Finite Group Databasebeta

← #26113808

Subgroups — #26113808

Order 12032.

436 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 95 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#20252494C211yesmaximal, characteristic, complemented
3#20242101C212yesmaximal
4#20241261C212yesmaximal, complemented
5C47×#1537C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7#15689489—22no—
8#15689471C414yes—
9#15689254—24no—
10#15689184—22no—
11#15689180C2212yes—
12#15689127—22no—
13C4×#10625329C2211yescharacteristic
14#15688866—24no—
15C47×#533C2211yescharacteristic
16C47×#432C2212yes—
17C47×#330—22no—
18C22×D188—21no—
19C22×D188—22no—
20C2×#10625329—21no—
21#13094021—21no—
22#13094021—22no—
23#13094021C2×C412yes—
24C4⋊(C47⋊C8)—24no—
25C4⋊(C47⋊C8)D414yes—
26#13093944—28no—
27C22⋊(C47⋊C8)—22no—
28#13093935—22no—
29#13093935—21no—
30#13093935—22no—
31C4×C47⋊C8—22no—
32C4×C47⋊C8C2×C412yes—
33C4⋊Q8×C47C2×C412yes—
34#125×C47D412yes—
35C22⋊Q8×C47—22no—
36C4×Q8×C47—22no—
37#117×C47D412yes—
38C2×C42×C47C2311yescharacteristic
39C4⋊C8×C47—22no—
40C22⋊C8×C47—22no—
41#97×C47—22no—
42C23×D47—41no—
43C2×Q8×C47—22no—
44C22×C4×C47C2×D411yescharacteristic
45C22×C4×C47#4112yes—
46C2×D188—22no—
47C2×D188—24no—
48C2×D188—44no—
49C2×D188—41no—
50C2×D188—42no—
51C2×D188—84no—
52C2×C8×C47—42no—
53C4⋊C4×C47—22no—
54C4⋊C4×C47—24no—
55C4⋊C4×C47C22⋊C414yes—
56C22⋊C4×C47—22no—
57C42×C47—22no—
58C42×C47C22×C411yescharacteristic
59C42×C47C2×D411yescharacteristic
60C42×C47C2×D412yes—
61#10625329—42no—
62C2×C47⋊C8—22no—
63C2×C47⋊C8—24no—
64C23×C47#12311yescharacteristic
65C22×D47—42no—
66C22×D47—82no—
67C22×D47—84no—
68Q8×C47—82no—
69C2×C4×C47—22no—
70C2×C4×C47—24no—
71C2×C4×C47—42no—
72C2×C4×C47C2×C22⋊C411yescharacteristic
73C2×C4×C47C4×D412yes—
74C2×C4×C47#12011yescharacteristic
75C2×C4×C47#12112yes—
76D188—44no—
77D188—84no—
78D188—84no—
79D188—82no—
80C8×C47—82no—
81C47⋊C8—42no—
82C47⋊C8—44no—
83#9424—471nomaximal, Sylow
84C22×C47—21no—
85C22×C47—22no—
86C22×C47#38611yescharacteristic
87C2×D47—82no—
88C2×D47—84no—
89C2×D47—164no—
90C4×C47—21no—
91C4×C47—22no—
92C4×C47—22no—
93C4×C47—44no—
94C4×C47#45412yes—
95#3138—471no—
96#2987—471no—
97#1555—472no—
98#1537D4711yescharacteristic, complemented
99#1291—472no—
100C2×C47—41no—
101C2×C47#188811yescharacteristic
102C2×C47#190312yes—
103D47—162no—
104#533C2×D4711yescharacteristic
105C2×#127—471no—
106#490—942no—
107#486—474no—
108#446—944no—
109#433—942no—
110#432—472no—
111#432C2×D4712yes—
112#418—942no—
113C4×M16—471no—
114#331—944no—
115#330—22no—
116C47#942411yescharacteristic, complemented, Sylow
117C22×D4—941no—
118C22×D4—942no—
119C2×M16—941no—
120C4⋊Q8C4×D4712yes—
121#127—472no—
122#127—942no—
123#127—941no—
124#125C22⋊D4712yes—
125C22⋊Q8—22no—
126C4×Q8—22no—
127#117D18812yes—
128C2×C42C22×D4711yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—474no—
131C4⋊C8—944no—
132#102—948no—
133C22⋊C8—22no—
134C22⋊C8—942no—
135#97—22no—
136#97—942no—
137#97—942no—
138#97—941no—
139C4×C8—472no—
140C4×C8—942no—
141C24—1881no—
142C2×Q8—22no—
143C2×D4—942no—
144C2×D4—944no—
145C2×D4—3764no—
146C2×D4—1884no—
147C2×D4—1881no—
148C2×D4—1882no—
149C22×C4C2×D18811yescharacteristic
150C22×C4#1062535212yes—
151M16—1882no—
152C2×C8—42no—
153C2×C8—942no—
154C2×C8—944no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4(C2×D47)⋊C414yes—
158C22⋊C4—22no—
159C42—22no—
160C42C2×C4×D4711yescharacteristic
161C42D4×D4712yes—
162C42C2×C22⋊D4711yescharacteristic
163C23—3764no—
164C23—3762no—
165C23—1882no—
166C23#1309400811yescharacteristic
167Q8—82no—
168D4—3764no—
169D4—3764no—
170D4—3762no—
171D4—1884no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4C2×(C2×D47)⋊C411yescharacteristic
176C2×C4(C22⋊D47)⋊C412yes—
177C2×C4#1309399411yescharacteristic
178C2×C4#1309399612yes—
179C8—82no—
180C8—1882no—
181C8—1884no—
182C22—21no—
183C22—22no—
184C22—3762no—
185C22—3764no—
186C22—7524no—
187C22#1568900011yescharacteristic
188C4—21no—
189C4—22no—
190C4—22no—
191C4—44no—
192C4#1568929812yes—
193C2—41no—
194C2—7522no—
195C2#2024378911yescharacteristic
196C2#2024386612yes—
subgroupG/H
centerC22#15689000
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1537C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1537D47
hypocenterC47#9424
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #20252494C2semidirect
4. #20241261C2semidirect
5. C47×#1537C2semidirect
98. #1537D47semidirect
116. C47#9424semidirect

Maximal subgroups

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

subgroupindex
#202537642
#202524942
#202421012
#202412612
C47×#15372
#942447