Finite Group Databasebeta

← #26129501

Subgroups — #26129501

Order 12032.

344 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 71 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#20245588C212yesmaximal, complemented
2#20244816C211yesmaximal, characteristic
3#20241400C212yesmaximal, complemented
4#20241312C211yesmaximal, characteristic
5C47×#3189C211yesmaximal, characteristic
6C2×#13093958C2211yescharacteristic
7#15688983—21no—
8#15688948—22no—
9#15688927—22no—
10C22⋊(C47⋊C16)C2212yes—
11#15688886—21no—
12#15688886C2211yescharacteristic
13C2×C22⋊Q8×C47—21no—
14C47×#484—22no—
15C47×#467—21no—
16C47×#467C2212yes—
17C47×#451—22no—
18C2×C22⋊C8×C47C2211yescharacteristic
19C2×C4⋊Dic47—21no—
20C2×C4⋊Dic47—21no—
21C2×C22⋊Dic47—41no—
22C2×C47⋊C16—22no—
23#13093958—22no—
24C47⋊#95—41no—
25C22×Q8×C47—21no—
26C23×C4×C47D411yescharacteristic
27C2×Q16×C47—21no—
28C2×Q16×C47D412yes—
29C22×C8×C47D411yescharacteristic
30C22×C8×C47C2311yescharacteristic
31C22⋊Q8×C47—22no—
32C22⋊Q8×C47—22no—
33C22⋊Q8×C47—41no—
34C22⋊Q8×C47—41no—
35C2×C4⋊C4×C47—21no—
36C2×C4⋊C4×C47—41no—
37C2×C22⋊C4×C47—41no—
38#107×C47—21no—
39#107×C47D412yes—
40#103×C47—22no—
41#103×C47—22no—
42C22⋊C8×C47—22no—
43C24×C47D811yescharacteristic
44C2×Q8×C47—22no—
45C2×Q8×C47—41no—
46C2×Q8×C47—81no—
47C2×Q8×C47—81no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—41no—
51C22×C4×C47—41no—
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47D811yescharacteristic
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—41no—
57C22×Dic47—41no—
58Q16×C47—42no—
59Q16×C47—81no—
60C2×C8×C47—21no—
61C2×C8×C47—41no—
62C2×C8×C47—41no—
63C2×C8×C47C2×D412yes—
64C4⋊C4×C47—22no—
65C4⋊C4×C47—41no—
66C4⋊C4×C47—42no—
67C4⋊C4×C47—42no—
68C22⋊C4×C47—42no—
69C22⋊C4×C47—42no—
70C22⋊Dic47—82no—
71C4⋊Dic47—41no—
72C4⋊Dic47—41no—
73C4⋊Dic47—41no—
74C4⋊Dic47—41no—
75C47⋊C16—42no—
76C23×C47—21no—
77C23×C47—21no—
78C23×C47—41no—
79C23×C47—41no—
80C23×C47Q3212yes—
81C23×C47C2×D811yescharacteristic
82Q8×C47—42no—
83Q8×C47—42no—
84Q8×C47—82no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—21no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47—42no—
92C2×C4×C47—42no—
93C2×C4×C47—81no—
94C2×C4×C47—82no—
95C2×C4×C47—82no—
96C2×C4×C47#12011yescharacteristic
97C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
98C2×Dic47—81no—
99C2×Dic47—81no—
100C2×Dic47—81no—
101C2×Dic47—81no—
102C2×Dic47—81no—
103C2×Dic47—81no—
104C8×C47—22no—
105C8×C47—41no—
106#12282—471nomaximal, Sylow
107C22×C47—21no—
108C22×C47—21no—
109C22×C47—22no—
110C22×C47—42no—
111C22×C47—41no—
112C22×C47—41no—
113C22×C47—81no—
114C22×C47#44711yescharacteristic
115C22×C47C2×Q3211yescharacteristic
116C22×C47#51011yescharacteristic
117C4×C47—21no—
118C4×C47—41no—
119C4×C47—82no—
120C4×C47—82no—
121C4×C47#45511yescharacteristic
122C4×C47#45811yescharacteristic
123Dic47—161no—
124Dic47—161no—
125#3189D4711yescharacteristic
126#2236—472no—
127#2076—471no—
128#1348—472no—
129#1314—471no—
130C2×C47—21no—
131C2×C47—21no—
132C2×C47—42no—
133C2×C47#160711yescharacteristic
134C2×C47#218811yescharacteristic
135C2×C47#219011yescharacteristic
136C2×C22⋊Q8—21no—
137#484—22no—
138#467—21no—
139#467C2×D4712yes—
140#451—22no—
141C2×#107—471no—
142C2×C22⋊C8C2×D4711yescharacteristic
143#380—941no—
144#366—942no—
145#358—942no—
146C22⋊C16—472no—
147#340—471no—
148#340—941no—
149C47#1228211yescharacteristic, complemented, Sylow
150C22×Q8—21no—
151C23×C4C22⋊D4711yescharacteristic
152C2×Q16—21no—
153C2×Q16C22⋊D4712yes—
154C22×C8C22⋊D4711yescharacteristic
155C22×C8C22×D4711yescharacteristic
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—941no—
163C2×C4⋊C4—941no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—1881no—
166C2×C16—942no—
167#107—21no—
168#107—942no—
169#107C22⋊D4712yes—
170#103—22no—
171#103—22no—
172C22⋊C8—22no—
173#95—1881no—
174C24#1062533411yescharacteristic
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—1881no—
186C22×C4—1881no—
187C22×C4#1062533411yescharacteristic
188C22×C4C2×C22⋊D4711yescharacteristic
189Q16—42no—
190Q16—81no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8C2×C22⋊D4712yes—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—1881no—
200C4⋊C4—1881no—
201C4⋊C4—1881no—
202C4⋊C4—1881no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—3762no—
206C16—1882no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1309397012yes—
212C23C2×#1062533411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—3761no—
228C2×C4—3761no—
229C2×C4—3761no—
230C2×C4—3761no—
231C2×C4—3761no—
232C2×C4—3761no—
233C2×C4#1309399311yescharacteristic
234C2×C4#1309405011yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1568925711yescharacteristic
245C22C2×#1309397011yescharacteristic
246C22#1568956411yescharacteristic
247C4—21no—
248C4—41no—
249C4—82no—
250C4—82no—
251C4—7521no—
252C4—7521no—
253C4#1568930111yescharacteristic
254C4#1568931311yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2024237411yescharacteristic
259C2#2024532211yescharacteristic
260C2#2024533411yescharacteristic
subgroupG/H
centerC22#15689257
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#3189C2
socleC22×C47#447
prime radicalC24×C47D8
hypercenter#3189D47
hypocenterC47#12282
LCSG ▷ C22×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. #20245588C2semidirect
3. #20241400C2semidirect
149. C47#12282semidirect

Maximal subgroups

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

subgroupindex
#202455882
#202448162
#202414002
#202413122
C47×#31892
#1228247