Finite Group Databasebeta

← #26135367

Subgroups — #26135367

Order 12032.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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#20253912C211yesmaximal, characteristic
2#20245108C212yesmaximal
3#20241495C212yesmaximal
4#20241472C211yesmaximal, characteristic
5C47×#1971C211yesmaximal, characteristic
6#15689680—21no—
7C2×#13093964C2212yes—
8#15689532—24no—
9#15689161—21no—
10#15689161C2212yes—
11#15688927—24no—
12C22×Q16×C47C2211yescharacteristic
13C2×#106×C47C2211yescharacteristic
14C2×#103×C47—21no—
15C2×#97×C47C2211yescharacteristic
16C47×#391—21no—
17#13094024—22no—
18#13094016—22no—
19#13094003—42no—
20C4×Dic94—42no—
21#13093983—22no—
22#13093964—22no—
23#13093964—22no—
24C2×C47⋊C16—22no—
25#13093958—22no—
26#13093958D414yes—
27#13093956—22no—
28#13093956—22no—
29C22×Q8×C47—21no—
30C2×Q16×C47—22no—
31C2×Q16×C47—22no—
32C2×Q16×C47—21no—
33C2×Q16×C47Q812yescomplemented
34C22×C8×C47D411yescharacteristic
35C22×C8×C47C2311yescharacteristic
36C2×C4⋊C4×C47—21no—
37C2×C42×C47D411yescharacteristic
38#106×C47—22no—
39#103×C47—42no—
40#97×C47—22no—
41#95×C47—42no—
42C2×Q8×C47—22no—
43C2×Q8×C47—41no—
44C2×Q8×C47—41no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C22×C4×C47—41no—
49C22×C4×C47—41no—
50C22×C4×C47D812yes—
51C22×C4×C47C2×D411yescharacteristic
52C2×Dic94—42no—
53Q16×C47—24no—
54Q16×C47—42no—
55C2×C8×C47—21no—
56C2×C8×C47—21no—
57C2×C8×C47SD1612yes—
58C2×C8×C47C2×D412yes—
59C2×C8×C47C2×Q811yescharacteristic
60C2×C8×C47#4112yes—
61C2×C8×C47#4111yescharacteristic
62C4⋊C4×C47—41no—
63C4⋊C4×C47—41no—
64C42×C47—21no—
65C42×C47—21no—
66C22⋊Dic47—42no—
67C4⋊Dic47—24no—
68C4⋊Dic47—42no—
69Dic47⋊C4—44no—
70C4×Dic47—42no—
71C47⋊C16—42no—
72C23×C47C2×D811yescharacteristic
73Q8×C47—42no—
74Q8×C47—42no—
75Q8×C47—81no—
76Q8×C47—81no—
77C2×C4×C47—22no—
78C2×C4×C47—42no—
79C2×C4×C47—42no—
80C2×C4×C47—81no—
81C2×C4×C47—81no—
82C2×C4×C47—82no—
83C2×C4×C47—81no—
84C2×C4×C47#12011yescharacteristic
85C2×C4×C47C22⋊Q811yescharacteristic
86C2×C4×C47#12411yescharacteristic
87C2×C4×C47C2×SD1611yescharacteristic
88C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C47#13711yescharacteristic
90C2×Dic47—44no—
91Dic94—84no—
92C8×C47—22no—
93C8×C47C22⋊Q812yes—
94C8×C47#12412yes—
95#13368—471nomaximal, Sylow
96C22×C47—22no—
97C22×C47#44711yescharacteristic
98C22×C47#47411yescharacteristic
99C22×C47#48611yescharacteristic
100C4×C47—42no—
101C4×C47—82no—
102C4×C47—161no—
103C4×C47#39311yescharacteristic
104C4×C47#44811yescharacteristic
105C4×C47#47611yescharacteristic
106C4×C47#48911yescharacteristic
107Dic47—84no—
108#3162—471no—
109#2141—472no—
110#1971D4711yescharacteristic
111#1386—472no—
112#1377—471no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47#202311yescharacteristic
116C2×C47#219112yes—
117C22×Q16C2×D4711yescharacteristic
118#533—941no—
119C2×M32—472no—
120#501—944no—
121#427—472no—
122#427—941no—
123C2×#106C2×D4711yescharacteristic
124C2×#103—21no—
125C2×#97C2×D4711yescharacteristic
126#391—21no—
127#358—944no—
128C47#1336811yescharacteristic, complemented, Sylow
129C22×Q8—21no—
130C2×Q16—21no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16Dic9412yescomplemented
134C22×C8C22⋊D4711yescharacteristic
135C22×C8C22×D4711yescharacteristic
136C4⋊Q8—942no—
137#125—942no—
138C22⋊Q8—1882no—
139C4×Q8—1882no—
140#117—942no—
141C2×C4⋊C4—21no—
142C2×C42C22⋊D4711yescharacteristic
143M32—942no—
144M32—942no—
145C2×C16—942no—
146#107—474no—
147#107—942no—
148#106—22no—
149#106—942no—
150#106—942no—
151#103—42no—
152#97—22no—
153#95—42no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—1882no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#1062533412yes—
164C22×C4C2×C22⋊D4711yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1062533512yes—
170C2×C8C2×Dic9411yescharacteristic
171C2×C8#1062535011yescharacteristic
172C2×C8D4×D4712yes—
173C2×C8#1062535212yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—944no—
177C4⋊C4—1884no—
178C4⋊C4—1882no—
179C22⋊C4—1882no—
180C42—21no—
181C42—21no—
182C42—1882no—
183C16—1882no—
184C23C2×#1062533411yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—3764no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—1884no—
198C2×C4#1309399211yescharacteristic
199C2×C4#1309400311yescharacteristic
200C2×C4#1309401111yescharacteristic
201C2×C4C2×#1062533511yescharacteristic
202C2×C4#1309404911yescharacteristic
203C2×C4#1309405411yescharacteristic
204C8—22no—
205C8#1309400112yes—
206C8#1309401012yes—
207C22—22no—
208C22#1568925611yescharacteristic
209C22#1568940311yescharacteristic
210C22#1568946911yescharacteristic
211C4—42no—
212C4—82no—
213C4—161no—
214C4—3764no—
215C4#1568903511yescharacteristic
216C4#1568926111yescharacteristic
217C4#1568941311yescharacteristic
218C4#1568948211yescharacteristic
219C2—21no—
220C2—21no—
221C2#2024452411yescharacteristic
222C2#2024533812yes—
subgroupG/H
centerC22#15689256
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#1971C2
socleC22×C47#447
prime radicalC23×C47C2×D8
hypercenter#1971D47
hypocenterC47#13368
LCSG ▷ C22×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1971

Complements

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

NH ≅ G/Ntype
33. C2×Q16×C47Q8semidirect
128. C47#13368semidirect
133. C2×Q16Dic94semidirect

Maximal subgroups

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

subgroupindex
#202539122
#202451082
#202414952
#202414722
C47×#19712
#1336847