Finite Group Databasebeta

← #26116253

Subgroups — #26116253

Order 12032.

412 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 119 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#20253711C211yesmaximal, characteristic
2#20241842C212yesmaximal, complemented
3#20241838C212yesmaximal
4C2×#15688844C211yesmaximal, characteristic, complemented
5C47×#1837C211yesmaximal, characteristic, complemented
6#15689680C2211yescharacteristic
7#15689673C2211yescharacteristic
8#15689478—22no—
9#15689474C412yes—
10#15689470C412yes—
11#15689120—22no—
12C2×C4×C47⋊C8C2212yes—
13#15688853—24no—
14#15688850—24no—
15#15688844—22no—
16C47×#423C2212yes—
17C2×C4×C8×C47C2211yescharacteristic
18C47×#341—22no—
19C2×#10625350—21no—
20C22×C47⋊C8—22no—
21#13094024C2×C411yescharacteristic
22#13094024C2×C412yes—
23#13094021C2×C411yescharacteristic
24#13094016—21no—
25#13094012—21no—
26#13094003—22no—
27#13093998—22no—
28C4×Dic94—22no—
29C4×D188—22no—
30#13093983—21no—
31#13093949—24no—
32#13093946—24no—
33C4×C47⋊C8—22no—
34C4×C47⋊C8D414yes—
35C2×M16×C47—22no—
36C22×C8×C47D412yes—
37C2×C42×C47C2311yescharacteristic
38C4⋊C8×C47—22no—
39C4⋊C8×C47—22no—
40C4×C8×C47C2×C412yes—
41C4×C8×C47D412yes—
42C22×C4×C47C2×D411yescharacteristic
43C22×C4×C47#4112yes—
44C2×C22⋊D47—41no—
45#10625350—44no—
46C2×D188—21no—
47C2×D188—22no—
48C2×C4×D47—41no—
49C2×Dic94—21no—
50C2×Dic94—22no—
51M16×C47—44no—
52C2×C8×C47—24no—
53C2×C8×C47—42no—
54C2×C8×C47C22⋊C414yes—
55C42×C47C22×C411yescharacteristic
56C42×C47C2×D411yescharacteristic
57C42×C47C2×D412yes—
58C22⋊Dic47—41no—
59(C2×D47)⋊C4—42no—
60C4⋊Dic47—22no—
61C4⋊Dic47—22no—
62C4⋊Dic47—22no—
63Dic47⋊C4—42no—
64C4×Dic47—41no—
65C2×C47⋊C8—24no—
66C2×C47⋊C8—44no—
67C23×C47#12411yescharacteristic
68C22×D47—41no—
69C2×C4×C47—24no—
70C2×C4×C47C2×C22⋊C411yescharacteristic
71C2×C4×C47C4×D412yes—
72C2×C4×C47#12112yes—
73C2×C4×C47#12711yescharacteristic
74C2×C4×C47#13514yes—
75C22⋊D47—82no—
76C2×Dic47—42no—
77C2×Dic47—41no—
78D188—42no—
79D188—44no—
80C4×D47—82no—
81Dic94—42no—
82Dic94—44no—
83C8×C47—24no—
84C8×C47—82no—
85C47⋊C8—44no—
86#9863—471nomaximal, Sylow
87C22×C47—22no—
88C22×C47#39011yescharacteristic
89C22×C47#48812yes—
90C2×D47—81no—
91C2×D47—82no—
92C4×C47—22no—
93C4×C47#41612yes—
94C4×C47#44414yes—
95C4×C47#49512yes—
96Dic47—81no—
97Dic47—82no—
98#3131—471no—
99#1837D4711yescharacteristic, complemented
100#1493—472no—
101#1492—472no—
102C2×#322—471no—
103C2×C47—22no—
104C2×C47#195811yescharacteristic
105C2×C47#197512yes—
106D47—161no—
107#533—471no—
108#532—471no—
109#488—942no—
110#487—472no—
111#486—472no—
112#423C2×D4712yes—
113#416—942no—
114C2×C4×C8—472no—
115C2×C4×C8C2×D4711yescharacteristic
116#341—22no—
117Q8⋊C8—944no—
118D4⋊C8—944no—
119#322—942no—
120C47#986311yescharacteristic, complemented, Sylow
121C2×#41—941no—
122C2×M16—22no—
123C22×C8—942no—
124C22×C8C22⋊D4712yes—
125C4⋊Q8—471no—
126C4⋊Q8—472no—
127#127—471no—
128#125—941no—
129#124—941no—
130C22⋊Q8—942no—
131#121—942no—
132C4×Q8—942no—
133C4×D4—942no—
134#117—941no—
135C2×C42C22×D4711yescharacteristic
136C4⋊C8—22no—
137C4⋊C8—22no—
138#103—944no—
139#102—944no—
140C4×C8—474no—
141C4×C8—942no—
142C4×C8C4×D4712yes—
143C4×C8D18812yes—
144#41—1884no—
145C2×Q8—941no—
146C2×Q8—942no—
147C2×D4—941no—
148C2×D4—942no—
149C2×D4—1881no—
150C22×C4—1881no—
151C22×C4#1062535412yes—
152C22×C4C2×C22⋊D4711yescharacteristic
153M16—44no—
154C2×C8—24no—
155C2×C8—42no—
156C2×C8—944no—
157C2×C8—1884no—
158C2×C8(C2×D47)⋊C414yes—
159C4⋊C4—942no—
160C4⋊C4—942no—
161C4⋊C4—942no—
162C4⋊C4—1882no—
163C22⋊C4—1881no—
164C22⋊C4—1882no—
165C42—1881no—
166C42C2×C4×D4711yescharacteristic
167C42C2×D18811yescharacteristic
168C42D4×D4712yes—
169C23—1881no—
170C23#1309401311yescharacteristic
171Q8—1884no—
172Q8—1882no—
173D4—3762no—
174D4—1884no—
175D4—1882no—
176C2×C4—24no—
177C2×C4—3762no—
178C2×C4—1882no—
179C2×C4—1881no—
180C2×C4C2×(C2×D47)⋊C411yescharacteristic
181C2×C4#1309398612yes—
182C2×C4#1309399912yes—
183C2×C4#1309402011yescharacteristic
184C2×C4#1309404414yes—
185C8—24no—
186C8—82no—
187C8—1884no—
188C22—22no—
189C22—3762no—
190C22—3761no—
191C22#1568902411yescharacteristic
192C22#1568948012yes—
193C4—22no—
194C4—3762no—
195C4—3761no—
196C4#1568911712yes—
197C4#1568924414yes—
198C4#1568950512yes—
199C2—22no—
200C2—7521no—
201C2#2024419711yescharacteristic
202C2#2024427912yes—
subgroupG/H
centerC2×C4#13094020
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1837C2
socleC22×C47#390
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1837D47
hypocenterC47#9863
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1837

Complements

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

NH ≅ G/Ntype
2. #20241842C2semidirect
4. C2×#15688844C2semidirect
5. C47×#1837C2semidirect
99. #1837D47semidirect
120. C47#9863semidirect

Maximal subgroups

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

subgroupindex
#202537112
#202418422
#202418382
C2×#156888442
C47×#18372
#986347