Finite Group Databasebeta

← #27141057

Subgroups — #27141057

Order 12416.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#20261483C211yesmaximal, characteristic
2#20261097C212yesmaximal, complemented
3#20260477C211yesmaximal, characteristic, complemented
4#20260474C212yesmaximal, complemented
5#491×C97C211yesmaximal, characteristic, complemented
6#15690798C2212yes—
7#15690795—22no—
8#15690788—22no—
9#15690785—22no—
10#15690733—22no—
11#15690730—22no—
12#15690721C2212yes—
13C4⋊Q8×C97C2211yescharacteristic
14#125×C97C2211yescharacteristic
15#103×C97—22no—
16#97×C97C2211yescharacteristic
17C2×Dic194D412yes—
18C2×D388—22no—
19C2×D388D412yes—
20C2×C4×D97—22no—
21C2×#11715895—22no—
22#13095490—22no—
23#13095488—22no—
24#13095488—22no—
25#13095488—22no—
26C2×Q8×C97—21no—
27C2×C8×C97—21no—
28C4⋊C4×C97—22no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97D412yes—
31C42×C97C2311yescharacteristic
32C22×D97—22no—
33Q8×C97—42no—
34C2×C4×C97—22no—
35C2×C4×C97—41no—
36C2×C4×C97C2×D412yes—
37C2×C4×C97C2×D411yescharacteristic
38C2×Dic97—22no—
39D388—42no—
40D388—82no—
41C4×D97—82no—
42Dic194—42no—
43C8×C97—41no—
44#11715895—42no—
45C22×C97#12011yescharacteristic
46C2×D97—42no—
47C2×D97—42no—
48C2×D97—42no—
49C4×C97—21no—
50C4×C97—22no—
51C4×C97—42no—
52C4×C97—81no—
53Dic97—42no—
54C2×C97#45411yescharacteristic
55C2×C97#45512yes—
56D97—82no—
57#1662—971nomaximal, Sylow
58C97#166211yescharacteristic, complemented, Sylow
59#562—971no—
60#491D9711yescharacteristic, complemented
61#489—972no—
62#330—971no—
63#329—972no—
64C4⋊Q8C2×D9711yescharacteristic
65#125C2×D9711yescharacteristic
66#124—972no—
67#123—1942no—
68C22⋊Q8—1942no—
69#121—1942no—
70#103—22no—
71#103—1942no—
72#102—1942no—
73#97—972no—
74#97C2×D9711yescharacteristic
75C2×Q8—21no—
76C2×Q8—972no—
77C2×D4—972no—
78C2×D4—1942no—
79C22×C4—1942no—
80C2×C8—21no—
81C2×C8—1942no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—1942no—
85C4⋊C4C22⋊D9712yes—
86C22⋊C4—1942no—
87C22⋊C4—1942no—
88C22⋊C4—1942no—
89C42C22×D9711yescharacteristic
90C23—1942no—
91Q8—42no—
92Q8—3882no—
93D4—7762no—
94D4—3882no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—7762no—
98C2×C4—1942no—
99C2×C4D4×D9712yes—
100C2×C4C2×C22⋊D9711yescharacteristic
101C8—41no—
102C8—3882no—
103C22—3882no—
104C22—3882no—
105C22—3882no—
106C22#1569077811yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—3882no—
112C2—7762no—
113C2#2026090711yescharacteristic
114C2#2026091112yes—
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#491×C97C2
socleC22×C97#120
prime radical#20261483C2
hypercenter#491D97
hypocenterC97#1662
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #20261097C2semidirect
3. #20260477C2semidirect
4. #20260474C2semidirect
5. #491×C97C2semidirect
58. C97#1662semidirect
60. #491D97semidirect

Maximal subgroups

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

subgroupindex
#202614832
#202610972
#202604772
#202604742
#491×C972
#166297