Finite Group Databasebeta

← #26111242

Subgroups — #26111242

Order 12032.

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

normal subgroups only

This group has 398 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#20253736C211yesmaximal, characteristic
2#20242053C211yesmaximal, characteristic, complemented
3#20241856C212yesmaximal
4#20241286C212yesmaximal, complemented
5C47×#2987C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689493C412yes—
8#15689489C412yes—
9#15689485—22no—
10#15689176C2212yes—
11#15689117—22no—
12C2×#13093935C2211yescharacteristic
13#15688869—22no—
14#15688850—24no—
15C47×#533C2211yescharacteristic
16C47×#446—24no—
17C47×#433—22no—
18C47×#432C2212yes—
19C2×#10625350—21no—
20C22×C47⋊C8—21no—
21#13094024C2×C411yescharacteristic
22#13094021C2×C411yescharacteristic
23#13094012—21no—
24#13093998—22no—
25C4×D188—22no—
26C4⋊(C47⋊C8)—22no—
27#13093947—24no—
28#13093944—24no—
29C22⋊(C47⋊C8)—22no—
30#13093935—21no—
31#13093935C2×C412yes—
32C4×C47⋊C8—22no—
33C4⋊Q8×C47D412yes—
34#125×C47C2×C412yes—
35C22⋊Q8×C47—22no—
36C4×Q8×C47—22no—
37#117×C47D412yes—
38C2×C42×C47C2311yescharacteristic
39C4⋊C8×C47—24no—
40C4⋊C8×C47D414yes—
41C22⋊C8×C47—22no—
42#97×C47—22no—
43#97×C47—22no—
44C4×C8×C47—22no—
45C2×Q8×C47—22no—
46C22×C4×C47C2×D411yescharacteristic
47C22×C4×C47#4112yes—
48C2×C22⋊D47—41no—
49#10625350—44no—
50C2×D188—21no—
51C2×D188—22no—
52C2×C4×D47—41no—
53C2×Dic94—21no—
54C2×C8×C47—24no—
55C4⋊C4×C47—24no—
56C4⋊C4×C47—22no—
57C4⋊C4×C47C22⋊C414yes—
58C22⋊C4×C47—22no—
59C42×C47—22no—
60C42×C47C22×C411yescharacteristic
61C42×C47C2×D411yescharacteristic
62C42×C47C2×D412yes—
63(C2×D47)⋊C4—42no—
64C4⋊Dic47—22no—
65C2×C47⋊C8—22no—
66C2×C47⋊C8—42no—
67C2×C47⋊C8—42no—
68C23×C47#12311yescharacteristic
69C22×D47—41no—
70Q8×C47—82no—
71C2×C4×C47—22no—
72C2×C4×C47—24no—
73C2×C4×C47—42no—
74C2×C4×C47C2×C22⋊C411yescharacteristic
75C2×C4×C47C4×D412yes—
76C2×C4×C47#12011yescharacteristic
77C2×C4×C47#12112yes—
78C22⋊D47—82no—
79C2×Dic47—41no—
80D188—44no—
81D188—42no—
82C4×D47—82no—
83Dic94—42no—
84C8×C47—44no—
85C47⋊C8—42no—
86C47⋊C8—82no—
87#9008—471nomaximal, Sylow
88C22×C47—22no—
89C22×C47—21no—
90C22×C47#38611yescharacteristic
91C2×D47—82no—
92C2×D47—81no—
93C4×C47—22no—
94C4×C47—22no—
95C4×C47—21no—
96C4×C47—44no—
97C4×C47#45412yes—
98Dic47—81no—
99#3135—471no—
100#2987D4711yescharacteristic, complemented
101#1544—471no—
102#1496—472no—
103#1302—472no—
104C2×C47—41no—
105C2×C47#188112yes—
106C2×C47#188811yescharacteristic
107D47—161no—
108#533C2×D4711yescharacteristic
109#532—471no—
110#491—472no—
111#490—472no—
112#489—942no—
113#446—24no—
114#433—22no—
115#432C2×D4712yes—
116#431—472no—
117#416—942no—
118C2×#97—471no—
119#332—942no—
120D4⋊C8—944no—
121C47#900811yescharacteristic, complemented, Sylow
122C2×#41—941no—
123C22×C8—941no—
124C4⋊Q8—471no—
125C4⋊Q8D18812yes—
126#127—471no—
127#125C4×D4712yes—
128#124—941no—
129C22⋊Q8—22no—
130#121—942no—
131C4×Q8—22no—
132C4×D4—942no—
133#117C22⋊D4712yes—
134C2×C42C22×D4711yescharacteristic
135C4⋊C8—24no—
136C4⋊C8—942no—
137C4⋊C8C22⋊D4714yes—
138#103—944no—
139#102—944no—
140C22⋊C8—22no—
141C22⋊C8—942no—
142#97—22no—
143#97—22no—
144#97—472no—
145#97—941no—
146C4×C8—22no—
147C4×C8—942no—
148#41—1884no—
149C2×Q8—22no—
150C2×Q8—941no—
151C2×D4—941no—
152C2×D4—942no—
153C2×D4—1881no—
154C22×C4—1881no—
155C22×C4#1062535012yes—
156C22×C4C2×C22⋊D4711yescharacteristic
157C2×C8—24no—
158C2×C8—942no—
159C2×C8—1882no—
160C2×C8—1882no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—942no—
164C4⋊C4(C2×D47)⋊C414yes—
165C22⋊C4—22no—
166C22⋊C4—1882no—
167C42—22no—
168C42C2×C4×D4711yescharacteristic
169C42C2×D18811yescharacteristic
170C42C2×C22⋊D4712yes—
171C23—1881no—
172C23#1309400711yescharacteristic
173Q8—82no—
174Q8—1882no—
175D4—3762no—
176D4—1884no—
177D4—1882no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—3762no—
182C2×C4—1881no—
183C2×C4C2×(C2×D47)⋊C411yescharacteristic
184C2×C4C4×C22⋊D4712yes—
185C2×C4#1309399311yescharacteristic
186C2×C4#1309399812yes—
187C8—44no—
188C8—3762no—
189C8—1882no—
190C22—22no—
191C22—21no—
192C22—3762no—
193C22—3761no—
194C22#1568900211yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—3761no—
200C4#1568929612yes—
201C2—41no—
202C2—7521no—
203C2#2024375012yes—
204C2#2024379111yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2987C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2987D47
hypocenterC47#9008
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
2. #20242053C2semidirect
4. #20241286C2semidirect
5. C47×#2987C2semidirect
100. #2987D47semidirect
121. C47#9008semidirect

Maximal subgroups

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

subgroupindex
#202537362
#202420532
#202418562
#202412862
C47×#29872
#900847