Finite Group Databasebeta

← #26116895

Subgroups — #26116895

Order 12032.

544 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 83 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#20254139C211yesmaximal, characteristic
2#20253328C212yesmaximal, complemented
3#20241747C212yesmaximal, complemented
4#20241265C211yesmaximal, characteristic, complemented
5C47×#1818C211yesmaximal, characteristic, complemented
6C2×#13093998C2212yes—
7#15689447—24no—
8#15689441—22no—
9#15689437—22no—
10#15689436—22no—
11#15689359—22no—
12#15689259—22no—
13#15689166—22no—
14#15689127—22no—
15#15689087C2212yes—
16#15688846—24no—
17C2×#117×C47C2211yescharacteristic
18C2×C22⋊C8×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2211yescharacteristic
20C47×#336—22no—
21C22×D188D412yes—
22C22×C22⋊D47—22no—
23C2×D376—22no—
24C2×#10625329—22no—
25#13093998—22no—
26#13093998—22no—
27#13093998—22no—
28#13093998—22no—
29C2×C4⋊Dic47D412yes—
30C2×(C2×D47)⋊C4—22no—
31#13093958—22no—
32#13093957—24no—
33#13093955—24no—
34#13093946—24no—
35#13093944—24no—
36#13093944—24no—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)D414yes—
39C23×C4×C47C2311yescharacteristic
40C22×C8×C47—21no—
41C22×C8×C47D412yes—
42#117×C47—22no—
43#117×C47—22no—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47—21no—
47C2×C42×C47—21no—
48C22⋊C8×C47—22no—
49C22⋊C8×C47—22no—
50C24×C47#4111yescharacteristic
51C23×D47—22no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—42no—
55C22×C4×C47C2×D412yes—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D412yes—
59C2×C22⋊D47—42no—
60C2×C22⋊D47—42no—
61C2×C22⋊D47—84no—
62C22×Dic47—22no—
63C2×D188—22no—
64C2×D188—22no—
65C2×D188—44no—
66C2×C8×C47—22no—
67C2×C8×C47—42no—
68C2×C8×C47—42no—
69C2×C8×C47—41no—
70C4⋊C4×C47—22no—
71C4⋊C4×C47—22no—
72C22⋊C4×C47—42no—
73C42×C47—41no—
74C42×C47—41no—
75(C2×D47)⋊C4—42no—
76(C2×D47)⋊C4—42no—
77C4⋊Dic47—22no—
78C4⋊Dic47—22no—
79#10625329—44no—
80C2×C47⋊C8—24no—
81D376—82no—
82C23×C47—22no—
83C23×C47—41no—
84C23×C47—41no—
85C23×C47#12112yes—
86C23×C47#12111yescharacteristic
87C22×D47—42no—
88C22×D47—44no—
89C22×D47—82no—
90C22×D47—82no—
91C2×C4×C47—24no—
92C2×C4×C47—22no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—42no—
99C2×C4×C47#12011yescharacteristic
100C2×C4×C47#12012yes—
101C2×C4×C47#12711yescharacteristic
102C22⋊D47—84no—
103C22⋊D47—84no—
104C2×Dic47—42no—
105C2×Dic47—82no—
106D188—44no—
107D188—44no—
108C8×C47—42no—
109C8×C47—81no—
110C47⋊C8—44no—
111#9977—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47—22no—
115C22×C47—22no—
116C22×C47—22no—
117C22×C47—82no—
118C22×C47#39211yescharacteristic
119C22×C47#47212yes—
120C2×D47—82no—
121C2×D47—82no—
122C2×D47—82no—
123C2×D47—164no—
124C4×C47—22no—
125C4×C47—21no—
126C4×C47—81no—
127C4×C47—81no—
128C4×C47#45812yes—
129Dic47—82no—
130#3186—471no—
131#3094—472no—
132#1818D4711yescharacteristic, complemented
133#1471—472no—
134#1293—471no—
135C2×C47—22no—
136C2×C47—42no—
137C2×C47#201711yescharacteristic
138C2×C47#201811yescharacteristic
139C2×C47#202211yescharacteristic
140D47—162no—
141C2×#121—472no—
142C2×#117C2×D4711yescharacteristic
143#482—944no—
144#481—942no—
145#480—942no—
146#480—942no—
147#466—942no—
148#447—942no—
149#428—942no—
150#418—942no—
151#407—472no—
152C2×C22⋊C8C2×D4711yescharacteristic
153C2×C22⋊C8C2×D4711yescharacteristic
154#336—22no—
155#323—944no—
156C47#997711yescharacteristic, complemented, Sylow
157C22×D4—472no—
158C22×D4—942no—
159C23×C4C22×D4711yescharacteristic
160C2×D8—942no—
161C2×M16—942no—
162C22×C8—21no—
163C22×C8C22⋊D4712yes—
164#121—942no—
165#121—942no—
166#121—942no—
167#121—942no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4—472no—
171C2×C4⋊C4D18811yescharacteristic
172C2×C4⋊C4D18811yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—942no—
175C2×C42—21no—
176#107—944no—
177#107—942no—
178#106—944no—
179#102—944no—
180#102—944no—
181#102—944no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—474no—
185C22⋊C8—942no—
186C24—942no—
187C24#1062535011yescharacteristic
188C2×D4—942no—
189C2×D4—942no—
190C2×D4—3764no—
191C2×D4—1884no—
192C2×D4—1882no—
193C2×D4—1882no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—942no—
198C22×C4C2×D18811yescharacteristic
199C22×C4D4×D4712yes—
200C22×C4D4×D4712yes—
201C22×C4C2×C22⋊D4711yescharacteristic
202D8—3762no—
203M16—1884no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—944no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—942no—
212C4⋊C4—942no—
213C22⋊C4—42no—
214C22⋊C4—1882no—
215C22⋊C4—1882no—
216C42—41no—
217C42—41no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—3762no—
222C23—3762no—
223C23—1884no—
224C23—1882no—
225C23#1309399712yes—
226C23#1309399811yescharacteristic
227D4—3764no—
228D4—3764no—
229D4—1884no—
230D4—1884no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—3762no—
240C2×C4—1882no—
241C2×C4#1309399211yescharacteristic
242C2×C4#1309399412yes—
243C2×C4#1309402011yescharacteristic
244C8—42no—
245C8—81no—
246C8—1884no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—82no—
253C22—3762no—
254C22—3762no—
255C22—3762no—
256C22—7524no—
257C22#1568903211yescharacteristic
258C22#1568939312yes—
259C4—21no—
260C4—22no—
261C4—81no—
262C4—81no—
263C4—3762no—
264C4#1568931212yes—
265C2—22no—
266C2—42no—
267C2—7522no—
268C2#2024449211yescharacteristic
269C2#2024449711yescharacteristic
270C2#2024451911yescharacteristic
subgroupG/H
centerC22#15689032
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1818C2
socleC22×C47#392
prime radical#20254139C2
hypercenter#1818D47
hypocenterC47#9977
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #20253328C2semidirect
3. #20241747C2semidirect
4. #20241265C2semidirect
5. C47×#1818C2semidirect
132. #1818D47semidirect
156. C47#9977semidirect

Maximal subgroups

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

subgroupindex
#202541392
#202533282
#202417472
#202412652
C47×#18182
#997747