Finite Group Databasebeta

← #26225753

Subgroups — #26225753

Order 12032.

914 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 235 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×#15689754C211yesmaximal, characteristic
2#20252023C212yesmaximal
3#20244379C214yesmaximal, complemented
4#20244375C214yesmaximal, complemented
5#20243469C212yesmaximal
6#20243468C211yesmaximal, characteristic
7C47×#1821C211yesmaximal, characteristic
8#15689754—28no—
9C2×#13094023C412yes—
10C2×#13094023C2212yes—
11C2×#13094015C412yes—
12C2×#13094015C2212yes—
13C2×#13094003C414yes—
14C2×#13093983C2211yescharacteristic
15C2×#13093983C2211yescharacteristic
16#15689181—28no—
17#15689146C2214yes—
18#15689087C2212yes—
19M16×Dic47C2214yes—
20#15688892C2214yes—
21#15688891C2214yes—
22#15688891C2214yes—
23C22×C4⋊C4×C47C2211yescharacteristic
24C47×#407C2212yes—
25C47×#343C2214yes—
26C22×Dic94C2×C414yes—
27C2×#10625329C2314yes—
28#13094023—28no—
29#13094023—24no—
30#13094023—24no—
31#13094015—28no—
32#13094015—24no—
33#13094015—24no—
34#13094003—216no—
35#13093983—24no—
36#13093983D418yes—
37C2×C4⋊Dic47C2×C414yes—
38C2×C4⋊Dic47C2×C411yescharacteristic
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×C4⋊Dic47C2311yescharacteristic
41C2×C4⋊Dic47C2311yescharacteristic
42C2×Dic47⋊C4C2×C414yes—
43C2×Dic47⋊C4C2×C414yes—
44C2×C22⋊Dic47C2×C412yes—
45C2×C22⋊Dic47C2×C412yes—
46C2×C4×Dic47C2×C412yes—
47C2×C4×Dic47C2312yes—
48Dic47⋊C8—28no—
49C22⋊(C47⋊C8)—24no—
50#13093934—24no—
51C8×Dic47—24no—
52C23×C4×C47C2×C412yes—
53C23×C4×C47C2311yescharacteristic
54C2×M16×C47C2314yes—
55C2×C4⋊C4×C47—24no—
56C2×C4⋊C4×C47C2×C412yes—
57C2×C4⋊C4×C47C2312yes—
58C22⋊C8×C47—24no—
59C24×C47C22×C411yescharacteristic
60C22×C4×C47—24no—
61C22×C4×C47—24no—
62C22×C4×C47—82no—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C22×C414yes—
65C22×C4×C47C2×D414yes—
66C22×C4×C47C2411yescharacteristic
67C22×Dic47C22×C412yes—
68C22×Dic47C22×C412yes—
69C22×Dic47C22×C412yes—
70C22×Dic47C22×C412yes—
71C2×Dic94—28no—
72C2×Dic94—84no—
73M16×C47—28no—
74C2×C8×C47—24no—
75C4⋊C4×C47—24no—
76C4⋊C4×C47—22no—
77C4⋊C4×C47—22no—
78C22⋊Dic47—24no—
79C22⋊Dic47C22⋊C418yes—
80C4⋊Dic47—22no—
81C4⋊Dic47—22no—
82C4⋊Dic47—28no—
83C4⋊Dic47C22⋊C414yes—
84C4⋊Dic47C22⋊C414yes—
85Dic47⋊C4—28no—
86Dic47⋊C4—28no—
87C4×Dic47—24no—
88C4×Dic47C2×D414yes—
89C4×Dic47C2×D414yes—
90#10625329—28no—
91C2×C47⋊C8—24no—
92C23×C47—22no—
93C23×C47—41no—
94C23×C47—41no—
95C23×C47C2×C22⋊C412yes—
96C23×C47C23×C411yescharacteristic
97C2×C4×C47—22no—
98C2×C4×C47—28no—
99C2×C4×C47—24no—
100C2×C4×C47—44no—
101C2×C4×C47—84no—
102C2×C4×C47C2×C22⋊C412yes—
103C2×C4×C47C22×D412yes—
104C2×C4×C472−1+414yes—
105C2×Dic47—22no—
106C2×Dic47—22no—
107C2×Dic47—42no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47C2×C22⋊C414yes—
112C2×Dic47C2×C22⋊C414yes—
113Dic94—88no—
114C8×C47—44no—
115C47⋊C8—44no—
116#21832—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—21no—
119C22×C47—22no—
120C22×C47—22no—
121C22×C47—22no—
122C22×C47—44no—
123C22×C47C22×C22⋊C411yescharacteristic
124C22×C47#51912yes—
125C4×C47—22no—
126C4×C47—44no—
127C4×C47#53614yes—
128Dic47—24no—
129Dic47—24no—
