Finite Group Databasebeta

← #26131466

Subgroups — #26131466

Order 12032.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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#20246108C211yesmaximal, characteristic
2C2×#15688922C211yesmaximal, characteristic
3#101×Dic47C212yesmaximal
4#20241514C212yesmaximal
5C2×C47×#356C211yesmaximal, characteristic
6C2×Q8×Dic47C411yescharacteristic
7C2×Q8×Dic47C2211yescharacteristic
8C2×#13093943C2212yes—
9M16×Dic47—22no—
10#15689030C411yescharacteristic
11#15689030C2211yescharacteristic
12#15689009—22no—
13#15689009—22no—
14#15688989C411yescharacteristic
15#15688969C411yescharacteristic
16#15688922C414yescomplemented
17#15688891—22no—
18C2×C4⋊Q8×C47C2211yescharacteristic
19C2×#101×C47C2212yes—
20C47×#356C414yescomplemented
21C2×#10625329—22no—
22Q8×Dic47—42no—
23Q8×Dic47—42no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47D412yes—
28C2×C4×Dic47—21no—
29C2×C4×Dic47C2×C411yescharacteristic
30C2×C4×Dic47C2×C411yescharacteristic
31C2×C4×Dic47D412yes—
32#13093943—22no—
33#13093943C2×C414yes—
34#13093934—42no—
35C8×Dic47—42no—
36C47⋊#95—22no—
37C47⋊#95—21no—
38C47⋊#95—21no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C22×Q8×C47C2×C411yescharacteristic
42C22×Q8×C47C2311yescharacteristic
43C2×M16×C47—22no—
44C4⋊Q8×C47—22no—
45C4⋊Q8×C47C2×C412yes—
46C4⋊Q8×C47C2×C412yes—
47C2×C4⋊C4×C47—22no—
48C2×C42×C47C2×C411yescharacteristic
49#101×C47—22no—
50#101×C47C2×C414yes—
51C2×Q8×C47—21no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47—41no—
55C2×Q8×C47—41no—
56C2×Q8×C47C4212yes—
57C2×Q8×C47C22×C411yescharacteristic
58C2×Q8×C47C22×C411yescharacteristic
59C2×Q8×C47#4112yes—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D412yes—
64C22×Dic47—21no—
65C22×Dic47—21no—
66C22×Dic47C22⋊C412yes—
67C22×Dic47C22⋊C411yescharacteristic
68C22×Dic47C22⋊C411yescharacteristic
69M16×C47—24no—
70M16×C47—42no—
71C2×C8×C47—42no—
72C4⋊C4×C47—24no—
73C4⋊C4×C47—24no—
74C42×C47—21no—
75C42×C47C4212yes—
76C4⋊Dic47—41no—
77C4⋊Dic47—42no—
78C4⋊Dic47—42no—
79C4⋊Dic47—41no—
80C4×Dic47—22no—
81C4×Dic47—22no—
82C4×Dic47—22no—
83C4×Dic47—42no—
84C4×Dic47—41no—
85#10625329—24no—
86#10625329—42no—
87C2×C47⋊C8—42no—
88C23×C47C2×C22⋊C411yescharacteristic
89Q8×C47—41no—
90Q8×C47—41no—
91Q8×C47—42no—
92Q8×C47—42no—
93Q8×C47—42no—
94C2×C4×C47—22no—
95C2×C4×C47—21no—
96C2×C4×C47—22no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—22no—
100C2×C4×C47—41no—
101C2×C4×C47—41no—
102C2×C4×C47—41no—
103C2×C4×C47—41no—
104C2×C4×C47C2×C4211yescharacteristic
105C2×C4×C47#11711yescharacteristic
106C2×C4×C47C4×D412yes—
107C2×C4×C47C4×D412yes—
108C2×Dic47—21no—
109C2×Dic47—22no—
110C2×Dic47—41no—
111C2×Dic47—42no—
112C2×Dic47—41no—
113C2×Dic47—41no—
114C2×Dic47—41no—
115C2×Dic47—42no—
116C2×Dic47—41no—
117C2×Dic47—41no—
118C2×Dic47#9912yes—
119C8×C47—44no—
120C47⋊C8—44no—
121#12631—471nomaximal, Sylow
122C22×C47—21no—
123C22×C47—21no—
124C22×C47C4×C22⋊C411yescharacteristic
125C22×C47C2×#9911yescharacteristic
126C22×C47#41011yescharacteristic
127C4×C47—22no—
128C4×C47—22no—
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—42no—
132C4×C47—42no—
133Dic47—22no—
134Dic47—41no—
135Dic47—42no—
136Dic47—81no—
137Dic47—81no—
138#2327—471no—
