Finite Group Databasebeta

← #26125672

Subgroups — #26125672

Order 12032.

510 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 99 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#20241722C212yesmaximal
2#20241322C214yesmaximal, complemented
3C47×#2310C211yesmaximal, characteristic
4#15689180—22no—
5#15688849C2214yes—
6#15688847—22no—
7C23×Q8×C47C411yescharacteristic
8C2×#117×C47C411yescharacteristic
9C2×#117×C47C2211yescharacteristic
10C47×#391—22no—
11C47×#391C412yes—
12C47×#391C2212yes—
13C47×#381—22no—
14C4⋊(C47⋊C8)—42no—
15C22⋊(C47⋊C8)—24no—
16#13093935—42no—
17C22×Q8×C47—24no—
18C22×Q8×C47—21no—
19C22×Q8×C47—42no—
20C22×Q8×C47—42no—
21C22×Q8×C47C2×C412yes—
22C23×C4×C47—21no—
23C23×C4×C47C2×C411yescharacteristic
24#117×C47—42no—
25#117×C47—42no—
26C2×C4⋊C4×C47—21no—
27C2×C4⋊C4×C47C2×C411yescharacteristic
28C2×C4⋊C4×C47C2311yescharacteristic
29C2×C22⋊C4×C47—21no—
30C2×C22⋊C4×C47D412yes—
31C2×C42×C47C2×C412yes—
32C2×C42×C47D412yes—
33#95×C47—24no—
34#95×C47—24no—
35C24×C47C22⋊C411yescharacteristic
36C2×Q8×C47—24no—
37C2×Q8×C47—22no—
38C2×Q8×C47—416no—
39C2×Q8×C47—42no—
40C2×Q8×C47—42no—
41C2×Q8×C47—42no—
42C2×Q8×C47—42no—
43C2×Q8×C47M1614yes—
44C22×C4×C47—22no—
45C22×C4×C47—22no—
46C22×C4×C47—22no—
47C22×C4×C47—24no—
48C22×C4×C47—22no—
49C22×C4×C47—22no—
50C22×C4×C47—42no—
51C22×C4×C47—42no—
52C22×C4×C47C22⋊C411yescharacteristic
53C22×C4×C47C22⋊C412yes—
54C22×C4×C47C22×C411yescharacteristic
55C22×C4×C47C2×D412yes—
56C4⋊C4×C47—41no—
57C4⋊C4×C47—41no—
58C4⋊C4×C47—42no—
59C22⋊C4×C47—42no—
60C22⋊C4×C47—42no—
61C42×C47—22no—
62C42×C47—22no—
63C42×C47—42no—
64C2×C47⋊C8—44no—
65C23×C47—21no—
66C23×C47—21no—
67C23×C47—41no—
68C23×C47—41no—
69C23×C47#10012yes—
70C23×C47C2×C22⋊C411yescharacteristic
71Q8×C47—44no—
72Q8×C47—48no—
73Q8×C47—44no—
74C2×C4×C47—28no—
75C2×C4×C47—21no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—24no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—48no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47#9912yes—
91C2×C4×C47C2×M1612yes—
92C47⋊C8—84no—
93#11510—471nomaximal, Sylow
94C22×C47—21no—
95C22×C47—21no—
96C22×C47—22no—
97C22×C47—42no—
98C22×C47—42no—
99C22×C47—41no—
100C22×C47—41no—
101C22×C47#40711yescharacteristic
102C22×C47C2×#9911yescharacteristic
103C22×C47C2×#10011yescharacteristic
104C4×C47—22no—
105C4×C47—22no—
106C4×C47—42no—
107C4×C47—42no—
108C4×C47—44no—
109C4×C47—82no—
110#2310D4711yescharacteristic
111#1464—472no—
112#1319—474no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47—42no—
116C2×C47#146211yescharacteristic
117C2×C47#212911yescharacteristic
118C2×C47#213511yescharacteristic
119C23×Q8Dic4711yescharacteristic
120C2×#117Dic4711yescharacteristic
121C2×#117C2×D4711yescharacteristic
122#432—942no—
123#391—22no—
124#391Dic4712yes—
125#391C2×D4712yes—
126#381—22no—
127#324—474no—
128#323—942no—
129C47#1151011yescharacteristic, complemented, Sylow
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C2×Dic4712yes—
135C23×C4—21no—
136C23×C4C2×Dic4711yescharacteristic
137#117—42no—
138#117—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C2×Dic4711yescharacteristic
141C2×C4⋊C4C22×D4711yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4C22⋊D4712yes—
144C2×C42C2×Dic4712yes—
145C2×C42C22⋊D4712yes—
146C4⋊C8—1882no—
147C22⋊C8—944no—
148#97—1882no—
149#95—24no—
150#95—24no—
151C24C22⋊Dic4711yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—416no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8#1062532914yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C22⋊Dic4712yes—
169C22×C4C22⋊Dic4711yescharacteristic
170C22×C4C22×Dic4711yescharacteristic
171C22×C4C2×C22⋊D4712yes—
172C2×C8—1884no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—22no—
179C42—22no—
180C42—42no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23#1309394112yes—
186C23C2×C22⋊Dic4711yescharacteristic
187Q8—44no—
188Q8—48no—
189Q8—44no—
190C2×C4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4#1309393912yes—
207C2×C4C2×#1062532912yes—
208C8—3764no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22#1568908711yescharacteristic
217C22C2×#1309393911yescharacteristic
218C22C2×#1309394111yescharacteristic
219C4—22no—
220C4—22no—
221C4—44no—
222C4—42no—
223C4—42no—
224C4—82no—
225C2—21no—
226C2—21no—
227C2—42no—
228C2#2024171511yescharacteristic
229C2#2024506311yescharacteristic
230C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2310C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2310D47
hypocenterC47#11510
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. #20241322C2semidirect
129. C47#11510semidirect

Maximal subgroups

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

subgroupindex
#202417222
#202413222
C47×#23102
#1151047