Finite Group Databasebeta

← #26113693

Subgroups — #26113693

Order 12032.

570 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 119 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#20248341C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20242072C212yesmaximal, complemented
4#20241290C212yesmaximal, complemented
5C47×#2975C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7C2×#13094003—22no—
8#15689147C2212yes—
9C2×#13093941—21no—
10#15689059C414yes—
11#15689026—21no—
12#15688992—21no—
13C4×C4⋊Dic47C2211yescharacteristic
14#15688892—21no—
15#15688870—24no—
16C2×C4×D4×C47C2211yescharacteristic
17C47×#436—24no—
18C47×#431C2212yes—
19C47×#407—22no—
20C22×Dic94—21no—
21C2×#10625329—22no—
22#13094024—22no—
23#13094024—22no—
24#13094003—48no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C412yes—
28C2×Dic47⋊C4—24no—
29C2×C22⋊Dic47—24no—
30C2×C4×Dic47—21no—
31C4⋊(C47⋊C8)—22no—
32C4⋊(C47⋊C8)—22no—
33#13093941—42no—
34C47⋊#95C2×C412yes—
35C22×D4×C47D411yescharacteristic
36C23×C4×C47D412yes—
37C2×M16×C47—24no—
38C4×D4×C47—22no—
39C4×D4×C47—22no—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C22⋊C4×C47C2×C412yes—
42C2×C42×C47C2311yescharacteristic
43C4⋊C8×C47D414yes—
44C22⋊C8×C47—24no—
45C22⋊C8×C47—22no—
46C4×C8×C47—22no—
47C24×C47C22⋊C412yes—
48C2×D4×C47—22no—
49C2×D4×C47—21no—
50C2×D4×C47—22no—
51C2×D4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—44no—
55C22×C4×C47C22⋊C412yes—
56C22×C4×C47C22×C411yescharacteristic
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×D411yescharacteristic
59C22×Dic47—22no—
60C22×Dic47—22no—
61C2×Dic94—44no—
62C2×Dic94—41no—
63C2×Dic94—41no—
64M16×C47—84no—
65C2×C8×C47—24no—
66C4⋊C4×C47—22no—
67C22⋊C4×C47—24no—
68C42×C47C2×D412yes—
69C42×C47#4112yes—
70C22⋊Dic47—48no—
71C4⋊Dic47—24no—
72C4⋊Dic47—24no—
73C4⋊Dic47—42no—
74Dic47⋊C4—48no—
75C4×Dic47—41no—
76C4×Dic47—41no—
77#10625329—42no—
78#10625329—42no—
79C2×C47⋊C8—42no—
80C23×C47—22no—
81C23×C47—22no—
82C23×C47—22no—
83C23×C47—42no—
84C23×C47—42no—
85C23×C47C2×C22⋊C411yescharacteristic
86D4×C47—42no—
87D4×C47—42no—
88C2×C4×C47—22no—
89C2×C4×C47—22no—
90C2×C4×C47—41no—
91C2×C4×C47—44no—
92C2×C4×C47—42no—
93C2×C4×C47—82no—
94C2×C4×C47C4×D412yes—
95C2×C4×C47#12011yescharacteristic
96C2×C4×C47#12112yes—
97C2×C4×C47#12311yescharacteristic
98C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
99C2×C4×C47#13712yes—
100C2×C4×C47#13711yescharacteristic
101C2×Dic47—24no—
102C2×Dic47—42no—
103C2×Dic47—42no—
104C2×Dic47—44no—
105C2×Dic47—42no—
106Dic94—82no—
107Dic94—82no—
108C8×C47—44no—
109C47⋊C8—82no—
110#9403—471nomaximal, Sylow
111C22×C47—21no—
112C22×C47—21no—
113C22×C47—44no—
114C22×C47—42no—
115C22×C47—42no—
116C22×C47—42no—
117C22×C47—82no—
118C22×C47#38611yescharacteristic
119C22×C47#41711yescharacteristic
120C22×C47#41911yescharacteristic
121C4×C47—21no—
122C4×C47—22no—
123C4×C47—42no—
124C4×C47#45312yes—
125C4×C47#46812yes—
126C4×C47#47012yes—
127Dic47—44no—
128Dic47—82no—
