Finite Group Databasebeta

← #27141039

Subgroups — #27141039

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#20261101C211yesmaximal, characteristic
3#20261100C212yesmaximal
4#20260476C212yesmaximal, complemented
5#330×C97C211yesmaximal, characteristic, complemented
6#15690807C2211yescharacteristic
7#15690802C2212yes—
8#15690801—22no—
9#15690800—22no—
10#15690800—22no—
11#15690732—22no—
12#15690732—22no—
13#15690730—22no—
14#15690721C2211yescharacteristic
15#125×C97C2211yescharacteristic
16#97×C97C2212yes—
17C2×D388—21no—
18C2×D388—22no—
19C2×#11715895—21no—
20C4⋊Dic97—22no—
21C4⋊Dic97D412yes—
22C4⋊Dic97D412yes—
23#13095490—22no—
24#13095490—22no—
25#13095490—22no—
26#13095490—22no—
27C4×Dic97—22no—
28C2×C8×C97—22no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97D412yes—
31C42×C97C2311yescharacteristic
32C22×D97—41no—
33C2×C4×C97—22no—
34C2×C4×C97C2×D412yes—
35C2×C4×C97C2×D411yescharacteristic
36C2×Dic97—22no—
37C2×Dic97—22no—
38D388—42no—
39D388—44no—
40C8×C97—42no—
41#11715895—41no—
42C22×C97#12011yescharacteristic
43C2×D97—82no—
44C2×D97—81no—
45C4×C97—21no—
46C4×C97—22no—
47C4×C97—42no—
48Dic97—42no—
49Dic97—42no—
50C2×C97#45412yes—
51C2×C97#45411yescharacteristic
52D97—161no—
53#1658—971nomaximal, Sylow
54C97#165811yescharacteristic, complemented, Sylow
55#564—971no—
56#490—971no—
57#490—972no—
58#330—972no—
59#330D9711yescharacteristic, complemented
60#127—971no—
61#125—972no—
62#125—1942no—
63#125—1942no—
64#125—1942no—
65#125C2×D9711yescharacteristic
66#102—1942no—
67#102—1942no—
68#102—1942no—
69#97—971no—
70#97C2×D9712yes—
71C2×D4—1942no—
72C2×D4—1941no—
73C2×C8—22no—
74C2×C8—1941no—
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⋊C4D38812yes—
84C42—1942no—
85C42C22×D9711yescharacteristic
86C23—3881no—
87D4—3884no—
88D4—3882no—
89C2×C4—22no—
90C2×C4—1942no—
91C2×C4—1942no—
92C2×C4D4×D9712yes—
93C2×C4C2×D38811yescharacteristic
94C8—42no—
95C8—3881no—
96C22—7762no—
97C22—7761no—
98C22#1569078011yescharacteristic
99C4—22no—
100C4—21no—
101C4—42no—
102C4—3882no—
103C4—3882no—
104C2—15521no—
105C2#2026090912yes—
106C2#2026091011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#330×C97C2
socleC22×C97#120
prime radical#15690807C22
hypercenter#330D97
hypocenterC97#1658
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. C97⋊#564C2semidirect
4. #20260476C2semidirect
5. #330×C97C2semidirect
54. C97#1658semidirect
59. #330D97semidirect

Maximal subgroups

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

subgroupindex
C97⋊#5642
#202611012
#202611002
#202604762
#330×C972
#165897