Finite Group Databasebeta

← #26131463

Subgroups — #26131463

Order 12032.

464 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 113 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#20246106C211yesmaximal, characteristic
2C2×#15688921C212yesmaximal
3#101×Dic47C212yesmaximal
4#20241515C211yesmaximal, characteristic
5C47×#1394C211yesmaximal, characteristic
6C2×#13094022C2211yescharacteristic
7C2×Q8×Dic47C2211yescharacteristic
8C2×#13093989C411yescharacteristic
9C2×#13093943C2212yes—
10M16×Dic47—22no—
11#15689029C411yescharacteristic
12#15689010—22no—
13#15689005—22no—
14#15688988C411yescharacteristic
15#15688969C411yescharacteristic
16#15688921C418yescomplemented
17#15688891—22no—
18C2×#101×C47C2212yes—
19C47×#391C2211yescharacteristic
20C22×Dic94C2×C411yescharacteristic
21C2×#10625329—22no—
22#13094022—21no—
23#13094022—21no—
24#13094022C2×C414yes—
25Q8×Dic47—42no—
26#13093989—42no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47D412yes—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—21no—
31C2×C4×Dic47—21no—
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47C2×C411yescharacteristic
34C2×C4×Dic47D412yes—
35#13093943—22no—
36#13093943C2×C414yes—
37#13093934—42no—
38C8×Dic47—42no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—22no—
42C47⋊#95—22no—
43C22×Q8×C47C2311yescharacteristic
44C2×M16×C47—22no—
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47#101×C47—22no—
48#101×C47C2×C414yes—
49#95×C47—22no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×Q8×C47C22×C412yes—
53C2×Q8×C47#4111yescharacteristic
54C2×Q8×C47#4111yescharacteristic
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47C22×C411yescharacteristic
58C22×C4×C47C2×D412yes—
59C22×Dic47—21no—
60C22×Dic47—21no—
61C22×Dic47C22⋊C412yes—
62C22×Dic47C22⋊C411yescharacteristic
63C22×Dic47C22⋊C411yescharacteristic
64C2×Dic94—21no—
65C2×Dic94—41no—
66C2×Dic94—41no—
67C2×Dic94C4212yes—
68M16×C47—24no—
69M16×C47—42no—
70C2×C8×C47—42no—
71C4⋊C4×C47—41no—
72C42×C47—41no—
73C4⋊Dic47—41no—
74C4⋊Dic47—42no—
75Dic47⋊C4—24no—
76Dic47⋊C4—24no—
77Dic47⋊C4—42no—
78C4×Dic47—21no—
79C4×Dic47—22no—
80C4×Dic47—22no—
81C4×Dic47—22no—
82C4×Dic47—42no—
83C4×Dic47C4212yes—
84#10625329—24no—
85#10625329—42no—
86C2×C47⋊C8—42no—
87C23×C47C2×C22⋊C411yescharacteristic
88Q8×C47—42no—
89Q8×C47—41no—
90Q8×C47—41no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—22no—
94C2×C4×C47—21no—
95C2×C4×C47—41no—
96C2×C4×C47—41no—
97C2×C4×C47—42no—
98C2×C4×C47—41no—
99C2×C4×C47—41no—
100C2×C4×C47C2×C4211yescharacteristic
101C2×C4×C47#11711yescharacteristic
102C2×C4×C47C4×D412yes—
103C2×C4×C47C4×D412yes—
104C2×Dic47—22no—
105C2×Dic47—22no—
106C2×Dic47—21no—
107C2×Dic47—22no—
108C2×Dic47—41no—
109C2×Dic47—41no—
110C2×Dic47—41no—
111C2×Dic47—41no—
112C2×Dic47—41no—
113C2×Dic47—42no—
114C2×Dic47—41no—
115C2×Dic47#9912yes—
116Dic94—42no—
117Dic94—42no—
118C8×C47—44no—
119C47⋊C8—44no—
120#12631—471nomaximal, Sylow
121C22×C47—21no—
122C22×C47—21no—
123C22×C47C4×C22⋊C411yescharacteristic
124C22×C47C2×#9911yescharacteristic
125C22×C47#41011yescharacteristic
126C4×C47—22no—
127C4×C47—22no—
128C4×C47—21no—
129C4×C47—21no—
130C4×C47—81no—
