Finite Group Databasebeta

← #26113800

Subgroups — #26113800

Order 12032.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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#20253770C211yesmaximal, characteristic
2#20242106C212yesmaximal
3#20242026C211yesmaximal, characteristic
4#20241259C212yesmaximal
5C47×#2987C211yesmaximal, characteristic
6C2×#13094024C2211yescharacteristic
7#15689493—22no—
8#15689475C414yes—
9#15689180C2212yes—
10#15689131—22no—
11C4×#10625329C2211yescharacteristic
12#15688870—24no—
13#15688865—22no—
14C47×#533C2211yescharacteristic
15C47×#446—24no—
16C47×#433—22no—
17C47×#432C2212yes—
18C22×Dic94—21no—
19C2×#10625329—21no—
20#13094024—22no—
21#13094024—21no—
22#13094024C2×C412yes—
23C2×C4⋊Dic47—22no—
24C4⋊(C47⋊C8)—22no—
25#13093947—28no—
26C22⋊(C47⋊C8)—22no—
27#13093935—21no—
28#13093935—22no—
29C4×C47⋊C8C2×C412yes—
30C4⋊Q8×C47C2×C412yes—
31#125×C47D412yes—
32C22⋊Q8×C47—22no—
33C4×Q8×C47—22no—
34#117×C47D412yes—
35C2×C42×C47C2311yescharacteristic
36C4⋊C8×C47—24no—
37C4⋊C8×C47D414yes—
38C22⋊C8×C47—22no—
39#97×C47—22no—
40#97×C47—22no—
41C4×C8×C47—22no—
42C2×Q8×C47—22no—
43C22×C4×C47C2×D411yescharacteristic
44C22×C4×C47#4112yes—
45C22×Dic47—41no—
46C2×Dic94—22no—
47C2×Dic94—44no—
48C2×Dic94—41no—
49C2×C8×C47—24no—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—24no—
52C4⋊C4×C47C22⋊C414yes—
53C22⋊C4×C47—22no—
54C42×C47—22no—
55C42×C47C22×C411yescharacteristic
56C42×C47C2×D411yescharacteristic
57C42×C47C2×D412yes—
58C4⋊Dic47—24no—
59C4⋊Dic47—42no—
60#10625329—42no—
61C2×C47⋊C8—22no—
62C2×C47⋊C8—42no—
63C23×C47#12311yescharacteristic
64Q8×C47—82no—
65C2×C4×C47—22no—
66C2×C4×C47—24no—
67C2×C4×C47—42no—
68C2×C4×C47C2×C22⋊C411yescharacteristic
69C2×C4×C47C4×D412yes—
70C2×C4×C47#12011yescharacteristic
71C2×C4×C47#12112yes—
72C2×Dic47—42no—
73C2×Dic47—82no—
74Dic94—44no—
75Dic94—82no—
76C8×C47—44no—
77C47⋊C8—42no—
78C47⋊C8—82no—
79#9422—471nomaximal, Sylow
80C22×C47—21no—
81C22×C47—22no—
82C22×C47#38611yescharacteristic
83C4×C47—21no—
84C4×C47—22no—
85C4×C47—22no—
86C4×C47—44no—
87C4×C47#45412yes—
88Dic47—82no—
89#3139—471no—
90#2987D4711yescharacteristic
91#1556—472no—
92#1537—471no—
93#1291—472no—
94C2×C47—41no—
95C2×C47#188811yescharacteristic
96C2×C47#190212yes—
97#533C2×D4711yescharacteristic
98C2×C4⋊Q8—471no—
99#491—942no—
100#487—474no—
101#446—24no—
102#433—22no—
103#432—472no—
104#432C2×D4712yes—
105#419—942no—
106C4×M16—471no—
107#332—944no—
108#330—942no—
109C47#942211yescharacteristic, complemented, Sylow
110C22×Q8—941no—
111C2×M16—941no—
112C4⋊Q8—472no—
113C4⋊Q8—941no—
114C4⋊Q8—942no—
115C4⋊Q8C4×D4712yes—
116#125D18812yes—
117C22⋊Q8—22no—
118C4×Q8—22no—
119#117C22⋊D4712yes—
120C2×C4⋊C4—942no—
121C2×C42C22×D4711yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—942no—
124C4⋊C8C22⋊D4714yes—
125#103—948no—
126C22⋊C8—22no—
127C22⋊C8—942no—
128#97—22no—
129#97—22no—
130#97—941no—
131#97—942no—
132C4×C8—22no—
133C4×C8—472no—
134C2×Q8—22no—
135C2×Q8—942no—
136C2×Q8—1884no—
137C2×Q8—1881no—
138C22×C4—1881no—
139C22×C4#1062535012yes—
140C22×C4C2×C22⋊D4711yescharacteristic
141M16—1882no—
142C2×C8—24no—
143C2×C8—942no—
144C2×C8—1882no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—944no—
148C4⋊C4—1882no—
149C4⋊C4(C2×D47)⋊C414yes—
150C22⋊C4—22no—
151C42—22no—
152C42C2×C4×D4711yescharacteristic
153C42C2×D18811yescharacteristic
154C42C2×C22⋊D4712yes—
155C23#1309400711yescharacteristic
156Q8—82no—
157Q8—3762no—
158Q8—1884no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—42no—
162C2×C4—3762no—
163C2×C4—1882no—
164C2×C4C2×(C2×D47)⋊C411yescharacteristic
165C2×C4C4×C22⋊D4712yes—
166C2×C4#1309399311yescharacteristic
167C2×C4#1309399812yes—
168C8—44no—
169C8—3762no—
170C8—1882no—
171C22—21no—
172C22—22no—
173C22#1568900211yescharacteristic
174C4—21no—
175C4—22no—
176C4—22no—
177C4—44no—
178C4—3762no—
179C4#1568929612yes—
180C2—41no—
181C2#2024379111yescharacteristic
182C2#2024386112yes—
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2987C2
socleC22×C47#386
prime radicalC23×C47#123
cosocleC2×C42×C47C23
hypercenter#2987D47
hypocenterC47#9422
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
109. C47#9422semidirect

Maximal subgroups

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

subgroupindex
#202537702
#202421062
#202420262
#202412592
C47×#29872
#942247