Finite Group Databasebeta

← #26113892

Subgroups — #26113892

Order 12032.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#20253764C211yesmaximal, characteristic
2#20242126C211yesmaximal, characteristic, complemented
3#20242055C212yesmaximal
4#20241279C212yesmaximal, complemented
5C47×#2986C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7#15689489C414yes—
8#15689471—22no—
9#15689147C2212yes—
10#15689127—22no—
11C4×#10625329C2211yescharacteristic
12#15688869—22no—
13#15688866—22no—
14#15688866—22no—
15C47×#533C2211yescharacteristic
16C47×#446—22no—
17C8×Q8×C47—22no—
18C47×#433—22no—
19C47×#431C2212yes—
20C22×D188—21no—
21C22×D188—22no—
22C2×#10625329—21no—
23C2×#10625329—22no—
24#13094021—22no—
25#13094021—21no—
26#13094021C2×C412yes—
27C4⋊(C47⋊C8)—22no—
28C4⋊(C47⋊C8)—22no—
29#13093944—28no—
30#13093935C2×C412yes—
31C4×C47⋊C8—21no—
32C4⋊Q8×C47D412yes—
33#125×C47C2×C412yes—
34C22⋊Q8×C47—22no—
35C4×Q8×C47—22no—
36#117×C47D412yes—
37C2×C42×C47C2311yescharacteristic
38C4⋊C8×C47—24no—
39C4⋊C8×C47D412yes—
40C4⋊C8×C47D412yes—
41C22⋊C8×C47—22no—
42#97×C47—22no—
43C4×C8×C47—22no—
44C4×C8×C47—22no—
45C23×D47—41no—
46C2×Q8×C47—22no—
47C22×C4×C47C2×D411yescharacteristic
48C22×C4×C47#4112yes—
49C2×D188—22no—
50C2×D188—24no—
51C2×D188—44no—
52C2×D188—41no—
53C2×D188—42no—
54C2×D188—82no—
55C2×D188—82no—
56C2×C8×C47—22no—
57C2×C8×C47—22no—
58C4⋊C4×C47—24no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47C22⋊C414yes—
61C22⋊C4×C47—22no—
62C42×C47—22no—
63C42×C47C22×C411yescharacteristic
64C42×C47C2×D411yescharacteristic
65C42×C47C2×D412yes—
66#10625329—42no—
67#10625329—82no—
68C2×C47⋊C8—22no—
69C2×C47⋊C8—42no—
70C23×C47#12311yescharacteristic
71C22×D47—42no—
72C22×D47—82no—
73C22×D47—84no—
74Q8×C47—42no—
75Q8×C47—42no—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—24no—
80C2×C4×C47C2×C22⋊C411yescharacteristic
81C2×C4×C47C4×D412yes—
82C2×C4×C47#12011yescharacteristic
83C2×C4×C47#12112yes—
84D188—44no—
85D188—84no—
86D188—82no—
87D188—82no—
88D188—82no—
89C8×C47—42no—
90C8×C47—42no—
91C47⋊C8—42no—
92C47⋊C8—82no—
93#9438—471nomaximal, Sylow
94C22×C47—21no—
95C22×C47—22no—
96C22×C47#38611yescharacteristic
97C2×D47—82no—
98C2×D47—84no—
99C2×D47—164no—
100C4×C47—21no—
101C4×C47—22no—
102C4×C47—44no—
103C4×C47#45412yes—
104C4×C47#47112yes—
105C4×C47#47212yes—
106#3138—471no—
107#2986D4711yescharacteristic, complemented
108#1561—471no—
109#1545—472no—
110#1299—472no—
111C2×C47—41no—
112C2×C47#188811yescharacteristic
113C2×C47#190712yes—
114D47—162no—
115#533C2×D4711yescharacteristic
116C2×#127—471no—
117#490—474no—
118#486—942no—
119#446—22no—
120C8×Q8—22no—
121#433—22no—
122#431C2×D4712yes—
123#423—472no—
124#418—942no—
125C4×M16—471no—
126#332—942no—
127#331—942no—
128#331—942no—
129C47#943811yescharacteristic, complemented, Sylow
130C22×D4—942no—
131C22×D4—941no—
132C2×M16—941no—
133C2×M16—942no—
134C4⋊Q8D18812yes—
135#127—472no—
136#127—941no—
137#127—942no—
138#125C4×D4712yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D4712yes—
142C2×C42C22×D4711yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—942no—
145C4⋊C8—942no—
146C4⋊C8C22⋊D4712yes—
147C4⋊C8C22⋊D4712yes—
148#102—948no—
149C22⋊C8—22no—
150#97—22no—
151#97—472no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—941no—
155C24—1881no—
156C2×Q8—22no—
157C2×D4—942no—
158C2×D4—944no—
159C2×D4—3762no—
160C2×D4—3762no—
161C2×D4—1884no—
162C2×D4—1881no—
163C2×D4—1882no—
164C22×C4#1062535012yes—
165C22×C4C2×C22⋊D4711yescharacteristic
166M16—3762no—
167M16—1882no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—942no—
171C2×C8—1882no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4(C2×D47)⋊C414yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D4711yescharacteristic
178C42C2×D18811yescharacteristic
179C42C2×C22⋊D4712yes—
180C23—3764no—
181C23—3762no—
182C23—1882no—
183C23#1309400711yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—3764no—
187D4—3762no—
188D4—3762no—
189D4—3762no—
190D4—1884no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×(C2×D47)⋊C411yescharacteristic
196C2×C4C4×C22⋊D4712yes—
197C2×C4#1309399311yescharacteristic
198C2×C4#1309399812yes—
199C8—42no—
200C8—42no—
201C8—3762no—
202C8—1882no—
203C22—21no—
204C22—22no—
205C22—3762no—
206C22—3764no—
207C22—7524no—
208C22#1568900211yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1568929612yes—
213C4#1568938612yes—
214C4#1568939312yes—
215C2—41no—
216C2—7522no—
217C2#2024379111yescharacteristic
218C2#2024389612yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2986C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#2986D47
hypocenterC47#9438
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #20242126C2semidirect
4. #20241279C2semidirect
5. C47×#2986C2semidirect
107. #2986D47semidirect
129. C47#9438semidirect

Maximal subgroups

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

subgroupindex
#202537642
#202421262
#202420552
#202412792
C47×#29862
#943847