131C4×C47—81no—
132Dic47—22no—
133Dic47—42no—
134Dic47—41no—
135Dic47—42no—
136Dic47—42no—
137#2327—471no—
138C2×#356—472no—
139C4×#101—472no—
140#1394—471no—
141#1394D4711yescharacteristic
142C2×C47—21no—
143C2×C47—21no—
144C2×C47C4×#9911yescharacteristic
145C2×C47#213311yescharacteristic
146C2×C47#213511yescharacteristic
147C2×C4⋊Q8—471no—
148C2×C4×Q8—471no—
149C2×C4×Q8—471no—
150C2×#101—472no—
151C2×#101C2×D4712yes—
152C4×M16—942no—
153#391—471no—
154#391C2×D4711yescharacteristic
155#387—942no—
156#387—942no—
157#382—471no—
158#376—471no—
159#356—478no—
160#343—942no—
161C47#1263111yescharacteristic, complemented, Sylow
162C22×Q8—471no—
163C22×Q8C22×D4711yescharacteristic
164C2×M16—22no—
165C2×M16—942no—
166C4⋊Q8—474no—
167C4⋊Q8—941no—
168C4⋊Q8—941no—
169C4×Q8—1882no—
170C4×Q8—1882no—
171C2×C4⋊C4—471no—
172C2×C4⋊C4—472no—
173C2×C4⋊C4—942no—
174C2×C4⋊C4—941no—
175C2×C4⋊C4C4×D4711yescharacteristic
176C2×C42—471no—
177C2×C42—471no—
178C2×C42—472no—
179C2×C42—941no—
180C2×C42C4×D4711yescharacteristic
181#101—22no—
182#101—474no—
183#101—942no—
184#101C4×D4714yes—
185#97—1882no—
186C4×C8—1882no—
187#95—22no—
188#95—941no—
189#95—941no—
190#95—942no—
191#95—942no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—472no—
195C2×Q8—941no—
196C2×Q8—1881no—
197C2×Q8—1881no—
198C2×Q8C2×C4×D4712yes—
199C2×Q8#1062535211yescharacteristic
200C2×Q8#1062535411yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—471no—
204C22×C4—472no—
205C22×C4—471no—
206C22×C4—941no—
207C22×C4—941no—
208C22×C4C2×C4×D4711yescharacteristic
209C22×C4D4×D4712yes—
210M16—24no—
211M16—42no—
212M16—944no—
213M16—1882no—
214C2×C8—42no—
215C2×C8—1882no—
216C4⋊C4—41no—
217C4⋊C4—944no—
218C4⋊C4—944no—
219C4⋊C4—1881no—
220C4⋊C4—1882no—
221C4⋊C4—1882no—
222C42—41no—
223C42—472no—
224C42—942no—
225C42—942no—
226C42—942no—
227C42—941no—
228C42—1882no—
229C23C22⋊C4×D4711yescharacteristic
230Q8—42no—
231Q8—41no—
232Q8—41no—
233Q8—1882no—
234Q8—1882no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—472no—
245C2×C4—942no—
246C2×C4—942no—
247C2×C4—941no—
248C2×C4—942no—
249C2×C4—1881no—
250C2×C4—1881no—
251C2×C4—1881no—
252C2×C4—1881no—
253C2×C4—1881no—
254C2×C4—1882no—
255C2×C4—1881no—
256C2×C4C42×D4711yescharacteristic
257C2×C4#1309398211yescharacteristic
258C2×C4(C22⋊D47)⋊C412yes—
259C2×C4#1309398612yes—
260C8—44no—
261C8—1884no—
262C22—21no—
263C22—21no—
264C22#1568897011yescharacteristic
265C22#99×D4711yescharacteristic
266C22#1568909511yescharacteristic
267C4—22no—
268C4—22no—
269C4—21no—
270C4—21no—
271C4—81no—
272C4—81no—
273C4—942no—
274C4—1882no—
275C4—1881no—
276C4—1882no—
277C4—1882no—
278C2—21no—
279C2—21no—
280C2#2024317511yescharacteristic
281C2#2024507411yescharacteristic
282C2#2024508111yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC47×#1394C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC22×Q8×C47C23
hypercenter#1394D47
hypocenterC47#12631
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
16. #15688921C4semidirect
161. C47#12631semidirect

Maximal subgroups

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

subgroupindex
#202461062
C2×#156889212
#101×Dic472
#202415152
C47×#13942
#1263147