Finite Group Databasebeta

← #26113890

Subgroups — #26113890

Order 12032.

452 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 103 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#20252490C211yesmaximal, characteristic, complemented
3#20242057C212yesmaximal
4#20241277C212yesmaximal, complemented
5C47×#1561C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7#15689489C414yes—
8#15689471—22no—
9#15689254—22no—
10Q8×C47⋊C8—22no—
11#15689184—22no—
12#15689176C2212yes—
13#15689127—22no—
14C4×#10625329C2211yescharacteristic
15#15688866—22no—
16#15688866—22no—
17C47×#533C2211yescharacteristic
18C47×#423C2212yes—
19C47×#332—22no—
20C22×D188—21no—
21C22×D188—22no—
22C2×#10625329—21no—
23#13094021—22no—
24#13094021—21no—
25#13094021C2×C412yes—
26C4⋊(C47⋊C8)—24no—
27C4⋊(C47⋊C8)D412yes—
28C4⋊(C47⋊C8)D412yes—
29#13093944—28no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935C2×C412yes—
33C4×C47⋊C8—22no—
34C4×C47⋊C8—22no—
35C4×C47⋊C8—21no—
36C2×M16×C47—22no—
37C4⋊Q8×C47D412yes—
38#125×C47C2×C412yes—
39C22⋊Q8×C47—22no—
40C4×Q8×C47—22no—
41#117×C47D412yes—
42C2×C42×C47C2311yescharacteristic
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—
56M16×C47—82no—
57C2×C8×C47—42no—
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—
67C2×C47⋊C8—22no—
68C2×C47⋊C8—22no—
69C2×C47⋊C8—22no—
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—82no—
90C47⋊C8—42no—
91C47⋊C8—42no—
92C47⋊C8—42no—
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#2986—471no—
108#1561D4711yescharacteristic, complemented
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—942no—
120C8×Q8—942no—
121#433—942no—
122#431—472no—
123#423C2×D4712yes—
124#418—942no—
125C4×M16—471no—
126#332—22no—
127#331—942no—
128#331—942no—
129C47#943811yescharacteristic, complemented, Sylow
130C22×D4—941no—
131C22×D4—942no—
132C2×M16—22no—
133C2×M16—941no—
134C4⋊Q8C22⋊D4712yes—
135#127—472no—
136#127—942no—
137#127—941no—
138#125C4×D4712yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117D18812yes—
142C2×C42C22×D4711yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—472no—
146C4⋊C8—472no—
147C4⋊C8—944no—
148#102—948no—
149C22⋊C8—942no—
150#97—472no—
151#97—942no—
152C4×C8—942no—
153C4×C8—942no—
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×C4C2×D18811yescharacteristic
165C22×C4#1062535212yes—
166M16—82no—
167M16—1882no—
168C2×C8—42no—
169C2×C8—942no—
170C2×C8—942no—
171C2×C8—942no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4(C2×D47)⋊C414yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D4711yescharacteristic
178C42D4×D4712yes—
179C42C2×C22⋊D4711yescharacteristic
180C23—3764no—
181C23—3762no—
182C23—1882no—
183C23#1309400811yescharacteristic
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×C4(C22⋊D47)⋊C412yes—
197C2×C4#1309399411yescharacteristic
198C2×C4#1309399612yes—
199C8—82no—
200C8—1882no—
201C8—1882no—
202C8—1882no—
203C22—21no—
204C22—22no—
205C22—3762no—
206C22—3764no—
207C22—7524no—
208C22#1568900011yescharacteristic
209C4—21no—
210C4—22no—
211C4—44no—
212C4#1568929812yes—
213C4#1568938812yes—
214C4#1568939112yes—
215C2—41no—
216C2—7522no—
217C2#2024378911yescharacteristic
218C2#2024389412yes—
subgroupG/H
centerC22#15689000
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1561C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1561D47
hypocenterC47#9438
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1561

Complements

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

NH ≅ G/Ntype
2. #20252490C2semidirect
4. #20241277C2semidirect
5. C47×#1561C2semidirect
108. #1561D47semidirect
129. C47#9438semidirect

Maximal subgroups

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

subgroupindex
#202537642
#202524902
#202420572
#202412772
C47×#15612
#943847