Finite Group Databasebeta

← #27141064

Subgroups — #27141064

Order 12416.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261145C212yesmaximal
2#20261105C211yesmaximal, characteristic
3#20260477C212yesmaximal
4#20260477C211yesmaximal, characteristic
5#564×C97C211yesmaximal, characteristic
6#15690810C2211yescharacteristic
7#15690743—22no—
8#15690741—22no—
9#15690733—22no—
10#15690721C2212yes—
11#15690721C2211yescharacteristic
12#125×C97—22no—
13#125×C97—22no—
14#125×C97—22no—
15#125×C97C2212yes—
16#125×C97C2211yescharacteristic
17C2×Dic194—21no—
18C2×#11715895—22no—
19C2×#11715895—21no—
20C4⋊Dic97—22no—
21C4⋊C4×C97—22no—
22C4⋊C4×C97—22no—
23C4⋊C4×C97—22no—
24C4⋊C4×C97—22no—
25C4⋊C4×C97—22no—
26C4⋊C4×C97—22no—
27C4⋊C4×C97D412yes—
28C4⋊C4×C97D412yes—
29C4⋊C4×C97D412yes—
30C42×C97—22no—
31C42×C97C2311yescharacteristic
32C2×C4×C97—22no—
33C2×C4×C97—22no—
34C2×C4×C97—22no—
35C2×C4×C97C2×D412yes—
36C2×C4×C97C2×D411yescharacteristic
37C2×Dic97—41no—
38Dic194—42no—
39#11715895—24no—
40#11715895—41no—
41C22×C97#12011yescharacteristic
42C4×C97—21no—
43C4×C97—22no—
44C4×C97—42no—
45C4×C97—42no—
46C4×C97—42no—
47Dic97—81no—
48C2×C97#45411yescharacteristic
49C2×C97#45612yes—
50#1663—971nomaximal, Sylow
51C97#166311yescharacteristic, complemented, Sylow
52#564D9711yescharacteristic
53#501—972no—
54#491—971no—
55#330—972no—
56#330—971no—
57C4⋊Q8—971no—
58#125—22no—
59#125—22no—
60#125—22no—
61#125C2×D9711yescharacteristic
62#125C2×D9712yes—
63#107—1942no—
64#106—1942no—
65#103—1942no—
66#97—971no—
67#97—972no—
68C2×Q8—1941no—
69C2×C8—1941no—
70C2×C8—1942no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—1942no—
78C4⋊C4C22⋊D9712yes—
79C4⋊C4C22⋊D9712yes—
80C4⋊C4C22⋊D9712yes—
81C42—22no—
82C42C22×D9711yescharacteristic
83Q8—3882no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—3881no—
88C2×C4C2×C22⋊D9712yes—
89C2×C4C2×C22⋊D9711yescharacteristic
90C8—3881no—
91C8—1944no—
92C22#1569077911yescharacteristic
93C4—21no—
94C4—22no—
95C4—42no—
96C4—42no—
97C4—42no—
98C4—7761no—
99C2#2026090811yescharacteristic
100C2#2026091812yes—
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#564×C97C2
socle, prime radicalC22×C97#120
hypercenter#564D97
hypocenterC97#1663
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #564

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
#202611452
#202611052
#202604772
#564×C972
#166397