Finite Group Databasebeta

← #26108819

Subgroups — #26108819

Order 12032.

656 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#15689734C211yesmaximal, characteristic
2#20242114C211yesmaximal, characteristic, complemented
3#20242014C212yesmaximal, complemented
4#20241720C212yesmaximal
5C47×#1558C211yesmaximal, characteristic, complemented
6#15689734—24no—
7#15689734—22no—
8#15689734—22no—
9C2×#13094019—22no—
10C2×#13094016C2211yescharacteristic
11C2×#13094014—22no—
12C2×#13094000C412yes—
13C2×Q8×Dic47C412yes—
14C2×#13093981C2211yescharacteristic
15#15689180C2212yes—
16#15688872—22no—
17#15688864—24no—
18#15688848—22no—
19#15688846—22no—
20C2×C4⋊Q8×C47C2211yescharacteristic
21C47×#432C2212yes—
22C47×#333—22no—
23C22×C4×D47C2×C411yescharacteristic
24#13094019—28no—
25#13094019—28no—
26#13094016—21no—
27#13094016—21no—
28#13094016D414yes—
29#13094014—28no—
30#13094014—28no—
31#13094000—24no—
32#13094000—24no—
33Q8×Dic47—24no—
34Q8×Dic47—24no—
35#13093981—21no—
36#13093981—22no—
37#13093981—21no—
38C2×C4⋊Dic47—22no—
39C2×C4⋊Dic47C2×C412yes—
40C2×Dic47⋊C4—22no—
41C2×Dic47⋊C4—22no—
42C2×Dic47⋊C4D412yes—
43C2×(C2×D47)⋊C4—22no—
44C2×(C2×D47)⋊C4D412yes—
45C2×C4×Dic47—22no—
46C2×C4×Dic47C2×C411yescharacteristic
47C4⋊(C47⋊C8)—22no—
48C22⋊(C47⋊C8)—22no—
49#13093935—22no—
50C22×Q8×C47C2×C412yes—
51C4⋊Q8×C47—22no—
52C4⋊Q8×C47D412yes—
53C4⋊Q8×C47D412yes—
54C2×C4⋊C4×C47—22no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C22⋊C8×C47—22no—
58#97×C47—22no—
59C23×D47C22⋊C411yescharacteristic
60C2×Q8×C47—22no—
61C2×Q8×C47—44no—
62C2×Q8×C47C22⋊C414yes—
63C22×C4×C47—22no—
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D412yes—
66C22×Dic47—22no—
67C22×Dic47C22⋊C411yescharacteristic
68C22×Dic47C22⋊C412yes—
69C2×C4×D47—21no—
70C2×C4×D47—21no—
71C2×C4×D47—42no—
72C2×C8×C47—42no—
73C4⋊C4×C47—24no—
74C4⋊C4×C47—24no—
75C42×C47C2×D411yescharacteristic
76C42×C47C2×D412yes—
77C42×C47C2×D411yescharacteristic
78(C2×D47)⋊C4—28no—
79(C2×D47)⋊C4—22no—
80(C2×D47)⋊C4—22no—
81C4⋊Dic47—28no—
82C4⋊Dic47—22no—
83C4⋊Dic47C22⋊C414yes—
84Dic47⋊C4—28no—
85Dic47⋊C4—24no—
86Dic47⋊C4—24no—
87Dic47⋊C4—22no—
88Dic47⋊C4—22no—
89C4×Dic47—28no—
90C4×Dic47—21no—
91C4×Dic47—21no—
92C2×C47⋊C8—42no—
93C23×C47C2×C22⋊C411yescharacteristic
94C22×D47—21no—
95C22×D47—22no—
96C22×D47—42no—
97Q8×C47—44no—
98Q8×C47—44no—
99C2×C4×C47—22no—
100C2×C4×C47—24no—
101C2×C4×C47—42no—
102C2×C4×C47—44no—
103C2×C4×C47C2×C22⋊C411yescharacteristic
104C2×C4×C47C2×C22⋊C411yescharacteristic
105C2×C4×C47#12012yes—
106C2×C4×C47#12012yes—
107C2×Dic47—22no—
108C2×Dic47—24no—
109C2×Dic47—21no—
110C2×Dic47—22no—
111C2×Dic47—44no—
112C2×Dic47—42no—
113C4×D47—42no—
114C4×D47—42no—
115C8×C47—82no—
116C47⋊C8—82no—
117#8527—471nomaximal, Sylow
118C22×C47—22no—
119C22×C47#37911yescharacteristic
120C22×C47#41312yes—
121C2×D47—42no—
122C2×D47—41no—
123C2×D47—82no—
124C4×C47—22no—
125C4×C47—22no—
126C4×C47—21no—
127C4×C47—21no—
128C4×C47—44no—
129Dic47—42no—
130Dic47—44no—
131Dic47—41no—
132C2×#541—471no—
133#1558—471no—
