Finite Group Databasebeta

← #356×Dic47 #26131462

Subgroups — #356×Dic47 #26131462

Order 12032.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C4⋊Q8×Dic47C211yesmaximal, 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#15689022—22no—
13#15688993C411yescharacteristic
14C4⋊C4×Dic47—22no—
15C42×Dic47C411yescharacteristic
16#15688922C414yescomplemented
17#15688891—22no—
18C2×C4⋊Q8×C47C2211yescharacteristic
19C2×#101×C47C2212yes—
20C47×#356C414yescomplemented
21C2×#10625329—22no—
22Q8×Dic47—41no—
23Q8×Dic47—41no—
24Q8×Dic47—41no—
25Q8×Dic47—41no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47D412yes—
31C2×C4×Dic47—21no—
32C2×C4×Dic47—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47C2×C411yescharacteristic
35C2×C4×Dic47C2×C411yescharacteristic
36C2×C4×Dic47D412yes—
37#13093943—22no—
38#13093943C2×C414yes—
39#13093934—42no—
40C8×Dic47—42no—
41C47⋊#95—22no—
42C47⋊#95—22no—
43C22×Q8×C47C2×C411yescharacteristic
44C22×Q8×C47C2311yescharacteristic
45C2×M16×C47—22no—
46C4⋊Q8×C47—22no—
47C4⋊Q8×C47C2×C412yes—
48C4⋊Q8×C47C2×C412yes—
49C2×C4⋊C4×C47—22no—
50C2×C42×C47C2×C411yescharacteristic
51#101×C47—22no—
52#101×C47C2×C414yes—
53C2×Q8×C47—21no—
54C2×Q8×C47—41no—
55C2×Q8×C47—41no—
56C2×Q8×C47—41no—
57C2×Q8×C47—41no—
58C2×Q8×C47C4212yes—
59C2×Q8×C47C22×C411yescharacteristic
60C2×Q8×C47C22×C411yescharacteristic
61C2×Q8×C47#4112yes—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D412yes—
66C22×Dic47—21no—
67C22×Dic47—21no—
68C22×Dic47C22⋊C411yescharacteristic
69C22×Dic47C22⋊C412yes—
70C22×Dic47C22⋊C411yescharacteristic
71M16×C47—24no—
72M16×C47—42no—
73C2×C8×C47—42no—
74C4⋊C4×C47—24no—
75C4⋊C4×C47—24no—
76C42×C47—21no—
77C42×C47C4212yes—
78C4⋊Dic47—41no—
79C4⋊Dic47—42no—
80C4⋊Dic47—42no—
81C4⋊Dic47—44no—
82C4⋊Dic47—41no—
83C4×Dic47—24no—
84C4×Dic47—22no—
85C4×Dic47—22no—
86C4×Dic47—21no—
87C4×Dic47—21no—
88C4×Dic47—42no—
89C4×Dic47—41no—
90C4×Dic47—41no—
91C4×Dic47—41no—
92#10625329—24no—
93#10625329—42no—
94C2×C47⋊C8—42no—
95C23×C47C2×C22⋊C411yescharacteristic
96Q8×C47—41no—
97Q8×C47—41no—
98Q8×C47—42no—
99Q8×C47—42no—
100Q8×C47—42no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—22no—
109C2×C4×C47—41no—
110C2×C4×C47—41no—
111C2×C4×C47C2×C4211yescharacteristic
112C2×C4×C47#11711yescharacteristic
113C2×C4×C47C4×D412yes—
114C2×C4×C47C4×D412yes—
115C2×Dic47—22no—
116C2×Dic47—22no—
117C2×Dic47—21no—
118C2×Dic47—22no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—41no—
125C2×Dic47#9911yescharacteristic
126C2×Dic47#9911yescharacteristic
127C8×C47—44no—
128C47⋊C8—44no—
129C4×#356—471nomaximal, Sylow
130C22×C47—21no—
131C22×C47—21no—
132C22×C47C4×C22⋊C411yescharacteristic
133C22×C47C2×#9911yescharacteristic
134C22×C47#41011yescharacteristic
135C4×C47—24no—
136C4×C47—22no—
137C4×C47—22no—
138C4×C47—21no—
139C4×C47—21no—
140C4×C47—42no—
141Dic47—21no—
142Dic47—42no—
143Dic47—41no—
144Dic47—42no—
145Dic47—81no—
146Dic47#35612yesdirect product factor
147C4×C4⋊Q8—471no—
148C2×#356—471no—
149C2×#356D4711yescharacteristic
150C4×#101—472no—
151#1394—472no—
