Finite Group Databasebeta

← #26109714

Subgroups — #26109714

Order 12032.

514 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 135 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#20245793C211yesmaximal, characteristic
2#20243365C212yesmaximal
3#20241272C212yesmaximal, complemented
4#20241271C211yesmaximal, characteristic, complemented
5C47×#1296C211yesmaximal, characteristic, complemented
6C2×#13093982C412yes—
7C2×C42×D47C2211yescharacteristic
8C2×C4⋊(C47⋊C8)C2212yes—
9C2×(C2×D47)⋊C8—22no—
10#15688993C2211yescharacteristic
11#15688988C412yes—
12#15688980—22no—
13C4⋊C4×Dic47—22no—
14#15688877—22no—
15C2×C4⋊C8×C47C2212yes—
16C47×#384C2211yescharacteristic
17C22×C4×D47C2×C411yescharacteristic
18C22×C4×D47D412yes—
19C22×C47⋊C8—22no—
20#13093982—44no—
21C2×C4⋊Dic47—22no—
22C2×C4⋊Dic47C2×C412yes—
23C2×(C2×D47)⋊C4—22no—
24C2×C4×Dic47—22no—
25C2×C4×Dic47C2×C411yescharacteristic
26C2×C4×Dic47D412yes—
27C42×D47—22no—
28C42×D47—21no—
29C42×D47—21no—
30C4⋊(C47⋊C8)—22no—
31C4⋊(C47⋊C8)—22no—
32(C2×D47)⋊C8—44no—
33C47⋊#95—24no—
34C22×C8×C47—22no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47C2×C412yes—
37C2×C42×C47C2311yescharacteristic
38C4⋊C8×C47—22no—
39C4⋊C8×C47—22no—
40C23×D47C22⋊C411yescharacteristic
41C22×C4×C47—22no—
42C22×C4×C47C22×C411yescharacteristic
43C22×C4×C47C2×D412yes—
44C22×Dic47—22no—
45C22×Dic47C22⋊C412yes—
46C22×Dic47C22⋊C411yescharacteristic
47C2×C4×D47—22no—
48C2×C4×D47—21no—
49C2×C4×D47—21no—
50C2×C4×D47—22no—
51C2×C4×D47—22no—
52C2×C4×D47—22no—
53C2×C4×D47—22no—
54C2×C4×D47—22no—
55C2×C4×D47—22no—
56C2×C4×D47—22no—
57C2×C8×C47—42no—
58C2×C8×C47—42no—
59C2×C8×C47—42no—
60C4⋊C4×C47—42no—
61C4⋊C4×C47—42no—
62C4⋊C4×C47—42no—
63C42×C47#4112yes—
64C42×C47#4111yescharacteristic
65C42×C47#4111yescharacteristic
66(C2×D47)⋊C4—44no—
67C4⋊Dic47—42no—
68C4⋊Dic47—42no—
69C4⋊Dic47—42no—
70C4×Dic47—21no—
71C4×Dic47—21no—
72C4×Dic47—22no—
73C4×Dic47—22no—
74C4×Dic47—44no—
75C2×C47⋊C8—42no—
76C2×C47⋊C8—42no—
77C2×C47⋊C8—42no—
78C23×C47C2×C22⋊C411yescharacteristic
79C22×D47—21no—
80C22×D47—21no—
81C22×D47—22no—
82C22×D47—22no—
83C22×D47#10112yes—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—42no—
87C2×C4×C47—44no—
88C2×C4×C47#11711yescharacteristic
89C2×C4×C47#11711yescharacteristic
90C2×C4×C47#12312yes—
91C2×C4×C47#12312yes—
92C2×C4×C47#13512yes—
93C2×C4×C47#13512yes—
94C2×C4×C47AGL(1,ℤ/8)12yes—
95C2×C4×C47#13712yes—
96C2×Dic47—21no—
97C2×Dic47—21no—
98C2×Dic47—22no—
99C2×Dic47—22no—
100C2×Dic47—22no—
101C2×Dic47—42no—
102C2×Dic47—44no—
103C2×Dic47#10012yes—
104C4×D47—24no—
105C4×D47—24no—
106C4×D47—42no—
107C4×D47—42no—
108C4×D47—42no—
109C4×D47—42no—
110C8×C47—82no—
111C47⋊C8—82no—
112#8728—471nomaximal, Sylow
113C22×C47#38111yescharacteristic
114C22×C47C2×#10011yescharacteristic
115C22×C47C2×#10111yescharacteristic
116C22×C47#41612yes—
117C22×C47#41712yes—
118C2×D47—24no—
119C2×D47—22no—
120C2×D47—41no—
121C2×D47—41no—
122C2×D47—42no—
123C4×C47—21no—
124C4×C47—21no—
125C4×C47—82no—
126C4×C47#48212yes—
127C4×C47#48212yes—
128C4×C47#48512yes—
129C4×C47#48512yes—
130Dic47—22no—
131Dic47—42no—
132Dic47—82no—
133#2285—471no—
134#1797—472no—
135#1296—472no—
