Finite Group Databasebeta

← #26113653

Subgroups — #26113653

Order 12032.

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

normal subgroups only

This group has 594 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#20248341C211yesmaximal, characteristic
2#20243380C211yesmaximal, characteristic
3#20242072C212yesmaximal, complemented
4#20241290C212yesmaximal, complemented
5C47×#2972C211yesmaximal, 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
17C8×D4×C47—24no—
18C47×#431C2212yes—
19C2×C22⋊C8×C47—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—
37C22×C8×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—22no—
50C2×D4×C47SD1612yes—
51C2×D4×C47Q1612yes—
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—
64C2×C8×C47—24no—
65C2×C8×C47—84no—
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—42no—
76#10625329—42no—
77#10625329—42no—
78C2×C47⋊C8—42no—
79C23×C47—22no—
80C23×C47—22no—
81C23×C47—42no—
82C23×C47—42no—
83C23×C47#10314yes—
84C23×C47C2×C22⋊C411yescharacteristic
85D4×C47—24no—
86D4×C47—24no—
87C2×C4×C47—22no—
88C2×C4×C47—22no—
89C2×C4×C47—41no—
90C2×C4×C47—44no—
91C2×C4×C47—42no—
92C2×C4×C47—82no—
93C2×C4×C47C4×D412yes—
94C2×C4×C47#12011yescharacteristic
95C2×C4×C47#12112yes—
96C2×C4×C47#12311yescharacteristic
97C2×C4×C47C2×SD1611yescharacteristic
98C2×C4×C47C2×Q1611yescharacteristic
99C2×C4×C47#13512yes—
100C2×Dic47—24no—
101C2×Dic47—42no—
102C2×Dic47—42no—
103C2×Dic47—44no—
104C2×Dic47—42no—
105Dic94—82no—
106Dic94—82no—
107C8×C47—44no—
108C47⋊C8—82no—
109#9395—471nomaximal, Sylow
110C22×C47—21no—
111C22×C47—21no—
112C22×C47—24no—
113C22×C47—44no—
114C22×C47—42no—
115C22×C47—42no—
116C22×C47—82no—
117C22×C47#38611yescharacteristic
118C22×C47C2×#10311yescharacteristic
119C22×C47#41611yescharacteristic
120C4×C47—21no—
121C4×C47—22no—
122C4×C47—42no—
123C4×C47#45312yes—
124C4×C47#46512yes—
125C4×C47#46712yes—
126Dic47—44no—
127Dic47—82no—
128#2972D4711yescharacteristic
129#2542—471no—
130#1800—471no—
131#1549—472no—
132#1303—472no—
133C2×C47—21no—
134C2×C47—21no—
135C2×C47—44no—
136C2×C47#188711yescharacteristic
137C2×C47#189211yescharacteristic
138C2×C47#191511yescharacteristic
139C2×C4⋊Q8—471no—
140C2×C22⋊Q8—942no—
141C2×C4×D4C2×D4711yescharacteristic
142C8×D4—24no—
143#431C2×D4712yes—
144#423—472no—
145C2×#100—941no—
146C2×C22⋊C8—22no—
147#399—474no—
148#390—941no—
149#384—941no—
150C4×C4⋊C4—471no—
151#343—941no—
152#332—944no—
153C47#939511yescharacteristic, complemented, Sylow
154C22×Q8—941no—
155C22×D4D18811yescharacteristic
156C23×C4C22⋊D4712yes—
157C2×M16—942no—
158C22×C8—24no—
159C4⋊Q8—942no—
160C4⋊Q8—942no—
161C22⋊Q8—1888no—
162C4×D4—22no—
163C4×D4—22no—
164C2×C4⋊C4—472no—
165C2×C4⋊C4—941no—
166C2×C4⋊C4—944no—
167C2×C4⋊C4—941no—
168C2×C4⋊C4D18811yescharacteristic
169C2×C22⋊C4—944no—
170C2×C22⋊C4C4×D4712yes—
171C2×C42—941no—
172C2×C42C22×D4711yescharacteristic
173C4⋊C8—942no—
174C4⋊C8—942no—
175C4⋊C8C22⋊D4714yes—
176#100—1882no—
177C22⋊C8—24no—
178C22⋊C8—22no—
179C4×C8—22no—
180#95—472no—
181C24(C2×D47)⋊C412yes—
182C2×Q8—1884no—
183C2×Q8—1881no—
184C2×Q8—1881no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4#1062532512yes—
188C2×D4Dic18812yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—44no—
192C22×C4—942no—
193C22×C4—942no—
194C22×C4(C2×D47)⋊C412yes—
195C22×C4C2×C4×D4711yescharacteristic
196C22×C4C2×D18811yescharacteristic
197C22×C4C2×C22⋊D4711yescharacteristic
198M16—1882no—
199M16—1882no—
200C2×C8—24no—
201C2×C8—84no—
202C2×C8—1882no—
203C4⋊C4—22no—
204C4⋊C4—944no—
205C4⋊C4—944no—
206C4⋊C4—1888no—
207C4⋊C4—1882no—
208C22⋊C4—24no—
209C22⋊C4—1888no—
210C42—1882no—
211C42#1062535012yes—
212C42C2×C22⋊D4712yes—
213C23—22no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23#1309394914yes—
218C23C2×(C2×D47)⋊C411yescharacteristic
219Q8—3762no—
220Q8—3762no—
221D4—24no—
222D4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—82no—
229C2×C4—944no—
230C2×C4—1884no—
231C2×C4—1882no—
232C2×C4—1882no—
233C2×C4—1882no—
234C2×C4C4×C22⋊D4712yes—
235C2×C4#1309399311yescharacteristic
236C2×C4#1309399812yes—
237C2×C4#1309400711yescharacteristic
238C2×C4C2×#1062532511yescharacteristic
239C2×C4C2×Dic18811yescharacteristic
240C2×C4#1309404612yes—
241C8—44no—
242C8—3762no—
243C22—21no—
244C22—21no—
245C22—24no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—82no—
250C22#1568900211yescharacteristic
251C22C2×#1309394911yescharacteristic
252C22#1568912011yescharacteristic
253C4—21no—
254C4—22no—
255C4—42no—
256C4—3762no—
257C4—1884no—
258C4#1568929112yes—
259C4#1568935212yes—
260C4#1568936412yes—
261C2—21no—
262C2—21no—
263C2—44no—
264C2#2024378611yescharacteristic
265C2#2024381111yescharacteristic
266C2#2024394011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2972C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#2972D47
hypocenterC47#9395
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. #20242072C2semidirect
4. #20241290C2semidirect
153. C47#9395semidirect

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×#29722
#939547