Finite Group Databasebeta

← #26132553

Subgroups — #26132553

Order 12032.

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

normal subgroups only

This group has 614 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
1#20243316C212yesmaximal
2#20241515C214yesmaximal, complemented
3C47×#2311C211yesmaximal, characteristic
4C2×#13093943C2214yes—
5C2×#13093941—22no—
6#15689087—22no—
7C22×#41×C47C2211yescharacteristic
8C2×#117×C47C411yescharacteristic
9C22×C4⋊C4×C47C411yescharacteristic
10C47×#391C2212yes—
11C47×#390C412yes—
12C47×#386—22no—
13C47×#380—22no—
14C2×#10625329—24no—
15#13093943—28no—
16#13093941—44no—
17C22⋊(C47⋊C8)—44no—
18C2×#41×C47—24no—
19C2×#41×C47—42no—
20C2×#41×C47—42no—
21C22×Q8×C47C2311yescharacteristic
22C22×D4×C47C2×C411yescharacteristic
23C22×D4×C47D412yes—
24C23×C4×C47—21no—
25C23×C4×C47C2×C411yescharacteristic
26C23×C4×C47D412yes—
27#117×C47—42no—
28C2×C4⋊C4×C47—21no—
29C2×C4⋊C4×C47—21no—
30C2×C4⋊C4×C47—42no—
31C2×C4⋊C4×C47C2×C412yes—
32C2×C22⋊C4×C47—21no—
33C2×C22⋊C4×C47—24no—
34C2×C42×C47C2×C412yes—
35#95×C47—24no—
36C24×C47C22⋊C412yes—
37C24×C47C22⋊C411yescharacteristic
38#41×C47—416no—
39C2×Q8×C47—41no—
40C2×Q8×C47—41no—
41C2×Q8×C47#4114yes—
42C2×D4×C47—24no—
43C2×D4×C47—22no—
44C2×D4×C47—24no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47—42no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—24no—
52C22×C4×C47—24no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—42no—
56C22×C4×C47—42no—
57C22×C4×C47—82no—
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22×C411yescharacteristic
60C22×C4×C47C2×D412yes—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—42no—
63C4⋊C4×C47—42no—
64C22⋊C4×C47—42no—
65C22⋊C4×C47—84no—
66C42×C47—42no—
67#10625329—44no—
68#10625329—44no—
69C2×C47⋊C8—44no—
70C23×C47—21no—
71C23×C47—21no—
72C23×C47—22no—
73C23×C47—22no—
74C23×C47—22no—
75C23×C47—41no—
76C23×C47—41no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47#9912yes—
80C23×C47C2×C22⋊C411yescharacteristic
81Q8×C47—44no—
82D4×C47—48no—
83D4×C47—44no—
84C2×C4×C47—28no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—22no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—48no—
92C2×C4×C47—42no—
93C2×C4×C47—44no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—84no—
98C2×C4×C47#9912yes—
99C2×C4×C47#11712yes—
100C2×C4×C47#12314yes—
101C47⋊C8—84no—
102#12863—471nomaximal, Sylow
103C22×C47—21no—
104C22×C47—21no—
105C22×C47—22no—
106C22×C47—42no—
107C22×C47—42no—
108C22×C47—44no—
109C22×C47—41no—
110C22×C47—41no—
111C22×C47—42no—
112C22×C47—42no—
113C22×C47—42no—
114C22×C47—82no—
115C22×C47#38111yescharacteristic
116C22×C47C2×#9911yescharacteristic
117C22×C47C2×#9911yescharacteristic
118C4×C47—24no—
119C4×C47—22no—
120C4×C47—22no—
121C4×C47—82no—
122C4×C47—82no—
123#2311D4711yescharacteristic
124#1785—472no—
125#1394—474no—
126C2×C47—21no—
127C2×C47—21no—
128C2×C47—42no—
129C2×C47—82no—
130C2×C47#181911yescharacteristic
131C2×C47#213411yescharacteristic
132C2×C47#213511yescharacteristic
133C22×#41C2×D4711yescharacteristic
134C2×#117Dic4711yescharacteristic
135C22×C4⋊C4Dic4711yescharacteristic
136C2×#101—474no—
137C2×#100—942no—
138#407—942no—
139#391C2×D4712yes—
140#390Dic4712yes—
141#386—22no—
142#380—22no—
143C47#1286311yescharacteristic, complemented, Sylow
144C2×#41—24no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8C22×D4711yescharacteristic
148C22×D4C2×Dic4711yescharacteristic
149C22×D4C22⋊D4712yes—
150C23×C4—21no—
151C23×C4C2×Dic4711yescharacteristic
152C23×C4C22⋊D4712yes—
153C2×M16—944no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×Dic4712yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C42C2×Dic4712yes—
162#101—948no—
163#100—1884no—
164C22⋊C8—1884no—
165#95—24no—
166C24C22⋊Dic4712yes—
167C24C22⋊Dic4711yescharacteristic
168#41—416no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8#1062535014yes—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—24no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic4711yescharacteristic
189C22×C4C22×Dic4711yescharacteristic
190C22×C4C2×C22⋊D4712yes—
191M16—1884no—
192M16—1884no—
193C2×C8—1884no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—84no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23#1309393912yes—
210C23C2×C22⋊Dic4711yescharacteristic
211Q8—44no—
212D4—48no—
213D4—44no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4#1309393912yes—
229C2×C4#1309398312yes—
230C2×C4#1309400714yes—
231C8—3764no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—42no—
236C22—42no—
237C22—44no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22#1568898611yescharacteristic
245C22C2×#1309393911yescharacteristic
246C22C2×#1309393911yescharacteristic
247C4—24no—
248C4—22no—
249C4—22no—
250C4—82no—
251C4—82no—
252C2—21no—
253C2—21no—
254C2—42no—
255C2—82no—
256C2#2024346211yescharacteristic
257C2#2024508011yescharacteristic
258C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#2311C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×Q8×C47C23
hypercenter#2311D47
hypocenterC47#12863
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #20241515C2semidirect
143. C47#12863semidirect

Maximal subgroups

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

subgroupindex
#202433162
#202415152
C47×#23112
#1286347