898411yescharacteristic
268C22#1568908911yescharacteristic
269C22#1568909111yescharacteristic
270C22C2×#1309393812yes—
271C22#1568910512yes—
272C22#1568918114yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—3762no—
277C4—3762no—
278C2—24no—
279C2—44no—
280C2#2024173412yes—
281C2#2024173712yes—
282C2#2024345711yescharacteristic
283C2#2024346011yescharacteristic
284C2#2024346911yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1779C2
socleC23×C47C2×C22⋊C4
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#1779D47
hypocenterC47#8702
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #20241230C2semidirect
158. C47#8702semidirect

Maximal subgroups

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

subgroupindex
C2×#156889862
#202432982
#202412302
C47×#17792
#870247