Finite Group Databasebeta

← #27141062

Subgroups — #27141062

Order 12416.

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

normal subgroups only

This group has 172 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
2#20261141C212yesmaximal
3#20260477C211yesmaximal, characteristic
4#20260476C212yesmaximal
5#491×C97C211yesmaximal, characteristic
6#15690802C2212yes—
7#15690801—22no—
8#15690800—22no—
9#15690800—22no—
10#15690743—22no—
11#15690741—22no—
12#15690721C2212yes—
13C4⋊Q8×C97C2211yescharacteristic
14#125×C97C2211yescharacteristic
15#103×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×Q8×C97—21no—
27C2×C8×C97—21no—
28C4⋊C4×C97—22no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97D412yes—
31C42×C97C2311yescharacteristic
32Q8×C97—42no—
33C2×C4×C97—22no—
34C2×C4×C97—41no—
35C2×C4×C97C2×D412yes—
36C2×C4×C97C2×D411yescharacteristic
37C2×Dic97—22no—
38C2×Dic97—22no—
39C8×C97—41no—
40#11715895—24no—
41C22×C97#12011yescharacteristic
42C4×C97—21no—
43C4×C97—22no—
44C4×C97—42no—
45C4×C97—81no—
46Dic97—42no—
47Dic97—42no—
48C2×C97#45411yescharacteristic
49C2×C97#45612yes—
50#1663—971nomaximal, Sylow
51C97#166311yescharacteristic, complemented, Sylow
52#564—971no—
53#501—972no—
54#491D9711yescharacteristic
55#330—971no—
56#330—972no—
57C4⋊Q8C2×D9711yescharacteristic
58#125—972no—
59#125—1942no—
60#125—1942no—
61#125—1942no—
62#125C2×D9711yescharacteristic
63#107—1942no—
64#106—1942no—
65#103—22no—
66#97—972no—
67#97C2×D9711yescharacteristic
68C2×Q8—21no—
69C2×C8—21no—
70C2×C8—1942no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—972no—
74C4⋊C4—972no—
75C4⋊C4—1942no—
76C4⋊C4—1942no—
77C4⋊C4—1942no—
78C4⋊C4—1942no—
79C4⋊C4—1942no—
80C4⋊C4C22⋊D9712yes—
81C42—1942no—
82C42C22×D9711yescharacteristic
83Q8—42no—
84C2×C4—22no—
85C2×C4—41no—
86C2×C4—1942no—
87C2×C4—1942no—
88C2×C4D4×D9712yes—
89C2×C4C2×C22⋊D9711yescharacteristic
90C8—41no—
91C8—1944no—
92C22#1569077811yescharacteristic
93C4—21no—
94C4—22no—
95C4—42no—
96C4—81no—
97C4—3882no—
98C4—3882no—
99C2#2026090711yescharacteristic
100C2#2026091612yes—
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#491×C97C2
socle, prime radicalC22×C97#120
hypercenter#491D97
hypocenterC97#1663
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
51. C97#1663semidirect

Maximal subgroups

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

subgroupindex
C97⋊#5642
#202611412
#202604772
#202604762
#491×C972
#166397