Finite Group Databasebeta

← #26109057

Subgroups — #26109057

Order 12032.

540 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 121 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#124×Dic47C211yesmaximal, characteristic
2#20243242C212yesmaximal, complemented
3C2×#15688863C211yesmaximal, characteristic
4#20241270C212yesmaximal, complemented
5C2×C47×#329C211yesmaximal, characteristic
6C2×Q8×Dic47C411yescharacteristic
7C2×D4×Dic47C411yescharacteristic
8C2×Dic47⋊C8—22no—
9C2×#13093935C2212yes—
10#15689030C411yescharacteristic
11#15689026C411yescharacteristic
12#15689016—22no—
13#15688989C2211yescharacteristic
14C22⋊C4×Dic47—22no—
15C42×Dic47C2211yescharacteristic
16#15688877—22no—
17#15688863—22no—
18C2×#124×C47C2211yescharacteristic
19C2×#97×C47C2212yes—
20C47×#329—22no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23Q8×Dic47—42no—
24D4×Dic47—42no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×C4×Dic47—21no—
31C2×C4×Dic47C2×C411yescharacteristic
32C2×C4×Dic47C2×C411yescharacteristic
33C2×C4×Dic47D412yes—
34C2×C4×Dic47D412yes—
35Dic47⋊C8—28no—
36#13093935—22no—
37#13093935—22no—
38C47⋊#95—22no—
39C47⋊#95—22no—
40C22×Q8×C47C2×C411yescharacteristic
41C22×D4×C47C2×C411yescharacteristic
42C22×C8×C47—22no—
43#124×C47—22no—
44#124×C47D412yes—
45#124×C47Q812yes—
46C2×C22⋊C4×C47—22no—
47C2×C42×C47C2311yescharacteristic
48#97×C47—22no—
49#97×C47—22no—
50C24×C47—21no—
51C2×Q8×C47—21no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47C4⋊C412yes—
55C2×D4×C47—21no—
56C2×D4×C47—41no—
57C2×D4×C47—41no—
58C2×D4×C47C4⋊C412yes—
59C22×C4×C47—21no—
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D412yes—
62C22×Dic47—21no—
63C22×Dic47—21no—
64C22×Dic47—44no—
65C22×Dic47—42no—
66C22×Dic47C22⋊C411yescharacteristic
67C22×Dic47C22⋊C411yescharacteristic
68C22×Dic47C22⋊C412yes—
69C2×C8×C47—24no—
70C2×C8×C47—42no—
71C2×C8×C47—42no—
72C22⋊C4×C47—24no—
73C22⋊C4×C47—24no—
74C42×C47C2×D411yescharacteristic
75C42×C47C2×Q811yescharacteristic
76C42×C47#4112yes—
77C22⋊Dic47—42no—
78C22⋊Dic47—44no—
79C4⋊Dic47—42no—
80C4⋊Dic47—41no—
81C4⋊Dic47—41no—
82C4×Dic47—22no—
83C4×Dic47—24no—
84C4×Dic47—41no—
85C4×Dic47—42no—
86C4×Dic47—42no—
87C2×C47⋊C8—24no—
88C2×C47⋊C8—42no—
89C2×C47⋊C8—42no—
90C23×C47—22no—
91C23×C47—41no—
92C23×C47—41no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47C2×C22⋊C411yescharacteristic
96Q8×C47—42no—
97Q8×C47—42no—
98D4×C47—42no—
99D4×C47—42no—
100C2×C4×C47—21no—
101C2×C4×C47—21no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47C2×C4⋊C411yescharacteristic
108C2×C4×C47#11711yescharacteristic
109C2×C4×C47#12112yes—
110C2×C4×C47C22⋊Q812yes—
111C2×Dic47—22no—
112C2×Dic47—22no—
113C2×Dic47—22no—
114C2×Dic47—21no—
115C2×Dic47—22no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120C2×Dic47—84no—
121C2×Dic47#10012yes—
122C2×Dic47#10414yes—
123C8×C47—44no—
124C47⋊C8—44no—
125#8583—471nomaximal, Sylow
126C22×C47—42no—
127C22×C47—44no—
128C22×C47—81no—
129C22×C47—82no—
130C22×C47—81no—
131C22×C47#38011yescharacteristic
132C22×C47C2×#10011yescharacteristic
133C22×C47#41311yescharacteristic
134C22×C47C2×#10412yes—
135C22×C47#42112yes—
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—22no—
139C4×C47—22no—
140C4×C47—42no—
141Dic47—24no—
142Dic47—22no—
143Dic47—41no—
144Dic47—81no—
145Dic47—81no—
