Finite Group Databasebeta

← #26111240

Subgroups — #26111240

Order 12032.

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

normal subgroups only

This group has 398 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#20253736C211yesmaximal, characteristic
2#20252494C211yesmaximal, characteristic, complemented
3#20241858C212yesmaximal
4#20241288C212yesmaximal, complemented
5C47×#1544C211yesmaximal, characteristic, complemented
6#15689673C2211yescharacteristic
7#15689493C412yes—
8#15689489C412yes—
9#15689485—22no—
10#15689254—24no—
11#15689184—22no—
12#15689180C2212yes—
13#15689117—22no—
14C2×#13093935C2211yescharacteristic
15#15688852—24no—
16C47×#533C2211yescharacteristic
17C47×#431C2212yes—
18C47×#332—22no—
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)—24no—
27C4⋊(C47⋊C8)D414yes—
28#13093947—24no—
29#13093944—24no—
30C22⋊(C47⋊C8)—22no—
31#13093935—22no—
32#13093935—21no—
33#13093935—22no—
34#13093935C2×C412yes—
35C4×C47⋊C8—22no—
36C4⋊Q8×C47D412yes—
37#125×C47C2×C412yes—
38C22⋊Q8×C47—22no—
39C4×Q8×C47—22no—
40#117×C47D412yes—
41C2×C42×C47C2311yescharacteristic
42C4⋊C8×C47—22no—
43C22⋊C8×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—42no—
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—24no—
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—82no—
85C47⋊C8—42no—
86C47⋊C8—44no—
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#2987—471no—
101#1544D4711yescharacteristic, complemented
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—944no—
114#433—942no—
115#432—472no—
116#431C2×D4712yes—
117#416—942no—
118C2×#97—471no—
119#332—22no—
120D4⋊C8—944no—
121C47#900811yescharacteristic, complemented, Sylow
122C2×#41—941no—
123C22×C8—941no—
124C4⋊Q8—471no—
125C4⋊Q8C22⋊D4712yes—
126#127—471no—
127#125C4×D4712yes—
128#124—941no—
129C22⋊Q8—22no—
130#121—942no—
131C4×Q8—22no—
132C4×D4—942no—
133#117D18812yes—
134C2×C42C22×D4711yescharacteristic
135C4⋊C8—22no—
136C4⋊C8—474no—
137C4⋊C8—944no—
138#103—944no—
139#102—944no—
140C22⋊C8—22no—
141C22⋊C8—942no—
142#97—472no—
143#97—942no—
144#97—942no—
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×C4C2×D18811yescharacteristic
156C22×C4#1062535212yes—
157C2×C8—42no—
158C2×C8—942no—
159C2×C8—944no—
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
169C42D4×D4712yes—
170C42C2×C22⋊D4711yescharacteristic
171C23—1881no—
172C23#1309400811yescharacteristic
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×C4(C22⋊D47)⋊C412yes—
185C2×C4#1309399411yescharacteristic
186C2×C4#1309399612yes—
187C8—82no—
188C8—1882no—
189C8—1884no—
190C22—22no—
191C22—21no—
192C22—3762no—
193C22—3761no—
194C22#1568900011yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—3761no—
200C4#1568929812yes—
201C2—41no—
202C2—7521no—
203C2#2024374812yes—
204C2#2024378911yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1544C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#1544D47
hypocenterC47#9008
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1544

Complements

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

NH ≅ G/Ntype
2. #20252494C2semidirect
4. #20241288C2semidirect
5. C47×#1544C2semidirect
101. #1544D47semidirect
121. C47#9008semidirect

Maximal subgroups

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

subgroupindex
#202537362
#202524942
#202418582
#202412882
C47×#15442
#900847