Finite Group Databasebeta

← #26117254

Subgroups — #26117254

Order 12032.

584 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 105 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#20253415C212yesmaximal, complemented
2#20244328C211yesmaximal, characteristic, complemented
3#20241787C212yesmaximal, complemented
4#20241219C211yesmaximal, characteristic, complemented
5C47×#3150C211yesmaximal, characteristic
6C2×C4×Dic94C2211yescharacteristic
7#15689439—22no—
8#15689405—22no—
9#15689320—22no—
10#15689261—22no—
11C4×#10625335—24no—
12#15689176C2212yes—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14#15688877—22no—
15D4⋊(C47⋊C8)—24no—
16C2×#135×C47—22no—
17C47×#532C2212yes—
18C47×#497—22no—
19C47×#496—22no—
20C2×#97×C47C2211yescharacteristic
21C22×Dic94D411yescharacteristic
22C2×#10625335—24no—
23C22×C47⋊C8—21no—
24C4×Dic94—22no—
25C4×Dic94—22no—
26C2×C4⋊Dic47D411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×C4×Dic47—21no—
29#13093955—24no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32#13093947—24no—
33#13093945—24no—
34C22⋊(C47⋊C8)—22no—
35C4×C47⋊C8D414yes—
36C2×#41×C47—22no—
37#135×C47—44no—
38#135×C47—44no—
39C2×Q16×C47—22no—
40C2×SD16×C47—24no—
41C2×D8×C47—22no—
42C22×C8×C47D412yes—
43C4⋊Q8×C47D412yes—
44#127×C47D412yes—
45#124×C47—22no—
46#121×C47—22no—
47#121×C47—22no—
48C4×D4×C47—22no—
49C4×D4×C47—22no—
50C2×C42×C47C2311yescharacteristic
51#97×C47—22no—
52#41×C47—44no—
53#41×C47—44no—
54C2×Q8×C47—22no—
55C2×D4×C47—22no—
56C2×D4×C47—22no—
57C2×D4×C47—22no—
58C2×D4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47#4111yescharacteristic
63C22×Dic47—21no—
64C22×Dic47—21no—
65C2×Dic94—22no—
66C2×Dic94—42no—
67Q16×C47—44no—
68SD16×C47—48no—
69D8×C47—44no—
70C2×C8×C47—22no—
71C2×C8×C47—22no—
72C2×C8×C47SD1614yescomplemented
73C4⋊C4×C47—22no—
74C4⋊C4×C47—22no—
75C22⋊C4×C47—42no—
76C22⋊C4×C47—42no—
77C42×C47C2×D412yes—
78C42×C47C2×D412yes—
79#10625335—48no—
80C4⋊Dic47—22no—
81Dic47⋊C4—42no—
82C4×Dic47—41no—
83C4×Dic47—41no—
84C2×C47⋊C8—24no—
85C2×C47⋊C8—41no—
86C2×C47⋊C8—42no—
87C23×C47—42no—
88C23×C47#12111yescharacteristic
89Q8×C47—44no—
90D4×C47—44no—
91D4×C47—44no—
92D4×C47—44no—
93D4×C47—84no—
94C2×C4×C47—41no—
95C2×C4×C47—41no—
96C2×C4×C47—42no—
97C2×C4×C47—82no—
98C2×C4×C47—82no—
99C2×C4×C47#12012yes—
100C2×C4×C47#12011yescharacteristic
101C2×C4×C47#12112yes—
102C2×C4×C47#12711yescharacteristic
103C2×C4×C47C2×SD1612yes—
104C2×C4×C47AGL(1,ℤ/8)12yes—
105C2×Dic47—22no—
106C2×Dic47—41no—
107C2×Dic47—41no—
108C2×Dic47—42no—
109C2×Dic47—41no—
110Dic94—24no—
111Dic94—42no—
112C8×C47—24no—
113C47⋊C8—44no—
114C47⋊C8—81no—
115#10034—471nomaximal, Sylow
116C22×C47—21no—
117C22×C47—21no—
118C22×C47—82no—
119C22×C47—82no—
120C22×C47—82no—
121C22×C47#39211yescharacteristic
122C22×C47#46012yes—
123C4×C47—22no—
124C4×C47—22no—
125C4×C47—82no—
126C4×C47#45012yes—
127C4×C47#49211yescharacteristic
128C4×C47#49611yescharacteristic
129Dic47—42no—
130Dic47—81no—
131#3150D4711yescharacteristic
132#3103—472no—
133#1985—471no—
134#1480—472no—
135#1275—471no—
136C2×C47—22no—
137C2×C47—162no—
138C2×C47#201011yescharacteristic