129#2975D4711yescharacteristic
130#2542—471no—
131#1800—471no—
132#1549—472no—
133#1303—472no—
134C2×C47—21no—
135C2×C47—21no—
136C2×C47—82no—
137C2×C47#188711yescharacteristic
138C2×C47#189411yescharacteristic
139C2×C47#191511yescharacteristic
140C2×C4⋊Q8—471no—
141C2×C22⋊Q8—942no—
142C2×C4×D4C2×D4711yescharacteristic
143#436—24no—
144#431C2×D4712yes—
145#423—472no—
146C2×#100—941no—
147#407—22no—
148#399—474no—
149#390—941no—
150#384—941no—
151C4×C4⋊C4—471no—
152#343—941no—
153#332—944no—
154C47#940311yescharacteristic, complemented, Sylow
155C22×Q8—941no—
156C22×D4D18811yescharacteristic
157C23×C4C22⋊D4712yes—
158C2×M16—24no—
159C2×M16—942no—
160C4⋊Q8—942no—
161C4⋊Q8—942no—
162C22⋊Q8—1888no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—472no—
166C2×C4⋊C4—941no—
167C2×C4⋊C4—944no—
168C2×C4⋊C4—941no—
169C2×C4⋊C4D18811yescharacteristic
170C2×C22⋊C4—944no—
171C2×C22⋊C4C4×D4712yes—
172C2×C42—941no—
173C2×C42C22×D4711yescharacteristic
174C4⋊C8—942no—
175C4⋊C8—942no—
176C4⋊C8C22⋊D4714yes—
177#100—1882no—
178C22⋊C8—24no—
179C22⋊C8—22no—
180C4×C8—22no—
181#95—472no—
182C24(C2×D47)⋊C412yes—
183C2×Q8—1884no—
184C2×Q8—1881no—
185C2×Q8—1881no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—942no—
194C22×C4—942no—
195C22×C4(C2×D47)⋊C412yes—
196C22×C4C2×C4×D4711yescharacteristic
197C22×C4C2×D18811yescharacteristic
198C22×C4C2×C22⋊D4711yescharacteristic
199M16—84no—
200M16—1882no—
201M16—1882no—
202C2×C8—24no—
203C2×C8—1882no—
204C4⋊C4—22no—
205C4⋊C4—944no—
206C4⋊C4—944no—
207C4⋊C4—1888no—
208C4⋊C4—1882no—
209C22⋊C4—24no—
210C22⋊C4—1888no—
211C42—1881no—
212C42—1881no—
213C42#1062535012yes—
214C42C2×C22⋊D4712yes—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23C2×(C2×D47)⋊C411yescharacteristic
221Q8—3762no—
222Q8—3762no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—944no—
232C2×C4—1884no—
233C2×C4—1882no—
234C2×C4—1882no—
235C2×C4—1882no—
236C2×C4C4×C22⋊D4712yes—
237C2×C4#1309399311yescharacteristic
238C2×C4#1309399812yes—
239C2×C4#1309400711yescharacteristic
240C2×C4#1309405211yescharacteristic
241C2×C4#1309405712yes—
242C2×C4#1309405711yescharacteristic
243C8—44no—
244C8—3762no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#1568900211yescharacteristic
253C22#1568912511yescharacteristic
254C22#1568913311yescharacteristic
255C4—21no—
256C4—22no—
257C4—42no—
258C4—3762no—
259C4—1884no—
260C4#1568929112yes—
261C4#1568936912yes—
262C4#1568938112yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2024378611yescharacteristic
267C2#2024382111yescharacteristic
268C2#2024394011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2975C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2975D47
hypocenterC47#9403
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #20242072C2semidirect
4. #20241290C2semidirect
154. C47#9403semidirect

Maximal subgroups

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

subgroupindex
#202483412
#202433802
#202420722
#202412902
C47×#29752
#940347