134#1558D4711yescharacteristic, complemented
135#1534—472no—
136#1464—472no—
137C2×C47—22no—
138C2×C47#178511yescharacteristic
139C2×C47#179612yes—
140D47—81no—
141#541—944no—
142#541—942no—
143#541—942no—
144C2×C4⋊Q8C2×D4711yescharacteristic
145C2×#126—942no—
146C2×#125—471no—
147C2×#125—942no—
148C2×C22⋊Q8—472no—
149C2×C4×Q8—472no—
150C2×#117—471no—
151#432—472no—
152#432C2×D4712yes—
153#333—22no—
154#333—942no—
155#330—944no—
156#324—942no—
157#323—942no—
158C47#852711yescharacteristic, complemented, Sylow
159C22×Q8C4×D4712yes—
160C23×C4—471no—
161C4⋊Q8—22no—
162C4⋊Q8D18812yes—
163C4⋊Q8C22⋊D4712yes—
164#126—948no—
165#126—948no—
166#125—474no—
167#125—948no—
168#125—948no—
169#125—941no—
170#125—941no—
171C22⋊Q8—944no—
172C22⋊Q8—944no—
173C4×Q8—944no—
174C4×Q8—944no—
175#117—941no—
176#117—942no—
177#117—941no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—472no—
180C2×C4⋊C4—472no—
181C2×C4⋊C4—942no—
182C2×C4⋊C4—942no—
183C2×C4⋊C4—942no—
184C2×C22⋊C4—472no—
185C2×C22⋊C4—942no—
186C2×C42—471no—
187C2×C42—942no—
188C2×C42C22×D4711yescharacteristic
189C4⋊C8—22no—
190C4⋊C8—942no—
191C22⋊C8—22no—
192C22⋊C8—942no—
193#97—22no—
194#97—942no—
195C24—471no—
196C2×Q8—22no—
197C2×Q8—44no—
198C2×Q8(C2×D47)⋊C414yes—
199C22×C4—22no—
200C22×C4—471no—
201C22×C4—472no—
202C22×C4—942no—
203C22×C4—941no—
204C22×C4—941no—
205C22×C4—1882no—
206C22×C4C2×C4×D4711yescharacteristic
207C22×C4D4×D4712yes—
208C2×C8—42no—
209C2×C8—1882no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—474no—
213C4⋊C4—948no—
214C4⋊C4—948no—
215C4⋊C4—942no—
216C4⋊C4—942no—
217C4⋊C4—944no—
218C4⋊C4—942no—
219C4⋊C4—944no—
220C22⋊C4—948no—
221C22⋊C4—942no—
222C22⋊C4—942no—
223C42—948no—
224C42—941no—
225C42—941no—
226C42C2×D18811yescharacteristic
227C42D4×D4712yes—
228C42C2×C22⋊D4711yescharacteristic
229C23—942no—
230C23—941no—
231C23—1882no—
232C23C22⋊C4×D4711yescharacteristic
233Q8—44no—
234Q8—44no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—942no—
240C2×C4—944no—
241C2×C4—941no—
242C2×C4—942no—
243C2×C4—1882no—
244C2×C4—1882no—
245C2×C4—1884no—
246C2×C4—1882no—
247C2×C4C22⋊C4×D4711yescharacteristic
248C2×C4C2×(C2×D47)⋊C411yescharacteristic
249C2×C4#1309399212yes—
250C2×C4#1309399412yes—
251C8—82no—
252C8—3762no—
253C22—22no—
254C22—3762no—
255C22—1881no—
256C22—1882no—
257C22#1568897811yescharacteristic
258C22#1568910412yes—
259C4—21no—
260C4—21no—
261C4—22no—
262C4—22no—
263C4—44no—
264C4—1881no—
265C4—1884no—
266C4—1882no—
267C2—22no—
268C2—3761no—
269C2#2024331311yescharacteristic
270C2#2024336012yes—
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1558C2
socleC22×C47#379
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1558D47
hypocenterC47#8527
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #20242114C2semidirect
3. #20242014C2semidirect
5. C47×#1558C2semidirect
134. #1558D47semidirect
158. C47#8527semidirect

Maximal subgroups

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

subgroupindex
C2×#156897342
#202421142
#202420142
#202417202
C47×#15582
#852747