139C2×C47#201411yescharacteristic
140C2×C47#201711yescharacteristic
141C2×#135—22no—
142#532C2×D4712yes—
143C2×C4×Q8—471no—
144#497—22no—
145#496—22no—
146#481—942no—
147#475—942no—
148#460—942no—
149#448—942no—
150C4×SD16—944no—
151#431—472no—
152C2×C4⋊C8—471no—
153C2×#97C2×D4711yescharacteristic
154#336—942no—
155D4⋊C8—944no—
156C47#1003411yescharacteristic, complemented, Sylow
157C2×#41—22no—
158C22×Q8—471no—
159#135—44no—
160#135—44no—
161C2×Q16—22no—
162C2×SD16—24no—
163C2×SD16—944no—
164C2×D8—22no—
165C22×C8—941no—
166C22×C8C22⋊D4712yes—
167C4⋊Q8C22⋊D4712yes—
168#127C22⋊D4712yes—
169#124—22no—
170#121—22no—
171#121—22no—
172C4×Q8—942no—
173C4×Q8—942no—
174C4×D4—22no—
175C4×D4—22no—
176C2×C4⋊C4—471no—
177C2×C4⋊C4—941no—
178C2×C42—941no—
179C2×C42C22×D4711yescharacteristic
180#106—944no—
181C4⋊C8—942no—
182C4⋊C8—942no—
183#103—944no—
184#102—944no—
185C22⋊C8—942no—
186#97—22no—
187C4×C8—474no—
188#41—44no—
189#41—44no—
190C2×Q8—22no—
191C2×Q8—942no—
192C2×Q8—1882no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—42no—
197C22×C4—42no—
198C22×C4—941no—
199C22×C4—941no—
200C22×C4D4×D4711yescharacteristic
201C22×C4#1062535211yescharacteristic
202C22×C4C2×C22⋊D4711yescharacteristic
203Q16—44no—
204SD16—48no—
205SD16—1888no—
206D8—44no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—944no—
210C2×C8—1881no—
211C2×C8—1882no—
212C2×C8#1062533514yescomplemented
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—942no—
216C4⋊C4—1882no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C42—1881no—
220C42—1881no—
221C42D4×D4712yes—
222C42C2×C22⋊D4712yes—
223C23—42no—
224C23#1309399511yescharacteristic
225Q8—44no—
226Q8—944no—
227Q8—1882no—
228D4—44no—
229D4—44no—
230D4—44no—
231D4—84no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—942no—
238C2×C4—1882no—
239C2×C4—1881no—
240C2×C4—1881no—
241C2×C4—1881no—
242C2×C4#1309399212yes—
243C2×C4#1309399311yescharacteristic
244C2×C4#1309399612yes—
245C2×C4#1309402011yescharacteristic
246C2×C4C2×#1062533512yes—
247C2×C4#1309405012yes—
248C8—24no—
249C8—3761no—
250C8—1884no—
251C22—21no—
252C22—21no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22#1568903111yescharacteristic
257C22#1568932012yes—
258C4—22no—
259C4—22no—
260C4—82no—
261C4—3761no—
262C4—1882no—
263C4#1568927412yes—
264C4#1568949511yescharacteristic
265C4#1568951011yescharacteristic
266C2—22no—
267C2—162no—
268C2#2024445211yescharacteristic
269C2#2024447611yescharacteristic
270C2#2024449111yescharacteristic
subgroupG/H
centerC2×C4#13094020
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3150C2
socleC22×C47#392
hypercenter#3150D47
hypocenterC47#10034
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3150

Complements

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

NH ≅ G/Ntype
1. #20253415C2semidirect
2. #20244328C2semidirect
3. #20241787C2semidirect
4. #20241219C2semidirect
72. C2×C8×C47SD16semidirect
156. C47#10034semidirect
212. C2×C8#10625335semidirect

Maximal subgroups

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

subgroupindex
#202534152
#202443282
#202417872
#202412192
C47×#31502
#1003447