Finite Group Databasebeta

← #26130042

Subgroups — #26130042

Order 12032.

346 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 67 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#20245559C212yesmaximal, complemented
2#20242987C211yesmaximal, characteristic, complemented
3#20241429C211yesmaximal, characteristic, complemented
4#20241361C212yesmaximal, complemented
5C47×#3291C211yesmaximal, characteristic
6#15689521C2211yescharacteristic, complemented
7#15689475—21no—
8#15689469—21no—
9#15688947—22no—
10#15688944C2212yescomplemented
11#15688933—22no—
12C8×C47⋊C8C2211yescharacteristic, complemented
13C47×#577—22no—
14C47×#492C2211yescharacteristic
15C47×#471—22no—
16C47×#456—21no—
17C47×#453—21no—
18C47×#443C2212yes—
19#13094024—21no—
20#13093964—22no—
21#13093956—22no—
22#13093947—42no—
23#13093945—42no—
24C4×C47⋊C8—21no—
252−1+4×C47—21no—
26H(5,𝔽2)×C47—21no—
27#137×C47—22no—
28#137×C47—42no—
29AGL(1,ℤ/8)×C47—22no—
30AGL(1,ℤ/8)×C47—42no—
31#135×C47—42no—
32#135×C47D412yes—
33C2×SD16×C47—22no—
34C2×SD16×C47—42no—
35#131×C47—22no—
36C4⋊Q8×C47D411yescharacteristic
37#127×C47D411yescharacteristic
38#108×C47D412yes—
39#104×C47—22no—
40#101×C47—21no—
41#100×C47—21no—
42C4×C8×C47C2311yescharacteristic
43#41×C47—22no—
44#41×C47—42no—
45#41×C47—42no—
46#41×C47—81no—
47C2×Q8×C47—21no—
48C2×Q8×C47—42no—
49C2×D4×C47—21no—
50C2×D4×C47—41no—
51C2×D4×C47—42no—
52C2×D4×C47—81no—
53C2×Dic94—41no—
54Q16×C47—42no—
55Q16×C47SD1612yes—
56SD16×C47—22no—
57SD16×C47—42no—
58SD16×C47—42no—
59SD16×C47—42no—
60SD16×C47—42no—
61D8×C47—42no—
62D8×C47SD1612yes—
63M16×C47—22no—
64M16×C47—42no—
65C2×C8×C47—42no—
66C2×C8×C47C2×D412yes—
67C4⋊C4×C47—41no—
68C42×C47C2×D411yescharacteristic
69C4⋊Dic47—41no—
70C4⋊Dic47—41no—
71C2×C47⋊C8—22no—
72C47⋊C16—42no—
73C23×C47—41no—
74C23×C47—81no—
75Q8×C47—22no—
76Q8×C47—42no—
77Q8×C47—81no—
78D4×C47—22no—
79D4×C47—42no—
80D4×C47—42no—
81D4×C47—82no—
82D4×C47—82no—
83D4×C47—81no—
84C2×C4×C47—21no—
85C2×C4×C47—42no—
86C2×C4×C47—41no—
87C2×C4×C47—81no—
88C2×C4×C47—82no—
89C2×C4×C47#12011yescharacteristic
90C2×Dic47—81no—
91Dic94—82no—
92C8×C47—42no—
93C8×C47C2×SD1612yes—
94C8×C47AGL(1,ℤ/8)12yes—
95C47⋊C8—42no—
96#12365—471nomaximal, Sylow
97C22×C47—42no—
98C22×C47—81no—
99C22×C47—161no—
100C22×C47—321no—
101C22×C47#45311yescharacteristic
102C4×C47—22no—
103C4×C47—42no—
104C4×C47—81no—
105C4×C47#45012yes—
106Dic47—161no—
107#3291D4711yescharacteristic
108#2230—472no—
109#1708—471no—
110#1359—471no—
111#1335—472no—
112C2×C47—21no—
113C2×C47—82no—
114C2×C47—161no—
115C2×C47#162911yescharacteristic
116#577—22no—
117#498—471no—
118#492C2×D4711yescharacteristic
119#487—941no—
120#486—941no—
121#471—22no—
122#456—21no—
123#453—21no—
124#443C2×D4712yes—
125#365—942no—
126#364—472no—
127#360—942no—
128C82—471no—
129C47#1236511yescharacteristic, complemented, Sylow
1302−1+4—21no—
131H(5,𝔽2)—21no—
132#137—22no—
133#137—42no—
134AGL(1,ℤ/8)—22no—
135AGL(1,ℤ/8)—42no—
136#135—42no—
137#135C22⋊D4712yes—
138C2×SD16—22no—
139C2×SD16—42no—
140#131—22no—
141C4⋊Q8—941no—
142C4⋊Q8C22⋊D4711yescharacteristic
143#127C22⋊D4711yescharacteristic
144M32—942no—
145#108C22⋊D4712yes—
146#106—942no—
147#104—22no—
148#103—1882no—
149#102—1882no—
150#101—21no—
151#100—21no—
152C4×C8—941no—
153C4×C8C22×D4711yescharacteristic
154#41—22no—
155#41—42no—
156#41—42no—
157#41—81no—
158C2×Q8—21no—
159C2×Q8—42no—
160C2×Q8—1881no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4—81no—
165Q16—42no—
166Q16#1062533512yes—
167SD16—22no—
168SD16—42no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172D8—42no—
173D8#1062533512yes—
174M16—22no—
175M16—42no—
176C2×C8—42no—
177C2×C8—942no—
178C2×C8C2×C22⋊D4712yes—
179C4⋊C4—41no—
180C4⋊C4—1881no—
181C4⋊C4—1881no—
182C42C2×C22⋊D4711yescharacteristic
183C16—1882no—
184C23—41no—
185C23—81no—
186Q8—22no—
187Q8—42no—
188Q8—81no—
189Q8—3762no—
190D4—22no—
191D4—42no—
192D4—42no—
193D4—82no—
194D4—82no—
195D4—81no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—3761no—
202C2×C4#1309399311yescharacteristic
203C8—42no—
204C8—1882no—
205C8C2×#1062533512yes—
206C8#1309405012yes—
207C22—42no—
208C22—81no—
209C22—161no—
210C22—321no—
211C22#1568929111yescharacteristic
212C4—22no—
213C4—42no—
214C4—81no—
215C4—7521no—
216C4#1568927412yes—
217C2—21no—
218C2—82no—
219C2—161no—
220C2#2024250411yescharacteristic
subgroupG/H
centerC2#20242504
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
Fitting, prime radicalC47×#3291C2
socleC2×C47#1629
hypercenter#3291D47
hypocenterC47#12365
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #3291

Complements

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

NH ≅ G/Ntype
1. #20245559C2semidirect
2. #20242987C2semidirect
3. #20241429C2semidirect
4. #20241361C2semidirect
6. #15689521C22semidirect
10. #15688944C22semidirect
12. C8×C47⋊C8C22semidirect
129. C47#12365semidirect

Maximal subgroups

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

subgroupindex
#202455592
#202429872
#202414292
#202413612
C47×#32912
#1236547