Finite Group Databasebeta

← #26104404

Subgroups — #26104404

Order 12032.

440 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 111 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#20245957C211yesmaximal, characteristic
2#20243530C212yesmaximal
3#20241214C211yesmaximal, characteristic
4#20241213C212yesmaximal
5C47×#1272C211yesmaximal, characteristic
6C2×#13094022C411yescharacteristic
7C2×Q8×Dic47C411yescharacteristic
8C2×#13093989C411yescharacteristic
9#15689029C2211yescharacteristic
10#15689020—22no—
11#15688992C411yescharacteristic
12#15688974—22no—
13C42×Dic47C2211yescharacteristic
14#15688878C2212yes—
15#15688877—22no—
16#15688877—22no—
17C47×#391C2211yescharacteristic
18C47×#336C2212yes—
19C22×Dic94C2×C411yescharacteristic
20C22×C47⋊C8—22no—
21#13094022—22no—
22#13094022C814yescomplemented
23Q8×Dic47—42no—
24#13093989—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47D412yes—
35C2×C4×Dic47D412yes—
36C47⋊#95—22no—
37C22×Q8×C47C2×C411yescharacteristic
38C22×C8×C47—22no—
39C2×C4⋊C4×C47C2×C411yescharacteristic
40C2×C42×C47C2311yescharacteristic
41#95×C47—22no—
42C2×Q8×C47—21no—
43C2×Q8×C47—41no—
44C2×Q8×C47—41no—
45C2×Q8×C47C2×C812yes—
46C22×C4×C47—21no—
47C22×C4×C47—21no—
48C22×C4×C47C22×C411yescharacteristic
49C22×C4×C47C2×D412yes—
50C22×Dic47—21no—
51C22×Dic47—21no—
52C22×Dic47C22⋊C411yescharacteristic
53C22×Dic47C22⋊C412yes—
54C22×Dic47C22⋊C411yescharacteristic
55C2×Dic94—21no—
56C2×Dic94—41no—
57C2×Dic94—41no—
58C2×Dic94C2×C812yes—
59C2×C8×C47—42no—
60C2×C8×C47—42no—
61C2×C8×C47—42no—
62C4⋊C4×C47—41no—
63C42×C47—41no—
64C4⋊Dic47—42no—
65C4⋊Dic47—41no—
66Dic47⋊C4—28no—
67Dic47⋊C4—44no—
68Dic47⋊C4—42no—
69C4×Dic47—22no—
70C4×Dic47—22no—
71C4×Dic47—22no—
72C4×Dic47—42no—
73C4×Dic47—42no—
74C4×Dic47—42no—
75C4×Dic47C2×C812yes—
76C4×Dic47M1612yes—
77C2×C47⋊C8—42no—
78C2×C47⋊C8—42no—
79C2×C47⋊C8—42no—
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—42no—
82Q8×C47—42no—
83C2×C4×C47—22no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—42no—
92C2×C4×C47C22×C811yescharacteristic
93C2×C4×C47C2×M1611yescharacteristic
94C2×C4×C47#13712yes—
95C2×C4×C47#13712yes—
96C2×Dic47—22no—
97C2×Dic47—22no—
98C2×Dic47—22no—
99C2×Dic47—22no—
100C2×Dic47—21no—
101C2×Dic47—22no—
102C2×Dic47—41no—
103C2×Dic47—41no—
104C2×Dic47—41no—
105C2×Dic47—42no—
106C2×Dic47C22⋊C814yes—
107C2×Dic47#10012yes—
108Dic94—42no—
109Dic94—42no—
110C8×C47—82no—
111C47⋊C8—82no—
112#7605—471nomaximal, Sylow
113C22×C47C2×C22⋊C811yescharacteristic
114C22×C47#41011yescharacteristic
115C22×C47C2×#10011yescharacteristic
116C22×C47#41912yes—
117C22×C47#42112yes—
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—22no—
121C4×C47—22no—
122C4×C47—42no—
123C4×C47—81no—
124Dic47—22no—
125Dic47—24no—
126Dic47—41no—
127Dic47—42no—
128Dic47—81no—
129C4×C4⋊Q8—471no—