152C2×C47—21no—
153C2×C47—21no—
154C2×C47C4×#9911yescharacteristic
155C2×C47C2×#35611yescharacteristic
156C2×C47#213211yescharacteristic
157C2×C4⋊Q8C2×D4711yescharacteristic
158C2×C4×Q8—471no—
159C2×C4×Q8—471no—
160C2×#101—472no—
161C2×#101C2×D4712yes—
162C4×M16—942no—
163#391—471no—
164#391—471no—
165#389—942no—
166#384—471no—
167C4×C4⋊C4—942no—
168C43—471no—
169#356—474no—
170#356Dic4714yesdirect product factor
171#343—942no—
172C47C4×#35611yescharacteristic, complemented, Sylow
173C22×Q8C4×D4711yescharacteristic
174C22×Q8C22×D4711yescharacteristic
175C2×M16—22no—
176C2×M16—942no—
177C4⋊Q8—22no—
178C4⋊Q8C4×D4712yes—
179C4⋊Q8C2×Dic4712yes—
180C4×Q8—1881no—
181C4×Q8—1881no—
182C4×Q8—1881no—
183C4×Q8—1881no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—471no—
186C2×C4⋊C4—472no—
187C2×C4⋊C4—471no—
188C2×C4⋊C4—941no—
189C2×C4⋊C4—942no—
190C2×C42—471no—
191C2×C42—472no—
192C2×C42—471no—
193C2×C42—941no—
194C2×C42—941no—
195C2×C42—941no—
196C2×C42C4×D4711yescharacteristic
197#101—22no—
198#101—474no—
199#101—942no—
200#101C2×Dic4714yes—
201#97—1882no—
202C4×C8—1882no—
203#95—942no—
204#95—942no—
205C2×Q8—21no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8C4×Dic4712yes—
211C2×Q8C2×C4×D4711yescharacteristic
212C2×Q8#1062535212yes—
213C2×Q8C22×Dic4711yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—471no—
217C22×C4—472no—
218C22×C4—471no—
219C22×C4—941no—
220C22×C4—941no—
221C22×C4C2×C4×D4711yescharacteristic
222C22×C4D4×D4712yes—
223M16—24no—
224M16—42no—
225M16—944no—
226M16—1882no—
227C2×C8—42no—
228C2×C8—1882no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—1881no—
232C4⋊C4—1882no—
233C4⋊C4—1881no—
234C4⋊C4—1884no—
235C4⋊C4—1882no—
236C42—21no—
237C42—942no—
238C42—942no—
239C42—941no—
240C42—941no—
241C42—944no—
242C42—1881no—
243C42—1882no—
244C42—1881no—
245C42—1881no—
246C42C4×Dic4712yes—
247C23C22⋊C4×D4711yescharacteristic
248Q8—41no—
249Q8—42no—
250Q8—41no—
251Q8—42no—
252Q8—42no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—471no—
264C2×C4—471no—
265C2×C4—942no—
266C2×C4—942no—
267C2×C4—941no—
268C2×C4—942no—
269C2×C4—1882no—
270C2×C4—1881no—
271C2×C4—1881no—
272C2×C4—1881no—
273C2×C4—1881no—
274C2×C4—1882no—
275C2×C4C2×C4×Dic4711yescharacteristic
276C2×C4#1309398011yescharacteristic
277C2×C4(C22⋊D47)⋊C412yes—
278C2×C4D4×Dic4712yes—
279C8—44no—
280C8—1884no—
281C22—21no—
282C22—21no—
283C22C22⋊C4×Dic4711yescharacteristic
284C22#99×D4711yescharacteristic
285C22#1568909511yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—472no—
293C4—941no—
294C4—3761no—
295C4—1882no—
296C4—1881no—
297C4—1882no—
298C2—21no—
299C2—21no—
300C2#99×Dic4711yescharacteristic
301C2#356×D4711yescharacteristic
302C2#2024507211yescharacteristic
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
hypocenterC47C4×#356
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
146. Dic47#356direct (N × H)
170. #356Dic47direct (N × H)
172. C47C4×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×Dic472
C2×#156889222
#101×Dic472
#202415142
C2×C47×#3562
C4×#35647