139C2×#356—471no—
140C2×#356D4711yescharacteristic
141C4×#101—472no—
142#1394—472no—
143C2×C47—21no—
144C2×C47—21no—
145C2×C47C4×#9911yescharacteristic
146C2×C47#213311yescharacteristic
147C2×C47#213511yescharacteristic
148C2×C4⋊Q8C2×D4711yescharacteristic
149C2×C4×Q8—471no—
150C2×C4×Q8—471no—
151C2×#101—472no—
152C2×#101C2×D4712yes—
153C4×M16—942no—
154#391—471no—
155#391—471no—
156#387—942no—
157#387—942no—
158#382—471no—
159#376—471no—
160#356—474no—
161#356Dic4714yescomplemented
162#343—942no—
163C47#1263111yescharacteristic, complemented, Sylow
164C22×Q8C4×D4711yescharacteristic
165C22×Q8C22×D4711yescharacteristic
166C2×M16—22no—
167C2×M16—942no—
168C4⋊Q8—22no—
169C4⋊Q8C4×D4712yes—
170C4⋊Q8C2×Dic4712yes—
171C4×Q8—1882no—
172C4×Q8—1882no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—471no—
175C2×C4⋊C4—472no—
176C2×C4⋊C4—471no—
177C2×C4⋊C4—941no—
178C2×C42—471no—
179C2×C42—472no—
180C2×C42—471no—
181C2×C42—941no—
182C2×C42C4×D4711yescharacteristic
183#101—22no—
184#101—474no—
185#101—942no—
186#101C2×Dic4714yes—
187#97—1882no—
188C4×C8—1882no—
189#95—941no—
190#95—941no—
191#95—942no—
192#95—942no—
193#95—942no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C4×Dic4712yes—
200C2×Q8C2×C4×D4711yescharacteristic
201C2×Q8#1062535212yes—
202C2×Q8C22×Dic4711yescharacteristic
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—471no—
206C22×C4—472no—
207C22×C4—471no—
208C22×C4—941no—
209C22×C4—941no—
210C22×C4C2×C4×D4711yescharacteristic
211C22×C4D4×D4712yes—
212M16—24no—
213M16—42no—
214M16—944no—
215M16—1882no—
216C2×C8—42no—
217C2×C8—1882no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—1881no—
221C4⋊C4—1882no—
222C4⋊C4—1881no—
223C4⋊C4—1882no—
224C42—21no—
225C42—942no—
226C42—942no—
227C42—942no—
228C42—1881no—
229C42—1882no—
230C42C4×Dic4712yes—
231C23C22⋊C4×D4711yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235Q8—42no—
236Q8—42no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—472no—
248C2×C4—941no—
249C2×C4—942no—
250C2×C4—1881no—
251C2×C4—1881no—
252C2×C4—1881no—
253C2×C4—1882no—
254C2×C4—1881no—
255C2×C4—1881no—
256C2×C4—1882no—
257C2×C4—1881no—
258C2×C4C2×C4×Dic4711yescharacteristic
259C2×C4#1309398011yescharacteristic
260C2×C4(C22⋊D47)⋊C412yes—
261C2×C4D4×Dic4712yes—
262C8—44no—
263C8—1884no—
264C22—21no—
265C22—21no—
266C22C22⋊C4×Dic4711yescharacteristic
267C22#99×D4711yescharacteristic
268C22#1568909511yescharacteristic
269C4—22no—
270C4—22no—
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—942no—
276C4—3761no—
277C4—3761no—
278C4—1881no—
279C4—1882no—
280C2—21no—
281C2—21no—
282C2#99×Dic4711yescharacteristic
283C2#2024507611yescharacteristic
284C2#2024508111yescharacteristic
subgroupG/H
centerC22#99×D47
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC2×C47×#356C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC22×Q8×C47C23
hypercenterC2×#356D47
hypocenterC47#12631
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
16. #15688922C4semidirect
20. C47×#356C4semidirect
161. #356Dic47semidirect
163. C47#12631semidirect

Maximal subgroups

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

subgroupindex
#202461082
C2×#156889222
#101×Dic472
#202415142
C2×C47×#3562
#1263147