146C4×#124—471no—
147#1767—472no—
148C2×#329—471no—
149C2×#329D4711yescharacteristic
150#1295—472no—
151C2×C47—82no—
152C2×C47#152511yescharacteristic
153C2×C47#153212yes—
154C2×C47#153311yescharacteristic
155C2×C47#180011yescharacteristic
156C2×C47#180612yes—
157C2×#124C2×D4711yescharacteristic
158C2×C4×Q8—471no—
159C2×C4×D4—471no—
160C2×C4⋊C8—942no—
161C2×#97—472no—
162C2×#97C2×D4712yes—
163#391—471no—
164#390—471no—
165#388—942no—
166#382—471no—
167C4×C22⋊C4—942no—
168C43—471no—
169#336—942no—
170#329—22no—
171#329—942no—
172C47#858311yescharacteristic, complemented, Sylow
173C22×Q8C4×D4711yescharacteristic
174C22×D4C4×D4711yescharacteristic
175C23×C4—941no—
176C22×C8—22no—
177C22×C8—942no—
178#124—22no—
179#124Dic9412yes—
180#124C22⋊D4712yes—
181C4×Q8—1882no—
182C4×D4—1882no—
183C2×C4⋊C4—471no—
184C2×C4⋊C4—471no—
185C2×C4⋊C4—941no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—941no—
188C2×C22⋊C4—942no—
189C2×C42—471no—
190C2×C42—472no—
191C2×C42—471no—
192C2×C42—472no—
193C2×C42—941no—
194C2×C42C22×D4711yescharacteristic
195C4⋊C8—948no—
196#97—22no—
197#97—22no—
198#97—942no—
199#97—942no—
200#95—942no—
201#95—942no—
202C24—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8Dic47⋊C412yes—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4Dic47⋊C412yes—
211C22×C4—21no—
212C22×C4—472no—
213C22×C4—471no—
214C22×C4—471no—
215C22×C4—941no—
216C22×C4—941no—
217C22×C4—1882no—
218C22×C4—1884no—
219C22×C4C2×C4×D4711yescharacteristic
220C22×C4D4×D4712yes—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—944no—
225C2×C8—1882no—
226C2×C8—1882no—
227C4⋊C4—1881no—
228C4⋊C4—1881no—
229C4⋊C4—1882no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—1884no—
233C22⋊C4—1882no—
234C42—944no—
235C42—942no—
236C42—1881no—
237C42—1882no—
238C42—1882no—
239C42C2×Dic9411yescharacteristic
240C42#1062535212yes—
241C42C2×C22⋊D4711yescharacteristic
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C22⋊C4×D4711yescharacteristic
248Q8—42no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—472no—
260C2×C4—474no—
261C2×C4—942no—
262C2×C4—942no—
263C2×C4—942no—
264C2×C4—941no—
265C2×C4—942no—
266C2×C4—3764no—
267C2×C4—1881no—
268C2×C4—1881no—
269C2×C4—1882no—
270C2×C4—1882no—
271C2×C4C2×Dic47⋊C411yescharacteristic
272C2×C4#1309398011yescharacteristic
273C2×C4#1309399612yes—
274C2×C4#1309400112yes—
275C8—44no—
276C8—1884no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22#1568898111yescharacteristic
283C22#100×D4711yescharacteristic
284C22#1568910411yescharacteristic
285C22#104×D4712yes—
286C22#1568913812yes—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—944no—
293C4—942no—
294C4—3761no—
295C4—3761no—
296C4—1881no—
297C2—82no—
298C2#2024197811yescharacteristic
299C2#2024200512yes—
300C2#2024201011yescharacteristic
301C2#2024337811yescharacteristic
302C2#2024340612yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#329C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D4×C47C2×C4
cosocleC2×C42×C47C23
hypercenterC2×#329D47
hypocenterC47#8583
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. #20243242C2semidirect
4. #20241270C2semidirect
172. C47#8583semidirect

Maximal subgroups

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

subgroupindex
#124×Dic472
#202432422
C2×#156888632
#202412702
C2×C47×#3292
#858347