Finite Group Databasebeta

← #26113684

Subgroups — #26113684

Order 12032.

534 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 123 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#20252429C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20242060C212yesmaximal, complemented
4#20241291C212yesmaximal, complemented
5C47×#2547C211yesmaximal, characteristic
6#15689201—24no—
7#15689176C2212yes—
8#15689147C2212yes—
9C2×#13093941—21no—
10#15689087—22no—
11#15688892—21no—
12#15688869—24no—
13C2×C4⋊Q8×C47C2211yescharacteristic
14C2×#124×C47—21no—
15C2×C4×D4×C47C2211yescharacteristic
16C47×#397C414yes—
17C47×#391—21no—
18C47×#380—22no—
19C4×C4⋊C4×C47C2211yescharacteristic
20C2×#10625329—24no—
21C2×#10625329—22no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)D414yes—
25#13093941—42no—
26C22⋊(C47⋊C8)—24no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—22no—
29C22×Q8×C47—21no—
30C22×D4×C47D411yescharacteristic
31C23×C4×C47D412yes—
32C4⋊Q8×C47—22no—
33C4⋊Q8×C47—22no—
34#124×C47—44no—
35C4×D4×C47—22no—
36C4×D4×C47—22no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47C2×C412yes—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47—24no—
42C2×C22⋊C4×C47C2×C412yes—
43C2×C42×C47—21no—
44C2×C42×C47C2311yescharacteristic
45#95×C47—24no—
46#95×C47C2×C412yes—
47C24×C47C22⋊C412yes—
48C2×Q8×C47—42no—
49C2×Q8×C47—42no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×D4×C47—22no—
53C2×D4×C47—22no—
54C2×D4×C47—21no—
55C2×D4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—44no—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47C22×C411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C4⋊C4×C47—22no—
66C4⋊C4×C47—24no—
67C4⋊C4×C47—42no—
68C4⋊C4×C47—41no—
69C4⋊C4×C47—41no—
70C22⋊C4×C47—24no—
71C22⋊C4×C47—48no—
72C42×C47—42no—
73C42×C47C2×D412yes—
74C42×C47#4112yes—
75#10625329—42no—
76#10625329—42no—
77#10625329—84no—
78C2×C47⋊C8—24no—
79C2×C47⋊C8—42no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—22no—
83C23×C47—42no—
84C23×C47—42no—
85C23×C47C2×C22⋊C411yescharacteristic
86Q8×C47—82no—
87Q8×C47—82no—
88D4×C47—42no—
89D4×C47—42no—
90C2×C4×C47—22no—
91C2×C4×C47—22no—
92C2×C4×C47—41no—
93C2×C4×C47—44no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—42no—
97C2×C4×C47—42no—
98C2×C4×C47—42no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—82no—
102C2×C4×C47C4×D412yes—
103C2×C4×C47#12011yescharacteristic
104C2×C4×C47#12112yes—
105C2×C4×C47#12311yescharacteristic
106C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
107C2×C4×C47AGL(1,ℤ/8)12yes—
108C2×C4×C47#13711yescharacteristic
109C47⋊C8—44no—
110C47⋊C8—82no—
111#9401—471nomaximal, Sylow
112C22×C47—21no—
113C22×C47—21no—
114C22×C47—44no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47—42no—
118C22×C47—82no—
119C22×C47#38611yescharacteristic
120C22×C47#41711yescharacteristic
121C22×C47#41811yescharacteristic
122C4×C47—21no—
123C4×C47—22no—
124C4×C47—42no—
125C4×C47—82no—
126C4×C47—82no—
127C4×C47#45512yes—
128C4×C47#46812yes—
129C4×C47#46912yes—
130#2975—471no—
131#2547D4711yescharacteristic
132#1800—471no—
133#1546—472no—
134#1303—472no—
135C2×C47—21no—
136C2×C47—21no—
137C2×C47—82no—
138C2×C47#188511yescharacteristic
139C2×C47#189511yescharacteristic
140C2×C47#191511yescharacteristic
141C2×C4⋊Q8C2×D4711yescharacteristic
142C2×#124—21no—
143C2×C4×D4C2×D4711yescharacteristic
144#436—944no—
145#431—472no—
146#423—472no—
147C2×#100—941no—
148#407—942no—
149#397Dic4714yes—
150#391—21no—
151#380—22no—
152C4×C4⋊C4C2×D4711yescharacteristic
153#343—941no—
154#332—944no—
155C47#940111yescharacteristic, complemented, Sylow
156C22×Q8—21no—
157C22×D4C22⋊D4711yescharacteristic
158C23×C4C22⋊D4712yes—
159C2×M16—944no—
160C2×M16—942no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163#124—44no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C2×Dic4712yes—
169C2×C4⋊C4C22⋊D4711yescharacteristic
170C2×C22⋊C4—24no—
171C2×C22⋊C4C2×Dic4712yes—
172C2×C42—21no—
173C2×C42C22×D4711yescharacteristic
174C4⋊C8—474no—
175C4⋊C8—942no—
176C4⋊C8—942no—
177#100—1882no—
178C22⋊C8—944no—
179C22⋊C8—942no—
180C4×C8—942no—
181#95—24no—
182#95C2×Dic4712yes—
183C24C22⋊Dic4712yes—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—44no—
197C22×C4C22⋊Dic4712yes—
198C22×C4C22×Dic4711yescharacteristic
199C22×C4C2×C22⋊D4711yescharacteristic
200C22×C4C2×C22⋊D4711yescharacteristic
201M16—3764no—
202M16—1882no—
203M16—1882no—
204C2×C8—944no—
205C2×C8—1882no—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C22⋊C4—24no—
212C22⋊C4—48no—
213C42—42no—
214C42D4×D4712yes—
215C42#1062535212yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×C22⋊Dic4711yescharacteristic
222Q8—82no—
223Q8—82no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4D4×Dic4712yes—
239C2×C4#1309399211yescharacteristic
240C2×C4#1309399612yes—
241C2×C4#1309400511yescharacteristic
242C2×C4#1309404911yescharacteristic
243C2×C4#1309405012yes—
244C2×C4#1309405411yescharacteristic
245C8—3762no—
246C8—1884no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1568900311yescharacteristic
255C22#1568912411yescharacteristic
256C22#1568912811yescharacteristic
257C4—21no—
258C4—22no—
259C4—42no—
260C4—82no—
261C4—82no—
262C4#1568930012yes—
263C4#1568936712yes—
264C4#1568937412yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#2024377711yescharacteristic
269C2#2024382711yescharacteristic
270C2#2024394111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2547C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2547D47
hypocenterC47#9401
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #20242060C2semidirect
4. #20241291C2semidirect
155. C47#9401semidirect

Maximal subgroups

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

subgroupindex
#202524292
#202433802
#202420602
#202412912
C47×#25472
#940147