Finite Group Databasebeta

← #27141037

Subgroups — #27141037

Order 12416.

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

normal subgroups only

This group has 182 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
1C97⋊#564C211yesmaximal, characteristic, complemented
2#20261099C212yesmaximal
3#20260477C211yesmaximal, characteristic, complemented
4#20260476C212yesmaximal, complemented
5#490×C97C211yesmaximal, characteristic
6#15690802C2212yes—
7#15690801—22no—
8#15690800—22no—
9#15690800—22no—
10#15690731—22no—
11#15690731—22no—
12#15690721C2212yes—
13#127×C97C2211yescharacteristic
14#125×C97C2211yescharacteristic
15#102×C97—22no—
16#97×C97C2211yescharacteristic
17C2×#11715895—22no—
18C4⋊Dic97—22no—
19C4⋊Dic97D412yes—
20C4⋊Dic97D412yes—
21#13095490—22no—
22#13095490—22no—
23#13095490—22no—
24#13095490—22no—
25C4×Dic97—22no—
26C2×D4×C97—21no—
27C2×D4×C97—22no—
28C2×C8×C97—21no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97D412yes—
31C42×C97C2311yescharacteristic
32C23×C97—41no—
33D4×C97—44no—
34D4×C97—42no—
35C2×C4×C97—22no—
36C2×C4×C97C2×D412yes—
37C2×C4×C97C2×D411yescharacteristic
38C2×Dic97—22no—
39C2×Dic97—22no—
40C8×C97—41no—
41#11715895—42no—
42C22×C97—82no—
43C22×C97—81no—
44C22×C97#12011yescharacteristic
45C4×C97—22no—
46C4×C97—21no—
47C4×C97—42no—
48Dic97—42no—
49Dic97—42no—
50C2×C97—161no—
51C2×C97#45412yes—
52C2×C97#45411yescharacteristic
53#1658—971nomaximal, Sylow
54C97#165811yescharacteristic, complemented, Sylow
55#564—971no—
56#490—972no—
57#490D9711yescharacteristic
58#330—971no—
59#330—972no—
60#127C2×D9711yescharacteristic
61#125—972no—
62#125—1942no—
63#125—1942no—
64#125—1942no—
65#125C2×D9711yescharacteristic
66#102—22no—
67#102—1942no—
68#102—1942no—
69#97—972no—
70#97C2×D9711yescharacteristic
71C2×D4—22no—
72C2×D4—21no—
73C2×C8—21no—
74C2×C8—1942no—
75C4⋊C4—22no—
76C4⋊C4—972no—
77C4⋊C4—972no—
78C4⋊C4—1942no—
79C4⋊C4—1942no—
80C4⋊C4—1942no—
81C4⋊C4—1942no—
82C4⋊C4—1942no—
83C4⋊C4C22⋊D9712yes—
84C42—1942no—
85C42C22×D9711yescharacteristic
86C23—41no—
87D4—44no—
88D4—42no—
89C2×C4—22no—
90C2×C4—1942no—
91C2×C4—1942no—
92C2×C4D4×D9712yes—
93C2×C4C2×C22⋊D9711yescharacteristic
94C8—41no—
95C8—3882no—
96C22—82no—
97C22—81no—
98C22#1569077811yescharacteristic
99C4—22no—
100C4—21no—
101C4—42no—
102C4—3882no—
103C4—3882no—
104C2—161no—
105C2#2026090612yes—
106C2#2026090711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#490×C97C2
socleC22×C97#120
prime radical#127×C97C22
hypercenter#490D97
hypocenterC97#1658
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. C97⋊#564C2semidirect
3. #20260477C2semidirect
4. #20260476C2semidirect
54. C97#1658semidirect

Maximal subgroups

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

subgroupindex
C97⋊#5642
#202610992
#202604772
#202604762
#490×C972
#165897