130#1839—472no—
131#1272—471no—
132#1272—472no—
133#1272D4711yescharacteristic
134C2×C47#146011yescharacteristic
135C2×C47#147912yes—
136C2×C47#151512yes—
137C2×C47#153611yescharacteristic
138C2×C47#155811yescharacteristic
139C2×C4⋊Q8—471no—
140C2×C4×Q8—471no—
141C2×C4×Q8—471no—
142#391—471no—
143#391C2×D4711yescharacteristic
144#389—942no—
145#384—471no—
146C4×C4⋊C4—942no—
147C43—471no—
148#336—472no—
149#336—942no—
150#336—942no—
151#336C2×D4712yes—
152C47#760511yescharacteristic, complemented, Sylow
153C22×Q8—471no—
154C22×Q8C4×D4711yescharacteristic
155C22×C8—22no—
156C22×C8—942no—
157C4⋊Q8—474no—
158C4⋊Q8—942no—
159C4×Q8—1882no—
160C4×Q8—1882no—
161C2×C4⋊C4—471no—
162C2×C4⋊C4—942no—
163C2×C4⋊C4—941no—
164C2×C4⋊C4—941no—
165C2×C4⋊C4—942no—
166C2×C4⋊C4C4×D4711yescharacteristic
167C2×C42—471no—
168C2×C42—471no—
169C2×C42—472no—
170C2×C42—472no—
171C2×C42—941no—
172C2×C42—941no—
173C2×C42C22×D4711yescharacteristic
174#95—22no—
175#95—942no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—472no—
180C2×Q8—941no—
181C2×Q8—1881no—
182C2×Q8—1881no—
183C2×Q8C8×D4712yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—471no—
187C22×C4—472no—
188C22×C4—471no—
189C22×C4—941no—
190C22×C4—941no—
191C22×C4C2×C4×D4711yescharacteristic
192C22×C4D4×D4712yes—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—1882no—
197C2×C8—1882no—
198C2×C8—1882no—
199C4⋊C4—41no—
200C4⋊C4—948no—
201C4⋊C4—1881no—
202C4⋊C4—1884no—
203C4⋊C4—1882no—
204C4⋊C4—1882no—
205C42—41no—
206C42—472no—
207C42—472no—
208C42—942no—
209C42—942no—
210C42—942no—
211C42—1882no—
212C42—1882no—
213C42—1882no—
214C23C22⋊C4×D4711yescharacteristic
215Q8—42no—
216Q8—42no—
217Q8—1882no—
218Q8—1882no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—474no—
229C2×C4—472no—
230C2×C4—942no—
231C2×C4—942no—
232C2×C4—942no—
233C2×C4—942no—
234C2×C4—941no—
235C2×C4—942no—
236C2×C4—1881no—
237C2×C4—1882no—
238C2×C4—1881no—
239C2×C4—1881no—
240C2×C4C2×C8×D4711yescharacteristic
241C2×C4M16×D4711yescharacteristic
242C2×C4#1309405312yes—
243C2×C4#1309405612yes—
244C8—82no—
245C8—3762no—
246C22C22⋊C8×D4711yescharacteristic
247C22#1568909511yescharacteristic
248C22#100×D4711yescharacteristic
249C22#1568913012yes—
250C22#1568913812yes—
251C4—22no—
252C4—22no—
253C4—21no—
254C4—21no—
255C4—42no—
256C4—81no—
257C4—944no—
258C4—942no—
259C4—3761no—
260C4—1882no—
261C4—1881no—
262C2#2024170711yescharacteristic
263C2#2024178112yes—
264C2#2024193712yes—
265C2#2024202111yescharacteristic
266C2#2024211211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7605
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
22. #13094022C8semidirect
152. C47#7605semidirect

Maximal subgroups

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

subgroupindex
#202459572
#202435302
#202412142
#202412132
C47×#12722
#760547