130Dic47—42no—
131Dic47—42no—
132C2×#544—471no—
133#2921—472no—
134#1998—474no—
135#1997—474no—
136#1822—472no—
137#1821—471no—
138#1821D4711yescharacteristic
139C2×C47—22no—
140C2×C47—42no—
141C2×C47#231011yescharacteristic
142C2×C47#288811yescharacteristic
143C2×C47#288911yescharacteristic
144#544—948no—
145C2×C4⋊Q8—472no—
146C2×C4⋊Q8—472no—
147C2×#125—472no—
148C2×#125—472no—
149C2×C22⋊Q8—474no—
150C2×#117—471no—
151C2×#117—471no—
152C22×C4⋊C4C2×D4711yescharacteristic
153#433—948no—
154#423—474no—
155#407—472no—
156#407C2×D4712yes—
157C4×M16—474no—
158#343—474no—
159#343—474no—
160#343—474no—
161#343C2×D4714yes—
162C47#2183211yescharacteristic, complemented, Sylow
163C22×Q8—474no—
164C23×C4C4×D4712yes—
165C23×C4C22×D4711yescharacteristic
166C2×M16—474no—
167C2×M16C22×D4714yes—
168C4⋊Q8—944no—
169C4⋊Q8—944no—
170C4⋊Q8—948no—
171#125—944no—
172#125—944no—
173#125—948no—
174C22⋊Q8—9416no—
175#117—478no—
176#117—944no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—471no—
179C2×C4⋊C4—471no—
180C2×C4⋊C4—474no—
181C2×C4⋊C4—474no—
182C2×C4⋊C4—474no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4C4×D4712yes—
186C2×C4⋊C4C22×D4712yes—
187C2×C22⋊C4—472no—
188C2×C22⋊C4—472no—
189C2×C42—472no—
190C2×C42—472no—
191C4⋊C8—948no—
192C22⋊C8—24no—
193C22⋊C8—944no—
194#97—944no—
195C4×C8—944no—
196C24C2×C4×D4711yescharacteristic
197C2×Q8—948no—
198C2×Q8—3764no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—82no—
202C22×C4—472no—
203C22×C4—472no—
204C22×C4—472no—
205C22×C4—472no—
206C22×C4C2×C4×D4711yescharacteristic
207C22×C4C2×C4×D4714yes—
208C22×C4D4×D4714yes—
209C22×C4C23×D4711yescharacteristic
210M16—28no—
211M16—948no—
212C2×C8—24no—
213C2×C8—944no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—474no—
218C4⋊C4—474no—
219C4⋊C4—948no—
220C4⋊C4—948no—
221C4⋊C4—948no—
222C4⋊C4—942no—
223C4⋊C4—942no—
224C22⋊C4—478no—
225C22⋊C4—944no—
226C42—474no—
227C42—474no—
228C42—944no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23C22⋊C4×D4712yes—
233C23C22×C4×D4711yescharacteristic
234Q8—3768no—
235C2×C4—22no—
236C2×C4—28no—
237C2×C4—24no—
238C2×C4—44no—
239C2×C4—84no—
240C2×C4—474no—
241C2×C4—474no—
242C2×C4—942no—
243C2×C4—942no—
244C2×C4—1882no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4C22⋊C4×D4712yes—
249C2×C4C2×D4×D4712yes—
250C2×C4#1309407114yes—
251C8—44no—
252C8—1884no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—44no—
259C22C2×C22⋊C4×D4711yescharacteristic
260C22#1568960012yes—
261C4—22no—
262C4—44no—
263C4—944no—
264C4—944no—
265C4—1882no—
266C4—1882no—
267C4#1568969514yes—
268C2—22no—
269C2—42no—
270C2#2024597311yescharacteristic
271C2#2025186411yescharacteristic
272C2#2025186811yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1821C2
socleC22×C47C22×C22⋊C4
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#1821D47
hypocenterC47#21832
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #20244379C2semidirect
4. #20244375C2semidirect
162. C47#21832semidirect

Maximal subgroups

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

subgroupindex
C2×#156897542
#202520232
#202443792
#202443752
#202434692
#202434682
C47×#18212
#2183247