Finite Group Databasebeta

← #26119417

Subgroups — #26119417

Order 12032.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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#20246693C211yesmaximal, characteristic, complemented
2#20244220C211yesmaximal, characteristic
3#20241884C212yesmaximal
4#20241290C212yesmaximal, complemented
5C47×#1274C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689147C2211yescharacteristic
8C2×#13093952—22no—
9#15689065—22no—
10#15689057—22no—
11C4×Dic47⋊C4—22no—
12C4×C4⋊Dic47C2212yes—
13#15688858—22no—
14#15688858—22no—
15C47×#431C2212yes—
16C47×#376C2211yescharacteristic
17C2×#10625350—21no—
18C2×#10625329—21no—
19#13094024Q811yescharacteristic
20#13094021Q811yescharacteristic
21#13094012—21no—
22#13093998—22no—
23C4×D188—22no—
24C2×C4⋊Dic47—22no—
25C2×Dic47⋊C4—22no—
26C2×Dic47⋊C4—22no—
27C2×C4×Dic47—22no—
28C4⋊(C47⋊C8)—22no—
29#13093952—44no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C47⋊#95D412yes—
33C47⋊#95D412yes—
34C2×C42×C47D412yes—
35C2×C42×C47C2311yescharacteristic
36C4⋊C8×C47—22no—
37C22⋊C8×C47—22no—
38C4×C8×C47—22no—
39#95×C47—22no—
40C22×C4×C47—22no—
41C22×C4×C47C2×D412yes—
42C22×C4×C47C2×D411yescharacteristic
43C2×C22⋊D47—41no—
44C22×Dic47—22no—
45C22×Dic47—22no—
46#10625350—42no—
47#10625350—42no—
48C2×D188—21no—
49C2×D188—22no—
50C2×C4×D47—41no—
51C2×Dic94—21no—
52C2×C8×C47—42no—
53C42×C47—22no—
54C42×C47—22no—
55C42×C47C2×Q811yescharacteristic
56C42×C47#4112yes—
57C42×C47#4111yescharacteristic
58(C2×D47)⋊C4—42no—
59C4⋊Dic47—22no—
60C4⋊Dic47—44no—
61Dic47⋊C4—44no—
62Dic47⋊C4—44no—
63C4×Dic47—42no—
64C4×Dic47—42no—
65#10625329—42no—
66C2×C47⋊C8—41no—
67C23×C47#12011yescharacteristic
68C22×D47—41no—
69C2×C4×C47—42no—
70C2×C4×C47—42no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47C22⋊Q812yes—
74C2×C4×C47C22⋊Q811yescharacteristic
75C2×C4×C47#12412yes—
76C2×C4×C47#12411yescharacteristic
77C2×C4×C47#13512yes—
78C2×C4×C47AGL(1,ℤ/8)12yes—
79C22⋊D47—82no—
80C2×Dic47—24no—
81C2×Dic47—42no—
82C2×Dic47—42no—
83C2×Dic47—42no—
84C2×Dic47—42no—
85C2×Dic47—41no—
86C2×Dic47—42no—
87D188—42no—
88D188—82no—
89C4×D47—81no—
90C4×D47—81no—
91Dic94—42no—
92C8×C47—82no—
93C47⋊C8—81no—
94#10372—471nomaximal, Sylow
95C22×C47—22no—
96C22×C47#39311yescharacteristic
97C22×C47#44912yes—
98C2×D47—81no—
99C2×D47—82no—
100C4×C47—22no—
101C4×C47—22no—
102C4×C47—82no—
103C4×C47#47812yes—
104C4×C47#48811yescharacteristic
105C4×C47#49011yescharacteristic
106Dic47—44no—
107Dic47—82no—
108Dic47—81no—
109#2402—471no—
110#1963—471no—
111#1502—472no—
112#1303—472no—
113#1274D4711yescharacteristic, complemented
114C2×C47—22no—
115C2×C47#202511yescharacteristic
116C2×C47#203112yes—
117D47—161no—
118#532—471no—
119#431C2×D4712yes—
120#423—471no—
121C2×#104—942no—
122#400—942no—
123#398—942no—
124C4×C4⋊C4—472no—
125C4×C4⋊C4—942no—
126#376C2×D4711yescharacteristic
127#327—942no—
128#327—942no—
129C47#1037211yescharacteristic, complemented, Sylow
130C2×#41—941no—
131C2×M16—941no—
132C4⋊Q8—471no—
133#127—471no—
134#124—941no—
135#121—942no—
136C4×D4—942no—
137C2×C4⋊C4—942no—
138C2×C4⋊C4—942no—
139C2×C4⋊C4—942no—
140C2×C42—942no—
141C2×C42D18812yes—
142C2×C42C22×D4711yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—942no—
145#104—1884no—
146C22⋊C8—22no—
147C4×C8—22no—
148#95—22no—
149#95—472no—
150#95—472no—
151#95—942no—
152#95—942no—
153#41—1882no—
154#41—1882no—
155C2×Q8—941no—
156C2×D4—942no—
157C2×D4—941no—
158C2×D4—1881no—
159C22×C4—22no—
160C22×C4—942no—
161C22×C4—942no—
162C22×C4—1881no—
163C22×C4C2×D18811yescharacteristic
164C22×C4D4×D4712yes—
165M16—1882no—
166C2×C8—42no—
167C2×C8—1881no—
168C4⋊C4—942no—
169C4⋊C4—1884no—
170C4⋊C4—1884no—
171C4⋊C4—1884no—
172C22⋊C4—1882no—
173C42—22no—
174C42—22no—
175C42—1882no—
176C42—1882no—
177C42#1062535012yes—
178C42#1062535211yescharacteristic
179C42Q8×D4711yescharacteristic
180C23—1881no—
181C23#1309399411yescharacteristic
182Q8—1882no—
183D4—3762no—
184D4—3762no—
185D4—1882no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—944no—
191C2×C4—3761no—
192C2×C4—3761no—
193C2×C4—1882no—
194C2×C4—1882no—
195C2×C4—1882no—
196C2×C4—1881no—
197C2×C4—1882no—
198C2×C4—1882no—
199C2×C4#1309400212yes—
200C2×C4#1309400411yescharacteristic
201C2×C4#1309401012yes—
202C2×C4#1309401211yescharacteristic
203C2×C4#1309404612yes—
204C2×C4#1309405212yes—
205C8—82no—
206C8—3761no—
207C22—22no—
208C22—3761no—
209C22—3762no—
210C22#1568903611yescharacteristic
211C22#1568927012yes—
212C4—22no—
213C4—22no—
214C4—82no—
215C4—3762no—
216C4—3761no—
217C4—1884no—
218C4#1568942412yes—
219C4#1568947811yescharacteristic
220C4#1568948811yescharacteristic
221C2—22no—
222C2—7521no—
223C2#2024453511yescharacteristic
224C2#2024455912yes—
subgroupG/H
centerC2×C4#13094012
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1274C2
socleC22×C47#393
prime radical#15689673C22
hypercenter#1274D47
hypocenterC47#10372
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #20246693C2semidirect
4. #20241290C2semidirect
5. C47×#1274C2semidirect
113. #1274D47semidirect
129. C47#10372semidirect

Maximal subgroups

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

subgroupindex
#202466932
#202442202
#202418842
#202412902
C47×#12742
#1037247