Finite Group Databasebeta

← #26104525

Subgroups — #26104525

Order 12032.

590 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 131 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#20243082C212yesmaximal
2#20241211C214yesmaximal, complemented
3C47×#2296C211yesmaximal, characteristic
4C22×#10625329—22no—
5#15688878—22no—
6#15688878C2214yes—
7C2×#117×C47C411yescharacteristic
8C2×#117×C47C411yescharacteristic
9C22×C42×C47C2211yescharacteristic
10C47×#389C412yes—
11C47×#389C2212yes—
12C47×#385—22no—
13C4×C4⋊C4×C47—22no—
14C2×#10625329—28no—
15C2×#10625329—44no—
16C2×#10625329—44no—
17C22×C47⋊C8—24no—
18C23×C4×C47C2×C411yescharacteristic
19C23×C4×C47D412yes—
20#117×C47—24no—
21#117×C47—42no—
22C2×C4⋊C4×C47—22no—
23C2×C4⋊C4×C47—22no—
24C2×C4⋊C4×C47C2×C412yes—
25C2×C4⋊C4×C47C2×C412yes—
26C2×C22⋊C4×C47—21no—
27C2×C22⋊C4×C47—21no—
28C2×C42×C47—21no—
29C2×C42×C47—21no—
30C2×C42×C47—42no—
31C2×C42×C47C2×C411yescharacteristic
32C2×C42×C47D412yes—
33C2×C42×C47C2311yescharacteristic
34#95×C47—22no—
35#95×C47—22no—
36C24×C47C22⋊C411yescharacteristic
37C22×C4×C47—22no—
38C22×C4×C47—24no—
39C22×C4×C47—22no—
40C22×C4×C47—24no—
41C22×C4×C47—22no—
42C22×C4×C47—22no—
43C22×C4×C47—22no—
44C22×C4×C47—22no—
45C22×C4×C47—22no—
46C22×C4×C47C22⋊C411yescharacteristic
47C22×C4×C47C22⋊C412yes—
48C22×C4×C47C22×C411yescharacteristic
49C22×C4×C47C2×D412yes—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—42no—
52C4⋊C4×C47—44no—
53C4⋊C4×C47—42no—
54C4⋊C4×C47—42no—
55C4⋊C4×C47M1614yescomplemented
56C22⋊C4×C47—24no—
57C22⋊C4×C47—42no—
58C42×C47—24no—
59C42×C47—41no—
60C42×C47—42no—
61C42×C47—41no—
62C42×C47—42no—
63#10625329—416no—
64C2×C47⋊C8—28no—
65C2×C47⋊C8—44no—
66C2×C47⋊C8—44no—
67C23×C47—21no—
68C23×C47—21no—
69C23×C47—22no—
70C23×C47—22no—
71C23×C47#9912yes—
72C23×C47C2×C22⋊C411yescharacteristic
73C2×C4×C47—28no—
74C2×C4×C47—28no—
75C2×C4×C47—28no—
76C2×C4×C47—21no—
77C2×C4×C47—22no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—22no—
81C2×C4×C47—21no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47—24no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47#10112yes—
91C2×C4×C47#10414yes—
92C2×C4×C47C2×M1612yes—
93C2×C4×C47AGL(1,ℤ/8)12yes—
94C2×C4×C47#13712yes—
95C47⋊C8—48no—
96#7632—471nomaximal, Sylow
97C22×C47—22no—
98C22×C47—24no—
99C22×C47—41no—
100C22×C47—42no—
101C22×C47—41no—
102C22×C47#40711yescharacteristic
103C22×C47C2×#9911yescharacteristic
104C22×C47C2×#10111yescharacteristic
105C22×C47#41712yes—
106C22×C47C2×#10412yes—
107C4×C47—22no—
108C4×C47—22no—
109C4×C47—22no—
110C4×C47—24no—
111C4×C47—22no—
112C4×C47—44no—
113C4×C47—82no—
114#2296D4711yescharacteristic
115#1728—472no—
116#1271—474no—
117C2×C47—42no—
118C2×C47#146311yescharacteristic
119C2×C47#148912yes—
120C2×C47#150412yes—
121C2×C47#152611yescharacteristic
122C2×C47#156111yescharacteristic
123C22×M16—942no—
124C2×#117Dic4711yescharacteristic
125C2×#117Dic4711yescharacteristic
126C22×C42C2×D4711yescharacteristic
127#389Dic4712yes—
128#389C2×D4712yes—
129#385—22no—
130C4×C4⋊C4—22no—
131#336—474no—
132#336—942no—
133C47#763211yescharacteristic, complemented, Sylow
134C23×C4C2×Dic4711yescharacteristic
135C23×C4C22⋊D4712yes—
136C2×M16—948no—
137C2×M16—1884no—
138C2×M16—1884no—
139C22×C8—944no—
140#117—24no—
141#117—42no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×Dic4712yes—
145C2×C4⋊C4C2×Dic4712yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—42no—
151C2×C42C2×Dic4711yescharacteristic
152C2×C42C22⋊D4712yes—
153C2×C42C22×D4711yescharacteristic
154#95—22no—
155#95—22no—
156C24C22⋊Dic4711yescharacteristic
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C22⋊Dic4711yescharacteristic
167C22×C4C22⋊Dic4712yes—
168C22×C4C22×Dic4711yescharacteristic
169C22×C4C2×C22⋊D4712yes—
170M16—18816no—
171C2×C8—948no—
172C2×C8—1884no—
173C2×C8—1884no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4#1062532914yescomplemented
180C22⋊C4—24no—
181C22⋊C4—42no—
182C42—24no—
183C42—41no—
184C42—42no—
185C42—41no—
186C42—42no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23#1309393912yes—
192C23C2×C22⋊Dic4711yescharacteristic
193C2×C4—28no—
194C2×C4—28no—
195C2×C4—28no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4#1309394312yes—
211C2×C4#1309395114yes—
212C2×C4C2×#1062532912yes—
213C2×C4#1309404912yes—
214C2×C4#1309405412yes—
215C8—1888no—
216C22—22no—
217C22—24no—
218C22—41no—
219C22—42no—
220C22—41no—
221C22#1568908711yescharacteristic
222C22C2×#1309393911yescharacteristic
223C22C2×#1309394311yescharacteristic
224C22#1568912412yes—
225C22C2×#1309395112yes—
226C4—22no—
227C4—22no—
228C4—22no—
229C4—24no—
230C4—22no—
231C4—44no—
232C4—82no—
233C2—42no—
234C2#2024171811yescharacteristic
235C2#2024182712yes—
236C2#2024189212yes—
237C2#2024198311yescharacteristic
238C2#2024212611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2296C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2296D47
hypocenterC47#7632
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #20241211C2semidirect
55. C4⋊C4×C47M16semidirect
133. C47#7632semidirect
179. C4⋊C4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202430822
#202412112
C47×#22962
#763247