136#1296—471no—
137#1296D4711yescharacteristic, complemented
138C2×C47#154311yescharacteristic
139C2×C47#155212yes—
140C2×C47#156011yescharacteristic
141C2×C47#182111yescharacteristic
142C2×C47#182512yes—
143D47—42no—
144C2×#117—472no—
145C22×C42—471no—
146C2×C4⋊C8—472no—
147C2×C4⋊C8C2×D4712yes—
148C2×C22⋊C8—942no—
149#384—471no—
150#384C2×D4711yescharacteristic
151#382—472no—
152#380—942no—
153C4×C4⋊C4—942no—
154#336—942no—
155C47#872811yescharacteristic, complemented, Sylow
156C23×C4—471no—
157C23×C4—472no—
158C22×C8—22no—
159C22×C8—942no—
160#117—1884no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—472no—
163C2×C4⋊C4—942no—
164C2×C4⋊C4C4×D4712yes—
165C2×C22⋊C4—942no—
166C2×C42—471no—
167C2×C42—472no—
168C2×C42—942no—
169C2×C42—941no—
170C2×C42—941no—
171C2×C42—942no—
172C2×C42C22×D4711yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—942no—
176C4⋊C8—942no—
177C22⋊C8—1884no—
178#95—944no—
179C24—471no—
180C22×C4—22no—
181C22×C4—471no—
182C22×C4—472no—
183C22×C4—942no—
184C22×C4—942no—
185C22×C4—941no—
186C22×C4—941no—
187C22×C4—942no—
188C22×C4—942no—
189C22×C4—942no—
190C22×C4—942no—
191C22×C4—942no—
192C22×C4—942no—
193C22×C4—942no—
194C22×C4C2×C4×D4711yescharacteristic
195C22×C4D4×D4712yes—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—1882no—
200C2×C8—1882no—
201C2×C8—1882no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—1882no—
206C4⋊C4—1882no—
207C4⋊C4—1882no—
208C22⋊C4—1884no—
209C42—941no—
210C42—941no—
211C42—942no—
212C42—942no—
213C42—1884no—
214C42#1062535012yes—
215C42#1062535211yescharacteristic
216C42#1062535411yescharacteristic
217C23—472no—
218C23—941no—
219C23—941no—
220C23—942no—
221C23—942no—
222C23C22⋊C4×D4711yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—472no—
228C2×C4—944no—
229C2×C4—944no—
230C2×C4—941no—
231C2×C4—941no—
232C2×C4—942no—
233C2×C4—942no—
234C2×C4—942no—
235C2×C4—1882no—
236C2×C4—1884no—
237C2×C4—1882no—
238C2×C4—1882no—
239C2×C4—1882no—
240C2×C4—1882no—
241C2×C4#1309398111yescharacteristic
242C2×C4#1309398211yescharacteristic
243C2×C4#1309400612yes—
244C2×C4#1309400912yes—
245C2×C4#1309404512yes—
246C2×C4#1309404712yes—
247C2×C4#1309404812yes—
248C2×C4#1309405312yes—
249C8—82no—
250C8—3762no—
251C22—942no—
252C22—944no—
253C22—1882no—
254C22—1881no—
255C22—1881no—
256C22#1568898511yescharacteristic
257C22#100×D4711yescharacteristic
258C22#101×D4711yescharacteristic
259C22#1568911812yes—
260C22#1568912112yes—
261C4—21no—
262C4—21no—
263C4—82no—
264C4—942no—
265C4—3762no—
266C4—1882no—
267C4#1568944612yes—
268C4#1568945012yes—
269C4#1568946312yes—
270C4#1568946712yes—
271C2—1882no—
272C2#2024204911yescharacteristic
273C2#2024208712yes—
274C2#2024212011yescharacteristic
275C2#2024346711yescharacteristic
276C2#2024348212yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1296D47
hypocenterC47#8728
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #20241272C2semidirect
4. #20241271C2semidirect
5. C47×#1296C2semidirect
137. #1296D47semidirect
155. C47#8728semidirect

Maximal subgroups

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

subgroupindex
#202457932
#202433652
#202412722
#202412712
C47×#12962
#872847