Finite Group Databasebeta

← #26104379

Subgroups — #26104379

Order 12032.

514 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 155 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#20243368C212yesmaximal
2#20241211C214yesmaximal, complemented
3C47×#2296C211yesmaximal, characteristic
4C2×C4⋊(C47⋊C8)—22no—
5C2×C22⋊(C47⋊C8)—22no—
6#15688878C2214yes—
7C2×#117×C47C411yescharacteristic
8C2×#117×C47C411yescharacteristic
9C22×C42×C47C2211yescharacteristic
10C47×#389C412yes—
11C47×#389C2212yes—
12C47×#385—22no—
13C4×C4⋊C4×C47—22no—
14C22×C47⋊C8—24no—
15C4⋊(C47⋊C8)—44no—
16C22⋊(C47⋊C8)—44no—
17C23×C4×C47C2×C411yescharacteristic
18C23×C4×C47D412yes—
19#117×C47—22no—
20#117×C47—42no—
21#117×C47C814yescomplemented
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—22no—
39C22×C4×C47—22no—
40C22×C4×C47—22no—
41C22×C4×C47—22no—
42C22×C4×C47—22no—
43C22×C4×C47—42no—
44C22×C4×C47—42no—
45C22×C4×C47C22⋊C412yes—
46C22×C4×C47C22⋊C411yescharacteristic
47C22×C4×C47C2×C812yes—
48C22×C4×C47M1612yes—
49C22×C4×C47C22×C411yescharacteristic
50C22×C4×C47C2×D412yes—
51C4⋊C4×C47—22no—
52C4⋊C4×C47—44no—
53C4⋊C4×C47—42no—
54C4⋊C4×C47—42no—
55C4⋊C4×C47—42no—
56C4⋊C4×C47C2×C814yes—
57C22⋊C4×C47—24no—
58C22⋊C4×C47—42no—
59C42×C47—24no—
60C42×C47—24no—
61C42×C47—41no—
62C42×C47—41no—
63C42×C47—42no—
64C2×C47⋊C8—44no—
65C2×C47⋊C8—44no—
66C2×C47⋊C8—44no—
67C23×C47—21no—
68C23×C47—21no—
69C23×C47—22no—
70C23×C47—22no—
71C23×C47C22⋊C812yes—
72C23×C47C2×C22⋊C411yescharacteristic
73C2×C4×C47—22no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—22no—
77C2×C4×C47—24no—
78C2×C4×C47—24no—
79C2×C4×C47—24no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—22no—
83C2×C4×C47—44no—
84C2×C4×C47—44no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—42no—
88C2×C4×C47—42no—
89C2×C4×C47—42no—
90C2×C4×C47C22⋊C812yes—
91C2×C4×C47#9912yes—
92C2×C4×C47#10112yes—
93C2×C4×C47C22×C811yescharacteristic
94C2×C4×C47C2×M1611yescharacteristic
95C2×C4×C47AGL(1,ℤ/8)12yes—
96C2×C4×C47#13712yes—
97C47⋊C8—84no—
98#7598—471nomaximal, Sylow
99C22×C47—22no—
100C22×C47—24no—
101C22×C47—41no—
102C22×C47—41no—
103C22×C47—42no—
104C22×C47C2×C22⋊C811yescharacteristic
105C22×C47C2×#9911yescharacteristic
106C22×C47C2×#10111yescharacteristic
107C22×C47#41712yes—
108C22×C47#42112yes—
109C4×C47—21no—
110C4×C47—21no—
111C4×C47—22no—
112C4×C47—22no—
113C4×C47—44no—
114C4×C47—42no—
115C4×C47—82no—
116C4×C47#32414yes—
117#2296D4711yescharacteristic
118#1797—472no—
119#1271—474no—
120C2×C47—42no—
121C2×C47C2×#32411yescharacteristic
122C2×C47#147712yes—
123C2×C47#151312yes—
124C2×C47#153711yescharacteristic
125C2×C47#156111yescharacteristic
126C2×#117Dic4711yescharacteristic
127C2×#117Dic4711yescharacteristic
128C22×C42C2×D4711yescharacteristic
129C2×C4⋊C8—942no—
130C2×C22⋊C8—942no—
131#389Dic4712yes—
132#389C2×D4712yes—
133#385—22no—
134C4×C4⋊C4—22no—
135#336—474no—
136C47#759811yescharacteristic, complemented, Sylow
137C23×C4C2×Dic4711yescharacteristic
138C23×C4C22⋊D4712yes—
139C22×C8—944no—
140#117—22no—
141#117—42no—
142#117C47⋊C814yescomplemented
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C2×Dic4712yes—
146C2×C4⋊C4C2×Dic4712yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—42no—
152C2×C42C2×Dic4711yescharacteristic
153C2×C42C22⋊D4712yes—
154C2×C42C22×D4711yescharacteristic
155C4⋊C8—1884no—
156C22⋊C8—1884no—
157#95—22no—
158#95—22no—
159C24C22⋊Dic4711yescharacteristic
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C2×C47⋊C812yes—
169C22×C4#1062532912yes—
170C22×C4C22⋊Dic4712yes—
171C22×C4C22⋊Dic4711yescharacteristic
172C22×C4C22×Dic4711yescharacteristic
173C22×C4C2×C22⋊D4712yes—
174C2×C8—1884no—
175C2×C8—1884no—
176C2×C8—1884no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4C2×C47⋊C814yes—
183C22⋊C4—24no—
184C22⋊C4—42no—
185C42—24no—
186C42—24no—
187C42—41no—
188C42—41no—
189C42—42no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23C22⋊(C47⋊C8)12yes—
195C23C2×C22⋊Dic4711yescharacteristic
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4C22⋊(C47⋊C8)12yes—
214C2×C4#1309393912yes—
215C2×C4#1309394312yes—
216C2×C4C22×C47⋊C811yescharacteristic
217C2×C4C2×#1062532911yescharacteristic
218C2×C4#1309404912yes—
219C2×C4#1309405412yes—
220C8—3764no—
221C22—22no—
222C22—24no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22C2×C22⋊(C47⋊C8)11yescharacteristic
227C22C2×#1309393911yescharacteristic
228C22C2×#1309394311yescharacteristic
229C22#1568912412yes—
230C22#1568914012yes—
231C4—22no—
232C4—22no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4#1568884914yes—
239C2—42no—
240C2C2×#1568884911yescharacteristic
241C2#2024177512yes—
242C2#2024193112yes—
243C2#2024202611yescharacteristic
244C2#2024212611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#2296C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#2296D47
hypocenterC47#7598
LCSG ▷ C2×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
21. #117×C47C8semidirect
136. C47#7598semidirect
142. #117C47⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202433682
#202412112
C47×#22962
#759847